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New Special Instructions
author Klaus Thoden <kthoden@mpiwg-berlin.mpg.de>
date Wed, 30 Jul 2014 15:58:21 +0200
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1 <?xml version="1.0" encoding="utf-8" ?>
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5 <html>
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7 <head><title></title>
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8 <meta http-equiv="Content-Type" content="text/html; charset=iso-8859-1" />
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9 <meta name="generator" content="TeX4ht (http://www.cse.ohio-state.edu/~gurari/TeX4ht/mn.html)" />
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10 <meta name="originator" content="TeX4ht (http://www.cse.ohio-state.edu/~gurari/TeX4ht/mn.html)" />
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12 <meta name="src" content="Einst_Grund_de_1916.tex" />
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13 <meta name="date" content="2005-03-10 16:49:00" />
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14 <link rel="stylesheet" type="text/css" href="Einst_Grund_de_1916.css" />
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15 </head><body >
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17 <!--l. 15--><p class="noindent"><pb/></p>
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18 <div class="center" >
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20 <!--l. 16--><p class="noindent">
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21 </p><!--l. 17--><p class="noindent"><span
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22 class="cmr-17x-x-120">ANNALEN DER PHYSIK.</span></p></div>
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23 <div class="center" >
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25 <!--l. 20--><p class="noindent">
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26 </p><!--l. 21--><p class="noindent"><span
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27 class="cmr-12x-x-120">VIERTE FOLGE. BAND 49.</span></p></div>
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28 <!--l. 26--><p class="noindent"><img
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29 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_19160x.png" alt="PICT" />
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30 </p>
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31 <div class="center" >
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33 <!--l. 32--><p class="noindent">
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34 </p><!--l. 33--><p class="noindent"><span
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35 class="cmr-12x-x-120">1.</span><span
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36 class="cmbxti-10x-x-144">Die Grundlage </span> <br/><span
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37 class="cmbxti-10x-x-144">der allgemeinen Relativit</span><span
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38 class="cmbxti-10x-x-144">ätstheorie;</span>
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39 <br/><span
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40 class="cmbxti-10x-x-144">von A. Einstein.</span></p></div>
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41 <div class="center" >
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43 <!--l. 38--><p class="noindent">
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44 </p><!--l. 39--><p class="noindent">--------</p></div>
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45 <!--l. 42--><p class="indent"> Die im nachfolgenden dargelegte Theorie bildet die denk-<br/>bar weitgehendste Verallgemeinerung der heute allgemein als <br/>,,Relativitätstheorie&#8220; bezeichneten Theorie; die letztere nenne <br/>ich im folgenden zur Unterscheidung von der ersteren ,,spezielle <br/>Relativitätstheorie&#8220; und setze sie als bekannt voraus. Die
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46 <br/>Verallgemeinerung der Relativitätstheorie wurde sehr er-<br/>leichtert durch die Gestalt, welche der speziellen Relativitäts-<br/>theorie durch Minkowski gegeben wurde, welcher Mathe-<br/>matiker zuerst die formale Gleichwertigkeit der räumlichen
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47 <br/>Koordinaten und der Zeitkoordinate klar erkannte und für <br/>den Aufbau der Theorie nutzbar machte. Die für die all-<br/>gemeine Relativitätstheorie nötigen mathematischen Hilfs-<br/>mittel lagen fertig bereit in dem ,,absoluten Differentialkalkül&#8220;, <br/>welcher auf den Forschungen von Gauss, Riemann und
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48 <br/>Christoffel über nichteuklidische Mannigfaltigkeiten ruht und <br/>von Ricci und Levi-Civita in ein System gebracht und <br/>bereits auf Probleme der theoretischen Physik angewendet <br/>wurde. Ich habe im Abschnitt B der vorliegenden Abhand-<br/>lung alle für uns nötigen, bei dem Physiker nicht als bekannt <br/>vorauszusetzenden mathematischen Hilfsmittel in möglichst <br/>einfacher und durchsichtiger Weise entwickelt, so daß ein <br/>Studium mathematischer Literatur für das Verständnis der
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49 <br/>vorliegenden Abhandlung nicht erforderlich ist. Endlich sei <br/>an dieser Stelle dankbar meines Freundes, des Mathematikers <br/>Grossmann, gedacht, der mir durch
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50 seine Hilfe nicht nur <br/>das Studium der einschlägigen mathematischen Literatur er-<br/>sparte, sondern mich auch beim Suchen nach den Feldgleichun-<br/>gen der Gravitation unterstützte.
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51 </p>
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52 <div class="center" >
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54 <!--l. 62--><p class="noindent">
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55 </p><!--l. 63--><p class="noindent">--------</p></div>
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56 <!--l. 65--><p class="noindent"><pb/>
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57 </p><!--l. 69--><p class="indent">
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59 </p>
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60 <div class="center" >
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62 <!--l. 70--><p class="noindent">
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63 </p><!--l. 71--><p class="noindent"><span
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64 class="cmbx-12">A. Prinzipielle Erw</span><span
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65 class="cmbx-12">ägungen zum Postulat der Relativit</span><span
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66 class="cmbx-12">ät.</span></p></div>
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67 <div class="center" >
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69 <!--l. 74--><p class="noindent">
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70 </p><!--l. 76--><p class="noindent"><span
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71 class="cmbsy-10x-x-120">§ </span><span
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72 class="cmbx-12">1. Bemerkungen zu der speziellen Relativit</span><span
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73 class="cmbx-12">ätstheorie.</span></p></div>
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74 <!--l. 80--><p class="indent"> Der speziellen Relativitätstheorie liegt folgendes Postulat <br/>zugrunde, welchem auch durch die Galilei-Newtonsche <br/>Mechanik Genüge geleistet wird: Wird ein Koordinatensystem <span
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75 class="cmmi-12">K </span> <br/>so gewählt, daß in bezug auf dasselbe die physikalischen Ge-<br/>setze in ihrer einfachsten Form gelten, so gelten <span
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76 class="cmti-12">dieselben </span> <br/>Gesetze auch in bezug auf jedes andere Koordinatens ystem <span
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77 class="cmmi-12">K</span><span
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78 class="cmsy-10x-x-120">'</span>, <br/>das relativ zu <span
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79 class="cmmi-12">K </span>in gleichförmiger Translationsbewegung be-<br/>griffen ist. Dieses Postulat nennen wir ,,spezielles Relativitäts-<br/>prinzip&#8220;. Durch das Wort ,,speziell&#8220; soll angedeutet werden,
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80 <br/>daß das Prinzip auf den Fall beschränkt ist, daß <span
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81 class="cmmi-12">K</span><span
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82 class="cmsy-10x-x-120">' </span>eine <span
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83 class="cmti-12">gleich- </span> <br/><span
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84 class="cmti-12">f</span><span
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85 class="cmti-12">örmige</span>
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86 <span
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87 class="cmti-12">Translationsbewegung </span>gegen <span
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88 class="cmmi-12">K </span>ausführt, daß sich <br/>aber die Gleichwertigkeit von <span
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89 class="cmmi-12">K</span><span
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90 class="cmsy-10x-x-120">'</span>
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91 und <span
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92 class="cmmi-12">K </span>nicht auf den Fall <br/><span
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93 class="cmti-12">ungleichf</span><span
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94 class="cmti-12">örmiger </span>Bewegung von <span
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95 class="cmmi-12">K</span><span
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96 class="cmsy-10x-x-120">' </span>gegen <span
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97 class="cmmi-12">K</span>
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98 erstreckt.
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99 </p><!--l. 99--><p class="indent"> Die spezielle Relativitätstheorie weicht also von der klas-<br/>sischen Mechanik nicht durch das Relativitätspostulat ab, <br/>sondern allein durch das Postulat von der Konstanz der <br/>Vakuum-Lichtgeschwindigkeit, aus welchem im Verein mit <br/>dem speziellen Relativitätsprinzip die Relativität der Gleich-<br/>zeitigkeit sowie die Lorentztransformation und die mit dieser <br/>verknüpften Gesetze über das Verhalten bewegter starrer <br/>Körper und Uhren in bekannter Weise folgen.
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100 </p><!--l. 110--><p class="indent"> Die Modifikation, welche die Theorie von Raum und Zeit <br/>durch die spezielle Relativitätstheorie erfahren hat, ist zwar <br/>eine tiefgehende; aber <span
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101 class="cmti-12">ein </span>wichtiger Punkt blieb unangetastet. <br/>Auch gemäß der speziellen Relativitätstheorie sind nämlich <br/>die Sätze der Geometrie unmittelbar als die Gesetze über <br/>die möglichen relativen Lagen (ruhender) fester Körper zu <br/>deuten, allgemeiner die Sätze der Kinematik als Sätze, welche <br/>das Verhalten von Meßkörpern und Uhren beschreiben. Zwei <br/>hervorgehobenen materiellen Punkten eines ruhenden (starren)
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102 <br/>Körpers entspricht hierbei stets eine Strecke von ganz be-<br/>stimmter Länge, unabhängig von Ort und Orientierung des <br/>Körpers sowie von der Zeit; zwei hervorgehobenen Zeiger-<br/>stellungen einer relativ zum (berechtigten) Bezugssystem ruhen-<br/>den Uhr entspricht stets eine Zeitstrecke von bestimmter Länge,
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103 <br/>unabhängig von Ort und Zeit. Es wird sich bald zeigen, daß <br/>die allgemeine Relativitätstheorie an dieser einfachen physika-<br/>lischen Deutung von Raum und Zeit nicht festhalten kann. <pb/>
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104 </p><!--l. 126--><p class="indent">
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105
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106 </p>
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107 <div class="center" >
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108
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109 <!--l. 127--><p class="noindent">
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110 </p><!--l. 130--><p class="noindent"><span
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111 class="cmbsy-10x-x-120">§ </span><span
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112 class="cmbx-12">2. </span><span
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113 class="cmbx-12">Über die Gr</span><span
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114 class="cmbx-12">ünde, welche eine Erweiterung des Relativit</span><span
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115 class="cmbx-12">äts-</span>
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116 <br/><span
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117 class="cmbx-12">postulates nahelegen.</span></p></div>
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118 <!--l. 134--><p class="indent"> Der klassischen Mechanik und nicht minder der speziellen <br/>Relativitätstheorie haftet ein erkenntnistheoretischer Mangel <br/>an, der vielleicht zum ersten Male von E. Mach klar hervor-<br/>gehoben wurde. Wir erläutern ihn am folgenden Beispiel.
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119 <br/>Zwei flüssige Körper von gleicher Größe und Art schweben <br/>frei im Raume in so großer Entfernung voneinander (und von <br/>allen übrigen Massen), daß nur diejenigen Gravitationskräfte <br/>berücksichtigt werden müssen, welche die Teile <span
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120 class="cmti-12">eines</span>
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121 dieser <br/>Körper aufeinander ausüben. Die Entfernung der Körper <br/>voneinander sei unveränderlich. Relative Bewegungen der <br/>Teile eines der Körper gegeneinander sollen nicht auftreten. <br/>Aber jede Masse soll -- von einem relativ zu der anderen Masse <br/>ruhenden Beobachter aus beurteilt -- um die Verbindungslinie <br/>der Massen mit konstanter Winkelgeschwindigkeit rotieren (es <br/>ist dies eine konstatierbare Relativbewegung beider Massen). <br/>Nun denken wir uns die Oberflächen beider Körper (<span
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122 class="cmmi-12">S</span><sub ><span
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123 class="cmr-8">1</span></sub> und <span
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124 class="cmmi-12">S</span><sub ><span
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125 class="cmr-8">2</span></sub>) <br/>mit Hilfe (relativ ruhender) Maßstäbe ausgemessen; es ergebe
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126 <br/>sich, daß die Oberfläche von <span
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127 class="cmmi-12">S</span><sub ><span
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128 class="cmr-8">1</span></sub> eine Kugel, die von <span
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129 class="cmmi-12">S</span><sub ><span
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130 class="cmr-8">2</span></sub> ein <br/>Rotationsellipsoid sei.
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131 </p><!--l. 159--><p class="indent"> Wir fragen nun: Aus welchem Grunde verhalten sich die <br/>Körper <span
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132 class="cmmi-12">S</span><sub ><span
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133 class="cmr-8">1</span></sub>
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134 und <span
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135 class="cmmi-12">S</span><sub ><span
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136 class="cmr-8">2</span></sub> verschieden? Eine Antwort auf diese Frage <br/>kann nur dann als erkenntnistheoretisch befriedigend<sup ><span
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137 class="cmr-8">1</span></sup>) an-<br/>erkannt werden, wenn die als Grund angegebene Sache eine <br/><span
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138 class="cmti-12">beobachtbare Erfahrungstatsache </span>ist; denn das Kausalitäts-<br/>gesetz hat nur dann den Sinn einer Aussage über die Er-<br/>fahrungswelt, wenn als Ursachen und Wirkungen letzten <br/>Endes nur <span
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139 class="cmti-12">beobachtbare Tatsachen</span>
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140 auftreten.
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141 </p><!--l. 169--><p class="indent"> Die Newtonsche Mechanik gibt auf diese Frage keine <br/>befriedigende Antwort. Sie sagt nämlich folgendes. Die Ge-<br/>setze der Mechanik gelten wohl für einen Raum <span
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142 class="cmmi-12">R</span><sub ><span
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143 class="cmr-8">1</span></sub>, gegen <br/>welchen der Körper <span
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144 class="cmmi-12">S</span><sub ><span
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145 class="cmr-8">1</span></sub> in Ruhe ist, nicht aber gegenüber einem
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146 <br/>Raume <span
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147 class="cmmi-12">R</span><sub ><span
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148 class="cmr-8">2</span></sub>, gegen welchen <span
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149 class="cmmi-12">S</span><sub ><span
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150 class="cmr-8">2</span></sub> in Ruhe ist. Der berechtigte <br/>Galileische Raum <span
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151 class="cmmi-12">R</span><sub ><span
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152 class="cmr-8">1</span></sub>, der hierbei eingeführt wird, ist aber <br/>eine <span
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153 class="cmti-12">blo</span><span
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154 class="cmmi-12"><img
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155 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi12-c.png" alt="b" class="cmmi-12x-x-c" align="middle" /> </span><span
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156 class="cmti-12">fingierte </span>Ursache, keine beobachtbare Sache. Es <br/>ist also klar, daß die Newtonsche Mechanik der Forderung <br/>----------
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157 </p><!--l. 177--><p class="indent"> 1) Eine derartige erkenntnistheoretisch befriedigende Antwort kann <br/>natürlich immer noch <span
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158 class="cmti-12">physikalisch </span>unzutreffend sein, falls sie mit anderen <br/>Erfahrungen im Widerspruch ist. <pb/>
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159 </p><!--l. 184--><p class="indent">
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160
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161 </p><!--l. 185--><p class="noindent">der Kausalität in dem betrachteten Falle nicht wirklich, son-<br/>dern nur scheinbar Genüge leistet, indem sie die bloß fin-<br/>gierte Ursache <span
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162 class="cmmi-12">R</span><sub ><span
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163 class="cmr-8">1</span></sub> für das beobachtbare verschiedene Ver-<br/>halten der Körper <span
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164 class="cmmi-12">S</span><sub ><span
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165 class="cmr-8">1</span></sub> und <span
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166 class="cmmi-12">S</span><sub ><span
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167 class="cmr-8">2</span></sub> verantwortlich macht.
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168 </p><!--l. 191--><p class="indent"> Eine befriedigende Antwort auf die oben aufgeworfene <br/>Frage kann nur so lauten: Das aus <span
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169 class="cmmi-12">S</span><sub ><span
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170 class="cmr-8">1</span></sub> und <span
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171 class="cmmi-12">S</span><sub ><span
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172 class="cmr-8">2</span></sub> bestehende <br/>physikalische System zeigt für sich allein keine denkbare Ur-<br/>sache, auf welche das verschiedene Verhalten von <span
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173 class="cmmi-12">S</span><sub ><span
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174 class="cmr-8">1</span></sub> und <span
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175 class="cmmi-12">S</span><sub ><span
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176 class="cmr-8">2</span></sub>
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177 <br/>zurückgeführt werden könnte. Die Ursache muß also <span
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178 class="cmti-12">au</span><span
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179 class="cmmi-12"><img
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180 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi12-c.png" alt="b" class="cmmi-12x-x-c" align="middle" /></span><span
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181 class="cmti-12">er- </span> <br/><span
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182 class="cmti-12">halb </span>dieses Systems liegen. Man gelangt zu der Auffassung, <br/>daß die allgemeinen Bewegungsgesetze, welche im speziellen <br/>die Gestalten von <span
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183 class="cmmi-12">S</span><sub ><span
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184 class="cmr-8">1</span></sub> und <span
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185 class="cmmi-12">S</span><sub ><span
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186 class="cmr-8">2</span></sub> bestimmen, derart sein müssen,
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187 <br/>daß das mechanische Verhalten von <span
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188 class="cmmi-12">S</span><sub ><span
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189 class="cmr-8">1</span></sub> und <span
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190 class="cmmi-12">S</span><sub ><span
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191 class="cmr-8">2</span></sub> ganz wesentlich <br/>durch ferne Massen mitbedingt werden muß, welche wir nicht zu <br/>dem betrachteten System gerechnet hatten. Diese fernen Massen <br/>(und ihre Relativbewegungen gegen die betrachteten Körper) <br/>sind dann als Träger prinzipiell beobachtbarer Ursachen für <br/>das verschiedene Verhalten unserer betrachteten Körper an-<br/>zusehen; sie übernehmen die Rolle der fingierten Ursache <span
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192 class="cmmi-12">R</span><sub ><span
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193 class="cmr-8">1</span></sub>. <br/>Von allen denkbaren, relativ zueinander beliebig bewegten <br/>Räumen <span
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194 class="cmmi-12">R</span><sub ><span
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195 class="cmr-8">1</span></sub>, <span
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196 class="cmmi-12">R</span><sub ><span
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197 class="cmr-8">2</span></sub> usw. darf a priori keiner als bevorzugt an-<br/>gesehen werden, wenn nicht der dargelegte erkenntnistheo-<br/>retische Einwand wieder aufleben soll. <span
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198 class="cmti-12">Die Gesetze der</span>
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199 <span
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200 class="cmti-12">Physik </span> <br/><span
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201 class="cmti-12">m</span><span
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202 class="cmti-12">üssen so beschaffen sein, da</span><span
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203 class="cmmi-12"><img
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204 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi12-c.png" alt="b" class="cmmi-12x-x-c" align="middle" /> </span><span
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205 class="cmti-12">sie in bezug auf beliebig bewegte</span>
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206 <br/><span
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207 class="cmti-12">Bezugssysteme gelten. </span>Wir gelangen also auf diesem Wege <br/>zu einer Erweiterung des Relativitätspostulates.
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208 </p><!--l. 219--><p class="indent"> Außer diesem schwerwiegenden erkenntnistheoretischen <br/>Argument spricht aber auch eine wohlbekannte physikalische <br/>Tatsache für eine Erweiterung der Relativitätstheorie. Es <br/>sei <span
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209 class="cmmi-12">K </span>ein Galileisches Bezugssystem, d. h. ein solches,
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210 <br/>relativ zu welchem (mindestens in dem betrachteten vier-<br/>dimensionalen Gebiete) eine von anderen hinlänglich ent-<br/>fernte Masse sich geradlinig und gleichförmig bewegt. Es <br/>sei <span
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211 class="cmmi-12">K</span><span
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212 class="cmsy-10x-x-120">' </span>ein zweites Koordinatensystem, welches relativ zu <span
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213 class="cmmi-12">K </span> <br/>in
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214 <span
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215 class="cmti-12">gleichf</span><span
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216 class="cmti-12">örmig beschleunigter </span>Translationsbewegung sei. Relativ <br/>zu <span
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217 class="cmmi-12">K</span><span
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218 class="cmsy-10x-x-120">' </span>führte dann eine von anderen hinreichend getrennte Masse <br/>eine beschleunigte Bewegung aus, derart, daß deren Beschleuni-<br/>gung und Beschleunigungsrichtung von ihrer stofflichen Zusam-<br/>mensetzung und ihrem physikalischen Zustande unabhängig ist.
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219 </p><!--l. 236--><p class="indent"> Kann ein relativ zu <span
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220 class="cmmi-12">K</span><span
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221 class="cmsy-10x-x-120">' </span>ruhender Beobachter hieraus <br/><pb/>
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222 </p><!--l. 239--><p class="indent">
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223
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224 </p><!--l. 240--><p class="noindent">den Schluß ziehen, daß er sich auf einem ,,wirklich&#8220; be-<br/>schleunigten Bezugssystem befindet? Diese Frage ist zu ver-<br/>neinen; denn das vorhin genannte Verhalten frei beweglicher <br/>Massen relativ zu <span
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225 class="cmmi-12">K</span><span
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226 class="cmsy-10x-x-120">' </span>kann ebensogut auf folgende Weise ge-<br/>deutet werden. Das Bezugssystem <span
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227 class="cmmi-12">K</span><span
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228 class="cmsy-10x-x-120">' </span>ist unbeschleunigt; in <br/>dem betrachteten zeiträumlichen Gebiete herrscht aber ein <br/>Gravitationsfeld, welches die beschleunigte Bewegung der <br/>Körper relativ zu <span
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229 class="cmmi-12">K</span><span
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230 class="cmsy-10x-x-120">' </span>erzeugt.
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231 </p><!--l. 250--><p class="indent"> Diese Auffassung wird dadurch ermöglicht, daß uns die <br/>Erfahrung die Existenz eines Kraftfeldes (nämlich des Gravi-<br/>tationsfeldes) gelehrt hat, welches die merkwürdige Eigen-<br/>schaft hat, allen Körpein dieselbe Beschleunigung zu erteilen.<sup ><span
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232 class="cmr-8">1</span></sup>) <br/>Das mechanische Verhalten der Körper relativ zu <span
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233 class="cmmi-12">K</span><span
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234 class="cmsy-10x-x-120">' </span>ist das-<br/>selbe, wie es gegenüber Systemen sich der Erfahrung dar-<br/>bietet, die wir als, ,,ruhende&#8220; bzw. als ,,berechtigte&#8220; Systeme <br/>anzusehen gewohnt sind; deshalb liegt es auch vom physi-<br/>kalischen Standpunkt nahe, anzunehmen, daß die Systeme <span
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235 class="cmmi-12">K </span> <br/>und <span
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236 class="cmmi-12">K</span><span
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237 class="cmsy-10x-x-120">' </span>beide mit demselben Recht als ,,ruhend&#8220; angesehen <br/>werden können, bzw. daß sie als Bezugssysteme für die physi-<br/>kalische Beschreibung der Vorgänge gleichberechtigt seien.
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238 </p><!--l. 266--><p class="indent"> Aus diesen Erwägungen sieht man, daß die Durchführung <br/>der allgemeinen Relativitätstheorie zugleich zu einer Theorie der <br/>Gravitation führen muß; denn man kann ein Gravitations-<br/>feld durch bloße Änderung des Koordinatensystems ,,erzeugen&#8220;. <br/>Ebenso sieht man unmittelbar, daß das Prinzip von der Kon-<br/>stanz der Vakuum-Lichtgeschwindigkeit eine Modifikation er-<br/>fahren muß. Denn man erkennt leicht, daß die Bahn eines <br/>Lichtstrahles in bezug auf <span
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239 class="cmmi-12">K</span><span
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240 class="cmsy-10x-x-120">' </span>im allgemeinen eine krumme <br/>sein muß, wenn sich das Licht in bezug auf <span
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241 class="cmmi-12">K </span>geradlinig und <br/>mit bestimmter, konstanter Geschwindigkeit fortpflanzt.
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242 </p>
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243 <div class="center" >
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244
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245 <!--l. 279--><p class="noindent">
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246 </p><!--l. 282--><p class="noindent"><span
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247 class="cmbsy-10x-x-120">§ </span><span
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248 class="cmbx-12">3. Das Raum-Zeit-Kontinuum. Forderung der allgemeinen</span>
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249 <br/><span
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250 class="cmbx-12">Kovarianz f</span><span
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251 class="cmbx-12">ür die die allgemeinen Naturgesetze ausdr</span><span
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252 class="cmbx-12">ückenden</span>
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253 <br/><span
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254 class="cmbx-12">Gleichungen.</span></p></div>
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255 <!--l. 286--><p class="indent"> In der klassischen Mechanik sowie in der speziellen Rela-<br/>tivitätstheorie haben die Koordinaten des Raumes und der <br/>Zeit eine unmittelbare physikalische Bedeutung. Ein Punkt-<br/>ereignis hat die <span
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256 class="cmmi-12">X</span><sub ><span
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257 class="cmr-8">1</span></sub>-Koordinate <span
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258 class="cmmi-12">x</span><sub ><span
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259 class="cmr-8">1</span></sub>, bedeutet: Die nach den
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260 <br/>----------
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261 </p><!--l. 293--><p class="indent"> 1) Daß das Gravitationsfeld diese Eigenschaft mit großer Genauig-<br/>keit besitzt, hat Eötvös experimentell bewiesen. <pb/>
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262 </p><!--l. 299--><p class="indent">
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263
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264 </p><!--l. 300--><p class="noindent">Regeln der Euklidischen Geometrie mittels starrer Stäbe er-<br/>mittelte Projektion des Punktereignisses auf die <span
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265 class="cmmi-12">X</span><sub ><span
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266 class="cmr-8">1</span></sub>-Achse <br/>wird erhalten, indem man einen bestimmten Stab, den Ein-<br/>heitsmaßstab, <span
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267 class="cmmi-12">x</span><sub ><span
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268 class="cmr-8">1</span></sub>mal vom Anfangspunkt des Koordinaten-<br/>körpers auf der (positiven) <span
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269 class="cmmi-12">X</span><sub ><span
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270 class="cmr-8">1</span></sub>-Achse abträgt. Ein Punkt <br/>hat die
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271 <span
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272 class="cmmi-12">X</span><sub ><span
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273 class="cmr-8">4</span></sub>-Koordinate <span
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274 class="cmmi-12">x</span><sub ><span
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275 class="cmr-8">4</span></sub> = <span
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276 class="cmmi-12">t</span>, bedeutet: Eine relativ zum <br/>Koordinatensystem ruhend angeordnete, mit dem Punkt-<br/>ereignis räumlich (praktisch) zusammenfallende Einheitsuhr, <br/>welche nach bestimmten Vorschriften gerichtet ist, hat <span
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277 class="cmmi-12">x</span><sub ><span
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278 class="cmr-8">4</span></sub> = <span
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279 class="cmmi-12">t</span>
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280 <br/>Perioden zurückgelegt beim Eintreten des Punktereignisses.<sup ><span
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281 class="cmr-8">1</span></sup>)
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282 </p><!--l. 314--><p class="indent"> Diese Auffassung von Raum und Zeit schwebte den Phy-<br/>sikern stets, wenn auch meist unbewußt, vor, wie aus der <br/>Rolle klar erkennbar ist, welche diese Begriffe in der messenden <br/>Physik spielen; diese Auffassung mußte der Leser auch der <br/>zweiten Betrachtung des letzten Paragraphen zugrunde legen, <br/>um mit diesen Ausführungen einen Sinn verbinden zu können. <br/>Aber wir wollen nun zeigen, daß man sie fallen lassen und <br/>durch eine allgemeinere ersetzen muß, um das Postulat der <br/>allgemeinen Relativität durchführen zu können, falls die <br/>spezielle Relativitätstheorie für den Grenzfall des Fehlens <br/>eines Gravitationsfeldes zutrifft.
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283 </p><!--l. 328--><p class="indent"> Wir führen in einem Raume, der frei sei von Gravitations-<br/>feldern, ein Galileisches Bezugssystem <span
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284 class="cmmi-12">K </span>(<span
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285 class="cmti-12">x, y, z, t</span>) ein, und <br/>außerdem ein relativ zu <span
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286 class="cmmi-12">K</span>
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287 gleichförmig rotierendes Koordi-<br/>natensystem <span
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288 class="cmmi-12">K</span><span
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289 class="cmsy-10x-x-120">'</span><img
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290 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_19161x.png" alt=" ' ' ' ' (x, y , z t)" class="left" align="middle" /> Die Anfangspunkte beider Sy-<br/>steme sowie deren <span
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291 class="cmmi-12">Z</span>-Achsen mögen dauernd zusammenfallen. <br/>Wir wollen zeigen, daß für eine Raum--Zeitmessung im <br/>System <span
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292 class="cmmi-12">K</span><span
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293 class="cmsy-10x-x-120">' </span>die obige Festsetzung für die physikalische Bedeu-<br/>tung von Längen und Zeiten nicht aufrecht erhalten werden <br/>kann. Aus Symmetriegründen ist klar, daß ein Kreis um den <br/>Anfangspunkt in der <span
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294 class="cmti-12">X-Y </span>-Ebene von <span
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295 class="cmmi-12">K </span>zugleich als Kreis in der
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296 <br/><span
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297 class="cmmi-12">X</span><span
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298 class="cmsy-10x-x-120">'</span>-<span
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299 class="cmmi-12">Y </span><span
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300 class="cmsy-10x-x-120">'</span>-Ebene von <span
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301 class="cmmi-12">K</span><span
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302 class="cmsy-10x-x-120">' </span>aufgefaßt werden kann. Wir denken uns <br/>nun Umfang und Durchmesser dieses Kreises mit einem (relativ <br/>zum Radius unendlich kleinen) Einheitsmaßstabe ausgemessen <br/>und den Quotienten beider Meßresultate gebildet. Würde man <br/>dieses Experiment mit einem relativ zum Galileischen System
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303 <br/>----------
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304 </p><!--l. 349--><p class="indent"> 1) Die Konstatierbarkeit der ,,Gleichzeitigkeit&#8220; für räumlich un-<br/>mittelbar benachbarte Ereignisse, oder -- präziser gesagt -- für das <br/>raumzeitliche unmittelbare Benachbartsein (Koinzidenz) nehmen wir an, <br/>ohne für diesen fundamentalen Begriff eine Definition zu geben. <pb/>
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305 </p><!--l. 358--><p class="indent">
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306
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307 </p><!--l. 359--><p class="noindent"><span
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308 class="cmmi-12">K </span>ruhenden Maßstabe ausführen, so würde man als Quotienten <br/>die Zahl <span
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309 class="cmmi-12"><img
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310 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi12-19.png" alt="p" class="12x-x-19" /></span>
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311 erhalten. Das Resultat der mit einem relativ zu <br/><span
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312 class="cmmi-12">K</span><span
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313 class="cmsy-10x-x-120">' </span>ruhenden Maßstabe ausgeführten Bestimmung würde eine <br/>Zahl sein, die größer ist als <span
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314 class="cmmi-12"><img
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315 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi12-19.png" alt="p" class="12x-x-19" /></span>. Man erkennt dies leicht, wenn <br/>man den ganzen Meßprozeß vom ,,ruhenden&#8220; System <span
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316 class="cmmi-12">K </span>aus
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317 <br/>beurteilt und berücksichtigt, daß der peripherisch angelegte <br/>Maßstab eine Lorentzverkürzung erleidet, der radial angelegte <br/>Maßstab aber nicht. Es gilt daher in bezug auf <span
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318 class="cmmi-12">K</span><span
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319 class="cmsy-10x-x-120">' </span>nicht die <br/>Euklidische Geometrie; der oben festgelegte Koordinaten-<br/>begriff, welcher die Gültigkeit der Euklidischen Geometrie
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320 <br/>voraussetzt, versagt also mit Bezug auf das System <span
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321 class="cmmi-12">K</span><span
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322 class="cmsy-10x-x-120">'</span>. Ebenso-<br/>wenig kann man in <span
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323 class="cmmi-12">K</span><span
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324 class="cmsy-10x-x-120">' </span>eine den physikalischen Bedürfnissen <br/>entsprechende Zeit einführen, welche durch relativ zu <span
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325 class="cmmi-12">K</span><span
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326 class="cmsy-10x-x-120">'</span> <br/>ruhende, gleich beschaffene Uhren angezeigt wird. Um dies
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327 <br/>einzusehen, denke man sich im Koordinatenursprung und an <br/>der Peripherie des Kreises je eine von zwei gleich beschaffenen <br/>Uhren angeordnet und vom ,,ruhenden&#8220; System <span
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328 class="cmmi-12">K </span>aus be-<br/>trachtet. Nach einem bekannten Resultat der speziellen Rela-<br/>tivitätstheorie geht -- von <span
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329 class="cmmi-12">K </span>aus beurteilt -- die auf der
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330 <br/>Kreisperipherie angeordnete Uhr langsamer als die im Anfangs-<br/>punkt angeordnete Uhr, weil erstere Uhr bewegt ist letztere <br/>aber nicht. Ein im gemeinsamen Koordinatenursprung be-<br/>findlicher Beobachter, welcher auch die an der Peripherie <br/>befindliche Uhr mittels des Lichtes zu beobachten fähig wäre,
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331 <br/>würde also die an der Peripherie angeordnete Uhr langsamer <br/>gehen sehen als die neben ihm angeordnete Uhr. Da er sich <br/>nicht dazu entschließen wird, die Lichtgeschwindigkeit auf <br/>dem in Betracht kommenden Wege explizite von der Zeit <br/>abhängen zu lassen, wird er seine Beobachtung dahin inter-<br/>pretieren, daß die Uhr an der Peripherie ,,wirklich&#8220; lang-<br/>samer gehe als die im Ursprung angeordnete. Er wird also <br/>nicht umhin können, die Zeit so zu definieren, daß die Gang-<br/>geschwindigkeit einer Uhr vom Orte abhängt.
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332 </p><!--l. 400--><p class="indent"> Wir gelangen also zu dem Ergebnis: In der allgemeinen <br/>Relativitätstheorie können Raum- und Zeitgrößen nicht so <br/>definiert werden, daß räumliche Koordinatendifferenzen un-<br/>mittelbar mit dem Einheitsmaßstab, zeitliche mit einer Normal-<br/>uhr gemessen werden könnten.
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333 </p><!--l. 407--><p class="indent"> Das bisherige Mittel, in das zeiträumliche Kontinuum <br/>in bestimmter Weise Koordinaten zu legen, versagt also, und <br/><pb/>
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334 </p><!--l. 412--><p class="indent">
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335
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336 </p><!--l. 413--><p class="noindent">es scheint sich auch kein <span
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337 class="cmti-12">anderer </span>Weg darzubieten, der ge-<br/>statten würde, der vierdimensionalen Welt Koordinatensysteme <br/>so anzupassen, daß bei ihrer Verwendung eine besonders <br/>einfache Formulierung der Naturgesetze zu erwarten wäre. <br/>Es bleibt daher nichts anderes übrig, als alle denkbaren<sup ><span
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338 class="cmr-8">1</span></sup>)
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339 <br/>Koordinatensysteme als für die Naturbeschreibung prinzipiell <br/>gleichberechtigt anzusehen. Dies kommt auf die Forderung <br/>hinaus:
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340 </p><!--l. 426--><p class="indent"> <span
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341 class="cmti-12">Die allgemeinen Naturgesetze sind durch Gleichungen aus- </span> <br/><span
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342 class="cmti-12">zudr</span><span
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343 class="cmti-12">ücken, die f</span><span
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344 class="cmti-12">ür</span>
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345 <span
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346 class="cmti-12">alle Koordinatensysteme gelten, d. h. die </span> <br/><span
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347 class="cmti-12">beliebigen Substitutionen gegen</span><span
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348 class="cmti-12">über</span>
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349 <span
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350 class="cmti-12">kovariant (allgemein ko- </span> <br/><span
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351 class="cmti-12">variant) sind.</span>
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352 </p><!--l. 428--><p class="indent"> Es ist klar, daß eine Physik, welche diesem Postulat ge-<br/>nügt, dem allgemeinen Relativitätspostulat gerecht wird. <br/>Denn in <span
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353 class="cmti-12">allen </span>Substitutionen sind jedenfalls auch diejenigen <br/>enthalten, welche allen Relativbewegungen der (dreidimen-<br/>sionalen) Koordinatensysteme entsprechen. Daß diese Forde-<br/>rung der allgemeinen Kovarianz, welche dem Raum und der <br/>Zeit den letzten Rest physikalischer Gegenständlichkeit nehmen, <br/>eine natürliche Forderung ist, geht aus folgender Überlegung <br/>hervor. Alle unsere zeiträumlichen Konstatierungen laufen <br/>stets auf die Bestimmung zeiträumlicher Koinzidenzen hinaus. <br/>Bestände beispielsweise das Geschehen nur in der Bewegung <br/>materieller Punkte, so wäre letzten Endes nichts beobachtbar <br/>als die Begegnungen zweier oder mehrerer dieser Punkte. <br/>Auch die Ergebnisse unserer Messungen sind nichts anderes <br/>als die Konstatierung derartiger Begegnungen materieller <br/>Punkte unserer Maßstäbe mit anderen materiellen Punkten <br/>bzw. Koinzidenzen zwischen Uhrzeigern, Zifferblattpunkten <br/>und ins Auge gefaßten, am gleichen Orte und zur gleichen <br/>Zeit stattfindenden Punktereignissen.
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354 </p><!--l. 451--><p class="indent"> Die Einführung eines Bezugssystems dient zu nichts <br/>anderem als zur leichteren Beschreibung der Gesamtheit <br/>solcher Koinzidenzen. Man ordnet der Welt vier zeiträum-<br/>liche Variable <span
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355 class="cmmi-12">x</span><sub ><span
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356 class="cmr-8">1</span></sub>, <span
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357 class="cmmi-12">x</span><sub ><span
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358 class="cmr-8">2</span></sub>, <span
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359 class="cmmi-12">x</span><sub ><span
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360 class="cmr-8">3</span></sub>, <span
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361 class="cmmi-12">x</span><sub ><span
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362 class="cmr-8">4</span></sub> zu, derart, daß jedem Punkt-<br/>ereignis ein Wertesystem der Variablen <span
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363 class="cmmi-12">x</span><sub ><span
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364 class="cmr-8">1</span></sub><span
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365 class="cmmi-12">...</span><span
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366 class="cmmi-12">.x</span><sub ><span
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367 class="cmr-8">4</span></sub> entspricht, <br/>Zwei koinzidierenden Punktereignissen entspricht dasselbe <br/>----------
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368 </p><!--l. 460--><p class="indent"> 1) Von gewissen Beschränkungen, welche der Forderung der ein-<br/>deutigen Zuordnung und derjenigen der Stetigkeit entsprechen, wollen <br/>wir hier nicht sprechen. <pb/>
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369 </p><!--l. 466--><p class="indent">
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370
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371 </p><!--l. 467--><p class="noindent">Wertesystem der Variablen <span
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372 class="cmmi-12">x</span><sub ><span
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373 class="cmr-8">1</span></sub><span
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374 class="cmmi-12">...</span><span
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375 class="cmmi-12">.x</span><sub ><span
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376 class="cmr-8">4</span></sub>; d. h. die Koinzidenz <br/>ist durch die Übereinstimmung der Koordinaten charak-<br/>terisiert. Führt man statt der Variablen <span
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377 class="cmmi-12">x</span><sub ><span
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378 class="cmr-8">1</span></sub><span
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379 class="cmmi-12">...</span><span
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380 class="cmmi-12">.x</span><sub ><span
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381 class="cmr-8">4</span></sub> beliebige <br/>Funktionen derselben, <span
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382 class="cmmi-12">x</span><sub ><span
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383 class="cmr-8">1</span></sub><sup ><span
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384 class="cmsy-8">'</span></sup>, <span
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385 class="cmmi-12">x</span><sub >
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386 <span
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387 class="cmr-8">2</span></sub><sup ><span
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388 class="cmsy-8">'</span></sup>, <span
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389 class="cmmi-12">x</span><sub >
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390 <span
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391 class="cmr-8">3</span></sub><sup ><span
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392 class="cmsy-8">'</span></sup>, <span
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393 class="cmmi-12">x</span><sub >
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394 <span
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395 class="cmr-8">4</span></sub><sup ><span
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396 class="cmsy-8">'</span></sup> als neues Koordinaten-<br/>system ein, so daß die Wertesysteme einander eindeutig zu-<br/>geordnet sind, so ist die Gleichheit aller vier Koordinaten <br/>auch im neuen System der Ausdruck für die raumzeitliche <br/>Koinzidenz zweier Punktereignisse. Da sich alle unsere physi-<br/>kalischen Erfahrungen letzten Endes auf solche Koinzidenzen <br/>zurückführen lassen, ist zunächst kein Grund vorhanden, <br/>gewisse Koordinatensysteme vor anderen zu bevorzugen, d. h. <br/>wir gelangen zu der Forderung der allgemeinen Kovarianz.
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397 </p>
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398 <div class="center" >
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399
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400 <!--l. 483--><p class="noindent">
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401 </p><!--l. 486--><p class="noindent"><span
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402 class="cmbsy-10x-x-120">§ </span><span
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403 class="cmbx-12">4. Beziehung der vier Koordinaten zu r</span><span
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404 class="cmbx-12">äumlichen und zeit-</span>
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405 <br/><span
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406 class="cmbx-12">lichen Meßergebnissen. </span> <br/><span
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407 class="cmbx-12">Analytischer Ausdruck f</span><span
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408 class="cmbx-12">ür das</span>
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409 <span
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410 class="cmbx-12">Gravitationsfeld.</span></p></div>
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411 <!--l. 490--><p class="indent"> Es kommt mir in dieser Abhandlung nicht darauf an, <br/>die allgemeine Relativitätstheorie als ein möglichst einfaches <br/>logisches System mit einem Minimum von Axiomen darzu-<br/>stellen. Sondern es ist mein Hauptziel, diese Theorie so zu <br/>entwickeln, daß der Leser die psychologische Natürlichkeit <br/>des eingeschlagenen Weges empfindet und daß die zugrunde <br/>gelegten Voraussetzungen durch die Erfahrung möglichst ge-<br/>sichert erscheinen. In diesem Sinne sei nun die Voraus-<br/>setzung eingeführt:
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412 </p><!--l. 501--><p class="indent"> Für unendlich kleine vierdimensionale Gebiete ist die <br/>Relativitätstheorie im engeren Sinne bei passender Koordi-<br/>natenwahl zutreffend.
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413 </p><!--l. 505--><p class="indent"> Der Beschleunigungszustand des unendlich kleinen (,,ört-<br/>lichen&#8220;) Koordinatensystems ist hierbei so zu wählen, daß <br/>ein Gravitationsfeld nicht auftritt; dies ist für ein unendlich <br/>kleines Gebiet möglich. <span
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414 class="cmmi-12">X</span><sub ><span
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415 class="cmr-8">1</span></sub>, <span
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416 class="cmmi-12">X</span><sub ><span
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417 class="cmr-8">2</span></sub>, <span
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418 class="cmmi-12">X</span><sub ><span
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419 class="cmr-8">3</span></sub> seien die räumlichen <br/>Koordinaten; <span
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420 class="cmmi-12">X</span><sub ><span
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421 class="cmr-8">4</span></sub> die zugehörige, in geeignetem Maßstabe ge-<br/>messene<sup ><span
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422 class="cmr-8">1</span></sup>) Zeitkoordinate. Diese Koordinaten haben, wenn <br/>ein starres Stäbchen als Einheitsmaßstab gegeben gedacht <br/>wird, bei gegebener Orientierung des Koordinatensystems <br/>eine unmittelbare physikalische Bedeutung im Sinne der
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423 <br/>speziellen Relativitätstheorie. Der Ausdruck</p>
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424 <table width="100%"
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425 class="equation"><tr><td><a
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426 id="x1-2r1"></a>
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427 <center class="math-display" >
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428 <img
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429 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_19162x.png" alt="d s2 = - dX12 - dX22 - dX32 + dX42 " class="math-display" /></center></td><td width="5%">(1)</td></tr></table>
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430 <!--l. 521--><p class="nopar"></p><!--l. 524--><p class="noindent">----------
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431 </p><!--l. 527--><p class="indent"> 1) Die Zeiteinheit ist so zu wählen, daß die Vakuum-Lichtgeschwindig-<br/>keit -- in dem ,,lokalen&#8220; Koordinatensystem gemessen -- gleich 1 wird.<pb/>
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432 </p><!--l. 532--><p class="indent">
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433
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434 </p><!--l. 533--><p class="noindent">hat dann nach der speziellen Relativitätstheorie einen von <br/>der Orientierung des lokalen Koordinatensystems unabhängigen, <br/>durch Raum--Zeitmessung ermittelbaren Wert. Wir nennen <br/><span
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435 class="cmmi-12">ds </span>die Größe des zu den unendlich benachbarten Punkten <br/>des vierdimensionalen Raumes gehörigen Linienelementes. Ist <br/>das zu dem Element <img
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436 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_19163x.png" alt="(d X1 ....d X4)" class="left" align="middle" /> gehörige <span
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437 class="cmmi-12">ds</span><sup ><span
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438 class="cmr-8">2</span></sup> positiv, <br/>so nennen wir mit Minkowski ersteres zeitartig, im entgegen-<br/>gesetzten Falle raumartig.
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439 </p><!--l. 545--><p class="indent"> Zu dem betrachteten ,,Linienelement&#8220; bzw. zu den beiden <br/>unendlich benachbarten Punktereignissen gehören auch be-<br/>stimmte Differentiale <span
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440 class="cmmi-12">dx</span><sub ><span
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441 class="cmr-8">1</span></sub><span
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442 class="cmmi-12">...</span><span
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443 class="cmmi-12">.dx</span><sub ><span
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444 class="cmr-8">4</span></sub>
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445 der vierdimensionalen Ko-<br/>ordinaten des gewählten Bezugssystems. Ist dieses sowie ein <br/>,,lokales&#8220; System obiger Art für die betrachtete Stelle gegeben, <br/>so werden sich hier die <span
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446 class="cmmi-12">dX</span><sub ><span
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447 class="cmmi-8"><img
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448 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub> durch bestimmte lineare homogene <br/>Ausdrücke der <span
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449 class="cmmi-12">dx</span><sub ><span
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450 class="cmmi-8"><img
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451 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-1b.png" alt="s" class="8x-x-1b" /></span></sub>
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452 darstellen lassen:</p>
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453 <table width="100%"
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454 class="equation"><tr><td><a
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455 id="x1-3r2"></a>
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456 <center class="math-display" >
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457 <img
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458 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_19164x.png" alt="d X = sum a d x . n ns s s " class="math-display" /></center></td><td width="5%">(2)</td></tr></table>
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459 <!--l. 558--><p class="nopar">
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460 </p><!--l. 561--><p class="noindent">Setzt man diese Ausdrücke in (1) ein, so erhält man</p>
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461 <table width="100%"
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462 class="equation"><tr><td><a
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463 id="x1-4r3"></a>
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464 <center class="math-display" >
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465 <img
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466 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_19165x.png" alt="ds2 = sum g dx dx , st s t st " class="math-display" /></center></td><td width="5%">(3)</td></tr></table>
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467 <!--l. 568--><p class="nopar">
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468
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469 </p><!--l. 571--><p class="noindent">wobei die <span
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470 class="cmmi-12">g</span><sub ><span
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471 class="cmmi-8"><img
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472 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-1b.png" alt="s" class="8x-x-1b" /><img
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473 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-1c.png" alt="t" class="8x-x-1c" /></span></sub> Funktionen der <span
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474 class="cmmi-12">x</span><sub ><span
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475 class="cmmi-8"><img
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476 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-1b.png" alt="s" class="8x-x-1b" /></span></sub> sein werden, die nicht mehr <br/>von der Orientierung und dem Bewegungszustand des ,,lokalen&#8220; <br/>Koordinatensystems abhängen können; denn <span
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477 class="cmmi-12">ds</span><sup ><span
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478 class="cmr-8">2</span></sup> ist eine <br/>durch Maßstab-Uhrenmessung ermittelbare, zu den betrach-<br/>teten, zeiträumlich unendlich benachbarten Punktereignissen <br/>gehörige, unabhängig von jeder besonderen Koordinatenwahl <br/>definierte Größe. Die <span
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479 class="cmmi-12">g</span><sub ><span
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482 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-1c.png" alt="t" class="8x-x-1c" /></span></sub> sind hierbei so zu wählen, daß <br/><span
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483 class="cmmi-12">g</span><sub ><span
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487 class="cmmi-12">g</span><sub ><span
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493 und <span
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495 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi12-1c.png" alt="t" class="12x-x-1c" /> </span> <br/>zu erstrecken, so daß die Summe aus 4 <span
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496 class="cmsy-10x-x-120">× </span>4 Summanden be-<br/>steht, von denen 12 paarweise gleich sind.
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497 </p><!--l. 586--><p class="indent"> Der Fall der gewöhnlichen Relativitätstheorie geht aus <br/>dem hier Betrachteten hervor, falls es, vermöge des beson-<br/>deren Verhaltens der <span
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498 class="cmmi-12">g</span><sub ><span
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499 class="cmmi-8"><img
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500 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-1b.png" alt="s" class="8x-x-1b" /><img
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501 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-1c.png" alt="t" class="8x-x-1c" /></span></sub> in einem endlichen Gebiete, möglich <br/>ist, in diesem das Bezugssystem so zu wählen, daß die <span
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502 class="cmmi-12">g</span><sub ><span
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503 class="cmmi-8"><img
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504 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-1b.png" alt="s" class="8x-x-1b" /><img
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505 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-1c.png" alt="t" class="8x-x-1c" /></span></sub> die
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506 <br/>konstanten Werte</p>
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507 <table width="100%"
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508 class="equation"><tr><td><a
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509 id="x1-5r4"></a>
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510 <center class="math-display" >
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511 <img
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512 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_19166x.png" alt=" - 1 0 0 0
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513 { 0 - 1 0 0 0 0 - 1 0
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514 0 0 0 +1 " class="math-display" /></center></td><td width="5%">(4)</td></tr></table>
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515 <!--l. 616--><p class="nopar">
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516 <pb/></p><!--l. 623--><p class="indent">
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517
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518 </p><!--l. 624--><p class="noindent">annehmen. Wir werden später sehen, daß die Wahl solcher Ko-<br/>ordinaten für endliche Gebiete im allgemeinen nicht möglich ist.
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519 </p><!--l. 628--><p class="indent"> Aus den Betrachtungen der <span
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520 class="cmsy-10x-x-120">§§ </span>2 und 3 geht hervor, <br/>daß die Größen <span
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521 class="cmmi-12">g</span><sub ><span
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522 class="cmmi-8"><img
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523 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-1b.png" alt="s" class="8x-x-1b" /><img
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524 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-1c.png" alt="t" class="8x-x-1c" /></span></sub> vom physikalischen Standpunkte aus als <br/>diejenigen Größen anzusehen sind, welche das Gravitations-<br/>feld in bezug auf das gewählte Bezugssystem beschreiben. <br/>Nehmen wir nämlich zunächst an, es sei für ein gewisses be-<br/>trachtetes vierdimensionales Gebiet bei geeigneter Wahl der <br/>Koordinaten die spezielle Relativitätstheorie gültig. Die <span
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525 class="cmmi-12">g</span><sub ><span
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526 class="cmmi-8"><img
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527 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-1b.png" alt="s" class="8x-x-1b" /><img
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528 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-1c.png" alt="t" class="8x-x-1c" /></span></sub> <br/>haben dann die in (4) angegebenen Werte. Ein freier materieller
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529 <br/>Punkt bewegt sich dann bezüglich dieses Systems geradlinig <br/>gleichförmig. Führt man nun durch eine beliebige Substitution <br/>neue Raum--Zeitkoordinaten <span
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530 class="cmmi-12">x</span><sub ><span
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531 class="cmr-8">1</span></sub><span
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532 class="cmmi-12">...</span><span
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533 class="cmmi-12">.x</span><sub ><span
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534 class="cmr-8">4</span></sub>
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535 ein, so werden in <br/>diesem neuen System die <span
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536 class="cmmi-12">g</span><sub ><span
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537 class="cmmi-8"><img
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538 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img
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539 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub> nicht mehr Konstante, sondern
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540 <br/>Raum--Zeitfunktionen sein. Gleichzeitig wird sich die Be-<br/>wegung des freien Massenpunktes in den neuen Koordinaten <br/>als eine krummlinige, nicht gleichförmige, darstellen, wobei <br/>dies Bewegungsgesetz unabhängig sein wird von der Natur <br/>des bewegten Massenpunktes. Wir werden also diese Be-<br/>wegung als eine solche unter dem Einfluß eines Gravitations-<br/>feldes deuten. Wir sehen das Auftreten eines Gravitations-<br/>feldes geknüpft an eine raumzeitliche Veränderlichkeit der <span
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541 class="cmmi-12">g</span><sub ><span
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542 class="cmmi-8"><img
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543 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-1b.png" alt="s" class="8x-x-1b" /><img
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544 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-1c.png" alt="t" class="8x-x-1c" /></span></sub><span
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545 class="cmmi-12">. </span> <br/>Auch in dem allgemeinen Falle, daß wir nicht in einem end-<br/>lichen Gebiete bei passender Koordinatenwahl die Gültigkeit <br/>der speziellen Relativitätstheorie herbeiführen können, werden <br/>wir an der Auffassung festzuhalten haben, daß die <span
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546 class="cmmi-12">g</span><sub ><span
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547 class="cmmi-8"><img
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548 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-1b.png" alt="s" class="8x-x-1b" /><img
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549 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-1c.png" alt="t" class="8x-x-1c" /></span></sub> das <br/>Gravitationsfeld beschreiben.
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550 </p><!--l. 660--><p class="indent"> Die Gravitation spielt also gemäß der allgemeinen Rela-<br/>tivitätstheorie eine Ausnahmerolle gegenüber den übrigen, ins-<br/>besondere den elektromagnetischen Kräften, indem die das <br/>Gravitationsfeld darstellenden 10 Funktionen <span
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551 class="cmmi-12">g</span><sub ><span
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552 class="cmmi-8"><img
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553 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-1b.png" alt="s" class="8x-x-1b" /><img
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554 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-1c.png" alt="t" class="8x-x-1c" /></span></sub>
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555 zugleich die <br/>metrischen Eigenschaften des vierdimensionalen Meßraumes
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556 <br/>bestimmen.
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557 </p>
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558 <div class="center" >
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559
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560 <!--l. 668--><p class="noindent">
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561 </p><!--l. 670--><p class="noindent"><span
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562 class="cmbx-12">B. Mathematische Hilfsmittel f</span><span
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563 class="cmbx-12">ür die Aufstellung allgemein</span>
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564 <br/><span
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565 class="cmbx-12">kovarianter Gleichungen.</span></p></div>
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566 <!--l. 674--><p class="indent"> Nachdem wir im vorigen gesehen haben, daß das all-<br/>gemeine Relativitätspostulat zu der Forderung führt, daß die <br/>Gleichungssysteme der Physik beliebigen Substitutionen der <br/>Koordinaten <span
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567 class="cmmi-12">x</span><sub ><span
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568 class="cmr-8">1</span></sub><span
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569 class="cmmi-12">...</span><span
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570 class="cmmi-12">.x</span><sub ><span
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571 class="cmr-8">4</span></sub> gegenüber kovariant sein müssen,
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572 <br/><pb/>
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573 </p><!--l. 681--><p class="indent">
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574
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575 </p><!--l. 682--><p class="noindent">haben wir zu überlegen, wie derartige allgemein kovariante <br/>Gleichungen gewonnen werden können. Dieser rein mathe-<br/>matischen Aufgabe wenden wir uns jetzt zu; es wird sich <br/>dabei zeigen, daß bei deren Lösung die in Gleichung (3) an-<br/>gegebene Invariante <span
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576 class="cmmi-12">ds </span>eine fundamentale Rolle spielt, welche <br/>wir in Anlehnung an die Gausssche Flächentheorie als ,,Linien-<br/>element&#8220; bezeichnet haben.
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577 </p><!--l. 691--><p class="indent"> Der Grundgedanke dieser allgemeinen Kovariantentheorie <br/>ist folgender. Es seien gewisse Dinge (,,Tensoren&#8220;) mit Bezug <br/>auf jedes Koordinatensystem definiert durch eine Anzahl <br/>Raumfunktionen, welche die ,,Komponenten&#8220; des Tensors <br/>genannt werden. Es gibt dann gewisse Regeln, nach welchen <br/>diese Komponenten für ein neues Koordinatensystem be-<br/>rechnet werden, wenn sie für das ursprüngliche System be-<br/>kannt sind, und wenn die beide Systeme verknüpfende Trans-<br/>formation bekannt ist. Die nachher als Tensoren bezeichneten <br/>Dinge sind ferner dadurch gekennzeichnet, daß die Trans-<br/>formationsgleichungen für ihre Komponenten linear und homo-<br/>gen sind. Demnach verschwinden sämtliche Komponenten im <br/>neuen System, wenn sie im ursprünglichen System sämtlich <br/>verschwinden. Wird also ein Naturgesetz durch das Null-<br/>setzen aller Komponenten eines Tensors formuliert, so ist es <br/>allgemein kovariant; indem wir die Bildungsgesetze der Ten-<br/>soren untersuchen, erlangen wir die Mittel zur Aufstellung all-<br/>gemein kovarianter Gesetze.
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578 </p>
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579 <div class="center" >
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580
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581 <!--l. 712--><p class="noindent">
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582 </p><!--l. 714--><p class="noindent"><span
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583 class="cmbsy-10x-x-120">§ </span><span
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584 class="cmbx-12">5. Kontravarianter und kovarianter Vierervektor.</span></p></div>
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585 <!--l. 718--><p class="indent"> <span
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586 class="cmti-12">Kontravarianter Vierervektor. </span>Das Linienelement ist defi-<br/>niert durch die vier ,,Komponenten&#8220; <span
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587 class="cmmi-12">dx</span><sub ><span
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588 class="cmmi-8"><img
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589 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub><span
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590 class="cmmi-12">, </span>deren Trans-<br/>formationsgesetz durch die Gleichung</p>
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591 <table width="100%"
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592 class="equation"><tr><td><a
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593 id="x1-6r5"></a>
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594 <center class="math-display" >
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595 <img
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596 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_19167x.png" alt=" sum @ xs' dxs'= -----d xn n @ xn " class="math-display" /></center></td><td width="5%">(5)</td></tr></table>
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597
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598 <!--l. 727--><p class="nopar">
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599 </p><!--l. 732--><p class="noindent">ausgedrückt wird. Die <span
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600 class="cmmi-12">dx</span><sub ><span
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601 class="cmmi-8"><img
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602 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-1b.png" alt="s" class="8x-x-1b" /></span></sub><sup ><span
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603 class="cmsy-8">'</span></sup> drücken sich linear und homogen <br/>durch die <span
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604 class="cmmi-12">dx</span><sub >
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605 <span
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606 class="cmmi-8"><img
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607 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub> aus; wir können diese Koordinatendifferentiale <br/><span
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608 class="cmmi-12">dx</span><sub ><span
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609 class="cmmi-8"><img
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610 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub> daher als die Komponenten eines ,,Tensors&#8220; ansehen, den <br/>wir speziell als kontravarianten Vierervektor bezeichnen. Jedes <br/>Ding, was bezüglich des Koordinatensystems durch vier <br/>Größen <span
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611 class="cmmi-12">A</span><sup ><span
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612 class="cmmi-8"><img
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613 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sup> definiert ist, die sich nach demselben Gesetz</p>
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614 <table width="100%"
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615 class="equation"><tr><td><a
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616 id="x1-7r6"></a>
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617 <center class="math-display" >
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618 <img
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619 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_19168x.png" alt=" s' sum @-xs' n A = @ xn A n " class="math-display" /></center></td><td width="5%">(5a)</td></tr></table>
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620 <!--l. 746--><p class="nopar">
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621 <pb/></p><!--l. 754--><p class="indent">
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622
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623 </p><!--l. 755--><p class="noindent">transformieren, bezeichnen wir ebenfalls als kontravarianten <br/>Vierervektor. Aus (5a) folgt sogleich, daß die Summen <img
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624 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_19169x.png" alt="(As ± Bs)" class="left" align="middle" /> <br/>ebenfalls Komponenten eines Vierervektors sind, wenn <span
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625 class="cmmi-12">A</span><sup ><span
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626 class="cmmi-8"><img
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627 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-1b.png" alt="s" class="8x-x-1b" /></span></sup> und <br/><span
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628 class="cmmi-12">B</span><sup ><span
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629 class="cmmi-8"><img
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630 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-1b.png" alt="s" class="8x-x-1b" /></span></sup> es sind. Entsprechendes gilt für alle später als ,,Tensoren&#8220; <br/>einzuführenden Systeme (Regel von der Addition und Sub-<br/>traktion der Tensoren).
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631 </p><!--l. 764--><p class="indent"> <span
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632 class="cmti-12">Kovarianter Vierervektor. </span>Vier Größen <span
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633 class="cmmi-12">A</span><sub ><span
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634 class="cmmi-8"><img
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635 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub> nennen wir die <br/>Komponenten eines kovarianten Vierervektors, wenn für jede <br/>beliebige Wahl des kontravarianten Vierervektors <span
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636 class="cmmi-12">B</span><sup ><span
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637 class="cmmi-8"><img
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638 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sup>
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639 </p>
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640 <table width="100%"
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641 class="equation"><tr><td><a
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642 id="x1-8r6"></a>
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643 <center class="math-display" >
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644 <img
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645 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_191610x.png" alt=" sum n An B = Invariante. n " class="math-display" /></center></td><td width="5%">(6)</td></tr></table>
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646 <!--l. 774--><p class="nopar">
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647 </p><!--l. 777--><p class="noindent">Aus dieser Definition folgt das Transformationsgesetz des <br/>kovarianten Vierervektors. Ersetzt man nämlich auf der <br/>rechten Seite der Gleichung
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648 </p>
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649 <center class="par-math-display" >
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650 <img
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651 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_191611x.png" alt=" sum ' sum
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652 As'Bs = An Bn s n " class="par-math-display" /></center>
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653 <!--l. 786--><p class="nopar">
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654 </p><!--l. 789--><p class="noindent"><span
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655 class="cmmi-12">B</span><sup ><span
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656 class="cmmi-8"><img
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657 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sup> durch den aus der Umkehrung der Gleichung (5a) folgenden <br/>Ausdruck
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658 </p>
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659 <center class="par-math-display" >
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660 <img
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661 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_191612x.png" alt=" sum
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662 @-xnBs', s @ xs' " class="par-math-display" /></center>
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663 <!--l. 797--><p class="nopar">
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664 </p><!--l. 801--><p class="noindent">so erhält man
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665 </p>
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666 <center class="par-math-display" >
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667 <img
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668 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_191613x.png" alt=" sum sum @ x sum
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669 Bs' ---n-An = Bs'As'. s n @ xs' s " class="par-math-display" /></center>
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670 <!--l. 809--><p class="nopar">
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671 </p><!--l. 812--><p class="noindent">Hieraus folgt aber, weil in dieser Gleichung die <span
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672 class="cmmi-12">B</span><sup ><span
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673 class="cmmi-8"><img
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674 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-1b.png" alt="s" class="8x-x-1b" /></span><sup ><span
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675 class="cmsy-6">'</span></sup></sup> unabhängig <br/>voneinander frei wählbar sind, das Transformationsgesetz</p>
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676 <table width="100%"
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677 class="equation"><tr><td><a
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678 id="x1-9r7"></a>
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679 <center class="math-display" >
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680 <img
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681 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_191614x.png" alt=" sum
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682 A '= @-xn-A . s @ xs' n " class="math-display" /></center></td><td width="5%">(7)</td></tr></table>
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683 <!--l. 821--><p class="nopar">
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684 </p><!--l. 826--><p class="noindent"><span
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685 class="cmbx-12">Bemerkungzur Vereinfachung der Schreibweise der Ausdr</span><span
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686 class="cmbx-12">ücke.</span> <br/></p><!--l. 829--><p class="indent">Ein Blick auf die Gleichungen dieses Paragraphen zeigt, <br/>daß über Indizes, die zweimal unter einem Summenzeichen <br/>auftreten [z. B. der Index <span
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687 class="cmmi-12"><img
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688 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi12-17.png" alt="n" class="12x-x-17" /></span>
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689 in (5)], stets summiert wird, <br/>und zwar <span
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690 class="cmti-12">nur </span>über zweimal auftretende Indizes. Es ist des-<br/>halb möglich, ohne die Klarheit zu beeinträchtigen, die
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691 <br/>Summenzeichen wegzulassen. Dafür führen wir die Vorschrift <br/>ein: Tritt ein Index in einem Term eines Ausdruckes zweimal <br/>auf, so ist über ihn stets zu summieren, wenn nicht ausdrück-<br/>lich das Gegenteil bemerkt ist.
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692 </p><!--l. 841--><p class="indent"> Der Unterschied zwischen dem kovarianten und kontra-<br/>varianten Vierervektor liegt in dem Transformationsgesetz <br/><pb/>
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693 </p><!--l. 846--><p class="indent">
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694
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695 </p><!--l. 847--><p class="noindent">[(7) bzw. (5)]. Beide Gebilde sind Tensoren im Sinne der <br/>obigen allgemeinen Bemerkung; hierin liegt ihre Bedeutung. <br/>Im Anschluß an Ricci und Levi-Civita wird der kontra-<br/>variante Charakter durch oberen, der kovariante durch unteren
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696 <br/>Index bezeichnet.
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697 </p>
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698 <div class="center" >
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699
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700 <!--l. 854--><p class="noindent">
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701 </p><!--l. 855--><p class="noindent"><span
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702 class="cmbsy-10x-x-120">§ </span><span
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703 class="cmbx-12">6. Tensoren zweiten und h</span><span
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704 class="cmbx-12">öheren Ranges.</span></p></div>
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705 <!--l. 859--><p class="indent"> <span
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706 class="cmti-12">Kontravarianter Tensor. </span>Bilden wir sämtliche 16 Produkte <br/><span
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707 class="cmmi-12">A</span><sup ><span
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708 class="cmmi-8"><img
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709 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img
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710 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sup> der Komponenten <span
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711 class="cmmi-12">A</span><sup ><span
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712 class="cmmi-8"><img
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713 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /></span></sup> und <span
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714 class="cmmi-12">B</span><sup ><span
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715 class="cmmi-8"><img
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716 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sup> zweier kontravarianten <br/>Vierervektoren
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717 </p>
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718 <table width="100%"
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719 class="equation"><tr><td><a
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720 id="x1-10r8"></a>
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721 <center class="math-display" >
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722 <img
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723 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_191615x.png" alt="Amn = Am Bn, " class="math-display" /></center></td><td width="5%">(8)</td></tr></table>
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724 <!--l. 866--><p class="nopar">
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725 </p><!--l. 871--><p class="noindent">so erfüllt <span
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726 class="cmmi-12">A</span><sup ><span
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727 class="cmmi-8"><img
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728 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img
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729 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sup> gemäß (8) und (5a) das Transformationsgesetz
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730 </p>
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731 <table width="100%"
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732 class="equation"><tr><td><a
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733 id="x1-11r9"></a>
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734 <center class="math-display" >
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735 <img
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736 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_191616x.png" alt=" ' ' Ast'= @-xs- @ xt-Am n. @ xm @ xn " class="math-display" /></center></td><td width="5%">(9)</td></tr></table>
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737 <!--l. 879--><p class="nopar">
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738
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739 </p><!--l. 884--><p class="indent"> Wir nennen ein Ding, das bezüglich eines jeden Bezugs-<br/>systems durch 16 Größen (Funktionen) beschrieben wird, die <br/>das Transformationsgesetz (9) erfüllen, einen kontravarianten <br/>Tensor zweiten Ranges. Nicht jeder solcher Tensor läßt sich
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740 <br/>gemäß (8) aus zwei Vierervektoren bilden. Aber es ist leicht <br/>zu beweisen, daß sich 16 beliebig gegebene <span
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741 class="cmmi-12">A</span><sup ><span
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742 class="cmmi-8"><img
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743 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img
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744 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sup> darstellen <br/>lassen als die Summe der <span
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745 class="cmmi-12">A</span><sup ><span
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747 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /></span></sup> <span
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748 class="cmmi-12">B</span><sup ><span
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749 class="cmmi-8"><img
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750 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sup> von vier geeignet gewählten <br/>Paaren von Vierervektoren. Deshalb kann man beinahe alle
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751 <br/>Sätze, die für den durch (9) definierten Tensor zweiten Ranges <br/>gelten, am einfachsten dadurch beweisen, daß man sie für <br/>spezielle Tensoren vom Typus (8) dartut.
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752 </p><!--l. 899--><p class="indent"> <span
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753 class="cmti-12">Kontravarianter Tensor beliebigen Ranges. </span>Es ist klar, daß <br/>man entsprechend (8) und (9) auch kontravariante Tensoren <br/>dritten und höheren Ranges definieren kann mit 4<sup ><span
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754 class="cmr-8">3</span></sup> usw. <br/>Komponenten. Ebenso erhellt aus (8) und (9), daß man in
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755 <br/>diesem Sinne den kontravarianten Vierervektor als kontra-<br/>varianten Tensor ersten Ranges auffassen kann.
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756 </p><!--l. 907--><p class="indent"> <span
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757 class="cmti-12">Kovarianter Tensor. </span>Bildet man andererseits die 16 Pro-<br/>dukte <span
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758 class="cmmi-12">A</span><sub ><span
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759 class="cmmi-8"><img
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760 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img
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761 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub> der Komponenten zweier <span
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762 class="cmti-12">kovarianter </span>Vierervektoren <br/><span
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763 class="cmmi-12">A</span><sub ><span
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764 class="cmmi-8"><img
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765 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /></span></sub> und <span
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766 class="cmmi-12">B</span><sub ><span
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767 class="cmmi-8"><img
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768 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub></p>
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769 <table width="100%"
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770 class="equation"><tr><td><a
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771 id="x1-12r10"></a>
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772 <center class="math-display" >
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773 <img
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774 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_191617x.png" alt="Amn = Am Bn, " class="math-display" /></center></td><td width="5%">(10)</td></tr></table>
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775 <!--l. 916--><p class="nopar">
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776 </p><!--l. 921--><p class="noindent">so gilt für diese das Transformationsgesetz</p>
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777 <table width="100%"
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778 class="equation"><tr><td><a
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779 id="x1-13r11"></a>
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780 <center class="math-display" >
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781 <img
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782 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_191618x.png" alt=" ' @ xm @ xn Ast = ----'----'Amn . @ xs @ xt " class="math-display" /></center></td><td width="5%">(11)</td></tr></table>
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783
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784 <!--l. 929--><p class="nopar">
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785 <pb/></p><!--l. 936--><p class="indent">
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786
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787 </p><!--l. 937--><p class="indent"> Durch dieses Transformationsgesetz wird der kovariante <br/>Tensor zweiten Ranges definiert. Alle Bemerkungen, welche <br/>vorher über die kontravarianten Tensoren gemacht wurden, <br/>gelten auch für die kovarianten Tensoren.
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788 </p><!--l. 942--><p class="indent"> <span
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789 class="cmti-12">Bemerkung. </span>Es ist bequem, den Skalar (Invariante) so-<br/>wohl als kontravarianten wie als kovarianten Tensor vom <br/>Range Null zu behandeln.
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790 </p><!--l. 946--><p class="indent"> <span
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791 class="cmti-12">Gemischter Tensor. </span>Man kann auch einen Tensor zweiten <br/>Ranges vom Typus</p>
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792 <table width="100%"
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793 class="equation"><tr><td><a
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794 id="x1-14r12"></a>
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795 <center class="math-display" >
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796 <img
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797 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_191619x.png" alt="A n = A Bn m m " class="math-display" /></center></td><td width="5%">(12)</td></tr></table>
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798 <!--l. 953--><p class="nopar">
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799 </p><!--l. 956--><p class="noindent">definieren, der bezüglich des Index <span
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800 class="cmmi-12"><img
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801 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi12-16.png" alt="m" class="cmmi-12x-x-16" align="middle" /> </span>kovariant, bezüglich <br/>des Index <span
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802 class="cmmi-12"><img
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803 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi12-17.png" alt="n" class="12x-x-17" /></span>
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804 kontravariant ist. Sein Transformationsgesetz ist</p>
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805 <table width="100%"
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806 class="equation"><tr><td><a
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807 id="x1-15r13"></a>
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808 <center class="math-display" >
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809 <img
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810 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_191620x.png" alt=" ' Ast'= @ xt-@-xa-Ab . @ xb @ xs' a " class="math-display" /></center></td><td width="5%">(13)</td></tr></table>
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811 <!--l. 964--><p class="nopar">
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812 </p><!--l. 969--><p class="indent"> Natürlich gibt es gemischte Tensoren mit beliebig vielen <br/>Indizes kovarianten und beliebig vielen Indizes kontravarianten <br/>Charakters. Der kovariante und der kontravariante Tensor <br/>können als spezielle Fälle des gemischten angesehen werden.
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813 </p><!--l. 975--><p class="indent"> <span
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814 class="cmti-12">Symmetrische Tensoren. </span>Ein kontravarianter bzw. ko-<br/>varianter Tensor zweiten oder höheren Ranges heißt <span
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815 class="cmti-12">sym- </span> <br/><span
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816 class="cmti-12">metrisch</span>, wenn zwei Komponenten, die durch Vertauschung <br/>irgend zweier Indizes auseinander hervorgehen, gleich sind. <br/>Der Tensor <span
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817 class="cmmi-12">A</span><sup ><span
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818 class="cmmi-8"><img
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819 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img
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820 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sup> bzw. <span
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821 class="cmmi-12">A</span><sub >
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822 <span
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823 class="cmmi-8"><img
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824 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img
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825 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub> ist also symmetrisch, wenn für jede <br/>Kombination der
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826 Indizes</p>
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827 <table width="100%"
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828 class="equation"><tr><td><a
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829 id="x1-16r14"></a>
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830 <center class="math-display" >
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831 <img
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832 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_191621x.png" alt=" mn nm A = A , " class="math-display" /></center></td><td width="5%">(14)</td></tr></table>
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833 <!--l. 986--><p class="nopar">
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834 bzw.
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835 </p>
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836 <table width="100%"
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837 class="equation"><tr><td><a
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838 id="x1-17r15"></a>
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839 <center class="math-display" >
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840 <img
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841 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_191622x.png" alt="Am n = Anm " class="math-display" /></center></td><td width="5%">(14a)</td></tr></table>
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842 <!--l. 992--><p class="nopar">
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843 </p><!--l. 995--><p class="noindent">ist.
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844 </p><!--l. 998--><p class="indent"> Es muß bewiesen werden, daß die so definierte Symmetrie <br/>eine vom Bezugssystem unabhängige Eigenschaft ist. (Aus (9) <br/>folgt in der Tat mit Rücksicht auf (14)</p>
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845 <center class="par-math-display" >
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846 <img
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847 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_191623x.png" alt=" st' @-xs'@-xt' mn @ xs'-@-xt' nm @-xt' @ xs' m n ts' A = @ xm @ xn A = @ xm @ xn A = @ xm @ xn A = A . " class="par-math-display" /></center>
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848 <!--l. 1013--><p class="nopar"></p><!--l. 1016--><p class="noindent">Die vorletzte Gleichsetzung beruht auf der Vertauschung der <br/>Summationsindizes <span
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849 class="cmmi-12"><img
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850 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi12-16.png" alt="m" class="cmmi-12x-x-16" align="middle" /></span>
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851 und <span
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852 class="cmmi-12"><img
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853 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi12-17.png" alt="n" class="12x-x-17" /> </span>(d. h. auf bloßer Änderung der <br/>Bezeichnungsweise).
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854 </p><!--l. 1020--><p class="indent"> <span
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855 class="cmti-12">Antisymmetrische Tensoren. </span>Ein kontravarianter bzw. ko-<br/>varianter Tenor zweiten, dritten oder vierten Ranges heißt <br/><pb/>
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856 </p><!--l. 1025--><p class="indent">
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857
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858 </p><!--l. 1026--><p class="noindent">antisymmetrisch, wenn zwei Komponenten, die durch Ver-<br/>tauschung irgend zweier Indizes auseinander hervorgehen, <br/><span
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859 class="cmti-12">entgegengesetzt gleich </span>sind. Der Tensor
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860 <span
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861 class="cmmi-12">A</span><sup ><span
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862 class="cmmi-8"><img
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863 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img
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864 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sup> bzw. <span
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865 class="cmmi-12">A</span><sub >
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866 <span
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867 class="cmmi-8"><img
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868 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img
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869 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub> ist also <br/>antisymmetrisch, wenn stets</p>
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870 <table width="100%"
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871 class="equation"><tr><td><a
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872 id="x1-18r15"></a>
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873 <center class="math-display" >
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874 <img
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875 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_191624x.png" alt=" mn nm A = - A , " class="math-display" /></center></td><td width="5%">(15)</td></tr></table>
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876 <!--l. 1035--><p class="nopar">
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877 </p><!--l. 1038--><p class="noindent">bzw.</p>
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878 <table width="100%"
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879 class="equation"><tr><td><a
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880 id="x1-19r16"></a>
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881 <center class="math-display" >
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882 <img
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883 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_191625x.png" alt="Am n = -An m " class="math-display" /></center></td><td width="5%">(15a)</td></tr></table>
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884 <!--l. 1045--><p class="nopar">
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885 </p><!--l. 1048--><p class="noindent">ist.
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886 </p><!--l. 1051--><p class="indent"> Von den 16 Komponenten <span
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887 class="cmmi-12">A</span><sup ><span
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888 class="cmmi-8"><img
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889 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img
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890 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sup> verschwinden die vier <br/>Komponenten <span
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891 class="cmmi-12">A</span><sup ><span
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892 class="cmmi-8"><img
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893 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img
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894 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /></span></sup>; die übrigen sind paarweise entgegengesetzt <br/>gleich, so daß nur 6 numerisch verschiedene Komponenten <br/>vorhanden sind (Sechservektor). Ebenso sieht man, daß der <br/>antisymmetrische Tensor <span
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895 class="cmmi-12">A</span><sup ><span
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896 class="cmmi-8"><img
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897 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img
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898 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-17.png" alt="n" class="8x-x-17" /> <img
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899 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-1b.png" alt="s" class="8x-x-1b" /></span></sup> (dritten Ranges) nur vier nume-<br/>risch verschiedene Komponenten hat, der antisymmetrische <br/>Tensor <span
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900 class="cmmi-12">A</span><sup ><span
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901 class="cmmi-8"><img
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902 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img
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903 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-17.png" alt="n" class="8x-x-17" /> <img
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904 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-1b.png" alt="s" class="8x-x-1b" /> <img
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905 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-1c.png" alt="t" class="8x-x-1c" /></span></sup> nur eine einzige. Symmetrische Tensoren höheren <br/>als vierten Ranges gibt es in einem Kontinuum von vier Dimen-<br/>sionen nicht.
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906 </p>
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907 <div class="center" >
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908
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909 <!--l. 1063--><p class="noindent">
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910 </p><!--l. 1064--><p class="noindent"><span
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911 class="cmbsy-10x-x-120">§ </span><span
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912 class="cmbx-12">7. Multiplikation der Tensoren.</span></p></div>
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913
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914 <!--l. 1068--><p class="indent"> <span
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915 class="cmti-12">Äu</span><span
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916 class="cmmi-12"><img
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917 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi12-c.png" alt="b" class="cmmi-12x-x-c" align="middle" /></span><span
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918 class="cmti-12">ere Multiplikation der Tensoren. </span>Man erhält aus den <br/>Komponenten eines Tensors vom Range <span
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919 class="cmmi-12">z </span>und eines solchen <br/>vom Range <span
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920 class="cmmi-12">z</span><span
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921 class="cmsy-10x-x-120">' </span>die Komponenten eines Tensors vom Range <br/><span
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922 class="cmmi-12">z </span>+ <span
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923 class="cmmi-12">z</span><span
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924 class="cmsy-10x-x-120">'</span>, indem man alle Komponenten des ersten mit allen <br/>Komponenten des zweiten paarweise multipliziert. So ent-<br/>stehen beispielsweise die Tensoren <span
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925 class="cmmi-12">T </span>aus den Tensoren <span
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926 class="cmmi-12">A </span> <br/>und <span
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927 class="cmmi-12">B </span>verschiedener Art
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928 </p>
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929 <center class="par-math-display" >
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930 <img
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931 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_191626x.png" alt="Tmns = Am nBs, ab gd a b gd T = A B , Tgd = Aa b Bg d. ab " class="par-math-display" /></center>
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932 <!--l. 1087--><p class="nopar">
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933 </p><!--l. 1089--><p class="indent"> Der Beweis des Tensorcharakters der <span
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934 class="cmmi-12">T </span>ergibt sich un-<br/>mittelbar aus den Darstellungen (8), (10), (12) oder aus den <br/>Transformationsregeln (9), (11), (13). Die Gleichungen (8), <br/>(10), (12) sind selbst Beispiele äußerer Multiplikation (von
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935 <br/>Tensoren ersten Ranges).
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936 </p><!--l. 1095--><p class="indent"> <span
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937 class="cmti-12">,,Verj</span><span
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938 class="cmti-12">üngung&#8220; eines gemischten Tensors. </span>Aus jedem ge-<br/>mischten Tensor kann ein Tensor von einem um zwei kleineren <br/>Range gebildet werden, indem man einen Index kovarianten <br/>und einen Index kontravarianten Charakters gleichsetzt und
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939 <br/><pb/>
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940 </p><!--l. 1102--><p class="indent">
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941
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942 </p><!--l. 1103--><p class="noindent">nach diesem Index summiert (,,Verjüngung&#8220;). Man gewinnt <br/>so z. B. aus dem gemischten Tensor vierten Ranges <span
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943 class="cmmi-12">A</span><sub><span
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944 class="cmmi-8"><img
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945 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-b.png" alt="a" class="8x-x-b" /><img
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946 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-c.png" alt="b" class="cmmi-8x-x-c" align="middle" /></span></sub><sup><span
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947 class="cmmi-8"><img
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948 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-d.png" alt="g" class="8x-x-d" /> <img
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949 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-e.png" alt="d" class="8x-x-e" /></span></sup> den <br/>gemischten Tensor zweiten Ranges
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950 </p>
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951 <center class="par-math-display" >
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952 <img
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953 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_191627x.png" alt=" d a d( sum a d) A b = Aa b = a Aa b " class="par-math-display" /></center>
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954 <!--l. 1114--><p class="nopar">
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955 </p><!--l. 1118--><p class="noindent">und aus diesem, abermals durch Verjüngung, den Tensor <br/>nullten Ranges
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956 <span
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957 class="cmmi-12">A </span>= <span
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958 class="cmmi-12">A</span><sub><span
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959 class="cmmi-8"><img
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960 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-c.png" alt="b" class="cmmi-8x-x-c" align="middle" /></span></sub><sup><span
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961 class="cmmi-8"><img
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962 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-c.png" alt="b" class="cmmi-8x-x-c" align="middle" /></span></sup> = <span
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963 class="cmmi-12">A</span><sub>
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964 <span
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965 class="cmmi-8"><img
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966 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-b.png" alt="a" class="8x-x-b" /><img
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967 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-c.png" alt="b" class="cmmi-8x-x-c" align="middle" /></span></sub><sup><span
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968 class="cmmi-8"><img
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969 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-b.png" alt="a" class="8x-x-b" /><img
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970 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-c.png" alt="b" class="cmmi-8x-x-c" align="middle" /></span></sup><span
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971 class="cmmi-12">.</span>
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972 </p><!--l. 1121--><p class="indent"> Der Beweis dafür, daß das Ergebnis der Verjüngung wirk-<br/>lich Tensorcharakter besitzt, ergibt sich entweder aus der <br/>Tensordarstellung gemäß der Verallgemeinerung von (12) in <br/>Verbindung mit (6) oder aus der Verallgemeinerung von (13). <br/></p><!--l. 1127--><p class="indent"><span
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973 class="cmti-12">Innere und gemischte Multiplikation der Tensoren. </span>Diese <br/>bestehen in der Kombination der äußeren Multiplikation mit <br/>der Verjüngung.
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974 </p><!--l. 1131--><p class="indent"> Beispiele. -- Aus dem kovarianten Tensor zweiten Ranges <br/><span
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975 class="cmmi-12">A</span><sub ><span
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976 class="cmmi-8"><img
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977 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img
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978 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub> und dem kontravarianten Tensor ersten Ranges <span
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
979 class="cmmi-12">B</span><sup ><span
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
980 class="cmmi-8"><img
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
981 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-1b.png" alt="s" class="8x-x-1b" /></span></sup> bilden <br/>wir durch äußere Multiplikation den gemischten Tensor
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
982 </p>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
983 <center class="par-math-display" >
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
984 <img
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
985 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_191628x.png" alt=" s s D mn = Amn B . " class="par-math-display" /></center>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
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diff changeset
986 <!--l. 1140--><p class="nopar">
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
987 </p><!--l. 1143--><p class="noindent">Durch Verjüngung nach den Indizes <span
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
988 class="cmmi-12"><img
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
989 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi12-17.png" alt="n" class="12x-x-17" />, <img
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
990 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi12-1b.png" alt="s" class="12x-x-1b" /> </span>entsteht der ko-<br/>variante Vierervektor</p>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
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diff changeset
991 <center class="par-math-display" >
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
992 <img
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
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diff changeset
993 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_191629x.png" alt="Dm = Dnmn = Amn Bn. " class="par-math-display" /></center>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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diff changeset
994 <!--l. 1151--><p class="nopar">
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
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diff changeset
995 </p><!--l. 1154--><p class="noindent">Diesen bezeichnen wir auch als inneres Produkt der Tensoren <br/><span
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
996 class="cmmi-12">A</span><sub ><span
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
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diff changeset
997 class="cmmi-8"><img
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
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diff changeset
998 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
999 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub> und <span
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
1000 class="cmmi-12">B</span><sup ><span
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
1001 class="cmmi-8"><img
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
1002 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-1b.png" alt="s" class="8x-x-1b" /></span></sup>. Analog bildet man aus den Tensoren <span
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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diff changeset
1003 class="cmmi-12">A</span><sub ><span
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
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diff changeset
1004 class="cmmi-8"><img
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
1005 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
1006 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub> und <br/><span
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
1007 class="cmmi-12">B</span><sup ><span
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
1008 class="cmmi-8"><img
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
1009 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-1b.png" alt="s" class="8x-x-1b" /><img
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
1010 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-1c.png" alt="t" class="8x-x-1c" /></span></sup> durch äußere Multiplikation und zweimalige Verjüngung <br/>das innere Produkt <span
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
1011 class="cmmi-12">A</span><sub ><span
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
1012 class="cmmi-8"><img
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
1013 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
1014 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub> <span
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
1015 class="cmmi-12">B</span><sup ><span
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
1016 class="cmmi-8"><img
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
1017 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
1018 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sup><span
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
1019 class="cmmi-12">. </span>Durch äußere Produktbildung <br/>und einmalige Verjüngung erhält man aus <span
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
1020 class="cmmi-12">A</span><sub ><span
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
1021 class="cmmi-8"><img
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
1022 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
1023 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub> und <span
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
1024 class="cmmi-12">B</span><sup ><span
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
1025 class="cmmi-8"><img
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
1026 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-1b.png" alt="s" class="8x-x-1b" /><img
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
1027 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-1c.png" alt="t" class="8x-x-1c" /></span></sup>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
1028 den <br/>gemischten Tensor zweiten Ranges <span
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
1029 class="cmmi-12">D</span><sub><span
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
1030 class="cmmi-8"><img
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
1031 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /></span></sub><sup><span
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
1032 class="cmmi-8"><img
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
1033 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-1c.png" alt="t" class="8x-x-1c" /></span></sup> = <span
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
1034 class="cmmi-12">A</span><sub >
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
1035 <span
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
1036 class="cmmi-8"><img
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
1037 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
1038 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub> <span
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
1039 class="cmmi-12">B</span><sup ><span
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
1040 class="cmmi-8"><img
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
1041 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-17.png" alt="n" class="8x-x-17" /><img
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
1042 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-1c.png" alt="t" class="8x-x-1c" /></span></sup><span
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
1043 class="cmmi-12">. </span>Man kann <br/>diese Operation passend als eine gemischte bezeichnen; denn <br/>sie ist eine äußere bezüglich der Indizes <span
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
1044 class="cmmi-12"><img
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
1045 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi12-16.png" alt="m" class="cmmi-12x-x-16" align="middle" /> </span>und <span
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
1046 class="cmmi-12"><img
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
1047 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi12-1c.png" alt="t" class="12x-x-1c" /></span>, eine innere <br/>bezüglich der Indizes <span
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
1048 class="cmmi-12"><img
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
1049 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi12-17.png" alt="n" class="12x-x-17" /> </span>und
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
1050 <span
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
1051 class="cmmi-12"><img
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
1052 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi12-1b.png" alt="s" class="12x-x-1b" /></span>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
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1053 </p><!--l. 1169--><p class="indent"> Wir beweisen nun einen Satz, der zum Nachweis des <br/>Tensorcharakters oft verwendbar ist. Nach dem soeben Dar-<br/>gelegten ist <span
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1054 class="cmmi-12">A</span><sub ><span
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1055 class="cmmi-8"><img
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1056 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img
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1057 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub> <span
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1058 class="cmmi-12">B</span><sup ><span
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1059 class="cmmi-8"><img
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1060 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img
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diff changeset
1061 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sup> ein Skalar, wenn <span
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1062 class="cmmi-12">A</span><sub >
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1063 <span
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1064 class="cmmi-8"><img
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1065 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img
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1066 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub>
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1067 und <span
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1068 class="cmmi-12">B</span><sup ><span
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1069 class="cmmi-8"><img
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1070 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-1b.png" alt="s" class="8x-x-1b" /><img
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1071 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-1c.png" alt="t" class="8x-x-1c" /></span></sup> Tensoren <br/>sind. Wir behaupten aber auch folgendes. Wenn <span
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1072 class="cmmi-12">A</span><sub >
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1073 <span
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1074 class="cmmi-8"><img
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1075 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img
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1076 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub> <span
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1077 class="cmmi-12">B</span><sup ><span
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1078 class="cmmi-8"><img
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1079 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img
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1080 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sup>
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1081 für <br/>jede Wahl <span
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1082 class="cmti-12">des Tensors </span><span
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1083 class="cmmi-12">B</span><sup ><span
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1084 class="cmmi-8"><img
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1085 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img
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1086 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sup> eine Invariante ist, so hat <span
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1087 class="cmmi-12">A</span><sub >
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1088 <span
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1089 class="cmmi-8"><img
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1090 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img
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1091 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub> Tensor-<br/>charakter.
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1092 </p><!--l. 1178--><p class="indent"> Beweis. -- Es ist nach Voraussetzung für eine beliebige <br/>Substitution</p>
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1093 <center class="par-math-display" >
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1094 <img
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1095 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_191630x.png" alt=" ' st' mn Ast B = Amn B . " class="par-math-display" /></center>
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1096 <!--l. 1186--><p class="nopar">
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1097 </p><!--l. 1192--><p class="indent">
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1098
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1099 </p><!--l. 1193--><p class="noindent">Nach der Umkehrung von (9) ist aber
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1100 </p>
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1101 <center class="par-math-display" >
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1102 <img
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1103 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_191631x.png" alt=" mn @-xm-@-xn- st' B = @ xs'@ xt'B . " class="par-math-display" /></center>
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1104 <!--l. 1202--><p class="nopar">
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1105 </p><!--l. 1204--><p class="noindent">Dies, eingesetzt in obige Gleichung, liefert:
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1106 </p>
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1107 <center class="par-math-display" >
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1108 <img
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1109 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_191632x.png" alt="( ) Ast '- -@ xm @-xn-Amn Bst'= 0 . @ xs' @ xt' " class="par-math-display" /></center>
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1110 <!--l. 1213--><p class="nopar">
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1111 </p><!--l. 1215--><p class="indent"> Dies kann bei beliebiger Wahl von <span
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1112 class="cmmi-12">B</span><sup ><span
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1113 class="cmmi-8"><img
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1114 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-1b.png" alt="s" class="8x-x-1b" /><img
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1115 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-1c.png" alt="t" class="8x-x-1c" /></span></sup><sup ><span
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1116 class="cmsy-8">'</span></sup> nur dann erfüllt <br/>sein, wenn die Klammer verschwindet, woraus mit Rück-<br/>sicht auf (11) die Behauptung folgt.
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1117 </p><!--l. 1220--><p class="indent"> Dieser Satz gilt entsprechend für Tensoren beliebigen <br/>Ranges und Charakters; der Beweis ist stets analog zu führen.
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1118 </p><!--l. 1224--><p class="indent"> Der Satz läßt sich ebenso beweisen in der Form: Sind <br/><span
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1119 class="cmmi-12">B</span><sup ><span
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1120 class="cmmi-8"><img
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1121 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /></span></sup> und <span
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1122 class="cmmi-12">C</span><sup ><span
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1123 class="cmmi-8"><img
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1124 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sup> beliebige Vektoren, und ist bei jeder Wahl der-<br/>selben das innere Produkt</p>
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1125 <center class="par-math-display" >
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1126 <img
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1127 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_191633x.png" alt="A Bm Cn mn " class="par-math-display" /></center>
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1128 <!--l. 1231--><p class="nopar">
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1129 </p><!--l. 1234--><p class="noindent">ein Skalar, so ist <span
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1130 class="cmmi-12">A</span><sub ><span
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1131 class="cmmi-8"><img
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1132 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img
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1133 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub> ein kovarianter Tensor. Dieser letztere <br/>Satz gilt auch dann noch, wenn nur die speziellere Aussage <br/>zutrifft, daß bei beliebiger Wahl des Vierervektors <span
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1134 class="cmmi-12">B</span><sup ><span
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1135 class="cmmi-8"><img
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1136 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /></span></sup> das <br/>skalare Produkt
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1137 </p>
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1138 <center class="par-math-display" >
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1139 <img
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1140 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_191634x.png" alt="Amn Bm Bn " class="par-math-display" /></center>
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1141 <!--l. 1243--><p class="nopar">
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1142 </p><!--l. 1245--><p class="noindent">ein Skalar ist, falls man außerdem weiß, daß <span
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1143 class="cmmi-12">A</span><sub ><span
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1144 class="cmmi-8"><img
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1145 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img
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1146 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub> der Sym-<br/>metriebedingung
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diff changeset
1147 <span
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1148 class="cmmi-12">A</span><sub ><span
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1149 class="cmmi-8"><img
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1150 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img
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diff changeset
1151 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub> = <span
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1152 class="cmmi-12">A</span><sub ><span
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1153 class="cmmi-8"><img
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diff changeset
1154 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-17.png" alt="n" class="8x-x-17" /><img
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parents:
diff changeset
1155 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /></span></sub> genügt. Denn auf dem vorhin <br/>angegebenen Wege beweist man den Tensorcharakter von <br/><img
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1156 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_191635x.png" alt="(Amn + Anm)" class="left" align="middle" /> <span
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1157 class="cmmi-12">, </span>woraus dann wegen der Symmetrieeigenschaft
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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1158 <br/>der Tensorcharakter von <span
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1159 class="cmmi-12">A</span><sub ><span
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1160 class="cmmi-8"><img
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diff changeset
1161 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
1162 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub> selbst folgt. Auch dieser Satz <br/>läßt sich leicht verallgemeinern auf den Fall kovarianter und <br/>kontravarianter Tensoren beliebigen Ranges.
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1163 </p><!--l. 1255--><p class="indent"> Endlich folgt aus dem Bewiesenen der ebenfalls auf be-<br/>liebige Tensoren zu
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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1164 verallgemeinernde Satz: Wenn die Größen <br/><span
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1165 class="cmmi-12">A</span><sub ><span
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1166 class="cmmi-8"><img
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1167 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img
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1168 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub> <span
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1169 class="cmmi-12">B</span><sup ><span
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1170 class="cmmi-8"><img
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
1171 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sup> bei beliebiger Wahl des Vierervektors <span
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1172 class="cmmi-12">B</span><sup ><span
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1173 class="cmmi-8"><img
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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1174 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sup> einen Tensor <br/>ersten Ranges bilden, so ist <span
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1175 class="cmmi-12">A</span><sub >
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1176 <span
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1177 class="cmmi-8"><img
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1178 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img
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1179 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub> ein Tensor zweiten Ranges. <br/>Ist nämlich <span
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1180 class="cmmi-12">C</span><sup ><span
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1181 class="cmmi-8"><img
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1182 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /></span></sup> ein beliebiger Vierervektor, so ist wegen des
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1183 <br/>Tensorcharakters <span
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1184 class="cmmi-12">A</span><sub ><span
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1185 class="cmmi-8"><img
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1186 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img
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1187 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub> <span
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1188 class="cmmi-12">B</span><sup ><span
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1189 class="cmmi-8"><img
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1190 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sup> das innere Produkt <span
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1191 class="cmmi-12">A</span><sub >
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1192 <span
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1193 class="cmmi-8"><img
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1194 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img
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1195 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub> <span
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1196 class="cmmi-12">C</span><sup ><span
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1197 class="cmmi-8"><img
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1198 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /></span></sup> <span
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1199 class="cmmi-12">B</span><sup ><span
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1200 class="cmmi-8"><img
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1201 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sup> bei <br/>beliebiger Wahl der beiden Vierervektoren <span
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1202 class="cmmi-12">C</span><sup ><span
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1203 class="cmmi-8"><img
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1204 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /></span></sup> und <span
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1205 class="cmmi-12">B</span><sup ><span
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1206 class="cmmi-8"><img
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1207 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sup> ein <br/>Skalar, woraus die Behauptung folgt.
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1208 </p>
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1209 <div class="center" >
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1210
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1211 <!--l. 1266--><p class="noindent">
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1212 </p><!--l. 1268--><p class="noindent"><span
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1213 class="cmbsy-10x-x-120">§ </span><span
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1214 class="cmbx-12">8. Einiges </span><span
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1215 class="cmbx-12">über den Fundamentaltensor der </span><span
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1216 class="cmmi-12">g</span><sub ><span
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1217 class="cmmi-8"><img
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1218 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img
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1219 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub><span
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1220 class="cmmi-12">.</span></p></div>
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1221 <!--l. 1272--><p class="indent"> <span
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1222 class="cmti-12">Der kovariante Fundamentaltensor. </span>In dem invarianten <br/>Ausdruck des Quadrates des Linienelementes</p>
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1223 <center class="par-math-display" >
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1224 <img
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1225 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_191636x.png" alt="ds2 = g mn d xmd xn " class="par-math-display" /></center>
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1226 <!--l. 1279--><p class="nopar">
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1227 </p><!--l. 1282--><p class="indent">
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1228
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1229 </p><!--l. 1283--><p class="noindent">spielt <span
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1230 class="cmmi-12">d x</span><sub ><span
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1231 class="cmmi-8"><img
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1232 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /></span></sub> die Rolle eines beliebig wählbaren kontravarianten <br/>Vektors. Da ferner
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1233 <span
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1234 class="cmmi-12">g</span><sub ><span
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1235 class="cmmi-8"><img
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1236 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img
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1237 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub> = <span
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1238 class="cmmi-12">g</span><sub ><span
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1239 class="cmmi-8"><img
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1240 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-17.png" alt="n" class="8x-x-17" /><img
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1241 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /></span></sub>, so folgt nach den Betrachtungen <br/>des letzten Paragraphen hieraus, daß <span
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1242 class="cmmi-12">g</span><sub ><span
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1243 class="cmmi-8"><img
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1244 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img
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1245 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub> ein kovarianter Tensor <br/>zweiten Ranges ist. Wir nennen ihn ,,Fundamentaltensor&#8220;. <br/>Im folgenden leiten wir einige Eigenschaften dieses Tensors
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1246 <br/>ab, die zwar jedem Tensor zweiten Ranges eigen sind; aber <br/>die besondere Rolle des Fundamentaltensors in unserer Theorie, <br/>welche in der Besonderheit der Gravitationswirkungen ihren <br/>physikalischen Grund hat, bringt es mit sich, daß die zu ent-<br/>wickelnden Relationen nur bei dem Fundamentaltensor für <br/>uns von Bedeutung sind.
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1247 </p><!--l. 1297--><p class="indent"> <span
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1248 class="cmti-12">Der kontravariante Fundamentaltensor. </span>Bildet man in dem <br/>Determinantenschema der <span
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1249 class="cmmi-12">g</span><sub ><span
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1250 class="cmmi-8"><img
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1251 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img
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1252 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub> zu jedem <span
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1253 class="cmmi-12">g</span><sub ><span
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1254 class="cmmi-8"><img
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1255 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img
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1256 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub> die Unterdetermi-<br/>nante und dividiert diese durch die Determinante <span
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1257 class="cmmi-12">g </span>= <img
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1258 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_191637x.png" alt="|g mn|" class="left" align="middle" /> der <br/><span
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1259 class="cmmi-12">g</span><sub ><span
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1260 class="cmmi-8"><img
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1261 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img
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1262 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub>, so erhält man gewisse Größen <span
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1263 class="cmmi-12">g</span><sup ><span
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1264 class="cmmi-8"><img
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1265 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img
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1266 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sup>(= <span
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1267 class="cmmi-12">g</span><sup ><span
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1268 class="cmmi-8"><img
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1269 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-17.png" alt="n" class="8x-x-17" /> <img
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1270 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /></span></sup>), von denen wir <br/>beweisen wollen, daß sie einen kontravarianten Tensor bilden.
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1271 </p><!--l. 1306--><p class="indent"> Nach einem bekannten Determinantensatze ist</p>
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1272 <table width="100%"
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1273 class="equation"><tr><td><a
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1274 id="x1-20r16"></a>
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1275 <center class="math-display" >
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1276 <img
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1277 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_191638x.png" alt="gm s gns = dmn, " class="math-display" /></center></td><td width="5%">(16)</td></tr></table>
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1278 <!--l. 1312--><p class="nopar">
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1279 </p><!--l. 1317--><p class="noindent">wobei das Zeichen <span
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1280 class="cmmi-12"><img
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1281 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi12-e.png" alt="d" class="12x-x-e" /></span><sub ><span
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1282 class="cmmi-8"><img
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1283 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /></span></sub><sup ><span
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1284 class="cmmi-8"><img
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1285 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sup> 1 oder 0 bedeutet, je nachdem <span
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1286 class="cmmi-12"><img
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1287 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi12-16.png" alt="m" class="cmmi-12x-x-16" align="middle" /> </span>= <span
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1288 class="cmmi-12"><img
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1289 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi12-17.png" alt="n" class="12x-x-17" /> </span> <br/>oder <span
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1290 class="cmmi-12"><img
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1291 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi12-16.png" alt="m" class="cmmi-12x-x-16" align="middle" /><img
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1292 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi12-17.png" alt="n" class="12x-x-17" /> </span>ist. Statt des obigen Ausdruckes für <span
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1293 class="cmmi-12">d s</span><sup ><span
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1294 class="cmr-8">2</span></sup> können <br/>wir auch
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1295 </p>
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1296 <center class="par-math-display" >
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1297 <img
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1298 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_191639x.png" alt="g ms dns d xm d xn, " class="par-math-display" /></center>
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1299 <!--l. 1327--><p class="nopar">
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1300 </p><!--l. 1329--><p class="noindent">oder nach (16) auch</p>
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1301 <center class="par-math-display" >
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1302 <img
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1303 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_191640x.png" alt="gms gnt gsn d xm d xn " class="par-math-display" /></center>
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1304 <!--l. 1336--><p class="nopar">
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1305 </p><!--l. 1338--><p class="noindent">schreiben. Nun bilden aber nach den Multiplikationsregeln <br/>des vorigen Paragraphen die Größen
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1306 </p>
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1307
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1308 <center class="par-math-display" >
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1309 <img
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1310 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_191641x.png" alt="d qs = g ms d xm " class="par-math-display" /></center>
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1311 <!--l. 1346--><p class="nopar">
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1312 </p><!--l. 1349--><p class="noindent">einen kovarianten Vierervektor, und zwar (wegen der will-<br/>kürlichen Wählbarkeit der <span
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1313 class="cmmi-12">d x</span><sub ><span
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1314 class="cmmi-8"><img
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1315 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /></span></sub>) einen beliebig wählbaren <br/>Vierervektor. Indem wir ihn in unseren Ausdruck einführen, <br/>erhalten wir</p>
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1316 <center class="par-math-display" >
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1317 <img
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1318 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_191642x.png" alt="d s2 = gst d qs d qt. " class="par-math-display" /></center>
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1319 <!--l. 1359--><p class="nopar">
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1320 </p><!--l. 1361--><p class="indent"> Da dies bei beliebiger Wahl des Vektors <span
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1321 class="cmmi-12">d <img
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1322 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi12-18.png" alt="q" class="cmmi-12x-x-18" align="middle" /></span><sub ><span
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1323 class="cmmi-8"><img
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1324 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-1b.png" alt="s" class="8x-x-1b" /></span></sub> ein Skalar <br/>ist und <span
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1325 class="cmmi-12">g</span><sup ><span
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1326 class="cmmi-8"><img
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1327 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-1b.png" alt="s" class="8x-x-1b" /><img
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1328 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-1c.png" alt="t" class="8x-x-1c" /></span></sup> nach seiner Definition in den Indizes <span
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1329 class="cmmi-12"><img
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1330 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi12-1b.png" alt="s" class="12x-x-1b" /> </span>und <span
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1331 class="cmmi-12"><img
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1332 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi12-1c.png" alt="t" class="12x-x-1c" /> </span>sym-<br/>metrisch ist, folgt aus den Ergebnissen des vorigen Para-<br/>graphen, daß <span
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1333 class="cmmi-12">g</span><sup ><span
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1334 class="cmmi-8"><img
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1335 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-1b.png" alt="s" class="8x-x-1b" /><img
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1336 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-1c.png" alt="t" class="8x-x-1c" /></span></sup> ein kontravarianter Tensor ist. Aus (16) <br/>folgt noch, daß auch <span
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1337 class="cmmi-12"><img
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1338 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi12-e.png" alt="d" class="12x-x-e" /></span><sub ><span
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1339 class="cmmi-8"><img
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1340 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /></span></sub><sup ><span
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1341 class="cmmi-8"><img
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1342 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sup> ein Tensor ist, den wir den gemischten <br/>Fundamentaltensor nennen können. <pb/>
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1343 </p><!--l. 1372--><p class="indent">
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1344
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1345 </p><!--l. 1373--><p class="indent"> <span
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1346 class="cmti-12">Determinante des Fundamentaltensors. </span>Nach dem Multi-<br/>plikationssatz der Determinanten ist
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1347 </p>
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1348 <center class="par-math-display" >
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1349 <img
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1350 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_191643x.png" alt="|gma gan| = |gma| |gan|. " class="par-math-display" /></center>
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1351 <!--l. 1382--><p class="nopar">
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1352 </p><!--l. 1385--><p class="noindent">Andererseits ist
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1353 </p>
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1354 <center class="par-math-display" >
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1355 <img
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1356 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_191644x.png" alt=" an n |g ma g |= |dm |= 1. " class="par-math-display" /></center>
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1357 <!--l. 1392--><p class="nopar">
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1358 </p><!--l. 1395--><p class="noindent">Also folgt</p>
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1359 <table width="100%"
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1360 class="equation"><tr><td><a
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1361 id="x1-21r17"></a>
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1362 <center class="math-display" >
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1363 <img
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1364 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_191645x.png" alt=" mn |gmn||g |= 1. " class="math-display" /></center></td><td width="5%">(17)</td></tr></table>
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1365 <!--l. 1401--><p class="nopar">
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1366 </p><!--l. 1406--><p class="indent"> <span
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1367 class="cmti-12">Invariante des Volumens. </span>Wir suchen zuerst das Trans-<br/>formationsgesetz der Determinante <span
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1368 class="cmmi-12">g </span>= <img
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1369 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_191646x.png" alt="|g | mn" class="left" align="middle" />. Gemäß (11) ist
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1370 </p>
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1371 <center class="par-math-display" >
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1372 <img
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1373 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_191647x.png" alt=" | | ' |@ xm @ xn | g = ||----'----'g mn||. @ xs @ xt " class="par-math-display" /></center>
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1374 <!--l. 1414--><p class="nopar">
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1375 </p><!--l. 1416--><p class="noindent">Hieraus folgt durch zweimalige Anwendung des Multiplikations-<br/>satzes der Determinanten
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1376 </p>
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1377 <center class="par-math-display" >
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1378 <img
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1379 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_191648x.png" alt=" | || | | |2 g'= ||@--xm||||-@-xn |||g |= ||-@-xm|| g, |@ xs'||@ xt'| mn |@ xs'| " class="par-math-display" /></center>
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1380 <!--l. 1426--><p class="nopar">
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1381 </p><!--l. 1429--><p class="noindent">oder</p>
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1382 <center class="par-math-display" >
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1383 <img
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1384 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_191649x.png" alt=" | | V~ -'- ||@-xm-|| V~ - g = |@ xs'| g. " class="par-math-display" /></center>
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1385 <!--l. 1436--><p class="nopar">
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1386 </p><!--l. 1438--><p class="noindent">Andererseits ist das Gesetz der Transformation des Volum-<br/>elementes</p>
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1387 <center class="par-math-display" >
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1388 <img
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1389 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_191650x.png" alt=" integral
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1390 ' d t = d x1 d x2 d x3 d x4 " class="par-math-display" /></center>
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1391 <!--l. 1446--><p class="nopar">
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1392 </p><!--l. 1450--><p class="noindent">nach dem bekannten Jakobischen Satze
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1393 </p>
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1394 <center class="par-math-display" >
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1395 <img
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1396 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_191651x.png" alt=" | '| d t'= ||@-xs-||d t. |@ xm | " class="par-math-display" /></center>
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1397 <!--l. 1457--><p class="nopar">
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1398 </p><!--l. 1459--><p class="noindent">Durch Multiplikation der beiden letzten Gleichungen erhält <br/>man</p>
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1399 <table width="100%"
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1400 class="equation"><tr><td><a
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1401 id="x1-22r18"></a>
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1402 <center class="math-display" >
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1403 <img
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1404 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_191652x.png" alt=" V~ --- g'd t'= V~ g-d t. " class="math-display" /></center></td><td width="5%">(18)</td></tr></table>
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1405 <!--l. 1466--><p class="nopar">
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1406 </p><!--l. 1470--><p class="noindent">Statt <img
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1407 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_191653x.png" alt=" V~ g" class="sqrt" /> wird im folgenden die Größe <img
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1408 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_191654x.png" alt=" V~ --g" class="sqrt" /> eingeführt, welche <br/>wegen des hyperbolischen Charakters des zeiträumlichen Kon-<br/>tinuums stets einen reellen Wert hat. Die Invariante <img
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1409 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_191655x.png" alt=" V~ ---- -g" class="sqrt" /><span
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1410 class="cmmi-12">d<img
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1411 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi12-1c.png" alt="t" class="12x-x-1c" /> </span> <br/>ist gleich der Größe des im ,,örtlichen
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1412 Bezugssystem&#8220; mit <br/>starren Maßstäben und Uhren im Sinne der speziellen Rela-<br/>tivitätstheorie gemessenen vierdimensionalen Volumelementes.
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1413 </p><!--l. 1481--><p class="indent"> <span
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1414 class="cmti-12">Bemerkung </span><span
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1415 class="cmti-12">über den Charakter des raumzeitlichen Kon- </span> <br/><span
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1416 class="cmti-12">tinuums. </span>Unsere Voraussetzung, daß im unendlich Kleinen <br/>stets die spezielle Relativitätstheorie gelte, bringt es mit sich, <br/><pb/>
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1417 </p><!--l. 1485--><p class="indent">
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1418
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1419 </p><!--l. 1486--><p class="noindent">daß sich <span
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1420 class="cmmi-12">d s</span><sup ><span
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1421 class="cmr-8">2</span></sup> immer gemäß (1) durch die reellen Größen <br/><span
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1422 class="cmmi-12">d X</span><sub >
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1423 <span
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1424 class="cmr-8">1</span></sub><img
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1425 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_191656x.png" alt="..." class="@cdots" /><span
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1426 class="cmsy-10x-x-120"><sup class="htf"><strong>.</strong></sup></span><span
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1427 class="cmmi-12">d X</span><sub ><span
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1428 class="cmr-8">4</span></sub> ausdrücken läßt. Nennen wir <span
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1429 class="cmmi-12">d <img
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1430 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi12-1c.png" alt="t" class="12x-x-1c" /></span><sub ><sub ><span
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1431 class="cmr-6">0</span></sub></sub> das ,,natür-<br/>liche&#8220; Volumelement <span
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1432 class="cmmi-12">d X</span><sub ><span
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1433 class="cmr-8">1</span></sub> <span
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1434 class="cmmi-12">d X</span><sub ><span
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1435 class="cmr-8">2</span></sub> <span
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1436 class="cmmi-12">d X</span><sub ><span
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1437 class="cmr-8">3</span></sub> <span
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1438 class="cmmi-12">d X</span><sub ><span
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1439 class="cmr-8">4</span></sub>, so ist also</p>
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1440 <table width="100%"
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1441 class="equation"><tr><td><a
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1442 id="x1-23r19"></a>
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1443 <center class="math-display" >
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1444 <img
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1445 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_191657x.png" alt=" V~ --- d t0 = - g d t. " class="math-display" /></center></td><td width="5%">(18a)</td></tr></table>
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1446 <!--l. 1496--><p class="nopar">
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1447 </p><!--l. 1499--><p class="indent"> Soll an einer Stelle des vierdimensionalen Kontinuums <br/><img
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1448 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_191658x.png" alt=" V~ --g" class="sqrt" /> verschwinden, so bedeutet dies, daß hier einem end-<br/>lichen Koordinatenvolumen ein unendlich kleines ,,natürliches&#8220; <br/>Volumen entspreche. Dies möge nirgends der Fall sein. Dann <br/>kann <span
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1449 class="cmmi-12">g </span>sein Vorzeichen nicht ändern; wir werden im Sinne
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1450 <br/>der speziellen Relativitätstheorie annehmen, daß <span
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1451 class="cmmi-12">g </span>stets einen <br/>endlichen negativen Wert habe. Es ist dies eine Hypothese <br/>über die physikalische Natur des betrachteten Kontinuums <br/>und gleichzeitig eine Festsetzung über die Koordinatenwahl.
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1452 </p><!--l. 1511--><p class="indent"> Ist aber <span
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1453 class="cmsy-10x-x-120">-</span><span
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1454 class="cmmi-12">g </span>stets positiv und endlich, so liegt es nahe, <br/>die Koordinatenwahl a posteriori so zu treffen, daß diese <br/>Größe gleich 1 wird. Wir werden später sehen, daß durch <br/>eine solche Beschränkung der Koordinatenwahl eine bedeutende
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1455 <br/>Vereinfachung der Naturgesetze erzielt werden kann. An Stelle <br/>von (18) tritt dann einfach</p>
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1456 <center class="par-math-display" >
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1457 <img
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1458 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_191659x.png" alt=" ' d t = d t, " class="par-math-display" /></center>
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1459 <!--l. 1523--><p class="nopar">
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1460 </p><!--l. 1525--><p class="noindent">woraus mit Rücksicht auf Jakobis Satz folgt</p>
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1461 <table width="100%"
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1462 class="equation"><tr><td><a
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1463 id="x1-24r19"></a>
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1464 <center class="math-display" >
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1465 <img
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1466 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_191660x.png" alt="| '| ||@-xs-|| = 1. |@ xm | " class="math-display" /></center></td><td width="5%">(19)</td></tr></table>
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1467 <!--l. 1532--><p class="nopar">
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1468 </p><!--l. 1536--><p class="noindent">Bei dieser Koordinatenwahl sind also nur Substitutionen der <br/>Koordinaten von der Determinante 1 zulässig.
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1469 </p><!--l. 1539--><p class="indent"> Es wäre aber irrtümlich, zu glauben, daß dieser Schritt <br/>einen partiellen Verzicht auf das allgemeine Relativitäts-<br/>postulat bedeute. Wir fragen nicht: ,,Wie heißen die Natur-<br/>gesetze, welche gegenüber allen Transformationen von der
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1470 <br/>Determinante 1 kovariant sind?&#8220; Sondern wir fragen: ,,Wie <br/>heißen die
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1471 <span
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1472 class="cmti-12">allgemein </span>kovarianten Naturgesetze?&#8220; Erst nach-<br/>dem wir diese aufgestellt haben, vereinfachen wir ihren Aus-<br/>druck durch eine besondere Wahl des Bezugssystems.
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1473 </p><!--l. 1551--><p class="indent"> <span
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1474 class="cmti-12">Bildung neuer Tensoren vermittelst des Fundamentaltensors. </span> <br/>Durch innere, äußere und gemischte Multiplikation eines <br/>Tensors mit dem Fundamentaltensor entstehen Tensoren <br/>anderen Charakters und Ranges. <pb/>
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1475 </p><!--l. 1558--><p class="indent">
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1476
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1477 </p><!--l. 1559--><p class="indent"> Beispiele:</p>
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1478 <center class="par-math-display" >
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1479 <img
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1480 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_191661x.png" alt="Am = gms As, mn A = gm n A . " class="par-math-display" /></center>
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1481 <!--l. 1568--><p class="nopar">
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1482 </p><!--l. 1570--><p class="noindent">Besonders sei auf folgende Bildungen hingewiesen:
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1483 </p>
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1484 <center class="par-math-display" >
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1485 <img
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1486 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_191662x.png" alt="Am n = gma gnb Aa b, a b Am n = gm a g nb A " class="par-math-display" /></center>
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1487 <!--l. 1581--><p class="nopar">
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1488 </p><!--l. 1584--><p class="noindent">(,,Ergänzung&#8220; des kovarianten bzw. kontravarianten Tensors) <br/>und</p>
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1489 <center class="par-math-display" >
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1490 <img
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1491 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_191663x.png" alt="Bm n = gmn ga b Aa b. " class="par-math-display" /></center>
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1492 <!--l. 1592--><p class="nopar">
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1493 </p><!--l. 1594--><p class="noindent">Wir nennen <span
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1494 class="cmmi-12">B</span><sub ><span
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1495 class="cmmi-8"><img
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1496 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img
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1497 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub> den zu <span
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1498 class="cmmi-12">A</span><sub ><span
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1499 class="cmmi-8"><img
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1500 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img
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1501 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub> gehörigen reduzierten Tensor. <br/>Analog</p>
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1502 <center class="par-math-display" >
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1503 <img
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1504 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_191664x.png" alt="Bm n = gmn g Aab. a b " class="par-math-display" /></center>
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1505 <!--l. 1602--><p class="nopar">
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1506 </p><!--l. 1604--><p class="noindent">Es sei bemerkt, daß <span
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1507 class="cmmi-12">g</span><sup ><span
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1508 class="cmmi-8"><img
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1509 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img
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1510 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sup> nichts anderes ist als die Ergänzung <br/>von <span
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1511 class="cmmi-12">g</span><sub >
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1512 <span
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1513 class="cmmi-8"><img
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1514 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img
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1515 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub><span
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1516 class="cmmi-12">. </span>Denn man hat
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1517 </p>
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1518 <center class="par-math-display" >
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1519 <img
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1520 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_191665x.png" alt="gma g nb g ab = g ma dan = gm n. " class="par-math-display" /></center>
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1521 <!--l. 1613--><p class="nopar">
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1522 </p>
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1523 <div class="center" >
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1524
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1525 <!--l. 1616--><p class="noindent">
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1526
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1527 </p><!--l. 1618--><p class="noindent"><span
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1528 class="cmbsy-10x-x-120">§ </span><span
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1529 class="cmbx-12">9. Gleichung der geod</span><span
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1530 class="cmbx-12">ätischen Linie (bzw. der Punkt-</span>
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1531 <br/><span
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1532 class="cmbx-12">bewegung).</span></p></div>
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1533 <!--l. 1622--><p class="indent"> Da das ,,Linienelement&#8220; <span
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1534 class="cmmi-12">d s </span>eine unabhängig vom Koordi-<br/>natensystem definierte Größe ist, hat auch die zwischen zwei <br/>Punkten <span
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1535 class="cmmi-12">P</span><sub ><span
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1536 class="cmr-8">1</span></sub> und <span
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1537 class="cmmi-12">P</span><sub ><span
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1538 class="cmr-8">2</span></sub> des vierdimensionalen Kontinuums ge-<br/>zogene Linie, für welche <span
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1539 class="cmsy-10x-x-120"><img
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1540 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmsy10-c-73.png" alt=" integral " class="10-120x-x-73" /></span> <span
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1541 class="cmmi-12">d s </span>ein Extremum ist (geodätische <br/>Linie), eine von der Koordinatenwahl unabhängige Bedeutung. <br/>Ihre Gleichung ist</p>
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1542 <table width="100%"
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1543 class="equation"><tr><td><a
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1544 id="x1-25r20"></a>
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1545 <center class="math-display" >
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1546 <img
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1547 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_191666x.png" alt=" { P integral 2 } d d s = 0.
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1548 P1 " class="math-display" /></center></td><td width="5%">(20)</td></tr></table>
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1549 <!--l. 1634--><p class="nopar">
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1550 </p><!--l. 1637--><p class="noindent">Aus dieser Gleichung findet man in bekannter Weise durch <br/>Ausführung der Variation vier totale Differentialgleichungen, <br/>welche diese geodätische Linie bestimmen; diese Ableitung <br/>soll der Vollständigkeit halber hier Platz finden. Es sei <span
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1551 class="cmmi-12"><img
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1552 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi12-15.png" alt="c" class="12x-x-15" /> </span>eine <br/>Funktion der Koordinaten <span
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1553 class="cmmi-12">x</span><sub ><span
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1554 class="cmmi-8"><img
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1555 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub>; diese definiert eine Schar von
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1556 <br/>Flächen, welche die gesuchte geodätische Linie sowie alle ihr <br/>unendlich benachbarten, durch die Punkte <span
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1557 class="cmmi-12">P</span><sub ><span
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1558 class="cmr-8">1</span></sub> und <span
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1559 class="cmmi-12">P</span><sub ><span
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1560 class="cmr-8">2</span></sub> gezoge-<br/>nen Linien schneiden. Jede solche Kurve kann dann dadurch <br/>gegeben gedacht werden, daß ihre Koordinaten <span
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1561 class="cmmi-12">x</span><sub ><span
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1562 class="cmmi-8"><img
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1563 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub> in Funk-<br/>tion von <span
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1564 class="cmmi-12"><img
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1565 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi12-15.png" alt="c" class="12x-x-15" /> </span>ausgedrückt werden. Das Zeichen <span
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1566 class="cmmi-12"><img
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1567 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi12-e.png" alt="d" class="12x-x-e" /></span>
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1568 entspreche <br/>dem Übergang von einem Punkte der gesuchten geodätischen
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1569 <br/><pb/>
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1570 </p><!--l. 1653--><p class="indent">
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1571
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1572 </p><!--l. 1654--><p class="noindent">Linie zu demjenigen Punkte einer benachbarten Kurve, welcher <br/>zu dem nämlichen
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1573 <span
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1574 class="cmmi-12"><img
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1575 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi12-15.png" alt="c" class="12x-x-15" /> </span>gehört. Dann läßt sich (20) durch</p>
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1576 <table width="100%"
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1577 class="equation"><tr><td><a
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1578 id="x1-26r21"></a>
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1579 <center class="math-display" >
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1580 <img
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1581 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_191667x.png" alt=" integral c2 d w d c = 0 {
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1582 c1 2 d xm d xn w = gmn---------- d c d c " class="math-display" /></center></td><td width="5%">(20a)</td></tr></table>
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1583 <!--l. 1666--><p class="nopar">
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1584 </p><!--l. 1669--><p class="noindent">ersetzen. Da aber
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1585 </p>
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1586 <center class="par-math-display" >
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1587 <img
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1588 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_191668x.png" alt=" { ( )} d w = 1- 1-@-gm-nd-xm-d-xn-d xs + g mnd-xm-d d-xn- , w 2 @ xs d c d c d c d c " class="par-math-display" /></center>
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1589 <!--l. 1678--><p class="nopar">
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1590 </p><!--l. 1680--><p class="noindent">so erhält man nach Einsetzen von <span
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1591 class="cmmi-12"><img
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1592 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi12-e.png" alt="d" class="12x-x-e" /> w </span>in (20a) mit Rücksicht <br/>darauf, daß</p>
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1593 <center class="par-math-display" >
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1594 <img
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1595 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_191669x.png" alt=" ( ) d-xn- d-d-xn d d c = d c , " class="par-math-display" /></center>
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1596 <!--l. 1686--><p class="nopar">
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1597 </p><!--l. 1688--><p class="noindent">nach partieller Integration</p>
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1598 <table width="100%"
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1599 class="equation"><tr><td><a
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1600 id="x1-27r21"></a>
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1601
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1602 <center class="math-display" >
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1603 <img
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1604 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_191670x.png" alt=" integral c2 d c xs d xs = 0 {
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1605 c1 { } -d-- g-mn d xm- -1--@ g-mnd-xm-d-xn xs = d c w @ c - 2w @ x d c d c . s " class="math-display" /></center></td><td width="5%">(20b)</td></tr></table>
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1606 <!--l. 1699--><p class="nopar">
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1607 </p><!--l. 1702--><p class="noindent">Hieraus folgt wegen der freien Wählbarkeit der <span
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1608 class="cmmi-12"><img
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1609 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi12-e.png" alt="d" class="12x-x-e" /> x</span><sub ><span
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1610 class="cmmi-8"><img
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1611 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-1b.png" alt="s" class="8x-x-1b" /></span></sub> das Ver-<br/>schwinden der <span
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1612 class="cmbxti-10x-x-120">x</span><sub ><span
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1613 class="cmmi-8"><img
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1614 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-1b.png" alt="s" class="8x-x-1b" /></span></sub>. Also sind</p>
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1615 <table width="100%"
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1616 class="equation"><tr><td><a
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1617 id="x1-28r21"></a>
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1618 <center class="math-display" >
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1619 <img
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1620 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_191671x.png" alt="xs = 0 " class="math-display" /></center></td><td width="5%">(20c)</td></tr></table>
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1621 <!--l. 1708--><p class="nopar">
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1622 </p><!--l. 1711--><p class="noindent">die Gleichungen der geodätischen Linie. Ist auf der betrach-<br/>teten geodätischen Linie nicht <span
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1623 class="cmmi-12">d s </span>= 0, so können wir als <br/>Parameter <span
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1624 class="cmmi-12"><img
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1625 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi12-15.png" alt="c" class="12x-x-15" /> </span>die auf der geodätischen Linie gemessene ,,Bogen-<br/>länge&#8220; <span
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1626 class="cmmi-12">s </span>wählen. Dann wird <span
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1627 class="cmmi-12">w </span>= 1, und man erhält an Stelle
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1628 <br/>von (20c)
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1629 </p>
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1630 <center class="par-math-display" >
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1631 <img
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1632 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_191672x.png" alt=" d2 x @ g d x d x 1 @ g d x d x gm n----m2-+ ----mn----s---m-- ------mn---m----n-= 0, d s @ xs d c d c 2 @ xs d c d c " class="par-math-display" /></center>
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1633 <!--l. 1726--><p class="nopar">
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1634
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1635 </p><!--l. 1728--><p class="noindent">oder durch bloße Änderung der Bezeichnungsweise</p>
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1636 <table width="100%"
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1637 class="equation"><tr><td><a
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1638 id="x1-29r21"></a>
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1639 <center class="math-display" >
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1640 <img
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1641 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_191673x.png" alt=" |_ _|
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1642 2 m n g d--xa-+ s d-xmd-xn-= 0, as d s2 |_ _| d s d s " class="math-display" /></center></td><td width="5%">(20d)</td></tr></table>
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1643 <!--l. 1737--><p class="nopar">
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1644 </p><!--l. 1740--><p class="noindent">wobei nach Christoffel gesetzt ist</p>
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1645 <table width="100%"
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1646 class="equation"><tr><td><a
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1647 id="x1-30r21"></a>
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1648 <center class="math-display" >
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1649 <img
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1650 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_191674x.png" alt=" |_ _|
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1651 m n ( ) s = 1- @-gms-+ @-gns-- @ g-mn . |_ _| 2 @ xn @ xm @ xs " class="math-display" /></center></td><td width="5%">(21)</td></tr></table>
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1652 <!--l. 1751--><p class="nopar">
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1653 </p><!--l. 1754--><p class="noindent">Multipliziert man endlich (20d) mit <span
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1654 class="cmmi-12">g</span><sup ><span
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1655 class="cmmi-8"><img
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1656 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-1b.png" alt="s" class="8x-x-1b" /><img
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1657 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-1c.png" alt="t" class="8x-x-1c" /></span></sup> (äußere Multiplikation <br/>bezüglich <span
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1658 class="cmmi-12"><img
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1659 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi12-1c.png" alt="t" class="12x-x-1c" /></span>, innere bezüglich <span
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1660 class="cmmi-12"><img
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1661 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi12-1b.png" alt="s" class="12x-x-1b" /></span>), so erhält man schließlich als <br/>endgültige Form der Gleichung der geodätischen Linie <pb/>
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1662 </p><!--l. 1762--><p class="indent">
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1663
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1664 </p>
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1665 <table width="100%"
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1666 class="equation"><tr><td><a
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1667 id="x1-31r22"></a>
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1668 <center class="math-display" >
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1669 <img
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1670 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_191675x.png" alt=" 2 { m n} d--xt-+ t d xm-d-xn-= 0. d s2 d s d s " class="math-display" /></center></td><td width="5%">(22)</td></tr></table>
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1671 <!--l. 1770--><p class="nopar">
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1672 </p><!--l. 1774--><p class="noindent">Hierbei ist nach Christoffel gesetzt</p>
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1673 <table width="100%"
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1674 class="equation"><tr><td><a
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1675 id="x1-32r23"></a>
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1676 <center class="math-display" >
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1677 <img
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1678 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_191676x.png" alt=" |_ _|
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1679 { m n } m n t = gta |_ a _| . " class="math-display" /></center></td><td width="5%">(23)</td></tr></table>
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1680 <!--l. 1786--><p class="nopar">
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1681 </p>
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1682 <div class="center" >
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1683
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1684 <!--l. 1791--><p class="noindent">
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1685 </p><!--l. 1793--><p class="noindent"><span
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1686 class="cmbsy-10x-x-120">§ </span><span
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1687 class="cmbx-12">10. Die Bildung von Tensoren durch Differentiation.</span></p></div>
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1688 <!--l. 1797--><p class="indent"> Gestützt auf die Gleichung der geodätischen Linie können <br/>wir nun leicht die Gesetze ableiten, nach welchen durch Diffe-<br/>rentiation aus Tensoren neue Tensoren gebildet werden können. <br/>Dadurch werden wir erst in den Stand gesetzt, allgemein ko-<br/>variante Differentialgleichungen aufzustellen. Wir erreichen <br/>dies Ziel durch wiederholte Anwendung des folgenden ein-<br/>fachen
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1689 Satzes.
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1690 </p><!--l. 1806--><p class="indent"> Ist in unserem Kontinuum eine Kurve gegeben, deren <br/>Punkte durch die Bogendistanz <span
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1691 class="cmmi-12">s </span>von einem Fixpunkt auf <br/>der Kurve charakterisiert sind, ist ferner <span
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1692 class="cmmi-12"><img
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1693 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi12-27.png" alt="f" class="cmmi-12x-x-27" align="middle" /> </span>eine invariante <br/>Raumfunktion, so ist auch <span
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1694 class="cmmi-12">d <img
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1695 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi12-27.png" alt="f" class="cmmi-12x-x-27" align="middle" /></span><img
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1696 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_191677x.png" alt="/" class="left" align="middle" /> <span
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1697 class="cmmi-12">d s</span> eine Invariante. Der Be-<br/>weis liegt darin, daß sowohl <span
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1698 class="cmmi-12">d<img
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1699 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi12-27.png" alt="f" class="cmmi-12x-x-27" align="middle" /> </span>als auch <span
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1700 class="cmmi-12">ds </span>Invariante sind.
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1701 </p><!--l. 1815--><p class="indent"> Da</p>
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1702 <center class="par-math-display" >
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1703 <img
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1704 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_191678x.png" alt="d f @ f d x ----= -------m-, d s @ xm d s " class="par-math-display" /></center>
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1705 <!--l. 1822--><p class="nopar">
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1706 </p><!--l. 1824--><p class="noindent">so ist auch</p>
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1707 <center class="par-math-display" >
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1708 <img
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1709 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_191679x.png" alt=" @ f d x y = --------m- @ xm d s " class="par-math-display" /></center>
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1710 <!--l. 1831--><p class="nopar">
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1711 </p><!--l. 1833--><p class="noindent">eine Invariante, und zwar für alle Kurven, die von einem <br/>Punkte des Kontinuums ausgehen, d. h. für beliebige Wahl <br/>des Vektors der <span
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1712 class="cmmi-12">d x</span><sub ><span
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1713 class="cmmi-8"><img
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1714 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /></span></sub><span
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1715 class="cmmi-12">. </span>Daraus folgt unmittelbar, daß</p>
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1716 <table width="100%"
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1717 class="equation"><tr><td><a
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1718 id="x1-33r24"></a>
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1719 <center class="math-display" >
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1720 <img
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1721 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_191680x.png" alt="Am = @-f-- @ xm " class="math-display" /></center></td><td width="5%">(24)</td></tr></table>
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1722 <!--l. 1842--><p class="nopar">
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1723 </p><!--l. 1847--><p class="noindent">ein kovarianter Vierervektor ist (Gradient von <span
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1724 class="cmmi-12"><img
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1725 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi12-27.png" alt="f" class="cmmi-12x-x-27" align="middle" /></span>).
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1726 </p><!--l. 1850--><p class="indent"> Nach unserem Satze ist ebenso der auf einer Kurve ge-<br/>nommene Differentialquotient</p>
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1727 <center class="par-math-display" >
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1728 <img
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1729 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_191681x.png" alt=" d y x = ---- d s " class="par-math-display" /></center>
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1730 <!--l. 1857--><p class="nopar">
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1731 </p><!--l. 1859--><p class="noindent">eine Invariante. Durch Einsetzen von <span
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1732 class="cmmi-12"><img
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1733 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi12-27.png" alt="f" class="cmmi-12x-x-27" align="middle" /> </span>erhalten wir zunächst
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1734 </p>
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1735 <center class="par-math-display" >
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1736 <img
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1737 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_191682x.png" alt=" @2 f d xm d xn @ f d 2 xm x = @-x--@-x--d-s--d s-+ @-x---d-s2-. m n m " class="par-math-display" /></center>
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1738 <!--l. 1869--><p class="nopar">
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1739 </p><!--l. 1871--><p class="noindent">Hieraus läßt sich zunächst die Existenz eines Tensors <br/>nicht ableiten. Setzen wir nun aber fest, daß die Kurve, <br/><pb/>
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1740 </p><!--l. 1876--><p class="indent">
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1741
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1742 </p><!--l. 1877--><p class="noindent">auf welcher wir differenziiert haben, eine geodätische Kurve <br/>sei, so erhalten wir nach (22) durch Ersetzen von <span
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1743 class="cmmi-12">d</span><sup ><span
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1744 class="cmr-8">2</span></sup> <span
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1745 class="cmmi-12">x</span><sub >
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1746 <span
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1747 class="cmmi-8"><img
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1748 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub><img
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1749 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_191683x.png" alt="/" class="left" align="middle" /> <span
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1750 class="cmmi-12">d s</span><sup ><span
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1751 class="cmr-8">2</span></sup>
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1752 </p>
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1753 <center class="par-math-display" >
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1754 <img
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1755 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_191684x.png" alt=" m n { --@2-f---- { } -@-f-} d-xm-d-xn- x = @ x @ x - t @ x d s d s . m n t " class="par-math-display" /></center>
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1756 <!--l. 1892--><p class="nopar">
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1757 </p><!--l. 1894--><p class="indent"> Aus der Vertauschbarkeit der Differentiationen nach <span
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1758 class="cmmi-12"><img
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1759 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi12-16.png" alt="m" class="cmmi-12x-x-16" align="middle" /> </span> <br/>und <span
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1760 class="cmmi-12"><img
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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1761 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi12-17.png" alt="n" class="12x-x-17" /> </span>und daraus, daß gemäß (23) und (21) die Klammer <img
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1762 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_191685x.png" alt=" mn { t }" class="left" align="middle" /> <br/>bezüglich <span
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1763 class="cmmi-12"><img
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1764 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi12-16.png" alt="m" class="cmmi-12x-x-16" align="middle" /> </span>und <span
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1765 class="cmmi-12"><img
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1766 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi12-17.png" alt="n" class="12x-x-17" /> </span>symmetrisch ist, folgt, daß der Klammer-<br/>ausdruck in <span
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1767 class="cmmi-12"><img
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1768 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi12-16.png" alt="m" class="cmmi-12x-x-16" align="middle" /> </span>und <span
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1769 class="cmmi-12"><img
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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1770 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi12-17.png" alt="n" class="12x-x-17" /> </span>symmetrisch ist. Da man von einem <br/>Punkt des Kontinuums aus in beliebiger Richtung eine geo-<br/>dätische Linie ziehen kann, <span
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1771 class="cmmi-12">d x</span><sub ><span
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1772 class="cmmi-8"><img
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1773 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /></span></sub><img
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1774 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_191686x.png" alt="/" class="left" align="middle" /> <span
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1775 class="cmmi-12">d s</span> also ein Vierervektor mit <br/>frei wählbarem Verhältnis der Komponenten ist, folgt nach <br/>den Ergebnissen des <span
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1776 class="cmsy-10x-x-120">§ </span>7, daß</p>
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1777 <table width="100%"
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1778 class="equation"><tr><td><a
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1779 id="x1-34r25"></a>
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1780 <center class="math-display" >
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1781 <img
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1782 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_191687x.png" alt=" @2 f { m n} @ f Am n = ----------- t -----: @ xm @ xm @ xt " class="math-display" /></center></td><td width="5%">(25)</td></tr></table>
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1783 <!--l. 1915--><p class="nopar">
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1784 </p><!--l. 1920--><p class="noindent">ein kovarianter Tensor zweiten Ranges ist. Wir haben also <br/>das Ergebnis gewonnen: Aus dem kovarianten Tensor ersten <br/>Ranges</p>
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1785 <center class="par-math-display" >
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1786 <img
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1787 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_191688x.png" alt=" @ f Am = ----- @ xm " class="par-math-display" /></center>
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1788 <!--l. 1928--><p class="nopar">
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1789 </p><!--l. 1930--><p class="noindent">können wir durch Differentiation einen kovarianten Tensor <br/>zweiten Ranges</p>
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1790 <table width="100%"
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1791 class="equation"><tr><td><a
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1792 id="x1-35r26"></a>
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1793
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1794 <center class="math-display" >
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1795 <img
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1796 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_191689x.png" alt=" { m n } A = @-Am- - t A mn @ xn t " class="math-display" /></center></td><td width="5%">(26)</td></tr></table>
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1797 <!--l. 1941--><p class="nopar">
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1798 </p><!--l. 1946--><p class="noindent">bilden. Wir nennen den Tensor <span
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1799 class="cmmi-12">A</span><sub ><span
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1800 class="cmmi-8"><img
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1801 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img
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1802 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub> die ,,<span
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1803 class="cmti-12">Erweiterung</span>&#8220; des <br/>Tensors <span
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1804 class="cmmi-12">A</span><sub ><span
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1805 class="cmmi-8"><img
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1806 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /></span></sub>. Zunächst können wir leicht zeigen, daß diese <br/>Bildung auch dann auf einen Tensor führt, wenn der Vektor <span
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1807 class="cmmi-12">A</span><sub ><span
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1808 class="cmmi-8"><img
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1809 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /></span></sub> <br/>nicht als ein Gradient darstellbar ist. Um dies einzusehen,
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1810 <br/>bemerken wir zunächst, daß</p>
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1811 <center class="par-math-display" >
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1812 <img
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1813 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_191690x.png" alt=" @ f y @-x-- m " class="par-math-display" /></center>
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1814 <!--l. 1958--><p class="nopar">
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1815 </p><!--l. 1960--><p class="noindent">ein kovarianter Vierervektor ist, wenn <span
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1816 class="cmmi-12"><img
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1817 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi12-27.png" alt="f" class="cmmi-12x-x-27" align="middle" /> </span>und <span
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1818 class="cmmi-12"><img
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1819 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi12-27.png" alt="f" class="cmmi-12x-x-27" align="middle" /> </span>Skalare sind. <br/>Dies ist auch der Fall für eine aus vier solchen Gliedern be-<br/>stehende Summe
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1820 </p>
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1821 <center class="par-math-display" >
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1822 <img
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1823 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_191691x.png" alt=" @ f(1) @ f(4) Sm = y(1)------+ .+ .+ y(4)------, @ xm @ xm " class="par-math-display" /></center>
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1824 <!--l. 1970--><p class="nopar">
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1825 </p><!--l. 1972--><p class="noindent">falls <span
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1826 class="cmmi-12"><img
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1827 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi12-20.png" alt="y" class="12x-x-20" /></span><sup ><span
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1828 class="cmr-8">(1)</span></sup> <span
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1829 class="cmmi-12"><img
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1830 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi12-27.png" alt="f" class="cmmi-12x-x-27" align="middle" /></span><sup ><span
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1831 class="cmr-8">(1)</span></sup><span
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1832 class="cmmi-12">...</span><span
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1833 class="cmmi-12">.<img
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1834 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi12-20.png" alt="y" class="12x-x-20" /></span><sup ><span
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1835 class="cmr-8">(4)</span></sup> <span
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1836 class="cmmi-12"><img
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1837 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi12-27.png" alt="f" class="cmmi-12x-x-27" align="middle" /></span><sup ><span
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1838 class="cmr-8">(4)</span></sup> Skalare sind. Nun ist aber klar, daß <br/>sich jeder kovariante Vierervektor in der Form <span
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1839 class="cmmi-12">S</span><sub ><span
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1840 class="cmmi-8"><img
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1841 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /></span></sub> darstellen <br/>läßt. Ist nämlich <span
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1842 class="cmmi-12">A</span><sub ><span
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1843 class="cmmi-8"><img
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1844 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /></span></sub> ein Vierervektor, dessen Komponenten <br/><pb/>
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1845 </p><!--l. 1977--><p class="indent">
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1846
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1847 </p><!--l. 1978--><p class="noindent">beliebig gegebene Funktionen der <span
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1848 class="cmmi-12">x</span><sub ><span
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1849 class="cmmi-8"><img
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1850 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub> sind, so hat man nur <br/>(bezüglich des gewählten Koordinatensystems) zu setzen
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1851 </p>
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1852 <center class="par-math-display" >
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1853 <img
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1854 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_191692x.png" alt="y(1) = A1, f(1) = x1, (2) (2) y = A2, f = x2, y(3) = A3, f(3) = x3, (4) (4) y = A4, f = x4, " class="par-math-display" /></center>
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1855 <!--l. 1995--><p class="nopar">
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1856 </p><!--l. 1999--><p class="noindent">um zu erreichen, daß <span
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1857 class="cmmi-12">S</span><sub ><span
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1858 class="cmmi-8"><img
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1859 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /></span></sub> gleich <span
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1860 class="cmmi-12">A</span><sub ><span
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1861 class="cmmi-8"><img
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1862 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /></span></sub> wird.
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1863 </p><!--l. 2002--><p class="indent"> Um daher zu beweisen, daß <span
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1864 class="cmmi-12">A</span><sub ><span
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1865 class="cmmi-8"><img
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1866 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img
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1867 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub> ein Tensor ist, wenn auf <br/>der rechten Seite für
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1868 <span
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1869 class="cmmi-12">A</span><sub ><span
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1870 class="cmmi-8"><img
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1871 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /></span></sub> ein beliebiger kovarianter Vierer-<br/>vektor eingesetzt wird, brauchen wir nur zu zeigen, daß dies <br/>für den Vierervektor <span
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1872 class="cmmi-12">S</span><sub ><span
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1873 class="cmmi-8"><img
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1874 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /></span></sub> zutrifft. Für letzteres ist es aber, <br/>wie ein Blick auf die rechte Seite von (26) lehrt, hinreichend, <br/>den Nachweis für den Fall</p>
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1875 <center class="par-math-display" >
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1876 <img
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1877 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_191693x.png" alt=" @ f Am = y ----- @ xm " class="par-math-display" /></center>
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1878 <!--l. 2014--><p class="nopar">
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1879 </p><!--l. 2018--><p class="noindent">zu führen. Es hat nun die mit <span
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1880 class="cmmi-12"><img
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1881 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi12-20.png" alt="y" class="12x-x-20" /> </span>multiplizierte rechte Seite <br/>von (25)</p>
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1882 <center class="par-math-display" >
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1883 <img
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1884 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_191694x.png" alt=" m n @2 f { } @ f y ---------- t y ----- @xm @ xn @ xt " class="par-math-display" /></center>
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1885 <!--l. 2029--><p class="nopar">
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1886 </p><!--l. 2031--><p class="noindent">Tensorcharakter. Ebenso ist</p>
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1887 <center class="par-math-display" >
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1888 <img
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1889 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_191695x.png" alt=" @ y @ f ---------- @ xm@ xn " class="par-math-display" /></center>
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1890 <!--l. 2038--><p class="nopar">
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1891 </p><!--l. 2043--><p class="noindent">ein Tensor (äußeres Produkt zweier Vierervektoren). Durch <br/>Addition folgt der Tensorcharakter von
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1892 </p>
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1893
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1894 <center class="par-math-display" >
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1895 <img
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1896 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_191696x.png" alt=" ( ) { m n} ( ) --@-- -@-f- @-f-- @ xn y@ xm - t y @ xt . " class="par-math-display" /></center>
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1897 <!--l. 2057--><p class="nopar">
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1898 </p><!--l. 2059--><p class="noindent">Damit ist, wie ein Blick auf (26) lehrt, der verlangte Nachweis <br/>für den Vierervektor
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1899 </p>
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1900 <center class="par-math-display" >
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1901 <img
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1902 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_191697x.png" alt="y -@-f-, @ xm " class="par-math-display" /></center>
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1903 <!--l. 2066--><p class="nopar">
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1904 </p><!--l. 2069--><p class="noindent">und daher nach dem vorhin Bewiesenen für jeden beliebigen <br/>Vierervektor <span
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1905 class="cmmi-12">A</span><sub ><span
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1906 class="cmmi-8"><img
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1907 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /></span></sub>
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1908 geführt. --
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1909 </p><!--l. 2072--><p class="indent"> Mit Hilfe der Erweiterung des Vierervektors kann man <br/>leicht die ,,Erweiterung&#8220; eines kovarianten Tensors beliebigen <br/>Ranges definieren; diese Bildung ist eine Verallgemeinerung <br/>der Erweiterung des Vierervektors. Wir beschränken uns auf <br/>die Aufstellung der Erweiterung des Tensors zweiten Ranges, <br/>da dieser das Bildungsgesetz bereits klar übersehen läßt.
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1910 <pb/>
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1911 </p><!--l. 2082--><p class="indent">
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1912
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1913 </p><!--l. 2083--><p class="indent"> Wie bereits bemerkt, läßt sich jeder kovariante Tensor <br/>zweiten Ranges darstellen<sup ><span
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1914 class="cmr-8">1</span></sup>) als eine Summe von Tensoren <br/>vom Typus <span
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1915 class="cmmi-12">A</span><sub >
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1916 <span
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1917 class="cmmi-8"><img
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1918 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /></span></sub> <span
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1919 class="cmmi-12">B</span><sub ><span
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1920 class="cmmi-8"><img
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1921 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub>. Es wird deshalb genügen, den Ausdruck <br/>der Erweiterung für einen solchen speziellen Tensor abzuleiten. <br/>Nach (26) haben die Ausdrücke
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1922 </p>
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1923 <center class="par-math-display" >
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1924 <img
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1925 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_191698x.png" alt="@ A { s m} ---m-- t At, @ xs
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1926 s n @-Bn- { } @ xs - t Bt " class="par-math-display" /></center>
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1927 <!--l. 2105--><p class="nopar">
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1928 </p><!--l. 2107--><p class="noindent">Tensorcharakter. Durch äußere Multiplikation des ersten mit <br/><span
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1929 class="cmmi-12">B</span><sub ><span
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1930 class="cmmi-8"><img
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1931 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub>, des zweiten mir
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1932 <span
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1933 class="cmmi-12">A</span><sub ><span
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1934 class="cmmi-8"><img
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1935 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /></span></sub> erhält man je einen Tensor dritten <br/>Ranges; deren Addition ergibt den Tensor dritten Ranges</p>
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1936 <table width="100%"
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1937 class="equation"><tr><td><a
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1938 id="x1-36r27"></a>
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1939 <center class="math-display" >
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1940 <img
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1941 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_191699x.png" alt=" { s m } { s n} Am ns = @-Am-n - t At n- t Am t, @ xs " class="math-display" /></center></td><td width="5%">(27)</td></tr></table>
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1942 <!--l. 2124--><p class="nopar">
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1943 </p><!--l. 2129--><p class="noindent">wobei <span
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1944 class="cmmi-12">A</span><sub ><span
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1945 class="cmmi-8"><img
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1946 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img
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1947 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub> = <span
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1948 class="cmmi-12">A</span><sub ><span
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1949 class="cmmi-8"><img
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1950 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /></span></sub> <span
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1951 class="cmmi-12">B</span><sub ><span
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1952 class="cmmi-8"><img
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1953 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub> gesetzt ist. Da die rechte Seite von (27) <br/>linear und homogen ist bezüglich der <span
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1954 class="cmmi-12">A</span><sub ><span
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1955 class="cmmi-8"><img
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1956 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img
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1957 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub> und deren ersten <br/>Ableitungen, führt dieses Bildungsgesetz nicht nur bei einem <br/>Tensor vom Typus <span
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1958 class="cmmi-12">A</span><sub ><span
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1959 class="cmmi-8"><img
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1960 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /></span></sub> <span
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1961 class="cmmi-12">B</span><sub ><span
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1962 class="cmmi-8"><img
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1963 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub>, sondern auch bei einer Summe
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1964 <br/>solcher Tensoren, d. h. bei einem beliebigen kovarianten <br/>Tensor zweiten Ranges, zu einem Tensor. Wir nennen <span
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1965 class="cmmi-12">A</span><sub ><span
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1966 class="cmmi-8"><img
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1967 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img
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1968 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-17.png" alt="n" class="8x-x-17" /> <img
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1969 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-1b.png" alt="s" class="8x-x-1b" /></span></sub> <br/>die Erweiterung des Tensors
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1970 <span
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1971 class="cmmi-12">A</span><sub ><span
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1972 class="cmmi-8"><img
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1973 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img
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1974 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub><span
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1975 class="cmmi-12">.</span>
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1976 </p><!--l. 2135--><p class="indent"> Es ist klar, daß (26) und (24) nur spezielle Fälle von (27) <br/>sind (Erweiterung
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1977 des Tensors ersten bzw. nullten Ranges). <br/>Überhaupt lassen sich alle speziellen Bildungsgesetze von <br/>Tensoren auf (27) in Verbindung mit Tensormultiplikationen
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1978 <br/>auffassen.
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1979 </p>
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1980 <div class="center" >
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1981
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1982 <!--l. 2141--><p class="noindent">
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1983 </p><!--l. 2143--><p class="noindent"><span
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1984 class="cmbsy-10x-x-120">§ </span><span
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1985 class="cmbx-12">11. Einige Spezialf</span><span
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1986 class="cmbx-12">älle von besonderer Bedeutung.</span></p></div>
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1987 <!--l. 2147--><p class="indent"> <span
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1988 class="cmti-12">Einige den Fundamentaltensor betreffende Hilfss</span><span
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1989 class="cmti-12">ätze. </span>Wir <br/>leiten zunächst einige im folgenden viel gebrauchte Hilfs-<br/></p><!--l. 2151--><p class="noindent">----------
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1990 </p><!--l. 2154--><p class="indent"> 1) Durch äußere Multiplikation der Vektoren mit den (beliebig <br/>gegebenen) Komponenten <span
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1991 class="cmmi-12">A</span><sub ><span
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1992 class="cmr-8">11</span></sub>, <span
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1993 class="cmmi-12">A</span><sub ><span
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1994 class="cmr-8">12</span></sub>, <span
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1995 class="cmmi-12">A</span><sub ><span
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1996 class="cmr-8">13</span></sub>, <span
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1997 class="cmmi-12">A</span><sub ><span
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1998 class="cmr-8">14</span></sub> bzw. 1, 0, 0, 0 entsteht <br/>ein Tensor mit den Komponenten
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1999 </p>
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2000 <center class="par-math-display" >
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2001 <img
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2002 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_1916100x.png" alt="A11 A12 A13 A14 0 0 0 0
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2003 0 0 0 0 0 0 0 0 " class="par-math-display" /></center>
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2004 <!--l. 2168--><p class="nopar">
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2005 </p><!--l. 2171--><p class="noindent">Durch Addition von vier Tensoren von diesem Typus erhält man den <br/>Tensor <span
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2006 class="cmmi-12">A</span><sub ><span
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2007 class="cmmi-8"><img
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2008 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img
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2009 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub>
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2010 mit beliebig vorgeschriebenen Komponenten. <pb/>
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2011 </p><!--l. 2177--><p class="indent">
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2012
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2013 </p><!--l. 2178--><p class="noindent">gleichungen ab. Nach der Regel von der Differentiation der <br/>Determinanten ist</p>
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2014 <table width="100%"
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2015 class="equation"><tr><td><a
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2016 id="x1-37r28"></a>
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2017 <center class="math-display" >
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2018 <img
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2019 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_1916101x.png" alt="d g = gm n g d g = - g g d gmn. mn m n " class="math-display" /></center></td><td width="5%">(28)</td></tr></table>
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2020 <!--l. 2185--><p class="nopar">
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2021 </p><!--l. 2189--><p class="noindent">Die letzte Form rechtfertigt sich durch die vorletzte, wenn <br/>man bedenkt, daß
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2022 <span
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2023 class="cmmi-12">g</span><sub ><span
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2024 class="cmmi-8"><img
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2025 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img
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2026 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub> <span
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2027 class="cmmi-12">g</span><sup ><span
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2028 class="cmmi-8"><img
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2029 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /></span><span
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2030 class="cmsy-8">'</span><span
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2031 class="cmmi-8"><img
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2032 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sup> = <span
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2033 class="cmmi-12"><img
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2034 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi12-e.png" alt="d" class="12x-x-e" /></span><sub >
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2035 <span
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2036 class="cmmi-8"><img
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2037 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /></span></sub><sup ><span
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2038 class="cmmi-8"><img
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2039 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /></span><span
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2040 class="cmsy-8">'</span></sup>, daß also <span
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2041 class="cmmi-12">g</span><sub >
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2042 <span
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2043 class="cmmi-8"><img
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2044 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img
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2045 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub> <span
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2046 class="cmmi-12">g</span><sup ><span
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2047 class="cmmi-8"><img
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2048 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img
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2049 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sup> = 4, folglich
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2050 </p>
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2051 <center class="par-math-display" >
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2052 <img
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2053 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_1916102x.png" alt="g d gmn + gmn d g = 0. mn mn " class="par-math-display" /></center>
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2054 <!--l. 2198--><p class="nopar">
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2055 </p><!--l. 2201--><p class="noindent">Aus (28) folgt</p>
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2056 <table width="100%"
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2057 class="equation"><tr><td><a
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2058 id="x1-38r29"></a>
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2059 <center class="math-display" >
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2060 <img
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2061 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_1916103x.png" alt=" V~ --- --1--@-----g 1-@-1g-(--g) 1- mn @-gmn- 1- @-gmn- V~ -g @ xs = 2 @ xs = 2 g @ xs = - 2 gm n@ xs . " class="math-display" /></center></td><td width="5%">(29)</td></tr></table>
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2062 <!--l. 2212--><p class="nopar">
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2063 </p><!--l. 2216--><p class="noindent">Aus</p>
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2064 <center class="par-math-display" >
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2065 <img
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2066 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_1916104x.png" alt="g ms gn s = dmn " class="par-math-display" /></center>
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2067 <!--l. 2222--><p class="nopar">
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2068 </p><!--l. 2224--><p class="noindent">folgt ferner durch Differentiation</p>
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2069 <table width="100%"
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2070 class="equation"><tr><td><a
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2071 id="x1-39r30"></a>
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2072 <center class="math-display" >
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2073 <img
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2074 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_1916105x.png" alt=" g d gns = -gn s d g { m s m s @-gns- n s@-gms- bzw. gm s@ xc = - g @ xc . " class="math-display" /></center></td><td width="5%">(30)</td></tr></table>
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2075 <!--l. 2236--><p class="nopar">
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2076 </p><!--l. 2240--><p class="noindent">Durch gemischte Multiplikation mit <span
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2077 class="cmmi-12">g</span><sup ><span
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2078 class="cmmi-8"><img
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2079 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-1b.png" alt="s" class="8x-x-1b" /><img
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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diff changeset
2080 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-1c.png" alt="t" class="8x-x-1c" /></span></sup> bzw. <span
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2081 class="cmmi-12">g</span><sub >
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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2082 <span
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2083 class="cmmi-8"><img
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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2084 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-17.png" alt="n" class="8x-x-17" /><img
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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diff changeset
2085 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-15.png" alt="c" class="8x-x-15" /></span></sub> erhält man <br/>hieraus (bei geänderter Bezeichnungsweise der Indizes)</p>
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2086 <table width="100%"
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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2087 class="equation"><tr><td><a
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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2088 id="x1-40r31"></a>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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2089 <center class="math-display" >
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2090 <img
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2091 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_1916106x.png" alt=" d gmn = - gma gn b d ga b, { mn ab @-g---= - gma gn b@-g--- @ xs @ xs " class="math-display" /></center></td><td width="5%">(31)</td></tr></table>
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2092 <!--l. 2255--><p class="nopar">
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2093 </p><!--l. 2260--><p class="noindent">bzw.</p>
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2094 <table width="100%"
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2095 class="equation"><tr><td><a
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2096 id="x1-41r32"></a>
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2097
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2098 <center class="math-display" >
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2099 <img
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2100 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_1916107x.png" alt=" d gmn = - gm a g nb d ga b { ab @-g-mn = - gma g nb@-g---. @ xs @ xs " class="math-display" /></center></td><td width="5%">(32)</td></tr></table>
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2101 <!--l. 2274--><p class="nopar">
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2102 </p><!--l. 2278--><p class="noindent">Die Beziehung (31) erlaubt eine Umformung, von der wir <br/>ebenfalls öfter Gebrauch zu machen haben. Gemäß (21) ist</p>
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2103 <table width="100%"
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2104 class="equation"><tr><td><a
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2105 id="x1-42r33"></a>
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2106 <center class="math-display" >
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2107 <img
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2108 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_1916108x.png" alt=" |_ _| |_ _|
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2109 a s b s @-ga-b @ x = |_ b _| + |_ a _| . s " class="math-display" /></center></td><td width="5%">(33)</td></tr></table>
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2110 <!--l. 2291--><p class="nopar">
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2111 </p><!--l. 2295--><p class="noindent">Setzt man dies in die zweite der Formeln (31) ein, so erhält <br/>man mit Rücksicht auf (23)</p>
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2112 <table width="100%"
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2113 class="equation"><tr><td><a
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2114 id="x1-43r34"></a>
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2115 <center class="math-display" >
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2116 <img
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2117 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_1916109x.png" alt=" ( ) { t s} { t s} @-gm-n = - gmt n + gnt m @ xa " class="math-display" /></center></td><td width="5%">(34)</td></tr></table>
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2118 <!--l. 2308--><p class="nopar">
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2119 </p><!--l. 2312--><p class="noindent">Durch Substitution der rechten Seite von (34) in (29) ergibt sich
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2120 </p>
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2121 <table width="100%"
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2122 class="equation"><tr><td><a
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2123 id="x1-44r35"></a>
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2124 <center class="math-display" >
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2125 <img
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2126 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_1916110x.png" alt=" 1 @ V~ --g { m s} V~ --- -------= m . - y @ xs " class="math-display" /></center></td><td width="5%">(29a)</td></tr></table>
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2127 <!--l. 2319--><p class="nopar">
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2128 <pb/></p><!--l. 2326--><p class="indent">
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2129
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2130 </p><!--l. 2327--><p class="indent"> <span
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2131 class="cmti-12">Divergenz des kontravarianten Vierervektors. </span>Multipliziert <br/>man (26) mit dem kontravarianten Fundamentaltensor <span
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2132 class="cmmi-12">g</span><sup ><span
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2133 class="cmmi-8"><img
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2134 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img
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2135 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sup> <br/>(innere Multiplikation), so nimmt die rechte Seite nach Um-<br/>formung des ersten Gliedes zunächst die Form an</p>
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2136 <center class="par-math-display" >
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2137 <img
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2138 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_1916111x.png" alt=" mn ( ma ) -@-- mn @g--- 1- ta @g--- @gna- @g-mn mn @xn (g Am) - Am @xn - 2 g @xn + @xm - @xa g At . " class="par-math-display" /></center>
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2139 <!--l. 2341--><p class="nopar">
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2140 </p><!--l. 2344--><p class="noindent">Das letzte Glied dieses Ausdruckes kann gemäß (31) und (29) <br/>in die Form
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2141 </p>
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2142 <center class="par-math-display" >
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2143 <img
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2144 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_1916112x.png" alt=" V~ -- 1-@-gt-n 1-@-gtm- --1-- @----g- mn 2 @ xn At + 2 @ xm At + V~ --g @ xa g At. " class="par-math-display" /></center>
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2145 <!--l. 2354--><p class="nopar">
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2146 </p><!--l. 2356--><p class="noindent">gebracht werden. Da es auf die Benennung der Summations-<br/>indizes nicht ankommt, heben sich die beiden ersten Glieder <br/>dieses Ausdruckes gegen das zweite des obigen weg; das letzte <br/>läßt sich mit dem ersten des obigen Ausdruckes vereinigen. <br/>Setzt man noch</p>
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2147 <center class="par-math-display" >
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2148 <img
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2149 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_1916113x.png" alt="gmn Am = An, " class="par-math-display" /></center>
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2150 <!--l. 2366--><p class="nopar">
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2151 </p><!--l. 2368--><p class="noindent">wobei <span
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2152 class="cmmi-12">A</span><sup ><span
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2153 class="cmmi-8"><img
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2154 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sup> ebenso wie <span
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2155 class="cmmi-12">A</span><sub >
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2156 <span
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2157 class="cmmi-8"><img
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2158 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /></span></sub> ein frei wählbarer Vektor ist, so er-<br/>hält man endlich</p>
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2159 <table width="100%"
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2160 class="equation"><tr><td><a
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2161 id="x1-45r35"></a>
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2162 <center class="math-display" >
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2163 <img
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2164 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_1916114x.png" alt="P = V~ -1---@--( V~ -g-An) . - g@ xn " class="math-display" /></center></td><td width="5%">(35)</td></tr></table>
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2165
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2166 <!--l. 2376--><p class="nopar">
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2167 </p><!--l. 2380--><p class="noindent">Dieser Skalar ist die <span
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2168 class="cmti-12">Divergenz </span>des kontravarianten Vierer-<br/>vektors <span
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2169 class="cmmi-12">A</span><sup ><span
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2170 class="cmmi-8"><img
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2171 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sup><span
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2172 class="cmmi-12">.</span>
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2173 </p><!--l. 2383--><p class="indent"> <span
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2174 class="cmti-12">,,Rotation&#8220; des </span>(<span
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2175 class="cmti-12">kovarianten</span>) <span
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2176 class="cmti-12">Vierervektors. </span>Das zweite <br/>Glied in (26) ist in den Indizes <span
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2177 class="cmmi-12"><img
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2178 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi12-16.png" alt="m" class="cmmi-12x-x-16" align="middle" /> </span>und <span
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2179 class="cmmi-12"><img
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2180 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi12-17.png" alt="n" class="12x-x-17" /> </span>symmetrisch. Es ist <br/>deshalb <span
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2181 class="cmmi-12">A</span><sub ><span
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2182 class="cmmi-8"><img
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2183 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img
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diff changeset
2184 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub> <span
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2185 class="cmsy-10x-x-120">-</span><span
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diff changeset
2186 class="cmmi-12">A</span><sub ><span
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2187 class="cmmi-8"><img
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2188 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-17.png" alt="n" class="8x-x-17" /><img
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diff changeset
2189 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /></span></sub> ein besonders einfach gebauter (anti-<br/>symmetrischer) Tensor. Man erhält</p>
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2190 <table width="100%"
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2191 class="equation"><tr><td><a
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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diff changeset
2192 id="x1-46r36"></a>
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2193 <center class="math-display" >
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2194 <img
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2195 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_1916115x.png" alt=" @-Am- @ An- Bm n = @ x - @ x . n m " class="math-display" /></center></td><td width="5%">(36)</td></tr></table>
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2196 <!--l. 2395--><p class="nopar">
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2197 </p><!--l. 2400--><p class="indent"> <span
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2198 class="cmti-12">Antisymmetrische Erweiterung eines Sechservektors. </span>Wendet <br/>man (27) auf einen antisymmetrischen Tensor zweiten Ranges <br/><span
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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2199 class="cmmi-12">A</span><sub ><span
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2200 class="cmmi-8"><img
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diff changeset
2201 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img
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diff changeset
2202 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub> an, bildet hierzu die beiden durch zyklische Vertauschung <br/>der Indizes <span
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2203 class="cmmi-12"><img
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2204 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi12-16.png" alt="m" class="cmmi-12x-x-16" align="middle" />, <img
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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diff changeset
2205 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi12-17.png" alt="n" class="12x-x-17" />, <img
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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diff changeset
2206 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi12-1b.png" alt="s" class="12x-x-1b" /> </span>entstehenden Gleichungen und addiert <br/>diese drei Gleichungen, so erhält man den Tensor dritten
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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2207 <br/>Ranges</p>
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2208 <table width="100%"
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2209 class="equation"><tr><td><a
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2210 id="x1-47r37"></a>
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2211 <center class="math-display" >
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2212 <img
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2213 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_1916116x.png" alt=" @-Am-n @-An-s @-Asm- Bm n s = Am ns + Ansm + Asm n = @ xs + @ xm + @ xn , " class="math-display" /></center></td><td width="5%">(37)</td></tr></table>
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2214 <!--l. 2413--><p class="nopar">
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2215 </p><!--l. 2418--><p class="noindent">von welchem leicht zu beweisen ist, daß er antisymmetrisch ist.
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2216 </p><!--l. 2420--><p class="indent"> <span
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2217 class="cmti-12">Divergenz des Sechservektors. </span>Multipliziert man (27) mit <br/><span
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2218 class="cmmi-12">g</span><sup ><span
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2219 class="cmmi-8"><img
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2220 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img
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2221 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-b.png" alt="a" class="8x-x-b" /></span></sup> <span
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2222 class="cmmi-12">g</span><sup ><span
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2223 class="cmmi-8"><img
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2224 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-17.png" alt="n" class="8x-x-17" /> <img
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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diff changeset
2225 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-c.png" alt="b" class="cmmi-8x-x-c" align="middle" /></span></sup> (gemischte Multiplikation), so erhält man ebenfalls <br/><pb/>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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2226 </p><!--l. 2424--><p class="indent">
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2227
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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2228 </p><!--l. 2425--><p class="noindent">einen Tensor. Das erste Glied der rechten Seite von (27) kann <br/>man in der Form
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2229 </p>
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2230 <center class="par-math-display" >
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2231 <img
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2232 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_1916117x.png" alt="-@--- m a nb ma @-gnb- nb @-gma- @ xs(g g Am n) - g @ xs Am n - g @ xa Am n " class="par-math-display" /></center>
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2233 <!--l. 2436--><p class="nopar">
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2234 </p><!--l. 2438--><p class="noindent">schreiben. Ersetzt man <span
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2235 class="cmmi-12">g</span><sup ><span
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2236 class="cmmi-8"><img
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2237 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img
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2238 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-b.png" alt="a" class="8x-x-b" /></span></sup> <span
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2239 class="cmmi-12">g</span><sup ><span
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2240 class="cmmi-8"><img
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2241 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-17.png" alt="n" class="8x-x-17" /> <img
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2242 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-c.png" alt="b" class="cmmi-8x-x-c" align="middle" /></span></sup> <span
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2243 class="cmmi-12">A</span><sub >
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2244 <span
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2245 class="cmmi-8"><img
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2246 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img
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2247 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-17.png" alt="n" class="8x-x-17" /> <img
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2248 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-1b.png" alt="s" class="8x-x-1b" /></span></sub> durch <span
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2249 class="cmmi-12">A</span><sub><span
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2250 class="cmmi-8"><img
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2251 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-1b.png" alt="s" class="8x-x-1b" /></span></sub><sup><span
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2252 class="cmmi-8"><img
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2253 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-b.png" alt="a" class="8x-x-b" /><img
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2254 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-c.png" alt="b" class="cmmi-8x-x-c" align="middle" /></span></sup><span
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2255 class="cmmi-12">,</span> g<sup ><span
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2257 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img
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2258 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-b.png" alt="a" class="8x-x-b" /></span></sup> <span
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2259 class="cmmi-12">g</span><sup ><span
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2260 class="cmmi-8"><img
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2261 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-17.png" alt="n" class="8x-x-17" /> <img
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2262 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-c.png" alt="b" class="cmmi-8x-x-c" align="middle" /></span></sup> <span
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2263 class="cmmi-12">A</span><sub >
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2264 <span
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2265 class="cmmi-8"><img
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2266 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img
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2267 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub> durch <br/><span
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2268 class="cmmi-12">A</span><sup ><span
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2269 class="cmmi-8"><img
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2270 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-b.png" alt="a" class="8x-x-b" /><img
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2271 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-c.png" alt="b" class="cmmi-8x-x-c" align="middle" /></span></sup> und ersetzt man in dem umgeformten ersten Gliede
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2272 </p>
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2273 <center class="par-math-display" >
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2274 <img
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2275 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_1916118x.png" alt="@-gnb- @-gma- @ xs und @ xs " class="par-math-display" /></center>
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2276 <!--l. 2447--><p class="nopar">
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2277 </p><!--l. 2451--><p class="noindent">vermittelst (34), so entsteht aus der rechten Seite von (27) <br/>ein siebengliedriger Ausdruck, von dem sich vier Glieder weg-<br/>heben. Es bleibt übrig</p>
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2278 <table width="100%"
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2279 class="equation"><tr><td><a
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2280 id="x1-48r38"></a>
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2281 <center class="math-display" >
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2282 <img
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2283 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_1916119x.png" alt=" @ Aa b { s x} { s x } Aasb = -------+ a Axb + b Aax . @ xs " class="math-display" /></center></td><td width="5%">(38)</td></tr></table>
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2284 <!--l. 2466--><p class="nopar">
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2285 </p><!--l. 2470--><p class="noindent">Es ist dies der Ausdruck für die Erweiterung eines kontra-<br/>varianten Tensors zweiten Ranges, der sich entsprechend auch <br/>für kontravariante Tensoren höheren und niedrigeren Ranges <br/>bilden läßt.
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2286 </p><!--l. 2475--><p class="indent"> Wir merken an, daß sich auf analogem Wege auch die <br/>Erweiterung eines gemischten Tensors <span
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2287 class="cmmi-12">A</span><sub><span
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2288 class="cmmi-8"><img
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2289 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /></span></sub><sup><span
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2290 class="cmmi-8"><img
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2291 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-b.png" alt="a" class="8x-x-b" /></span></sup> bilden läßt:
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2292 </p>
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2293 <table width="100%"
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2294 class="equation"><tr><td><a
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2295 id="x1-49r39"></a>
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2296
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2297 <center class="math-display" >
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2298 <img
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2299 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_1916120x.png" alt=" a { s m} { s t} Aa = @-Am-- t Aa + a At . m s @ xs t m " class="math-display" /></center></td><td width="5%">(39)</td></tr></table>
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2300 <!--l. 2490--><p class="nopar">
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2301 </p><!--l. 2494--><p class="noindent">Durch Verjüngung von (38) bezüglich der Indizes <span
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2302 class="cmmi-12"><img
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2303 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi12-c.png" alt="b" class="cmmi-12x-x-c" align="middle" /> </span>und <span
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2304 class="cmmi-12"><img
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2305 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi12-1b.png" alt="s" class="12x-x-1b" /> </span> <br/>(innere Multiplikation mit <span
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2306 class="cmmi-12"><img
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2307 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi12-e.png" alt="d" class="12x-x-e" /></span><sub><span
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2308 class="cmmi-8"><img
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2309 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-c.png" alt="b" class="cmmi-8x-x-c" align="middle" /></span></sub><sup><span
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2310 class="cmmi-8"><img
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2311 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-1b.png" alt="s" class="8x-x-1b" /></span></sup>) erhält man den kontravarianten <br/>Vierervektor
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2312 </p>
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2313 <center class="par-math-display" >
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2314 <img
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2315 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_1916121x.png" alt=" a b { b x } { b x} Aa = @-A----+ b Aa x + a Ax b. @ xb n " class="par-math-display" /></center>
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2316 <!--l. 2510--><p class="nopar">
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2317 </p><!--l. 2513--><p class="noindent">Wegen der Symmetrie von <img
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2318 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_1916122x.png" alt="{ } b x a" class="left" align="middle" /> bezüglich der Indizes <span
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2319 class="cmmi-12"><img
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2320 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi12-c.png" alt="b" class="cmmi-12x-x-c" align="middle" /> </span>und <span
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2321 class="cmmi-12">x</span><span
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2322 class="cmbxti-10x-x-120">x</span> <br/>verschwindet das dritte Glied der rechten Seite, falls <span
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2323 class="cmmi-12">A</span><sup ><span
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2324 class="cmmi-8"><img
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2325 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-b.png" alt="a" class="8x-x-b" /><img
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2326 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-c.png" alt="b" class="cmmi-8x-x-c" align="middle" /></span></sup> ein <br/>antisymmetrischer Tensor ist, was wir annehmen wollen; das <br/>zweite Glied läßt sich gemäß (29a) umformen. Man erhält also
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2327 </p>
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2328 <table width="100%"
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2329 class="equation"><tr><td><a
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2330 id="x1-50r40"></a>
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2331 <center class="math-display" >
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2332 <img
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2333 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_1916123x.png" alt=" 1 @ ( V~ --g Aa b) Aa = V~ -----------------. - g @ xb " class="math-display" /></center></td><td width="5%">(40)</td></tr></table>
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2334 <!--l. 2521--><p class="nopar">
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2335 </p><!--l. 2525--><p class="noindent">Dies ist der Ausdruck der Divergenz eines kontravarianten <br/>Sechservektors.
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2336 </p><!--l. 2528--><p class="indent"> <span
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2337 class="cmti-12">Divergenz des gemischten Tensors zweiten Ranges. </span>Bilden <br/>wir die Verjüngung von (39) bezüglich der Indizes <span
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2338 class="cmmi-12"><img
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2339 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi12-b.png" alt="a" class="12x-x-b" /> </span>und <span
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2340 class="cmmi-12"><img
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2341 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi12-1b.png" alt="s" class="12x-x-1b" /></span>, <br/>so erhalten wir mit Rücksicht auf (29a)
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2342 <pb/>
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2343 </p><!--l. 2535--><p class="indent">
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2344
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2345 </p>
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2346 <table width="100%"
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2347 class="equation"><tr><td><a
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2348 id="x1-51r41"></a>
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2349 <center class="math-display" >
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2350 <img
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2351 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_1916124x.png" alt=" ( V~ --- s) { s m } V~ -g-A = @------g-A-m-- t V~ --g As . m @ xs t " class="math-display" /></center></td><td width="5%">(41)</td></tr></table>
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2352 <!--l. 2545--><p class="nopar">
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2353 </p><!--l. 2549--><p class="noindent">Führt man im letzten Gliede den kontravarianten Tensor <br/><span
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2354 class="cmmi-12">A</span><sup ><span
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2355 class="cmmi-8"><img
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2356 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-25.png" alt="r" class="cmmi-8x-x-25" align="middle" /><img
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2357 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-1b.png" alt="s" class="8x-x-1b" /></span></sup> = <span
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2358 class="cmmi-12">g</span><sup ><span
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2359 class="cmmi-8"><img
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2360 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-25.png" alt="r" class="cmmi-8x-x-25" align="middle" /><img
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2361 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-1c.png" alt="t" class="8x-x-1c" /></span></sup> <span
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2362 class="cmmi-12">A</span><sub >
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2363 <span
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2364 class="cmmi-8"><img
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2365 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-1c.png" alt="t" class="8x-x-1c" /></span></sub><sup ><span
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2366 class="cmmi-8"><img
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2367 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-1b.png" alt="s" class="8x-x-1b" /></span></sup> ein, so nimmt es die Form an
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2368 </p>
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2369 <center class="par-math-display" >
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2370 <img
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2371 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_1916125x.png" alt=" |_ _|
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2372 s m V~ --- rs - |_ r _| -g A . " class="par-math-display" /></center>
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2373 <!--l. 2559--><p class="nopar">
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2374 </p><!--l. 2561--><p class="noindent">Ist ferner der Tensor <span
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2375 class="cmmi-12">A</span><sup ><span
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2376 class="cmmi-8"><img
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2377 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-25.png" alt="r" class="cmmi-8x-x-25" align="middle" /><img
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2378 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-1b.png" alt="s" class="8x-x-1b" /></span></sup> ein symmetrischer, so reduziert sich <br/>dies auf
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2379 </p>
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2380 <center class="par-math-display" >
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2381 <img
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2382 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_1916126x.png" alt=" 1 V~ --- @-grs- r s - 2 - g @ xm A . " class="par-math-display" /></center>
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2383 <!--l. 2569--><p class="nopar">
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2384 </p><!--l. 2571--><p class="noindent">Hätte man statt <span
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2385 class="cmmi-12">A</span><sup ><span
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2386 class="cmmi-8"><img
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2387 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-25.png" alt="r" class="cmmi-8x-x-25" align="middle" /><img
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2388 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-1b.png" alt="s" class="8x-x-1b" /></span></sup> den ebenfalls symmetrischen kovarianten <br/>Tensor
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2389 <span
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2390 class="cmmi-12">A</span><sub ><span
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2391 class="cmmi-8"><img
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2392 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-25.png" alt="r" class="cmmi-8x-x-25" align="middle" /><img
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2393 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-1b.png" alt="s" class="8x-x-1b" /></span></sub> = <span
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2394 class="cmmi-12">g</span><sub ><span
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2397 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-b.png" alt="a" class="8x-x-b" /></span></sub> <span
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2398 class="cmmi-12">g</span><sub ><span
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2399 class="cmmi-8"><img
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2400 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-1b.png" alt="s" class="8x-x-1b" /><img
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2401 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-c.png" alt="b" class="cmmi-8x-x-c" align="middle" /></span></sub> <span
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2402 class="cmmi-12">A</span><sup ><span
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2403 class="cmmi-8"><img
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2404 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-b.png" alt="a" class="8x-x-b" /><img
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2405 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-c.png" alt="b" class="cmmi-8x-x-c" align="middle" /></span></sup> eingeführt, so würde das letzte Glied <br/>vermöge (31) die Form</p>
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2406 <center class="par-math-display" >
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2407 <img
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2408 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_1916127x.png" alt=" V~ --- @ grs 12 - g ------Ars @ xm " class="par-math-display" /></center>
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2409 <!--l. 2582--><p class="nopar">
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2410
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2411 </p><!--l. 2586--><p class="noindent">annehmen. In dem betrachteten Symmetriefalle kann also <br/>(41) auch durch die beiden Formen</p>
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2412 <table width="100%"
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2413 class="equation"><tr><td><a
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2414 id="x1-52r42"></a>
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2415 <center class="math-display" >
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2416 <img
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2417 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_1916128x.png" alt=" ( V~ --- ) V~ --- @ -g Asm 1 @ gr s V~ --- rs - g Am = ------------- - -------- - g A @ xs 2 @ xm " class="math-display" /></center></td><td width="5%">(41a)</td></tr></table>
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2418 <!--l. 2596--><p class="nopar">
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2419 </p><!--l. 2601--><p class="noindent">und</p>
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2420 <table width="100%"
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2421 class="equation"><tr><td><a
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2422 id="x1-53r42"></a>
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2423 <center class="math-display" >
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2424 <img
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2425 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_1916129x.png" alt=" ( V~ --- s) r s V~ --g Am = @------g A-m-+ 1-@-g--- V~ --g Asr @ xs 2 @ xm " class="math-display" /></center></td><td width="5%">(41b)</td></tr></table>
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2426 <!--l. 2610--><p class="nopar">
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2427 </p><!--l. 2613--><p class="noindent">ersetzt werden, von denen wir im folgenden Gebrauch zu <br/>machen haben.
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2428 </p>
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2429 <div class="center" >
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2430
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2431 <!--l. 2616--><p class="noindent">
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2432 </p><!--l. 2617--><p class="noindent"><span
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2433 class="cmbsy-10x-x-120">§ </span><span
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2434 class="cmbx-12">12. Der Riemann-Christoffelsche Tensor.</span></p></div>
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2435
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2436 <!--l. 2621--><p class="indent"> Wir fragen nun nach denjenigen Tensoren, welche aus <br/>dem Fundamentaltensor der <span
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2437 class="cmmi-12">g</span><sub ><span
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2438 class="cmmi-8"><img
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2439 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img
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2440 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub> <span
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2441 class="cmti-12">allein </span>durch Differentiation <br/>gewonnen werden können. Die Antwort scheint zunächst auf <br/>der Hand zu liegen. Man setzt in (27) statt des beliebig ge-<br/>gebenen Tensors <span
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2442 class="cmmi-12">A</span><sub ><span
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2443 class="cmmi-8"><img
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2444 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img
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2445 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub> den Fundamentaltensor der <span
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2446 class="cmmi-12">g</span><sub ><span
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2447 class="cmmi-8"><img
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2448 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img
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2449 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub> ein und <br/>erhält dadurch einen neuen Tensor, nämlich die Erweiterung <br/>des Fundamentaltensors. Man überzeugt sich jedoch leicht, <br/>daß diese letztere identisch verschwindet. Man gelangt jedoch <br/>auf folgendem Wege zum Ziel. Man setze in (27)
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2450 </p>
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2451 <center class="math-display" >
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2452 <img
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2453 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_1916130x.png" alt=" m n @-Am- { } Am n = @ x - r Ar , n " class="math-display" /></center>
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2454 <!--l. 2638--><p class="nopar"> <pb/>
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2455 </p><!--l. 2645--><p class="indent">
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2456
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2457 </p><!--l. 2646--><p class="noindent">d. h. die Erweiterung des Vierervektors <span
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2458 class="cmmi-12">A</span><sub ><span
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2459 class="cmmi-8"><img
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2460 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub> ein. Dann erhält <br/>man (bei etwas geänderter Benennung der Indizes) den Tensor <br/>dritten Ranges</p>
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2461 <center class="par-math-display" >
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2462 <img
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2463 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_1916131x.png" alt=" @2 A Am st = ------m--- @ xs @ xt { m s } { m t } { s t} @-Ar- @-Ar- @-Am- - r @ xt - r @ xs - r @ xr
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2464 |_ _|
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2465 @ { m s} { m t} { a s } { s t} { a m} + |_ ------ r + a r + a r _| Ar . @ xt " class="par-math-display" /></center>
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2466 <!--l. 2686--><p class="nopar">
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2467 </p><!--l. 2689--><p class="noindent">Dieser Ausdruck ladet zur Bildung des Tensors <span
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2468 class="cmmi-12">A</span><sub ><span
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2469 class="cmmi-8"><img
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2473 class="cmsy-10x-x-120">- </span><span
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2474 class="cmmi-12">A</span><sub ><span
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2479 class="cmmi-12">A</span><sub ><span
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2484 class="cmmi-12">A</span><sub ><span
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2488 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-1b.png" alt="s" class="8x-x-1b" /></span><span
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2489 class="cmmi-8"> </span></sub> weg: das erste Glied, das vierte <br/>Glied, sowie das dem letzten Term in der eckigen Klammer <br/>entsprechende Glied; denn alle diese sind in <span
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2490 class="cmmi-12"><img
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2491 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi12-1b.png" alt="s" class="12x-x-1b" /> </span>und <span
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2492 class="cmmi-12"><img
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2493 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi12-1c.png" alt="t" class="12x-x-1c" /> </span>symme-<br/>trisch. Gleiches gilt von der Summe des zweiten und dritten <br/>Gliedes. Wir erhalten also</p>
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2494 <table width="100%"
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2495 class="equation"><tr><td><a
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2496 id="x1-54r42"></a>
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2497 <center class="math-display" >
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2498 <img
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2499 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_1916132x.png" alt=" r Am st- Amts = B mst Ar, " class="math-display" /></center></td><td width="5%">(42)</td></tr></table>
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2500 <!--l. 2705--><p class="nopar"></p>
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2501 <table width="100%"
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2502 class="equation"><tr><td><a
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2503 id="x1-55r43"></a>
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2504
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2505 <center class="math-display" >
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2506 <img
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2507 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_1916133x.png" alt=" m s m t r @ { } @ { } Bms t = - @-x-- r + @-x-- r t s {
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2508 { m s } { a t} { m t} { a s} - a r + a r . " class="math-display" /></center></td><td width="5%">(43)</td></tr></table>
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2509 <!--l. 2737--><p class="nopar">
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2510 </p><!--l. 2741--><p class="noindent">Wesentlich ist an diesem Resultat, daß auf der rechten Seite <br/>von (42) nur die <span
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2511 class="cmmi-12">A</span><sub ><span
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2512 class="cmmi-8"><img
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2513 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-25.png" alt="r" class="cmmi-8x-x-25" align="middle" /></span></sub>, aber nicht mehr ihre Ableitungen auf-<br/>treten. Aus dem Tensorcharakter von
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2514 <span
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2515 class="cmmi-12">A</span><sub ><span
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2516 class="cmmi-8"><img
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2517 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img
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2518 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-1b.png" alt="s" class="8x-x-1b" /> <img
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2519 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-1c.png" alt="t" class="8x-x-1c" /></span></sub> <span
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2520 class="cmsy-10x-x-120">- </span><span
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2521 class="cmmi-12">A</span><sub ><span
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2522 class="cmmi-8"><img
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2523 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img
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2524 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-1c.png" alt="t" class="8x-x-1c" /> <img
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2525 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-1b.png" alt="s" class="8x-x-1b" /></span></sub> in Ver-<br/>bindung damit, daß <span
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2526 class="cmmi-12">A</span><sub ><span
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2527 class="cmmi-8"><img
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2528 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-25.png" alt="r" class="cmmi-8x-x-25" align="middle" /></span></sub> ein frei wählbarer Vierervektor ist, <br/>folgt, vermöge der Resultate des <span
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2529 class="cmsy-10x-x-120">§ </span>7, daß <span
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2530 class="cmmi-12">B</span><sub><span
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2531 class="cmmi-8"><img
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2532 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img
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2533 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-1b.png" alt="s" class="8x-x-1b" /> <img
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2534 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-1c.png" alt="t" class="8x-x-1c" /></span></sub><sup><span
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2535 class="cmmi-8"><img
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2536 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-25.png" alt="r" class="cmmi-8x-x-25" align="middle" /></span></sup> ein Tensor <br/>ist (Riemann-Christoffelscher Tensor).
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2537 </p><!--l. 2751--><p class="indent"> Die mathematische Bedeutung dieses Tensors liegt im <br/>folgenden. Wenn das Kontinuum so beschaffen ist, daß es <br/>ein Koordinatensystem gibt, bezüglich dessen die <span
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2538 class="cmmi-12">g</span><sub ><span
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2539 class="cmmi-8"><img
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2540 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img
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2541 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub> Kon-<br/>stanten sind, so verschwinden alle <span
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2542 class="cmmi-12">R</span><sub><span
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2543 class="cmmi-8"><img
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2544 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img
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2545 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-1b.png" alt="s" class="8x-x-1b" /> <img
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2546 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-1c.png" alt="t" class="8x-x-1c" /></span></sub><sup><span
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2547 class="cmmi-8"><img
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2548 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-25.png" alt="r" class="cmmi-8x-x-25" align="middle" /></span></sup>. Wählt man statt des <br/>ursprünglichen Koordinatensystems ein beliebiges neues, so <br/>werden die auf letzteres bezogenen <span
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2549 class="cmmi-12">g</span><sub ><span
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2550 class="cmmi-8"><img
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2551 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img
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2552 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub> nicht Konstanten sein. <br/>Der Tensorcharakter von <span
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2553 class="cmmi-12">R</span><sub><span
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2554 class="cmmi-8"><img
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2555 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img
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2556 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-1b.png" alt="s" class="8x-x-1b" /> <img
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2557 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-1c.png" alt="t" class="8x-x-1c" /></span></sub><sup><span
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2558 class="cmmi-8"><img
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2559 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-25.png" alt="r" class="cmmi-8x-x-25" align="middle" /></span></sup> bringt es aber mit sich, daß <br/>diese Komponenten auch in dem beliebig gewählten Bezugs-<br/>system sämtlich verschwinden. Das Verschwinden des Rie-<br/>mannschen Tensors ist also eine notwendige Bedingung dafür, <br/>daß durch geeignete Wahl des Bezugssystems die Konstanz
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2560 <br/><pb/>
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2561 </p><!--l. 2768--><p class="indent">
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2562
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2563 </p><!--l. 2769--><p class="noindent">der <span
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2564 class="cmmi-12">g</span><sub ><span
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2565 class="cmmi-8"><img
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2566 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img
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2567 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub> herbeigeführt werden kann.<sup ><span
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2568 class="cmr-8">1</span></sup>) In unserem Problem <br/>entspricht dies dem Falle, daß bei passender Wahl des Ko-<br/>ordinatensystems in endlichen Gebieten die spezielle Rela-<br/>tivitätstheorie gilt.
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2569 </p><!--l. 2775--><p class="indent"> Durch Verjüngung von (43) bezüglich der Indizes <span
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2570 class="cmmi-12"><img
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2571 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi12-1c.png" alt="t" class="12x-x-1c" /> </span>und <span
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2572 class="cmmi-12"><img
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2573 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi12-25.png" alt="r" class="cmmi-12x-x-25" align="middle" /> </span> <br/>erhält man den kovarianten Tensor zweiten Ranges</p>
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2574 <table width="100%"
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2575 class="equation"><tr><td><a
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2576 id="x1-56r44"></a>
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2577 <center class="math-display" >
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2578 <img
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2579 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_1916134x.png" alt=" Bm n = Rm n + Sm n m n m a n b @ { } { } { } Rm n = - @-x-- a + b a { a
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2580 V~ --- { m n} V~ --- Sm n = @-lg----g-- a @-lg-----g-. @ xm @ xn @ xa " class="math-display" /></center></td><td width="5%">(44)</td></tr></table>
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2581 <!--l. 2803--><p class="nopar">
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2582 </p><!--l. 2808--><p class="indent"> <span
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2583 class="cmti-12">Bemerkung </span><span
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2584 class="cmti-12">über die Koordinatenwahl. </span>Es ist schon in <span
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2585 class="cmsy-10x-x-120">§ </span>8 <br/>im Anschluß an Gleichung (18a) bemerkt worden, daß die <br/>Koordinatenwahl mit Vorteil so getroffen werden kann, daß <br/><img
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2586 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_1916135x.png" alt=" V~ --g" class="sqrt" /> = 1 wird. Ein Blick auf die in den beiden letzten Para-<br/>graphen erlangten Gleichungen zeigt, daß durch eine solche
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2587 <br/>Wahl die Bildungsgesetze der Tensoren eine bedeutende Ver-<br/>einfachung erfahren. Besonders gilt dies für den soeben ent-<br/>wickelten Tensor <span
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2588 class="cmmi-12">B</span><sub ><span
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2589 class="cmmi-8"><img
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2590 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img
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2591 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub>, welcher in der darzulegenden Theorie <br/>eine fundamentale Rolle spielt. Die ins Auge gefaßte Speziali-<br/>sierung der Koordinatenwahl bringt nämlich das Ver-<br/>schwinden von <span
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2592 class="cmmi-12">S</span><sub ><span
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2593 class="cmmi-8"><img
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2594 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img
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2595 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub> mit sich, so daß sich der Tensor <span
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2596 class="cmmi-12">B</span><sub ><span
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2597 class="cmmi-8"><img
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2598 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img
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2599 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub> auf <br/><span
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2600 class="cmmi-12">R</span><sub ><span
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2601 class="cmmi-8"><img
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2602 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img
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2603 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub>
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2604 reduziert.
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2605 </p><!--l. 2823--><p class="indent"> Ich will deshalb im folgenden alle Beziehungen in der <br/>vereinfachten Form angeben, welche die genannte Speziali-<br/>sierung der Koordinatenwahl mit sich bringt. Es ist dann <br/>ein Leichtes, auf die <span
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2606 class="cmti-12">allgemein </span>kovarianten Gleichungen zu-<br/>rückzugreifen, falls dies in einem speziellen Falle erwünscht
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2607 <br/>erscheint.
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2608 </p>
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2609 <div class="center" >
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2610
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2611
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2612 <!--l. 2830--><p class="noindent">
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2613 </p><!--l. 2831--><p class="noindent"><span
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2614 class="cmbx-12">C. Theorie des Gravitationsfeldes.</span></p></div>
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2615 <div class="center" >
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2616
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2617 <!--l. 2834--><p class="noindent">
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2618 </p><!--l. 2837--><p class="noindent"><span
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2619 class="cmbsy-10x-x-120">§ </span><span
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2620 class="cmbx-12">13. Bewegungsgleichung des materiellen Punktes </span> <br/><span
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2621 class="cmbx-12">im</span>
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2622 <span
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2623 class="cmbx-12">Gravitationsfeld. </span> <br/><span
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2624 class="cmbx-12">Ausdruck f</span><span
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2625 class="cmbx-12">ür die Feldkomponenten der</span>
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2626 <span
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2627 class="cmbx-12">Gravitation.</span></p></div>
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2628 <!--l. 2841--><p class="indent"> Ein frei beweglicher, äußeren Kräften nicht unterworfener <br/>Körper bewegt sich nach der speziellen Relativitätstheorie <br/>geradlinig und gleichförmig. Dies gilt auch nach der allgemeinen <br/>----------
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2629 </p><!--l. 2847--><p class="indent"> 1) Die Mathematiker haben bewiesen, daß diese Bedingung auch <br/>eine
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2630 <span
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2631 class="cmti-12">hinreichende </span>ist. <pb/>
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2632 </p><!--l. 2852--><p class="indent">
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2633
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2634 </p><!--l. 2853--><p class="noindent">Relativitätstheorie für einen Teil des vierdimensionalen Raumes, <br/>in welchem das Koordinatensystem <span
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2635 class="cmmi-12">K</span><sub ><span
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2636 class="cmr-8">0</span></sub> so wählbar und so <br/>gewählt ist, daß die <span
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2637 class="cmmi-12">g</span><sub ><span
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2638 class="cmmi-8"><img
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2639 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img
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2640 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub> die in (4) gegebenen speziellen kon-<br/>stanten Werte haben.
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2641 </p><!--l. 2859--><p class="indent"> Betrachten wir eben diese Bewegung von einem beliebig <br/>gewählten Koordinatensystem <span
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2642 class="cmmi-12">K</span><sub ><span
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2643 class="cmr-8">1</span></sub> aus, so bewegt er sich von <br/><span
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2644 class="cmmi-12">K</span><sub ><span
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2645 class="cmr-8">1</span></sub> aus, beurteilt nach den Überlegungen des <span
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2646 class="cmsy-10x-x-120">§ </span>2 in einem <br/>Gravitationsfelde. Das Bewegungsgesetz mit Bezug auf <span
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2647 class="cmmi-12">K</span><sub ><span
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2648 class="cmr-8">1</span></sub> <br/>ergibt sich leicht aus folgender Überlegung. Mit Bezug auf <br/><span
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2649 class="cmmi-12">K</span><sub ><span
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2650 class="cmr-8">0</span></sub> ist das Bewegungsgesetz eine vierdimensionale Gerade, <br/>also eine geodätische Linie. Da nun die geodätische Linie <br/>unabhängig vom Bezugssystem definiert ist, wird ihre Glei-<br/>chung auch die Bewegungsgleichung des materiellen Punktes <br/>in bezug auf
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2651 <span
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2652 class="cmmi-12">K</span><sub ><span
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2653 class="cmr-8">1</span></sub> sein. Setzen wir</p>
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2654 <table width="100%"
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2655 class="equation"><tr><td><a
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2656 id="x1-57r45"></a>
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2657 <center class="math-display" >
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2658 <img
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2659 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_1916136x.png" alt=" { m n } t t G mn = - , " class="math-display" /></center></td><td width="5%">(45)</td></tr></table>
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2660 <!--l. 2872--><p class="nopar">
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2661 </p><!--l. 2875--><p class="noindent">so lautet also die Gleichung der Punktbewegung inbezug auf <span
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2662 class="cmmi-12">K</span><sub ><span
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2663 class="cmr-8">1</span></sub></p>
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2664 <table width="100%"
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2665 class="equation"><tr><td><a
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2666 id="x1-58r46"></a>
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2667 <center class="math-display" >
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2668 <img
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2669 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_1916137x.png" alt="d2 xt d xn d xn ----2-= Gtmn----------. d s d s d s " class="math-display" /></center></td><td width="5%">(46)</td></tr></table>
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2670 <!--l. 2880--><p class="nopar">
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2671
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2672 </p><!--l. 2884--><p class="noindent">Wir machen nun die sehr naheliegende Annahme, daß dieses <br/>allgemein kovariante Gleichungssystem die Bewegung des <br/>Punktes im Gravitationsfeld auch in dem Falle bestimmt, <br/>daß kein Bezugssystem <span
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2673 class="cmmi-12">K</span><sub ><span
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2674 class="cmr-8">0</span></sub> existiert, bezüglich dessen in end-<br/>lichen Räumen die spezielle Relativitätstheorie gilt. Zu dieser <br/>Annahme sind wir um so berechtigter, als (46) nur <span
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2675 class="cmti-12">erste </span>Ab-<br/>leitungen der <span
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2676 class="cmmi-12">g</span><sub ><span
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2677 class="cmmi-8"><img
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2678 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img
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2679 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub> enthält, zwischen denen auch im Spezial-<br/>falle der Existenz von <span
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2680 class="cmmi-12">K</span><sub ><span
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2681 class="cmr-8">0</span></sub> keine Beziehungen bestehen.<sup ><span
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2682 class="cmr-8">1</span></sup>)
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2683 </p><!--l. 2895--><p class="indent"> Verschwinden die <img
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2684 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmr12-0.png" alt="G" class="12x-x-0" /><sub><span
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2685 class="cmmi-8"><img
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2686 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img
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2687 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub><sup><span
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2688 class="cmmi-8"><img
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2689 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-1c.png" alt="t" class="8x-x-1c" /></span></sup>, so bewegt sich der Punkt gerad-<br/>linig und gleichförmig; diese Größen bedingen also die Ab-<br/>weichung der Bewegung von der Gleichförmigkeit. Sie sind <br/>die Komponenten des Gravitationsfeldes.
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2690 </p>
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2691 <div class="center" >
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2692
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2693 <!--l. 2901--><p class="noindent">
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2694 </p><!--l. 2903--><p class="noindent"><span
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2695 class="cmbsy-10x-x-120">§ </span><span
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2696 class="cmbx-12">14. Die Feldgleichungen der Gravitation bei Abwesenheit </span> <br/><span
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2697 class="cmbx-12">von</span>
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2698 <span
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2699 class="cmbx-12">Materie.</span></p></div>
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2700 <!--l. 2907--><p class="indent"> Wir unterscheiden im folgenden zwischen ,,Gravitations-<br/>feld&#8220; und ,,Materie&#8220;, in dem Sinne, daß alles außer dem <br/>Gravitationsfeld als ,,Materie&#8220; bezeichnet wird, also nicht nur <br/>----------
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2701 </p><!--l. 2913--><p class="indent"> 1) Erst zwischen den zweiten (und ersten) Ableitungen bestehen <br/>gemäß <span
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2702 class="cmsy-10x-x-120">§ </span>12 die Beziehungen <span
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2703 class="cmmi-12">B</span><sub><span
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2704 class="cmmi-8"><img
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2705 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img
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2706 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-1b.png" alt="s" class="8x-x-1b" /> <img
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2707 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-1c.png" alt="t" class="8x-x-1c" /></span></sub><sup><span
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2708 class="cmmi-8"><img
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2709 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-25.png" alt="r" class="cmmi-8x-x-25" align="middle" /></span></sup> = 0. <pb/>
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2710 </p><!--l. 2919--><p class="indent">
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2711
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2712 </p><!--l. 2920--><p class="noindent">die ,,Materie&#8220; im üblichen Sinne, sondern auch das elektro-<br/>magnetische Feld.
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2713 </p><!--l. 2924--><p class="indent"> Unsere nächste Aufgabe ist es, die Feldgleichungen der <br/>Gravitation bei Abwesenheit von Materie aufzusuchen. Dabei <br/>verwenden wir wieder dieselbe Methode wie im vorigen Para-<br/>graphen bei der Aufstellung der Bewegungsgleichung des <br/>materiellen Punktes. Ein Spezialfall, in welchem die gesuchten <br/>Feldgleichungen jedenfalls erfüllt sein müssen, ist der der <br/>ursprünglichen Relativitätstheorie, in dem die <span
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2714 class="cmmi-12">g</span><sub ><span
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2715 class="cmmi-8"><img
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2716 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img
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2717 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub> gewisse <br/>konstante Werte haben. Dies sei der Fall in einem gewissen <br/>endlichen Gebiete in bezug auf ein bestimmtes Koordinaten-<br/>system <span
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2718 class="cmmi-12">K</span><sub ><span
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2719 class="cmr-8">0</span></sub>. In bezug auf dies System verschwinden sämtliche
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2720 <br/>Komponenten <span
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2721 class="cmmi-12">B</span><sub><span
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2722 class="cmmi-8"><img
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2723 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img
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2724 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-1b.png" alt="s" class="8x-x-1b" /> <img
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2725 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-1c.png" alt="t" class="8x-x-1c" /></span></sub><sup><span
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2726 class="cmmi-8"><img
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2727 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-25.png" alt="r" class="cmmi-8x-x-25" align="middle" /></span></sup> des Riemannschen Tensors [Gleichung (43)]. <br/>Diese verschwinden dann für das betrachtete Gebiet auch be-<br/>züglich jedes anderen Koordinatensystems.
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2728 </p><!--l. 2942--><p class="indent"> Die gesuchten Gleichungen des materiefreien Gravitations-<br/>feldes müssen also jedenfalls erfüllt sein, wenn alle <span
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2729 class="cmmi-12">B</span><sub><span
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2731 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img
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2732 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-1b.png" alt="s" class="8x-x-1b" /> <img
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2733 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-1c.png" alt="t" class="8x-x-1c" /></span></sub><sup><span
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2734 class="cmmi-8"><img
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2735 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-25.png" alt="r" class="cmmi-8x-x-25" align="middle" /></span></sup> ver-<br/>schwinden. Aber diese Bedingung ist jedenfalls eine zu weit-<br/>gehende. Denn es ist klar, daß z. B. das von einem Massen-<br/>punkte in seiner Umgebung erzeugte Gravitationsfeld sicher-<br/>lich durch keine Wahl des Koordinatensystems ,,wegtrans-<br/>formiert&#8220;, d. h. auf den Fall konstanter
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2736 <span
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2737 class="cmmi-12">g</span><sub ><span
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2738 class="cmmi-8"><img
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2739 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img
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2740 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub> transformiert <br/>werden kann.
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2741 </p><!--l. 2954--><p class="indent"> Deshalb liegt es nahe, für das materiefreie Gravitations-<br/>feld das Verschwinden des aus dem Tensor <span
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2742 class="cmmi-12">B</span><sub><span
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2747 class="cmmi-8"><img
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2748 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-25.png" alt="r" class="cmmi-8x-x-25" align="middle" /></span></sup> abgeleiteten <br/>symmetrischen Tensors <span
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2749 class="cmmi-12">B</span><sub >
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2750 <span
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2751 class="cmmi-8"><img
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2752 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img
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2753 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub> zu verlangen. Man erhält so <br/>10 Gleichungen für die 10 Größen <span
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2754 class="cmmi-12">g</span><sub ><span
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2756 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img
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2757 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub>, welche im speziellen <br/>erfüllt sind, wenn sämtliche <span
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2758 class="cmmi-12">B</span><sub><span
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2759 class="cmmi-8"><img
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2760 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img
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2761 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-1b.png" alt="s" class="8x-x-1b" /> <img
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2762 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-1c.png" alt="t" class="8x-x-1c" /></span></sub><sup><span
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2763 class="cmmi-8"><img
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2764 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-25.png" alt="r" class="cmmi-8x-x-25" align="middle" /></span></sup> verschwinden. Diese Glei-<br/>chungen lauten mit Rücksicht auf (44) bei der von uns ge-<br/>troffenen Wahl für das Koordinatensystem für das materie-<br/>freie Feld</p>
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2765 <table width="100%"
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2766 class="equation"><tr><td><a
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2767 id="x1-59r47"></a>
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2768 <center class="math-display" >
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2769 <img
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2770 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_1916138x.png" alt=" @ Ga { ---mn-+ Gamb Gbna = 0 @ xa V~ --- -g = 1. " class="math-display" /></center></td><td width="5%">(47)</td></tr></table>
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2771 <!--l. 2974--><p class="nopar">
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2772 </p><!--l. 2977--><p class="indent"> Es muß darauf hingewiesen werden, daß der Wahl dieser <br/>Gleichungen ein Minimum von Willkür anhaftet. Denn es <br/>gibt außer <span
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2773 class="cmmi-12">B</span><sub ><span
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2774 class="cmmi-8"><img
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2775 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img
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2776 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub> keinen Tensor zweiten
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2777 Ranges, der aus den <br/><pb/>
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2778 </p><!--l. 2983--><p class="indent">
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2779
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2780 </p><!--l. 2984--><p class="noindent"><span
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2781 class="cmmi-12">g</span><sub ><span
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2782 class="cmmi-8"><img
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2783 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img
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2784 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub> und deren Ableitungen gebildet ist, keine höheren als <br/>zweite Ableitungen enthält und in letzteren linear ist.<sup ><span
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2785 class="cmr-8">1</span></sup>)
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2786 </p><!--l. 2987--><p class="indent"> Daß diese aus der Forderung der allgemeinen Relativität <br/>auf rein mathematischem Wege fließenden Gleichungen in <br/>Verbindung mit den Bewegungsgleichungen (46) in erster Nähe-<br/>rung das Newtonsche Attraktionsgesetz, in zweiter Nähe-<br/>rung die Erklärung der von Leverrier entdeckten (nach
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2787 <br/>Anbringung der Störungskorrektionen übrigbleibenden) Perihel-<br/>bewegung des Merkur liefern, muß nach meiner Ansicht von <br/>der physikalischen Richtigkeit der Theorie überzeugen.
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2788 </p>
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2789 <div class="center" >
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2790
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2791 <!--l. 2998--><p class="noindent">
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2792 </p><!--l. 3000--><p class="noindent"><span
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2793 class="cmbsy-10x-x-120">§ </span><span
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2794 class="cmbx-12">15. Hamiltonsche Funktion f</span><span
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2795 class="cmbx-12">ür das Gravitationsfeld,</span>
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2796 <br/><span
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2797 class="cmbx-12">Impulsenergiesatz.</span></p></div>
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2798 <!--l. 3004--><p class="indent"> Um zu zeigen, daß die Feldgleichungen dem Impuls-<br/>energiesatz entsprechen, ist es am bequemsten, sie in folgender <br/>Hamilton scher Form zu schreiben:</p>
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2799 <table width="100%"
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2800 class="equation"><tr><td><a
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2801 id="x1-60r48"></a>
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2802 <center class="math-display" >
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2803 <img
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2804 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_1916139x.png" alt=" { integral }
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2805 d H d t = 0 { mn a b H = g-- Gmb G na V~ - g = 1 . " class="math-display" /></center></td><td width="5%">(47a)</td></tr></table>
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2806 <!--l. 3017--><p class="nopar">
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2807 </p><!--l. 3021--><p class="noindent">Dabei verschwinden die Variationen an den Grenzen des be-<br/>trachteten begrenzten vierdimensionalen Integrationsraumes.
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2808 </p><!--l. 3025--><p class="indent"> Es ist zunächst zu zeigen, daß die Form (47a) den Glei-<br/>chungen (47) äquivalent ist. Zu diesem Zweck betrachten <br/>wir <span
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2809 class="cmmi-12">H </span>als Funktion der <span
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2810 class="cmmi-12">g</span><sup ><span
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2811 class="cmmi-8"><img
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2812 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img
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2813 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sup> und der
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2814 </p>
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2815
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2816 <center class="par-math-display" >
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2817 <img
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2818 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_1916140x.png" alt=" ( ) mn @-gmn- gs = @ xs . " class="par-math-display" /></center>
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2819 <!--l. 3035--><p class="nopar">
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2820 </p><!--l. 3039--><p class="noindent">Dann ist zunächst
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2821 </p>
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2822 <center class="par-math-display" >
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2823 <img
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2824 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_1916141x.png" alt="dH = Gam b Gbna d gm n + 2gmn Gamb d Gbna a b m n a ( mn b ) = - Gm b G na d g + 2G mb d g Gn a . " class="par-math-display" /></center>
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2825 <!--l. 3054--><p class="nopar">
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2826 </p><!--l. 3058--><p class="noindent">Nun ist aber
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2827 </p>
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2828 <center class="par-math-display" >
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2829 <img
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2830 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_1916142x.png" alt=" ( ) [ ( ac )] d gm n Gbna = - 1 d gm n gb c @-gnc-+ @-g---- @-gan- . 2 @ xa @ xn @ xc " class="par-math-display" /></center>
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2831 <!--l. 3070--><p class="nopar">
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2832 </p><!--l. 3073--><p class="noindent">----------
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2833 </p><!--l. 3076--><p class="indent"> 1) Eigentlich läßt sich dies nur von dem Tensor <span
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2834 class="cmmi-12">B</span><sub ><span
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2835 class="cmmi-8"><img
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2836 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img
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2837 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub> + <span
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2838 class="cmmi-12"><img
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2839 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi12-15.png" alt="c" class="12x-x-15" /> g</span><sub ><span
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2840 class="cmmi-8"><img
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2841 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img
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2842 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub>(<span
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2843 class="cmmi-12">g</span><sup ><span
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2844 class="cmmi-8"><img
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2845 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-b.png" alt="a" class="8x-x-b" /><img
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2846 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-c.png" alt="b" class="cmmi-8x-x-c" align="middle" /></span></sup> <span
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2847 class="cmmi-12">B</span><sub >
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2848 <span
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2849 class="cmmi-8"><img
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2850 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-b.png" alt="a" class="8x-x-b" /><img
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2851 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-c.png" alt="b" class="cmmi-8x-x-c" align="middle" /></span></sub>)
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2852 <br/>behaupten, wobei <span
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2853 class="cmmi-12"><img
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2854 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi12-15.png" alt="c" class="12x-x-15" /> </span>eine Konstante ist. Setzt man jedoch diesen = 0, <br/>so kommt man wieder zu den Gleichungen <span
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2855 class="cmmi-12">B</span><sub ><span
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2856 class="cmmi-8"><img
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2857 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img
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2858 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub> = 0. <pb/>
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2859 </p><!--l. 3086--><p class="indent">
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2860
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2861 </p><!--l. 3087--><p class="noindent">Die aus den beiden letzten Termen der runden Klammer hervor-<br/>gehenden Terme sind von verschiedenem Vorzeichen und <br/>gehen auseinander (da die Benennung der Summationsindizes <br/>belanglos ist) durch Vertauschung der Indizes <span
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2862 class="cmmi-12"><img
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2863 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi12-16.png" alt="m" class="cmmi-12x-x-16" align="middle" /> </span>und <span
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2864 class="cmmi-12"><img
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2865 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi12-c.png" alt="b" class="cmmi-12x-x-c" align="middle" /> </span>hervor.
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2866 <br/>Sie heben einander im Ausdruck für <span
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2867 class="cmmi-12"><img
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2868 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi12-e.png" alt="d" class="12x-x-e" /></span><span
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2869 class="cmti-12">H </span>weg, weil sie mit <br/>der bezüglich der Indizes
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2870 <span
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2871 class="cmmi-12"><img
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2872 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi12-16.png" alt="m" class="cmmi-12x-x-16" align="middle" /> </span>und <span
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2873 class="cmmi-12"><img
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2874 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi12-c.png" alt="b" class="cmmi-12x-x-c" align="middle" /> </span>symmetrischen Größe <img
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2875 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmr12-0.png" alt="G" class="12x-x-0" /><sub><span
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2876 class="cmmi-8"><img
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2877 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img
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2878 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-c.png" alt="b" class="cmmi-8x-x-c" align="middle" /></span></sub><sup><span
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2879 class="cmmi-8"><img
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2880 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-b.png" alt="a" class="8x-x-b" /></span></sup> <br/>multipliziert werden. Es bleibt also nur das erste Glied der <br/>runden Klammer zu berücksichtigen, so daß man mit Rück-<br/>sicht auf (31) erhält
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2881 </p>
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2882 <center class="par-math-display" >
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2883 <img
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2884 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_1916143x.png" alt="dH = - Ga Gb d gmn - Ga dgm b . m b na m b a " class="par-math-display" /></center>
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2885 <!--l. 3107--><p class="nopar">
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2886 </p><!--l. 3109--><p class="noindent">Es ist also</p>
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2887 <table width="100%"
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2888 class="equation"><tr><td><a
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2889 id="x1-61r48"></a>
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2890 <center class="math-display" >
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2891 <img
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2892 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_1916144x.png" alt=" @ H --m-n-= - Gam b Gbn a { @ g @-H--- s @ gms n = G mn . " class="math-display" /></center></td><td width="5%">(48)</td></tr></table>
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2893 <!--l. 3120--><p class="nopar">
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2894 </p><!--l. 3124--><p class="noindent">Die Ausführung der Variation in (47a) ergibt zunächst das <br/>Gleichungssystem</p>
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2895 <table width="100%"
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2896 class="equation"><tr><td><a
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2897 id="x1-62r49"></a>
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2898 <center class="math-display" >
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2899 <img
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2900 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_1916145x.png" alt=" ( ) -@-- -@-H-- - -@-H-- = 0 , @xa @ gman @ gmn " class="math-display" /></center></td><td width="5%">(47b)</td></tr></table>
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2901 <!--l. 3133--><p class="nopar">
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2902 </p><!--l. 3138--><p class="noindent">welches wegen (48) mit (47) übereinstimmt, was zu beweisen <br/>war. -- Multipliziert man (47b) mit <span
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2903 class="cmmi-12">g</span><sub><span
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2904 class="cmmi-8"><img
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2905 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-b.png" alt="a" class="8x-x-b" /></span></sub><sup><span
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2906 class="cmmi-8"><img
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2907 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img
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2908 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sup>, so erhält man, weil
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2909 </p>
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2910 <center class="par-math-display" >
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2911 <img
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2912 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_1916146x.png" alt=" mn mn @-gs--= @-ga-- @ xa @ xs " class="par-math-display" /></center>
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2913 <!--l. 3147--><p class="nopar">
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2914 </p><!--l. 3149--><p class="noindent">und folglich
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2915 </p>
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2916 <center class="par-math-display" >
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2917 <img
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2918 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_1916147x.png" alt=" @ ( @ H ) @ ( @ H ) @ H @ gmn gms n ----- ---mn- = ----- gms n---m-n - ---m-n---a-- @ xa @ ga @ xa @ ga @ ga @ xs " class="par-math-display" /></center>
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2919 <!--l. 3161--><p class="nopar">
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2920 </p><!--l. 3163--><p class="noindent">die Gleichung
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2921 </p>
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2922 <center class="par-math-display" >
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2923 <img
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2924 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_1916148x.png" alt=" @ ( @ H ) @ H ----- gmsn ---mn- - -----= 0 @ xa @ ga @ xs " class="par-math-display" /></center>
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2925 <!--l. 3171--><p class="nopar">
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2926 </p><!--l. 3174--><p class="noindent">oder<sup ><span
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2927 class="cmr-8">1</span></sup>)
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2928 </p>
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2929 <table width="100%"
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2930 class="equation"><tr><td><a
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2931 id="x1-63r49"></a>
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2932 <center class="math-display" >
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2933 <img
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2934 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_1916149x.png" alt=" a @-ts-= 0 { @ xa @ H - 2 x tas = gmsn---m-n- das H, @ ga " class="math-display" /></center></td><td width="5%">(49)</td></tr></table>
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2935 <!--l. 3186--><p class="nopar">
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2936 </p><!--l. 3190--><p class="noindent">----------
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2937 </p><!--l. 3193--><p class="indent"> 1) Der Grund der Einführung des Faktors <span
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2938 class="cmsy-10x-x-120">-</span>2 <span
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2939 class="cmbxti-10x-x-120">x</span> wird später deut-<br/>lich werden.
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2940 <pb/>
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2941 </p><!--l. 3198--><p class="indent">
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2942
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2943 </p><!--l. 3199--><p class="noindent">oder, wegen (48), der zweiten Gleichung (47) und (34)</p>
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2944 <table width="100%"
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2945 class="equation"><tr><td><a
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2946 id="x1-64r50"></a>
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2947 <center class="math-display" >
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2948 <img
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2949 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_1916150x.png" alt="x tas = 1das gmn Gam b Gbna- gm n Gamb Gbns . 2 " class="math-display" /></center></td><td width="5%">(50)</td></tr></table>
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2950 <!--l. 3207--><p class="nopar">
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2951 </p><!--l. 3212--><p class="indent"> Es ist zu beachten, daß <span
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2952 class="cmmi-12">t</span><sub ><span
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2953 class="cmmi-8"><img
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2954 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-1b.png" alt="s" class="8x-x-1b" /></span></sub><sup ><span
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2955 class="cmmi-8"><img
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2956 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-b.png" alt="a" class="8x-x-b" /></span></sup> kein Tensor ist; dagegen gilt <br/>(49) für alle Koordinatensysteme, für welche <img
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2957 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_1916151x.png" alt=" V~ ---- -g" class="sqrt" /> = 1 ist. <br/>Diese Gleichung drückt den Erhaltungssatz des Impulses und <br/>der Energie für das Gravitationsfeld aus. In der Tat liefert <br/>die Integration dieser Gleichung über ein <span
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2958 class="cmti-12">dreidimensionales </span> <br/>Volumen <span
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2959 class="cmmi-12">V</span>
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2960 die vier Gleichungen</p>
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2961 <table width="100%"
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2962 class="equation"><tr><td><a
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2963 id="x1-65r51"></a>
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2964 <center class="math-display" >
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2965 <img
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2966 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_1916152x.png" alt=" { integral } integral
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2967 -d-- t 4d V = (t 1 a + t2 a + t 3 a )d S , dx4 s s 1 s 2 s 3 " class="math-display" /></center></td><td width="5%">(49a)</td></tr></table>
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2968 <!--l. 3227--><p class="nopar">
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2969 </p><!--l. 3232--><p class="noindent">wobei <span
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2970 class="cmmi-12"><img
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2971 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi12-b.png" alt="a" class="12x-x-b" /></span><sub ><span
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2972 class="cmr-8">1</span></sub><span
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2973 class="cmmi-12">, <img
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2974 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi12-b.png" alt="a" class="12x-x-b" /></span><sub ><span
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2975 class="cmr-8">2</span></sub><span
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2976 class="cmmi-12">, <img
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2977 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi12-b.png" alt="a" class="12x-x-b" /></span><sub ><span
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2978 class="cmr-8">3</span></sub> der Richtungskosinus der nach innen ge-<br/>richteten Normale eines Flächenelementes der Begrenzung <br/>von der Größe <span
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2979 class="cmmi-12">dS </span>(im Sinne der euklidischen Geometrie) be-<br/>deuten. Man erkennt hierin den Ausdruck der Erhaltungs-<br/>sätze in üblicher Fassung. Die Größen <span
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2980 class="cmmi-12">t</span><sub ><span
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2981 class="cmmi-8"><img
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2982 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-1b.png" alt="s" class="8x-x-1b" /></span></sub><sup ><span
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2983 class="cmmi-8"><img
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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diff changeset
2984 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-b.png" alt="a" class="8x-x-b" /></span></sup> bezeichnen wir als <br/>die ,,Energiekomponenten&#8220; des Gravitationsfeldes.
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2985 </p><!--l. 3238--><p class="indent"> Ich will nun die Gleichungen (47) noch in einer dritten <br/>Form angeben, die einer lebendigen Erfassung unseres Gegen-<br/>standes besonders dienlich ist. Durch Multiplikation der <br/>Feldgleichungen (47) mit <span
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2986 class="cmmi-12">g</span><sup ><span
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2987 class="cmmi-8"><img
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2988 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-17.png" alt="n" class="8x-x-17" /> <img
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2989 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-1b.png" alt="s" class="8x-x-1b" /></span></sup> ergeben sich diese in der ,,ge-<br/>mischten&#8220; Form. Beachtet man, daß
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2990 </p>
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2991
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2992 <center class="par-math-display" >
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2993 <img
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2994 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_1916153x.png" alt=" ns @-Gamn- -@---( ns a ) @ gn-s a g @ x = @ x g Gm n - @ x G mn , a a a " class="par-math-display" /></center>
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2995 <!--l. 3253--><p class="nopar">
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2996 </p><!--l. 3255--><p class="noindent">welche Größe wegen (34) gleich
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2997 </p>
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2998 <center class="par-math-display" >
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2999 <img
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3000 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_1916154x.png" alt=" @ ( ) ----- gn s Gamn - gnb Gsab Gamn - gsb GnbaGamn , @ xa " class="par-math-display" /></center>
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3001 <!--l. 3264--><p class="nopar">
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3002 </p><!--l. 3266--><p class="noindent">oder (nach geänderter Benennung der Summationsindizes) gleich</p>
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3003 <center class="par-math-display" >
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3004 <img
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3005 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_1916155x.png" alt="--@--(gs b Ga ) - gm n Gs Gb - gns Ga Gb . @ xa mb m b n m mb na " class="par-math-display" /></center>
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3006 <!--l. 3275--><p class="nopar">
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3007 </p><!--l. 3279--><p class="noindent">Das dritte Glied dieses Ausdrucks hebt sich weg gegen das <br/>aus dem zweiten Glied der Feldgleichungen (47) entstehende; <br/>an Stelle des zweiten Gliedes dieses Ausdruckes läßt sich nach <br/>Beziehung (50)</p>
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3008 <center class="par-math-display" >
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3009 <img
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3010 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_1916156x.png" alt="x (t s - 1d st) m 2 m " class="par-math-display" /></center>
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3011 <!--l. 3288--><p class="nopar">
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3012 </p><!--l. 3291--><p class="noindent">setzen (<span
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3013 class="cmmi-12">t </span>= <span
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3014 class="cmmi-12">t</span><sub ><span
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3015 class="cmmi-8"><img
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3016 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-1b.png" alt="s" class="8x-x-1b" /></span></sub><sup ><span
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3017 class="cmmi-8"><img
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3018 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-b.png" alt="a" class="8x-x-b" /></span></sup>). Man erhält also an Stelle der Gleichungen (47)
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3019 </p>
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3020 <table width="100%"
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3021 class="equation"><tr><td><a
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3022 id="x1-66r51"></a>
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3023 <center class="math-display" >
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3024 <img
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3025 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_1916157x.png" alt=" --@--(gsb Ga ) = - x (ts - 1d st) { @ xa mb m 2 m V~ ---g = 1. " class="math-display" /></center></td><td width="5%">(51)</td></tr></table>
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3026 <!--l. 3301--><p class="nopar">
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3027 <pb/>
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3028 </p><!--l. 3308--><p class="indent">
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3029
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3030 </p>
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3031 <div class="center" >
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3032
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3033 <!--l. 3309--><p class="noindent">
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3034 </p><!--l. 3311--><p class="noindent"><span
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3035 class="cmbsy-10x-x-120">§ </span><span
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3036 class="cmbx-12">16. Allgemeine Fassung der Feldgleichungen der Gravitation.</span></p></div>
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3037 <!--l. 3315--><p class="indent"> Die im vorigen Paragraphen aufgestellten Feldgleichungen <br/>für materiefreie Räume sind mit der Feldgleichung
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3038 </p>
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3039 <center class="par-math-display" >
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3040 <img
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3041 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_1916158x.png" alt="D f = 0 " class="par-math-display" /></center>
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3042 <!--l. 3323--><p class="nopar">
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3043 </p><!--l. 3326--><p class="noindent">der Newtonschen Theorie zu vergleichen. Wir haben die <br/>Gleichungen aufzusuchen, welche der Poissonschen Gleichung
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3044 </p>
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3045 <center class="math-display" >
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3046 <img
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3047 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_1916159x.png" alt="D f = 4 p xr " class="math-display" /></center>
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3048 <!--l. 3330--><p class="nopar">
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3049 </p><!--l. 3332--><p class="noindent">entspricht, wobei <span
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3050 class="cmmi-12"><img
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3051 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi12-25.png" alt="r" class="cmmi-12x-x-25" align="middle" /> </span>die Dichte der Materie bedeutet.
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3052 </p><!--l. 3335--><p class="indent"> Die spezielle Relativitätstheorie hat zu dem Ergebnis <br/>geführt, daß die träge Masse nichts anderes ist als Energie, <br/>welche ihren vollständigen mathematischen Ausdruck in einem <br/>symmetrischen Tensor zweiten Ranges, dem Energietensor, <br/>findet. Wir werden daher auch in der allgemeinen Relativitäts-<br/>theorie einen Energietensor der Materie <span
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3053 class="cmmi-12">T</span><sub ><span
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3055 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-1b.png" alt="s" class="8x-x-1b" /></span></sub><sup ><span
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3056 class="cmmi-8"><img
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3057 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-b.png" alt="a" class="8x-x-b" /></span></sup> einzuführen haben, <br/>der wie die Energiekomponenten <span
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3058 class="cmmi-12">t</span><sub ><span
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3060 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-1b.png" alt="s" class="8x-x-1b" /></span></sub><sup ><span
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3061 class="cmmi-8"><img
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3062 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-b.png" alt="a" class="8x-x-b" /></span></sup> [Gleichungen (49) und (50)] <br/>des Gravitationsfeldes gemischten Charakter haben wird, aber <br/>zu einem symmetrischen kovarianten Tensor gehören wird<sup ><span
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3063 class="cmr-8">1</span></sup>).
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3064 </p><!--l. 3348--><p class="indent"> Wie dieser Energietensor(entsprechend der Dichte <span
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3065 class="cmmi-12"><img
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3066 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi12-25.png" alt="r" class="cmmi-12x-x-25" align="middle" /> </span>in <br/>der Poissonschen Gleichung) in die Feldgleichungen der <br/>Gravitation einzuführen ist, lehrt das Gleichungssystem (51). <br/>Betrachtet man nämlich ein vollständiges System (z. B. das <br/>Sonnensystem), so wird die Gesamtmasse des Systems, also <br/>auch seine gesamte gravitierende Wirkung, von der Gesamt-<br/>energie des Systems, also von der ponderablen und Gravi-<br/>tationsenergie zusammen, abhängen. Dies wird sich dadurch <br/>ausdrücken lassen, daß man in (51) an Stelle der Energie-<br/>komponenten
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3067 <span
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3068 class="cmmi-12">t</span><sub ><span
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3070 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /></span></sub><sup ><span
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3071 class="cmmi-8"><img
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3072 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-1b.png" alt="s" class="8x-x-1b" /></span></sup> des Gravitationsfeldes allein die Summen <br/><span
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3073 class="cmmi-12">t</span><sub >
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3074 <span
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3079 class="cmmi-12">T</span><sub >
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3080 <span
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3081 class="cmmi-8"><img
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3082 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /></span></sub><sup ><span
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3083 class="cmmi-8"><img
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3084 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-1b.png" alt="s" class="8x-x-1b" /></span></sup> der Energiekomponenten von Materie und Gravi-<br/>tationsfeld einführt. Man erhält so statt (51) die Tensor-<br/>gleichung</p>
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3085 <table width="100%"
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3086 class="equation"><tr><td><a
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3087 id="x1-67r52"></a>
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3088
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3089 <center class="math-display" >
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3090 <img
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3091 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_1916160x.png" alt=" -@--(gs b Ga ) = - x [(t s + T s)- 1 d s (t + T )] { @ xa m b m m 2 m V~ ---- - g = 1, " class="math-display" /></center></td><td width="5%">(52)</td></tr></table>
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3092 <!--l. 3373--><p class="nopar">
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3093 </p><!--l. 3378--><p class="noindent">wobei <span
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3094 class="cmmi-12">T </span>= <span
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3095 class="cmmi-12">T</span><sub ><span
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3097 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /></span></sub><sup ><span
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3098 class="cmmi-8"><img
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3099 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /></span></sup> gesetzt ist (Lauescher Skalar). Dies sind die <br/>gesuchten allgemeinen Feldgleichungen der Gravitation in ge-<br/>----------
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3100 </p><!--l. 3383--><p class="indent"> 1) <span
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3101 class="cmmi-12">g</span><sub ><span
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3104 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-1c.png" alt="t" class="8x-x-1c" /></span></sub> <span
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3105 class="cmmi-12">T</span><sub ><span
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3112 class="cmmi-8"><img
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3113 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-1b.png" alt="s" class="8x-x-1b" /><img
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3114 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-1c.png" alt="t" class="8x-x-1c" /></span></sub> und <span
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3115 class="cmmi-12">g</span><sup ><span
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3116 class="cmmi-8"><img
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3117 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-1b.png" alt="s" class="8x-x-1b" /> <img
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3118 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-c.png" alt="b" class="cmmi-8x-x-c" align="middle" /></span></sup> <span
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3119 class="cmmi-12">T</span><sub >
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3120 <span
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3121 class="cmmi-8"><img
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3122 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-1b.png" alt="s" class="8x-x-1b" /></span></sub><sup ><span
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3123 class="cmmi-8"><img
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3124 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-b.png" alt="a" class="8x-x-b" /></span></sup> = <span
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3125 class="cmmi-12">T</span><sup ><span
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3126 class="cmmi-8"><img
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3127 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-b.png" alt="a" class="8x-x-b" /><img
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3128 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-c.png" alt="b" class="cmmi-8x-x-c" align="middle" /></span></sup> sollen symmetrische Tensoren <br/>sein.
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3129 <pb/>
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3130 </p><!--l. 3390--><p class="indent">
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3131
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3132 </p><!--l. 3391--><p class="noindent">mischter Form. An Stelle von (47) ergibt sich daraus rück-<br/>wärts das System</p>
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3133 <table width="100%"
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3134 class="equation"><tr><td><a
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3135 id="x1-68r53"></a>
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3136 <center class="math-display" >
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3137 <img
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3138 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_1916161x.png" alt=" @ Gamn a b 1 { -@-x-- + Gn b G na = - x(Tm n - 2 gmn T), a V~ ----- - g = 1. " class="math-display" /></center></td><td width="5%">(53)</td></tr></table>
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3139 <!--l. 3403--><p class="nopar">
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3140 </p><!--l. 3408--><p class="indent"> Es muß zugegeben werden, daß diese Einführung des <br/>Energietensors der Materie durch das Relativitätspostulat <br/>allein nicht gerechtfertigt wird; deshalb haben wir sie im <br/>vorigen aus der Forderung abgeleitet, daß die Energie des
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3141 <br/>Gravitationsfeldes in gleicher Weise gravitierend wirken soll, <br/>wie jegliche Energie anderer Art. Der stärkste Grund für <br/>die Wahl der vorstehenden Gleichungen liegt aber darin, daß <br/>sie zur Folge haben, daß für die Komponenten der Total-<br/>energie Erhaltungsgleichungen (des Impulses und der Energie) <br/>gelten, welche den Gleichungen (49) und (49a) genau ent-<br/>sprechen. Dies soll im folgenden dargetan werden.
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3142 </p>
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3143 <div class="center" >
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3144
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3145 <!--l. 3422--><p class="noindent">
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3146 </p><!--l. 3424--><p class="noindent"><span
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3147 class="cmbsy-10x-x-120">§ </span><span
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3148 class="cmbx-12">17. Die Erhaltungss</span><span
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3149 class="cmbx-12">ätze im allgemeinen Falle.</span></p></div>
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3150 <!--l. 3428--><p class="indent"> Die Gleichung (52) ist leicht so umzuformen, daß auf <br/>der rechten Seite das zweite Glied wegfällt. Man verjünge (52) <br/>nach den Indizes <span
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3151 class="cmmi-12"><img
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3152 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi12-16.png" alt="m" class="cmmi-12x-x-16" align="middle" /> </span>und <span
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3153 class="cmmi-12"><img
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3154 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi12-1b.png" alt="s" class="12x-x-1b" /> </span>und subtrahiere die so erhaltene, <br/>mit <span
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3155 class="cmr-8">1</span>
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3156 <span
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3157 class="cmr-8">2</span> <span
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3158 class="cmmi-12"><img
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3159 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi12-e.png" alt="d" class="12x-x-e" /></span><sub ><span
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3160 class="cmmi-8"><img
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3161 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /></span></sub><sup ><span
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3162 class="cmmi-8"><img
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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3163 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-1b.png" alt="s" class="8x-x-1b" /></span></sup> multiplizierte Gleichung von (52). Es ergibt sich</p>
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3164 <table width="100%"
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3165 class="equation"><tr><td><a
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3166 id="x1-69r54"></a>
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3167 <center class="math-display" >
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3168
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3169 <img
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3170 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_1916162x.png" alt="--@-- sb a 1 s cb a s s @ xa (g Gm b - 2 @m g Gcb) = -x (tm + Tm ) . " class="math-display" /></center></td><td width="5%">(52a)</td></tr></table>
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3171 <!--l. 3441--><p class="nopar">
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3172 </p><!--l. 3445--><p class="noindent">An dieser Gleichung bilden wir die Operation <span
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3173 class="cmmi-12"><img
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3174 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi12-40.png" alt="@" class="12x-x-40" /></span><img
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3175 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_1916163x.png" alt="/" class="left" align="middle" /> <span
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3176 class="cmmi-12"><img
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3177 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi12-40.png" alt="@" class="12x-x-40" />x</span><sub ><span
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3178 class="cmmi-8"><img
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3179 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-1b.png" alt="s" class="8x-x-1b" /></span></sub>. Es ist <br/></p>
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3180 <center class="par-math-display" >
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3181 <img
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3182 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_1916164x.png" alt=" @2 ----------(gsbGamb) @ xa @ xs [ ( )] 1-----@2---- s b ac @-gmc- @-gbc- @-g-mb = - 2 @ xa @ xs g g @ xb + @ xm - @ xc . " class="par-math-display" /></center>
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3183 <!--l. 3461--><p class="nopar">
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3184 </p><!--l. 3465--><p class="noindent">Das erste und das dritte Glied der runden Klammer liefern <br/>Beiträge, die einander wegheben, wie man erkennt, wenn <br/>man im Beitrage des dritten Gliedes die Summationsindizes <br/><span
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3185 class="cmmi-12"><img
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3186 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi12-b.png" alt="a" class="12x-x-b" /> </span>und <span
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3187 class="cmmi-12"><img
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3188 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi12-1b.png" alt="s" class="12x-x-1b" /> </span>einerseits, <span
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3189 class="cmmi-12"><img
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3190 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi12-c.png" alt="b" class="cmmi-12x-x-c" align="middle" /> </span>und <span
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3191 class="cmmi-12"><img
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3192 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi12-15.png" alt="c" class="12x-x-15" /> </span>andererseits vertauscht. Das <br/>zweite Glied läßt sich nach (31) umformen, so daß man erhält</p>
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3193 <table width="100%"
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3194 class="equation"><tr><td><a
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3195 id="x1-70r54"></a>
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3196 <center class="math-display" >
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3197 <img
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3198 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_1916165x.png" alt=" 2 3 ab ----@-----(gsb Ga ) = 1-----@--g-------. @ xa @ xs m b 2 @ xa @ xb @ xm " class="math-display" /></center></td><td width="5%">(54)</td></tr></table>
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3199 <!--l. 3478--><p class="nopar">
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3200 </p><!--l. 3482--><p class="noindent">Das zweite Glied der linken Seite von (52a) liefert zunächst
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3201 </p>
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3202 <center class="par-math-display" >
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3203 <img
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3204 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_1916166x.png" alt=" 2 - 1----@-----(gc b Ga ) 2 @ xa @ xm cb " class="par-math-display" /></center>
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3205 <!--l. 3489--><p class="nopar"> <pb/>
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3206 </p><!--l. 3496--><p class="indent">
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3207
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3208 </p><!--l. 3497--><p class="noindent">oder
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3209 </p>
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3210 <center class="par-math-display" >
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3211 <img
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3212 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_1916167x.png" alt=" 2 [ ( )] 1- ---@----- gc bgad @ g-dc-+ @-g-db - @-g-cb . 4 @ xa@ xm @ xb @ xc @ xd " class="par-math-display" /></center>
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3213 <!--l. 3506--><p class="nopar">
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3214 </p><!--l. 3510--><p class="noindent">Das vom letzten Glied der runden Klammer herrührende <br/>Glied verschwindet wegen (29) bei der von uns getroffenen <br/>Koordinatenwahl. Die beiden anderen lassen sich zusammen-<br/>fassen und liefern wegen (31) zusammen
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3215 </p>
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3216 <center class="par-math-display" >
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3217 <img
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3218 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_1916168x.png" alt=" 1-----@3ga-b---- - 2 @ x @ x @ x , a b m " class="par-math-display" /></center>
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3219 <!--l. 3520--><p class="nopar">
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3220 </p><!--l. 3525--><p class="noindent">so daß mit Rücksicht auf (54) die Identität</p>
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3221 <table width="100%"
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3222 class="equation"><tr><td><a
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3223 id="x1-71r55"></a>
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3224 <center class="math-display" >
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3225 <img
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3226 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_1916169x.png" alt="----@2--- s b a 1 s cb a @ xa @ xs(g G mb - 2 dm g Gcb) =_ 0 " class="math-display" /></center></td><td width="5%">(55)</td></tr></table>
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3227 <!--l. 3534--><p class="nopar">
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3228 </p><!--l. 3539--><p class="noindent">besteht. Aus (55) und (52a) folgt</p>
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3229 <table width="100%"
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3230 class="equation"><tr><td><a
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3231 id="x1-72r56"></a>
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3232 <center class="math-display" >
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3233 <img
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3234 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_1916170x.png" alt=" s s @-(tm--+--Tm-)- = 0 . @ xs " class="math-display" /></center></td><td width="5%">(56)</td></tr></table>
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3235 <!--l. 3546--><p class="nopar">
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3236 </p><!--l. 3551--><p class="indent"> Aus unseren Feldgleichungen der Gravitation geht also <br/>hervor, daß den Erhaltungssätzen des Impulses und der Energie <br/>Genüge geleistet ist. Man sieht dies am einfachsten nach <br/>der Betrachtung ein, die zu Gleichung (49a) führt; nur hat <br/>man hier an Stelle der Energiekomponenten <span
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3237 class="cmmi-12">t</span><sub ><span
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3238 class="cmmi-8"><img
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3239 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /></span></sub><sup ><span
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3240 class="cmmi-8"><img
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3241 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-1b.png" alt="s" class="8x-x-1b" /></span></sup> des Gravi-<br/>tationsfeldes die Gesamtenergiekomponenten von Materie und <br/>Gravitationsfeld einzuführen.
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3242 </p>
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3243 <div class="center" >
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3244
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3245 <!--l. 3560--><p class="noindent">
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3246 </p><!--l. 3562--><p class="noindent"><span
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3247 class="cmbsy-10x-x-120">§ </span><span
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3248 class="cmbx-12">18. Der Impulsenergiesatz f</span><span
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3249 class="cmbx-12">ür die Materie als Folge der</span>
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3250 <br/><span
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3251 class="cmbx-12">Feldgleichungen.</span></p></div>
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3252 <!--l. 3566--><p class="indent"> Multipliziert man (53) mit <span
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3253 class="cmmi-12"><img
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3254 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi12-40.png" alt="@" class="12x-x-40" />g</span><sup ><span
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3255 class="cmmi-8"><img
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3256 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img
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3257 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sup><img
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3258 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_1916171x.png" alt="/" class="left" align="middle" /> <span
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3259 class="cmmi-12"><img
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3260 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi12-40.png" alt="@" class="12x-x-40" />x</span><sub >
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3261 <span
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3262 class="cmmi-8"><img
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3263 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-1b.png" alt="s" class="8x-x-1b" /></span></sub>, so erhält man auf <br/>dem in <span
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3264 class="cmsy-10x-x-120">§ </span>15 eingeschlagenen Wege mit Rücksicht auf das <br/>Verschwinden von</p>
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3265 <center class="par-math-display" >
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3266 <img
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3267 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_1916172x.png" alt=" @ gmn g mn------ @ xs " class="par-math-display" /></center>
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3268 <!--l. 3578--><p class="nopar">
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3269 </p><!--l. 3582--><p class="noindent">die Gleichung</p>
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3270 <center class="par-math-display" >
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3271 <img
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3272 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_1916173x.png" alt="@ tsa 1 @ gmn ----- + --------Tm n = 0, @ xa 2 @ xs " class="par-math-display" /></center>
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3273 <!--l. 3590--><p class="nopar">
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3274 </p><!--l. 3593--><p class="noindent">oder mit Rücksicht auf (56)</p>
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3275 <table width="100%"
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3276 class="equation"><tr><td><a
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3277 id="x1-73r57"></a>
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3278 <center class="math-display" >
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3279 <img
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3280 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_1916174x.png" alt=" a mn @-Ts-- + 1-@-g---T = 0 . @ xa 2 @ xs mn " class="math-display" /></center></td><td width="5%">(57)</td></tr></table>
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3281 <!--l. 3601--><p class="nopar">
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3282 </p><!--l. 3606--><p class="indent"> Ein Vergleich mit (41 b) zeigt, daß diese Gleichung bei <br/>der getroffenen Wahl für das Koordinatensystem nichts anderes <br/><pb/>
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3283 </p><!--l. 3610--><p class="indent">
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3284
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3285 </p><!--l. 3611--><p class="noindent">aussagt als das Verschwinden der Divergenz des Tensors der <br/>Energiekomponenten der Materie. Physikalisch zeigt das Auf-<br/>treten des zweiten Gliedes der linken Seite, daß für die Materie <br/>allein Erhaltungssätze des Impulses und der Energie im eigent-<br/>lichen Sinne nicht, bzw. nur dann gelten, wenn die <span
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3286 class="cmmi-12">g</span><sup ><span
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3287 class="cmmi-8"><img
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3288 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img
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3289 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sup> kon-<br/>stant sind, d. h. wenn die Feldstärken der Gravitation ver-<br/>schwinden. Dies zweite Glied ist ein Ausdruck für Impuls <br/>bzw. Energie, welche pro Volumen und Zeiteinheit vom Gravi-<br/>tationsfelde auf die Materie übertragen werden. Dies tritt <br/>noch klarer hervor, wenn man statt (57) im Sinne von (41)
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3290 <br/>schreibt</p>
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3291 <table width="100%"
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3292 class="equation"><tr><td><a
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3293 id="x1-74r58"></a>
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3294 <center class="math-display" >
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3295 <img
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3296 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_1916175x.png" alt=" a @-Ts-- = - Ga T b . @ xa s b a " class="math-display" /></center></td><td width="5%">(57a)</td></tr></table>
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3297 <!--l. 3629--><p class="nopar">
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3298 </p><!--l. 3633--><p class="noindent">Die rechte Seite drückt die energetische Einwirkung des Gravi-<br/>tationsfeldes auf die Materie aus.
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3299 </p><!--l. 3636--><p class="indent"> Die Feldgleichungen der Gravitation enthalten also gleich-<br/>zeitig vier Bedingungen, welchen der materielle Vorgang zu <br/>genügen hat. Sie liefern die Gleichungen des materiellen Vor-<br/>ganges vollständig, wenn letzterer durch vier voneinander <br/>unabhängige Differentialgleichungen charakterisierbar ist.<sup ><span
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3300 class="cmr-8">1</span></sup>)
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3301 </p>
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3302 <div class="center" >
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3303
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3304 <!--l. 3643--><p class="noindent">
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3305 </p><!--l. 3644--><p class="noindent"><span
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3306 class="cmbx-12">D. Die ,,materiellen&#8220; Vorg</span><span
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3307 class="cmbx-12">änge.</span></p></div>
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3308 <!--l. 3648--><p class="indent"> Die unter B entwickelten mathematischen Hilfsmittel <br/>setzen uns ohne weiteres in den Stand, die physikalischen <br/>Gesetze der Materie (Hydrodynamik, Maxwellsche Elektro-<br/>dynamik), wie sie in der speziellen Relativitätstheorie formu-<br/>liert vorliegen, so zu verallgemeinern, daß sie in die allgemeine
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3309 <br/>Relativitätstheorie hineinpassen. Dabei ergibt das allgemeine <br/>Relativitätsprinzip zwar keine weitere Einschränkung der <br/>Möglichkeiten; aber es lehrt den Einfluß des Gravitations-<br/>feldes auf alle Prozesse exakt kennen, ohne daß irgendwelche <br/>neue Hypothese eingeführt werden müßte.
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3310 </p><!--l. 3661--><p class="indent"> Diese Sachlage bringt es mit sich, daß über die physi-<br/>kalische Natur der Materie (im engeren Sinne) nicht notwendig <br/>bestimmte Voraussetzungen eingeführt werden müssen. Ins-<br/>besondere kann die Frage offen bleiben, ob die Theorie des <br/>elektromagnetischen Feldes und des Gravitationsfeldes zu-<br/>----------
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3311 </p><!--l. 3669--><p class="indent"> 1) Vgl. hierüber D. Hilbert, Nachr. d. K. Gesellsch. d. Wiss. zu <br/>Göttingen, Math.-phys. Klasse. p. 3. 1915. <pb/>
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3312 </p><!--l. 3675--><p class="indent">
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3313
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3314 </p><!--l. 3676--><p class="noindent">sammen eine hinreichende Basis für die Theorie der Materie <br/>liefern oder nicht. Das allgemeine Relativitätspostulat kann <br/>uns hierüber im Prinzip nichts lehren. Es muß sich bei dem <br/>Ausbau der Theorie zeigen, ob Elektromagnetik und Gravi-<br/>tationslehre zusammen leisten können, was ersterer allein <br/>nicht gelingen will.
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3315 </p>
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3316 <div class="center" >
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3317
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3318 <!--l. 3684--><p class="noindent">
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3319 </p><!--l. 3686--><p class="noindent"><span
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3320 class="cmbsy-10x-x-120">§ </span><span
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3321 class="cmbx-12">19. Eulersche Gleichungen f</span><span
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3322 class="cmbx-12">ür reibungslose adiabatische</span>
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3323 <br/><span
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3324 class="cmbx-12">Fl</span><span
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3325 class="cmbx-12">üssigkeiten.</span></p></div>
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3326 <!--l. 3690--><p class="indent"> Es seien <span
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3327 class="cmmi-12">p </span>und <span
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3328 class="cmmi-12"><img
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3329 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi12-25.png" alt="r" class="cmmi-12x-x-25" align="middle" /> </span>zwei Skalare, von denen wir ersteren <br/>als den ,,Druck&#8220;, letzteren als die ,,Dichte&#8220; einer Flüssigkeit <br/>bezeichnen; zwischen ihnen bestehe eine Gleichung. Der <br/>kontravariante symmetrische Tensor</p>
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3330 <table width="100%"
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3331 class="equation"><tr><td><a
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3332 id="x1-75r58"></a>
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3333 <center class="math-display" >
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3334 <img
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3335 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_1916176x.png" alt=" ab ab d xa d xb T = - g p + r----- ----- d s d s " class="math-display" /></center></td><td width="5%">(58)</td></tr></table>
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3336 <!--l. 3700--><p class="nopar">
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3337 </p><!--l. 3705--><p class="noindent">sei der kontravariante Energietensor der Flüssigkeit. Zu ihm <br/>gehört der kovariante Tensor</p>
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3338 <table width="100%"
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3339 class="equation"><tr><td><a
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3340 id="x1-76r59"></a>
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3341 <center class="math-display" >
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3342 <img
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3343 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_1916177x.png" alt=" d xa- d-xb- Tm n = - gm np + gma d s gmb d s r , " class="math-display" /></center></td><td width="5%">(58a)</td></tr></table>
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3344
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3345 <!--l. 3715--><p class="nopar">
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3346 </p><!--l. 3720--><p class="noindent">sowie der gemischte Tensor<sup ><span
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3347 class="cmr-8">1</span></sup>)
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3348 </p>
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3349 <table width="100%"
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3350 class="equation"><tr><td><a
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3351 id="x1-77r59"></a>
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3352 <center class="math-display" >
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3353 <img
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3354 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_1916178x.png" alt=" a a d xb d xa Ts = -ds p + gsb ----------r . d s d s " class="math-display" /></center></td><td width="5%">(58b)</td></tr></table>
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3355 <!--l. 3728--><p class="nopar">
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3356 </p><!--l. 3732--><p class="noindent">Setzt man die rechte Seite von (58b) in (57a) ein, so erhält <br/>man die Eulerschen hydrodynamischen Gleichungen der all-<br/>gemeinen Relativitätstheorie. Diese lösen das Bewegungs-<br/>problem im Prinzip vollständig; denn die vier Gleichungen (57a) <br/>zusammen mit der gegebenen Gleichung zwischen <span
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3357 class="cmmi-12">p </span>und <span
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3358 class="cmmi-12"><img
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3359 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi12-25.png" alt="r" class="cmmi-12x-x-25" align="middle" /> </span>und <br/>der Gleichung</p>
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3360 <center class="par-math-display" >
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3361 <img
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3362 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_1916179x.png" alt=" d xa d xb gsb----- -----= 1 d s d s " class="par-math-display" /></center>
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3363 <!--l. 3745--><p class="nopar">
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3364 </p><!--l. 3750--><p class="noindent">genügen bei gegebenen <span
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3365 class="cmmi-12">g</span><sub ><span
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3366 class="cmmi-8"><img
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3367 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-b.png" alt="a" class="8x-x-b" /><img
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3368 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-c.png" alt="b" class="cmmi-8x-x-c" align="middle" /></span></sub> zur Bestimmung der 6 Unbekannten
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3369 </p>
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3370 <center class="par-math-display" >
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3371 <img
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3372 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_1916180x.png" alt=" d x1 d x2 d x3 d x4 p, r, ----, ---- , ----, ---- . d s d s d s d s " class="par-math-display" /></center>
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3373 <!--l. 3756--><p class="nopar">
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3374 </p><!--l. 3758--><p class="noindent">----------
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3375 </p><!--l. 3761--><p class="indent"> 1) Für einen mitbewegten Beobachter, der im unendlich Kleinen <br/>ein Bezugssystem im Sinne der speziellen Relativitätstheorie benutzt, <br/>ist die Energiedichte <span
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3376 class="cmmi-12">T</span><sub ><span
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3377 class="cmr-8">4</span></sub><sup ><span
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3378 class="cmr-8">4</span></sup> gleich <span
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3379 class="cmmi-12"><img
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3380 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi12-25.png" alt="r" class="cmmi-12x-x-25" align="middle" /> </span><span
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3381 class="cmsy-10x-x-120">- </span><span
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3382 class="cmmi-12">p</span>. Hierin liegt die Definition von <span
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3383 class="cmmi-12"><img
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3384 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi12-25.png" alt="r" class="cmmi-12x-x-25" align="middle" /></span>. <br/>Es ist also <span
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3385 class="cmmi-12"><img
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3386 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi12-25.png" alt="r" class="cmmi-12x-x-25" align="middle" /></span>; nicht konstant für eine inkompressible Flüssigkeit. <pb/>
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3387 </p><!--l. 3770--><p class="indent">
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3388
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3389 </p><!--l. 3771--><p class="noindent">Sind auch die <span
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3390 class="cmmi-12">g</span><sub ><span
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3391 class="cmmi-8"><img
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3392 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img
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3393 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub> unbekannt, so kommen hierzu noch die <br/>Gleichungen (53). Dies sind 11 Gleichungen zur Bestimmung <br/>der 10 Funktionen <span
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3394 class="cmmi-12">g</span><sub ><span
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3395 class="cmmi-8"><img
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3396 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img
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3397 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub>, so daß diese überbestimmt scheinen. <br/>Es ist indessen zu beachten, daß die Gleichungen (57a) in
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3398 <br/>den Gleichungen (53) bereits enthalten sind, so daß letztere <br/>nur mehr 7 unabhöngige Gleichungen repräsentieren. Diese <br/>Unbestimmtheit hat ihren guten Grund darin, daß die weit-<br/>gehende Freiheit in der Wahl der Koordinaten es mit sich <br/>bringt, daß das Problem mathematisch in solchem Grade
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3399 <br/>unbestimmt bleibt, daß drei der Raumfunktionen beliebig <br/>gewählt werden können.<sup ><span
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3400 class="cmr-8">1</span></sup>)
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3401 </p>
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3402 <div class="center" >
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3403
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3404 <!--l. 3785--><p class="noindent">
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3405 </p><!--l. 3787--><p class="noindent"><span
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3406 class="cmbsy-10x-x-120">§ </span><span
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3407 class="cmbx-12">20. Maxwellsche elektromagnetische Feldgleichungen </span> <br/><span
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3408 class="cmbx-12">f</span><span
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3409 class="cmbx-12">ür das</span>
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3410 <span
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3411 class="cmbx-12">Vakuum.</span></p></div>
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3412 <!--l. 3791--><p class="indent"> Es seien <span
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3413 class="cmmi-12"><img
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3414 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi12-27.png" alt="f" class="cmmi-12x-x-27" align="middle" /></span><sub ><span
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3415 class="cmmi-8"><img
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3416 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub> die Komponenten eines kovarianten Vierer-<br/>vektors, des Vierervektors des elektromagnetischen Potentials. <br/>Aus ihnen bilden wir gemäß (36) die Komponenten <span
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3417 class="cmmi-12">F</span><sub ><span
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3418 class="cmmi-8"><img
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3419 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-25.png" alt="r" class="cmmi-8x-x-25" align="middle" /><img
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3420 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-1b.png" alt="s" class="8x-x-1b" /></span></sub> des <br/>kovarianten Sechservektors des elektromagnetischen Feldes <br/>gemäß dem Gleichungssystem</p>
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3421 <table width="100%"
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3422 class="equation"><tr><td><a
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3423 id="x1-78r59"></a>
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3424 <center class="math-display" >
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3425 <img
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3426 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_1916181x.png" alt=" @ fr @ fs Fr s = ----- - -----. @ xs @ xr " class="math-display" /></center></td><td width="5%">(59)</td></tr></table>
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3427 <!--l. 3804--><p class="nopar">
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3428 </p><!--l. 3808--><p class="noindent">Aus (59) folgt, daß das Gleichungssystem</p>
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3429 <table width="100%"
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3430 class="equation"><tr><td><a
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3431 id="x1-79r60"></a>
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3432 <center class="math-display" >
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3433 <img
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diff changeset
3434 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_1916182x.png" alt="@-Frs-+ @-Fst- + @-Ftr- = 0 @ xt @ xr @ xr " class="math-display" /></center></td><td width="5%">(60)</td></tr></table>
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3435 <!--l. 3817--><p class="nopar">
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3436 </p><!--l. 3822--><p class="noindent">erfüllt ist, dessen linke Seite gemäß (37) ein antisymmetrischer <br/>Tensor dritten Ranges ist. Das System (60) enthält also im <br/>wesentlichen 4 Gleichungen, die ausgeschrieben wie folgt lauten:</p>
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3437 <table width="100%"
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3438 class="equation"><tr><td><a
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diff changeset
3439 id="x1-80r61"></a>
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3440 <center class="math-display" >
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3441 <img
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3442 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_1916183x.png" alt=" @ F23 @ F34 @ F42 -@ x--+ -@-x--+ @-x---= 0 4 2 3 @-F34-+ @-F41-+ @-F13-= 0 { @ x1 @ x3 @ x4 @ F41 @ F12 @ F24 ------+ ------+ ------= 0 @ x2 @ x4 @ x1 @-F12-+ @-F23-+ @-F31-= 0 . @ x3 @ x1 @ x2 " class="math-display" /></center></td><td width="5%">(60a)</td></tr></table>
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3443 <!--l. 3848--><p class="nopar">
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3444 </p><!--l. 3852--><p class="noindent">----------
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3445 </p><!--l. 3855--><p class="indent"> 1) Bei Verzicht auf die Koordinatenwahl gemäß <span
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3446 class="cmmi-12">g </span>= <span
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3447 class="cmsy-10x-x-120">- </span>1 blieben
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3448 <br/><span
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3449 class="cmti-12">vier </span>Raumfunktionen frei wählbar, entsprechend den vier willkürlichen
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3450 <br/>Funktionen, über die man bei der Koordinatenwahl frei verfügen kann.
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3451 <pb/>
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3452 </p><!--l. 3861--><p class="indent">
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3453
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3454 </p><!--l. 3862--><p class="indent"> Dieses Gleichungssystem entspricht dem zweiten Glei-<br/>chungssystem Maxwells. Man erkennt dies sofort, indem <br/>man setzt</p>
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3455 <table width="100%"
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3456 class="equation"><tr><td><a
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3457 id="x1-81r61"></a>
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3458 <center class="math-display" >
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3459 <img
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3460 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_1916184x.png" alt=" F = h F = e { 23 x 14 x F31 = hy F24 = ey F12 = hz F34 = ez . " class="math-display" /></center></td><td width="5%">(61)</td></tr></table>
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3461 <!--l. 3875--><p class="nopar">
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3462 </p><!--l. 3879--><p class="noindent">Dann kann man statt (60a) in üblicher Schreibweise der drei-<br/>dimensionalen Vektoranalyse setzen</p>
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3463 <table width="100%"
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3464 class="equation"><tr><td><a
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3465 id="x1-82r62"></a>
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3466 <center class="math-display" >
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3467 <img
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3468 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_1916185x.png" alt=" @ h { ----+ rot e = 0 @ t divh = 0 . " class="math-display" /></center></td><td width="5%">(60b)</td></tr></table>
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3469 <!--l. 3890--><p class="nopar">
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3470 </p><!--l. 3895--><p class="indent"> Das erste Maxwellsche System erhalten wir durch Ver-<br/>allgemeinerung der von Minkowski angegebenen Form. Wir <br/>führen den zu <span
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3471 class="cmmi-12">F</span><sub ><span
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3472 class="cmmi-8"><img
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3473 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-b.png" alt="a" class="8x-x-b" /><img
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3474 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-c.png" alt="b" class="cmmi-8x-x-c" align="middle" /></span></sub> gehörigen kontravarianten Sechservektor</p>
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3475 <table width="100%"
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3476 class="equation"><tr><td><a
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3477 id="x1-83r62"></a>
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3478
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3479 <center class="math-display" >
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3480 <img
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3481 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_1916186x.png" alt="F mn = gm a qn b Fa b " class="math-display" /></center></td><td width="5%">(62)</td></tr></table>
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3482 <!--l. 3905--><p class="nopar">
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3483 </p><!--l. 3910--><p class="noindent">ein sowie den kontravarianten Vierervektor <span
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3484 class="cmmi-12">J</span><sup ><span
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3485 class="cmmi-8"><img
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3486 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /></span></sup> der elektrischen <br/>Vakuumstromdichte; dann kann man das mit Rücksicht auf <br/>(40) gegenüber beliebigen Substitutionen von der Determinante 1 <br/>(gemäß der von uns getroffenen Koordinatenwahl) invariante <br/>Gleichungssystem ansetzen:</p>
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3487 <table width="100%"
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3488 class="equation"><tr><td><a
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3489 id="x1-84r63"></a>
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3490 <center class="math-display" >
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3491 <img
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3492 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_1916187x.png" alt="@-F-mn m @ x = J . n " class="math-display" /></center></td><td width="5%">(63)</td></tr></table>
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3493 <!--l. 3922--><p class="nopar">
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3494 </p><!--l. 3926--><p class="noindent">Setzt man nämlich</p>
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3495 <table width="100%"
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3496 class="equation"><tr><td><a
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3497 id="x1-85r64"></a>
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3498 <center class="math-display" >
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3499 <img
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3500 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_1916188x.png" alt=" F23 = hx' F 14 = - ex' { F31 = h ' F 24 = - e ' 12 y' 34 y' F = hz F = - ez ,
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3501 " class="math-display" /></center></td><td width="5%">(64)</td></tr></table>
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3502 <!--l. 3939--><p class="nopar">
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3503 </p><!--l. 3944--><p class="noindent">welche Größen im Spezialfall der speziellen Relativitätstheorie <br/>den Größen
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3504 <span
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3505 class="cmmi-12">h</span><sub ><span
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3506 class="cmmi-8">x</span></sub> <span
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3507 class="cmmi-12">. . . . e</span><sub ><span
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3508 class="cmmi-8">z</span></sub> gleich sind, und außerdem
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3509 </p>
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3510 <center class="par-math-display" >
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3511 <img
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3512 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_1916189x.png" alt=" 1 2 3 4 J = ix, J = iy, J = iz, J = r, " class="par-math-display" /></center>
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3513 <!--l. 3952--><p class="nopar">
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3514 </p><!--l. 3956--><p class="noindent">so erhält man an Stelle von (63)</p>
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3515 <table width="100%"
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3516 class="equation"><tr><td><a
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diff changeset
3517 id="x1-86r65"></a>
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diff changeset
3518 <center class="math-display" >
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diff changeset
3519 <img
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diff changeset
3520 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_1916190x.png" alt=" ' @-e' { rot h - @ t = i ' div e = r. " class="math-display" /></center></td><td width="5%">(63a)</td></tr></table>
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3521 <!--l. 3966--><p class="nopar">
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diff changeset
3522 </p><!--l. 3971--><p class="indent"> Die Gleichungen (60), (62) und (63) bilden also die <br/>Verallgemeinerung der Maxwellschen Feldgleichungen des <br/><pb/>
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3523 </p><!--l. 3976--><p class="indent">
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3524
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diff changeset
3525 </p><!--l. 3977--><p class="noindent">Vakuums bei der von uns bezüglich der Koordinatenwahl <br/>getroffenen Festsetzung.
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3526 </p><!--l. 3981--><p class="indent"> <span
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diff changeset
3527 class="cmti-12">Die Energiekomponenten des elektromagnetischen Feldes</span>. <br/>Wir bilden das innere Produkt</p>
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diff changeset
3528 <table width="100%"
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diff changeset
3529 class="equation"><tr><td><a
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diff changeset
3530 id="x1-87r65"></a>
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diff changeset
3531 <center class="math-display" >
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diff changeset
3532 <img
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diff changeset
3533 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_1916191x.png" alt="xs = Fsm Jm . " class="math-display" /></center></td><td width="5%">(65)</td></tr></table>
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3534 <!--l. 3987--><p class="nopar">
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3535 </p><!--l. 3991--><p class="noindent">Seine Komponenten lauten gemäß (61) in dreidimensionaler <br/>Schreibweise</p>
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3536 <table width="100%"
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3537 class="equation"><tr><td><a
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3538 id="x1-88r66"></a>
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3539 <center class="math-display" >
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3540 <img
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diff changeset
3541 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_1916192x.png" alt=" x1 = r ex + [i, h]x . . . . . . . {
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diff changeset
3542 . . . . . . . x = - (i, e) . 4 " class="math-display" /></center></td><td width="5%">(65a)</td></tr></table>
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3543 <!--l. 4003--><p class="nopar">
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3544 </p><!--l. 4008--><p class="indent"> Es ist <span
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3545 class="cmbxti-10x-x-120">x</span><sub ><span
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3546 class="cmmi-8"><img
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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diff changeset
3547 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-1b.png" alt="s" class="8x-x-1b" /></span></sub> ein kovarianter Vierervektor, dessen Kompo-<br/>nenten gleich sind dem negativen Impuls bzw. der Energie, <br/>welche pro Zeit- und Volumeinheit auf das elektromagnetische <br/>Feld von den elektrischen Massen übertragen werden. Sind <br/>die elektrischen Massen frei, d. h. unter dem alleinigen Ein-<br/>fluß des elektromagnetischen Feldes, so wird der kovariante <br/>Vierervektor <span
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3548 class="cmbxti-10x-x-120">x</span><sub ><span
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3549 class="cmmi-8"><img
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3550 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-1b.png" alt="s" class="8x-x-1b" /></span></sub>
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diff changeset
3551 verschwinden.
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3552 </p><!--l. 4014--><p class="indent"> Um die Energiekomponenten <span
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diff changeset
3553 class="cmmi-12">T</span><sub ><span
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3554 class="cmmi-8"><img
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diff changeset
3555 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-1b.png" alt="s" class="8x-x-1b" /></span></sub><sup ><span
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3556 class="cmmi-8"><img
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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diff changeset
3557 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sup> des elektromagnetischen <br/>Feldes zu erhalten, brauchen wir nur der Gleichung <span
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3558 class="cmbxti-10x-x-120">x</span><sub ><span
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3559 class="cmmi-8"><img
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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diff changeset
3560 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-1b.png" alt="s" class="8x-x-1b" /></span></sub> = 0 <br/>die Gestalt der Gleichung (57) zu geben. Aus (63) und (65) <br/>ergibt sich zunächst
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3561 </p>
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3562 <center class="par-math-display" >
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3563 <img
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3564 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_1916193x.png" alt=" @-F-mn -@--- mn mn@-Fsm- xs = Fsm @ x = @ x (Fsm F ) - F @ x . n n n " class="par-math-display" /></center>
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3565 <!--l. 4024--><p class="nopar">
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3566 </p><!--l. 4026--><p class="noindent">Das zweite Glied der rechten Seite gestattet vermöge (60) <br/>die Umformung
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3567 </p>
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3568 <center class="par-math-display" >
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3569 <img
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3570 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_1916194x.png" alt="Fm n @-Fsm = - 1-F mn @-Fmn- = - 1-gmagn b F @-Fm-n , @ xn 2 @ xs 2 ab @ xs " class="par-math-display" /></center>
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3571 <!--l. 4035--><p class="nopar">
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3572 </p><!--l. 4038--><p class="noindent">welch letzterer Ausdruck aus Symmetriegründen auch in der <br/>Form</p>
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3573 <center class="par-math-display" >
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3574 <img
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3575 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_1916195x.png" alt=" [ ] - 1- gmagn b F @-Fmn- + gmagnb @-Fa-b F 4 ab @ xs @ xs m n " class="par-math-display" /></center>
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3576 <!--l. 4048--><p class="nopar">
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3577 </p><!--l. 4053--><p class="noindent">geschrieben werden kann. Dafür aber läßt sich setzen
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3578 </p>
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3579 <center class="par-math-display" >
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3580 <img
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diff changeset
3581 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_1916196x.png" alt=" 1 @ 1 @ - -- ---(gm agnb Fa b Fm n) + --Fa b Fmn ---(gm agnb) . 4 @xs 4 @xs " class="par-math-display" /></center>
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3582 <!--l. 4062--><p class="nopar">
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3583 </p><!--l. 4066--><p class="noindent">Das erste dieser Glieder lautet in kürzerer Schreibweise
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3584 </p>
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3585 <center class="par-math-display" >
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3586 <img
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3587 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_1916197x.png" alt="- 1- -@--(F mn F ) , 4 @ xs mn " class="par-math-display" /></center>
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3588 <!--l. 4074--><p class="nopar"> <pb/>
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3589 </p><!--l. 4081--><p class="indent">
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3590
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3591 </p><!--l. 4082--><p class="noindent">das zweite ergibt nach Ausführung der Differentiation nach <br/>einiger Umformung</p>
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3592 <center class="par-math-display" >
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3593 <img
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3594 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_1916198x.png" alt=" 1 mt nr @ gs t - --F Fmn g ------. 2 @ xs " class="par-math-display" /></center>
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3595 <!--l. 4090--><p class="nopar">
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3596 </p><!--l. 4093--><p class="noindent">Nimmt man alle drei berechneten Glieder zusammen, so erhält <br/>man die Relation</p>
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3597 <table width="100%"
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3598 class="equation"><tr><td><a
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3599 id="x1-89r66"></a>
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3600 <center class="math-display" >
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3601 <img
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3602 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_1916199x.png" alt=" @-Tsn- 1- tm @-gmn- n xs = @ xn - 2 g @ xs Tt , " class="math-display" /></center></td><td width="5%">(66)</td></tr></table>
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3603 <!--l. 4101--><p class="nopar">
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3604 </p><!--l. 4106--><p class="noindent">wobei</p>
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3605 <table width="100%"
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3606 class="equation"><tr><td><a
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3607 id="x1-90r67"></a>
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3608 <center class="math-display" >
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3609 <img
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3610 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_1916200x.png" alt=" n na 1- n ab Ts = - Fsa F + 4 ds Fab F . " class="math-display" /></center></td><td width="5%">(66a)</td></tr></table>
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3611 <!--l. 4114--><p class="nopar">
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3612 </p><!--l. 4119--><p class="indent"> Die Gleichung (66) ist für verschwindendes <span
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3613 class="cmbxti-10x-x-120">x</span><sub ><span
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3614 class="cmmi-8"><img
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3615 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-1b.png" alt="s" class="8x-x-1b" /></span></sub> wegen (30) <br/>mit (57) bzw. (57a) gleichwertig. Es sind also die <span
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3616 class="cmmi-12">T</span><sub ><span
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3617 class="cmmi-8"><img
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3618 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-1b.png" alt="s" class="8x-x-1b" /></span></sub><sup ><span
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3619 class="cmmi-8"><img
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3620 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sup> die <br/>Energiekomponenten des elektromagnetischen Feldes. Mit <br/>Hilfe von (61) und (64) zeigt man leicht, daß diese Energie-<br/>komponenten des elektromagnetischen Feldes im Falle der <br/>speziellen Relativitätstheorie die wohlbekannten Maxwell-<br/>Pointingschen Ausdrücke
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3621 ergeben.
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3622 </p><!--l. 4125--><p class="indent"> Wir haben nun die allgemeinsten Gesetze abgeleitet, <br/>welchen das Gravitationsfeld und die Materie genügen, indem <br/>wir uns konsequent eines Koordinatensystems bedienten, für <br/>welches <img
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3623 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_1916201x.png" alt=" V~ ---g-" class="sqrt" /> = 1 wird. Wir erzielten dadurch eine erhebliche
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3624 <br/>Vereinfachung der Formeln und Rechnungen, ohne daß wir <br/>auf die Forderung der allgemeinen Kovarianz verzichtet hätten: <br/>denn wir fanden unsere Gleichungen durch Spezialisierung <br/>des Koordinatensystems aus allgemein kovarianten Gleichungen.
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3625 </p><!--l. 4136--><p class="indent"> Immerhin ist die Frage nicht ohne formales Interesse, <br/>ob bei entsprechend verallgemeinerter Definition der Energie-<br/>komponenten des Gravitationsfeldes und der Materie auch <br/>ohne Spezialisierung des Koordinatensystems Erhaltungssätze
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3626 <br/>von der Gestalt der Gleichung (56) sowie Feldgleichungen der <br/>Gravitation von der Art der Gleichungen (52) bzw. (52a) <br/>gelten, derart, daß links eine Divergenz (im gewöhnlichen <br/>Sinne), rechts die Summe der Energiekomponenten der Materie <br/>und der Gravitation steht. Ich habe gefunden, daß beides <br/>in der Tat der Fall ist. Doch glaube ich, daß sich eine Mit-<br/>teilung meiner ziemlich umfangreichen Betrachtungen über <br/>diesen Gegenstand nicht lohnen würde, da doch etwas sach-<br/>lich Neues dabei nicht herauskommt. <pb/>
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3627 </p><!--l. 4154--><p class="indent">
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3628
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3629 </p>
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3630 <div class="center" >
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3631
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3632 <!--l. 4155--><p class="noindent">
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3633 </p><!--l. 4157--><p class="noindent"><span
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3634 class="cmbx-12">E. </span><span
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3635 class="cmbsy-10x-x-120">§ </span><span
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3636 class="cmbx-12">21. Newtons Theorie als erste N</span><span
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3637 class="cmbx-12">äherung.</span></p></div>
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3638 <!--l. 4161--><p class="indent"> Wie schon mehrfach erwähnt, ist die spezielle Relativitäts-<br/>theorie als Spezialfall der allgemeinen dadurch charakterisiert, <br/>daß die <span
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3639 class="cmmi-12">g</span><sub ><span
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3640 class="cmmi-8"><img
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3641 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img
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3642 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub> die konstanten Werte (4) haben. Dies bedeutet <br/>nach dem Vorherigen eine völlige Vernachlässigung der Gravi-<br/>tationswirkungen. Eine der Wirklichkeit näher liegende Ap-<br/>proximation erhalten wir, indem wir den Fall betrachten, daß <br/>die <span
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3643 class="cmmi-12">g</span><sub ><span
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3644 class="cmmi-8"><img
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3645 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img
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3646 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub>
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3647 von den Werten (4) nur um (gegen 1) kleine Größen <br/>abweichen, wobei wir kleine Größen zweiten und höheren <br/>Grades vernachlässigen. (Erster Gesichtspunkt der Ap-<br/>proximation.)
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3648 </p><!--l. 4175--><p class="indent"> Ferner soll angenommen werden, daß in dem betrach-<br/>teten zeiträumlichen Gebiete die <span
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3649 class="cmmi-12">g</span><sub ><span
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3650 class="cmmi-8"><img
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3651 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img
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3652 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub> im räumlich Unendlichen <br/>bei passender Wahl der Koordinaten den Werten (4) zustreben; <br/>d. h. wir betrachten Gravitationsfelder, welche als ausschließ-<br/>lich durch im Endlichen befindliche Materie erzeugt betrachtet <br/>werden können.
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3653 </p><!--l. 4183--><p class="indent"> Man könnte annehmen, daß diese Vernachlässigungen auf <br/>Newtons Theorie hinführen müßten. Indessen bedarf es <br/>hierfür noch der approximativen Behandlung der Grund-<br/>gleichungen nach einem zweiten Gesichtspunkte. Wir fassen <br/>die Bewegung eines Massenpunktes gemäß den Gleichungen (46) <br/>ins Auge. Im Falle der speziellen Relativitätstheorie können <br/>die Komponenten</p>
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3654 <center class="par-math-display" >
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3655 <img
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3656 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_1916202x.png" alt="d x1 d-x2 d-x3 d s , d s , d s " class="par-math-display" /></center>
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3657 <!--l. 4197--><p class="nopar">
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3658 </p><!--l. 4201--><p class="noindent">beliebige Werte annehmen; dies bedeutet, daß beliebige Ge-<br/>schwindigkeiten
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3659 </p>
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3660 <center class="par-math-display" >
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3661 <img
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3662 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_1916203x.png" alt=" V~ ------------------------- d-x12 d-x22 d-x32 v = d x4 + d x4 + d x4 " class="par-math-display" /></center>
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3663 <!--l. 4209--><p class="nopar">
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3664 </p><!--l. 4213--><p class="noindent">auftreten können, die kleiner sind als die Vakuumlichtgeschwin-<br/>digkeit (<span
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3665 class="cmmi-12">v &lt; </span>1). Will man sich auf den fast ausschließlich <br/>der Erfahrung sich darbietenden Fall beschränken, daß <span
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3666 class="cmmi-12">v </span> <br/>gegen die Lichtgeschwindigkeit klein ist, so bedeutet dies, <br/>daß die Komponenten
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3667 </p>
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3668 <center class="par-math-display" >
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3669 <img
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3670 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_1916204x.png" alt="d x d x d x ---1, ---2, ---3 d s d s d s " class="par-math-display" /></center>
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3671 <!--l. 4225--><p class="nopar">
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3672 </p><!--l. 4229--><p class="noindent">als kleine Größen zu behandeln sind, während <span
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3673 class="cmmi-12">dx</span><sub ><span
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3674 class="cmr-8">4</span></sub><img
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3675 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_1916205x.png" alt="/" class="left" align="middle" /> <span
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3676 class="cmmi-12">ds</span> bis <br/>auf Größen zweiter Ordnung gleich 1 ist (zweiter Gesichts-<br/>punkt der Approximation).
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3677 <pb/>
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3678 </p><!--l. 4237--><p class="indent">
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3679
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3680 </p><!--l. 4238--><p class="indent"> Nun beachten wir, daß nach dem ersten Gesichtspunkte <br/>der Approximation die Größen <img
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3681 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmr12-0.png" alt="G" class="12x-x-0" /><sub><span
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3682 class="cmmi-8"><img
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3683 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img
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3684 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub><sup><span
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3685 class="cmmi-8"><img
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3686 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-1c.png" alt="t" class="8x-x-1c" /></span></sup> alle kleine Größen mindestens <br/>erster Ordnung sind. Ein Blick auf (46) lehrt also, daß in dieser <br/>Gleichung nach dem zweiten Gesichtspunkt der Approximation <br/>nur Glieder zu berücksichtigen sind, für welche <span
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3687 class="cmmi-12"><img
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3688 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi12-16.png" alt="m" class="cmmi-12x-x-16" align="middle" /> </span>= <span
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3689 class="cmmi-12"><img
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3690 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi12-17.png" alt="n" class="12x-x-17" /> </span>= 4 <br/>ist. Bei Beschränkung auf Glieder niedrigster Ordnung erhält <br/>man an Stelle von (46) zunächst die Gleichungen
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3691 </p>
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3692 <center class="par-math-display" >
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3693 <img
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3694 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_1916206x.png" alt="d-2 xt t d t2 = G44 , " class="par-math-display" /></center>
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3695 <!--l. 4254--><p class="nopar">
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3696 </p><!--l. 4259--><p class="noindent">wobei <span
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3697 class="cmmi-12">ds </span>= <span
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3698 class="cmmi-12">dx</span><sub ><span
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3699 class="cmr-8">4</span></sub> = <span
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3700 class="cmmi-12">dt </span>gesetzt ist, oder unter Beschränkung <br/>auf Glieder, die nach dem ersten Gesichtspunkte der Ap-<br/>proximation erster Ordnung sind:
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3701 </p>
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3702 <center class="par-math-display" >
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3703 <img
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3704 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_1916207x.png" alt=" |_ _|
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3705 44 d-2 xt d t2 = |_ t _| (t = 1, 2, 3)
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3706 |_ _|
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3707 d 2 x 44 ----4- = - |_ 4 _| . d t2 " class="par-math-display" /></center>
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3708 <!--l. 4276--><p class="nopar">
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3709 </p><!--l. 4278--><p class="noindent">Setzt man außerdem voraus, daß das Gravitationsfeld ein <br/>quasi statisches sei, indem man sich auf den Fall beschränkt, <br/>daß die das Gravitationsfeld erzeugende Materie nur langsam <br/>(im Vergleich mit der Fortpflanzungsgeschwindigkeit des
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3710 <br/>Lichtes) bewegt ist, so kann man auf der rechten Seite Ab-<br/>leitungen nach der Zeit neben solchen nach den örtlichen <br/>Koordinaten vernachlässigen, so daß man erhält</p>
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3711 <table width="100%"
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3712 class="equation"><tr><td><a
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3713 id="x1-91r67"></a>
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3714 <center class="math-display" >
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3715 <img
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3716 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_1916208x.png" alt=" 2 d--xt- = - 1-@-g44(t = 1, 2, 3). d t2 2 @ xt " class="math-display" /></center></td><td width="5%">(67)</td></tr></table>
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3717 <!--l. 4293--><p class="nopar">
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3718 </p><!--l. 4297--><p class="noindent">Dies ist die Bewegungsgleichung des materiellen Punktes nach <br/>Newtons Theorie, wobei <span
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3719 class="cmmi-12">g</span><sub ><span
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3720 class="cmr-8">44</span></sub><img
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3721 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_1916209x.png" alt="/" class="left" align="middle" /> 2 die Rolle des Gravitations-<br/>potentiales spielt. Das Merkwürdige an diesem Resultat ist, <br/>daß nur die Komponente <span
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3722 class="cmmi-12">g</span><sub ><span
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3723 class="cmr-8">44</span></sub> des Fundamentaltensors allein <br/>in erster Näherung die Bewegung des materiellen Punktes <br/>bestimmt.
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3724 </p><!--l. 4306--><p class="indent"> Wir wenden uns nun zu den Feldgleichungen (53). Dabei <br/>ist zu berücksichtigen, daß der Energietensor der ,,Materie&#8220; <br/>fast ausschließlich durch die Dichte <span
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3725 class="cmmi-12"><img
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3726 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi12-25.png" alt="r" class="cmmi-12x-x-25" align="middle" /> </span>der Materie im engeren <br/>Sinne bestimmt wird, d. h. durch das zweite Glied der rechten <br/>Seite von (58) [bzw. (58a) oder (58b)]. Bildet man die uns
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3727 <br/>interessierende Näherung, so verschwinden alle Komponenten <br/>bis auf die Komponente</p>
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3728 <center class="par-math-display" >
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3729 <img
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3730 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_1916210x.png" alt="T44 = r = T . " class="par-math-display" /></center>
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3731 <!--l. 4319--><p class="nopar"> <pb/>
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3732 </p><!--l. 4326--><p class="indent">
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3733
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3734 </p><!--l. 4327--><p class="noindent">Auf der linken Seite von (53) ist das zweite Glied klein von <br/>zweiter Ordnung; das erste liefert in der uns interessierenden <br/>Näherung</p>
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3735 <center class="par-math-display" >
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3736 <img
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3737 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_1916211x.png" alt=" |_ _| |_ _| |_ _| |_ _|
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3738 @ m n @ m n @ m n @ mn + ---- |_ 1 _| + ---- |_ 2 _| + ---- |_ 3 _| - ---- |_ 4 _| . @ x1 @ x2 @ x3 @ x4 " class="par-math-display" /></center>
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3739 <!--l. 4350--><p class="nopar">
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3740 </p><!--l. 4352--><p class="noindent">Dies liefert für <span
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3741 class="cmmi-12"><img
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3742 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi12-16.png" alt="m" class="cmmi-12x-x-16" align="middle" /> </span>= <span
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3743 class="cmmi-12"><img
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3744 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi12-17.png" alt="n" class="12x-x-17" /> </span>= 4 bei Weglassung von nach der Zeit <br/>differenzierten Gliedern
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3745 </p>
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3746 <center class="par-math-display" >
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3747 <img
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3748 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_1916212x.png" alt=" ( 2 2 2 ) - 1- @-g-44+ @--g44 + @--g44 = - 1 D g . 2 @ x12 @ x22 @ x32 2 44 " class="par-math-display" /></center>
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3749 <!--l. 4362--><p class="nopar">
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3750 </p><!--l. 4365--><p class="noindent">Die letzte der Gleichungen (53) liefert also</p>
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3751 <table width="100%"
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3752 class="equation"><tr><td><a
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3753 id="x1-92r68"></a>
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3754 <center class="math-display" >
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diff changeset
3755 <img
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3756 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_1916213x.png" alt="D g = x r . 44 " class="math-display" /></center></td><td width="5%">(68)</td></tr></table>
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3757 <!--l. 4371--><p class="nopar">
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3758 </p><!--l. 4375--><p class="noindent">Die Gleichungen (67) und (68) zusammen sind äquivalent <br/>dem Newtonschen Gravitationsgesetz.
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3759 </p><!--l. 4378--><p class="indent"> Für das Gravitationspotential ergibt sich nach (67) und <br/>(68) der Ausdruck</p>
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3760 <table width="100%"
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3761 class="equation"><tr><td><a
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3762 id="x1-93r69"></a>
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3763 <center class="math-display" >
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3764
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3765 <img
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3766 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_1916214x.png" alt=" integral
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3767 -x- r-d-t - 8p r , " class="math-display" /></center></td><td width="5%">(68a)</td></tr></table>
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3768 <!--l. 4385--><p class="nopar">
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3769 </p><!--l. 4390--><p class="noindent">während Newtons Theorie bei der von uns gewählten Zeit-<br/>einheit</p>
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3770 <center class="par-math-display" >
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3771 <img
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3772 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_1916215x.png" alt=" K integral r d t - -2- ----- c r " class="par-math-display" /></center>
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3773 <!--l. 4397--><p class="nopar">
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3774 </p><!--l. 4402--><p class="noindent">ergibt, wobei <span
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3775 class="cmmi-12">K </span>die gewöhnlich als Gravitationskonstante <br/>bezeichnete Konstante 6,7 . 10<sup ><span
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3776 class="cmsy-8">-</span><span
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3777 class="cmr-8">8</span></sup> bedeutet. Durch Vergleich <br/>ergibt sich</p>
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3778 <table width="100%"
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3779 class="equation"><tr><td><a
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3780 id="x1-94r69"></a>
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3781 <center class="math-display" >
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3782 <img
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3783 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_1916216x.png" alt="x = 8-p-K- = 1,87 .10-27. c2 " class="math-display" /></center></td><td width="5%">(69)</td></tr></table>
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3784 <!--l. 4411--><p class="nopar">
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3785 </p>
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3786 <div class="center" >
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3787
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3788 <!--l. 4416--><p class="noindent">
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3789 </p><!--l. 4419--><p class="noindent"><span
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3790 class="cmbsy-10x-x-120">§ </span><span
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3791 class="cmbx-12">22. Verhalten von Masst</span><span
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3792 class="cmbx-12">äben und Uhren im statischen</span>
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3793 <br/><span
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3794 class="cmbx-12">Gravitationsfelde. Kr</span><span
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3795 class="cmbx-12">ümmung der Lichtstrahlen.</span> <br/><span
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3796 class="cmbx-12">Perihelbewegung</span>
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3797 <span
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3798 class="cmbx-12">der Planetenbahnen.</span></p></div>
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3799 <!--l. 4423--><p class="indent"> Um die Newton sche Theorie als erste Näherung zu er-<br/>halten, brauchten wir von den 10 Komponenten des Gravi-<br/>tationspotentials <span
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3800 class="cmmi-12">g</span><sub ><span
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3801 class="cmmi-8"><img
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3802 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img
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3803 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub> nur <span
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3804 class="cmmi-12">g</span><sub ><span
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3805 class="cmr-8">44</span></sub> zu berechnen, da nur diese Kom-<br/>ponente in die erste Näherung (67) der Bewegungsgleichung
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3806 <br/>des materiellen Punktes im Gravitationsfelde eingeht. Man <br/>sieht indessen schon daraus, daß noch andere Komponenten <br/>der <span
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3807 class="cmmi-12">g</span><sub ><span
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3808 class="cmmi-8"><img
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3809 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img
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3810 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub> von den in (4) angegebenen
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3811 Werten in erster Näherung <br/>abweichen müssen, daß letzteres durch die Bedingung
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3812 <span
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3813 class="cmmi-12">g </span>= <span
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3814 class="cmsy-10x-x-120">- </span>1 <br/>verlangt wird. <pb/>
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3815 </p><!--l. 4437--><p class="indent">
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3816
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3817 </p><!--l. 4438--><p class="indent"> Für einen im Anfangspunkt des Koordinatensystems be-<br/>findlichen felderzeugenden Massenpunkt erhält man in erster <br/>Näherung die radialsymmetrische Lösung</p>
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3818 <table width="100%"
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3819 class="equation"><tr><td><a
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3820 id="x1-95r70"></a>
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3821 <center class="math-display" >
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3822 <img
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3823 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_1916217x.png" alt=" xr-xs- grs = - drs - a r3 (r und s zwischen 1 und 3) { g = g = 0 (r zwischen 1 und 3) r4 4r g44 = 1 - a- . r " class="math-display" /></center></td><td width="5%">(70)</td></tr></table>
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3824 <!--l. 4454--><p class="nopar">
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3825 </p><!--l. 4458--><p class="noindent"><span
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3826 class="cmmi-12"><img
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3827 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi12-e.png" alt="d" class="12x-x-e" /></span><sub ><span
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3828 class="cmmi-8"><img
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3829 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-25.png" alt="r" class="cmmi-8x-x-25" align="middle" /><img
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3830 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-1b.png" alt="s" class="8x-x-1b" /></span></sub> ist dabei 1 bzw. 0, je nachdem <span
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3831 class="cmmi-12"><img
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3832 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi12-25.png" alt="r" class="cmmi-12x-x-25" align="middle" /> </span>= <span
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3833 class="cmmi-12"><img
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diff changeset
3834 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi12-1b.png" alt="s" class="12x-x-1b" /> </span>oder <span
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3835 class="cmmi-12"><img
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diff changeset
3836 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi12-25.png" alt="r" class="cmmi-12x-x-25" align="middle" /> <img
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3837 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi12-1b.png" alt="s" class="12x-x-1b" />, r </span>ist die <br/>Größe</p>
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3838 <center class="par-math-display" >
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3839 <img
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3840 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_1916218x.png" alt=" V~ ------------------ + x12 + x22 + x32 . " class="par-math-display" /></center>
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3841 <!--l. 4466--><p class="nopar">
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3842 </p><!--l. 4469--><p class="noindent">Dabei ist wegen (68a)</p>
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3843 <table width="100%"
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3844 class="equation"><tr><td><a
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3845 id="x1-96r71"></a>
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3846 <center class="math-display" >
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3847 <img
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3848 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_1916219x.png" alt=" x-M-- a = 8 p , " class="math-display" /></center></td><td width="5%">(70a)</td></tr></table>
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3849
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3850 <!--l. 4475--><p class="nopar">
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3851 </p><!--l. 4480--><p class="noindent">wenn mit <span
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3852 class="cmmi-12">M </span>die felderzeugende Masse bezeichnet wird. Daß <br/>durch diese Lösung die Feldgleichungen (außerhalb der Masse) <br/>in erster Näherung erfüllt werden, ist leicht zu verifizieren.
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3853 </p><!--l. 4485--><p class="indent"> Wir untersuchen nun die Beeinflussung, welche die metri-<br/>schen Eigenschaften des Raumes durch das Feld der Masse <span
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3854 class="cmmi-12">M </span> <br/>erfahren. Stets gilt zwischen den ,,lokal&#8220; (<span
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3855 class="cmsy-10x-x-120">§ </span>4) gemessenen <br/>Längen und Zeiten <span
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3856 class="cmmi-12">ds </span>einerseits und den Koordinatendifferenzen <br/><span
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3857 class="cmmi-12">d x</span><sub ><span
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3858 class="cmmi-8">v</span></sub> andererseits die Beziehung
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3859 </p>
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3860 <center class="par-math-display" >
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3861 <img
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3862 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_1916220x.png" alt="d s2 = gm n d xm d xn . " class="par-math-display" /></center>
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3863 <!--l. 4496--><p class="nopar">
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3864 </p><!--l. 4500--><p class="indent"> Für einen ,,parallel&#8220; der <span
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3865 class="cmmi-12">x</span>-Achse gelegten Einheitsmaßstab <br/>wäre beispielsweise zu setzen
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3866 </p>
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3867 <center class="par-math-display" >
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3868 <img
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3869 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_1916221x.png" alt="d s2 = - 1; d x2 = d x3 = d x4 = 0 , " class="par-math-display" /></center>
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3870 <!--l. 4508--><p class="nopar">
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3871 </p><!--l. 4512--><p class="noindent">also</p>
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3872 <center class="par-math-display" >
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3873 <img
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3874 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_1916222x.png" alt="-1 = g11 d x12 . " class="par-math-display" /></center>
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3875 <!--l. 4518--><p class="nopar">
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3876 </p><!--l. 4521--><p class="noindent">Liegt der Einheitsmaßstab außerdem auf der <span
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3877 class="cmmi-12">x</span>-Achse, so <br/>ergibt die erste der Gleichungen (70)
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3878 </p>
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3879 <center class="par-math-display" >
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3880 <img
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3881 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_1916223x.png" alt=" ( a ) g11 = - 1 + -- . r " class="par-math-display" /></center>
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3882 <!--l. 4528--><p class="nopar">
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3883 </p><!--l. 4531--><p class="noindent">Aus beiden Relationen folgt in erster Näherung genau</p>
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3884 <table width="100%"
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3885 class="equation"><tr><td><a
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3886 id="x1-97r71"></a>
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3887
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3888 <center class="math-display" >
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3889 <img
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3890 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_1916224x.png" alt=" -a- d x = 1 - 2 r . " class="math-display" /></center></td><td width="5%">(71)</td></tr></table>
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3891 <!--l. 4537--><p class="nopar">
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3892 <pb/></p><!--l. 4544--><p class="indent">
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3893
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3894 </p><!--l. 4545--><p class="noindent">Der Einheitsmaßstab erscheint also mit Bezug auf das Ko-<br/>ordinatensystem in dem gefundenen Betrage durch das Vor-<br/>handensein des Gravitationsfeldes verkürzt, wenn er radial <br/>angelegt wird.
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3895 </p><!--l. 4550--><p class="indent"> Analog erhält man seine Koordinatenlänge in tangentialer <br/>Richtung, indem man beispielsweise setzt</p>
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3896 <center class="par-math-display" >
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3897 <img
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3898 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_1916225x.png" alt="d s2 = - 1; d x = d x = d x = 0 ; x = r, x = x = 0 . 1 3 4 1 2 3 " class="par-math-display" /></center>
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3899 <!--l. 4558--><p class="nopar">
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3900 </p><!--l. 4561--><p class="noindent">Es ergibt sich</p>
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3901 <table width="100%"
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3902 class="equation"><tr><td><a
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3903 id="x1-98r72"></a>
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3904 <center class="math-display" >
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3905 <img
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3906 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_1916226x.png" alt=" 2 2 - 1 = g22 d x2 = - d x2 . " class="math-display" /></center></td><td width="5%">(71a)</td></tr></table>
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3907 <!--l. 4567--><p class="nopar">
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3908 </p><!--l. 4571--><p class="noindent">Bei tangentialer Stellung hat also das Gravitationsfeld des <br/>Massenpunktes keinen Einfluß auf die Stablänge.
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3909 </p><!--l. 4575--><p class="indent"> Es gilt also die Euklidische Geometrie im Gravitations-<br/>felde nicht einmal in erster Näherung, falls man einen und <br/>denselben Stab unabhängig von seinem Ort und seiner Orien-<br/>tierung als Realisierung derselben Strecke auffassen will.
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3910 <br/>Allerdings zeigt ein Blick auf (70a) und (69), daß die zu er-<br/>wartenden Abweichungen viel zu gering sind, um sich bei <br/>der Vermessung der Erdoberfläche bemerkbar machen zu <br/>können.
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3911 </p><!--l. 4584--><p class="indent"> Es werde ferner die auf die Zeitkoordinate untersuchte <br/>Ganggeschwindigkeit einer Einheitsuhr untersucht, welche in <br/>einem statischen Gravitationsfelde ruhend angeordnet ist. Hier <br/>gilt für eine Uhrperiode
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3912 </p>
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3913 <center class="par-math-display" >
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3914 <img
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3915 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_1916227x.png" alt="d s = 1; d x1 = d x2 = d x3 = 0 . " class="par-math-display" /></center>
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3916 <!--l. 4593--><p class="nopar">
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3917
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3918 </p><!--l. 4596--><p class="noindent">Also ist</p>
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3919 <center class="par-math-display" >
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3920 <img
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3921 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_1916228x.png" alt=" 1 = g44 dx42 ;
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3922 d x4 = V~ 1-- = V~ -------1--------- = 1 - g-44----1 g 44 1 + (g 44 - 1) 2 " class="par-math-display" /></center>
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3923 <!--l. 4606--><p class="nopar">
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3924 </p><!--l. 4611--><p class="noindent">oder</p>
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3925 <table width="100%"
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3926 class="equation"><tr><td><a
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3927 id="x1-99r72"></a>
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3928 <center class="math-display" >
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3929 <img
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3930 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_1916229x.png" alt=" integral
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3931 d x4 = 1 + -x-- r-d-t-. 8 p r " class="math-display" /></center></td><td width="5%">(72)</td></tr></table>
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3932 <!--l. 4617--><p class="nopar">
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3933 </p><!--l. 4621--><p class="noindent">Die Uhr läuft also langsamer, wenn sie in der Nähe ponde-<br/>rabler Massen aufgestellt ist. Es folgt daraus, daß die Spektral-<br/>linien von der Oberfläche großer Sterne zu uns gelangenden <br/>Lichtes nach dem roten Spektralende verschoben erscheinen <br/>müssen.<sup ><span
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3934 class="cmr-8">1</span></sup>)
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3935 </p><!--l. 4628--><p class="noindent">----------
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3936 </p><!--l. 4631--><p class="indent"> 1) Für das Bestehen eines derartigen Effektes sprechen nach <br/>E. Freundlich spektrale Beobachtungen an Fixsternen bestimmter <br/>Typen. Eine endgültige Prüfung dieser Konsequenz steht indes noch aus. <pb/>
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3937 </p><!--l. 4637--><p class="indent">
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3938
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3939 </p><!--l. 4638--><p class="indent"> Wir untersuchen ferner den Gang der Lichtstrahlen im <br/>statischen Gravitationsfeld. Gemäß der speziellen Relativitäts-<br/>theorie ist die Lichtgeschwindigkeit durch die Gleichung
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3940 </p>
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3941 <center class="par-math-display" >
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3942 <img
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3943 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_1916230x.png" alt=" 2 2 2 2 - d x1 - d x2 - d x3 + d x4 = 0 " class="par-math-display" /></center>
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3944 <!--l. 4647--><p class="nopar">
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3945 </p><!--l. 4649--><p class="noindent">gegeben, also gemäß der allgemeinen Relativitätstheorie durch <br/>die Gleichung</p>
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3946 <table width="100%"
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3947 class="equation"><tr><td><a
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3948 id="x1-100r73"></a>
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3949 <center class="math-display" >
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3950 <img
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3951 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_1916231x.png" alt="d s2 = gmn dxm d xn = 0. " class="math-display" /></center></td><td width="5%">(73)</td></tr></table>
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3952 <!--l. 4655--><p class="nopar">
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3953 </p><!--l. 4658--><p class="noindent">Ist die Richtung, d. h. das Verhältnis <span
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3954 class="cmmi-12">d x</span><sub ><span
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3955 class="cmr-8">1</span></sub> : <span
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3956 class="cmmi-12">d x</span><sub ><span
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3957 class="cmr-8">2</span></sub> : <span
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3958 class="cmmi-12">d x</span><sub ><span
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3959 class="cmr-8">3</span></sub> ge-<br/>geben, so liefert die Gleichung (73) die Größen
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3960 </p>
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3961 <center class="par-math-display" >
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3962 <img
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3963 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_1916232x.png" alt="d x1 d x2 d x3 ----, ----, ---- dx4 d x4 d x4 " class="par-math-display" /></center>
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3964 <!--l. 4665--><p class="nopar">
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3965 </p><!--l. 4667--><p class="noindent">und somit die Geschwindigkeit
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3966 </p>
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3967 <center class="par-math-display" >
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3968 <img
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3969 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_1916233x.png" alt=" V~ ----------------------------------- ( )2 ( )2 ( )2 d-x1 + dx2- + d-x3 = g , d x4 dx4 d x4 " class="par-math-display" /></center>
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3970 <!--l. 4674--><p class="nopar">
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3971 </p><!--l. 4676--><p class="noindent">im Sinne der Euklidischen Geometrie definiert. Man erkennt <br/>leicht, daß die Lichtstrahlen gekrümmt verlaufen müssen mit <br/>Bezug auf das Koordinatensystem, falls die <span
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3972 class="cmmi-12">g</span><sub ><span
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3973 class="cmmi-8"><img
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3974 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img
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3975 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub> nicht konstant <br/>sind. Ist <span
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3976 class="cmmi-12">n </span>eine Richtung senkrecht zur
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3977 Lichtfortpflanzung, <br/>so ergibt das Huggenssche Prinzip, daß der Lichtstrahl [in <br/>der Ebene (<span
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3978 class="cmmi-12"><img
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3979 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi12-d.png" alt="g" class="12x-x-d" />, n</span>) betrachtet] die Krümmung <span
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3980 class="cmsy-10x-x-120">-</span><span
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3981 class="cmmi-12">d<img
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3982 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi12-d.png" alt="g" class="12x-x-d" /></span><img
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3983 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_1916234x.png" alt="/" class="left" align="middle" /> <span
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3984 class="cmmi-12"><img
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3985 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmmi12-40.png" alt="@" class="12x-x-40" /> n</span> besitzt. <img
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3986 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_1916235x.png" alt="PIC" class="graphics" width="244.6948pt" height="170.72084pt" /><!--tex4ht:graphics
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3987 name="img/Einst_Grund_de_1916235x.png" src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/Einst_Grund_de_1916_001.eps"
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3988 -->
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3989 </p><!--l. 4689--><p class="indent"> Wir untersuchen die Krümmung, welche ein Lichtstrahl <br/>erleidet, der im Abstand <img
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3990 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/cmr12-1.png" alt="D" class="12x-x-1" /> an einer Masse <span
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3991 class="cmmi-12">M </span>vorbeigeht. <br/>Wählt man das Koordinatensystem gemäß der vorstehenden <br/>Skizze, so ist die gesamte Biegung <span
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3992 class="cmmi-12">B </span>des Lichtstrahles (positiv <br/>gerechnet, wenn sie nach dem Ursprung hin konkav ist) in <br/>genügender Näherung gegeben durch
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3993 </p>
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3994 <center class="math-display" >
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3995 <img
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3996 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_1916236x.png" alt=" + integral oo
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3997 B = @-g--d x , @ x1 2 - oo
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3998 " class="math-display" /></center>
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3999 <!--l. 4700--><p class="nopar"> <pb/>
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4000 </p><!--l. 4703--><p class="indent">
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4001
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4002 </p><!--l. 4704--><p class="noindent">während (73) und (70) ergeben
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4003 </p>
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4004 <center class="par-math-display" >
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4005 <img
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4006 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_1916237x.png" alt=" V~ ------ ( ) g-44- -a- x22- g = - g 22 = 1 + 2 r 1 + r2 . " class="par-math-display" /></center>
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4007 <!--l. 4711--><p class="nopar">
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4008 </p><!--l. 4713--><p class="noindent">Die Ausrechnung ergibt</p>
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4009 <table width="100%"
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4010 class="equation"><tr><td><a
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4011 id="x1-101r74"></a>
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4012 <center class="math-display" >
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4013 <img
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4014 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_1916238x.png" alt="B = 2a- = x-M--. D 4p D " class="math-display" /></center></td><td width="5%">(74)</td></tr></table>
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4015 <!--l. 4719--><p class="nopar">
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4016 </p><!--l. 4723--><p class="noindent">Ein an der Sonne vorbeigehender Lichtstrahl erfährt dem-<br/>nach eine Biegung von 1<span
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4017 class="cmmi-12">, </span>7<span
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4018 class="cmsy-10x-x-120">''</span>, ein am Planeten Jupiter vorbei-<br/>gehender eine solche von etwa 0<span
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4019 class="cmmi-12">, </span>02<span
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4020 class="cmsy-10x-x-120">''</span>.
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4021 </p><!--l. 4728--><p class="indent"> Berechnet man das Gravitationsfeld um eine Größen-<br/>ordnung genauer, und ebenso mit entsprechender Genauig-<br/>keit die Bahnbewegung eines materiellen Punktes von relativ <br/>unendlich kleiner Masse, so erhält man gegenüber den Kepler-<br/>Newtonschen Gesetzen der Planetenbewegung eine Abwei-<br/>chung von folgender Art. Die Bahnellipse eines Planeten er-<br/>fährt in Richtung der Bahnbewegung eine langsame Drehung <br/>vom Betrage</p>
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4022 <table width="100%"
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4023 class="equation"><tr><td><a
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4024 id="x1-102r75"></a>
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4025 <center class="math-display" >
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4026 <img
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4027 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Grund_de_1916/fulltext/img/Einst_Grund_de_1916239x.png" alt=" 3------a2------ e = 24 p T 2 c2(1 - e2)
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4028 " class="math-display" /></center></td><td width="5%">(75)</td></tr></table>
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4029 <!--l. 4742--><p class="nopar">
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4030 </p><!--l. 4747--><p class="noindent">pro Umlauf. In dieser Formel bedeutet <span
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4031 class="cmmi-12">a </span>die große Halbachse, <br/><span
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4032 class="cmmi-12">c </span>die Lichtgeschwindigkeit in üblichem Maße, <span
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4033 class="cmmi-12">e </span>die Exzentrizität, <br/><span
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4034 class="cmmi-12">T </span>die Umlaufszeit in Sekunden.<sup ><span
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4035 class="cmr-8">1</span></sup>)
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4036 </p><!--l. 4752--><p class="indent"> Die Rechnung ergibt für den Planeten Merkur eine Drehung <br/>der Bahn um 43<span
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4037 class="cmsy-10x-x-120">''</span>
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4038 pro Jahrhundert, genau entsprechend der <br/>Konstatierung der Astronomen (Leverrier); diese fanden <br/>nämlich einen durch Störungen der übrigen Planeten nicht <br/>erklärbaren Rest der Perihelbewegung dieses Planeten von <br/>der angegebenen Größe.
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4039 </p><!--l. 4760--><p class="noindent">----------
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4040 </p><!--l. 4763--><p class="indent"> 1) Bezüglich der Rechnung verweise ich auf die Originalabhand-<br/>lungen A. Einstein, Sitzungsber. d. Preuß. Akad. d. Wiss. <span
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4041 class="cmbx-12">47. </span>p. 831. <br/>1915. -- K. Schwarzschild, Sitzungsber. d. Preuß. Akad. d. Wiss. <span
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4042 class="cmbx-12">7. </span> <br/>p. 189. 1916.
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4043 </p>
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4044 <div class="center" >
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4045
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4046 <!--l. 4769--><p class="noindent">
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4047 </p><!--l. 4770--><p class="noindent">(Eingegangen 20. März 1916.)</p></div>
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4048 <div class="center" >
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4049
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4050 <!--l. 4775--><p class="noindent">
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4051 </p><!--l. 4776--><p class="noindent">----------</p></div>
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4052
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4053 </body></html>
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4054
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4055
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4056