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Removing DESpecs directory which deserted to git
author Klaus Thoden <kthoden@mpiwg-berlin.mpg.de>
date Wed, 29 Nov 2017 16:55:37 +0100
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1 <?xml version="1.0" encoding="utf-8" ?>
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4 <!--http://www.w3.org/TR/xhtml1/DTD/xhtml1-transitional.dtd-->
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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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11 <!-- xhtml,html -->
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12 <meta name="src" content="Einst_Ueber_de_1911.tex" />
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13 <meta name="date" content="2005-03-10 17:55:00" />
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14 <link rel="stylesheet" type="text/css" href="Einst_Ueber_de_1911.css" />
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15 </head><body >
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17 <!--l. 16--><p class="noindent"><pb/></p>
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18 <div class="center" >
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20 <!--l. 17--><p class="noindent">
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21 </p><!--l. 18--><p class="noindent"><span
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22 class="cmr-12x-x-120">4. </span><span
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23 class="cmbxti-10x-x-144">Über den Einflu</span><span
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24 class="cmmi-12x-x-120"><img
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25 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1911/fulltext/img/cmmi12-b-c.png" alt="b" class="cmmi-12-120x-x-c" align="middle" /> </span> <br/><span
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26 class="cmbxti-10x-x-144">der Schwerkraft auf die</span>
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27 <span
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28 class="cmbxti-10x-x-144">Ausbreitung des Lichtes; </span> <br/><span
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29 class="cmbxti-10x-x-144">von A. Einstein.</span></p></div>
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30 <div class="center" >
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32 <!--l. 23--><p class="noindent">
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33 </p><!--l. 24--><p class="noindent">----------</p></div>
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34 <!--l. 28--><p class="indent"> Die Frage, ob die Ausbreitung des Lichtes durch die <br/>Schwere beinflußt wird, habe ich schon an einer vor 3 Jahren <br/>erschienenen Abhandlung zu beantworten gesucht.<sup ><span
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35 class="cmr-8">1</span></sup>) Ich komme <br/>auf dies Thema wieder zurück, weil mich meine damalige
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36 <br/>Darstellung des Gegenstandes nicht befriedigt, noch mehr <br/>aber, weil ich nun nachträglich einsehe, daß eine der wichtigsten <br/>Konsequenzen jener Betrachtung der experimentellen Prüfung <br/>zugänglich ist. Es ergibt sich nämlich, daß Lichtstrahlen, die <br/>in der Nähe der Sonne vorbeigehen, durch das Gravitationsfeld
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37 <br/>derselben nach der vorzubringenden Theorie eine Ablenkung <br/>erfahren, so daß eine scheinbare Vergrößerung des Winkel-<br/>abstandes eines nahe an der Sonne erscheinenden Fixsternes <br/>von dieser im Betrage von fast einer Bogensekunde eintritt.
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38 </p><!--l. 45--><p class="indent"> Es haben sich bei der Durchführung der Überlegungen <br/>auch noch weitere Resultate ergeben, die sich auf die Gravi-<br/>tation beziehen. Da aber die Darlegung der ganzen Be-<br/>trachtung ziemlich unübersichtlich würde, sollen im folgenden <br/>nur einige ganz elementare Überlegungen gegeben werden, aus
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39 <br/>denen man sich bequem über die Voraussetzungen und den <br/>Gedankengang der Theorie orientieren kann. Die hier ab-<br/>geleiteten Beziehungen sind, auch wenn die theoretische Grund-<br/>lage zutrifft, nur in erster Näherung gültig.
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40 </p>
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41 <div class="center" >
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43 <!--l. 57--><p class="noindent">
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44 </p><!--l. 58--><p class="noindent"><span
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45 class="cmsy-10x-x-120">§ </span>1. Hypothese über die physikalische Natur <br/>des Gravitationsfeldes.</p></div>
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46 <!--l. 63--><p class="indent"> In einem homogenen Schwerefeld (Schwerebeschleunigung <span
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47 class="cmmi-12"><img
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48 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1911/fulltext/img/cmmi12-d.png" alt="g" class="12x-x-d" /></span>) <br/>befinde sich ein ruhendes Koordinatensystem <span
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49 class="cmmi-12">K</span>, das so orien-<br/>tiert sei, daß die Kraftlinien des Schwerefeldes in Richtung <br/>----------
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50 </p><!--l. 70--><p class="indent"> 1) A. Einstein, Jahrb. f. Radioakt. n. Elektronik IV. 4. <pb/>
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51 </p><!--l. 75--><p class="indent">
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52
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53 </p><!--l. 76--><p class="noindent">der negativen <span
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54 class="cmmi-12">z</span>-Achse verlaufen. In einem von Gravitations-<br/>feldern freien Raume befinde sich ein zweites Koordinaten-<br/>system <span
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55 class="cmmi-12">K</span><span
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56 class="cmsy-10x-x-120">'</span>, das in Richtung seiner positiven
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57 <span
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58 class="cmmi-12">z</span>-Achse eine <br/>gleichförmig beschleunigte Bewegung (Beschleunigung <span
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59 class="cmmi-12"><img
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60 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1911/fulltext/img/cmmi12-d.png" alt="g" class="12x-x-d" /></span>) aus-<br/>führe. Um die Betrachtung nicht unnütz zu komplizieren, <br/>sehen wir dabei von der Relativitätstheorie vorläufig ab, be-<br/>trachten also beide Systeme nach der gewohnten Kinematik <br/>und in denselben stattfindende Bewegungen nach der gewöhn-<br/>lichen Mechanik.
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61 </p><!--l. 88--><p class="indent"> Relativ zu <span
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62 class="cmmi-12">K</span>, sowie relativ zu <span
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63 class="cmmi-12">K</span><span
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64 class="cmsy-10x-x-120">'</span>, bewegen sich materielle <br/>Punkte, die der Einwirkung anderer materieller Punkte nicht <br/>unterliegen, nach den Gleichungen:
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65 </p>
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66 <center class="par-math-display" >
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67 <img
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68 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1911/fulltext/img/Einst_Ueber_de_19110x.png" alt=" 2 2 2 d-xn-= 0, d--yn = 0 , d-zn-= - g . d t2 d t2 d t2 " class="par-math-display" /></center>
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69 <!--l. 97--><p class="nopar">
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70 </p><!--l. 101--><p class="noindent">Dies folgt für das beschleunigte System <span
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71 class="cmmi-12">K</span><span
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72 class="cmsy-10x-x-120">' </span>direkt aus dem <br/>Galileischen Prinzip, für das in einem homogenen Gravi-<br/>tationsfeld ruhende System <span
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73 class="cmmi-12">K </span>aber aus der Erfahrung, daß <br/>in einem solchen Felde alle Körper gleich stark und gleich-<br/>mäßig beschleunigt werden. Diese Erfahrung vom gleichen <br/>Fallen aller Körper im Gravitationsfelde ist eine der all-<br/>gemeinsten, welche die Naturbeobachtung uns geliefert hat; <br/>trotzdem hat dieses Gesetz in den Fundamenten unseres
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74 <br/>physikalischen Weltbildes keinen Platz erhalten.
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75 </p><!--l. 113--><p class="indent"> Wir gelangen aber zu einer sehr befriedigenden Inter-<br/>pretation des Erfahrungssatzes, wenn wir annehmen, daß die <br/>Systeme <span
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76 class="cmmi-12">K </span>und <span
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77 class="cmmi-12">K</span><span
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78 class="cmsy-10x-x-120">' </span>physikalisch genau gleichwertig sind, d. h. <br/>wenn wir annehmen, man könne das System <span
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79 class="cmmi-12">K</span>
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80 ebenfalls als <br/>in einem von einem Schwerefeld freien Raume befindlich an-<br/>nehmen; dafür müssen wir <span
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81 class="cmmi-12">K </span>dann aber als gleichförmig be-<br/>schleunigt betrachten. Man kann bei dieser Auffassung ebenso-<br/>wenig von der <span
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82 class="cmti-12">absoluten</span>
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83 <span
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84 class="cmti-12">Beschleunigung </span>des Bezugssystems <br/>sprechen, wie man nach der gewöhnlichen Relativitätstheorie <br/>von der <span
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85 class="cmti-12">absoluten Geschwindigkeit </span>eines Systems reden kann.<sup ><span
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86 class="cmr-8">1</span></sup>)
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87 <br/>----------
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88 </p><!--l. 127--><p class="indent"> 1) Natürlich kann man ein <span
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89 class="cmti-12">beliebiges </span>Schwerefeld nicht durch einen
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90 <br/>Bewegungszustand des Systems ohne Gravitationsfeld ersetzen, ebenso-<br/>wenig, als man durch eine Relativitätstransformation alle Punkte eines <br/>beliebig bewegten Mediums auf Ruhe trausformieren kann. <pb/>
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91 </p><!--l. 136--><p class="indent">
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92
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93 </p><!--l. 137--><p class="noindent">Bei dieser Auffassung ist das gleiche Fallen aller Körper <br/>in einem Gravitationsfelde selbstverständlich.
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94 </p><!--l. 141--><p class="indent"> Solange wir uns auf rein mechanische Vorgänge aus dem <br/>Gültigkeitsbereich von Newtons Mechanik beschränken, sind <br/>wir der Gleichwertigkeit der Systeme <span
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95 class="cmmi-12">K </span>und <span
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96 class="cmmi-12">K</span><span
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97 class="cmsy-10x-x-120">' </span>sicher. Unsere <br/>Auffassung wird jedoch nur dann tiefere Bedeutung haben, <br/>wenn die Systeme <span
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98 class="cmmi-12">K </span>und <span
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99 class="cmmi-12">K</span><span
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100 class="cmsy-10x-x-120">' </span>in bezug auf alle physikalischen
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101 <br/>Vorgänge gleichwertig sind, d. h. wenn die Naturgesetze in <br/>bezug auf <span
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102 class="cmmi-12">K</span>
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103 mit denen in bezug auf <span
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104 class="cmmi-12">K</span><span
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105 class="cmsy-10x-x-120">' </span>vollkommen über-<br/>einstimmen. Indem wir dies annehmen, erhalten wir ein <br/>Prinzip, das, falls es wirklich zutrifft, eine große heuristische <br/>Bedeutung besitzt. Denn wir erhalten durch die theoretische <br/>Betrachtung der Vorgänge, die sich relativ zu einem gleich-<br/>förmig beschleunigten Bezugssystem abspielen, Aufschluß über <br/>den Verlauf der Vorgänge in einem homogenen Gravitations-<br/>felde.<sup ><span
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106 class="cmr-8">1</span></sup>) Im folgenden soll zunächst gezeigt werden, inwiefern <br/>unserer Hypothese vom Standpunkte der gewöhnlichen Rela-<br/>tivitätstheorie aus eine beträchtliche Wahrscheinlichkeit zu-<br/>kommt.
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107 </p>
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108 <div class="center" >
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109
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110 <!--l. 163--><p class="noindent">
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111 </p><!--l. 164--><p class="noindent"><span
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112 class="cmsy-10x-x-120">§ </span>2. Über die Schwere der Energie.</p></div>
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113 <!--l. 168--><p class="indent"> Die Relativitätstheorie hat ergeben, daß die träge Masse <br/>eines Körpers mit dem Energieinhalt desselben wächst; <br/>beträgt der Energiezuwachs <span
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114 class="cmmi-12">E</span>, so ist der Zuwachs an träger <br/>Masse gleich <span
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115 class="cmmi-12">E/c</span><sup ><span
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116 class="cmr-8">2</span></sup>, wenn <span
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117 class="cmmi-12">c </span>die Lichtgeschwindigkeit bedeutet.
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118 <br/>Entspricht nun aber diesem Zuwachs an träger Masse auch <br/>ein Zuwachs an gravitierender Masse? Wenn nicht, so fiele <br/>ein Körper in demselben Schwerefelde mit verschiedener Be-<br/>schleunigung je nach dem Energieinhalte des Körpers. Das <br/>so befriedigende Resultat der Relativitätstheorie, nach welchem <br/>der Satz von der Erhaltung der Masse in dem Satze von der
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119 <br/>Erhaltung der Energie aufgeht, wäre nicht aufrecht zu er-<br/>halten; denn so wäre der Satz von der Erhaltung der Masse <br/>zwar für die <span
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120 class="cmti-12">tr</span><span
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121 class="cmti-12">äge </span>Masse in der alten Fassung aufzugeben, <br/>für die gravitierende Masse aber aufrecht zu erhalten.
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122 </p><!--l. 186--><p class="noindent">----------
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123 </p><!--l. 188--><p class="indent"> 1) In einer späteren Abhandlung wird gezeigt werden, daß das hier <br/>in Betracht kommende Gravitationsfeld nur in erster Annäherung <br/>homogen ist.
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124 <pb/>
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125 </p><!--l. 195--><p class="indent">
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126
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127 </p><!--l. 196--><p class="indent"> Dies muß als sehr unwahrscheinlich betrachtet werden. <br/>Andererseits liefert uns die gewöhnliche Relativitätstheorie <br/>kein Argument, aus dem wir folgern könnten, daß das Gewicht <br/>eines Körpers von dessen Energieinhalt abhängt. Wir werden <br/>aber zeigen, daß unsere Hypothese von der Äquivalenz der
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128 <br/>Systeme <span
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129 class="cmmi-12">K </span>und <span
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130 class="cmmi-12">K</span><span
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131 class="cmsy-10x-x-120">' </span>die Schwere der Energie als notwendige <br/>Konsequenz liefert. Es mögen sich die beiden mit Meßinstrumenten versehenen <br/>körperlichen Systeme <span
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132 class="cmmi-12">S</span><sub ><span
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133 class="cmr-8">1</span></sub> und <span
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134 class="cmmi-12">S</span><sub ><span
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135 class="cmr-8">2</span></sub> in der Entfernung <span
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136 class="cmmi-12">h </span>von-<br/>einander auf der <span
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137 class="cmmi-12">z</span>-Achse von <span
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138 class="cmmi-12">K </span>befinden <sup ><span
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139 class="cmr-8">1</span></sup>), derart, daß das <br/>Gravitationspotential in <span
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140 class="cmmi-12">S</span><sub >
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141 <span
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142 class="cmr-8">2</span></sub> um <span
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143 class="cmmi-12"><img
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144 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1911/fulltext/img/cmmi12-d.png" alt="g" class="12x-x-d" /> </span>. <span
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145 class="cmmi-12">h </span>größer ist, als das in <span
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146 class="cmmi-12">S</span><sub ><span
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147 class="cmr-8">1</span></sub>. Es <br/>wurde von <span
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148 class="cmmi-12">S</span><sub ><span
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149 class="cmr-8">2</span></sub> gegen <span
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150 class="cmmi-12">S</span><sub ><span
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151 class="cmr-8">1</span></sub> eine bestimmte Energie-<br/>menge
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152 <span
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153 class="cmmi-12">E </span>in Form von Strahlung gesendet. Die <br/>Energiemengen mögen dabei in <span
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154 class="cmmi-12">S</span><sub ><span
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155 class="cmr-8">1</span></sub> und <span
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156 class="cmmi-12">S</span><sub ><span
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157 class="cmr-8">2</span></sub> mit <br/>Vorrichtungen gemessen werden, die -- an <span
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158 class="cmti-12">einen </span> <br/>Ort des Systems <span
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159 class="cmmi-12">z </span>gebracht und dort mit-<br/>einander verglichen -- vollkommen gleich seien. <br/>Über den Vorgang dieser Energieübertragung
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160 <br/>durch Strahlung läßt sich a priori nichts aus-<br/>sagen, weil wir den Einfluß des Schwerefeldes <br/>auf die Strahlung und die Meßinstrumente in <span
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161 class="cmmi-12">S</span><sub ><span
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162 class="cmr-8">1</span></sub> und <span
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163 class="cmmi-12">S</span><sub ><span
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164 class="cmr-8">2</span></sub> nicht <br/>kennen. <img
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165 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1911/fulltext/img/Einst_Ueber_de_19111x.png" alt="PIC" class="graphics" width="71.13188pt" height="124.48445pt" /><!--tex4ht:graphics
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166 name="img/Einst_Ueber_de_19111x.png" src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1911/fulltext/Einst_Ueber_de_1911_001.EPS"
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167 --></p><!--l. 226--><p class="indent">Nach unserer Voraussetzung von der Äquivalenz von <span
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168 class="cmmi-12">K </span> <br/>und <span
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169 class="cmmi-12">K</span><span
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170 class="cmsy-10x-x-120">' </span>können wir aber an Stelle des im homogenen Schwere-<br/>felde befindlichen Systems <span
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171 class="cmmi-12">K </span>das schwerefreie, im Sinne der <br/>positiven <span
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172 class="cmmi-12">z </span>gleichförmig beschleunigt bewegte System
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173 <span
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174 class="cmmi-12">K</span><span
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175 class="cmsy-10x-x-120">' </span>setzen, <br/>mit dessen <span
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176 class="cmmi-12">z</span>-Achse die körperlichen Systeme <span
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177 class="cmmi-12">S</span><sub ><span
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178 class="cmr-8">1</span></sub> und <span
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179 class="cmmi-12">S</span><sub ><span
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180 class="cmr-8">2</span></sub> fest <br/>verbunden sind.
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181 </p><!--l. 234--><p class="indent"> Den Vorgang der Energieübertragung durch Strahlung <br/>von <span
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182 class="cmmi-12">S</span><sub ><span
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183 class="cmr-8">2</span></sub> auf <span
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184 class="cmmi-12">S</span><sub ><span
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185 class="cmr-8">1</span></sub>
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186 beurteilen wir von einem System <span
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187 class="cmmi-12">K</span><sub ><span
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188 class="cmr-8">0</span></sub> aus, das <br/>beschleunigungsfrei sei. In bezug auf
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189 <span
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190 class="cmmi-12">K</span><sub ><span
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191 class="cmr-8">0</span></sub> besitze <span
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192 class="cmmi-12">K</span><span
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193 class="cmsy-10x-x-120">' </span>in dem <br/>Augenblick die Geschwindigkeit Null, in welchem die Strah-<br/>lungsenergie <span
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194 class="cmmi-12">E</span><sub ><span
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195 class="cmr-8">2</span></sub> von <span
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196 class="cmmi-12">S</span><sub ><span
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197 class="cmr-8">2</span></sub> gegen <span
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198 class="cmmi-12">S</span><sub ><span
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199 class="cmr-8">1</span></sub> abgesendet wird. Die Strah-<br/>lung wird in <span
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200 class="cmmi-12">S</span><sub ><span
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201 class="cmr-8">1</span></sub>
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202 ankommen, wenn die Zeit <span
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203 class="cmmi-12">h/c </span>verstrichen ist <br/>(in erster Annäherung). In diesem Momente besitzt aber <span
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204 class="cmmi-12">S</span><sub ><span
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205 class="cmr-8">1</span></sub> <br/>in bezug auf <span
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206 class="cmmi-12">K</span><sub ><span
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207 class="cmr-8">0</span></sub> die Geschwindigkeit
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208 <span
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209 class="cmmi-12"><img
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210 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1911/fulltext/img/cmmi12-d.png" alt="g" class="12x-x-d" /> .h/c </span>= <span
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211 class="cmmi-12">v</span>. Deshalb <br/>besitzt nach der gewöhnlichen Relativitätstheorie die in <span
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212 class="cmmi-12">S</span><sub ><span
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213 class="cmr-8">1</span></sub>
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214 <br/>----------
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215 </p><!--l. 247--><p class="indent"> 1) <span
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216 class="cmmi-12">S</span><sub ><span
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217 class="cmr-8">1</span></sub> und <span
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218 class="cmmi-12">S</span><sub ><span
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219 class="cmr-8">2</span></sub> werden als gegenüber <span
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220 class="cmmi-12">h </span>unendlich klein betrachtet. <pb/>
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221 </p><!--l. 253--><p class="indent">
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222
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223 </p><!--l. 254--><p class="noindent">ankommende Strahlung nicht die Energie <span
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224 class="cmmi-12">E</span><sub ><span
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225 class="cmr-8">2</span></sub>, sondern eine <br/>größere Energie
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226 <span
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227 class="cmmi-12">E</span><sub ><span
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228 class="cmr-8">1</span></sub>, welche mit <span
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229 class="cmmi-12">E</span><sub ><span
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230 class="cmr-8">2</span></sub> in erster Annäherung durch <br/>die Gleichung verknüpft ist<sup ><span
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231 class="cmr-8">1</span></sup>):
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232 </p>
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233 <table width="100%"
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234 class="equation"><tr><td><a
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235 id="x1-2r1"></a>
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236 <center class="math-display" >
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237 <img
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238 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1911/fulltext/img/Einst_Ueber_de_19112x.png" alt=" ( v) ( g h) E1 = E2 1 + -- = E2 1 + --2 . c c " class="math-display" /></center></td><td width="5%">(1)</td></tr></table>
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239 <!--l. 263--><p class="nopar">
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240 </p><!--l. 266--><p class="indent"> Nach unserer Annahme gilt genau die gleiche Beziehung, <br/>falls derselbe Vorgang in dem nicht beschleunigten, aber mit <br/>Gravitationsfeld versehenen System <span
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241 class="cmmi-12">K </span>stattfindet. In diesem <br/>Falle können wir <span
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242 class="cmmi-12"><img
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243 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1911/fulltext/img/cmmi12-d.png" alt="g" class="12x-x-d" /></span><span
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244 class="cmsy-10x-x-120"><sup class="htf"><strong>.</strong></sup></span><span
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245 class="cmmi-12">h </span>ersetzen durch das Potential
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246 <img
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247 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1911/fulltext/img/cmr12-8.png" alt="P" class="12x-x-8" /> des <br/>Gravitationsvektors in <span
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248 class="cmmi-12">S</span><sub ><span
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249 class="cmr-8">2</span></sub>, wenn die willkürliche Konstante <br/>von <img
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250 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1911/fulltext/img/cmr12-8.png" alt="P" class="12x-x-8" /> in <span
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251 class="cmmi-12">S</span><sub ><span
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252 class="cmr-8">1</span></sub>
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253 gleich Null gesetzt wird. Es gilt also die <br/>Gleichung:</p>
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254 <table width="100%"
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255 class="equation"><tr><td><a
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256 id="x1-3r2"></a>
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257 <center class="math-display" >
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258 <img
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259 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1911/fulltext/img/Einst_Ueber_de_19113x.png" alt=" E2- E1 = E2 + c2 P . " class="math-display" /></center></td><td width="5%">(1 a)</td></tr></table>
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260 <!--l. 278--><p class="nopar">
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261 </p><!--l. 282--><p class="noindent">Diese Gleichung spricht den Energiesatz für den ins Auge <br/>gefaßten Vorgang aus. Die in <span
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262 class="cmmi-12">S</span><sub ><span
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263 class="cmr-8">1</span></sub> ankommende Energie <span
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264 class="cmmi-12">E</span><sub ><span
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265 class="cmr-8">1</span></sub> ist <br/>größer als die mit gleichen Mitteln gemessene Energie <span
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266 class="cmmi-12">E</span><sub ><span
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267 class="cmr-8">2</span></sub>, <br/>welche in <span
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268 class="cmmi-12">S</span><sub ><span
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269 class="cmr-8">2</span></sub> emittiert wurde, und zwar um die potentielle <br/>Energie der Masse <span
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270 class="cmmi-12">E</span><sub ><span
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271 class="cmr-8">2</span></sub><span
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272 class="cmmi-12">/c</span><sup ><span
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273 class="cmr-8">2</span></sup> im Schwerefelde. Es zeigt sich
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274 <br/>also, daß man, damit das Energieprinzip erfüllt sei, der <br/>Energie <span
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275 class="cmmi-12">E </span>vor ihrer Aussendung in <span
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276 class="cmmi-12">S</span><sub ><span
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277 class="cmr-8">2</span></sub> eine potentielle Energie <br/>der Schwere zuschreiben
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278 muß, die der (schweren) Masse <span
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279 class="cmmi-12">E/c</span><sup ><span
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280 class="cmr-8">2</span></sup> <br/>entspricht. Unsere Annahme der Äquivalenz von <span
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281 class="cmmi-12">K </span>und <span
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282 class="cmmi-12">K</span><span
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283 class="cmsy-10x-x-120">'</span> <br/>hebt also die am Anfang dieses Paragraphen dargelegte Schwierig-<br/>keit, welche die gewöhnliche Relativitätstheorie übrig läßt.
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284 </p><!--l. 297--><p class="indent"> Besonders deutlich zeigt sich der Sinn dieses Resultates <br/>bei Betrachtung des folgenden Kreisprozesses:
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285 </p><!--l. 300--><p class="indent"> 1. Man sendet die Energie <span
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286 class="cmmi-12">E </span>(in <span
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287 class="cmmi-12">S</span><sub ><span
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288 class="cmr-8">2</span></sub> gemessen) in Form <br/>von Strahlung in <span
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289 class="cmmi-12">S</span><sub ><span
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290 class="cmr-8">2</span></sub> ab nach <span
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291 class="cmmi-12">S</span><sub ><span
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292 class="cmr-8">1</span></sub>, wo nach dem soeben er-<br/>langten Resultat die Energie <span
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293 class="cmmi-12">E</span>(1 + <span
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294 class="cmmi-12"><img
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295 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1911/fulltext/img/cmmi12-d.png" alt="g" class="12x-x-d" /> h/c</span><sup ><span
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296 class="cmr-8">2</span></sup>) aufgenommen wird <br/>(in <span
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297 class="cmmi-12">S</span><sub ><span
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298 class="cmr-8">1</span></sub> gemessen).
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299 </p><!--l. 305--><p class="indent"> 2. Man senkt einen Körper <span
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300 class="cmmi-12">W </span>von der Masse <span
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301 class="cmmi-12">M </span>von <span
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302 class="cmmi-12">S</span><sub ><span
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303 class="cmr-8">2</span></sub> <br/>nach <span
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304 class="cmmi-12">S</span><sub ><span
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305 class="cmr-8">1</span></sub>, wobei die Arbeit <span
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306 class="cmmi-12">M <img
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307 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1911/fulltext/img/cmmi12-d.png" alt="g" class="12x-x-d" /> h </span>nach außen abgegeben wird.
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308 </p><!--l. 309--><p class="indent"> 3. Man überträgt die Energie <span
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309 class="cmmi-12">E </span>von <span
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310 class="cmmi-12">S</span><sub ><span
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311 class="cmr-8">1</span></sub> auf den Körper <span
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312 class="cmmi-12">W</span>, <br/>während sich <span
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313 class="cmmi-12">W </span>in
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314 <span
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315 class="cmmi-12">S</span><sub ><span
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316 class="cmr-8">1</span></sub> befindet. Dadurch ädere sich die <br/>schwere Masse <span
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317 class="cmmi-12">M</span>, so daß sie den Wert <span
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318 class="cmmi-12">M</span><span
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319 class="cmsy-10x-x-120">'</span>
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320 erhält.
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321 </p><!--l. 314--><p class="noindent">----------
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322 </p><!--l. 316--><p class="indent"> 1) A. Einstein, Ann. d. Phys. <span
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323 class="cmbx-12">17. </span>p. 913 u. 914. 1905. <pb/>
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324 </p><!--l. 321--><p class="indent">
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325
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326 </p><!--l. 322--><p class="indent"> 4. Man hebe <span
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327 class="cmmi-12">W </span>wieder nach <span
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328 class="cmmi-12">S</span><sub ><span
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329 class="cmr-8">2</span></sub>, wobei die Arbeit <span
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330 class="cmmi-12">M</span><span
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331 class="cmsy-10x-x-120">'</span><span
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332 class="cmmi-12"><img
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333 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1911/fulltext/img/cmmi12-d.png" alt="g" class="12x-x-d" /> h </span> <br/>aufzuwenden ist.
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334 </p><!--l. 325--><p class="indent"> 5. Man übertrage <span
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335 class="cmmi-12">E </span>von <span
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336 class="cmmi-12">W </span>wieder auf <span
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337 class="cmmi-12">S</span><sub ><span
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338 class="cmr-8">2</span></sub>.
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339 </p><!--l. 327--><p class="indent"> Der Effekt dieses Kreisprozesses besteht einzig darin, <br/>daß <span
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340 class="cmmi-12">S</span><sub ><span
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341 class="cmr-8">1</span></sub> den Energiezuwachs
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342 <span
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343 class="cmmi-12">E</span>(<span
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344 class="cmmi-12"><img
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345 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1911/fulltext/img/cmmi12-d.png" alt="g" class="12x-x-d" /> h/c</span><sup ><span
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346 class="cmr-8">2</span></sup>) erlitten hat, und daß <br/>dem System die Energiemenge
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347 </p>
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348 <center class="par-math-display" >
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349 <img
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350 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1911/fulltext/img/Einst_Ueber_de_19114x.png" alt=" ' M g h- M g h " class="par-math-display" /></center>
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351 <!--l. 334--><p class="nopar"></p><!--l. 338--><p class="noindent">in Form von mechanischer Arbeit zugeführt wurde. Nach <br/>dem Energieprinzip muß also
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352 </p>
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353 <center class="par-math-display" >
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354 <img
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355 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1911/fulltext/img/Einst_Ueber_de_19115x.png" alt=" g h E -2-= M 'g h- M g h c " class="par-math-display" /></center>
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356 <!--l. 344--><p class="nopar"></p><!--l. 348--><p class="noindent">oder</p>
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357 <table width="100%"
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358 class="equation"><tr><td><a
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359 id="x1-4r2"></a>
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360 <center class="math-display" >
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361 <img
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362 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1911/fulltext/img/Einst_Ueber_de_19116x.png" alt=" ' E- M - M = c2 " class="math-display" /></center></td><td width="5%">(1 b)</td></tr></table>
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363 <!--l. 354--><p class="nopar">
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364 </p><!--l. 358--><p class="noindent">sein. Der Zuwachs an <span
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365 class="cmti-12">schwerer </span>Masse ist also gleich <span
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366 class="cmmi-12">E/c</span><sup ><span
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367 class="cmr-8">2</span></sup>, <br/>also gleich dem aus der Relativitätstheorie sich ergebenden <br/>Zuwachs an <span
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368 class="cmti-12">tr</span><span
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369 class="cmti-12">äger </span>Masse.
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370 </p><!--l. 363--><p class="indent"> Noch unmittelbarer ergibt sich das Resultat aus der <br/>Äquivalenz der Systeme <span
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371 class="cmmi-12">K</span>
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372 und <span
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373 class="cmmi-12">K</span><span
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374 class="cmsy-10x-x-120">'</span>, nach welcher die <span
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375 class="cmti-12">schwere </span> <br/>Masse in bezug auf <span
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376 class="cmmi-12">K </span>der <span
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377 class="cmti-12">tr</span><span
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378 class="cmti-12">ägen </span>Masse in bezug auf <span
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379 class="cmmi-12">K</span><span
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380 class="cmsy-10x-x-120">' </span>voll-<br/>kommen gleich ist; es muß deshalb die Energie eine <span
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381 class="cmti-12">schwere</span>
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382 <br/>Masse besitzen, die ihrer <span
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383 class="cmti-12">tr</span><span
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384 class="cmti-12">ägen </span>Masse gleich ist. Hängt man <br/>im System <span
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385 class="cmmi-12">K</span><span
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386 class="cmsy-10x-x-120">' </span>eine Masse <span
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387 class="cmmi-12">M</span><sub ><span
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388 class="cmr-8">0</span></sub> an einer Federwaage auf, so <br/>wird letztere wegen der Trägheit von <span
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389 class="cmmi-12">M</span><sub ><span
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390 class="cmr-8">0</span></sub>
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391 das scheinbare Ge-<br/>wicht <span
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392 class="cmmi-12">M</span><sub ><span
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393 class="cmr-8">0</span></sub> <span
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394 class="cmmi-12"><img
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395 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1911/fulltext/img/cmmi12-d.png" alt="g" class="12x-x-d" /> </span>anzeigen. Überträgt man die Energiemenge <span
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396 class="cmmi-12">E</span>
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397 <br/>auf <span
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398 class="cmmi-12">M</span><sub ><span
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399 class="cmr-8">0</span></sub>, so wird die Federwaage nach dem Satz von der Träg-<br/>heit der Energie <img
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400 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1911/fulltext/img/Einst_Ueber_de_19117x.png" alt="( ) M0 + -E c2" class="left" align="middle" /><span
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401 class="cmmi-12"><img
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402 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1911/fulltext/img/cmmi12-d.png" alt="g" class="12x-x-d" /> </span>anzeigen. Nach unserer Grund-<br/>annahme muß ganz
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403 dasselbe eintreten bei Wiederholung des <br/>Versuches im System <span
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404 class="cmmi-12">K</span>, d. h. im Gravitationsfelde.
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405 </p>
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406 <div class="center" >
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407
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408 <!--l. 382--><p class="noindent">
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409 </p><!--l. 383--><p class="noindent"><span
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410 class="cmsy-10x-x-120">§ </span>3. Zeit und Lichtgeschwindigkeit im Schwerefelde.</p></div>
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411 <!--l. 387--><p class="indent"> Wenn die im gleichförmig beschleunigten System <span
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412 class="cmmi-12">K</span><span
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413 class="cmsy-10x-x-120">' </span>in <br/><span
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414 class="cmmi-12">S</span><sub ><span
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415 class="cmr-8">2</span></sub> gegen <span
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416 class="cmmi-12">S</span><sub ><span
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417 class="cmr-8">1</span></sub>
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418 emittierte Strahlung mit Bezug auf die in <span
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419 class="cmmi-12">S</span><sub ><span
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420 class="cmr-8">2</span></sub> be-<br/>findliche Uhr die Frequenz <span
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421 class="cmmi-12"><img
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422 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1911/fulltext/img/cmmi12-17.png" alt="n" class="12x-x-17" /></span><sub ><span
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423 class="cmr-8">2</span></sub>
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424 besaß, so besitzt sie in bezug <br/>auf <span
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425 class="cmmi-12">S</span><sub ><span
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426 class="cmr-8">1</span></sub> bei ihrer Ankunft in <span
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427 class="cmmi-12">S</span><sub ><span
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428 class="cmr-8">1</span></sub> in bezug auf die in <span
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429 class="cmmi-12">S</span><sub ><span
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430 class="cmr-8">1</span></sub> befindliche <br/>gleich beschaffene Uhr nicht mehr die Frequenz
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431 <span
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432 class="cmmi-12"><img
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433 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1911/fulltext/img/cmmi12-17.png" alt="n" class="12x-x-17" /></span><sub ><span
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434 class="cmr-8">2</span></sub> sondern <br/>eine größere Frequenz <span
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435 class="cmmi-12"><img
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436 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1911/fulltext/img/cmmi12-17.png" alt="n" class="12x-x-17" /></span><sub ><span
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437 class="cmr-8">1</span></sub>, derart, daß in erster Annäherung
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438 <br/></p>
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439 <table width="100%"
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440 class="equation"><tr><td><a
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441 id="x1-5r2"></a>
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442 <center class="math-display" >
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443 <img
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444 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1911/fulltext/img/Einst_Ueber_de_19118x.png" alt=" ( ) g-h n1 = n2 1 + c2 . " class="math-display" /></center></td><td width="5%">(2)</td></tr></table>
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445 <!--l. 401--><p class="nopar">
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446 <pb/>
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447 </p><!--l. 408--><p class="indent">
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448
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449 </p><!--l. 409--><p class="noindent">Führt man nämlich wieder das beschleunigungsfreie Bezugs-<br/>system <span
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450 class="cmmi-12">K</span><sub ><span
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451 class="cmr-8">0</span></sub> ein, relativ zu welchem <span
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452 class="cmmi-12">K</span><span
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453 class="cmsy-10x-x-120">' </span>zur Zeit der Lichtaus-<br/>sendung keine Geschwindigkeit besitzt, so hat <span
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454 class="cmmi-12">S</span><sub ><span
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455 class="cmr-8">1</span></sub> in bezug auf <span
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456 class="cmmi-12">K</span><sub ><span
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457 class="cmr-8">0</span></sub> <br/>zur Zeit der Ankunft der Strahlung in <span
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458 class="cmmi-12">S</span><sub ><span
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459 class="cmr-8">1</span></sub> die Geschwindigkeit <br/><span
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460 class="cmmi-12"><img
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461 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1911/fulltext/img/cmmi12-d.png" alt="g" class="12x-x-d" /></span>(<span
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462 class="cmmi-12">h/c</span>), woraus sich die angegebene Beziehung vermöge des
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463 <br/>Dopplerschen Prinzipes unmittelbar ergibt.
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464 </p><!--l. 418--><p class="indent"> Nach unserer Voraussetzung von der Äquivalenz der <br/>Systeme <span
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465 class="cmmi-12">K</span><span
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466 class="cmsy-10x-x-120">' </span>und
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467 <span
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468 class="cmmi-12">K </span>gilt diese Gleichung auch für das ruhende, <br/>mit einem gleichförmigen Schwerefeld versehene Koordinaten-<br/>system <span
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469 class="cmmi-12">K</span>, falls in diesem die geschilderte Strahlungsüber-<br/>tragung stattfindet. Es ergibt sich also, daß ein bei be-<br/>stimmtem Schwerepotential in <span
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470 class="cmmi-12">S</span><sub ><span
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471 class="cmr-8">2</span></sub> emittierter Lichtstrahl, der <br/>bei seiner Emission -- mit einer in <span
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472 class="cmmi-12">S</span><sub ><span
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473 class="cmr-8">2</span></sub> befindlichen Uhr ver-<br/>glichen -- die Frequenz <span
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474 class="cmmi-12"><img
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475 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1911/fulltext/img/cmmi12-17.png" alt="n" class="12x-x-17" /></span><sub ><span
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476 class="cmr-8">2</span></sub> besitzt, bei seiner Ankunft in <span
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477 class="cmmi-12">S</span><sub ><span
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478 class="cmr-8">1</span></sub> <br/>eine andere Frequenz <span
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479 class="cmmi-12"><img
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480 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1911/fulltext/img/cmmi12-17.png" alt="n" class="12x-x-17" /></span><sub ><span
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481 class="cmr-8">1</span></sub> esitzt, falls letztere mittels einer <br/>in <span
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482 class="cmmi-12">S</span><sub ><span
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483 class="cmr-8">1</span></sub>
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484 befindlichen gleich beschaffenen Uhr gemessen wird. <br/>Wir ersetzen <span
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485 class="cmmi-12"><img
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486 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1911/fulltext/img/cmmi12-d.png" alt="g" class="12x-x-d" /> h </span>durch das Schwerepotential <img
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487 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1911/fulltext/img/cmr12-8.png" alt="P" class="12x-x-8" /> von <span
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488 class="cmmi-12">S</span><sub ><span
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489 class="cmr-8">2</span></sub> in <br/>bezug auf <span
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490 class="cmmi-12">S</span><sub ><span
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491 class="cmr-8">1</span></sub> als Nullpunkt und nehmen an, daß unsere für <br/>das <span
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492 class="cmti-12">homogene </span>Gravitationsfeld abgeleitete Beziehung auch für <br/>anders gestaltete Felder gelte; es ist dann</p>
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493 <table width="100%"
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494 class="equation"><tr><td><a
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495 id="x1-6r3"></a>
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496 <center class="math-display" >
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497 <img
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498 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1911/fulltext/img/Einst_Ueber_de_19119x.png" alt=" ( ) P- n1 = n2 1 + c2 . " class="math-display" /></center></td><td width="5%">(2 a)</td></tr></table>
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499 <!--l. 441--><p class="nopar">
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500 </p><!--l. 445--><p class="noindent">Dies (nach unserer Ableitung in erster Näherung gültige) Resul-<br/>tat gestattet zunächst folgende Anwendung. Es sei <span
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501 class="cmmi-12"><img
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502 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1911/fulltext/img/cmmi12-17.png" alt="n" class="12x-x-17" /></span><sub ><span
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503 class="cmr-8">0</span></sub> die <br/>Schwingungszahl eines elementaren Lichterzeugers, gemessen <br/>mit einer an demselben Orte gemessenen Uhr <span
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504 class="cmmi-12">U</span>. Diese
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505 <br/>Schwingungszahl ist dann unabhängig davon, wo der Licht-<br/>erzeuger samt der Uhr aufgestellt wird. Wir wollen uns beide <br/>etwa an der Sonnenoberfläche angeordnet denken (dort befindet <br/>sich unser <span
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506 class="cmmi-12">S</span><sub ><span
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507 class="cmr-8">2</span></sub>). Von dem dort emittierten Lichte gelangt ein
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508 <br/>Teil zur Erde (<span
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509 class="cmmi-12">S</span><sub ><span
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510 class="cmr-8">1</span></sub>), wo wir mit einer Uhr <span
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511 class="cmmi-12">U </span>von genau gleicher <br/>Beschaffenheit als der soeben genannten die Frequenz <span
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512 class="cmmi-12"><img
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513 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1911/fulltext/img/cmmi12-17.png" alt="n" class="12x-x-17" /> </span>des <br/>ankommenden Lichtes messen Dann ist nach (2a)
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514 </p>
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515 <center class="par-math-display" >
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516 <img
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517 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1911/fulltext/img/Einst_Ueber_de_191110x.png" alt=" ( ) n = n0 1 + P- , c2 " class="par-math-display" /></center>
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518 <!--l. 463--><p class="nopar">
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519 </p><!--l. 467--><p class="noindent">wobei <img
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520 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1911/fulltext/img/cmr12-8.png" alt="P" class="12x-x-8" /> die (negative) Gravitationspotentialdifferenz zwischen <br/>Sonnenoberfläche und Erde bedeutet. Nach unserer Auffassung <br/><pb/>
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521 </p><!--l. 473--><p class="indent">
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522
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523 </p><!--l. 474--><p class="noindent">müssen also die Spektrallinien des Sonnenlichtes gegenüber <br/>den entsprechenden Spektrallinien irdischer Lichtquellen etwas <br/>nach dem Rot verschoben sein, und zwar um den relativen <br/>Betrag
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524 </p>
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525 <center class="par-math-display" >
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526 <img
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527 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1911/fulltext/img/Einst_Ueber_de_191111x.png" alt="n0---n-= --P- = 2 .10-6 . n0 c2 " class="par-math-display" /></center>
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528 <!--l. 483--><p class="nopar">
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529 </p><!--l. 487--><p class="noindent">Wenn die Bedingungen, unter welchen die Sonnenlinien ent-<br/>stehen, genau bekannt wären, wäre diese Verschiebung noch <br/>der Messung zugänglich. Da aber anderweitige Einflüsse <br/>(Druck, Temperatur) die Lage des Schwerpunktes der Spektral-<br/>linien beeinflussen, ist es schwer zu konstatieren, ob der hier <br/>abgeleitete Einfluß des Gravitationspotentials wirklich existiert.<sup ><span
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530 class="cmr-8">1</span></sup>)
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531 </p><!--l. 495--><p class="indent"> Bei oberflächlicher Betrachtung scheint Gleichung (2) <br/>bzw. (2a) eine Absurdität auszusagen. Wie kann bei bestän-<br/>diger Lichtübertragung von <span
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532 class="cmmi-12">S</span><sub ><span
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533 class="cmr-8">2</span></sub> nach
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534 <span
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535 class="cmmi-12">S</span><sub ><span
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536 class="cmr-8">1</span></sub> in <span
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537 class="cmmi-12">S</span><sub ><span
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538 class="cmr-8">1</span></sub> eine andere An-<br/>zahl von Perioden pro Sekunde ankommen, als in <span
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539 class="cmmi-12">S</span><sub ><span
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540 class="cmr-8">2</span></sub>
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541 emittiert <br/>wird? Die Antwort ist aber einfach. Wir können <span
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542 class="cmmi-12"><img
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543 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1911/fulltext/img/cmmi12-17.png" alt="n" class="12x-x-17" /></span><sub ><span
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544 class="cmr-8">2</span></sub> bzw. <span
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545 class="cmmi-12"><img
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546 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1911/fulltext/img/cmmi12-17.png" alt="n" class="12x-x-17" /></span><sub ><span
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547 class="cmr-8">1</span></sub> <br/>nicht als Frequenzen schlechthin (als Anzahl Perioden pro <br/>Sekunde) ansehen, da wir eine Zeit im System <span
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548 class="cmmi-12">K </span>noch nicht <br/>festgelegt haben. <span
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549 class="cmmi-12"><img
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550 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1911/fulltext/img/cmmi12-17.png" alt="n" class="12x-x-17" /></span><sub ><span
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551 class="cmr-8">2</span></sub> bedeutet die Anzahl Perioden, bezogen <br/>auf die Zeiteinheit der Uhr <span
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552 class="cmmi-12">U </span>in <span
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553 class="cmmi-12">S</span><sub ><span
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554 class="cmr-8">2</span></sub><span
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555 class="cmmi-12">, <img
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556 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1911/fulltext/img/cmmi12-17.png" alt="n" class="12x-x-17" /></span><sub ><span
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557 class="cmr-8">1</span></sub> die Anzahl Perioden, <br/>bezogen auf die Zeiteinheit der gleich beschaffenen Uhr <span
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558 class="cmmi-12">U</span>
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559 <br/>in <span
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560 class="cmmi-12">S</span><sub ><span
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561 class="cmr-8">1</span></sub>. Nichts zwingt uns zu der Annahme, daß die in ver-<br/>schiedenen Gravitationspotentialen befindlichen Uhren <span
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562 class="cmmi-12">U </span>als <br/>gleich rasch gehend aufgefaßt werden müssen. Dagegen müssen <br/>wir die Zeit in <span
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563 class="cmmi-12">K </span>sicher so definieren, daß die Anzahl der <br/>Wellenberge und Wellentäler, die sich zwischen <span
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564 class="cmmi-12">S</span><sub ><span
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565 class="cmr-8">2</span></sub> und <span
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566 class="cmmi-12">S</span><sub ><span
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567 class="cmr-8">1</span></sub> be-<br/>finden, von dem Absolutwerte der Zeit unabhängig ist; denn <br/>der ins Auge gefaßte Prozeß ist seiner Natur nach ein statio-<br/>närer. Würden wir diese Bedingung nicht erfüllen, so kämen <br/>wir zu einer Zeitdefinition, bei deren Anwendung die Zeit <br/>explizite in die Naturgesetze einginge, was sicher unnatürlich <br/>und unzweckmäßig wäre. Die Uhren in <span
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568 class="cmmi-12">S</span><sub ><span
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569 class="cmr-8">1</span></sub> und <span
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570 class="cmmi-12">S</span><sub ><span
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571 class="cmr-8">2</span></sub> geben also
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572 <br/>----------
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573 </p><!--l. 525--><p class="indent"> 1) L. F. Jewell (Journ. de phys. <span
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574 class="cmbx-12">6. </span>p. 84. 1897) und insbesondere <br/>Ch. Fabry u. H. Boisson (Compt. rend. <span
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575 class="cmbx-12">148. </span>p. 688--690. 1909) haben <br/>derartige Verschiebungen feiner Spektrallinien nach dem roten Ende des <br/>Spektrums von der hier berechneten Größenordnung tatsächlich kon-<br/>statiert, aber einer Wirkung des Druckes in der absorbierenden Schicht <br/>zugeschrichen.
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576 <pb/>
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577 </p><!--l. 537--><p class="indent">
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578
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579 </p><!--l. 538--><p class="noindent">nicht beide die ,,Zeit&#8220; richtig an. Messen wir die Zeit in <span
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580 class="cmmi-12">S</span><sub ><span
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581 class="cmr-8">1</span></sub> <br/>mit der Uhr <span
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582 class="cmti-12">U, so</span>
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583 <span
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584 class="cmti-12">m</span><span
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585 class="cmti-12">üssen wir die Zeit in S</span><sub ><span
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586 class="cmr-8">2</span></sub> <span
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587 class="cmti-12">mit einer Uhr </span> <br/><span
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588 class="cmti-12">messen, die </span>1 + <img
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589 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1911/fulltext/img/cmr12-8.png" alt="P" class="12x-x-8" /><span
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590 class="cmmi-12">/c</span><sup ><span
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591 class="cmr-8">2</span></sup> <span
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592 class="cmti-12">mal langsamer l</span><span
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593 class="cmti-12">äuft</span>
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594 <span
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595 class="cmti-12">als die Uhr U, falls </span> <br/><span
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596 class="cmti-12">sie mit der Uhr U an derselben Stelle verglichen wird</span>. Denn
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597 <br/>mit einer solchen Uhr gemessen ist die Frequenz des oben <br/>betrachteten Lichtstrahles bei seiner Aussendung in <span
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598 class="cmmi-12">S</span><sub ><span
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599 class="cmr-8">2</span></sub>
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600 </p>
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601 <center class="par-math-display" >
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602 <img
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603 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1911/fulltext/img/Einst_Ueber_de_191112x.png" alt=" ( ) n2 1 + P- , c2 " class="par-math-display" /></center>
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604 <!--l. 550--><p class="nopar">
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605 </p><!--l. 554--><p class="noindent">also nach (2a) gleich der Frequenz <span
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606 class="cmmi-12"><img
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607 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1911/fulltext/img/cmmi12-17.png" alt="n" class="12x-x-17" /></span><sub ><span
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608 class="cmr-8">1</span></sub> desselben Lichtstrahles <br/>bei dessen Ankunft in <span
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609 class="cmmi-12">S</span><sub ><span
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610 class="cmr-8">1</span></sub>.
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611 </p><!--l. 558--><p class="indent"> Hieraus ergibt sich eine Konsequenz von für diese Theorie <br/>fundamentaler Bedeutung. Mißt man nämlich in dem be-<br/>schleunigten, gravitationsfeldfreien System <span
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612 class="cmmi-12">K</span><span
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613 class="cmsy-10x-x-120">' </span>an verschiedenen <br/>Orten die Lichtgeschwindigkeit unter Benutzung gleich be-<br/>schaffener Uhren <span
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614 class="cmmi-12">U</span>, so erhält man überall dieselbe Größe. <br/>Dasselbe gilt nach unserer Grundannahme auch für das <br/>System <span
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615 class="cmmi-12">K</span>. Nach dem soeben Gesagten müssen wir aber an <br/>Stellen verschiedenen Gravitationspotentials uns verschieden <br/>beschaffener Uhren zur Zeitmessung bedienen. Wir müssen <br/>zur Zeitmessung an einem Orte, der relativ zum Koordinaten-<br/>ursprung das Gravitationspotential <img
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616 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1911/fulltext/img/cmr12-8.png" alt="P" class="12x-x-8" /> besitzt, eine Uhr ver-<br/>wenden, die -- an den Koordinatenursprung versetzt --<br/>(1 + <img
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617 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1911/fulltext/img/cmr12-8.png" alt="P" class="12x-x-8" /><span
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618 class="cmmi-12">/c</span><sup ><span
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619 class="cmr-8">2</span></sup>) mal langsamer läuft als jene Uhr, mit welcher am <br/>Koordinatenursprung die Zeit gemessen wird. Nennen wir <span
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620 class="cmmi-12">c</span><sub ><span
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621 class="cmr-8">0</span></sub> <br/>die Lichtgeschwindigkeit im Koordinatenanfangspunkt, so wird <br/>daher die Lichtgeschwindigkeit <span
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622 class="cmmi-12">c </span>in einem Orte vom Gravi-<br/>tationspotential <img
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623 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1911/fulltext/img/cmr12-8.png" alt="P" class="12x-x-8" /> durch die Beziehung</p>
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624 <table width="100%"
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625 class="equation"><tr><td><a
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626 id="x1-7r3"></a>
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627 <center class="math-display" >
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628 <img
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629 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1911/fulltext/img/Einst_Ueber_de_191113x.png" alt=" ( P ) c = c0 1 + -- c2 " class="math-display" /></center></td><td width="5%">(3)</td></tr></table>
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630 <!--l. 584--><p class="nopar">
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631 </p><!--l. 588--><p class="noindent">gegeben sein. Das Prinzip von der Konstanz der Licht-<br/>geschwindigkeit gilt nach dieser Theorie nicht in derjenigen <br/>Fassung, wie es der gewöhnlichen
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632 Relativitätstheorie zugrunde <br/>gelegt zu werden pflegt.
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633 </p>
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634 <div class="center" >
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635
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636 <!--l. 595--><p class="noindent">
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637 </p><!--l. 596--><p class="noindent"><span
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638 class="cmsy-10x-x-120">§ </span>4. Krümmung der Lichtstrahlen im Gravitationsfeld.</p></div>
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639 <!--l. 600--><p class="indent"> Aus dem soeben bewiesenen Satze, daß die Lichtgeschwin-<br/>digkeit im Schwerefelde eine Funktion des Ortes ist, läßt sich <br/>leicht mittels des Huygensschen Prinzipes schließen, daß quer <br/><pb/>
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640 </p><!--l. 607--><p class="indent">
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641
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642 </p><!--l. 608--><p class="noindent">zu einem Schwerefeld sich fortpflanzende Lichtstrahlen eine <br/>Krümmung erfahren müssen. Sei nämlich <span
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643 class="cmmi-12"><img
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644 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1911/fulltext/img/cmmi12-22.png" alt="e" class="12x-x-22" /> </span>eine Ebene gleicher <br/>Phase einer ebenen Lichtwelle zur Zeit
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645 <span
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646 class="cmmi-12">t</span>, <span
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647 class="cmmi-12">P</span><sub ><span
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648 class="cmr-8">1</span></sub> und <span
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649 class="cmmi-12">P</span><sub ><span
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650 class="cmr-8">2</span></sub> zwei <br/>Punkte in ihr, welche den Abstand 1 besitzen. <span
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651 class="cmmi-12">P</span><sub ><span
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652 class="cmr-8">1</span></sub> und <span
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653 class="cmmi-12">P</span><sub ><span
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654 class="cmr-8">2</span></sub> <br/>liegen in der Papierebene, die so gewählt ist, daß der in der <br/>Richtung ihrer Normale genommene Differentialquotient von <img
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655 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1911/fulltext/img/cmr12-8.png" alt="P" class="12x-x-8" /> <br/>also auch von <span
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656 class="cmmi-12">c </span>verschwindet. Die entsprechende Ebene <br/>gleicher Phase bzw. deren Schnitt mit der Papierebene, zur <br/>Zeit <span
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657 class="cmmi-12">t </span>+ <span
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658 class="cmmi-12">dt </span>erhalten wir, indem wir um die Punkte <span
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659 class="cmmi-12">P</span><sub ><span
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660 class="cmr-8">1</span></sub> und <span
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661 class="cmmi-12">P</span><sub ><span
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662 class="cmr-8">2</span></sub> <br/>mit den Radien <span
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663 class="cmmi-12">c</span><sub ><span
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664 class="cmr-8">1</span></sub> <span
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665 class="cmmi-12">dt </span>bzw. <span
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666 class="cmmi-12">c</span><sub ><span
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667 class="cmr-8">2</span></sub> <span
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668 class="cmmi-12">dt </span>Kreise und an diese die <br/>Tangente legen, wobei <span
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669 class="cmmi-12">c</span><sub ><span
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670 class="cmr-8">1</span></sub> bzw. <span
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671 class="cmmi-12">c</span><sub ><span
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672 class="cmr-8">2</span></sub> die Lichtgeschwindigkeit in <br/>den Punkten <span
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673 class="cmmi-12">P</span><sub ><span
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674 class="cmr-8">1</span></sub> bzw. <span
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675 class="cmmi-12">P</span><sub ><span
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676 class="cmr-8">2</span></sub>
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677 bedeutet. Der Krümmungswinkel <br/>des Lichtstrahles auf dem Wege <span
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678 class="cmti-12">c dt </span>ist also
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679 </p>
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680 <center class="par-math-display" >
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681 <img
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682 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1911/fulltext/img/Einst_Ueber_de_191114x.png" alt="(c1---c2)dt- -@-c 1 = - @ n'd t, " class="par-math-display" /></center>
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683 <!--l. 630--><p class="nopar">
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684 </p><!--l. 634--><p class="noindent">falls wir den Krümmungswinkel positiv rechnen, wenn der <br/>Lichtstrahl nach der Seite der wachsenden <span
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685 class="cmmi-12">n</span><span
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686 class="cmsy-10x-x-120">' </span>hin gekrümmt <img
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687 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1911/fulltext/img/Einst_Ueber_de_191115x.png" alt="PIC" class="graphics" width="250.38434pt" height="87.25682pt" /><!--tex4ht:graphics
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688 name="img/Einst_Ueber_de_191115x.png" src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1911/fulltext/Einst_Ueber_de_1911_002.EPS"
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689 -->
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690 </p><!--l. 644--><p class="noindent">wird. Der Krümmungswinkel pro Wegeinheit des Lichtstrahles <br/>ist also</p>
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691 <center class="par-math-display" >
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692 <img
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693 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1911/fulltext/img/Einst_Ueber_de_191116x.png" alt=" 1 @ c - -----' c @ n " class="par-math-display" /></center>
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694 <!--l. 650--><p class="nopar">
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695 </p><!--l. 654--><p class="noindent">oder nach (3) gleich</p>
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696 <center class="par-math-display" >
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697 <img
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698 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1911/fulltext/img/Einst_Ueber_de_191117x.png" alt="- -1 @-P- . c2 @ n' " class="par-math-display" /></center>
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699 <!--l. 661--><p class="nopar">
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700 </p><!--l. 665--><p class="noindent">Endlich erhalten wir für die Ablenkung <span
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701 class="cmmi-12"><img
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702 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1911/fulltext/img/cmmi12-b.png" alt="a" class="12x-x-b" /></span>, welche ein Licht-<br/>strahl auf einem beliebigen Wege (<span
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703 class="cmmi-12">s</span>) nach der Seite <span
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704 class="cmmi-12">n</span><span
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705 class="cmsy-10x-x-120">' </span>er-<br/>leidet, den Ausdruck</p>
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706 <table width="100%"
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707 class="equation"><tr><td><a
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708 id="x1-8r4"></a>
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709
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710 <center class="math-display" >
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711 <img
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712 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1911/fulltext/img/Einst_Ueber_de_191118x.png" alt=" integral
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713 a = - 1- @-P-d s. c2 @ n' " class="math-display" /></center></td><td width="5%">(4)</td></tr></table>
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714 <!--l. 674--><p class="nopar">
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715 </p><!--l. 678--><p class="noindent">Dasselbe Resultat hätten wir erhalten können durch unmittel-<br/>bare Betrachtung der Fortpflanzung eines Lichtstrahles in <br/>dem gleichförmig beschleunigten System
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716 <span
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717 class="cmmi-12">K</span><span
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718 class="cmsy-10x-x-120">' </span>und Übertragung <br/>des Resultates auf das System <span
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719 class="cmmi-12">K </span>und von hier auf den Fall,
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720 <br/>daß das Gravitationsfeld beliebig gestaltet ist. <pb/>
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721 </p><!--l. 688--><p class="indent">
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722
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723 </p><!--l. 689--><p class="indent"> Nach Gleichung (4) erleidet ein an einem Himmelskörper <br/>vorbeigehender Lichtstrahl eine Ablenkung nach der Seite <br/>sinkenden Gravitationspotentials, also nach der dem Himmels-<br/>körper zugewandten Seite von der Größe
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724 </p>
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725 <center class="par-math-display" >
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726 <img
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727 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1911/fulltext/img/Einst_Ueber_de_191119x.png" alt=" p h= integral + 2 a = 1- k-M--cos h .d s = 2-kM-, c2 r2 c2D h=- p2 " class="par-math-display" /></center>
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728 <!--l. 700--><p class="nopar">
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729 wobei <span
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730 class="cmmi-12">k </span>die Gravitationskonstante, <span
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731 class="cmmi-12">M </span>die Masse des Himmels-<br/>körpers,
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732 <img
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733 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1911/fulltext/img/cmr12-1.png" alt="D" class="12x-x-1" /> den Abstand des Lichtstrahles vom Mittelpunkt <br/>des Himmelskörpers bedeutet.
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734 <span
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735 class="cmti-12">Ein an der Sonne vorbeigehender </span> <br/><span
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736 class="cmti-12">Lichtstrahl erlitte demnach eine Ablenkung</span>
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737 <span
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738 class="cmti-12">vom Betrage</span> 4 <span
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739 class="cmsy-10x-x-120"><sup class="htf"><strong>.</strong></sup> </span>10<sup ><span
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740 class="cmsy-8">-</span><span
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741 class="cmr-8">6</span></sup> <br/>= 0<span
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742 class="cmmi-12">, </span>83 <span
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743 class="cmti-12">Bogensekunden</span>. Um diesen Betrag er-<br/>scheint
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744 <img
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745 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1911/fulltext/img/Einst_Ueber_de_191120x.png" alt="PIC" class="graphics" width="73.97733pt" height="146.8141pt" /><!--tex4ht:graphics
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746 name="img/Einst_Ueber_de_191120x.png" src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1911/fulltext/Einst_Ueber_de_1911_003.EPS"
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747 --> die Winkeldistanz des Sternes vom Sonnen-<br/>mittelpunkt durch die Krümmung des Strahles <br/>vergrößert. Da die Fixsterne der der Sonne <br/>zugewandten Himmelspartien bei totalen Sonnen-<br/>finsternissen sichtbar werden, ist diese Kon-<br/>sequenz der Theorie mit der Erfahrung ver-<br/>gleichbar. Beim Planeten Jupiter erreicht die <br/>zu erwartende Verschiebung etwa <span
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748 class="cmr-8">1</span> <span
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749 class="cmmi-12">/</span><span
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750 class="cmr-8">100</span> des an-<br/>gegebenen Betrages. Es wäre dringend zu <br/>wünschen, daß sich Astronomen der hier auf-<br/>gerollten Frage annähmen, auch wenn die im vorigen ge-<br/>gebenen Überlegungen ungenügend fundiert oder gar aben-<br/>teuerlich erscheinen sollten. Denn abgesehen von jeder Theorie <br/>muß man sich fragen, ob mit den heutigen Mitteln ein Einfluß
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751 <br/>der Gravitationsfelder auf die Ausbreitung des Lichtes sich <br/>konstatieren läßt.
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752 </p><!--l. 734--><p class="indent"> Prag, Juni 1911.
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753 </p>
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754 <div class="center" >
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755
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756 <!--l. 737--><p class="noindent">
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757 </p><!--l. 738--><p class="noindent">(Eingegangen 21. Juni 1914.)</p></div>
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758 <div class="center" >
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759
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760 <!--l. 742--><p class="noindent">
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761 </p><!--l. 743--><p class="noindent">----------</p></div>
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762
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763 </body></html>
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764
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765
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766