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Version vom 2009-02-14
author Klaus Thoden <kthoden@mpiwg-berlin.mpg.de>
date Thu, 02 May 2013 11:08:12 +0200
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1 <?xml version="1.0" encoding="utf-8" ?>
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5 <html>
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7 <head><title></title>
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9 <meta name="generator" content="TeX4ht (http://www.cse.ohio-state.edu/~gurari/TeX4ht/mn.html)" />
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10 <meta name="originator" content="TeX4ht (http://www.cse.ohio-state.edu/~gurari/TeX4ht/mn.html)" />
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12 <meta name="src" content="Einst_Bemer_de_1909_01.tex" />
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13 <meta name="date" content="2005-03-01 19:48:00" />
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14 <link rel="stylesheet" type="text/css" href="Einst_Bemer_de_1909_01.css" />
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15 </head><body >
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17 <!--l. 11--><p class="noindent"><pb/></p>
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18 <div class="center" >
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20 <!--l. 12--><p class="noindent">
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21 </p><!--l. 13--><p class="noindent"><span
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22 class="cmr-12">5. </span><span
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23 class="cmbxti-10x-x-120">Bemerkung zu der Arbeit von D. Mirimanoff </span> <br/><span
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24 class="cmbxti-10x-x-120">,,</span><span
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25 class="cmbxti-10x-x-120">Über</span>
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26 <span
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27 class="cmbxti-10x-x-120">die Grundgleichungen...&#8220;; </span> <br/><span
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28 class="cmbxti-10x-x-120">von A. Einstein.</span></p></div>
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29 <div class="center" >
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31 <!--l. 18--><p class="noindent">
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32 </p><!--l. 19--><p class="noindent">--------</p></div>
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33 <!--l. 22--><p class="indent"> 1. Das in dieser Arbeit<sup ><span
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34 class="cmr-7">1</span></sup>) angegebene System von Diffe-<br/>rentialgleichungen und Transformationsgleichungen unter-<br/>scheidet sich von dem Minkowskis in keiner Weise bzw. <br/><span
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35 class="cmti-10">nur </span>dadurch, daß derjenige Vektor, welcher gewöhnlich mit H <br/>bezeichnet wird (magnetische Kraft), vom Verfasser mit
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36 </p>
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37 <center class="par-math-display" >
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38 <img
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39 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Bemer_de_1909_01/fulltext/img/Einst_Bemer_de_1909_010x.png" alt=" 1 Q = H - c[Pw] " class="par-math-display" /></center>
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40 <!--l. 32--><p class="nopar">
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41 </p><!--l. 36--><p class="noindent">bezeichnet wurde.
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42 </p><!--l. 39--><p class="indent"> Differentialgleichung (I) ist nämlich bei Einführung von Q, <br/>wie der Verfasser selbst zeigt, identisch mit der betreffenden <br/>Gleichung Minkowskis, während die übrigen drei Differential-<br/>gleichungen H nicht enthalten und bereits die Form der ent-<br/>sprechenden Gleichungen Minkowskis haben. Der Verfasser <br/>sagt auch selbst, daß sich seine Vektoren G, D, Q, B trans-<br/>formieren, wie die gewöhnlich mit G, D, H, B bezeichneten <br/>Vektoren.
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43 </p><!--l. 49--><p class="indent"> 2. Auch die Beziehungen zwischen den Vektoren, welche <br/>Materialkonstanten (<span
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44 class="cmmi-10"><img
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45 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Bemer_de_1909_01/fulltext/img/cmmi10-22.png" alt="e" class="10x-x-22" /></span>, <span
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46 class="cmmi-10"><img
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47 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Bemer_de_1909_01/fulltext/img/cmmi10-16.png" alt="m" class="cmmi-10x-x-16" align="middle" /> </span>und <span
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48 class="cmmi-10"><img
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49 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Bemer_de_1909_01/fulltext/img/cmmi10-1b.png" alt="s" class="10x-x-1b" /></span>) enthalten, unterscheiden sich <br/>nicht von den entsprechenden Minkowskis. Der Verfasser geht <br/>nämlich davon aus, daß für ein relativ zu dem betrachteten <br/>Systempunkt momentan ruhendes Koordinatensystem die Glei-<br/>chungen
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50 </p>
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51 <center class="par-math-display" >
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52 <img
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53 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Bemer_de_1909_01/fulltext/img/Einst_Bemer_de_1909_011x.png" alt=" 1 D = eG, H = --B, J = sG m " class="par-math-display" /></center>
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54 <!--l. 61--><p class="nopar">
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55 </p><!--l. 65--><p class="noindent">gelten sollen; bedenkt man nun, daß der Vektor H (des Ver-<br/>fassers) für <span
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56 class="cmmi-10">w </span>= 0 mit dem Vektor Q identisch ist, und daß Q <br/>in den Differentialgleichungen des Verfassers und in dessen <br/>Transformationsgleichungen genau dieselbe Rolle spielt, wie m <br/>in Minkowskis Gleichungen (gewöhnlich mit H bezeichnet),
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57 <br/>
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58 </p><!--l. 73--><p class="noindent">----------
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59 </p><!--l. 76--><p class="indent"> 1) D. Mirimanoff, Ann. d. Phys. <span
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60 class="cmbx-10">28. </span>p. 192. 1909. <pb/>
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61 </p><!--l. 82--><p class="indent">
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62
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63 </p><!--l. 83--><p class="noindent">so ersieht man, daß auch diese Gleichungen mit den ent-<br/>sprechenden Minkowskis übereinstimmen, bis auf den Um-<br/>stand, daß die Bezeichnung H durch die Bezeichnung Q er-<br/>setzt ist.
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64 </p><!--l. 88--><p class="indent"> 3. Es ist also gezeigt, daß die Größe Q Mirimanoffs in <br/>dessen sämtlichen Gleichungen dieselbe Rolle spielt wie die-<br/>jenige Größe, welche man gewöhnlich mit H bezeichnet und <br/>,,magnetische Kraft&#8220; oder ,,magnetische Feldstärke&#8220; nennt.
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65 <br/>Trotzdem hätten die Gleichungen Mirimanoffs einen anderen <br/>Inhalt als die Gleichungen Minkowskis, wenn die Größe Q <br/>Mirimanoffs definitionsgemäß eine andere physikalische Be-<br/>deutung hätte als die gewöhnlich mit H bezeichnete Größe.
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66 </p><!--l. 100--><p class="indent"> Um hierüber ein Urteil zu gewinnen, fragen wir uns zu-<br/>nächst, was in den Minkowskischen Gleichungen
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67 </p>
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68 <table width="100%"
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69 class="equation"><tr><td><a
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70 id="x1-2r1"></a>
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71 <center class="math-display" >
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72 <img
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73 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Bemer_de_1909_01/fulltext/img/Einst_Bemer_de_1909_012x.png" alt=" 1@D- curlH = c @t +i, { 1 @B- curlG = -c @t , divD = r, div B = 0 " class="math-display" /></center></td><td width="5%">(A)</td></tr></table>
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74 <!--l. 114--><p class="nopar">
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75 </p><!--l. 118--><p class="noindent">die Vektoren G, D, H, B für eine Bedeutung haben. Man <br/>muß zugeben, daß diese Vektoren für den Fall, daß die Ge-<br/>schwindigkeit w der Materie von Null abweicht, bisher nicht <br/>eigens definiert worden sind; Definitionen, auf welchen (ideale)
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76 <br/>Messungen dieser Größen basiert werden könnten, besitzen <br/>wir nur für den Fall, daß w verschwindet, und zwar denke <br/>ich an jene Definitionen, welche aus der Elektrodynamik <br/>ruhender Körper wohlbekannt sind. Wenn daher unter Be-<br/>nutzung der Minkowskischen Gleichungen gefunden ist, daß <br/>in einem bestimmten, mit der Geschwindigkeit w bewegten <br/>Volumelement des Körpers die Feldvektoren zu einer gewissen <br/>Zeit die bestimmten (Vektor-) Werte G, D, H, B haben, so <br/>müssen wir diese Feldvektoren erst auf ein mit Bezug auf <br/>das betreffende Volumelement ruhendes Bezugssystem trans-<br/>formieren. Die so erhaltenen Vektoren <span
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77 class="cmmi-10">G</span><span
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78 class="cmsy-10">'</span><span
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79 class="cmmi-10">,D</span><span
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80 class="cmsy-10">'</span><span
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81 class="cmmi-10">,H</span><span
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82 class="cmsy-10">'</span><span
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83 class="cmmi-10">,B</span><span
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84 class="cmsy-10">' </span>haben <br/>erst eine bestimmte physikalische Bedeutung, die aus der <br/>Elektrodynamik ruhender Körper bekannt ist. <pb/>
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85
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86 </p><!--l. 142--><p class="indent">
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87
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88 </p><!--l. 143--><p class="indent"> Die Minkowskischen Differentialgleichungen sagen <br/>also für Punkte, in denen <span
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89 class="cmmi-10">w</span><img
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90 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Bemer_de_1909_01/fulltext/img/Einst_Bemer_de_1909_013x.png" alt="/=" class="neq" align="middle" />0 ist, für sich allein noch gar <br/>nichts aus, wohl aber die Minkowskischen Differentialglei-<br/>chungen zusammen mit den Minkowskischen Transformations-<br/>gleichungen und mit der Bestimmung, daß für den Fall <span
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91 class="cmmi-10">w </span>= 0 <br/>die Definitionen der Elektrodynamik ruhender Körper für die <br/>Feldvektoren gelten sollen.
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92 </p><!--l. 152--><p class="indent"> Wir haben nun zu fragen: Ist der Vektor Q Mirimanoffs <br/>in anderer Weise definiert als der von uns soeben mit H be-<br/>zeichnete Vektor? Dies ist nicht der Fall, und zwar aus <br/>folgenden Gründen:
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93 </p><!--l. 157--><p class="indent"> 1. Für die Feldvektoren G, D, Q, B Mirimanoffs <br/>gelten dieselben Differentialgleichungen und Transformations-<br/>gleichungen wie für die Vektoren G, D, H, B der Minkowski-<br/>schen Gleichungen (A).
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94 </p><!--l. 162--><p class="indent"> 2. Sowohl Mirimanoffs Vektor Q als auch der Vektor H <br/>von (A) sind nur für den Fall <span
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95 class="cmmi-10">w </span>= 0 definiert. In diesem <br/>Falle ist aber wegen Mirimanoffs Gleichung
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96 </p>
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97 <center class="par-math-display" >
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98 <img
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99 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Bemer_de_1909_01/fulltext/img/Einst_Bemer_de_1909_014x.png" alt=" 1 Q = H - c[Pw] " class="par-math-display" /></center>
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100 <!--l. 170--><p class="nopar">
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101 </p><!--l. 173--><p class="noindent"><span
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102 class="cmmi-10">Q </span>= <span
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103 class="cmmi-10">H </span>= Feldstärke zu setzen; für den Vektor H der Glei-<br/>chungen (A) gilt genau in gleicher Weise, daß er im Falle <span
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104 class="cmmi-10">w </span>= 0 <br/>mit der Feldstärke im Sinne der Elektrodynamik ruhender <br/>Körper gleichbedeutend ist.
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105 </p><!--l. 178--><p class="indent"> Aus diesen beiden Argumenten folgt, daß der Vektor Q <br/>Mirimanoffs und der Vektor H von (A) durchaus gleich-<br/>wertig sind.
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106 </p><!--l. 182--><p class="indent"> 4. Um seine Resultate bezüglich der Wilsonschen An-<br/>ordnung mit den von Hrn. Laub und mir erhaltenen zu ver-<br/>gleichen, hätte der Verfasser die Betrachtung so weit durch-<br/>führen müssen, daß er zu Beziehungen zwischen definierten, <br/>d. h. wenigstens prinzipiell der Erfahrung zugänglichen Größen ge-<br/>langt wäre. Er hätte zu diesem Zwecke nur die seinem Gleichungs-<br/>system entsprechenden Grenzbedingungen anzuwenden gehabt. <br/>Nach dem Vorigen hätte er so zu genau denselben Folgerungen
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107 <br/>gelangen müssen wie wir, da seine Theorie mit der von Min-<br/>kowski identisch ist.
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108 </p><!--l. 195--><p class="indent"> Schließlich möchte ich noch hinweisen auf die Bedeutung <br/><pb/>
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109 </p><!--l. 200--><p class="indent">
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110
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111 </p><!--l. 201--><p class="noindent">der neulich erschienenen Arbeit von Ph. Frank<sup ><span
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112 class="cmr-7">1</span></sup>), welche die <br/>Übereinstimmung zwischen der Lorentzschen elektronen-<br/>theoretischen und der Minkowskischen Behandlung der <br/>Elektrodynamik bewegter Körper durch Berücksichtigung der
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113 <br/>Lorentzkontraktion wiederherstellt. Der Vorzug der elektronen-<br/>theoretischen Behandlungsweise liegt einerseits darin, daß sie <br/>eine anschauliche Deutung der Feldvektoren liefert, anderer-<br/>seits darin, daß sie auskommt ohne die willkürliche Voraus-<br/>setzung, daß die Differentialquotienten der Geschwindigkeit der <br/>Materie in den Differentialgleichungen nicht auftreten.
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114 </p><!--l. 214--><p class="indent"> Bern, Januar 1909.
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115 </p><!--l. 216--><p class="noindent">----------
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116 </p><!--l. 219--><p class="indent"> 1) Ph. Frank, Ann. d. Phys. <span
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117 class="cmbx-10">27. </span>p. 1059. 1908.
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118 </p>
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119 <div class="center" >
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120
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121 <!--l. 221--><p class="noindent">
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122 </p><!--l. 222--><p class="noindent">(Eingegangen 22. Januar 1909.)</p></div>
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123 <div class="center" >
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125 <!--l. 225--><p class="noindent">
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126 </p><!--l. 226--><p class="noindent">----------</p></div>
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128 </body></html>
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