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annotate texts/XML/diverse/en/078_A_1916.xml @ 14:c3c84bdd1fdd

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Chinesische Version und XML-Version hinzugef?gt
author Klaus Thoden <kthoden@mpiwg-berlin.mpg.de>
date Thu, 02 May 2013 11:38:23 +0200
parents 22d6a63640c6
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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="078_A_1916.tex" />
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13 <meta name="date" content="2005-03-10 19:24:00" />
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14 <link rel="stylesheet" type="text/css" href="078_A_1916.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="cmbx-12x-x-120">THE FOUNDATION OF THE GENERAL </span> <br/><span
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23 class="cmbx-12x-x-120">THEORY</span>
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24 <span
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25 class="cmbx-12x-x-120">OF RELATIVITY</span></p></div>
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26 <div class="center" >
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28 <!--l. 23--><p class="noindent">
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29 </p><!--l. 24--><p class="noindent"><span
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30 class="cmcsc-10x-x-144"><small
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31 class="small-caps">b</small><small
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32 class="small-caps">y</small></span></p></div>
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33 <div class="center" >
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36 </p><!--l. 29--><p class="noindent"><span
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37 class="cmbx-12x-x-120">A. EINSTEIN</span></p></div>
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38 <div class="center" >
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40 <!--l. 32--><p class="noindent">
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41 </p><!--l. 33--><p class="noindent"><span
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42 class="cmti-12">Translated from &#8220;Die Grundlage der allgemeinen Rela- </span> <br/><span
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43 class="cmti-12">tivit</span><span
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44 class="cmti-12">ätstheorie,&#8221;</span>
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45 <span
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46 class="cmti-12">Annalen der Physik, </span>49, 1916.</p></div>
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47 <!--l. 36--><p class="noindent"><pb/>
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48 </p><!--l. 40--><p class="indent">
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50 </p>
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51 <div class="center" >
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53 <!--l. 41--><p class="noindent">
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54 </p><!--l. 42--><p class="noindent">THE FOUNDATION OF THE GENERAL THEORY <br/>OF RELATIVITY</p></div>
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57 <!--l. 45--><p class="noindent">
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58 </p><!--l. 46--><p class="noindent">B<span
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59 class="cmcsc-10x-x-120"><small
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60 class="small-caps">y</small> </span>A. EINSTEIN</p></div>
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61 <div class="center" >
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63 <!--l. 49--><p class="noindent">
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64 </p><!--l. 50--><p class="noindent">A. F<span
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66 class="small-caps">u</small><small
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67 class="small-caps">n</small><small
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70 class="small-caps">m</small><small
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75 class="small-caps">l</small> </span>C<span
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83 class="small-caps">r</small><small
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89 class="small-caps">s</small> <small
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94 class="small-caps">e</small> </span>P<span
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100 class="small-caps">l</small><small
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101 class="small-caps">a</small><small
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103 class="small-caps">e</small> <small
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104 class="small-caps">o</small><small
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105 class="small-caps">f</small> </span> <br/>R<span
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106 class="cmcsc-10x-x-120"><small
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107 class="small-caps">e</small><small
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108 class="small-caps">l</small><small
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109 class="small-caps">a</small><small
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110 class="small-caps">t</small><small
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111 class="small-caps">i</small><small
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112 class="small-caps">v</small><small
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113 class="small-caps">i</small><small
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114 class="small-caps">t</small><small
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115 class="small-caps">y</small></span>
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116 </p></div>
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117 <div class="center" >
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119 <!--l. 55--><p class="noindent">
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120 </p><!--l. 56--><p class="noindent"><span
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121 class="cmsy-10x-x-120">§ </span><span
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122 class="cmbx-12">1. Observations on the Special Theory of Relativity</span></p></div>
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123 <!--l. 61--><p class="indent"> <sub ><span
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124 class="cmmi-12x-x-172">T</span></sub>HE special theory of relativity is based on the <br/>following postulate, which is also satisfied by the <br/>mechanics of Galileo and Newton.
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125 </p><!--l. 65--><p class="indent"> If a system of co-ordinates K is chosen so that, in re-<br/>lation to it, physical laws hold good in their simplest form, <br/>the <span
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126 class="cmti-12">same </span>laws also hold good in relation to any other system <br/>of co-ordinates <span
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127 class="cmmi-12">K</span><span
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128 class="cmsy-10x-x-120">' </span>moving in uniform translation relatively <br/>to K. This postulate we call the &#8220; special principle of <br/>relativity. &#8221; The word &#8220; special &#8221; is meant to intimate <br/>that the principle is restricted to the case when <span
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129 class="cmmi-12">K</span><span
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130 class="cmsy-10x-x-120">' </span>has a
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131 <br/>motion of uniform translation relatively to K, but that the <br/>equivalence of <span
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132 class="cmmi-12">K</span><span
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133 class="cmsy-10x-x-120">' </span>and K does not extend to the case of non-<br/>uniform motion of <span
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134 class="cmmi-12">K</span><span
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135 class="cmsy-10x-x-120">' </span>relatively to K.
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136 </p><!--l. 78--><p class="indent"> Thus the special theory of relativity does not depart from <br/>classical mechanics through the postulate of relativity, but <br/>through the postulate of the constancy of the velocity of light <br/><span
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137 class="cmti-12">in vacuo</span>, from which, in combination with the special prin-<br/>ciple of relativity, there follow, in the well-known way, the <br/>relativity of simultaneity, the Lorentzian transformation, and <br/>the related laws for the behaviour of moving bodies and <br/>clocks.
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138 </p><!--l. 88--><p class="indent"> The modification to which the special theory of relativity <br/>has subjected the theory of space and time is indeed far-<br/>reaching, but one important point has remained unaffected. <br/><pb/>
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139 </p><!--l. 94--><p class="indent">
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140
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141 </p><!--l. 95--><p class="noindent">For the laws of geometry, even according to the special theory <br/>of relativity, are to be interpreted directly as laws relating to <br/>the possible relative positions of solid bodies at rest; and, in <br/>a more general way, the laws of kinematics are to be inter-<br/>preted as laws which describe the relations of measuring <br/>bodies and clocks. To two selected material points of a <br/>stationary rigid body there always corresponds a distance of <br/>quite definite length, which is independent of the locality and
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142 <br/>orientation of the body, and is also independent of the time. <br/>To two selected positions of the hands of a clock at rest <br/>relatively to the privileged system of reference there always <br/>corresponds an interval of time of a definite length, which is
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143 <br/>independent of place and time. We shall soon see that the <br/>general theory of relativity cannot adhere to this simple <br/>physical interpretation of space and time.
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144 </p>
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145 <div class="center" >
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146
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147 <!--l. 115--><p class="noindent">
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148 </p><!--l. 116--><p class="noindent"><span
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149 class="cmsy-10x-x-120">§ </span><span
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150 class="cmbx-12">2. The Need for an Extension of the Postulate of </span> <br/><span
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151 class="cmbx-12">Relativity</span></p></div>
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152 <!--l. 121--><p class="indent"> In classical mechanics, and no less in the special theory <br/>of relativity, there is an inherent epistemological defect which <br/>was, perhaps for the first time, clearly pointed out by Ernst <br/>Mach. We will elucidate it by the following example:--Two <br/>fluid bodies of the same size and nature hover freely in space <br/>at so great a distance from each other and from all other <br/>masses that only those gravitational forces need be taken into <br/>account which arise from the interaction of different parts of <br/>the same body. Let the distance between the two bodies be <br/>invariable, and in neither of the bodies let there be any <br/>relative movements of the parts with respect to one another. <br/>But let either mass, as judged by an observer at rest <br/>relatively to the other mass, rotate with constant angular <br/>velocity about the line joining the masses. This is a verifi-<br/>able relative motion of the two bodies. Now let us imagine <br/>that each of the bodies has been surveyed by means of <br/>measuring instruments at rest relatively to itself, and let the <br/>surface of S<sub ><span
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153 class="cmr-8">1</span></sub> prove to be a sphere, and that of S<sub ><span
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154 class="cmr-8">2</span></sub> an ellipsoid <br/>of revolution. Thereupon we put the question--What is the <br/>reason for this difference in the two bodies? No answer can
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155 <br/><pb/>
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156 </p><!--l. 146--><p class="indent">
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157
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158 </p><!--l. 147--><p class="noindent">be admitted as epistemologically satisfactory,<sup ><span
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159 class="cmsy-8">*</span></sup> unless the <br/>reason given is an
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160 <span
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161 class="cmti-12">observable fact of experience. </span>The law of <br/>causality has not the significance of a statement as to the <br/>world of experience, except when <span
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162 class="cmti-12">observable facts </span>ultimately
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163 <br/>appear as causes and effects.
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164 </p><!--l. 154--><p class="indent"> Newtonian mechanics does not give a satisfactory answer <br/>to this question. It pronounces as follows:--The laws of <br/>mechanics apply to the space R<sub ><span
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165 class="cmr-8">1</span></sub>, in respect to which the body <br/>S<sub ><span
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166 class="cmr-8">1</span></sub> is at rest, but not to the space R<sub ><span
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167 class="cmr-8">2</span></sub>, in respect to which the
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168 <br/>body S<sub ><span
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169 class="cmr-8">2</span></sub> is at rest. But the privileged space R<sub ><span
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170 class="cmr-8">1</span></sub> of Galileo, <br/>thus introduced, is a merely <span
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171 class="cmti-12">factitious </span>cause, and not a thing <br/>that can be observed. It is therefore clear that Newton&#8217;s <br/>mechanics does not really satisfy the requirement of causality <br/>in the case under consideration, but only apparently does so, <br/>since it makes the factitious cause R<sub ><span
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172 class="cmr-8">1</span></sub> responsible for the ob-<br/>servable difference in the bodies S<sub ><span
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173 class="cmr-8">1</span></sub> and S<sub ><span
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174 class="cmr-8">2</span></sub>.
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175 </p><!--l. 168--><p class="indent"> The only satisfactory answer must be that the physical <br/>system consisting of S<sub ><span
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176 class="cmr-8">1</span></sub>
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177 and S<sub ><span
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178 class="cmr-8">2</span></sub> reveals within itself no imagin-<br/>able cause to which the differing behaviour of S<sub ><span
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179 class="cmr-8">1</span></sub> and S<sub ><span
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180 class="cmr-8">2</span></sub> can <br/>be referred. The cause must therefore lie <span
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181 class="cmti-12">outside</span>
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182 this system. <br/>We have to take it that the general laws of motion, which in <br/>particular determine the shapes of S<sub ><span
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183 class="cmr-8">1</span></sub> and S<sub ><span
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184 class="cmr-8">2</span></sub>, must be such <br/>that the mechanical behaviour of S<sub ><span
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185 class="cmr-8">1</span></sub> and S<sub ><span
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186 class="cmr-8">2</span></sub> is partly con-<br/>ditioned, in quite essential respects, by distant masses which <br/>we have not included in the system under consideration. <br/>These distant masses and their motions relative to S<sub ><span
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187 class="cmr-8">1</span></sub> and <br/>S<sub ><span
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188 class="cmr-8">2</span></sub> must then be regarded as the seat of the causes (which <br/>must be susceptible to observation) of the different behaviour <br/>of our two bodies S<sub ><span
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189 class="cmr-8">1</span></sub> and S<sub ><span
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190 class="cmr-8">2</span></sub>. They take over the rôle of the <br/>factitious cause R<sub ><span
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191 class="cmr-8">1</span></sub>. Of all imaginable spaces R<sub ><span
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192 class="cmr-8">1</span></sub>, R<sub ><span
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193 class="cmr-8">2</span></sub>, etc., in <br/>any kind of motion relatively to one another, there is none
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194 <br/>which we may look upon as privileged a <span
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195 class="cmti-12">priori </span>without re-<br/>viving the above-mentioned epistemological objection. <span
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196 class="cmti-12">The </span> <br/><span
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197 class="cmti-12">laws of physics must be</span>
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198 <span
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199 class="cmti-12">of such a nature that they apply to </span> <br/><span
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200 class="cmti-12">systems of reference in any kind of</span>
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201 <span
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202 class="cmti-12">motion. </span>Along this road <br/>we arrive at an extension of the postulate of relativity.
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203 </p><!--l. 193--><p class="indent"> In addition to this weighty argument from the theory of <br/>
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204 </p><!--l. 196--><p class="indent"> <sup ><span
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205 class="cmsy-8">*</span></sup> Of course an answer may be satisfactory from the point of view of episte-<br/>mology, and yet be unsound physically, if it is in conflict with other experi-<br/>ences.
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206 <pb/>
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207 </p><!--l. 203--><p class="indent">
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208
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209 </p><!--l. 204--><p class="noindent">knowledge, there is a well-known physical fact which favours <br/>an extension of the theory of relativity. Let K be a Galilean <br/>system of reference, i.e. a system relatively to which (at least <br/>in the four-dimensional region under consideration) a mass, <br/>sufficiently distant from other masses, is moving with uniform <br/>motion in a straight line. Let <span
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210 class="cmmi-12">K</span><span
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211 class="cmsy-10x-x-120">' </span>be a second system of <br/>reference which is moving relatively to K in <span
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212 class="cmti-12">uniformly </span> <br/><span
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213 class="cmti-12">accelerated </span>translation. Then, relatively to <span
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214 class="cmmi-12">K</span><span
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215 class="cmsy-10x-x-120">'</span>, a mass <br/>sufficiently distant from other masses would have an acceler-<br/>ated motion such that its acceleration and direction of <br/>acceleration are independent of the material composition and <br/>physical state of the mass.
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216 </p><!--l. 219--><p class="indent"> Does this permit an observer at rest relatively to <span
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217 class="cmmi-12">K</span><span
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218 class="cmsy-10x-x-120">' </span>to <br/>infer that he is on a &#8220; really &#8221; accelerated system of reference? <br/>The answer is in the negative; for the above-mentioned <br/>relation of freely movable masses to <span
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219 class="cmmi-12">K</span><span
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220 class="cmsy-10x-x-120">' </span>may be interpreted
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221 <br/>equally well in the following way. The system of reference <br/><span
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222 class="cmmi-12">K</span><span
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223 class="cmsy-10x-x-120">' </span>is unaccelerated, but the space-time territory in question <br/>is under the sway of a gravitational field, which generates the <br/>accelerated motion of the bodies relatively to
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224 <span
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225 class="cmmi-12">K</span><span
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226 class="cmsy-10x-x-120">'</span>.
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227 </p><!--l. 230--><p class="indent"> This view is made possible for us by the teaching of <br/>experience as to the existence of a field of force, namely, the <br/>gravitational field, which possesses the remarkable property <br/>of imparting the same acceleration to all bodies.<sup ><span
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228 class="cmsy-8">*</span></sup> The
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229 <br/>mechanical behaviour of bodies relatively to <span
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230 class="cmmi-12">K</span><span
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231 class="cmsy-10x-x-120">' </span>is the same <br/>as presents itself to experience in the case of systems which <br/>we are wont to regard as &#8220; stationary &#8221; or as &#8220; privileged.&#8221; <br/>Therefore, from the physical standpoint, the assumption <br/>readily suggests itself that the systems K and <span
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232 class="cmmi-12">K</span><span
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233 class="cmsy-10x-x-120">' </span>may both <br/>with equal right be looked upon as &#8220; stationary,&#8221; that is to <br/>say, they have an equal title as systems of reference for the <br/>physical description of phenomena.
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234 </p><!--l. 245--><p class="indent"> It will be seen from these reflexions that in pursuing the <br/>general theory of relativity we shall be led to a theory of <br/>gravitation, since we are able to &#8220; produce &#8221; a gravitational <br/>field merely by changing the system of co-ordinates. It will <br/>also be obvious that the principle of the constancy of the <br/>velocity of light <span
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235 class="cmti-12">in</span>
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236 <span
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237 class="cmti-12">vacuo </span>must be modified, since we easily <br/>
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238 </p><!--l. 255--><p class="indent"> <sup ><span
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239 class="cmsy-8">*</span></sup> Eötvös has proved experimentally that the gravitational field has this
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240 <br/>property in great accuracy. <pb/>
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241 </p><!--l. 261--><p class="indent">
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242
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243 </p><!--l. 262--><p class="noindent">recognize that the path of a ray of light with respect to <span
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244 class="cmmi-12">K</span><span
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245 class="cmsy-10x-x-120">'</span> <br/>must in general be curvilinear, if with respect to K light is <br/>propagated in a straight line with a definite constant velocity.
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246 </p>
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247 <div class="center" >
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248
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249 <!--l. 268--><p class="noindent">
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250 </p><!--l. 269--><p class="noindent"><span
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251 class="cmsy-10x-x-120">§ </span><span
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252 class="cmbx-12">3. The Space-Time Continuum. Requirement of General</span>
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253 <br/><span
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254 class="cmbx-12">Co-Variance for the Equations Expressing General </span> <br/><span
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255 class="cmbx-12">Laws of Nature</span></p></div>
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256 <!--l. 275--><p class="indent"> In classical mechanics, as well as in the special theory of <br/>relativity, the co-ordinates of space and time have a direct <br/>physical meaning. To say that a point-event has the X<sub ><span
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257 class="cmr-8">1</span></sub> co-<br/>ordinate <span
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258 class="cmmi-12">x</span><sub ><span
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259 class="cmr-8">1</span></sub> means that the projection of the point-event on the <br/>axis of X<sub ><span
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260 class="cmr-8">1</span></sub>, determined by rigid rods and in accordance with the <br/>rules of Euclidean geometry, is obtained by measuring off a <br/>given rod (the unit of length) <span
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261 class="cmmi-12">x</span><sub ><span
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262 class="cmr-8">1</span></sub> times from the origin of co-<br/>ordinates along the axis of X<sub ><span
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263 class="cmr-8">1</span></sub>. To say that a point-event <br/>has the X<sub ><span
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264 class="cmr-8">4</span></sub> co-ordinate <span
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265 class="cmmi-12">x</span><sub ><span
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266 class="cmr-8">4</span></sub> = <span
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267 class="cmmi-12">t</span>, means that a standard clock,
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268 <br/>made to measure time in a definite unit period, and which is <br/>stationary relatively to the system of co-ordinates and practic-<br/>ally coincident in space with the point-event,<sup ><span
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269 class="cmsy-8">*</span></sup> will have <br/>measured off <span
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270 class="cmmi-12">x</span><sub >
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271 <span
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272 class="cmr-8">4</span></sub> = <span
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273 class="cmmi-12">t </span>periods at the occurrence of the event.
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274 </p><!--l. 292--><p class="indent"> This view of space and time has always been in the minds <br/>of physicists, even if, as a rule, they have been unconscious <br/>of it. This is clear from the part which these concepts play <br/>in physical measurements; it must also have underlain the
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275 <br/>reader&#8217;s reflexions on the preceding paragraph (<span
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276 class="cmsy-10x-x-120">§ </span>2) for <br/>him to connect any meaning with what he there read. But <br/>we shall now show that we must put it aside and replace it <br/>by a more general view, in order to be able to carry through <br/>the postulate of general relativity, if the special theory of
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277 <br/>relativity applies to the special case of the absence of a gravi-<br/>tational field.
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278 </p><!--l. 305--><p class="indent"> In a space which is free of gravitational fields we introduce <br/>a Galilean system of reference <span
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279 class="cmmi-12">K</span>(<span
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280 class="cmmi-12">x, y, z, t</span>)<span
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281 class="cmmi-12">, </span>and also a system <br/>of co-ordinates <span
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282 class="cmmi-12">K</span><span
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283 class="cmsy-10x-x-120">' </span>(<span
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284 class="cmmi-12">x</span><span
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285 class="cmsy-10x-x-120">'</span><span
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286 class="cmmi-12">, y</span><span
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287 class="cmsy-10x-x-120">'</span><span
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288 class="cmmi-12">, z</span><span
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289 class="cmsy-10x-x-120">'</span><span
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290 class="cmmi-12">, t</span><span
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291 class="cmsy-10x-x-120">'</span>) in uniform rotation relatively <br/>to K. Let the origins of both systems, as well as their axes <br/>
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292 </p><!--l. 312--><p class="indent"> <sup ><span
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293 class="cmsy-8">*</span></sup> We assume the possibility of verifying &#8220; simultaneity &#8221; for events im-<br/>mediately proximate in space, or--to speak more precisely--for immediate
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294 <br/>proximity or coincidence in space-time, without giving a definition of this
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295 <br/>fundamental concept. <pb/>
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296 </p><!--l. 320--><p class="indent">
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297
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298 </p><!--l. 321--><p class="noindent">of Z, permanently coincide. We shall show that for a space-<br/>time measurement in the system <span
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299 class="cmmi-12">K</span><span
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300 class="cmsy-10x-x-120">' </span>the above definition of <br/>the physical meaning of lengths and times cannot be main-<br/>tained. For reasons of symmetry it is clear that a circle <br/>around the origin in the X, Y plane of K may at the same <br/>time be regarded as a circle in the <span
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301 class="cmmi-12">K</span><span
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302 class="cmsy-10x-x-120">'</span><span
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303 class="cmmi-12">,</span> Y&#8217; plane of <span
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304 class="cmmi-12">K</span><span
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305 class="cmsy-10x-x-120">'</span>. We <br/>suppose that the circumference and diameter of this circle <br/>have been measured with a unit measure infinitely small <br/>compared with the radius, and that we have the quotient of <br/>the two results. If this experiment were performed with a <br/>measuring-rod at rest relatively to the Galilean system K, the <br/>quotient would be <span
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306 class="cmmi-12"><img
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307 src="img/cmmi12-19.png" alt="p" class="12x-x-19" /></span>. With a measuring-rod at rest relatively <br/>to
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308 <span
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309 class="cmmi-12">K</span><span
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310 class="cmsy-10x-x-120">'</span>, the quotient would be greater than <span
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311 class="cmmi-12"><img
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312 src="img/cmmi12-19.png" alt="p" class="12x-x-19" /></span>. This is readily <br/>understood if we envisage the whole process of measuring <br/>from the &#8220; stationary &#8221; system K, and take into consideration <br/>that the measuring-rod applied to the periphery undergoes <br/>a Lorentzian contraction, while the one applied along the <br/>radius does not. Hence Euclidean geometry does not apply <br/>to
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313 <span
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314 class="cmmi-12">K</span><span
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315 class="cmsy-10x-x-120">'</span>. The notion of co-ordinates defined above, which pre-<br/>supposes the validity of Euclidean geometry, therefore breaks <br/>down in relation to the system <span
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316 class="cmmi-12">K</span><span
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317 class="cmsy-10x-x-120">'</span>. So, too, we are unable <br/>to introduce a time corresponding to physical requirements <br/>in <span
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318 class="cmmi-12">K</span><span
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319 class="cmsy-10x-x-120">'</span>, indicated by clocks at rest relatively to <span
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320 class="cmmi-12">K</span><span
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321 class="cmsy-10x-x-120">'</span>. To
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322 <br/>convince ourselves of this impossibility, let us imagine two <br/>clocks of identical constitution placed, one at the origin of <br/>co-ordinates, and the other at the circumference of the <br/>circle, and both envisaged from the &#8220; stationary &#8221; system <br/>K. By a familiar result of the special theory of relativity, <br/>the clock at the circumference--judged from K--goes more <br/>slowly than the other, because the former is in motion and <br/>the latter at rest. An observer at the common origin of
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323 <br/>co-ordinates, capable of observing the clock at the circum-<br/>ference by means of light, would therefore see it lagging be-<br/>hind the clock beside him. As he will not make up his mind <br/>to let the velocity of light along the path in question depend <br/>explicitly on the time, he will interpret his observations as
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324 <br/>showing that the clock at the circumference &#8220; really &#8221; goes <br/>more slowly than the clock at the origin. So he will be <br/>obliged to define time in such a way that the rate of a clock <br/>depends upon where the clock may be.
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325 <pb/>
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326 </p><!--l. 370--><p class="indent">
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327
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328 </p><!--l. 371--><p class="indent"> We therefore reach this result:--In the general theory of <br/>relativity, space and time cannot be defined in such a way <br/>that differences of the spatial co-ordinates can be directly <br/>measured by the unit measuring-rod, or differences in the <br/>time co-ordinate by a standard clock.
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329 </p><!--l. 378--><p class="indent"> The method hitherto employed for laying co-ordinates <br/>into the space-time continuum in a definite manner thus breaks <br/>down, and there seems to be no other way which would allow <br/>us to adapt systems of co-ordinates to the four-dimensional <br/>universe so that we might expect from their application a
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330 <br/>particularly simple formulation of the laws of nature. So <br/>there is nothing for it but to regard all imaginable systems <br/>of co-ordinates, on principle, as equally suitable for the <br/>description of nature. This comes to requiring that:--
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331 </p><!--l. 389--><p class="indent"> <span
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332 class="cmti-12">The general laws of nature are to be expressed by equations </span> <br/><span
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333 class="cmti-12">which hold good for</span>
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334 <span
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335 class="cmti-12">all systems of co-ordinates, that is, are </span> <br/><span
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336 class="cmti-12">co-variant with respect to any substitutions</span>
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337 <span
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338 class="cmti-12">whatever (generally </span> <br/><span
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339 class="cmti-12">co-variant)</span>.
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340 </p><!--l. 394--><p class="indent"> It is clear that a physical theory which satisfies this <br/>postulate will also be suitable for the general postulate of <br/>relativity. For the sum of <span
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341 class="cmti-12">all</span>
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342 substitutions in any case in-<br/>cludes those which correspond to all relative motions of three-<br/>dimensional systems of co-ordinates. That this requirement
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343 <br/>of general co-variance, which takes away from space and <br/>time the last remnant of physical objectivity, is a natural <br/>one, will be seen from the following reflexion. All our <br/>space-time verifications invariably amount to a determination <br/>of space-time coincidences. If, for example, events consisted
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344 <br/>merely in the motion of material points, then ultimately <br/>nothing would be observable but the meetings of two or more <br/>of these points. Moreover, the results of our measurings are <br/>nothing but verifications of such meetings of the material <br/>points of our measuring instruments with other material
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345 <br/>points, coincidences between the hands of a clock and points <br/>on the clock dial, and observed point-events happening at the <br/>same place at the same time.
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346 </p><!--l. 415--><p class="indent"> The introduction of a system of reference serves no other <br/>purpose than to facilitate the description of the totality of such <br/>coincidences. We allot to the universe four space-time vari-<br/>ables <span
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347 class="cmmi-12">x</span><sub ><span
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348 class="cmr-8">1</span></sub> <span
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349 class="cmmi-12">, x</span><sub ><span
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350 class="cmr-8">2</span></sub> <span
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351 class="cmmi-12">, x</span><sub ><span
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352 class="cmr-8">3</span></sub> <span
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353 class="cmmi-12">, x</span><sub ><span
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354 class="cmr-8">4</span></sub> in such a way that for every point-event <br/><pb/>
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355 </p><!--l. 425--><p class="indent">
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356
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357 </p><!--l. 426--><p class="noindent">there is a corresponding system of values of the variables <br/><span
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358 class="cmmi-12">x</span><sub ><span
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359 class="cmr-8">1</span></sub> <span
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360 class="cmmi-12">...</span><span
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361 class="cmmi-12">x</span><sub ><span
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362 class="cmr-8">4</span></sub>. To two coincident point-events there corre-<br/>sponds one system of values of the variables
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363 <span
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364 class="cmmi-12">x</span><sub ><span
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365 class="cmr-8">1</span></sub> <span
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366 class="cmmi-12">...</span><span
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367 class="cmmi-12">x</span><sub ><span
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368 class="cmr-8">4</span></sub>, i.e. <br/>coincidence is characterized by the identity of the co-ordinates. <br/>If, in place of the variables <span
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369 class="cmmi-12">x</span><sub ><span
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370 class="cmr-8">1</span></sub> <span
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371 class="cmmi-12">...</span><span
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372 class="cmmi-12">x</span><sub ><span
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373 class="cmr-8">4</span></sub>, we introduce functions <br/>of them, <span
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374 class="cmmi-12">x</span><span
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375 class="cmsy-10x-x-120">'</span><sub ><span
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376 class="cmr-8">1</span></sub><span
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377 class="cmmi-12">, x</span><span
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378 class="cmsy-10x-x-120">'</span><sub ><span
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379 class="cmr-8">2</span></sub><span
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380 class="cmmi-12">, x</span><span
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381 class="cmsy-10x-x-120">'</span><sub ><span
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382 class="cmr-8">3</span></sub><span
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383 class="cmmi-12">, x</span><span
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384 class="cmsy-10x-x-120">'</span><sub ><span
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385 class="cmr-8">4</span></sub><span
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386 class="cmmi-12">,</span>
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387 as a new system of co-ordinates, so <br/>that the systems of values are made to correspond to one <br/>another without ambiguity, the equality of all four co-ordin-<br/>ates in the new system will also serve as an expression for <br/>the space-time coincidence of the two point-events. As all <br/>our physical experience can be ultimately reduced to such <br/>coincidences, there is no immediate reason for preferring <br/>certain systems of co-ordinates to others, that is to say, we <br/>arrive at the requirement of general co-variance.
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388 </p>
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389 <div class="center" >
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390
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391 <!--l. 446--><p class="noindent">
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392 </p><!--l. 447--><p class="noindent"><span
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393 class="cmbsy-10x-x-120">§ </span><span
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394 class="cmbx-12">4. The Relation of the Four Co-ordinates to Measure- </span> <br/><span
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395 class="cmbx-12">ment in</span>
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396 <span
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397 class="cmbx-12">Space and Time</span></p></div>
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398 <!--l. 452--><p class="indent"> It is not my purpose in this discussion to represent the <br/>general theory of relativity as a system that is as simple and <br/>logical as possible, and with the minimum number of axioms; <br/>but my main object is to develop this theory in such a way <br/>that the reader will feel that the path we have entered upon <br/>is psychologically the natural one, and that the underlying <br/>assumptions will seem to have the highest possible degree <br/>of security. With this aim in view let it now be granted <br/>that:--
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399 </p><!--l. 462--><p class="indent"> For infinitely small four-dimensional regions the theory <br/>of relativity in the restricted sense is appropriate, if the co-<br/>ordinates are suitably chosen.
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400 </p><!--l. 466--><p class="indent"> For this purpose we must choose the acceleration of the <br/>infinitely small (&#8220; local &#8221;) system of co-ordinates so that no <br/>gravitational field occurs; this is possible for an infinitely <br/>small region. Let X<sub ><span
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401 class="cmr-8">1</span></sub> <span
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402 class="cmmi-12">, X</span><sub ><span
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403 class="cmr-8">2</span></sub> <span
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404 class="cmmi-12">, X</span><sub ><span
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405 class="cmr-8">3</span></sub>, be the co-ordinates of space, <br/>and <span
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406 class="cmmi-12">X</span><sub ><span
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407 class="cmr-8">4</span></sub> the appertaining co-ordinate of time measured in the <br/>appropriate unit.<sup ><span
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408 class="cmsy-8">*</span></sup> If a rigid rod is imagined to be given as <br/>the unit measure, the co-ordinates, with a given orientation <br/>of the system of co-ordinates, have a direct physical meaning <br/>
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409 </p><!--l. 478--><p class="indent"> <sup ><span
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410 class="cmsy-8">*</span></sup> The unit of time is to be chosen so that the velocity of light <span
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411 class="cmti-12">in vacuo </span>as
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412 <br/>measured in the &#8220; local &#8221; system of co-ordinates is to be equal to unity.
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413 <pb/>
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414 </p><!--l. 485--><p class="indent">
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415
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416 </p><!--l. 486--><p class="noindent">in the sense of the special theory of relativity. By the <br/>special theory of relativity the expression
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417 </p>
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418 <table width="100%"
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419 class="equation"><tr><td><a
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420 id="x1-2r1"></a>
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421 <center class="math-display" >
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422 <img
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423 src="img/078_A_19160x.png" alt=" 2 2 2 2 2 ds = - dX 1- dX 2- dX 3 + dX 4 " class="math-display" /></center></td><td width="5%">(1)</td></tr></table>
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424 <!--l. 492--><p class="nopar">
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425 </p><!--l. 496--><p class="noindent">then has a value which is independent of the orientation of <br/>the local system of co-ordinates, and is ascertainable by <br/>measurements of space and time. The magnitude of the <br/>linear element pertaining to points of the four-dimensional
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426 <br/>continuum in infinite proximity, we call <span
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427 class="cmti-12">ds</span>. If the <span
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428 class="cmti-12">ds </span>belong-<br/>ing to the element
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429 <span
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430 class="cmmi-12">dX</span><sub ><span
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431 class="cmr-8">1</span></sub> <span
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432 class="cmmi-12">...</span><span
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433 class="cmmi-12">dX</span><sub ><span
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434 class="cmr-8">4</span></sub> is positive, we follow <br/>Minkowski in calling it time-like; if it is negative, we call it <br/>space-like.
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435 </p><!--l. 506--><p class="indent"> To the &#8220; linear element &#8221; in question, or to the two infin-<br/>itely proximate point-events, there will also correspond <br/>definite differentials <span
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436 class="cmmi-12">dx</span><sub ><span
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437 class="cmr-8">1</span></sub> <span
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438 class="cmmi-12">...</span><span
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439 class="cmmi-12">dx</span><sub ><span
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440 class="cmr-8">4</span></sub> of the four-dimensional <br/>co-ordinates of any chosen system of reference. If this <br/>system, as well as the &#8220; local &#8221; system, is given for the region <br/>under consideration, the <span
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441 class="cmmi-12">dX</span><sub ><span
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442 class="cmmi-8"><img
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443 src="img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub>
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444 will allow themselves to be <br/>represented here by definite linear homogeneous expressions <br/>of the <span
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445 class="cmmi-12">dx</span><sub ><span
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446 class="cmmi-8"><img
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447 src="img/cmmi8-1b.png" alt="s" class="8x-x-1b" /></span></sub>:--
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448 </p>
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449 <table width="100%"
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450 class="equation"><tr><td><a
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451 id="x1-3r2"></a>
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452 <center class="math-display" >
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453 <img
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454 src="img/078_A_19161x.png" alt="dXn = S ansdxs s " class="math-display" /></center></td><td width="5%">(2)</td></tr></table>
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455 <!--l. 520--><p class="nopar">
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456 </p><!--l. 523--><p class="noindent">Inserting these expressions in (1), we obtain
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457 </p>
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458
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459 <table width="100%"
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460 class="equation"><tr><td><a
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461 id="x1-4r3"></a>
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462 <center class="math-display" >
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463 <img
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464 src="img/078_A_19162x.png" alt=" 2 ds = Stsgstdxsdxt , " class="math-display" /></center></td><td width="5%">(3)</td></tr></table>
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465 <!--l. 530--><p class="nopar">
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466 </p><!--l. 535--><p class="noindent">where the <span
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467 class="cmmi-12">g</span><sub ><span
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468 class="cmmi-8"><img
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469 src="img/cmmi8-1b.png" alt="s" class="8x-x-1b" /><img
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470 src="img/cmmi8-1c.png" alt="t" class="8x-x-1c" /></span></sub> will be functions of the <span
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471 class="cmmi-12">x</span><sub ><span
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472 class="cmmi-8"><img
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473 src="img/cmmi8-1b.png" alt="s" class="8x-x-1b" /></span></sub>. These can no <br/>longer be dependent on the orientation and the state of <br/>motion of the &#8220; local &#8221; system of co-ordinates, for <span
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474 class="cmmi-12">ds</span><sup ><span
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475 class="cmr-8">2</span></sup> is a <br/>quantity ascertainable by rod-clock measurement of point-<br/>events infinitely proximate in space-time, and defined inde-<br/>pendently of any particular choice of co-ordinates. The <span
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476 class="cmmi-12">g</span><sub ><span
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477 class="cmmi-8"><img
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478 src="img/cmmi8-1b.png" alt="s" class="8x-x-1b" /><img
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479 src="img/cmmi8-1c.png" alt="t" class="8x-x-1c" /></span></sub> <br/>are to be chosen here so that <span
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480 class="cmmi-12">g</span><sub ><span
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481 class="cmmi-8"><img
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482 src="img/cmmi8-1b.png" alt="s" class="8x-x-1b" /><img
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483 src="img/cmmi8-1c.png" alt="t" class="8x-x-1c" /></span></sub> = <span
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484 class="cmmi-12">g</span><sub ><span
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485 class="cmmi-8"><img
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486 src="img/cmmi8-1c.png" alt="t" class="8x-x-1c" /><img
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487 src="img/cmmi8-1b.png" alt="s" class="8x-x-1b" /></span></sub> ; the summation is <br/>to extend over all values of <span
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488 class="cmmi-12"><img
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489 src="img/cmmi12-1b.png" alt="s" class="12x-x-1b" /> </span>and
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490 <span
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491 class="cmmi-12"><img
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492 src="img/cmmi12-1c.png" alt="t" class="12x-x-1c" /></span>, so that the sum consists <br/>of 4 <span
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493 class="cmsy-10x-x-120">× </span>4 terms, of which twelve are equal in pairs.
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494 </p><!--l. 548--><p class="indent"> The case of the ordinary theory of relativity arises out of <br/>the case here considered, if it is possible, by reason of the <br/>particular relations of the <span
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495 class="cmmi-12">g</span><sub ><span
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496 class="cmmi-8"><img
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497 src="img/cmmi8-1b.png" alt="s" class="8x-x-1b" /><img
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498 src="img/cmmi8-1c.png" alt="t" class="8x-x-1c" /></span></sub> in a finite region, to choose the <br/>system of reference in the finite region in such a way that <br/>the <span
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499 class="cmmi-12">g</span><sub ><span
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500 class="cmmi-8"><img
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501 src="img/cmmi8-1b.png" alt="s" class="8x-x-1b" /><img
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502 src="img/cmmi8-1c.png" alt="t" class="8x-x-1c" /></span></sub> assume the constant values <pb/>
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503 </p><!--l. 557--><p class="indent">
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504
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505 </p>
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506 <table width="100%"
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507 class="equation"><tr><td><a
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508 id="x1-5r4"></a>
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509 <center class="math-display" >
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510 <img
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511 src="img/078_A_19163x.png" alt="- 1 0 0 0 0 - 1 0 0 } 0 0 - 1 0 0 0 0 + 1 " class="math-display" /></center></td><td width="5%">(4)</td></tr></table>
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512 <!--l. 566--><p class="nopar">
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513 </p><!--l. 569--><p class="noindent">We shall find hereafter that the choice of such co-ordinates <br/>is, in general, not possible for a finite region.
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514 </p><!--l. 573--><p class="indent"> From the considerations of <span
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515 class="cmsy-10x-x-120">§ </span>2 and <span
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516 class="cmsy-10x-x-120">§ </span>3 it follows that <br/>the quantities <span
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517 class="cmmi-12">g</span><sub ><span
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518 class="cmmi-8"><img
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519 src="img/cmmi8-1c.png" alt="t" class="8x-x-1c" /><img
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520 src="img/cmmi8-1b.png" alt="s" class="8x-x-1b" /></span></sub> are to be regarded from the physical stand-<br/>point as the quantities which describe the gravitational <br/>field in relation to the chosen system of reference. For, if <br/>we now assume the special theory of relativity to apply to a <br/>certain four-dimensional region with the co-ordinates properly <br/>chosen, then the <span
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521 class="cmmi-12">g</span><sub ><span
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522 class="cmmi-8"><img
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523 src="img/cmmi8-1b.png" alt="s" class="8x-x-1b" /><img
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524 src="img/cmmi8-1c.png" alt="t" class="8x-x-1c" /></span></sub> have the values given in (4). A free <br/>material point then moves, relatively to this system, with
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525 <br/>uniform motion in a straight line. Then if we introduce new <br/>space-time co-ordinates <span
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526 class="cmmi-12">x</span><sub ><span
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527 class="cmr-8">1</span></sub><span
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528 class="cmmi-12">, x</span><sub ><span
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529 class="cmr-8">2</span></sub><span
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530 class="cmmi-12">, x</span><sub ><span
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531 class="cmr-8">3</span></sub><span
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532 class="cmmi-12">, x</span><sub ><span
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533 class="cmr-8">4</span></sub><span
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534 class="cmmi-12">, </span>by means of any substi-<br/>tution we choose, the <span
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535 class="cmmi-12">g</span><sup ><span
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536 class="cmmi-8"><img
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537 src="img/cmmi8-1b.png" alt="s" class="8x-x-1b" /><img
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538 src="img/cmmi8-1c.png" alt="t" class="8x-x-1c" /></span></sup> in this new system will no longer <br/>be constants, but functions of space and time. At the same <br/>time the motion of the free material point will present itself <br/>in the new co-ordinates as a curvilinear non-uniform motion, <br/>and the law of this motion will be independent of the nature <br/>of the moving particle. We shall therefore interpret this <br/>motion as a motion under the influence of a gravitational <br/>field. We thus find the occurrence of a gravitational field <br/>connected with a space-time variability of the <span
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539 class="cmmi-12">g</span><sub ><span
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541 src="img/cmmi8-1b.png" alt="s" class="8x-x-1b" /></span></sub>. So, too, <br/>in the general case, when we are no longer able by a suitable
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542 <br/>choice of co-ordinates to apply the special theory of relativity <br/>to a finite region, we shall hold fast to the view that the <span
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546 src="img/cmmi8-1c.png" alt="t" class="8x-x-1c" /></span></sub> <br/>describe the gravitational field.
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547 </p><!--l. 601--><p class="indent"> Thus, according to the general theory of relativity, gravi-<br/>tation occupies an exceptional position with regard to other <br/>forces, particularly the electromagnetic forces, since the ten <br/>functions representing the gravitational field at the same time
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548 <br/>define the metrical properties of the space measured.
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549 </p>
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550
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551 <div class="center" >
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553 <!--l. 609--><p class="noindent">
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554 </p><!--l. 610--><p class="noindent">B. <span
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555 class="cmcsc-10x-x-120">M<small
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566 class="small-caps">l</small> A<small
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614 <!--l. 616--><p class="indent"> Having seen in the foregoing that the general postulate <br/>of relativity leads to the requirement that the equations of <br/><pb/>
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615 </p><!--l. 621--><p class="indent">
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617 </p><!--l. 622--><p class="noindent">physics shall be covariant in the face of any substitution of <br/>the co-ordinates
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620 class="cmr-8">1</span></sub> <span
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623 class="cmr-8">4</span></sub>, we have to consider how such <br/>generally covariant equations can be found. We now turn <br/>to this purely mathematical task, and we shall find that in its
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624 <br/>solution a fundamental rôle is played by the invariant <span
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625 class="cmti-12">ds </span> <br/>given in equation (3), which, borrowing from Gauss&#8217;s theory <br/>of surfaces, we have called the &#8220; linear element. &#8221;
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626 </p><!--l. 632--><p class="indent"> The fundamental idea of this general theory of covariants <br/>is the following:--Let certain things (&#8220; tensors &#8221;) be defined <br/>with respect to any system of co-ordinates by a number of <br/>functions of the co-ordinates, called the &#8220; components &#8221; of <br/>the tensor. There are then certain rules by which these <br/>components can be calculated for a new system of co-ordin-<br/>ates, if they are known for the original system of co-ordinates, <br/>and if the transformation connecting the two systems is <br/>known. The things hereafter called tensors are further
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627 <br/>characterized by the fact that the equations of transformation <br/>for their components are linear and homogeneous. Accord-<br/>ingly, all the components in the new system vanish, if they <br/>all vanish in the original system. If, therefore, a law of <br/>nature is expressed by equating all the components of a tensor <br/>to zero, it is generally covariant. By examining the laws <br/>of the formation of tensors, we acquire the means of formu-<br/>lating generally covariant laws.
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628 </p>
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629 <div class="center" >
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631 <!--l. 653--><p class="noindent">
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632 </p><!--l. 654--><p class="noindent"><span
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633 class="cmbsy-10x-x-120">§ </span><span
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634 class="cmbx-12">5. Contravariant and Covariant Four-vectors</span></p></div>
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635 <!--l. 658--><p class="indent"> <span
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636 class="cmti-12">Contravariant Four-vectors.</span>--The linear element is de-<br/>fined by the four &#8220; components &#8221; <span
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637 class="cmmi-12">dx</span><sub ><span
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638 class="cmmi-8"><img
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639 src="img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub>, for which the law of <br/>transformation is expressed by the equation
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640 </p>
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641 <table width="100%"
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642 class="equation"><tr><td><a
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643 id="x1-6r5"></a>
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644 <center class="math-display" >
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645 <img
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646 src="img/078_A_19164x.png" alt=" ' @x's dxs = Sn @x--dxn n " class="math-display" /></center></td><td width="5%">(5)</td></tr></table>
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647 <!--l. 668--><p class="nopar">
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648
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649 </p><!--l. 671--><p class="noindent">The <span
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650 class="cmmi-12">dx</span><span
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651 class="cmsy-10x-x-120">'</span><sub ><span
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652 class="cmmi-8"><img
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653 src="img/cmmi8-1b.png" alt="s" class="8x-x-1b" /></span></sub> are expressed as linear and homogeneous functions <br/>of the <span
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654 class="cmmi-12">dx</span><sub ><span
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655 class="cmmi-8"><img
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656 src="img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub>. Hence we may look upon these co-ordinate differ-<br/>entials as the components of a &#8220; tensor &#8221; of the particular <br/>kind which we call a contravariant four-vector. Any thing <br/>which is defined relatively to the system of co-ordinates by <br/>four quantities <span
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657 class="cmmi-12">A</span><sup ><span
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658 class="cmmi-8"><img
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659 src="img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sup>, and which is transformed by the same law
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660 </p>
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661 <table width="100%"
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662 class="equation"><tr><td><a
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663 id="x1-7r6"></a>
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664 <center class="math-display" >
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665 <img
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666 src="img/078_A_19165x.png" alt=" 's @x's n A = S ----A , n @xn " class="math-display" /></center></td><td width="5%">(5a)</td></tr></table>
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667 <!--l. 685--><p class="nopar">
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668 <pb/>
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669 </p><!--l. 692--><p class="indent">
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670
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671 </p><!--l. 693--><p class="noindent">we also call a contravariant four-vector. From (5a) it <br/>follows at once that the sums
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672 <span
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673 class="cmmi-12">A</span><sup ><span
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674 class="cmmi-8"><img
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675 src="img/cmmi8-1b.png" alt="s" class="8x-x-1b" /></span></sup> <span
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676 class="cmsy-10x-x-120">± </span><span
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677 class="cmmi-12">B</span><sup ><span
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678 class="cmmi-8"><img
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679 src="img/cmmi8-1b.png" alt="s" class="8x-x-1b" /></span></sup> are also components <br/>of a four-vector, if <span
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680 class="cmmi-12">A</span><sup ><span
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681 class="cmmi-8"><img
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682 src="img/cmmi8-1b.png" alt="s" class="8x-x-1b" /></span></sup> and <span
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683 class="cmmi-12">B</span><sup ><span
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684 class="cmmi-8"><img
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685 src="img/cmmi8-1b.png" alt="s" class="8x-x-1b" /></span></sup> are such. Corresponding rela-<br/>tions hold for all &#8220; tensors &#8221; subsequently to be introduced.
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686 <br/>(Rule for the addition and subtraction of tensors.)
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687 </p><!--l. 701--><p class="indent"> <span
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688 class="cmti-12">Covariant Four-vectors.</span>--We call four quantities <span
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689 class="cmmi-12">A</span><sub ><span
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690 class="cmmi-8"><img
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691 src="img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub> the <br/>components of a covariant four-vector, if for any arbitrary <br/>choice of the contravariant four-vector
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692 <span
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693 class="cmmi-12">B</span><sup ><span
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694 class="cmmi-8"><img
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695 src="img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sup>
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696 </p>
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697 <table width="100%"
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698 class="equation"><tr><td><a
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699 id="x1-8r6"></a>
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700 <center class="math-display" >
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701 <img
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702 src="img/078_A_19166x.png" alt=" n Sn AnB = Invariant " class="math-display" /></center></td><td width="5%">(6)</td></tr></table>
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703 <!--l. 710--><p class="nopar">
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704 </p><!--l. 713--><p class="noindent">The law of transformation of a covariant four-vector follows <br/>from this definition. For if we replace <span
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705 class="cmmi-12">B</span><sup ><span
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706 class="cmmi-8"><img
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707 src="img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sup> on the right-hand <br/>side of the equation
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708 </p>
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709 <center class="par-math-display" >
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710 <img
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711 src="img/078_A_19167x.png" alt="S A'sB's = S AnBn s n " class="par-math-display" /></center>
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712 <!--l. 722--><p class="nopar">
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713 </p><!--l. 726--><p class="noindent">by the expression resulting from the inversion of (5a),
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714 </p>
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715 <center class="par-math-display" >
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716 <img
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717 src="img/078_A_19168x.png" alt="S @xn-B's , s @x's " class="par-math-display" /></center>
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718 <!--l. 732--><p class="nopar">
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719 </p><!--l. 736--><p class="noindent">we obtain
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720 </p>
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721 <center class="par-math-display" >
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722 <img
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723 src="img/078_A_19169x.png" alt=" @x S B's S ---n' An = S B'sA's . s n@x s s
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724 " class="par-math-display" /></center>
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725 <!--l. 745--><p class="nopar">
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726 </p><!--l. 748--><p class="noindent">Since this equation is true for arbitrary values of the <span
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727 class="cmmi-12">B</span><span
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728 class="cmsy-10x-x-120">'</span><sup ><span
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729 class="cmmi-8"><img
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730 src="img/cmmi8-1b.png" alt="s" class="8x-x-1b" /></span></sup>, it <br/>follows that the law of transformation is
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731 </p>
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732 <table width="100%"
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733 class="equation"><tr><td><a
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734 id="x1-9r7"></a>
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735 <center class="math-display" >
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736 <img
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737 src="img/078_A_191610x.png" alt=" ' @xn A s = Sn @x'-An s " class="math-display" /></center></td><td width="5%">(7)</td></tr></table>
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738 <!--l. 756--><p class="nopar">
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739 </p><!--l. 760--><p class="indent"> <span
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740 class="cmti-12">Note on a Simplified Way of Writing the Expressions.</span>--<br/>A glance at the equations of this paragraph shows that there <br/>is always a summation with respect to the indices which <br/>occur twice under a sign of summation (e.g. the index <span
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741 class="cmmi-12"><img
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742 src="img/cmmi12-17.png" alt="n" class="12x-x-17" /> </span>in
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743 <br/>(5)), and only with respect to indices which occur twice. It <br/>is therefore possible, without loss of clearness, to omit the sign <br/>of summation. In its place we introduce the convention:--<br/>If an index occurs twice in one term of an expression, it is <br/>always to be summed unless the contrary is expressly stated.
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744 </p><!--l. 771--><p class="indent"> The difference between covariant and contravariant four-<br/>vectors lies in the law of transformation ((7) or (5) respectively). <br/>Both forms are tensors in the sense of the general remark <br/>above. Therein lies their importance. Following Ricci and
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745 <br/><pb/>
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746 </p><!--l. 778--><p class="indent">
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747
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748 </p><!--l. 779--><p class="noindent">Levi-Civita, we denote the contravariant character by placing <br/>the index above, the covariant by placing it below.
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749 </p>
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750 <div class="center" >
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751
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752 <!--l. 784--><p class="noindent">
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753 </p><!--l. 785--><p class="noindent"><span
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754 class="cmbsy-10x-x-120">§ </span><span
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755 class="cmbx-12">6. Tensors of the Second and Higher Ranks</span></p></div>
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756 <!--l. 789--><p class="indent"> <span
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757 class="cmti-12">Contravariant Tensors</span>.--If we form all the sixteen pro-<br/>ducts <span
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758 class="cmmi-12">A</span><sup ><span
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759 class="cmmi-8"><img
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760 src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img
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761 src="img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sup> of the components <span
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762 class="cmmi-12">A</span><sup ><span
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763 class="cmmi-8"><img
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764 src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /></span></sup> and <span
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765 class="cmmi-12">B</span><sup ><span
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766 class="cmmi-8"><img
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767 src="img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sup> of two contravariant <br/>four-vectors
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768 </p>
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769 <table width="100%"
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770 class="equation"><tr><td><a
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771 id="x1-10r8"></a>
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772 <center class="math-display" >
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773 <img
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774 src="img/078_A_191611x.png" alt=" mn m n A = A B " class="math-display" /></center></td><td width="5%">(8)</td></tr></table>
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775 <!--l. 797--><p class="nopar">
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776 </p><!--l. 801--><p class="noindent">then by (8) and (5a) <span
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777 class="cmmi-12">A</span><sup ><span
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778 class="cmmi-8"><img
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779 src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img
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780 src="img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sup> satisfies the law of transformation
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781 </p>
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782 <table width="100%"
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783 class="equation"><tr><td><a
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784 id="x1-11r9"></a>
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785 <center class="math-display" >
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786 <img
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787 src="img/078_A_191612x.png" alt=" 'st @x's @x't mn A = @x . @x A m n " class="math-display" /></center></td><td width="5%">(9)</td></tr></table>
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788 <!--l. 809--><p class="nopar">
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789 </p><!--l. 813--><p class="indent"> We call a thing which is described relatively to any system <br/>of reference by sixteen quantities, satisfying the law of trans-<br/>formation (9), a contravariant tensor of the second rank. Not <br/>every such tensor allows itself to be formed in accordance <br/>with (8) from two four-vectors, but it is easily shown that <br/>any
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790 given sixteen <span
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791 class="cmmi-12">A</span><sup ><span
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792 class="cmmi-8"><img
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793 src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img
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794 src="img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sup> can be represented as the sums of the <br/><span
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795 class="cmmi-12">A</span><sup ><span
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796 class="cmmi-8"><img
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797 src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /></span></sup><span
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798 class="cmmi-12">B</span><sup ><span
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799 class="cmmi-8"><img
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800 src="img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sup> of four appropriately selected pairs of four-vectors. <br/>Hence we can prove nearly all the laws which apply to the <br/>tensor of the second rank defined by (9) in the simplest <br/>manner by demonstrating them for the special tensors of the <br/>type (8).
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801 </p><!--l. 826--><p class="indent"> <span
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802 class="cmti-12">Contravariant Tensors of Any Rank</span>.--It is clear that, on <br/>the lines of (8) and (9), contravariant tensors of the third and <br/>higher ranks may also be defined with 4<sup ><span
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803 class="cmr-8">3</span></sup> components, and so <br/>on. In the same way it follows from (8) and (9) that the
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804 <br/>contravariant four-vector may be taken in this sense as a <br/>contravariant tensor of the first rank.
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805 </p><!--l. 834--><p class="indent"> <span
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806 class="cmti-12">Covariant Tensors</span>.--On the other hand, if we take the <br/>sixteen products <span
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807 class="cmmi-12">A</span><sub ><span
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808 class="cmmi-8"><img
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809 src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img
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810 src="img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub> of two covariant four-vectors <span
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811 class="cmmi-12">A</span><sub ><span
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812 class="cmmi-8"><img
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813 src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /></span></sub> and <span
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814 class="cmmi-12">B</span><sub ><span
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815 class="cmmi-8"><img
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816 src="img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub>,
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817 </p>
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818 <table width="100%"
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819 class="equation"><tr><td><a
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820 id="x1-12r10"></a>
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821 <center class="math-display" >
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822 <img
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823 src="img/078_A_191613x.png" alt="Amn = AmBn, " class="math-display" /></center></td><td width="5%">(10)</td></tr></table>
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824 <!--l. 841--><p class="nopar">
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825 </p><!--l. 845--><p class="noindent">the law of transformation for these is
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826 </p>
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827 <table width="100%"
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828 class="equation"><tr><td><a
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829 id="x1-13r11"></a>
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830 <center class="math-display" >
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831 <img
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832 src="img/078_A_191614x.png" alt="A' = @xm-@xn-A st @x's @x't mn " class="math-display" /></center></td><td width="5%">(11)</td></tr></table>
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833 <!--l. 853--><p class="nopar">
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834 </p><!--l. 857--><p class="indent"> This law of transformation defines the covariant tensor of <br/>the second rank. All our previous remarks on contravariant <br/>tensors apply equally to covariant tensors.
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835 <pb/>
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836
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837 </p><!--l. 864--><p class="indent">
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838
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839 </p><!--l. 865--><p class="indent"> <span
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840 class="cmcsc-10x-x-120">N<small
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841 class="small-caps">o</small><small
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842 class="small-caps">t</small><small
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843 class="small-caps">e</small></span>.--It is convenient to treat the scalar (or invariant) <br/>both as a contravariant and a covariant tensor of zero rank.
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844 </p><!--l. 869--><p class="indent"> <span
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845 class="cmti-12">Mixed Tensors</span>.--We may also define a tensor of the <br/>second rank of the type
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846 </p>
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847 <table width="100%"
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848 class="equation"><tr><td><a
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849 id="x1-14r12"></a>
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850 <center class="math-display" >
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851 <img
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852 src="img/078_A_191615x.png" alt="An = AmBn m " class="math-display" /></center></td><td width="5%">(12)</td></tr></table>
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853 <!--l. 875--><p class="nopar">
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854 </p><!--l. 879--><p class="noindent">which is covariant with respect to the index <span
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855 class="cmmi-12"><img
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856 src="img/cmmi12-16.png" alt="m" class="cmmi-12x-x-16" align="middle" /></span>, and contra-<br/>variant with respect to the index <span
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857 class="cmmi-12"><img
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858 src="img/cmmi12-17.png" alt="n" class="12x-x-17" /></span>. Its law of transforma-<br/>tion is
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859 </p>
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860 <table width="100%"
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861 class="equation"><tr><td><a
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862 id="x1-15r13"></a>
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863 <center class="math-display" >
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864 <img
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865 src="img/078_A_191616x.png" alt=" ' A't = @x-t@xm-An s @xn @x's m " class="math-display" /></center></td><td width="5%">(13)</td></tr></table>
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866 <!--l. 889--><p class="nopar">
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867 </p><!--l. 893--><p class="indent"> Naturally there are mixed tensors with any number of <br/>indices of covariant character, and any number of indices of <br/>contravariant character. Covariant and contravariant tensors <br/>may be looked upon as special cases of mixed tensors.
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868 </p><!--l. 899--><p class="indent"> <span
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869 class="cmti-12">Symmetrical Tensors</span>.--A contravariant, or a covariant <br/>tensor, of the second or higher rank is said to be symmetrical <br/>if two components, which are obtained the one from the other <br/>by the interchange of two indices, are equal. The tensor <span
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870 class="cmmi-12">A</span><sup ><span
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871 class="cmmi-8"><img
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872 src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img
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873 src="img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sup>,
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874 <br/>or the tensor <span
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875 class="cmmi-12">A</span><sub ><span
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876 class="cmmi-8"><img
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877 src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img
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878 src="img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub>, is thus symmetrical if for any combination <br/>of the indices <span
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879 class="cmmi-12"><img
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880 src="img/cmmi12-16.png" alt="m" class="cmmi-12x-x-16" align="middle" /></span>,
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881 <span
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882 class="cmmi-12"><img
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883 src="img/cmmi12-17.png" alt="n" class="12x-x-17" /></span>,
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884 </p>
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885
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886 <table width="100%"
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887 class="equation"><tr><td><a
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888 id="x1-16r14"></a>
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889 <center class="math-display" >
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890 <img
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891 src="img/078_A_191617x.png" alt=" mn nm A = A , " class="math-display" /></center></td><td width="5%">(14)</td></tr></table>
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892 <!--l. 910--><p class="nopar">
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893 </p><!--l. 914--><p class="noindent">or respectively,</p>
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894 <table width="100%"
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895 class="equation"><tr><td><a
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896 id="x1-17r15"></a>
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897 <center class="math-display" >
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898 <img
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899 src="img/078_A_191618x.png" alt="Amn = Anm. " class="math-display" /></center></td><td width="5%">(14a)</td></tr></table>
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900 <!--l. 920--><p class="nopar">
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901 </p><!--l. 924--><p class="indent"> It has to be proved that the symmetry thus defined is a <br/>property which is independent of the system of reference. <br/>It follows in fact from (9), when (14) is taken into consider-<br/>ation, that</p>
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902 <center class="par-math-display" >
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903 <img
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904 src="img/078_A_191619x.png" alt=" 'st @x's @x't- mn @x's-@x't nm @x's@x't mn 'ts A = @xm @xv A = @xm @xv A = @xn @xm A = A . " class="par-math-display" /></center>
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905 <!--l. 939--><p class="nopar">
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906 </p><!--l. 942--><p class="noindent">The last equation but one depends upon the interchange of <br/>the summation indices
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907 <span
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908 class="cmmi-12"><img
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909 src="img/cmmi12-16.png" alt="m" class="cmmi-12x-x-16" align="middle" /> </span>and <span
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910 class="cmmi-12"><img
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911 src="img/cmmi12-17.png" alt="n" class="12x-x-17" /></span>, i.e. merely on a change of <br/>notation.
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912 </p><!--l. 946--><p class="indent"> <span
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913 class="cmti-12">Antisymmetrical Tensors</span>.--A contravariant or a covariant <br/>tensor of the second, third, or fourth rank is said to be anti-<br/>symmetrical if two components, which are obtained the one <br/>from the other by the interchange of two indices, are equal <br/>and of opposite sign. The tensor <span
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914 class="cmmi-12">A</span><sup ><span
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915 class="cmmi-8"><img
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916 src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img
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917 src="img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sup>, or the tensor <span
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918 class="cmmi-12">A</span><sub >
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919 <span
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920 class="cmmi-8"><img
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921 src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img
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922 src="img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub>, is <br/>therefore
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923 antisymmetrical, if always <pb/>
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924 </p><!--l. 956--><p class="indent">
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925
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926 </p>
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927 <table width="100%"
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928 class="equation"><tr><td><a
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929 id="x1-18r15"></a>
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930 <center class="math-display" >
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931 <img
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932 src="img/078_A_191620x.png" alt="Amn = - Anm, " class="math-display" /></center></td><td width="5%">(15)</td></tr></table>
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933 <!--l. 959--><p class="nopar">
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934 </p><!--l. 963--><p class="noindent">or respectively,</p>
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935 <table width="100%"
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936 class="equation"><tr><td><a
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937 id="x1-19r16"></a>
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938 <center class="math-display" >
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939 <img
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940 src="img/078_A_191621x.png" alt="Amn = - Anm " class="math-display" /></center></td><td width="5%">(15a)</td></tr></table>
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941 <!--l. 968--><p class="nopar">
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942 </p><!--l. 972--><p class="indent"> Of the sixteen components <span
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943 class="cmmi-12">A</span><sup ><span
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944 class="cmmi-8"><img
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945 src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img
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946 src="img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sup>, the four components <span
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947 class="cmmi-12">A</span><sup ><span
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948 class="cmmi-8"><img
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949 src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img
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950 src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /></span></sup> <br/>vanish; the rest are equal and of opposite sign in pairs, so <br/>that there are only six components numerically different (a <br/>six-vector). Similarly we see that the antisymmetrical tensor <br/>of the third rank <span
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951 class="cmmi-12">A</span><sup ><span
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952 class="cmmi-8"><img
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953 src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img
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954 src="img/cmmi8-17.png" alt="n" class="8x-x-17" /><img
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955 src="img/cmmi8-1b.png" alt="s" class="8x-x-1b" /></span></sup> has only four numerically different
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956 <br/>components, while the antisymmetrical tensor <span
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957 class="cmmi-12">A</span><sup ><span
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958 class="cmmi-8"><img
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959 src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img
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960 src="img/cmmi8-17.png" alt="n" class="8x-x-17" /><img
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961 src="img/cmmi8-1b.png" alt="s" class="8x-x-1b" /><img
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962 src="img/cmmi8-1c.png" alt="t" class="8x-x-1c" /></span></sup> has only <br/>one. There are no antisymmetrical tensors of higher rank <br/>than the fourth in a continuum of four dimensions.
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963 </p>
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964 <div class="center" >
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965
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966 <!--l. 984--><p class="noindent">
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967 </p><!--l. 985--><p class="noindent"><span
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968 class="cmbsy-10x-x-120">§ </span><span
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969 class="cmbx-12">7. Multiplication of Tensors</span></p></div>
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970
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971 <!--l. 989--><p class="indent"> <span
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972 class="cmti-12">Outer Multiplication of Tensors</span>.--We obtain from the <br/>components of a tensor of rank <span
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973 class="cmmi-12">n </span>and of a tensor of rank <span
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974 class="cmmi-12">m </span> <br/>the components of a tensor of rank <span
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975 class="cmmi-12">n </span>+ <span
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976 class="cmmi-12">m </span>by multiplying <br/>each component of the one tensor by each component of the <br/>other. Thus, for example, the tensors T arise out of the <br/>tensors A and B of different kinds,
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977 </p>
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978 <center class="par-math-display" >
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979 <img
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980 src="img/078_A_191622x.png" alt="Tmns = AmnBs, Tmnst = AmnBst , st sn Tmn = AmnB . " class="par-math-display" /></center>
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981 <!--l. 1005--><p class="nopar">
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982 </p><!--l. 1009--><p class="indent"> The proof of the tensor character of T is given directly <br/>by the representations (8), (10), (12), or by the laws of trans-<br/>formation (9), (11), (13). The equations (8), (10), (12) are <br/>themselves examples of outer multiplication of tensors of the
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983 <br/>first rank.
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984 </p><!--l. 1015--><p class="indent"> &#8220; <span
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985 class="cmti-12">Contraction </span>&#8221; <span
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986 class="cmti-12">of a Mixed Tensor</span>.--From any mixed <br/>tensor we may form a tensor whose rank is less by two, by <br/>equating an index of covariant with one of contravariant <br/>character, and summing with respect to this index (&#8220; con-<br/>traction &#8221;). Thus, for example, from the mixed tensor of the <br/>fourth rank <span
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987 class="cmmi-12">A</span><sub><span
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988 class="cmmi-8"><img
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989 src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img
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990 src="img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub><sup><span
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991 class="cmmi-8"><img
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992 src="img/cmmi8-1b.png" alt="s" class="8x-x-1b" /><img
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993 src="img/cmmi8-1c.png" alt="t" class="8x-x-1c" /></span></sup>, we obtain the mixed tensor of the second <br/>rank,
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994 </p>
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995 <center class="par-math-display" >
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996 <img
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997 src="img/078_A_191623x.png" alt=" t mt mt A n = A mn (= Sm A mn), " class="par-math-display" /></center>
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998 <!--l. 1030--><p class="nopar">
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999 </p><!--l. 1034--><p class="noindent">and from this, by a second contraction, the tensor of zero <br/>rank,
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1000 </p>
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1001 <center class="par-math-display" >
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1002 <img
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1003 src="img/078_A_191624x.png" alt=" n mn A = A n = A mn . " class="par-math-display" /></center>
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1004 <!--l. 1040--><p class="nopar"> <pb/>
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1005 </p><!--l. 1047--><p class="indent">
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1006
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1007 </p><!--l. 1048--><p class="indent"> The proof that the result of contraction really possesses <br/>the tensor character is given either by the representation of a <br/>tensor according to the generalization of (12) in combination <br/>with (6), or by the generalization of (13).
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1008 </p><!--l. 1054--><p class="indent"> <span
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1009 class="cmti-12">Inner and Mixed Multiplication of Tensors</span>.--These consist <br/>in a combination of outer multiplication with contraction.
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1010 </p><!--l. 1058--><p class="indent"> <span
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1011 class="cmti-12">Examples</span>.--From the covariant tensor of the second rank <br/><span
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1012 class="cmmi-12">A</span><sub ><span
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1013 class="cmmi-8"><img
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1014 src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img
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1015 src="img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub> and the contravariant tensor of the first rank <span
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1016 class="cmmi-12">B</span><sup ><span
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1017 class="cmmi-8"><img
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1018 src="img/cmmi8-1b.png" alt="s" class="8x-x-1b" /></span></sup> we form <br/>by outer multiplication the mixed tensor
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1019 </p>
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1020 <center class="par-math-display" >
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1021 <img
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1022 src="img/078_A_191625x.png" alt="Dsmn = AmnBs . " class="par-math-display" /></center>
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1023 <!--l. 1066--><p class="nopar">
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1024 </p><!--l. 1069--><p class="noindent">On contraction with respect to the indices <span
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1025 class="cmmi-12"><img
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1026 src="img/cmmi12-17.png" alt="n" class="12x-x-17" /> </span>and <span
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1027 class="cmmi-12"><img
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1028 src="img/cmmi12-1b.png" alt="s" class="12x-x-1b" /></span>, we obtain <br/>the covariant four-vector
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1029 </p>
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1030 <center class="par-math-display" >
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1031 <img
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1032 src="img/078_A_191626x.png" alt="Dm = Dnmn = AmnBn . " class="par-math-display" /></center>
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1033 <!--l. 1076--><p class="nopar">
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1034 </p><!--l. 1080--><p class="noindent">This we call the inner product of the tensors <span
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1035 class="cmmi-12">A</span><sub ><span
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1036 class="cmmi-8"><img
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1037 src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img
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1038 src="img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub> and <span
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1039 class="cmmi-12">B</span><sup ><span
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1040 class="cmmi-8"><img
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1041 src="img/cmmi8-1b.png" alt="s" class="8x-x-1b" /></span></sup>. <br/>Analogously we form from the tensors <span
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1042 class="cmmi-12">A</span><sub ><span
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1043 class="cmmi-8"><img
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1044 src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img
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1045 src="img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub> and <span
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1046 class="cmmi-12">B</span><sup ><span
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1047 class="cmmi-8"><img
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1048 src="img/cmmi8-1b.png" alt="s" class="8x-x-1b" /><img
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1049 src="img/cmmi8-1c.png" alt="t" class="8x-x-1c" /></span></sup>, by outer <br/>multiplication and double contraction, the inner product <br/><span
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1050 class="cmmi-12">A</span><sub ><span
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1051 class="cmmi-8"><img
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1052 src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img
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1053 src="img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub><span
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1054 class="cmmi-12">B</span><sup ><span
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1055 class="cmmi-8"><img
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1056 src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img
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1057 src="img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sup>. By outer multiplication and one contraction, we <br/>obtain from <span
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1058 class="cmmi-12">A</span><sub ><span
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1059 class="cmmi-8"><img
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1060 src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img
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1061 src="img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub> and <span
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1062 class="cmmi-12">B</span><sup ><span
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1063 class="cmmi-8"><img
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1064 src="img/cmmi8-1b.png" alt="s" class="8x-x-1b" /><img
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1065 src="img/cmmi8-1c.png" alt="t" class="8x-x-1c" /></span></sup> the mixed tensor of the second rank <br/>D<sub>
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1066 <span
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1067 class="cmmi-8"><img
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1068 src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /></span></sub><sup><span
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1069 class="cmmi-8"><img
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1070 src="img/cmmi8-1c.png" alt="t" class="8x-x-1c" /></span></sup> = <span
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1071 class="cmmi-12">A</span><sub >
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1072 <span
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1073 class="cmmi-8"><img
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1074 src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img
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1075 src="img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub><span
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1076 class="cmmi-12">B</span><sup ><span
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1077 class="cmmi-8"><img
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1078 src="img/cmmi8-17.png" alt="n" class="8x-x-17" /><img
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1079 src="img/cmmi8-1c.png" alt="t" class="8x-x-1c" /></span></sup>. This operation may be aptly characterized as <br/>a mixed one, being &#8220; outer &#8221; with respect to the indices <span
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1080 class="cmmi-12"><img
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1081 src="img/cmmi12-16.png" alt="m" class="cmmi-12x-x-16" align="middle" /> </span> <br/>and <span
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1082 class="cmmi-12"><img
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1083 src="img/cmmi12-1c.png" alt="t" class="12x-x-1c" /></span>, and &#8220; inner &#8221; with respect to the indices <span
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1084 class="cmmi-12"><img
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1085 src="img/cmmi12-17.png" alt="n" class="12x-x-17" /> </span>and
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1086 <span
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1087 class="cmmi-12"><img
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1088 src="img/cmmi12-1b.png" alt="s" class="12x-x-1b" /></span>.
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1089 </p><!--l. 1094--><p class="indent"> We now prove a proposition which is often useful as evi-<br/>dence of tensor character. From what has just been ex-<br/>plained, <span
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1090 class="cmmi-12">A</span><sub ><span
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1091 class="cmmi-8"><img
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1092 src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img
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1093 src="img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub><span
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1094 class="cmmi-12">B</span><sup ><span
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1095 class="cmmi-8"><img
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1096 src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img
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1097 src="img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sup> is a scalar if <span
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1098 class="cmmi-12">A</span><sub >
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1099 <span
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1100 class="cmmi-8"><img
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1101 src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img
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1102 src="img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub> and <span
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1103 class="cmmi-12">B</span><sup ><span
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1104 class="cmmi-8"><img
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1105 src="img/cmmi8-1b.png" alt="s" class="8x-x-1b" /><img
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1106 src="img/cmmi8-1c.png" alt="t" class="8x-x-1c" /></span></sup>
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1107 are tensors. But <br/>we may also make the following assertion: If <span
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1108 class="cmmi-12">A</span><sub ><span
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1109 class="cmmi-8"><img
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1110 src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img
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1111 src="img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub><span
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1112 class="cmmi-12">B</span><sup ><span
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1113 class="cmmi-8"><img
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1114 src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img
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1115 src="img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sup> is <br/>a scalar
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1116 <span
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1117 class="cmti-12">for any choice of the tensor </span><span
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1118 class="cmmi-12">B</span><sup ><span
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1119 class="cmmi-8"><img
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1120 src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img
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1121 src="img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sup>, then <span
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1122 class="cmmi-12">A</span><sub >
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1123 <span
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1124 class="cmmi-8"><img
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1125 src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img
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1126 src="img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub> has tensor <br/>character. For, by hypothesis, for any substitution,
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1127 </p>
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1128 <center class="par-math-display" >
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1129 <img
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1130 src="img/078_A_191627x.png" alt="A' B'st = A Bmn . st mn " class="par-math-display" /></center>
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1131 <!--l. 1108--><p class="nopar">
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1132 </p><!--l. 1111--><p class="noindent">But by an inversion of (9)
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1133 </p>
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1134 <center class="par-math-display" >
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1135
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1136 <img
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1137 src="img/078_A_191628x.png" alt="Bmn = @xm-@xn-B'st . @x's@x't " class="par-math-display" /></center>
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1138 <!--l. 1118--><p class="nopar">
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1139 </p><!--l. 1121--><p class="noindent">This, inserted in the above equation, gives
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1140 </p>
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1141 <center class="par-math-display" >
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1142 <img
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1143 src="img/078_A_191629x.png" alt="( ) A' - @xm-@xn-A B'st = 0. st @x's @x't mn " class="par-math-display" /></center>
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1144 <!--l. 1129--><p class="nopar">
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1145 </p><!--l. 1132--><p class="noindent">This can only be satisfied for arbitrary values of <span
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1146 class="cmmi-12">B</span><span
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1147 class="cmsy-10x-x-120">'</span><sup ><span
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1148 class="cmmi-8"><img
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1149 src="img/cmmi8-1b.png" alt="s" class="8x-x-1b" /><img
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1150 src="img/cmmi8-1c.png" alt="t" class="8x-x-1c" /></span></sup> if the <br/><pb/>
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1151 </p><!--l. 1136--><p class="indent">
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1152
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1153 </p><!--l. 1137--><p class="noindent">bracket vanishes. The result then follows by equation (11). <br/>This rule applies correspondingly to tensors of any rank and <br/>character, and the proof is analogous in all cases.
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1154 </p><!--l. 1142--><p class="indent"> The rule may also be demonstrated in this form: If <span
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1155 class="cmmi-12">B</span><sup ><span
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1156 class="cmmi-8"><img
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1157 src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /></span></sup> <br/>and C<sup ><span
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1158 class="cmmi-8"><img
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1159 src="img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sup> are any vectors, and if, for all values of these, the <br/>inner product <span
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1160 class="cmmi-12">A</span><sub ><span
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1161 class="cmmi-8"><img
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1162 src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img
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1163 src="img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub><span
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1164 class="cmmi-12">B</span><sup ><span
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1165 class="cmmi-8"><img
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1166 src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /></span></sup>C<sup ><span
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1167 class="cmmi-8"><img
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1168 src="img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sup> is a scalar, then <span
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1169 class="cmmi-12">A</span><sub >
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1170 <span
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1171 class="cmmi-8"><img
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1172 src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img
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1173 src="img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub> is a covariant <br/>tensor. This latter proposition also holds good even if only <br/>the more special assertion is correct, that with any choice of <br/>the four-vector <span
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1174 class="cmmi-12">B</span><sup ><span
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1175 class="cmmi-8"><img
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1176 src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /></span></sup> the inner product <span
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1177 class="cmmi-12">A</span><sub ><span
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1178 class="cmmi-8"><img
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1179 src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img
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1180 src="img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub><span
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1181 class="cmmi-12">B</span><sup ><span
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1182 class="cmmi-8"><img
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1183 src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /></span></sup><span
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1184 class="cmmi-12">B</span><sup ><span
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1185 class="cmmi-8"><img
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1186 src="img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sup> is a scalar, if <br/>in addition it is known that <span
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1187 class="cmmi-12">A</span><sub >
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1188 <span
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1189 class="cmmi-8"><img
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1190 src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img
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1191 src="img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub>
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1192 satisfies the condition of <br/>symmetry <span
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1193 class="cmmi-12">A</span><sub ><span
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1194 class="cmmi-8"><img
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1195 src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img
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1196 src="img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub> = <span
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1197 class="cmmi-12">A</span><sub ><span
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1198 class="cmmi-8"><img
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1199 src="img/cmmi8-17.png" alt="n" class="8x-x-17" /><img
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1200 src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /></span></sub> For by the method given above we <br/>prove the tensor character of (<span
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1201 class="cmmi-12">A</span><sub ><span
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1202 class="cmmi-8"><img
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1203 src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img
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1204 src="img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub> + <span
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1205 class="cmmi-12">A</span><sub ><span
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1206 class="cmmi-8"><img
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1207 src="img/cmmi8-17.png" alt="n" class="8x-x-17" /><img
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1208 src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /></span></sub>), and from this the
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1209 <br/>tensor character of <span
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1210 class="cmmi-12">A</span><sub ><span
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1211 class="cmmi-8"><img
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1212 src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img
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1213 src="img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub> follows on account of symmetry. <br/>This also can be easily generalized to the case of covariant <br/>and contravariant tensors of any rank.
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1214 </p><!--l. 1160--><p class="indent"> Finally, there follows from what has been proved, this <br/>law, which may also be generalized for any tensors: If for <br/>any choice of the four-vector <span
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1215 class="cmmi-12">B</span><sup ><span
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1216 class="cmmi-8"><img
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1217 src="img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sup> the quantities <span
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1218 class="cmmi-12">A</span><sub ><span
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1219 class="cmmi-8"><img
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1220 src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img
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1221 src="img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub><span
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1222 class="cmmi-12">B</span><sup ><span
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1223 class="cmmi-8"><img
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1224 src="img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sup> form a <br/>tensor of the first rank, then <span
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1225 class="cmmi-12">A</span><sub >
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1226 <span
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1227 class="cmmi-8"><img
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1228 src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img
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1229 src="img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub> is a tensor of the second <br/>rank. For, if C<sup ><span
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1230 class="cmmi-8"><img
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1231 src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /></span></sup> is any four-vector, then on account of the
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1232 <br/>tensor character of <span
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1233 class="cmmi-12">A</span><sub ><span
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1234 class="cmmi-8"><img
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1235 src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img
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1236 src="img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub><span
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1237 class="cmmi-12">B</span><sup ><span
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1238 class="cmmi-8"><img
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1239 src="img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sup>, the inner product <span
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1240 class="cmmi-12">A</span><sub >
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1241 <span
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1242 class="cmmi-8"><img
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1243 src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img
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1244 src="img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub><span
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1245 class="cmmi-12">B</span><sup ><span
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1246 class="cmmi-8"><img
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1247 src="img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sup>C<sup ><span
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1248 class="cmmi-8"><img
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1249 src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /></span></sup> is a <br/>scalar for any choice of the two four-vectors <span
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1250 class="cmmi-12">B</span><sup ><span
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1251 class="cmmi-8"><img
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1252 src="img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sup> and C<sup ><span
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1253 class="cmmi-8"><img
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1254 src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /></span></sup>. From <br/>which the proposition follows.
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1255 </p>
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1256 <div class="center" >
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1257
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1258 <!--l. 1173--><p class="noindent">
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1259 </p><!--l. 1174--><p class="noindent"><span
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1260 class="cmbsy-10x-x-120">§ </span><span
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1261 class="cmbx-12">8. Some Aspects of the Fundamental Tensor </span><span
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1262 class="cmmi-12">g</span><sub ><span
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1263 class="cmmi-8"><img
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1264 src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img
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1265 src="img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub></p></div>
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1266 <!--l. 1179--><p class="indent"> <span
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1267 class="cmti-12">The Covariant Fundamental Tensor</span>.--In the invariant <br/>expression for the square of the linear element,
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1268 </p>
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1269 <center class="par-math-display" >
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1270 <img
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1271 src="img/078_A_191630x.png" alt=" 2 ds = gmndxmdxn , " class="par-math-display" /></center>
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1272 <!--l. 1186--><p class="nopar">
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1273 </p><!--l. 1190--><p class="noindent">the part played by the <span
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1274 class="cmmi-12">dx</span><sub ><span
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1275 class="cmmi-8"><img
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1276 src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /></span></sub> is that of a contravariant vector <br/>which may be chosen at will. Since further, <span
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1277 class="cmmi-12">g</span><sub ><span
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1278 class="cmmi-8"><img
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1279 src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img
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1280 src="img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub> = <span
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1281 class="cmmi-12">g</span><sub ><span
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1282 class="cmmi-8"><img
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1283 src="img/cmmi8-17.png" alt="n" class="8x-x-17" /><img
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1284 src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /></span></sub>, it <br/>follows from the considerations of the preceding paragraph <br/>that <span
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1285 class="cmmi-12">g</span><sub ><span
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1286 class="cmmi-8"><img
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1287 src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img
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1288 src="img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub> is a covariant tensor of the second rank. We call <br/>it the &#8220; fundamental tensor.&#8221; In what follows we deduce <br/>some properties of this tensor which, it is true, apply to any <br/>tensor of the second rank. But as the fundamental tensor <br/>plays a special part in our theory, which has its physical basis
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1289 <br/>in the peculiar effects of gravitation, it so happens that the <br/>relations to be developed are of importance to us only in the <br/>case of the fundamental tensor.
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1290 <pb/>
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1291 </p><!--l. 1206--><p class="indent">
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1292
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1293 </p><!--l. 1207--><p class="indent"> <span
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1294 class="cmti-12">The Contravariant Fundamental Tensor.</span>--If in the deter-<br/>minant formed by the elements <span
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1295 class="cmmi-12">g</span><sub ><span
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1296 class="cmmi-8"><img
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1297 src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img
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1298 src="img/cmmi8-17.png" alt="n" class="8x-x-17" />,</span></sub> we take the co-factor of <br/>each of the <span
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1299 class="cmmi-12">g</span><sub ><span
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1300 class="cmmi-8"><img
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1301 src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img
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1302 src="img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub> and divide it by the determinant <span
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1303 class="cmmi-12">g </span>= <img
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1304 src="img/078_A_191631x.png" alt="|gmn |" class="left" align="middle" /><span
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1305 class="cmmi-12">, </span> <br/>we obtain certain quantities <span
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1306 class="cmmi-12">g</span><sup ><span
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1307 class="cmmi-8"><img
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1308 src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img
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1309 src="img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sup><img
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1310 src="img/078_A_191632x.png" alt="(= gnm)" class="left" align="middle" /> which, as we shall
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1311 <br/>demonstrate, form a contravariant tensor.
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1312 </p><!--l. 1215--><p class="indent"> By a known property of determinants
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1313 </p>
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1314 <table width="100%"
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1315 class="equation"><tr><td><a
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1316 id="x1-20r16"></a>
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1317 <center class="math-display" >
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1318 <img
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1319 src="img/078_A_191633x.png" alt=" ns n gmsg = dm " class="math-display" /></center></td><td width="5%">(16)</td></tr></table>
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1320 <!--l. 1221--><p class="nopar">
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1321 </p><!--l. 1224--><p class="noindent">where the symbol <span
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1322 class="cmmi-12"><img
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1323 src="img/cmmi12-e.png" alt="d" class="12x-x-e" /></span><sub><span
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1324 class="cmmi-8"><img
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1325 src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /></span></sub><sup><span
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1326 class="cmmi-8"><img
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1327 src="img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sup> denotes 1 or 0, according as <span
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1328 class="cmmi-12"><img
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1329 src="img/cmmi12-16.png" alt="m" class="cmmi-12x-x-16" align="middle" /> </span>= <span
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1330 class="cmmi-12"><img
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1331 src="img/cmmi12-17.png" alt="n" class="12x-x-17" /> </span>or <br/><span
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1332 class="cmmi-12"><img
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1333 src="img/cmmi12-16.png" alt="m" class="cmmi-12x-x-16" align="middle" /></span><img
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1334 src="img/078_A_191634x.png" alt="/=" class="neq" align="middle" /><span
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1335 class="cmmi-12"><img
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1336 src="img/cmmi12-17.png" alt="n" class="12x-x-17" />.</span>
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1337 </p><!--l. 1227--><p class="indent"> Instead of the above expression for <span
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1338 class="cmti-12">ds</span><sup ><span
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1339 class="cmr-8">2</span></sup> we may thus write
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1340 </p>
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1341 <center class="par-math-display" >
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1342 <img
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1343 src="img/078_A_191635x.png" alt="gmsdsndxmdxn " class="par-math-display" /></center>
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1344 <!--l. 1233--><p class="nopar">
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1345 </p><!--l. 1237--><p class="noindent">or, by (16)
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1346 </p>
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1347 <center class="par-math-display" >
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1348 <img
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1349 src="img/078_A_191636x.png" alt=" st gmsgntg dxmdxn. " class="par-math-display" /></center>
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1350 <!--l. 1243--><p class="nopar">
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1351 </p><!--l. 1246--><p class="noindent">But, by the multiplication rules of the preceding paragraphs, <br/>the quantities
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1352 </p>
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1353 <center class="par-math-display" >
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1354 <img
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1355 src="img/078_A_191637x.png" alt="dqs = gmsdxm " class="par-math-display" /></center>
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1356 <!--l. 1253--><p class="nopar">
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1357 </p><!--l. 1257--><p class="noindent">form a covariant four-vector, and in fact an arbitrary vector, <br/>since the <span
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1358 class="cmmi-12">dx</span><sub ><span
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1359 class="cmmi-8"><img
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1360 src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /></span></sub> are arbitrary. By introducing this into our ex-<br/>pression we obtain
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1361 </p>
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1362
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1363 <center class="par-math-display" >
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1364 <img
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1365 src="img/078_A_191638x.png" alt="ds2 = gstdqsdqt . " class="par-math-display" /></center>
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1366 <!--l. 1264--><p class="nopar">
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1367 </p><!--l. 1267--><p class="noindent">Since this, with the arbitrary choice of the vector <span
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1368 class="cmmi-12">d<img
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1369 src="img/cmmi12-18.png" alt="q" class="cmmi-12x-x-18" align="middle" /></span><sub ><span
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1370 class="cmmi-8"><img
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1371 src="img/cmmi8-1b.png" alt="s" class="8x-x-1b" /></span></sub><span
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1372 class="cmmi-12">, </span>is a <br/>scalar, and <span
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1373 class="cmmi-12">g</span><sup ><span
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1374 class="cmmi-8"><img
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1375 src="img/cmmi8-1b.png" alt="s" class="8x-x-1b" /><img
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1376 src="img/cmmi8-1c.png" alt="t" class="8x-x-1c" /></span></sup> by its definition is symmetrical in the indices <br/><span
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1377 class="cmmi-12"><img
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1378 src="img/cmmi12-1b.png" alt="s" class="12x-x-1b" /> </span>and <span
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1379 class="cmmi-12"><img
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1380 src="img/cmmi12-1c.png" alt="t" class="12x-x-1c" />, </span>it follows from the results of the preceding paragraph <br/>that <span
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1381 class="cmmi-12">g</span><sup ><span
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1382 class="cmmi-8"><img
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1383 src="img/cmmi8-1b.png" alt="s" class="8x-x-1b" /><img
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1384 src="img/cmmi8-1c.png" alt="t" class="8x-x-1c" /></span></sup> is a contravariant tensor.
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1385 </p><!--l. 1275--><p class="indent"> It further follows from (16) that <span
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1386 class="cmmi-12"><img
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1387 src="img/cmmi12-e.png" alt="d" class="12x-x-e" /></span><sub ><span
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1388 class="cmmi-8"><img
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1389 src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /></span></sub> is also a tensor, which <br/>we may call the mixed fundamental tensor.
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1390 </p><!--l. 1278--><p class="indent"> <span
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1391 class="cmti-12">The Determinant of the Fundamental Tensor.</span>--By the <br/>rule for the multiplication of determinants
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1392 </p>
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1393 <center class="par-math-display" >
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1394 <img
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1395 src="img/078_A_191639x.png" alt="|g gan |= |g |× |gan |. ma ma " class="par-math-display" /></center>
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1396 <!--l. 1286--><p class="nopar">
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1397 </p><!--l. 1289--><p class="noindent">On the other hand
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1398 </p>
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1399 <center class="par-math-display" >
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1400 <img
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1401 src="img/078_A_191640x.png" alt=" | | |gmagan |= |dn |= 1. m " class="par-math-display" /></center>
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1402 <!--l. 1297--><p class="nopar">
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1403 </p><!--l. 1300--><p class="noindent">It therefore follows that
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1404 </p>
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1405 <table width="100%"
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1406 class="equation"><tr><td><a
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1407 id="x1-21r17"></a>
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1408 <center class="math-display" >
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1409 <img
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1410 src="img/078_A_191641x.png" alt=" mn |gmn |× |g |= 1 " class="math-display" /></center></td><td width="5%">(17)</td></tr></table>
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1411 <!--l. 1308--><p class="nopar">
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1412 </p><!--l. 1311--><p class="indent"> <span
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1413 class="cmti-12">The Volume Scalar.</span>--We seek first the law of transfor-<br/><pb/>
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1414 </p><!--l. 1315--><p class="indent">
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1415
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1416 </p><!--l. 1316--><p class="noindent">mation of the determinant <span
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1417 class="cmmi-12">g </span>= <img
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1418 src="img/078_A_191642x.png" alt="|gmn |" class="left" align="middle" /><span
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1419 class="cmmi-12">. </span>In accordance with <br/>(11)
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1420 </p>
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1421 <center class="par-math-display" >
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1422 <img
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1423 src="img/078_A_191643x.png" alt=" | | ' ||@xm--@x- || g = |@x' @x' gmn |. s t " class="par-math-display" /></center>
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1424 <!--l. 1324--><p class="nopar">
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1425 </p><!--l. 1327--><p class="noindent">Hence, by a double application of the rule for the multipli-<br/>cation of determinants, it follows that
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1426 </p>
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1427 <center class="par-math-display" >
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1428 <img
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1429 src="img/078_A_191644x.png" alt=" ||@x || ||@x || ||@x ||2 g'= ||--m'-|| . ||--n'-|| . |gmn | = ||--m'-|| g, @x s @xt @x s " class="par-math-display" /></center>
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1430 <!--l. 1338--><p class="nopar">
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1431 </p><!--l. 1342--><p class="noindent">or
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1432 </p>
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1433 <center class="par-math-display" >
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1434 <img
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1435 src="img/078_A_191645x.png" alt=" | | V~ -g'= ||@xm-|| V~ g. |@x's | " class="par-math-display" /></center>
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1436 <!--l. 1349--><p class="nopar">
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1437 </p><!--l. 1352--><p class="noindent">On the other hand, the law of transformation of the element <br/>of volume
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1438 </p>
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1439 <center class="par-math-display" >
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1440 <img
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1441 src="img/078_A_191646x.png" alt=" integral
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1442
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1443 dt = dx1dx2dx3dx4 " class="par-math-display" /></center>
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1444 <!--l. 1359--><p class="nopar">
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1445 </p><!--l. 1363--><p class="noindent">is, in accordance with the theorem of Jacobi,
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1446 </p>
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1447 <center class="par-math-display" >
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1448 <img
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1449 src="img/078_A_191647x.png" alt=" | | ' |@x's | dt = ||-@x- ||dt. m " class="par-math-display" /></center>
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1450 <!--l. 1370--><p class="nopar">
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1451 </p><!--l. 1373--><p class="noindent">By multiplication of the last two equations, we obtain
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1452 </p>
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1453 <table width="100%"
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1454 class="equation"><tr><td><a
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1455 id="x1-22r18"></a>
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1456
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1457 <center class="math-display" >
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1458 <img
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1459 src="img/078_A_191648x.png" alt=" V~ --- V~ -- g'dt '= g dt " class="math-display" /></center></td><td width="5%">(18)</td></tr></table>
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1460 <!--l. 1379--><p class="nopar">
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1461 </p><!--l. 1382--><p class="noindent">Instead of <img
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1462 src="img/078_A_191649x.png" alt=" V~ -- g" class="sqrt" /><span
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1463 class="cmmi-12">, </span>we introduce in what follows the quantity <br/><img
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1464 src="img/078_A_191650x.png" alt=" V~ ---- - g" class="sqrt" /><span
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1465 class="cmmi-12">, </span>which is always real on account of the hyperbolic <br/>character of the space-time continuum. Theinvariant <img
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1466 src="img/078_A_191651x.png" alt=" V~ ---- - g" class="sqrt" /><span
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1467 class="cmmi-12">d<img
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1468 src="img/cmmi12-1c.png" alt="t" class="12x-x-1c" /> </span> <br/>is equal to the magnitude of the four-dimensional element
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1469 <br/>of volume in the &#8220; local &#8221; system of reference, as measured <br/>with rigid rods and clocks in the sense of the special theory <br/>of relativity.
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1470 </p><!--l. 1391--><p class="indent"> <span
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1471 class="cmti-12">Note on the Character of the Space-time Continuum.</span>--Our <br/>assumption that the special theory of relativity can always <br/>be applied to an infinitely small region, implies that <span
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1472 class="cmti-12">ds</span><sup ><span
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1473 class="cmr-8">2</span></sup> can <br/>always be expressed in accordance with (1) by means of real
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1474 <br/>quantities <span
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1475 class="cmmi-12">dX</span><sub ><span
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1476 class="cmr-8">1</span></sub> <span
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1477 class="cmmi-12">...</span><span
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1478 class="cmmi-12">dX</span><sub ><span
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1479 class="cmr-8">4</span></sub><span
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1480 class="cmmi-12">. </span>If we denote by <span
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1481 class="cmmi-12">d<img
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1482 src="img/cmmi12-1c.png" alt="t" class="12x-x-1c" /></span><sub ><span
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1483 class="cmr-8">0</span></sub> the &#8220; natural &#8221; <br/>element of volume
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1484 <span
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1485 class="cmmi-12">dX</span><sub ><span
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1486 class="cmr-8">1</span></sub><span
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1487 class="cmmi-12">, dX</span><sub ><span
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1488 class="cmr-8">2</span></sub><span
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1489 class="cmmi-12">, dX</span><sub ><span
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1490 class="cmr-8">3</span></sub><span
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1491 class="cmmi-12">, dX</span><sub ><span
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1492 class="cmr-8">4</span></sub><span
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1493 class="cmmi-12">, </span>then
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1494 </p>
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1495 <table width="100%"
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1496 class="equation"><tr><td><a
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1497 id="x1-23r19"></a>
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1498 <center class="math-display" >
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1499 <img
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1500 src="img/078_A_191652x.png" alt=" V~ ---- dt0 = -gdt " class="math-display" /></center></td><td width="5%">(18a)</td></tr></table>
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1501 <!--l. 1404--><p class="nopar">
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1502 <pb/>
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1503 </p><!--l. 1411--><p class="indent">
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1504
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1505 </p><!--l. 1412--><p class="indent"> If <img
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1506 src="img/078_A_191653x.png" alt=" V~ ---- - g" class="sqrt" /> were to vanish at a point of the four-dimensional <br/>continuum, it would mean that at this point an infinitely small <br/>&#8220; natural &#8221; volume would correspond to a finite volume in <br/>the co-ordinates. Let us assume that this is never the case. <br/>Then <span
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1507 class="cmmi-12">g </span>cannot change sign. We will assume that, in the
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1508 <br/>sense of the special theory of relativity, <span
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1509 class="cmmi-12">g </span>always has a finite <br/>negative value. This is a hypothesis as to the physical <br/>nature of the continuum under consideration, and at the same <br/>time a convention as to the choice of co-ordinates.
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1510 </p><!--l. 1423--><p class="indent"> But if <span
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1511 class="cmsy-10x-x-120">-</span><span
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1512 class="cmmi-12">g </span>is always finite and positive, it is natural to settle <br/>the choice of co-ordinates <span
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1513 class="cmti-12">a posteriori </span>in such a way that this <br/>quantity is always equal to unity. We shall see later that <br/>by such a restriction of the choice of co-ordinates it is possible <br/>to achieve an important simplification of the laws of nature.
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1514 </p><!--l. 1430--><p class="indent"> In place of (18), we then have simply <span
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1515 class="cmmi-12">d<img
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1516 src="img/cmmi12-1c.png" alt="t" class="12x-x-1c" /></span><span
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1517 class="cmsy-10x-x-120">' </span>= <span
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1518 class="cmmi-12">d<img
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1519 src="img/cmmi12-1c.png" alt="t" class="12x-x-1c" />, </span>from <br/>which, in view of Jacobi&#8217;s theorem, it follows that
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1520 </p>
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1521 <table width="100%"
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1522 class="equation"><tr><td><a
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1523 id="x1-24r19"></a>
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1524 <center class="math-display" >
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1525 <img
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1526 src="img/078_A_191654x.png" alt="| | |@x' | ||---s|| = 1 @xm " class="math-display" /></center></td><td width="5%">(19)</td></tr></table>
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1527 <!--l. 1437--><p class="nopar">
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1528 </p><!--l. 1440--><p class="noindent">Thus, with this choice of co-ordinates, only substitutions for <br/>which the determinant is unity are permissible.
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1529 </p><!--l. 1445--><p class="indent"> But it would be erroneous to believe that this step indicates <br/>a partial abandonment of the general postulate of relativity. <br/>We do not ask &#8220; What are the laws of nature which are co-<br/>variant in face of all substitutions for which the determinant <br/>is unity? &#8221; but our question is &#8220; What are the generally co-<br/>variant laws of nature?&#8221; It is not until we have formulated <br/>these that we simplify their expression by a particular choice <br/>of the system of reference.
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1530 </p><!--l. 1455--><p class="indent"> <span
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1531 class="cmti-12">The Formation of New Tensors by Means of the Funda- </span> <br/><span
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1532 class="cmti-12">mental Tensor.</span>--Inner, outer, and mixed multiplication of a <br/>tensor by the fundamental tensor give tensors of different <br/>character and rank. For example,
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1533 </p>
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1534 <center class="par-math-display" >
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1535 <img
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1536 src="img/078_A_191655x.png" alt=" m ms A = g As, A = gmnAmn. " class="par-math-display" /></center>
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1537 <!--l. 1468--><p class="nopar">
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1538 </p><!--l. 1471--><p class="noindent">The following forms may be specially noted:--
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1539 </p>
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1540 <center class="par-math-display" >
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1541 <img
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1542 src="img/078_A_191656x.png" alt="Amn = gmagnbAab, ab Amn = gmagnbA " class="par-math-display" /></center>
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1543 <!--l. 1482--><p class="nopar"> <pb/>
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1544 </p><!--l. 1489--><p class="indent">
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1545
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1546 </p><!--l. 1490--><p class="noindent">(the &#8220;complements&#8221; of covariant and contravariant tensors <br/>respectively), and
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1547 </p>
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1548 <center class="par-math-display" >
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1549 <img
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1550 src="img/078_A_191657x.png" alt="B = g gabA . mn mn ab " class="par-math-display" /></center>
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1551 <!--l. 1497--><p class="nopar">
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1552 </p><!--l. 1500--><p class="noindent">We call <span
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1553 class="cmmi-12">B</span><sub ><span
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1554 class="cmmi-8"><img
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1555 src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img
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1556 src="img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub> the reduced tensor associated with <span
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1557 class="cmmi-12">A</span><sub ><span
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1558 class="cmmi-8"><img
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1559 src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img
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1560 src="img/cmmi8-17.png" alt="n" class="8x-x-17" /></span><span
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1561 class="cmsy-8"><sup class="htf"><strong>.</strong></sup></span></sub> Similarly,
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1562 </p>
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1563 <center class="par-math-display" >
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1564 <img
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1565 src="img/078_A_191658x.png" alt="Bmn = gmng Aab . ab " class="par-math-display" /></center>
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1566 <!--l. 1509--><p class="nopar">
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1567 </p><!--l. 1512--><p class="noindent">It may be noted that <span
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1568 class="cmmi-12">g</span><sup ><span
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1569 class="cmmi-8"><img
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1570 src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img
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1571 src="img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sup> is nothing more than the comple-<br/>ment of <span
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1572 class="cmmi-12">g</span><sub >
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1573 <span
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1574 class="cmmi-8"><img
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1575 src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img
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1576 src="img/cmmi8-17.png" alt="n" class="8x-x-17" />,</span></sub>
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1577 since
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1578 </p>
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1579 <center class="par-math-display" >
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1580 <img
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1581 src="img/078_A_191659x.png" alt="gmagnbg = gmadn = gmn. ab a " class="par-math-display" /></center>
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1582 <!--l. 1520--><p class="nopar">
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1583 </p>
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1584 <div class="center" >
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1585
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1586 <!--l. 1525--><p class="noindent">
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1587 </p><!--l. 1526--><p class="noindent"><span
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1588 class="cmsy-10x-x-120">§ </span>9 <span
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1589 class="cmbx-12">. The Equation of the Geodetic Line. The Motion of a </span> <br/><span
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1590 class="cmbx-12">Particle</span></p></div>
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1591 <!--l. 1531--><p class="indent"> As the linear element <span
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1592 class="cmti-12">ds </span>is defined independently of the <br/>system of co-ordinates, the line drawn between two points P <br/>and <span
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1593 class="cmmi-12">P</span><span
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1594 class="cmsy-10x-x-120">' </span>of the four-dimensional continuum in such a way that <br/><span
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1595 class="cmsy-10x-x-120"><img
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1596 src="img/cmsy10-c-73.png" alt=" integral " class="10-120x-x-73" /></span> <span
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1597 class="cmmi-12">ds </span>is stationary--a geodetic line--has a meaning which also <br/>is independent of the choice of co-ordinates. Its equation is
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1598 </p>
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1599 <table width="100%"
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1600 class="equation"><tr><td><a
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1601 id="x1-25r20"></a>
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1602 <center class="math-display" >
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1603 <img
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1604 src="img/078_A_191660x.png" alt=" integral P' d ds = 0 P " class="math-display" /></center></td><td width="5%">(20)</td></tr></table>
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1605 <!--l. 1541--><p class="nopar">
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1606 </p><!--l. 1544--><p class="noindent">Carrying out the variation in the usual way, we obtain <br/>from this equation four differential equations which define the <br/>geodetic line; this operation will be inserted here for the sake <br/>of completeness. Let <span
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1607 class="cmmi-12"><img
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1608 src="img/cmmi12-15.png" alt="c" class="12x-x-15" /> </span>be a function of the co-ordinates <span
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1609 class="cmmi-12">x</span><sub ><span
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1610 class="cmmi-8"><img
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1611 src="img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub><span
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1612 class="cmmi-12">, </span> <br/>and let this define a family of surfaces which intersect the <br/>required geodetic line as well as all the lines in immediate <br/>proximity to it which are drawn through the points P and <span
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1613 class="cmmi-12">P</span><span
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1614 class="cmsy-10x-x-120">'</span><span
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1615 class="cmmi-12">. </span> <br/>Any such line may then be supposed to be given by expres-<br/>sing its co-ordinates <span
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1616 class="cmmi-12">x</span><sub ><span
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1617 class="cmmi-8"><img
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1618 src="img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub> as functions of <span
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1619 class="cmmi-12"><img
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1620 src="img/cmmi12-15.png" alt="c" class="12x-x-15" />. </span>Let the symbol <span
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1621 class="cmmi-12"><img
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1622 src="img/cmmi12-e.png" alt="d" class="12x-x-e" /></span>
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1623 <br/>indicate the transition from a point of the required geodetic <br/>to the point corresponding to the same <span
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1624 class="cmmi-12"><img
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1625 src="img/cmmi12-15.png" alt="c" class="12x-x-15" /> </span>on a neighbouring <br/>line. Then for (20) we may substitute
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1626 </p>
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1627 <table width="100%"
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1628 class="equation"><tr><td><a
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1629 id="x1-26r21"></a>
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1630 <center class="math-display" >
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1631 <img
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1632 src="img/078_A_191661x.png" alt=" integral c2 dwdc = 0 c1 } 2 dxm-dxn- w = gmndc dc " class="math-display" /></center></td><td width="5%">(20a)</td></tr></table>
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1633 <!--l. 1567--><p class="nopar">
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1634 </p><!--l. 1570--><p class="noindent">But since <pb/>
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1635 </p><!--l. 1575--><p class="indent">
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1636
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1637 </p>
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1638 <center class="par-math-display" >
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1639 <img
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1640 src="img/078_A_191662x.png" alt=" { ( )} 1 1@gmn dxm dxn dxm dxn dw = -- --------------dxs + gmn----d ---- , w 2 @xs dc dc dc dc " class="par-math-display" /></center>
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1641 <!--l. 1583--><p class="nopar">
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1642 </p><!--l. 1587--><p class="noindent">and
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1643 </p>
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1644 <center class="par-math-display" >
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1645 <img
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1646 src="img/078_A_191663x.png" alt=" ( ) dxn- -d- d dc = dc (dxn) , " class="par-math-display" /></center>
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1647 <!--l. 1595--><p class="nopar">
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1648 </p><!--l. 1599--><p class="noindent">we obtain from (20a), after a partial integration,
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1649 </p>
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1650 <center class="par-math-display" >
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1651 <img
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1652 src="img/078_A_191664x.png" alt=" integral c 3 c ksdxsdc = 0, 1 " class="par-math-display" /></center>
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1653 <!--l. 1607--><p class="nopar">
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1654 </p><!--l. 1611--><p class="noindent">where
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1655 </p>
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1656 <table width="100%"
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1657 class="equation"><tr><td><a
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1658 id="x1-27r21"></a>
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1659 <center class="math-display" >
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1660 <img
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1661 src="img/078_A_191665x.png" alt=" { } d gmn dxm 1 @gmn dxm dxn ks= --- -------- - ---------------- dc w dc 2w @xs dc dc " class="math-display" /></center></td><td width="5%">(20b)</td></tr></table>
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1662 <!--l. 1622--><p class="nopar">
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1663 </p><!--l. 1625--><p class="noindent">Since the values of <span
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1664 class="cmmi-12"><img
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1665 src="img/cmmi12-e.png" alt="d" class="12x-x-e" />x</span><sub ><span
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1666 class="cmmi-8"><img
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1667 src="img/cmmi8-1b.png" alt="s" class="8x-x-1b" /></span></sub> are arbitrary, it follows from this that
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1668 </p>
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1669 <table width="100%"
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1670 class="equation"><tr><td><a
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1671 id="x1-28r21"></a>
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1672
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1673 <center class="math-display" >
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1674 <img
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1675 src="img/078_A_191666x.png" alt="ks = 0 " class="math-display" /></center></td><td width="5%">(20c)</td></tr></table>
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1676 <!--l. 1633--><p class="nopar">
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1677 </p><!--l. 1637--><p class="noindent">are the equations of the geodetic line.
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1678 </p><!--l. 1640--><p class="indent"> If <span
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1679 class="cmti-12">ds </span>does not vanish along the geodetic line we may <br/>choose the &#8220; length of the arc &#8221; <span
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1680 class="cmmi-12">s</span>, measured along the geodetic <br/>line, for the parameter <span
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1681 class="cmmi-12"><img
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1682 src="img/cmmi12-15.png" alt="c" class="12x-x-15" /></span>. Then <span
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1683 class="cmmi-12">w </span>= 1<span
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1684 class="cmmi-12">, </span>and in place of (20c) <br/>we obtain
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1685 </p>
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1686 <center class="par-math-display" >
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1687 <img
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1688 src="img/078_A_191667x.png" alt=" 2 d-xm- @gmndxs-dxm- 1-@gmn-dxmdxn- gmn ds2 + @xs ds ds - 2 @xs ds ds = 0 " class="par-math-display" /></center>
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1689 <!--l. 1652--><p class="nopar">
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1690 </p><!--l. 1656--><p class="noindent">or, by a mere change of notation,
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1691 </p>
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1692 <table width="100%"
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1693 class="equation"><tr><td><a
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1694 id="x1-29r21"></a>
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1695 <center class="math-display" >
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1696 <img
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1697 src="img/078_A_191668x.png" alt=" d2xa- dxm-dxn- gas ds2 + [mn, s] ds ds = 0 " class="math-display" /></center></td><td width="5%">(20d)</td></tr></table>
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1698 <!--l. 1664--><p class="nopar">
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1699 </p><!--l. 1668--><p class="noindent">where, following Christoffel, we have written
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1700 </p>
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1701 <table width="100%"
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1702 class="equation"><tr><td><a
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1703 id="x1-30r21"></a>
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1704
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1705 <center class="math-display" >
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1706 <img
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1707 src="img/078_A_191669x.png" alt=" ( ) [mn, s] = 1- @gms-+ @gns-- @gmn- 2 @xn @xm @xs " class="math-display" /></center></td><td width="5%">(21)</td></tr></table>
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1708 <!--l. 1678--><p class="nopar">
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1709 </p><!--l. 1682--><p class="noindent">Finally, if we multiply (20d) by <span
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1710 class="cmmi-12">g</span><sup ><span
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1711 class="cmmi-8"><img
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1712 src="img/cmmi8-1b.png" alt="s" class="8x-x-1b" /><img
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1713 src="img/cmmi8-1c.png" alt="t" class="8x-x-1c" /></span></sup> (outer multiplication with <br/>respect to <span
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1714 class="cmmi-12"><img
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1715 src="img/cmmi12-1c.png" alt="t" class="12x-x-1c" />, </span>inner with respect to <span
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1716 class="cmmi-12"><img
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1717 src="img/cmmi12-1b.png" alt="s" class="12x-x-1b" /></span>), we obtain the equations <br/>of the geodetic line in the form
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1718 </p>
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1719 <table width="100%"
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1720 class="equation"><tr><td><a
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1721 id="x1-31r22"></a>
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1722 <center class="math-display" >
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1723 <img
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1724 src="img/078_A_191670x.png" alt="d2xt dxm dxn -ds2-+ {mn, t} -ds--ds-= 0 " class="math-display" /></center></td><td width="5%">(22)</td></tr></table>
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1725 <!--l. 1692--><p class="nopar">
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1726 </p><!--l. 1696--><p class="noindent">where, following Christoffel, we have set
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1727 </p>
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1728 <table width="100%"
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1729 class="equation"><tr><td><a
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1730 id="x1-32r23"></a>
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1731 <center class="math-display" >
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1732 <img
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1733 src="img/078_A_191671x.png" alt="{mn, t}= gta [mn, a] " class="math-display" /></center></td><td width="5%">(23)</td></tr></table>
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1734 <!--l. 1704--><p class="nopar">
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1735 <pb/>
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1736 </p><!--l. 1711--><p class="indent">
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1737
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1738 </p>
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1739 <div class="center" >
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1740
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1741 <!--l. 1712--><p class="noindent">
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1742 </p><!--l. 1713--><p class="noindent"><span
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1743 class="cmsy-10x-x-120">§ </span>10 <span
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1744 class="cmbx-12">. The Formation of Tensors by Differentiation</span></p></div>
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1745 <!--l. 1718--><p class="indent"> With the help of the equation of the geodetic line we can <br/>now easily deduce the laws by which new tensors can be <br/>formed from old by differentiation. By this means we are <br/>able for the first time to formulate generally covariant <br/>differential equations. We reach this goal by repeated appli-<br/>cation of the following simple law:--
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1746 </p><!--l. 1725--><p class="indent"> If in our continuum a curve is given, the points of which <br/>are specified by the arcual distance <span
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1747 class="cmmi-12">s </span>measured from a fixed <br/>point on the curve, and if, further, <span
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1748 class="cmmi-12"><img
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1749 src="img/cmmi12-1e.png" alt="f" class="12x-x-1e" /> </span>is an invariant function <br/>of space, then <span
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1750 class="cmmi-12">d<img
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1751 src="img/cmmi12-1e.png" alt="f" class="12x-x-1e" /></span><img
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1752 src="img/078_A_191672x.png" alt="/" class="left" align="middle" /> <span
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1753 class="cmmi-12">ds</span> is also an invariant. The proof lies in <br/>this, that <span
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1754 class="cmti-12">ds </span>is an invariant as well as <span
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1755 class="cmmi-12">d<img
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1756 src="img/cmmi12-1e.png" alt="f" class="12x-x-1e" />.</span>
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1757 </p><!--l. 1734--><p class="indent"> As
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1758 </p>
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1759 <center class="par-math-display" >
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1760 <img
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1761 src="img/078_A_191673x.png" alt="df- -@f-dxm- ds = @xm ds " class="par-math-display" /></center>
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1762 <!--l. 1741--><p class="nopar">
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1763 </p><!--l. 1745--><p class="noindent">therefore
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1764 </p>
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1765 <center class="par-math-display" >
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1766 <img
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1767 src="img/078_A_191674x.png" alt=" @f dx y = ------m- dxm ds " class="par-math-display" /></center>
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1768 <!--l. 1751--><p class="nopar">
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1769 </p><!--l. 1755--><p class="noindent">is also an invariant, and an invariant for all curves starting <br/>from a point of the continuum, that is, for any choice of the <br/>vector <span
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1770 class="cmmi-12">dx</span><sub ><span
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1771 class="cmmi-8"><img
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1772 src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /></span></sub>. Hence it immediately follows that
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1773 </p>
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1774 <table width="100%"
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1775 class="equation"><tr><td><a
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1776 id="x1-33r24"></a>
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1777 <center class="math-display" >
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1778 <img
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1779 src="img/078_A_191675x.png" alt=" -@f- Am = @xm
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1780 " class="math-display" /></center></td><td width="5%">(24)</td></tr></table>
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1781 <!--l. 1763--><p class="nopar">
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1782 </p><!--l. 1767--><p class="noindent">is a covariant four-vector--the &#8220; gradient &#8221; of <span
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1783 class="cmmi-12"><img
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1784 src="img/cmmi12-1e.png" alt="f" class="12x-x-1e" />.</span>
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1785 </p><!--l. 1770--><p class="indent"> According to our rule, the differential quotient
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1786 </p>
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1787 <center class="par-math-display" >
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1788 <img
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1789 src="img/078_A_191676x.png" alt=" dy x = --- ds " class="par-math-display" /></center>
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1790 <!--l. 1775--><p class="nopar">
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1791 </p><!--l. 1779--><p class="noindent">taken on a curve, is similarly an invariant. Inserting the <br/>value of <span
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1792 class="cmmi-12"><img
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1793 src="img/cmmi12-20.png" alt="y" class="12x-x-20" />, </span>we obtain in the first place
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1794 </p>
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1795 <center class="par-math-display" >
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1796 <img
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1797 src="img/078_A_191677x.png" alt=" 2 2 x = -@--f--dxm-dxn-+ @f--d-xm-. @xm@xn ds ds @xm ds2 " class="par-math-display" /></center>
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1798 <!--l. 1787--><p class="nopar">
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1799 </p><!--l. 1790--><p class="noindent">The existence of a tensor cannot be deduced from this forth-<br/>with. But if we may take the curve along which we have <br/>differentiated to be a geodetic, we obtain on substitution for <br/><span
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1800 class="cmmi-12">d</span><sup ><span
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1801 class="cmr-8">2</span></sup><span
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1802 class="cmmi-12">x</span><sub >
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1803 <span
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1804 class="cmmi-8"><img
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1805 src="img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub><img
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1806 src="img/078_A_191678x.png" alt="/" class="left" align="middle" /> <span
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1807 class="cmmi-12">ds</span><sup ><span
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1808 class="cmr-8">2</span></sup> from (22),
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1809 </p>
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1810 <center class="par-math-display" >
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1811 <img
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1812 src="img/078_A_191679x.png" alt=" ( ) @2f @f dxm dxn x = --------- {mn, t} ---- --------. @xm@xn @xt ds ds " class="par-math-display" /></center>
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1813 <!--l. 1802--><p class="nopar">
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1814 </p><!--l. 1806--><p class="noindent">Since we may interchange the order of the differentiations, <br/><pb/>
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1815 </p><!--l. 1811--><p class="indent">
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1816
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1817 </p><!--l. 1812--><p class="noindent">and since by (23) and (21) <img
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1818 src="img/078_A_191680x.png" alt="{mn, t}" class="left" align="middle" /> is symmetrical in <span
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1819 class="cmmi-12"><img
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1820 src="img/cmmi12-16.png" alt="m" class="cmmi-12x-x-16" align="middle" /> </span>and <span
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1821 class="cmmi-12"><img
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1822 src="img/cmmi12-17.png" alt="n" class="12x-x-17" /></span>, <br/>it follows that the expression in brackets is symmetrical in <span
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1823 class="cmmi-12"><img
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1824 src="img/cmmi12-16.png" alt="m" class="cmmi-12x-x-16" align="middle" /> </span> <br/>and <span
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1825 class="cmmi-12"><img
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1826 src="img/cmmi12-17.png" alt="n" class="12x-x-17" /></span>. Since a geodetic line can be drawn in any direction <br/>from a point of the continuum, and therefore <span
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1827 class="cmmi-12">dx</span><sub ><span
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1828 class="cmmi-8"><img
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1829 src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /></span></sub><span
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1830 class="cmmi-12">/ds </span>is a four-<br/>vector with the ratio of its components arbitrary, it follows <br/>from the results of <span
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1831 class="cmsy-10x-x-120">§ </span>7 that
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1832 </p>
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1833 <table width="100%"
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1834 class="equation"><tr><td><a
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1835 id="x1-34r25"></a>
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1836 <center class="math-display" >
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1837 <img
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1838 src="img/078_A_191681x.png" alt=" @2f @f Amn = -------- - {mn,t} ---- @xm@xn @xt " class="math-display" /></center></td><td width="5%">(25)</td></tr></table>
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1839 <!--l. 1827--><p class="nopar">
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1840 </p><!--l. 1831--><p class="noindent">is a covariant tensor of the second rank. We have therefore <br/>come to this result: from the covariant tensor of the first <br/>rank
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1841 </p>
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1842 <center class="par-math-display" >
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1843 <img
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1844 src="img/078_A_191682x.png" alt="Am = -@f- @xm " class="par-math-display" /></center>
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1845 <!--l. 1839--><p class="nopar">
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1846 </p><!--l. 1843--><p class="noindent">we can, by differentiation, form a covariant tensor of the <br/>second rank
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1847 </p>
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1848 <table width="100%"
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1849 class="equation"><tr><td><a
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1850 id="x1-35r26"></a>
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1851 <center class="math-display" >
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1852 <img
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1853 src="img/078_A_191683x.png" alt="Amn = @Am--- {mn, t}At @xn " class="math-display" /></center></td><td width="5%">(26)</td></tr></table>
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1854 <!--l. 1851--><p class="nopar">
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1855
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1856 </p><!--l. 1854--><p class="noindent">We call the tensor <span
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1857 class="cmmi-12">A</span><sub ><span
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1858 class="cmmi-8"><img
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1859 src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img
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1860 src="img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub> the &#8220; extension &#8221; (covariant derivative) <br/>of the tensor <span
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1861 class="cmmi-12">A</span><sub ><span
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1862 class="cmmi-8"><img
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1863 src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /></span></sub> In the first place we can readily show that <br/>the operation leads to a tensor, even if the vector <span
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1864 class="cmmi-12">A</span><sub ><span
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1865 class="cmmi-8"><img
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1866 src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /></span></sub> cannot <br/>be represented as a gradient. To see this, we first observe
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1867 <br/>that
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1868 </p>
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1869 <center class="par-math-display" >
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1870 <img
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1871 src="img/078_A_191684x.png" alt=" @f y ---- @xm " class="par-math-display" /></center>
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1872 <!--l. 1864--><p class="nopar">
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1873 </p><!--l. 1868--><p class="noindent">is a covariant vector, if <span
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1874 class="cmmi-12"><img
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1875 src="img/cmmi12-20.png" alt="y" class="12x-x-20" /> </span>and <span
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1876 class="cmmi-12"><img
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1877 src="img/cmmi12-1e.png" alt="f" class="12x-x-1e" /> </span>are scalars. The sum of <br/>four such terms
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1878 </p>
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1879 <center class="par-math-display" >
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1880 <img
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1881 src="img/078_A_191685x.png" alt=" (1)f@(1) (4)@f(4) Sm = y -----+ .+ .+ y -----, @xm @xm " class="par-math-display" /></center>
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1882 <!--l. 1877--><p class="nopar">
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1883 </p><!--l. 1881--><p class="noindent">is also a covariant vector, if <span
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1884 class="cmmi-12"><img
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1885 src="img/cmmi12-20.png" alt="y" class="12x-x-20" /></span><sup ><img
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1886 src="img/078_A_191686x.png" alt="(1)" class="left" align="middle" /></sup><span
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1887 class="cmmi-12">, <img
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1888 src="img/cmmi12-1e.png" alt="f" class="12x-x-1e" /></span><sup ><img
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1889 src="img/078_A_191687x.png" alt="(1)" class="left" align="middle" /></sup> <img
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1890 src="img/078_A_191688x.png" alt=" ..." class="@cdots" /><span
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1891 class="cmmi-12"><img
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1892 src="img/cmmi12-20.png" alt="y" class="12x-x-20" /></span><sup ><img
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1893 src="img/078_A_191689x.png" alt="(4)" class="left" align="middle" /></sup><span
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1894 class="cmmi-12">, <img
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1895 src="img/cmmi12-1e.png" alt="f" class="12x-x-1e" /></span><sup ><img
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1896 src="img/078_A_191690x.png" alt="(4)" class="left" align="middle" /></sup> are scalars. <br/>But it is clear that any covariant vector can be represented <br/>in the form S<sub ><span
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1897 class="cmmi-8"><img
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1898 src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /></span></sub>. For, if <span
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1899 class="cmmi-12">A</span><sub ><span
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1900 class="cmmi-8"><img
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1901 src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /></span></sub> is a vector whose components are <br/>any given functions of the <span
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1902 class="cmmi-12">x</span><sub ><span
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1903 class="cmmi-8"><img
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1904 src="img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub>, we have only to put (in terms <br/>of the selected system of co-ordinates)
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1905 </p>
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1906 <center class="par-math-display" >
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1907 <img
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1908 src="img/078_A_191691x.png" alt=" (1) (1) y = A1, f = x1, y(2) = A , f(2) = x , 2 2 y(3) = A3, f(3) = x3, (4) (4) y = A4, f = x4, " class="par-math-display" /></center>
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1909 <!--l. 1903--><p class="nopar">
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1910 </p><!--l. 1907--><p class="noindent">in order to ensure that S<sub ><span
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1911 class="cmmi-8"><img
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1912 src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /></span></sub> shall be equal to <span
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1913 class="cmmi-12">A</span><sub ><span
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1914 class="cmmi-8"><img
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1915 src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /></span></sub>. <pb/>
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1916 </p><!--l. 1912--><p class="indent">
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1917
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1918 </p><!--l. 1913--><p class="indent"> Therefore, in order to demonstrate that <span
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1919 class="cmmi-12">A</span><sub ><span
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1920 class="cmmi-8"><img
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1921 src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img
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1922 src="img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub> is a tensor if <br/><span
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1923 class="cmti-12">any </span>covariant vector is inserted on the right-hand side for <span
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1924 class="cmmi-12">A</span><sub ><span
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1925 class="cmmi-8"><img
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1926 src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /></span></sub>, <br/>we only need show that this is so for the vector S<sub ><span
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1927 class="cmmi-8"><img
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1928 src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /></span></sub>. But for <br/>this latter purpose it is sufficient, as a glance at the right-<br/>hand side of (26) teaches us, to furnish the proof for the case
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1929 </p>
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1930 <center class="par-math-display" >
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1931 <img
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1932 src="img/078_A_191692x.png" alt="A = y @f-. m @xm " class="par-math-display" /></center>
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1933 <!--l. 1925--><p class="nopar">
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1934 </p><!--l. 1928--><p class="noindent">Now the right-hand side of (25) multiplied by <span
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1935 class="cmmi-12"><img
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1936 src="img/cmmi12-20.png" alt="y" class="12x-x-20" /></span>,
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1937 </p>
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1938 <center class="par-math-display" >
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1939 <img
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1940 src="img/078_A_191693x.png" alt=" 2 --@-f--- @f-- y@xm@xn - {mn,t}y @xt " class="par-math-display" /></center>
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1941 <!--l. 1935--><p class="nopar">
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1942 </p><!--l. 1939--><p class="noindent">is a tensor. Similarly
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1943 </p>
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1944 <center class="par-math-display" >
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1945 <img
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1946 src="img/078_A_191694x.png" alt="-@y--@f- @xm @xn " class="par-math-display" /></center>
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1947 <!--l. 1947--><p class="nopar">
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1948 </p><!--l. 1951--><p class="noindent">being the outer product of two vectors, is a tensor. By ad-<br/>dition, there follows the tensor character of
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1949 </p>
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1950 <center class="par-math-display" >
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1951 <img
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1952 src="img/078_A_191695x.png" alt=" ( ) ( ) -@-- y @f-- - {mn, t } y-@f- . @xn @xm @xt " class="par-math-display" /></center>
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1953 <!--l. 1960--><p class="nopar">
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1954 </p><!--l. 1964--><p class="noindent">As a glance at (26) will show, this completes the demon-<br/>stration for the vector</p>
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1955 <center class="par-math-display" >
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1956 <img
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1957 src="img/078_A_191696x.png" alt=" @f y ---- @xm
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1958 " class="par-math-display" /></center>
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1959 <!--l. 1971--><p class="nopar">
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1960 </p><!--l. 1975--><p class="noindent">and consequently, from what has already been proved, for any <br/>vector
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1961 <span
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1962 class="cmmi-12">A</span><sub ><span
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1963 class="cmmi-8"><img
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1964 src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /></span></sub>.
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1965 </p><!--l. 1978--><p class="indent"> By means of the extension of the vector, we may easily <br/>define the &#8220; extension &#8221; of a covariant tensor of any rank. <br/>This operation is a generalization of the extension of a vector. <br/>We restrict ourselves to the case of a tensor of the second <br/>rank, since this suffices to give a clear idea of the law of
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1966 <br/>formation.
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1967 </p><!--l. 1985--><p class="indent"> As has already been observed, any covariant tensor of the <br/>second rank can be represented <sup ><span
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1968 class="cmsy-8">*</span></sup> as the sum of tensors of the <br/>
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1969 </p><!--l. 1989--><p class="indent"> <sup ><span
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1970 class="cmsy-8">*</span></sup>By outer multiplication of the vector with arbitrary components A<sub >
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1971 <span
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1972 class="cmr-8">11</span></sub>, A<sub ><span
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1973 class="cmr-8">12</span></sub>,
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1974 <br/>A<sub ><span
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1975 class="cmr-8">13</span></sub>, A<sub ><span
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1976 class="cmr-8">14</span></sub> by the vector with components 1, 0, 0, 0, we produce a tensor with
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1977 <br/>components
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1978 </p>
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1979 <center class="par-math-display" >
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1980 <img
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1981 src="img/078_A_191697x.png" alt="A11 A12 A13 A14 0 0 0 0
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1982 0 0 0 0 0 0 0 0. " class="par-math-display" /></center>
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1983 <!--l. 2003--><p class="nopar">
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1984 </p><!--l. 2006--><p class="noindent">By the addition of four tensors of this type, we obtain the tensor <span
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1985 class="cmmi-12">A</span><sub ><span
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1986 class="cmmi-8"><img
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1987 src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img
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1988 src="img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub> with any
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1989 <br/>ssigned components. <pb/>
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1990 </p><!--l. 2012--><p class="indent">
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1991
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1992 </p><!--l. 2013--><p class="noindent">type <span
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1993 class="cmmi-12">A</span><sub ><span
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1994 class="cmmi-8"><img
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1995 src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /></span></sub><span
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1996 class="cmmi-12">B</span><sub ><span
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1997 class="cmmi-8"><img
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1998 src="img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub><span
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1999 class="cmmi-12">. </span>It will therefore be sufficient to deduce the ex-<br/>pression for the extension of a tensor of this special type. <br/>By (26) the expressions
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2000 </p>
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2001 <center class="par-math-display" >
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2002 <img
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2003 src="img/078_A_191698x.png" alt="@Am-- @x - {sm, t}At , s @Bn- @x - {sn, t}Bt , s " class="par-math-display" /></center>
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2004 <!--l. 2027--><p class="nopar">
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2005 </p><!--l. 2031--><p class="noindent">are tensors. On outer multiplication of the first by <span
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2006 class="cmmi-12">B</span><sub ><span
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2007 class="cmmi-8"><img
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2008 src="img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub>, and <br/>of the second by <span
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2009 class="cmmi-12">A</span><sub ><span
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2010 class="cmmi-8"><img
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2011 src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /></span></sub>, we obtain in each case a tensor of the <br/>third rank. By adding these, we have the tensor of the third <br/>rank
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2012 </p>
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2013 <table width="100%"
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2014 class="equation"><tr><td><a
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2015 id="x1-36r27"></a>
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2016 <center class="math-display" >
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2017 <img
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2018 src="img/078_A_191699x.png" alt=" @Amn- Amns = @xs - {sm, t }Atn - {sn, t}Amt " class="math-display" /></center></td><td width="5%">(27)</td></tr></table>
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2019 <!--l. 2042--><p class="nopar">
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2020 </p><!--l. 2046--><p class="noindent">where we have put <span
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2021 class="cmmi-12">A</span><sub ><span
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2022 class="cmmi-8"><img
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2023 src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img
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2024 src="img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub> = <span
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2025 class="cmmi-12">A</span><sub ><span
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2026 class="cmmi-8"><img
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2027 src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /></span></sub><span
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2028 class="cmmi-12">B</span><sub ><span
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2029 class="cmmi-8"><img
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2030 src="img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub><span
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2031 class="cmmi-12">. </span>As the right-hand side <br/>of (27) is linear and homogeneous in the <span
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2032 class="cmmi-12">A</span><sub ><span
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2033 class="cmmi-8"><img
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2034 src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img
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2035 src="img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub> and their first <br/>derivatives, this law of formation leads to a tensor, not only <br/>in the case of a tensor of the type <span
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2036 class="cmmi-12">A</span><sub ><span
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2037 class="cmmi-8"><img
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2038 src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /></span></sub><span
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2039 class="cmmi-12">B</span><sub ><span
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2040 class="cmmi-8"><img
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2041 src="img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub>, but also in the case <br/>of a sum of such tensors, i.e. in the case of any covariant <br/>tensor of the second rank. We call <span
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2042 class="cmmi-12">A</span><sub ><span
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2043 class="cmmi-8"><img
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2044 src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img
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2045 src="img/cmmi8-17.png" alt="n" class="8x-x-17" /><img
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2046 src="img/cmmi8-1b.png" alt="s" class="8x-x-1b" /></span></sub> the extension of the <br/>tensor <span
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2047 class="cmmi-12">A</span><sub ><span
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2048 class="cmmi-8"><img
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2049 src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img
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2050 src="img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub><span
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2051 class="cmmi-12">.</span>
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2052 </p><!--l. 2056--><p class="indent"> It is clear that (26) and (24) concern only special cases <br/>of extension (the extension of the tensors of rank one and <br/>zero respectively).
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2053 </p><!--l. 2060--><p class="indent"> In general, all special laws of formation of tensors are in-<br/>cluded in (27) in combination with the multiplication of <br/>tensors.
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2054 </p>
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2055 <div class="center" >
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2056
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2057 <!--l. 2065--><p class="noindent">
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2058 </p><!--l. 2066--><p class="noindent"><span
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2059 class="cmbsy-10x-x-120">§ </span><span
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2060 class="cmbx-12">11. Some Cases of Special Importance</span></p></div>
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2061
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2062 <!--l. 2070--><p class="indent"> <span
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2063 class="cmti-12">The Fundamental Tensor.</span>--We will first prove some <br/>lemmas which will be useful hereafter. By the rule for the <br/>differentiation of determinants
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2064 </p>
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2065 <table width="100%"
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2066 class="equation"><tr><td><a
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2067 id="x1-37r28"></a>
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2068 <center class="math-display" >
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2069 <img
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2070 src="img/078_A_1916100x.png" alt=" mn mn dg = g gdgmn = - gmngdg " class="math-display" /></center></td><td width="5%">(28)</td></tr></table>
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2071 <!--l. 2078--><p class="nopar">
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2072 </p><!--l. 2081--><p class="noindent">The last member is obtained from the last but one, if we bear <br/>in mind that
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2073 <span
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2074 class="cmmi-12">g</span><sub ><span
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2075 class="cmmi-8"><img
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2076 src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img
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2077 src="img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub><span
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2078 class="cmmi-12">g</span><sup ><sub ><span
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2079 class="cmmi-6"><img
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2080 src="img/cmmi6-16.png" alt="m" class="cmmi-6x-x-16" align="middle" /></span><span
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2081 class="cmsy-6">'</span><span
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2082 class="cmmi-6"><img
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2083 src="img/cmmi6-17.png" alt="n" class="6x-x-17" /></span></sub></sup> = <span
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2084 class="cmmi-12"><img
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2085 src="img/cmmi12-e.png" alt="d" class="12x-x-e" /></span><sub><span
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2086 class="cmmi-8"><img
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2087 src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /></span></sub><sup><span
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2088 class="cmmi-8"><img
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2089 src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /></span><span
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diff changeset
2090 class="cmsy-8">'</span></sup><span
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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diff changeset
2091 class="cmmi-12">, </span>so that <span
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diff changeset
2092 class="cmmi-12">g</span><sub ><span
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diff changeset
2093 class="cmmi-8"><img
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diff changeset
2094 src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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diff changeset
2095 src="img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub><span
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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diff changeset
2096 class="cmmi-12">g</span><sup ><span
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parents:
diff changeset
2097 class="cmmi-8"><img
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diff changeset
2098 src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img
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diff changeset
2099 src="img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sup> = 4<span
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diff changeset
2100 class="cmmi-12">, </span>and conse-<br/>quently
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diff changeset
2101 </p>
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diff changeset
2102 <center class="par-math-display" >
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diff changeset
2103 <img
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diff changeset
2104 src="img/078_A_1916101x.png" alt="gmndgmn + gmndg = 0. mn " class="par-math-display" /></center>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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diff changeset
2105 <!--l. 2090--><p class="nopar"> <pb/>
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diff changeset
2106 </p><!--l. 2097--><p class="indent">
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diff changeset
2107
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2108 </p><!--l. 2098--><p class="noindent">From (28), it follows that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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diff changeset
2109 </p>
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diff changeset
2110 <table width="100%"
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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diff changeset
2111 class="equation"><tr><td><a
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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diff changeset
2112 id="x1-38r29"></a>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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diff changeset
2113 <center class="math-display" >
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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diff changeset
2114 <img
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diff changeset
2115 src="img/078_A_1916102x.png" alt=" 1 @ V~ --g @ log(- g) @g @gmn V~ -----------= 12-----------= 12gmn --mn-= 12gmn-----. - g @xs @xs @xs @xs " class="math-display" /></center></td><td width="5%">(29)</td></tr></table>
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2116 <!--l. 2110--><p class="nopar">
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2117 </p><!--l. 2113--><p class="noindent">Further, from <span
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2118 class="cmmi-12">g</span><sub ><span
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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diff changeset
2119 class="cmmi-8"><img
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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2120 src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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diff changeset
2121 src="img/cmmi8-1b.png" alt="s" class="8x-x-1b" /></span></sub><span
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diff changeset
2122 class="cmmi-12">g</span><sup ><span
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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diff changeset
2123 class="cmmi-8"><img
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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diff changeset
2124 src="img/cmmi8-17.png" alt="n" class="8x-x-17" /><img
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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diff changeset
2125 src="img/cmmi8-1b.png" alt="s" class="8x-x-1b" /></span></sup> = <span
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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diff changeset
2126 class="cmmi-12"><img
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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diff changeset
2127 src="img/cmmi12-e.png" alt="d" class="12x-x-e" /></span><sub>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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diff changeset
2128 <span
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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diff changeset
2129 class="cmmi-8"><img
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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2130 src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /></span></sub><sup><span
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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diff changeset
2131 class="cmmi-8"><img
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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diff changeset
2132 src="img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sup><span
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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2133 class="cmmi-12">, </span>it follows on differentiation that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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2134 </p>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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diff changeset
2135 <table width="100%"
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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diff changeset
2136 class="equation"><tr><td><a
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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diff changeset
2137 id="x1-39r30"></a>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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diff changeset
2138 <center class="math-display" >
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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diff changeset
2139 <img
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2140 src="img/078_A_1916103x.png" alt="gms dgns = - gnsdgms ns } gms@g---= - gns@gms- @xc @xc " class="math-display" /></center></td><td width="5%">(30)</td></tr></table>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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2141 <!--l. 2127--><p class="nopar">
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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2142 </p><!--l. 2130--><p class="noindent">From these, by mixed multiplication by <span
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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diff changeset
2143 class="cmmi-12">g</span><sup ><span
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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2144 class="cmmi-8"><img
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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2145 src="img/cmmi8-1b.png" alt="s" class="8x-x-1b" /><img
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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diff changeset
2146 src="img/cmmi8-1c.png" alt="t" class="8x-x-1c" /></span></sup> and <span
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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diff changeset
2147 class="cmmi-12">g</span><sub >
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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2148 <span
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diff changeset
2149 class="cmmi-8"><img
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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2150 src="img/cmmi8-17.png" alt="n" class="8x-x-17" /><img
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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2151 src="img/cmmi8-15.png" alt="c" class="8x-x-15" /></span></sub> re-<br/>spectively, and a change of notation for the indices, we have
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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diff changeset
2152 </p>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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diff changeset
2153 <table width="100%"
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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diff changeset
2154 class="equation"><tr><td><a
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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diff changeset
2155 id="x1-40r31"></a>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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2156 <center class="math-display" >
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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2157 <img
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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2158 src="img/078_A_1916104x.png" alt=" mn ma nb dg = - g g dgab @gmn ma nb@gab } @x---= - g g @x--- s s " class="math-display" /></center></td><td width="5%">(31)</td></tr></table>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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2159 <!--l. 2144--><p class="nopar">
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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2160 </p><!--l. 2148--><p class="noindent">and</p>
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2161 <table width="100%"
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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2162 class="equation"><tr><td><a
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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2163 id="x1-41r32"></a>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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2164 <center class="math-display" >
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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2165 <img
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2166 src="img/078_A_1916105x.png" alt=" ab dgmn = - gmagnbdg @gmn @gab} -----= -gmagnb ----- @xs @xs " class="math-display" /></center></td><td width="5%">(32)</td></tr></table>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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2167 <!--l. 2162--><p class="nopar">
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2168 </p><!--l. 2166--><p class="noindent">The relation (31) admits of a transformation, of which we <br/>also have frequently to make use. From (21)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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2169 </p>
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2170 <table width="100%"
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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2171 class="equation"><tr><td><a
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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2172 id="x1-42r33"></a>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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2173 <center class="math-display" >
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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2174 <img
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2175 src="img/078_A_1916106x.png" alt="@g ---ab = [as, b] + [bs, a] @xs " class="math-display" /></center></td><td width="5%">(33)</td></tr></table>
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2176 <!--l. 2176--><p class="nopar">
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2177 </p><!--l. 2180--><p class="noindent">Inserting this in the second formula of (31), we obtain, in <br/>view of (23)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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2178 </p>
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2179 <table width="100%"
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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2180 class="equation"><tr><td><a
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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2181 id="x1-43r34"></a>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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diff changeset
2182
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diff changeset
2183 <center class="math-display" >
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diff changeset
2184 <img
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diff changeset
2185 src="img/078_A_1916107x.png" alt=" mn @g---= - gmt {ts, n} - gnt {ts, m} @xs " class="math-display" /></center></td><td width="5%">(34)</td></tr></table>
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2186 <!--l. 2190--><p class="nopar">
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diff changeset
2187 </p><!--l. 2194--><p class="noindent">Substituting the right-hand side of (34) in (29), we have
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diff changeset
2188 </p>
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diff changeset
2189 <table width="100%"
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diff changeset
2190 class="equation"><tr><td><a
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diff changeset
2191 id="x1-44r35"></a>
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2192 <center class="math-display" >
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diff changeset
2193 <img
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diff changeset
2194 src="img/078_A_1916108x.png" alt=" 1 @V ~ ---g V~ ---------- = {ms, m} - g @xs " class="math-display" /></center></td><td width="5%">(29a)</td></tr></table>
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2195 <!--l. 2202--><p class="nopar">
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diff changeset
2196 </p><!--l. 2206--><p class="indent"> <span
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diff changeset
2197 class="cmti-12">The &#8220;Divergence&#8221; of a Contravariant Vector.</span>--If we <br/>take the inner product of (26) by the contravariant funda-<br/>mental tensor <span
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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diff changeset
2198 class="cmmi-12">g</span><sup ><span
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diff changeset
2199 class="cmmi-8"><img
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diff changeset
2200 src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img
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diff changeset
2201 src="img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sup><span
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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diff changeset
2202 class="cmmi-12">, </span>the right-hand side, after a transformation <br/>of the first term, assumes the form</p>
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2203 <center class="par-math-display" >
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2204 <img
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diff changeset
2205 src="img/078_A_1916109x.png" alt=" mn ( ) --@- mn @g--- 1 ta @gma- @gna- @gmn- mn @xn (g Am) - Am @xn - 2g @xn + @xm - @xa g At. " class="par-math-display" /></center>
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2206 <!--l. 2222--><p class="nopar"> <pb/>
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2207 </p><!--l. 2229--><p class="indent">
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2208
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2209 </p><!--l. 2230--><p class="noindent">In accordance with (31) and (29), the last term of this ex-<br/>pression may be written
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2210 </p>
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2211 <center class="par-math-display" >
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2212 <img
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2213 src="img/078_A_1916110x.png" alt=" V~ ---- 1@gtn 1@gtm 1 @ -g mn 2-----At + 2-----At + V~ -----------g At. @xn @xm -g @xa " class="par-math-display" /></center>
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2214 <!--l. 2243--><p class="nopar">
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2215 </p><!--l. 2246--><p class="noindent">As the symbols of the indices of summation are immaterial, <br/>the first two terms of this expression cancel the second of the <br/>one above. If we then write <span
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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2216 class="cmmi-12">g</span><sup ><span
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2217 class="cmmi-8"><img
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2218 src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img
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2219 src="img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sup><span
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2220 class="cmmi-12">A</span><sub >
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2221 <span
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2222 class="cmmi-8"><img
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2223 src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /></span></sub> = <span
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2224 class="cmmi-12">A</span><sup ><span
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2225 class="cmmi-8"><img
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2226 src="img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sup><span
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2227 class="cmmi-12">, </span>so that <span
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2228 class="cmmi-12">A</span><sup ><span
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2229 class="cmmi-8"><img
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2230 src="img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sup> like <span
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2231 class="cmmi-12">A</span><sub >
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2232 <span
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2233 class="cmmi-8"><img
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2234 src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /></span></sub> <br/>is an arbitrary vector, we finally obtain
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2235 </p>
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2236 <table width="100%"
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2237 class="equation"><tr><td><a
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2238 id="x1-45r35"></a>
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2239 <center class="math-display" >
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2240 <img
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2241 src="img/078_A_1916111x.png" alt="P = V~ -1--@--( V~ --gAn) - g@xn " class="math-display" /></center></td><td width="5%">(35)</td></tr></table>
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2242 <!--l. 2258--><p class="nopar">
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2243 </p><!--l. 2261--><p class="noindent">This scalar is the <span
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2244 class="cmti-12">divergence </span>of the contravariant vector <span
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2245 class="cmmi-12">A</span><sup ><span
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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2246 class="cmmi-8"><img
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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2247 src="img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sup>.
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diff changeset
2248 </p><!--l. 2265--><p class="indent"> <span
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2249 class="cmti-12">The &#8220;Curl&#8221; of a Covariant Vector.</span>--The second term in <br/>(26) is symmetrical in the indices <span
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2250 class="cmmi-12"><img
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2251 src="img/cmmi12-16.png" alt="m" class="cmmi-12x-x-16" align="middle" /> </span>and <span
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2252 class="cmmi-12"><img
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2253 src="img/cmmi12-17.png" alt="n" class="12x-x-17" /></span>. Therefore <br/><span
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2254 class="cmmi-12">A</span><sub ><span
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2255 class="cmmi-8"><img
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2256 src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img
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2257 src="img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub> <span
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diff changeset
2258 class="cmsy-10x-x-120">- </span><span
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2259 class="cmmi-12">A</span><sub ><span
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2260 class="cmmi-8"><img
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2261 src="img/cmmi8-17.png" alt="n" class="8x-x-17" /><img
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2262 src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /></span></sub> is a particularly simply constructed antisym-<br/>metrical tensor. We obtain
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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2263 </p>
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2264 <table width="100%"
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2265 class="equation"><tr><td><a
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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2266 id="x1-46r36"></a>
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2267 <center class="math-display" >
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2268 <img
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2269 src="img/078_A_1916112x.png" alt=" @Am-- @An- Bmn = @x - @x n m " class="math-display" /></center></td><td width="5%">(36)</td></tr></table>
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2270
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2271 <!--l. 2276--><p class="nopar">
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2272 </p><!--l. 2280--><p class="indent"> <span
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2273 class="cmti-12">Antisymmetrical Extension of a Six-vector.</span>--Applying <br/>(27) to an antisymmetrical tensor of the second rank <span
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2274 class="cmmi-12">A</span><sub ><span
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2275 class="cmmi-8"><img
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2276 src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img
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2277 src="img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub><span
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2278 class="cmmi-12">, </span> <br/>forming in addition the two equations which arise through <br/>cyclic permutations of the indices, and adding these three <br/>equations, we obtain the tensor of the third rank
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2279 </p>
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2280 <table width="100%"
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2281 class="equation"><tr><td><a
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2282 id="x1-47r37"></a>
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2283 <center class="math-display" >
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2284 <img
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2285 src="img/078_A_1916113x.png" alt=" @Amn @Ans @Asm Bmns = Amns + Ansm + Asmn = ----- + -----+ ------ @xs @xm @xn " class="math-display" /></center></td><td width="5%">(37)</td></tr></table>
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2286 <!--l. 2294--><p class="nopar">
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2287 </p><!--l. 2298--><p class="noindent">which it is easy to prove is antisymmetrical.
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2288 </p><!--l. 2301--><p class="indent"> <span
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2289 class="cmti-12">The Divergence of a Six-vector.</span>--Taking the mixed pro-<br/>duct of (27) by
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2290 <span
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2291 class="cmmi-12">g</span><sup ><span
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2292 class="cmmi-8"><img
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2293 src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img
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2294 src="img/cmmi8-b.png" alt="a" class="8x-x-b" /></span></sup><span
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2295 class="cmmi-12">g</span><sup ><span
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2296 class="cmmi-8"><img
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2297 src="img/cmmi8-17.png" alt="n" class="8x-x-17" /><img
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2298 src="img/cmmi8-c.png" alt="b" class="cmmi-8x-x-c" align="middle" /></span></sup><span
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2299 class="cmmi-12">, </span>we also obtain a tensor. The first <br/>term on the right-hand side of (27) may be written in the <br/>form
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2300 </p>
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2301 <center class="par-math-display" >
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2302 <img
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2303 src="img/078_A_1916114x.png" alt=" nb ma -@-(gmagnbAmn) - gma@g---Amn - gnb@g---Amn. @xs @xs @xs " class="par-math-display" /></center>
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2304 <!--l. 2314--><p class="nopar">
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2305 </p><!--l. 2317--><p class="noindent">If we write <span
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2306 class="cmmi-12">A</span><sub><span
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2307 class="cmmi-8"><img
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2308 src="img/cmmi8-1b.png" alt="s" class="8x-x-1b" /></span></sub><sup><span
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2309 class="cmmi-8"><img
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2310 src="img/cmmi8-b.png" alt="a" class="8x-x-b" /><img
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2311 src="img/cmmi8-c.png" alt="b" class="cmmi-8x-x-c" align="middle" /></span></sup> for <span
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2312 class="cmmi-12">g</span><sup ><span
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2313 class="cmmi-8"><img
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2314 src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img
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2315 src="img/cmmi8-b.png" alt="a" class="8x-x-b" /></span></sup><span
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2316 class="cmmi-12">g</span><sup ><span
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2318 src="img/cmmi8-17.png" alt="n" class="8x-x-17" /><img
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2319 src="img/cmmi8-c.png" alt="b" class="cmmi-8x-x-c" align="middle" /></span></sup><span
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2320 class="cmmi-12">A</span><sub >
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2321 <span
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2322 class="cmmi-8"><img
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2323 src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img
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2324 src="img/cmmi8-17.png" alt="n" class="8x-x-17" /><img
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2325 src="img/cmmi8-1b.png" alt="s" class="8x-x-1b" /></span></sub> and <span
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2326 class="cmmi-12">A</span><sup ><span
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2327 class="cmmi-8"><img
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2328 src="img/cmmi8-b.png" alt="a" class="8x-x-b" /><img
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2329 src="img/cmmi8-c.png" alt="b" class="cmmi-8x-x-c" align="middle" /></span></sup> for <span
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2330 class="cmmi-12">g</span><sup ><span
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2331 class="cmmi-8"><img
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2332 src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img
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2333 src="img/cmmi8-b.png" alt="a" class="8x-x-b" /></span></sup><span
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2334 class="cmmi-12">g</span><sup ><span
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2335 class="cmmi-8"><img
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2336 src="img/cmmi8-17.png" alt="n" class="8x-x-17" /><img
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2337 src="img/cmmi8-c.png" alt="b" class="cmmi-8x-x-c" align="middle" /></span></sup><span
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2338 class="cmmi-12">A</span><sub >
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2339 <span
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2340 class="cmmi-8"><img
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2341 src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img
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2342 src="img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub><span
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2343 class="cmmi-12">, </span>and in <br/>the transformed first term replace
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2344 </p>
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2345 <center class="par-math-display" >
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2346 <img
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2347 src="img/078_A_1916115x.png" alt="@gnb @gma -----and ----- @xs @xs " class="par-math-display" /></center>
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2348 <!--l. 2328--><p class="nopar"> <pb/>
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2349 </p><!--l. 2335--><p class="indent">
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2350
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2351 </p><!--l. 2336--><p class="noindent">by their values as given by (34), there results from the right-<br/>hand side of (27) an expression consisting of seven terms, of <br/>which four cancel, and there remains
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2352 </p>
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2353 <table width="100%"
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2354 class="equation"><tr><td><a
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2355 id="x1-48r38"></a>
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2356 <center class="math-display" >
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2357 <img
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2358 src="img/078_A_1916116x.png" alt=" ab @Aab-- gb ag A s = @x + {sg, a}A + {sg, b}A s " class="math-display" /></center></td><td width="5%">(38)</td></tr></table>
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2359 <!--l. 2346--><p class="nopar">
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2360 </p><!--l. 2350--><p class="noindent">This is the expression for the extension of a contravariant <br/>tensor of the second rank, and corresponding expressions for <br/>the extension of contravariant tensors of higher and lower <br/>rank may also be formed.
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2361 </p><!--l. 2357--><p class="indent"> We note that in an analogous way we may also form the <br/>extension of a mixed tensor:--
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2362 </p>
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2363 <table width="100%"
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diff changeset
2364 class="equation"><tr><td><a
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diff changeset
2365 id="x1-49r39"></a>
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diff changeset
2366 <center class="math-display" >
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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diff changeset
2367 <img
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diff changeset
2368 src="img/078_A_1916117x.png" alt=" a Aa = @A-m-+ {sm, t}Aa + {st, a}At ms @xs t m " class="math-display" /></center></td><td width="5%">(39)</td></tr></table>
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2369 <!--l. 2365--><p class="nopar">
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diff changeset
2370 </p><!--l. 2369--><p class="indent"> On contracting (38) with respect to the indices <span
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diff changeset
2371 class="cmmi-12"><img
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diff changeset
2372 src="img/cmmi12-c.png" alt="b" class="cmmi-12x-x-c" align="middle" /> </span>and <span
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diff changeset
2373 class="cmmi-12"><img
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diff changeset
2374 src="img/cmmi12-1b.png" alt="s" class="12x-x-1b" /> </span> <br/>(inner multiplication by <span
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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diff changeset
2375 class="cmmi-12"><img
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diff changeset
2376 src="img/cmmi12-e.png" alt="d" class="12x-x-e" /></span><sub><span
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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diff changeset
2377 class="cmmi-8"><img
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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diff changeset
2378 src="img/cmmi8-c.png" alt="b" class="cmmi-8x-x-c" align="middle" /></span></sub><sup><span
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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diff changeset
2379 class="cmmi-8"><img
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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diff changeset
2380 src="img/cmmi8-1b.png" alt="s" class="8x-x-1b" /></span></sup>), we obtain the vector
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diff changeset
2381 </p>
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diff changeset
2382 <center class="par-math-display" >
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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diff changeset
2383 <img
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diff changeset
2384 src="img/078_A_1916118x.png" alt=" ab Aa = @A----+ {bg, b}Aag + {bg, a}Agb. @xb " class="par-math-display" /></center>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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diff changeset
2385 <!--l. 2379--><p class="nopar">
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2386 </p><!--l. 2382--><p class="noindent">On account of the symmetry of <img
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diff changeset
2387 src="img/078_A_1916119x.png" alt="{bg, a}" class="left" align="middle" /> with respect to the in-<br/>dices <span
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diff changeset
2388 class="cmmi-12"><img
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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diff changeset
2389 src="img/cmmi12-c.png" alt="b" class="cmmi-12x-x-c" align="middle" /> </span>and <span
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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diff changeset
2390 class="cmmi-12"><img
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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diff changeset
2391 src="img/cmmi12-d.png" alt="g" class="12x-x-d" /></span>, the third term on the right-hand side vanishes, <br/>if <span
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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diff changeset
2392 class="cmmi-12">A</span><sup ><span
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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diff changeset
2393 class="cmmi-8"><img
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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diff changeset
2394 src="img/cmmi8-b.png" alt="a" class="8x-x-b" /><img
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diff changeset
2395 src="img/cmmi8-c.png" alt="b" class="cmmi-8x-x-c" align="middle" /></span></sup> is, as we will assume, an antisymmetrical tensor. The <br/>second term allows itself to be transformed in accordance <br/>with (29a). Thus we obtain
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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diff changeset
2396 </p>
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diff changeset
2397 <table width="100%"
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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diff changeset
2398 class="equation"><tr><td><a
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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diff changeset
2399 id="x1-50r40"></a>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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diff changeset
2400 <center class="math-display" >
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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diff changeset
2401 <img
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diff changeset
2402 src="img/078_A_1916120x.png" alt=" ( V~ ---- ) a --1--@------gAab-- A = V~ --g @xb " class="math-display" /></center></td><td width="5%">(40)</td></tr></table>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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2403 <!--l. 2396--><p class="nopar">
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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2404 </p><!--l. 2400--><p class="noindent">This is the expression for the divergence of a contravariant <br/>six-vector.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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diff changeset
2405 </p><!--l. 2404--><p class="indent"> <span
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2406 class="cmti-12">The Divergence of a Mixed Tensor of the Second Rank.</span>--<br/>Contracting (39) with respect to the indices <span
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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diff changeset
2407 class="cmmi-12"><img
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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diff changeset
2408 src="img/cmmi12-b.png" alt="a" class="12x-x-b" /> </span>and <span
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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diff changeset
2409 class="cmmi-12"><img
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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2410 src="img/cmmi12-1b.png" alt="s" class="12x-x-1b" /></span>, and <br/>taking (29a) into consideration, we obtain
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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diff changeset
2411 </p>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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diff changeset
2412 <table width="100%"
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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diff changeset
2413 class="equation"><tr><td><a
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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diff changeset
2414 id="x1-51r41"></a>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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diff changeset
2415 <center class="math-display" >
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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2416 <img
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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2417 src="img/078_A_1916121x.png" alt=" ( V~ ---- s) V~ --gA = @------gA-m- - {sm, t}V ~ --gAs m @xs t " class="math-display" /></center></td><td width="5%">(41)</td></tr></table>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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2418 <!--l. 2414--><p class="nopar">
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2419 </p><!--l. 2417--><p class="noindent">If we introduce the contravariant tensor <span
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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diff changeset
2420 class="cmmi-12">A</span><sup ><span
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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diff changeset
2421 class="cmmi-8"><img
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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2422 src="img/cmmi8-1a.png" alt="r" class="cmmi-8x-x-1a" align="middle" /><img
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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2423 src="img/cmmi8-1b.png" alt="s" class="8x-x-1b" /></span></sup> = <span
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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diff changeset
2424 class="cmmi-12">g</span><sup ><span
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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2425 class="cmmi-8"><img
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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2426 src="img/cmmi8-1a.png" alt="r" class="cmmi-8x-x-1a" align="middle" /><img
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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diff changeset
2427 src="img/cmmi8-1c.png" alt="t" class="8x-x-1c" /></span></sup><span
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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diff changeset
2428 class="cmmi-12">A</span><sub>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
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diff changeset
2429 <span
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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diff changeset
2430 class="cmmi-8"><img
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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diff changeset
2431 src="img/cmmi8-1c.png" alt="t" class="8x-x-1c" /></span></sub><sup><span
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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diff changeset
2432 class="cmmi-8"><img
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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2433 src="img/cmmi8-1b.png" alt="s" class="8x-x-1b" /></span></sup> in the <br/>last term, it assumes the form
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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2434 </p>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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diff changeset
2435
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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2436 <center class="par-math-display" >
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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2437 <img
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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2438 src="img/078_A_1916122x.png" alt="- [sm,r] V~ --gArs. " class="par-math-display" /></center>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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2439 <!--l. 2425--><p class="nopar"> <pb/>
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2440 </p><!--l. 2432--><p class="indent">
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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2441
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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2442 </p><!--l. 2433--><p class="noindent">If, further, the tensor <span
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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diff changeset
2443 class="cmmi-12">A</span><sup ><span
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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2444 class="cmmi-8"><img
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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2445 src="img/cmmi8-1a.png" alt="r" class="cmmi-8x-x-1a" align="middle" /><img
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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diff changeset
2446 src="img/cmmi8-1b.png" alt="s" class="8x-x-1b" /></span></sup> is symmetrical, this reduces to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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diff changeset
2447 </p>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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2448 <center class="par-math-display" >
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2449 <img
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2450 src="img/078_A_1916123x.png" alt=" 1 V~ ----@grs- rs - 2 -g @x A . m " class="par-math-display" /></center>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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2451 <!--l. 2442--><p class="nopar">
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2452 </p><!--l. 2445--><p class="noindent">Had we introduced, instead of <span
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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2453 class="cmmi-12">A</span><sup ><span
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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2454 class="cmmi-8"><img
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2455 src="img/cmmi8-1a.png" alt="r" class="cmmi-8x-x-1a" align="middle" /><img
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2456 src="img/cmmi8-1b.png" alt="s" class="8x-x-1b" /></span></sup>, the covariant tensor <br/><span
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2457 class="cmmi-12">A</span><sub >
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2458 <span
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2459 class="cmmi-8"><img
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2460 src="img/cmmi8-1a.png" alt="r" class="cmmi-8x-x-1a" align="middle" /><img
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2461 src="img/cmmi8-1b.png" alt="s" class="8x-x-1b" /></span></sub> = <span
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2462 class="cmmi-12">g</span><sub ><span
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2463 class="cmmi-8"><img
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2464 src="img/cmmi8-1a.png" alt="r" class="cmmi-8x-x-1a" align="middle" /><img
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2465 src="img/cmmi8-b.png" alt="a" class="8x-x-b" /></span></sub><span
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2466 class="cmmi-12">g</span><sub ><span
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2467 class="cmmi-8"><img
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2468 src="img/cmmi8-1b.png" alt="s" class="8x-x-1b" /><img
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2469 src="img/cmmi8-c.png" alt="b" class="cmmi-8x-x-c" align="middle" /></span></sub><span
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2470 class="cmmi-12">A</span><sup ><span
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2471 class="cmmi-8"><img
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2472 src="img/cmmi8-b.png" alt="a" class="8x-x-b" /><img
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2473 src="img/cmmi8-c.png" alt="b" class="cmmi-8x-x-c" align="middle" /></span></sup><span
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2474 class="cmmi-12">, </span>which is also symmetrical, the last term, by <br/>virtue of (31), would assume the form
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2475 </p>
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2476 <center class="par-math-display" >
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2477 <img
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2478 src="img/078_A_1916124x.png" alt=" V~ ----@grs - 12 -g -----Ars. @xm " class="par-math-display" /></center>
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2479 <!--l. 2456--><p class="nopar">
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2480 </p><!--l. 2459--><p class="noindent">In the case of symmetry in question, (41) may therefore be <br/>replaced by the two forms
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2481 </p>
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2482 <table width="100%"
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2483 class="equation"><tr><td><a
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2484 id="x1-52r42"></a>
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2485 <center class="math-display" >
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2486 <img
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2487 src="img/078_A_1916125x.png" alt=" V~ ---- @( V~ --gAs ) @g V~ ---- - gAm = ---------m--- 12 --rs- -gArs @xs @xm " class="math-display" /></center></td><td width="5%">(41a)</td></tr></table>
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2488 <!--l. 2469--><p class="nopar">
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2489 </p>
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2490 <table width="100%"
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2491 class="equation"><tr><td><a
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2492 id="x1-53r42"></a>
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2493 <center class="math-display" >
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2494
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2495 <img
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2496 src="img/078_A_1916126x.png" alt=" V~ ---- s V~ ---- @(----gA-m)- 1 @grs V~ ---- - gAm = @xs + 2 @xm -gArs " class="math-display" /></center></td><td width="5%">(41b)</td></tr></table>
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2497 <!--l. 2478--><p class="nopar">
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2498 </p><!--l. 2482--><p class="noindent">which we have to employ later on.
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2499 </p>
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2500 <div class="center" >
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2501
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2502 <!--l. 2487--><p class="noindent">
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2503 </p><!--l. 2488--><p class="noindent"><span
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2504 class="cmsy-10x-x-120">§ </span><span
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2505 class="cmbx-12">12. The Riemann-Christoffel Tensor</span></p></div>
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2506 <!--l. 2492--><p class="indent"> We now seek the tensor which can be obtained from the <br/>fundamental tensor
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2507 <span
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2508 class="cmti-12">alone, </span>by differentiation. At first sight <br/>the solution seems obvious. We place the fundamental <br/>tensor of the <span
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2509 class="cmmi-12">g</span><sub ><span
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2510 class="cmmi-8"><img
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2511 src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img
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2512 src="img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub> in (27) instead of any given tensor
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2513 <span
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2514 class="cmmi-12">A</span><sub ><span
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2515 class="cmmi-8"><img
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2516 src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img
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2517 src="img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub>, and <br/>thus have a new tensor, namely, the extension of the funda-<br/>mental tensor. But we easily convince ourselves that this <br/>extension vanishes identically. We reach our goal, however, <br/>in the following way. In (27) place
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2518 </p>
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2519 <center class="par-math-display" >
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2520 <img
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2521 src="img/078_A_1916127x.png" alt=" @A Amn = ---m- {mn, r}Ar, @xn " class="par-math-display" /></center>
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2522 <!--l. 2507--><p class="nopar">
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2523 </p><!--l. 2511--><p class="noindent">i.e. the extension of the four-vector A<sub ><span
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2524 class="cmmi-8"><img
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2525 src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /></span></sub>. Then (with a some-<br/>what different naming of the indices) we get the tensor of the <br/>third rank</p>
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2526 <center class="par-math-display" >
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2527 <img
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2528 src="img/078_A_1916128x.png" alt=" @2Am @Ar @Ar @Am Amst = --------- {ms, r}---- - {mt, r} ----- {st, r}----- @xs@[xt @xt @xs @xr ] -@-- + - @x {ms, r}+ {mt, a}{as, r}+ {st, a}{am, r} Ar. t " class="par-math-display" /></center>
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2529 <!--l. 2529--><p class="nopar"> <pb/>
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2530 </p><!--l. 2536--><p class="indent">
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2531
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2532 </p><!--l. 2537--><p class="noindent">This expression suggests forming the tensor <span
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2533 class="cmmi-12">A</span><sub ><span
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2534 class="cmmi-8"><img
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2535 src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img
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2536 src="img/cmmi8-1b.png" alt="s" class="8x-x-1b" /><img
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2537 src="img/cmmi8-1c.png" alt="t" class="8x-x-1c" /></span></sub> <span
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2538 class="cmsy-10x-x-120">- </span><span
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2539 class="cmmi-12">A</span><sub ><span
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2540 class="cmmi-8"><img
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2541 src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img
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2542 src="img/cmmi8-1c.png" alt="t" class="8x-x-1c" /><img
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2543 src="img/cmmi8-1b.png" alt="s" class="8x-x-1b" /></span></sub>. <br/>For, if we do so, the following terms of the expression for <br/><span
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2544 class="cmmi-12">A</span><sub ><span
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2545 class="cmmi-8"><img
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2546 src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img
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2547 src="img/cmmi8-1b.png" alt="s" class="8x-x-1b" /><img
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2548 src="img/cmmi8-1c.png" alt="t" class="8x-x-1c" /></span></sub> cancel those of <span
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2549 class="cmmi-12">A</span><sub ><span
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2550 class="cmmi-8"><img
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2551 src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img
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2552 src="img/cmmi8-1c.png" alt="t" class="8x-x-1c" /><img
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2553 src="img/cmmi8-1b.png" alt="s" class="8x-x-1b" /></span></sub>, the first, the fourth, and the <br/>member corresponding to the last term in square brackets;
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2554 <br/>because all these are symmetrical in <span
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2555 class="cmmi-12"><img
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2556 src="img/cmmi12-1b.png" alt="s" class="12x-x-1b" /> </span>and <span
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2557 class="cmmi-12"><img
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2558 src="img/cmmi12-1c.png" alt="t" class="12x-x-1c" /></span>. The same <br/>holds good for the sum of the second and third terms. Thus <br/>we obtain
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2559 </p>
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2560 <table width="100%"
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2561 class="equation"><tr><td><a
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2562 id="x1-54r42"></a>
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2563 <center class="math-display" >
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2564 <img
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2565 src="img/078_A_1916129x.png" alt=" r Amst - Amts = B mstAr " class="math-display" /></center></td><td width="5%">(42)</td></tr></table>
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2566 <!--l. 2551--><p class="nopar">
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2567 </p><!--l. 2555--><p class="noindent">where
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2568 </p>
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2569 <center class="par-math-display" >
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2570 <img
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2571 src="img/078_A_1916130x.png" alt=" r @ @ B mst = - @x-{ms, r}+ @x-{mt, r} - {ms, a}{at, r} t s " class="par-math-display" /></center>
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2572 <!--l. 2564--><p class="nopar">
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2573 </p>
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2574 <table width="100%"
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2575 class="equation"><tr><td><a
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2576 id="x1-55r43"></a>
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2577 <center class="math-display" >
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2578 <img
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2579 src="img/078_A_1916131x.png" alt="+{mt, a}{as, r} " class="math-display" /></center></td><td width="5%">(43)</td></tr></table>
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2580 <!--l. 2571--><p class="nopar">
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2581
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2582 </p><!--l. 2576--><p class="noindent">The essential feature of the result is that on the right side of <br/>(42) the A<sub ><span
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2583 class="cmmi-8"><img
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2584 src="img/cmmi8-1a.png" alt="r" class="cmmi-8x-x-1a" align="middle" /></span></sub> occur alone, without their derivatives. From the <br/>tensor character of
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2585 <span
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2586 class="cmmi-12">A</span><sub ><span
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2587 class="cmmi-8"><img
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2588 src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img
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2589 src="img/cmmi8-1b.png" alt="s" class="8x-x-1b" /><img
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2590 src="img/cmmi8-1c.png" alt="t" class="8x-x-1c" /></span></sub> <span
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2591 class="cmsy-10x-x-120">- </span><span
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2592 class="cmmi-12">A</span><sub ><span
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2593 class="cmmi-8"><img
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2594 src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img
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2595 src="img/cmmi8-1c.png" alt="t" class="8x-x-1c" /><img
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2596 src="img/cmmi8-1b.png" alt="s" class="8x-x-1b" /></span></sub> in conjunction with the fact <br/>that A<sub ><span
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2597 class="cmmi-8"><img
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2598 src="img/cmmi8-1a.png" alt="r" class="cmmi-8x-x-1a" align="middle" /></span></sub> is an arbitrary vector, it follows, by reason of <span
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2599 class="cmsy-10x-x-120">§ </span>7, <br/>that <span
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2600 class="cmmi-12">B</span><sub><span
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2601 class="cmmi-8"><img
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2602 src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img
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2603 src="img/cmmi8-1b.png" alt="s" class="8x-x-1b" /><img
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2604 src="img/cmmi8-1c.png" alt="t" class="8x-x-1c" /></span></sub><sup><span
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2605 class="cmmi-8"><img
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2606 src="img/cmmi8-1a.png" alt="r" class="cmmi-8x-x-1a" align="middle" /></span></sup> is a tensor (the Riemann-Christoffel tensor).
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2607 </p><!--l. 2584--><p class="indent"> The mathematical importance of this tensor is as follows: <br/>If the continuum is of such a nature that there is a co-ordinate <br/>system with reference to which the <span
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2608 class="cmmi-12">g</span><sub ><span
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2609 class="cmmi-8"><img
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2610 src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img
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2611 src="img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub>
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2612 are constants, then <br/>all the <span
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2613 class="cmmi-12">B</span><sub><span
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2614 class="cmmi-8"><img
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2615 src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img
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2616 src="img/cmmi8-1b.png" alt="s" class="8x-x-1b" /><img
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2617 src="img/cmmi8-1c.png" alt="t" class="8x-x-1c" /></span></sub><sup><span
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2618 class="cmmi-8"><img
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2619 src="img/cmmi8-1a.png" alt="r" class="cmmi-8x-x-1a" align="middle" /></span></sup> vanish. If we choose any new system of co-<br/>ordinates in place of the original ones, the <span
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2620 class="cmmi-12">g</span><sub ><span
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2621 class="cmmi-8"><img
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2622 src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img
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2623 src="img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub> referred <br/>thereto will not be constants, but in consequence of its tensor <br/>nature, the transformed components of <span
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2624 class="cmmi-12">B</span><sub><span
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2625 class="cmmi-8"><img
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2626 src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img
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2627 src="img/cmmi8-1b.png" alt="s" class="8x-x-1b" /><img
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2628 src="img/cmmi8-1c.png" alt="t" class="8x-x-1c" /></span></sub><sup><span
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2629 class="cmmi-8"><img
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2630 src="img/cmmi8-1a.png" alt="r" class="cmmi-8x-x-1a" align="middle" /></span></sup> will still vanish <br/>in the new system. Thus the vanishing of the Riemann <br/>tensor is a necessary condition that, by an appropriate choice <br/>of the system of reference, the <span
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2631 class="cmmi-12">g</span><sub ><span
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2632 class="cmmi-8"><img
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2633 src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img
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2634 src="img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub> may be constants. In our <br/>problem this corresponds to the case in which,<sup ><span
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2635 class="cmsy-8">*</span></sup> with a <br/>suitable choice of the system of reference, the special <br/>theory of relativity holds good for a <span
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2636 class="cmti-12">finite </span>region of the
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2637 <br/>continuum.
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2638 </p><!--l. 2602--><p class="indent"> Contracting (43) with respect to the indices <span
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2639 class="cmmi-12"><img
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2640 src="img/cmmi12-1c.png" alt="t" class="12x-x-1c" /> </span>and <span
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2641 class="cmmi-12"><img
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2642 src="img/cmmi12-1a.png" alt="r" class="cmmi-12x-x-1a" align="middle" /> </span>we <br/>obtain the covariant tensor of second rank
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2643 </p><!--l. 2607--><p class="indent"> <sup ><span
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2644 class="cmsy-8">*</span></sup> The mathematicians have proved that this is also a <span
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2645 class="cmti-12">sufficient </span>condition.
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2646 <pb/>
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2647 </p><!--l. 2612--><p class="indent">
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2648
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2649 </p>
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2650 <table width="100%"
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2651 class="equation"><tr><td><a
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2652 id="x1-56r44"></a>
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2653 <center class="math-display" >
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2654 <img
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2655 src="img/078_A_1916132x.png" alt=" r Gmn = B mnr = Rmn + Smn where
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2656 Rmn = - -@--{mn, a}+ {ma, b}{nb, a} } @xa @2 log V~ --g- @ log V~ --g Smn = ------------- {mn, a} ----------- @xm@xn @xa " class="math-display" /></center></td><td width="5%">(44)</td></tr></table>
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2657 <!--l. 2626--><p class="nopar">
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2658 </p><!--l. 2630--><p class="indent"> <span
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2659 class="cmti-12">Note on the Choice of Co-ordinates.</span>--It has already been <br/>observed in <span
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2660 class="cmsy-10x-x-120">§ </span>8, in connexion with equation (18a), that the <br/>choice of co-ordinates may with advantage be made so that <br/><img
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2661 src="img/078_A_1916133x.png" alt=" V~ ---- - g" class="sqrt" /> = 1. A glance at the equations obtained in the last <br/>two sections shows that by such a choice the laws of forma-<br/>tion of tensors undergo an important simplification. This <br/>applies particularly to G<sub ><span
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2662 class="cmmi-8"><img
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2663 src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img
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2664 src="img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub>, the tensor just developed, which <br/>plays a fundamental part in the theory to be set forth. For <br/>this specialization of the choice of co-ordinates brings about <br/>the vanishing of S<sub ><span
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2665 class="cmmi-8"><img
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2666 src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img
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2667 src="img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub>, so that the tensor G<sub ><span
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2668 class="cmmi-8"><img
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2669 src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img
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2670 src="img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub> reduces to R<sub ><span
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2671 class="cmmi-8"><img
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2672 src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img
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2673 src="img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub>.
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2674 </p><!--l. 2643--><p class="indent"> On this account I shall hereafter give all relations in the <br/>simplified form which this specialization of the choice of co-<br/>ordinates brings with it. It will then be an easy matter to <br/>revert to the <span
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2675 class="cmti-12">generally </span>covariant equations, if this seems <br/>desirable in a special case.
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2676 </p>
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2677 <div class="center" >
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2678
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2679 <!--l. 2650--><p class="noindent">
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2680 </p><!--l. 2651--><p class="noindent"><span
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2681 class="cmcsc-10x-x-120">C. T<small
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2682 class="small-caps">h</small><small
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2683 class="small-caps">e</small><small
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2684 class="small-caps">o</small><small
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2685 class="small-caps">r</small><small
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2686 class="small-caps">y</small> <small
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2687 class="small-caps">o</small><small
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2688 class="small-caps">f</small> <small
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2689 class="small-caps">t</small><small
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2690 class="small-caps">h</small><small
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2691 class="small-caps">e</small> G<small
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2692 class="small-caps">r</small><small
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2693 class="small-caps">a</small><small
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2694 class="small-caps">v</small><small
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2695 class="small-caps">i</small><small
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2696 class="small-caps">t</small><small
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2697 class="small-caps">a</small><small
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2698 class="small-caps">t</small><small
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2699 class="small-caps">i</small><small
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2700 class="small-caps">o</small><small
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2701 class="small-caps">n</small><small
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2702 class="small-caps">a</small><small
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2703 class="small-caps">l</small> F<small
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2704 class="small-caps">i</small><small
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2705 class="small-caps">e</small><small
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2706 class="small-caps">l</small><small
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2707 class="small-caps">d</small></span></p></div>
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2708 <div class="center" >
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2709
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2710 <!--l. 2655--><p class="noindent">
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2711 </p><!--l. 2656--><p class="noindent"><span
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2712 class="cmsy-10x-x-120">§ </span><span
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2713 class="cmbx-12">13. Equations of Motion of a Material Point in the </span> <br/><span
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2714 class="cmbx-12">Gravitational</span>
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2715 <span
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2716 class="cmbx-12">Field. Expression for the Field-com- </span> <br/><span
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2717 class="cmbx-12">ponents of Gravitation</span></p></div>
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2718
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2719 <!--l. 2662--><p class="indent"> A freely movable body not subjected to external forces <br/>moves, according to the special theory of relativity, in a <br/>straight line and uniformly. This is also the case, according <br/>to the general theory of relativity, for a part of four-di-<br/>mensional space in which the system of co-ordinates K<sub ><span
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2720 class="cmr-8">0</span></sub>, may
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2721 <br/>be, and is, so chosen that they have the special constant <br/>values given in (4).
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2722 </p><!--l. 2670--><p class="indent"> If we consider precisely this movement from any chosen <br/>system of co-ordinates K<sub ><span
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2723 class="cmr-8">1</span></sub>, the body, observed from K<sub ><span
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2724 class="cmr-8">1</span></sub>, moves, <br/>according to the considerations in <span
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2725 class="cmsy-10x-x-120">§ </span>2, in a gravitational field. <br/>The law of motion with respect to K<sub ><span
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2726 class="cmr-8">1</span></sub> results without diffi-<br/><pb/>
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2727 </p><!--l. 2677--><p class="indent">
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2728
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2729 </p><!--l. 2678--><p class="noindent">culty from the following consideration. With respect to K<sub ><span
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2730 class="cmr-8">0</span></sub> <br/>the law of motion corresponds to a four-dimensional straight <br/>line, i.e. to a geodetic line. Now since the geodetic line <br/>is defined independently of the system of reference, its <br/>equations will also be the equation of motion of the material <br/>point with respect to K<sub ><span
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2731 class="cmr-8">1</span></sub>. If we set
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2732 </p>
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2733 <table width="100%"
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2734 class="equation"><tr><td><a
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2735 id="x1-57r45"></a>
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2736 <center class="math-display" >
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2737 <img
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2738 src="img/078_A_1916134x.png" alt=" t G mn = -{mn, t } " class="math-display" /></center></td><td width="5%">(45)</td></tr></table>
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2739 <!--l. 2689--><p class="nopar">
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2740 </p><!--l. 2693--><p class="noindent">the equation of the motion of the point with respect to K<sub ><span
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2741 class="cmr-8">1</span></sub>, <br/>becomes
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2742 </p>
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2743 <table width="100%"
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2744 class="equation"><tr><td><a
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2745 id="x1-58r46"></a>
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2746 <center class="math-display" >
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2747 <img
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2748 src="img/078_A_1916135x.png" alt="d2xt- tdxm-dxn- ds2 = G mn ds ds " class="math-display" /></center></td><td width="5%">(46)</td></tr></table>
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2749 <!--l. 2701--><p class="nopar">
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2750 </p><!--l. 2705--><p class="noindent">We now make the assumption, which readily suggests itself, <br/>that this covariant system of equations also defines the motion <br/>of the point in the gravitational field in the case when there <br/>is no system of reference K<sub ><span
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2751 class="cmr-8">0</span></sub>, with respect to which the
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2752 <br/>special theory of relativity holds good in a finite region. <br/>We have all the more justification for this assumption as (46) <br/>contains only <span
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2753 class="cmti-12">first </span>derivatives of the <span
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2754 class="cmmi-12">g</span><sub ><span
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2755 class="cmmi-8"><img
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2756 src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img
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2757 src="img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub>, between which even <br/>in the special case of the existence of K<sub ><span
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2758 class="cmr-8">0</span></sub>, no relations sub-<br/>sist.<sup ><span
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2759 class="cmsy-8">*</span></sup>
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2760 </p><!--l. 2717--><p class="indent"> If the <img
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2761 src="img/cmr12-0.png" alt="G" class="12x-x-0" /><sub><span
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2762 class="cmmi-8"><img
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2763 src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img
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2764 src="img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub><sup><span
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2765 class="cmmi-8"><img
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2766 src="img/cmmi8-1c.png" alt="t" class="8x-x-1c" /></span></sup> vanish, then the point moves uniformly in a <br/>straight line. These quantities therefore condition the devi-<br/>ation of the motion from uniformity. They are the com-<br/>ponents of the gravitational field.
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2767 </p>
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2768 <div class="center" >
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2769
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2770 <!--l. 2725--><p class="noindent">
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2771 </p><!--l. 2726--><p class="noindent"><span
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2772 class="cmsy-10x-x-120">§ </span><span
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2773 class="cmbx-12">14. The Field Equations of Gravitation in the Absence </span> <br/><span
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2774 class="cmbx-12">of Matter</span></p></div>
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2775 <!--l. 2731--><p class="indent"> We make a distinction hereafter between &#8220; gravitational <br/>field &#8221; and &#8220; matter &#8221; in this way, that we denote everything <br/>but the gravitational field as &#8220; matter. &#8221; Our use of the word <br/>therefore includes not only matter in the ordinary sense, but
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2776 <br/>the electromagnetic field as well.
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2777 </p><!--l. 2738--><p class="indent"> Our next task is to find the field equations of gravitation <br/>in the absence of matter. Here we again apply the method <br/>
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2778 </p><!--l. 2742--><p class="indent"> <sup ><span
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2779 class="cmsy-8">*</span></sup> It is only between the second (and first) derivatives that, by <span
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2780 class="cmsy-10x-x-120">§ </span>12, the
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2781 <br/>relations <span
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2782 class="cmmi-12">B</span><sub><span
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2783 class="cmmi-8"><img
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2784 src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img
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2785 src="img/cmmi8-1b.png" alt="s" class="8x-x-1b" /><img
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2786 src="img/cmmi8-1c.png" alt="t" class="8x-x-1c" /></span></sub><sup><span
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2787 class="cmmi-8"><img
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2788 src="img/cmmi8-1a.png" alt="r" class="cmmi-8x-x-1a" align="middle" /></span></sup> = 0 subsist. <pb/>
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2789 </p><!--l. 2748--><p class="indent">
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2790
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2791 </p><!--l. 2749--><p class="noindent">employed in the preceding paragraph in formulating the <br/>equations of motion of the material point. A special case in <br/>which the required equations must in any case be satisfied is <br/>that of the special theory of relativity, in which the <span
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2792 class="cmmi-12">g</span><sub ><span
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2793 class="cmmi-8"><img
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2794 src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img
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2795 src="img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub> have
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2796 <br/>certain constant values. Let this be the case in a certain <br/>finite space in relation to a definite system of co-ordinates K<sub ><span
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2797 class="cmr-8">0</span></sub>. <br/>Relatively to this system all the components of the Riemann <br/>tensor <span
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2798 class="cmmi-12">B</span><sub><span
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2799 class="cmmi-8"><img
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2800 src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img
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2801 src="img/cmmi8-1b.png" alt="s" class="8x-x-1b" /><img
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2802 src="img/cmmi8-1c.png" alt="t" class="8x-x-1c" /></span></sub><sup> <span
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2803 class="cmmi-8"><img
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2804 src="img/cmmi8-1a.png" alt="r" class="cmmi-8x-x-1a" align="middle" /></span></sup> defined in (43), vanish. For the space under <br/>consideration they then vanish, also in any other system of
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2805 <br/>co-ordinates.
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2806 </p><!--l. 2762--><p class="indent"> Thus the required equations of the matter-free gravita-<br/>tional field must in any case be satisfied if all <span
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2807 class="cmmi-12">B</span><sub><span
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2808 class="cmmi-8"><img
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2809 src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img
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2810 src="img/cmmi8-1b.png" alt="s" class="8x-x-1b" /><img
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2811 src="img/cmmi8-1c.png" alt="t" class="8x-x-1c" /></span></sub><sup> <span
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2812 class="cmmi-8"><img
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2813 src="img/cmmi8-1a.png" alt="r" class="cmmi-8x-x-1a" align="middle" /></span></sup> vanish. <br/>But this condition goes too far. For it is clear that, e.g., the <br/>gravitational field generated by a material point in its environ-<br/>ment certainly cannot be &#8220; transformed away &#8221; by any choice <br/>of the system of co-ordinates, i.e. it cannot be transformed to <br/>the case of constant
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2814 <span
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2815 class="cmmi-12">g</span><sub ><span
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2816 class="cmmi-8"><img
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2817 src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img
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2818 src="img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub>.
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2819 </p><!--l. 2772--><p class="indent"> This prompts us to require for the matter-free gravitational <br/>field that the symmetrical tensor <span
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2820 class="cmmi-12">g</span><sub ><span
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2821 class="cmmi-8"><img
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2822 src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img
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2823 src="img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub>, derived from the tensor <br/><span
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2824 class="cmmi-12">B</span><sub><span
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2825 class="cmmi-8"><img
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2826 src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img
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2827 src="img/cmmi8-17.png" alt="n" class="8x-x-17" /><img
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2828 src="img/cmmi8-1c.png" alt="t" class="8x-x-1c" /></span></sub><sup> <span
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2829 class="cmmi-8"><img
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2830 src="img/cmmi8-1a.png" alt="r" class="cmmi-8x-x-1a" align="middle" /></span></sup>, shall vanish. Thus we obtain ten equations for the ten <br/>quantities <span
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2831 class="cmmi-12">g</span><sub ><span
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2832 class="cmmi-8"><img
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2833 src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img
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2834 src="img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub>, which are satisfied in the special case of the <br/>vanishing of all <span
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2835 class="cmmi-12">B</span><sub><span
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2836 class="cmmi-8"><img
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2837 src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img
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2838 src="img/cmmi8-17.png" alt="n" class="8x-x-17" /><img
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2839 src="img/cmmi8-1c.png" alt="t" class="8x-x-1c" /></span></sub><sup> <span
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2840 class="cmmi-8"><img
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2841 src="img/cmmi8-1a.png" alt="r" class="cmmi-8x-x-1a" align="middle" /></span></sup>. With the choice which we have made <br/>of a system of co-ordinates, and taking (44) into considera-<br/>tion, the equations for the matter-free field are
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2842 </p>
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2843 <table width="100%"
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2844 class="equation"><tr><td><a
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2845 id="x1-59r47"></a>
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2846 <center class="math-display" >
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2847 <img
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2848 src="img/078_A_1916136x.png" alt=" a @G-mn a b @xa + GmbG na = 0} V~ ---- - g = 1 " class="math-display" /></center></td><td width="5%">(47)</td></tr></table>
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2849 <!--l. 2791--><p class="nopar">
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2850 </p><!--l. 2795--><p class="indent"> It must be pointed out that there is only a minimum of <br/>arbitrariness in the choice of these equations. For besides <br/>G<sub ><span
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2851 class="cmmi-8"><img
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2852 src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img
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2853 src="img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub> there is no tensor of second rank which is formed from <br/>the <span
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2854 class="cmmi-12">g</span><sub ><span
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2855 class="cmmi-8"><img
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2856 src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img
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2857 src="img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub> and its derivatives, contains no derivations higher than
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2858 <br/>second, and is linear in these derivatives.<sup ><span
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2859 class="cmsy-8">*</span></sup>
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2860 </p><!--l. 2802--><p class="indent"> These equations, which proceed, by the method of pure
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2861 </p><!--l. 2806--><p class="indent"> <sup ><span
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2862 class="cmsy-8">*</span></sup> Properly speaking, this can be affirmed only of the tensor
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2863 </p>
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2864
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2865 <center class="par-math-display" >
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2866 <img
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2867 src="img/078_A_1916137x.png" alt="Gmn + cgmngabGab, " class="par-math-display" /></center>
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2868 <!--l. 2813--><p class="nopar"></p><!--l. 2816--><p class="noindent"><span
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2869 class="cmr-10x-x-109">where </span><span
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2870 class="cmmi-10x-x-109"><img
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2871 src="img/cmmi10-15.png" alt="c" class="10-109x-x-15" /> </span><span
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2872 class="cmr-10x-x-109">is a constant. If, however, we set this</span> tensor = 0<span
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2873 class="cmr-10x-x-109">, we come back again </span> <br/><span
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2874 class="cmr-10x-x-109">to the</span>
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2875 <span
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2876 class="cmr-10x-x-109">equations</span> G<sub ><span
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2877 class="cmmi-8"><img
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2878 src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img
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2879 src="img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub> = 0<span
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2880 class="cmr-10x-x-109">. </span><pb/>
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2881 </p><!--l. 2823--><p class="indent">
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2882
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2883 </p><!--l. 2824--><p class="noindent">mathematics, from the requirement of the general theory of <br/>relativity, give us, in combination with the equations of <br/>motion (46), to a first approximation Newton&#8217;s law of at-<br/>traction, and to a second approximation the explanation of <br/>the motion of the perihelion of the planet Mercury discovered <br/>by Leverrier (as it remains after corrections for perturbation <br/>have been made). These facts must, in my opinion, be <br/>taken as a convincing proof of the correctness of the theory.
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2884 </p>
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2885 <div class="center" >
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2886
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2887 <!--l. 2836--><p class="noindent">
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2888 </p><!--l. 2837--><p class="noindent"><span
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2889 class="cmbsy-10x-x-120">§ </span><span
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2890 class="cmbx-12">15. The Hamiltonian Function for the Gravitational </span> <br/><span
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2891 class="cmbx-12">Field. Laws</span>
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2892 <span
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2893 class="cmbx-12">of Momentum and Energy</span></p></div>
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2894 <!--l. 2842--><p class="indent"> To show that the field equations correspond to the laws of <br/>momentum and energy, it is most convenient to write them <br/>in the following Hamiltonian form:--
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2895 </p>
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2896 <table width="100%"
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2897 class="equation"><tr><td><a
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2898 id="x1-60r48"></a>
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2899 <center class="math-display" >
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2900 <img
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2901 src="img/078_A_1916138x.png" alt=" integral
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2902 d Hdt = 0 } H = gmnGa Gb V~ ---- mb na - g = 1 " class="math-display" /></center></td><td width="5%">(47a)</td></tr></table>
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2903 <!--l. 2855--><p class="nopar">
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2904 </p><!--l. 2859--><p class="noindent">where, on the boundary of the finite four-dimensional region <br/>of integration which we have in view, the variations vanish.
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2905 </p><!--l. 2863--><p class="indent"> We first have to show that the form (47a) is equivalent <br/>to the equations (47). For this purpose we regard H as a <br/>function of the <span
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2906 class="cmmi-12">g</span><sub ><span
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2907 class="cmmi-8"><img
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2908 src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img
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2909 src="img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub> and the <span
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2910 class="cmmi-12">g</span><sub><span
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2911 class="cmmi-8"><img
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2912 src="img/cmmi8-1b.png" alt="s" class="8x-x-1b" /></span></sub><sup><span
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2913 class="cmmi-8"><img
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2914 src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img
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2915 src="img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sup>(= <span
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2916 class="cmmi-12"><img
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2917 src="img/cmmi12-40.png" alt="@" class="12x-x-40" />g</span><sup ><span
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2918 class="cmmi-8"><img
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2919 src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img
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2920 src="img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sup><img
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2921 src="img/078_A_1916139x.png" alt="/" class="left" align="middle" /> <span
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2922 class="cmmi-12"><img
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2923 src="img/cmmi12-40.png" alt="@" class="12x-x-40" />x</span><sub >
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2924 <span
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2925 class="cmmi-8"><img
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2926 src="img/cmmi8-1b.png" alt="s" class="8x-x-1b" /></span></sub>).
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2927 </p><!--l. 2870--><p class="noindent">Then in the first place
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2928 </p>
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2929 <center class="par-math-display" >
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2930 <img
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2931 src="img/078_A_1916140x.png" alt=" a b mn mn a b dH = G mbGnadg + 2g G mbdGna a b mn a mn b = - GmbG nadg + 2Gmbd(g G na). " class="par-math-display" /></center>
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2932 <!--l. 2885--><p class="nopar">
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2933 </p><!--l. 2888--><p class="noindent">But
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2934 </p>
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2935 <center class="par-math-display" >
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2936 <img
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2937 src="img/078_A_1916141x.png" alt=" [ ( )] ( mn b ) 1 mn bc @gnc- @gac- @gan- d g Gna = - 2d g g @xa + @xn + @xc . " class="par-math-display" /></center>
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2938 <!--l. 2899--><p class="nopar">
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2939 </p><!--l. 2902--><p class="noindent">The terms arising from the last two terms in round brackets <br/>are of different sign, and result from each other (since the de-<br/>nomination of the summation indices is immaterial) through <br/>interchange of the indices <span
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2940 class="cmmi-12"><img
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2941 src="img/cmmi12-16.png" alt="m" class="cmmi-12x-x-16" align="middle" /> </span>and <span
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2942 class="cmmi-12"><img
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2943 src="img/cmmi12-c.png" alt="b" class="cmmi-12x-x-c" align="middle" /></span>. They cancel each other <br/>in the expression for <span
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2944 class="cmmi-12"><img
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2945 src="img/cmmi12-e.png" alt="d" class="12x-x-e" /></span>H, because they are multiplied by the
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2946 <br/><pb/>
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2947 </p><!--l. 2911--><p class="indent">
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2948
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2949 </p><!--l. 2912--><p class="noindent">quantity <img
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2950 src="img/cmr12-0.png" alt="G" class="12x-x-0" /><sub><span
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2951 class="cmmi-8"><img
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2952 src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img
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2953 src="img/cmmi8-c.png" alt="b" class="cmmi-8x-x-c" align="middle" /></span></sub><sup><span
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2954 class="cmmi-8"><img
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2955 src="img/cmmi8-b.png" alt="a" class="8x-x-b" /></span></sup>, which is symmetrical with respect to the in-<br/>dices <span
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2956 class="cmmi-12"><img
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2957 src="img/cmmi12-16.png" alt="m" class="cmmi-12x-x-16" align="middle" /> </span>and <span
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2958 class="cmmi-12"><img
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2959 src="img/cmmi12-c.png" alt="b" class="cmmi-12x-x-c" align="middle" /></span>. Thus there remains only the first term in <br/>round brackets to be considered, so that, taking (31) into ac-<br/>count, we obtain
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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2960 </p>
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2961 <center class="par-math-display" >
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2962 <img
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2963 src="img/078_A_1916142x.png" alt=" a b mn a mb dH = - GmbG nadg + Gmbdga . " class="par-math-display" /></center>
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2964 <!--l. 2924--><p class="nopar">
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2965 </p><!--l. 2927--><p class="noindent">Thus
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2966 </p>
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2967 <table width="100%"
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2968 class="equation"><tr><td><a
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2969 id="x1-61r48"></a>
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2970 <center class="math-display" >
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2971 <img
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2972 src="img/078_A_1916143x.png" alt="-@H-- = - Ga Gb @gmn mb na } @H ---mn = Gsmn @g s " class="math-display" /></center></td><td width="5%">(48)</td></tr></table>
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2973 <!--l. 2939--><p class="nopar">
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2974 </p><!--l. 2942--><p class="noindent">Carrying out the variation in (47a), we get in the first place
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2975 </p>
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2976 <table width="100%"
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2977 class="equation"><tr><td><a
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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2978 id="x1-62r49"></a>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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2979 <center class="math-display" >
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2980 <img
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2981 src="img/078_A_1916144x.png" alt=" ( ) -@-- @H--- - -@H-- = 0, @xa @gman @gmn " class="math-display" /></center></td><td width="5%">(47b)</td></tr></table>
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2982 <!--l. 2950--><p class="nopar">
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2983
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2984 </p><!--l. 2954--><p class="noindent">which, on account of (48), agrees with (47), as was to be <br/>proved.
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2985 </p><!--l. 2957--><p class="indent"> If we multiply (47b) by <span
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2986 class="cmmi-12">g</span><sub><span
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2987 class="cmmi-8"><img
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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2988 src="img/cmmi8-1b.png" alt="s" class="8x-x-1b" /></span></sub><sup><span
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2989 class="cmmi-8"><img
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2990 src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img
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diff changeset
2991 src="img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sup>, then because
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2992 </p>
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2993 <center class="par-math-display" >
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2994 <img
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2995 src="img/078_A_1916145x.png" alt=" mn mn @gs--= @ga-- @xa @xs " class="par-math-display" /></center>
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2996 <!--l. 2963--><p class="nopar">
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2997 </p><!--l. 2967--><p class="noindent">and, consequently,
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2998 </p>
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2999 <center class="par-math-display" >
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3000 <img
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3001 src="img/078_A_1916146x.png" alt=" @ ( @H ) @ ( @H ) @H @gmn gmsn---- --mn- = ---- gmsn --mn- - --mn---a-, @xa @ga @xa @ga @ga @xs " class="par-math-display" /></center>
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3002 <!--l. 2978--><p class="nopar">
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3003 </p><!--l. 2982--><p class="noindent">we obtain the equation
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3004 </p>
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3005 <center class="par-math-display" >
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3006 <img
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3007 src="img/078_A_1916147x.png" alt=" @ ( @H ) @H ---- gmsn---mn - ---- = 0 @xa @ga @xs " class="par-math-display" /></center>
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3008 <!--l. 2990--><p class="nopar">
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3009 </p><!--l. 2994--><p class="noindent">or <sup ><span
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3010 class="cmsy-8">*</span></sup>
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3011 </p>
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3012 <table width="100%"
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3013 class="equation"><tr><td><a
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3014 id="x1-63r49"></a>
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3015 <center class="math-display" >
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3016 <img
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3017 src="img/078_A_1916148x.png" alt=" a -@ts = 0 @xa } a mn @H a - 2kts = gs --mn-- dsH @ga " class="math-display" /></center></td><td width="5%">(49)</td></tr></table>
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3018 <!--l. 3004--><p class="nopar">
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3019
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3020 </p><!--l. 3008--><p class="noindent">where, on account of (48), the second equation of (47), and <br/>(34)
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3021 </p>
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3022 <table width="100%"
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3023 class="equation"><tr><td><a
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3024 id="x1-64r50"></a>
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3025 <center class="math-display" >
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3026 <img
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3027 src="img/078_A_1916149x.png" alt=" a 1 a mn c b mn a b kts = 2ds g GmbG nc - g G mbGns " class="math-display" /></center></td><td width="5%">(50)</td></tr></table>
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3028 <!--l. 3017--><p class="nopar">
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3029 </p><!--l. 3021--><p class="indent"> <sup ><span
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3030 class="cmsy-8">*</span></sup> The reason for the introduction of the factor <span
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3031 class="cmsy-10x-x-120">-</span>2<span
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3032 class="cmmi-12"><img
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3033 src="img/cmmi12-14.png" alt="k" class="12x-x-14" /> </span>will be apparent later.
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3034 <pb/>
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3035 </p><!--l. 3026--><p class="indent">
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3036
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3037 </p><!--l. 3027--><p class="indent"> It is to be noticed that <span
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3038 class="cmmi-12">t</span><sub><span
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3039 class="cmmi-8"><img
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3040 src="img/cmmi8-1b.png" alt="s" class="8x-x-1b" /></span></sub><sup><span
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3041 class="cmmi-8"><img
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3042 src="img/cmmi8-b.png" alt="a" class="8x-x-b" /></span></sup> is not a tensor; on the other <br/>hand (49) applies to all systems of co-ordinates for which <br/><img
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3043 src="img/078_A_1916150x.png" alt=" V~ ---- -g" class="sqrt" /> = 1. This equation expresses the law of conservation <br/>of momentum and of energy for the gravitational field. <br/>Actually the integration of this equation over a three-<br/>dimensional volume V yields the four equations
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3044 </p>
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3045 <table width="100%"
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3046 class="equation"><tr><td><a
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3047 id="x1-65r51"></a>
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3048 <center class="math-display" >
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3049 <img
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3050 src="img/078_A_1916151x.png" alt=" d integral integral
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3051 ---- t4sdV = (lt1s + mt2s + nt3s)dS dx4 " class="math-display" /></center></td><td width="5%">(49a)</td></tr></table>
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3052 <!--l. 3040--><p class="nopar">
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3053 </p><!--l. 3044--><p class="noindent">where <span
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3054 class="cmmi-12">l, m, n </span>denote the direction-cosines of direction of the <br/>in ward drawn normal at the element <span
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3055 class="cmmi-12">d</span>S of the bounding sur-<br/>face (in the sense of Euclidean geometry). We recognize in <br/>this the expression of the laws of conservation in their usual <br/>form. The quantities <span
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3056 class="cmmi-12">t</span><sub><span
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3057 class="cmmi-8"><img
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3058 src="img/cmmi8-1b.png" alt="s" class="8x-x-1b" /></span></sub><sup><span
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3059 class="cmmi-8"><img
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3060 src="img/cmmi8-b.png" alt="a" class="8x-x-b" /></span></sup> we call the &#8220; energy components &#8221; <br/>of the gravitational field.
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3061 </p><!--l. 3052--><p class="indent"> I will now give equations (47) in a third form, which is <br/>particularly useful for a vivid grasp of our subject. By <br/>multiplication of the field equations (47) by <span
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3062 class="cmmi-12">g</span><sup ><span
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3063 class="cmmi-8"><img
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3064 src="img/cmmi8-17.png" alt="n" class="8x-x-17" /><img
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3065 src="img/cmmi8-1b.png" alt="s" class="8x-x-1b" /></span></sup>
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3066 these are ob-<br/>tained in the &#8220; mixed &#8221; form. Note that
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3067 </p>
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3068 <center class="par-math-display" >
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3069 <img
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3070 src="img/078_A_1916152x.png" alt=" ns@Gamn @ ( ns a ) @gns a g -@x-- = @x-- g Gmn - -@x--Gmn, a a a " class="par-math-display" /></center>
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3071 <!--l. 3065--><p class="nopar">
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3072 </p><!--l. 3069--><p class="noindent">which quantity, by reason of (34), is equal to
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3073 </p>
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3074 <center class="par-math-display" >
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3075 <img
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3076 src="img/078_A_1916153x.png" alt=" @ ( ) ---- gnsGamn - gnbGsabGamn - gsbGnbaGamn, @xa " class="par-math-display" /></center>
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3077 <!--l. 3079--><p class="nopar">
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3078 </p><!--l. 3083--><p class="noindent">or (with different symbols for the summation indices)
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3079 </p>
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3080
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3081 <center class="par-math-display" >
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3082 <img
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diff changeset
3083 src="img/078_A_1916154x.png" alt="-@--( sb a ) gd s b ns a b @xa g G mb - g GgbGdm - g GmbG na. " class="par-math-display" /></center>
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3084 <!--l. 3093--><p class="nopar">
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3085 </p><!--l. 3096--><p class="noindent">The third term of this expression cancels with the one aris-<br/>ing from the second term of the field equations (47); using <br/>relation (50), the second term may be written
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3086 </p>
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3087 <center class="par-math-display" >
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3088 <img
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3089 src="img/078_A_1916155x.png" alt="k(tsm- 12dsmt), " class="par-math-display" /></center>
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3090 <!--l. 3105--><p class="nopar">
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3091 </p><!--l. 3109--><p class="noindent">where <span
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3092 class="cmmi-12">t </span>= <span
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3093 class="cmmi-12">t</span><sub><span
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3094 class="cmmi-8"><img
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3095 src="img/cmmi8-b.png" alt="a" class="8x-x-b" /></span></sub><sup><span
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3096 class="cmmi-8"><img
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3097 src="img/cmmi8-b.png" alt="a" class="8x-x-b" /></span></sup>. Thus instead of equations (47) we obtain
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3098 </p>
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3099 <table width="100%"
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3100 class="equation"><tr><td><a
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3101 id="x1-66r51"></a>
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3102 <center class="math-display" >
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3103 <img
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3104 src="img/078_A_1916156x.png" alt=" @ ( ) ---- gsbGamb = -k(tsm - 12dsmt)} @xa V~ ---- - g = 1 " class="math-display" /></center></td><td width="5%">(51)</td></tr></table>
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3105 <!--l. 3120--><p class="nopar">
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3106 <pb/>
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3107 </p><!--l. 3127--><p class="indent">
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3108
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3109 </p>
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3110 <div class="center" >
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3111
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3112 <!--l. 3128--><p class="noindent">
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3113 </p><!--l. 3129--><p class="noindent"><span
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3114 class="cmsy-10x-x-120">§ </span><span
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3115 class="cmbx-12">16. The General Form of the Field Equations of </span> <br/><span
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3116 class="cmbx-12">Gravitation</span></p></div>
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3117 <!--l. 3134--><p class="indent"> The field equations for matter-free space formulated in <br/><span
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3118 class="cmsy-10x-x-120">§ </span>15 are to be compared with the field equation
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3119 </p>
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3120 <center class="par-math-display" >
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3121 <img
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3122 src="img/078_A_1916157x.png" alt=" 2 \~/ f = 0 " class="par-math-display" /></center>
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3123 <!--l. 3140--><p class="nopar">
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3124 </p><!--l. 3144--><p class="noindent">of Newton&#8217;s theory. We require the equation corresponding <br/>to Poisson&#8217;s equation
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3125 </p>
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3126 <center class="par-math-display" >
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3127 <img
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3128 src="img/078_A_1916158x.png" alt=" 2 \~/ f = 4pkr, " class="par-math-display" /></center>
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3129 <!--l. 3150--><p class="nopar">
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3130 </p><!--l. 3154--><p class="noindent">where <span
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3131 class="cmmi-12"><img
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3132 src="img/cmmi12-1a.png" alt="r" class="cmmi-12x-x-1a" align="middle" /> </span>denotes the density of matter.
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3133 </p><!--l. 3156--><p class="indent"> The special theory of relativity has led to the conclusion <br/>that inert mass is nothing more or less than energy, which <br/>finds its complete mathematical expression in a symmetrical <br/>tensor of second rank, the energy-tensor. Thus in the
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3134 <br/>general theory of relativity we must introduce a correspond-<br/>ing energy-tensor of matter T<sub><span
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3135 class="cmmi-8"><img
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3136 src="img/cmmi8-1b.png" alt="s" class="8x-x-1b" /></span></sub><sup><span
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3137 class="cmmi-8"><img
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3138 src="img/cmmi8-b.png" alt="a" class="8x-x-b" /></span></sup>, which, like the energy-com-<br/>ponents <span
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3139 class="cmmi-12">t</span><sub >
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3140 <span
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3141 class="cmmi-8"><img
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3142 src="img/cmmi8-1b.png" alt="s" class="8x-x-1b" /></span></sub> [equations (49) and (50)] of the gravitational field, <br/>will have mixed character, but will pertain to a symmetrical <br/>covariant tensor.<sup ><span
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3143 class="cmsy-8">*</span></sup>
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3144 </p><!--l. 3168--><p class="indent"> The system of equation (51) shows how this energy-tensor <br/>(corresponding to the density <span
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3145 class="cmmi-12"><img
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3146 src="img/cmmi12-1a.png" alt="r" class="cmmi-12x-x-1a" align="middle" /> </span>in Poisson&#8217;s equation) is to <br/>be introduced into the field equations of gravitation. For if <br/>we consider a complete system (e.g. the solar system), the <br/>total mass of the system, and therefore its total gravitating <br/>action as well, will depend on the total energy of the system, <br/>and therefore on the ponderable energy together with the <br/>gravitational energy. This will allow itself to be expressed <br/>by introducing into (51), in place of the energy-components <br/>of the gravitational field alone, the sums <span
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3147 class="cmmi-12">t</span><sub><span
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3148 class="cmmi-8"><img
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3149 src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /></span></sub><sup><span
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3150 class="cmmi-8"><img
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3151 src="img/cmmi8-1b.png" alt="s" class="8x-x-1b" /></span></sup> + <span
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3152 class="cmmi-12">T</span><sub>
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3153 <span
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3154 class="cmmi-8"><img
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3155 src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /></span></sub><sup><span
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3156 class="cmmi-8"><img
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3157 src="img/cmmi8-1b.png" alt="s" class="8x-x-1b" /></span></sup> of the energy-<br/>components of matter and of gravitational field. Thus instead <br/>of (51) we obtain the tensor equation
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3158 </p>
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3159 <table width="100%"
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3160 class="equation"><tr><td><a
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3161 id="x1-67r52"></a>
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3162
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3163 <center class="math-display" >
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3164 <img
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3165 src="img/078_A_1916159x.png" alt="-@--(gsbT a ) = - k[(ts+ Ts) - 1ds(t + T )], @xa mb m m 2 m } V~ --g-= 1 " class="math-display" /></center></td><td width="5%">(52)</td></tr></table>
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3166 <!--l. 3191--><p class="nopar">
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3167 </p><!--l. 3195--><p class="noindent">where we have set <span
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3168 class="cmmi-12">T </span>= <span
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3169 class="cmmi-12">T</span><sub><span
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3170 class="cmmi-8"><img
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3171 src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /></span></sub><sup><span
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3172 class="cmmi-8"><img
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3173 src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /></span></sup> (Laue&#8217;s scalar). These are the <br/>
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3174 </p><!--l. 3198--><p class="indent"> <sup ><span
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3175 class="cmsy-8">*</span></sup> <span
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3176 class="cmmi-12">g</span><sub >
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3177 <span
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3178 class="cmmi-8"><img
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3179 src="img/cmmi8-b.png" alt="a" class="8x-x-b" /><img
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3180 src="img/cmmi8-1c.png" alt="t" class="8x-x-1c" /></span></sub><span
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3181 class="cmmi-12">T</span><sub><span
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3182 class="cmmi-8"><img
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3183 src="img/cmmi8-1b.png" alt="s" class="8x-x-1b" /></span></sub><sup><span
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3184 class="cmmi-8"><img
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3185 src="img/cmmi8-b.png" alt="a" class="8x-x-b" /></span></sup> = <span
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3186 class="cmmi-12">T</span><sub >
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3187 <span
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3188 class="cmmi-8"><img
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3189 src="img/cmmi8-b.png" alt="a" class="8x-x-b" /><img
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3190 src="img/cmmi8-1c.png" alt="t" class="8x-x-1c" /></span></sub> and <span
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3191 class="cmmi-12">g</span><sup ><span
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3192 class="cmmi-8"><img
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3193 src="img/cmmi8-1b.png" alt="s" class="8x-x-1b" /><img
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3194 src="img/cmmi8-c.png" alt="b" class="cmmi-8x-x-c" align="middle" /></span></sup><span
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3195 class="cmmi-12">T</span><sub>
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3196 <span
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3197 class="cmmi-8"><img
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3198 src="img/cmmi8-1b.png" alt="s" class="8x-x-1b" /></span></sub><sup><span
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3199 class="cmmi-8"><img
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3200 src="img/cmmi8-b.png" alt="a" class="8x-x-b" /></span></sup> = <span
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3201 class="cmmi-12">T</span><sup ><span
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3202 class="cmmi-8"><img
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3203 src="img/cmmi8-b.png" alt="a" class="8x-x-b" /><img
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3204 src="img/cmmi8-c.png" alt="b" class="cmmi-8x-x-c" align="middle" /></span></sup> are to be symmetrical tensors.
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3205 <pb/>
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3206 </p><!--l. 3204--><p class="indent">
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3207
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3208 </p><!--l. 3205--><p class="noindent">required general field equations of gravitation in mixed form. <br/>Working back from these, we have in place of (47)
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3209 </p>
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3210 <table width="100%"
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3211 class="equation"><tr><td><a
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3212 id="x1-68r53"></a>
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3213 <center class="math-display" >
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3214 <img
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3215 src="img/078_A_1916160x.png" alt="-@--Ga + Ga Gb = -k(T - 1g T ), @xa mn mb na mn 2 mn } V~ --g = 1 " class="math-display" /></center></td><td width="5%">(53)</td></tr></table>
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3216 <!--l. 3218--><p class="nopar">
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3217 </p><!--l. 3222--><p class="indent"> It must be admitted that this introduction of the energy-<br/>tensor of matter is not justified by the relativity postulate <br/>alone. For this reason we have here deduced it from the <br/>requirement that the energy of the gravitational field shall <br/>act gravitatively in the same way as any other kind of energy. <br/>But the strongest reason for the choice of these equations <br/>lies in their consequence, that the equations of conservation <br/>of momentum and energy, corresponding exactly to equations <br/>(49) and (49a), hold good for the components of the total <br/>energy. This will be shown in <span
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3218 class="cmsy-10x-x-120">§ </span>17.
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3219 </p>
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3220 <div class="center" >
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3221
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3222 <!--l. 3236--><p class="noindent">
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3223 </p><!--l. 3237--><p class="noindent"><span
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3224 class="cmsy-10x-x-120">§ </span><span
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3225 class="cmbx-12">17. The Laws of Conservation in the General Case</span></p></div>
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3226 <!--l. 3241--><p class="indent"> Equation (52) may readily be transformed so that the <br/>second term on the right-hand side vanishes. Contract (52) <br/>with respect to the indices <span
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3227 class="cmmi-12"><img
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3228 src="img/cmmi12-16.png" alt="m" class="cmmi-12x-x-16" align="middle" /> </span>and <span
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3229 class="cmmi-12"><img
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3230 src="img/cmmi12-1b.png" alt="s" class="12x-x-1b" /></span>, and after multiplying the <br/>resulting equation by <span
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3231 class="cmr-8">1</span>
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3232 <span
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3233 class="cmr-8">2</span><span
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3234 class="cmmi-12"><img
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3235 src="img/cmmi12-e.png" alt="d" class="12x-x-e" /></span><sub><span
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3236 class="cmmi-8"><img
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3237 src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /></span></sub><sup><span
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3238 class="cmmi-8"><img
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3239 src="img/cmmi8-1b.png" alt="s" class="8x-x-1b" /></span></sup>, subtract it from equation (52).
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3240 <br/>This gives
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3241 </p>
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3242 <table width="100%"
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3243 class="equation"><tr><td><a
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3244 id="x1-69r54"></a>
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3245 <center class="math-display" >
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3246
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3247 <img
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3248 src="img/078_A_1916161x.png" alt="-@--(gsbGa - 1dsgcbGa ) = - k(ts+ T s). @xa mb 2 m cb m m " class="math-display" /></center></td><td width="5%">(52a)</td></tr></table>
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3249 <!--l. 3254--><p class="nopar">
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3250 </p><!--l. 3257--><p class="noindent">On this equation we perform the operation <span
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3251 class="cmmi-12"><img
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3252 src="img/cmmi12-40.png" alt="@" class="12x-x-40" /></span><img
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3253 src="img/078_A_1916162x.png" alt="/" class="left" align="middle" /> <span
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3254 class="cmmi-12"><img
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3255 src="img/cmmi12-40.png" alt="@" class="12x-x-40" />x</span><sub ><span
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3256 class="cmmi-8"><img
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3257 src="img/cmmi8-1b.png" alt="s" class="8x-x-1b" /></span></sub><span
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3258 class="cmmi-12">. </span>We have</p>
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3259 <center class="par-math-display" >
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3260 <img
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3261 src="img/078_A_1916163x.png" alt=" @2 ( ) @2 [ (@gmc @gbc @gmb )] -------- gsGabm = - 12-------- gsbgac -----+ ------ ----- . @xa@xs @xa@xs @xb @xm @xc " class="par-math-display" /></center>
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3262 <!--l. 3273--><p class="nopar">
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3263 </p><!--l. 3276--><p class="noindent">The first and third terms of the round brackets yield con-<br/>tributions which cancel one another, as may be seen by <br/>interchanging, in the contribution of the third term, the <br/>summation indices <span
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3264 class="cmmi-12"><img
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3265 src="img/cmmi12-b.png" alt="a" class="12x-x-b" /> </span>and <span
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3266 class="cmmi-12"><img
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3267 src="img/cmmi12-1b.png" alt="s" class="12x-x-1b" /> </span>on the one hand, and <span
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3268 class="cmmi-12"><img
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3269 src="img/cmmi12-c.png" alt="b" class="cmmi-12x-x-c" align="middle" /> </span>and <span
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3270 class="cmmi-12"><img
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3271 src="img/cmmi12-15.png" alt="c" class="12x-x-15" /></span>
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3272 <br/>on the other. The second term may be re-modelled by (31), <br/>so that we have
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3273 </p>
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3274 <table width="100%"
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3275 class="equation"><tr><td><a
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3276 id="x1-70r54"></a>
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3277 <center class="math-display" >
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3278 <img
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3279 src="img/078_A_1916164x.png" alt=" @2 ( ) @3gab -------- gsbGamb = 1------------ @xa@xs 2@xa@xb@xm " class="math-display" /></center></td><td width="5%">(54)</td></tr></table>
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3280 <!--l. 3291--><p class="nopar">
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3281 </p><!--l. 3294--><p class="noindent">The second term on the left-hand side of (52a) yields in the <br/><pb/>
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3282 </p><!--l. 3299--><p class="indent">
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3283
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3284 </p><!--l. 3300--><p class="noindent">first place
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3285 </p>
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3286 <center class="par-math-display" >
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3287 <img
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3288 src="img/078_A_1916165x.png" alt=" 1 @2 ( cb a ) - 2 -------- g Gcb @xa@xm " class="par-math-display" /></center>
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3289 <!--l. 3308--><p class="nopar">
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3290 </p><!--l. 3312--><p class="noindent">or
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3291 </p>
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3292 <center class="par-math-display" >
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3293 <img
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3294 src="img/078_A_1916166x.png" alt=" 2 [ ( )] 1---@---- gcbgad @gdc-+ @gdb-- @gcb- . 4@xa@xm @xb @xc @xd " class="par-math-display" /></center>
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3295 <!--l. 3324--><p class="nopar">
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3296 </p><!--l. 3328--><p class="noindent">With the choice of co-ordinates which we have made, the <br/>term deriving from the last term in round brackets disappears <br/>by reason of (29). The other two may be combined, and <br/>together, by (31), they give
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3297 </p>
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3298 <center class="par-math-display" >
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3299 <img
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3300 src="img/078_A_1916167x.png" alt=" 1 --@3gab----- - 2 @x @x @x , a b m " class="par-math-display" /></center>
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3301 <!--l. 3338--><p class="nopar">
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3302 </p><!--l. 3342--><p class="noindent">so that in consideration of (54), we have the identity
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3303 </p>
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3304 <table width="100%"
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3305 class="equation"><tr><td><a
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3306 id="x1-71r55"></a>
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3307 <center class="math-display" >
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3308 <img
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3309 src="img/078_A_1916168x.png" alt=" @2 ( ) -------- grbGmb - 12@smgcbGacb =_ 0 @xa@xs " class="math-display" /></center></td><td width="5%">(55)</td></tr></table>
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3310 <!--l. 3352--><p class="nopar">
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3311 </p><!--l. 3355--><p class="noindent">From (55) and (52a), it follows that
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3312 </p>
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3313 <table width="100%"
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3314 class="equation"><tr><td><a
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3315 id="x1-72r56"></a>
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3316
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3317 <center class="math-display" >
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3318 <img
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3319 src="img/078_A_1916169x.png" alt=" s s @(tm-+-T-m) = 0. @xs " class="math-display" /></center></td><td width="5%">(56)</td></tr></table>
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3320 <!--l. 3363--><p class="nopar">
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3321 </p><!--l. 3367--><p class="indent"> Thus it results from our field equations of gravitation <br/>that the laws of conservation of momentum and energy are <br/>satisfied. This may be seen most easily from the consider-<br/>ation which leads to equation (49a); except that here, instead <br/>of the energy components <span
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3322 class="cmmi-12">t</span><sup ><span
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3323 class="cmmi-8"><img
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3324 src="img/cmmi8-1b.png" alt="s" class="8x-x-1b" /></span></sup> of the gravitational field, we have <br/>to introduce the totality of the energy components of matter <br/>and gravitational field.
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3325 </p>
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3326 <div class="center" >
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3327
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3328 <!--l. 3377--><p class="noindent">
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3329 </p><!--l. 3378--><p class="noindent"><span
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3330 class="cmbsy-10x-x-120">§ </span><span
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3331 class="cmbx-12">18. The Laws of Momentum and Energy for Matter, as </span> <br/><span
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3332 class="cmbx-12">a</span>
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3333 <span
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3334 class="cmbx-12">Consequence of the Field Equations</span></p></div>
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3335 <!--l. 3383--><p class="indent"> Multiplying (53) by <span
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3336 class="cmmi-12"><img
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3337 src="img/cmmi12-40.png" alt="@" class="12x-x-40" />g</span><sup ><span
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3338 class="cmmi-8"><img
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3339 src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img
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3340 src="img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sup><img
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3341 src="img/078_A_1916170x.png" alt="/" class="left" align="middle" /> <span
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3342 class="cmmi-12"><img
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3343 src="img/cmmi12-40.png" alt="@" class="12x-x-40" />x</span><sub >
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3344 <span
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3345 class="cmmi-8"><img
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3346 src="img/cmmi8-1b.png" alt="s" class="8x-x-1b" /></span></sub><span
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3347 class="cmmi-12">, </span>we obtain, by the method <br/>adopted in <span
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3348 class="cmsy-10x-x-120">§ </span>15, in view of the vanishing of
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3349 </p>
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3350 <center class="par-math-display" >
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3351 <img
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3352 src="img/078_A_1916171x.png" alt=" @gmn- gmn@xs , " class="par-math-display" /></center>
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3353 <!--l. 3392--><p class="nopar">
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3354 </p><!--l. 3396--><p class="noindent">the equation
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3355 </p>
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3356 <center class="par-math-display" >
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3357 <img
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3358 src="img/078_A_1916172x.png" alt="@tas- 1@gmn- @x + 2@x Tmn = 0, a s " class="par-math-display" /></center>
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3359 <!--l. 3404--><p class="nopar"> <pb/>
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3360 </p><!--l. 3411--><p class="indent">
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3361
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3362 </p><!--l. 3412--><p class="noindent">or, in view of (56),
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3363 </p>
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3364 <table width="100%"
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3365 class="equation"><tr><td><a
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3366 id="x1-73r57"></a>
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3367 <center class="math-display" >
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3368 <img
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3369 src="img/078_A_1916173x.png" alt=" a mn @Ts--+ 1@g---Tmn = 0 @xa 2 @xs " class="math-display" /></center></td><td width="5%">(57)</td></tr></table>
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3370 <!--l. 3420--><p class="nopar">
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3371 </p><!--l. 3424--><p class="indent"> Comparison with (41b) shows that with the choice of <br/>system of co-ordinates which we have made, this equation <br/>predicates nothing more or less than the vanishing of di-<br/>vergence of the material energy-tensor. Physically, the <br/>occurrence of the second term on the left-hand side shows <br/>that laws of conservation of momentum and energy do not <br/>apply in the strict sense for matter alone, or else that they <br/>apply only when the <span
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3372 class="cmmi-12">g</span><sup ><span
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3373 class="cmmi-8"><img
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3374 src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img
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3375 src="img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sup> are constant, i.e. when the field in-<br/>tensities of gravitation vanish. This second term is an ex-<br/>pression for momentum, and for energy, as transferred per <br/>unit of volume and time from the gravitational field to matter. <br/>This is brought out still more clearly by re-writing (57) in the <br/>sense of (41) as
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3376 </p>
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3377 <table width="100%"
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3378 class="equation"><tr><td><a
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3379 id="x1-74r58"></a>
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3380 <center class="math-display" >
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3381 <img
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3382 src="img/078_A_1916174x.png" alt="@T a ---s-= - GbasTba @xa " class="math-display" /></center></td><td width="5%">(57a)</td></tr></table>
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3383 <!--l. 3443--><p class="nopar">
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3384 </p><!--l. 3446--><p class="noindent">The right side expresses the energetic effect of the gravitational <br/>field on matter.
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3385 </p><!--l. 3450--><p class="indent"> Thus the field equations of gravitation contain four con-<br/>ditions which govern the course of material phenomena. <br/>They give the equations of material phenomena completely, <br/>if the latter is capable of being characterized by four differ-<br/>ential equations independent of one another.<sup ><span
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3386 class="cmsy-8">*</span></sup>
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3387 </p>
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3388 <div class="center" >
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3389
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3390 <!--l. 3459--><p class="noindent">
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3391 </p><!--l. 3460--><p class="noindent">D. <span
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3392 class="cmcsc-10x-x-120">M<small
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3393 class="small-caps">a</small><small
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3394 class="small-caps">t</small><small
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3395 class="small-caps">e</small><small
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3396 class="small-caps">r</small><small
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3397 class="small-caps">i</small><small
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3398 class="small-caps">a</small><small
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3399 class="small-caps">l</small> P<small
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3400 class="small-caps">h</small><small
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3401 class="small-caps">e</small><small
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3402 class="small-caps">n</small><small
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3403 class="small-caps">o</small><small
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3404 class="small-caps">m</small><small
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3405 class="small-caps">e</small><small
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3406 class="small-caps">n</small><small
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3407 class="small-caps">a</small></span></p></div>
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3408 <!--l. 3464--><p class="indent"> The mathematical aids developed in part B enable us <br/>forthwith to generalize the physical laws of matter (hydro-<br/>dynamics, Maxwell&#8217;s electrodynamics), as they are formulated <br/>in the special theory of relativity, so that they will fit in with <br/>the general theory of relativity. When this is done, the <br/>general principle of relativity does not indeed afford us a <br/>further limitation of possibilities; but it makes us acquainted <br/>with the influence of the gravitational field on all processes,
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3409 <br/>
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3410 </p><!--l. 3475--><p class="indent"> <sup ><span
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3411 class="cmsy-8">*</span></sup> On this question of. H. Hilbert, Nachr. d. K. Gesellsch. d. Wiss. zu
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3412 <br/>Göttingen, Math.-phys. Klasse, 1915, p. 3. <pb/>
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3413 </p><!--l. 3481--><p class="indent">
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3414
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3415 </p><!--l. 3482--><p class="noindent">without our having to introduce any new hypothesis what-<br/>ever.
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3416 </p><!--l. 3485--><p class="indent"> Hence it comes about that it is not necessary to introduce <br/>definite assumptions as to the physical nature of matter (in <br/>the narrower sense). In particular it may remain an open <br/>question whether the theory of the electromagnetic field in
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3417 <br/>conjunction with that of the gravitational field furnishes a <br/>sufficient basis for the theory of matter or not. The general <br/>postulate of relativity is unable on principle to tell us anything <br/>about this. It must remain to be seen, during the working <br/>out of the theory, whether electromagnetics and the doctrine <br/>of gravitation are able in collaboration to perform what the <br/>former by itself is unable to do.
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3418 </p>
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3419 <div class="center" >
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3420
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3421 <!--l. 3500--><p class="noindent">
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3422 </p><!--l. 3501--><p class="noindent"><span
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3423 class="cmbsy-10x-x-120">§ </span><span
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3424 class="cmbx-12">19. Euler&#8217;s Equations for a Frictionless Adiabatic Fluid</span></p></div>
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3425 <!--l. 3506--><p class="indent"> Let <span
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3426 class="cmmi-12">p </span>and <span
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3427 class="cmmi-12"><img
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3428 src="img/cmmi12-1a.png" alt="r" class="cmmi-12x-x-1a" align="middle" /> </span>be two scalars, the former of which we call <br/>the &#8220; pressure,&#8221; the latter the &#8220; density &#8221; of a fluid; and let <br/>an equation subsist between them. Let the contravariant <br/>symmetrical tensor
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3429 </p>
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3430 <table width="100%"
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3431 class="equation"><tr><td><a
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3432 id="x1-75r58"></a>
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3433 <center class="math-display" >
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3434 <img
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3435 src="img/078_A_1916175x.png" alt=" ab ab dxa- dxb- T = - g p + r ds ds " class="math-display" /></center></td><td width="5%">(58)</td></tr></table>
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3436 <!--l. 3517--><p class="nopar">
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3437 </p><!--l. 3521--><p class="noindent">be the contravariant energy-tensor of the fluid. To it belongs <br/>the covariant tensor
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3438 </p>
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3439 <table width="100%"
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3440 class="equation"><tr><td><a
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3441 id="x1-76r59"></a>
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3442 <center class="math-display" >
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3443 <img
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3444 src="img/078_A_1916176x.png" alt=" dx dx Tmn = -gmnp + gmagmb --a----br, ds ds " class="math-display" /></center></td><td width="5%">(58a)</td></tr></table>
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3445 <!--l. 3529--><p class="nopar">
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3446 </p><!--l. 3533--><p class="noindent">as well as the mixed tensor <sup ><span
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3447 class="cmsy-8">*</span></sup>
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3448 </p>
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3449 <table width="100%"
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3450 class="equation"><tr><td><a
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3451 id="x1-77r59"></a>
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3452 <center class="math-display" >
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3453 <img
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3454 src="img/078_A_1916177x.png" alt=" dx dx Tas = -das p + gsb--b----ar ds ds " class="math-display" /></center></td><td width="5%">(58ba)</td></tr></table>
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3455 <!--l. 3540--><p class="nopar">
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3456 </p><!--l. 3546--><p class="noindent">Inserting the right-hand side of (58b) in (57a), we obtain the <br/>Eulerian hydrodynamical equations of the general theory of <br/>relativity. They give, in theory, a complete solution of the <br/>problem of motion, since the four equations (57a), together <br/>
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3457 </p><!--l. 3554--><p class="indent"> <sup ><span
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3458 class="cmsy-8">*</span></sup> For an observer using a system of reference in the sense of the special <br/>theory of relativity for an infinitely small region, and moving with it, the <br/>density of energy <span
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3459 class="cmmi-12">T</span><sub><span
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3460 class="cmr-8">4</span></sub><sup><span
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3461 class="cmr-8">4</span></sup> equals <span
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3462 class="cmmi-12"><img
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3463 src="img/cmmi12-1a.png" alt="r" class="cmmi-12x-x-1a" align="middle" /> </span><span
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3464 class="cmsy-10x-x-120">- </span><span
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3465 class="cmmi-12">p</span>. This gives the definition of <span
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3466 class="cmmi-12"><img
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3467 src="img/cmmi12-1a.png" alt="r" class="cmmi-12x-x-1a" align="middle" /></span>. Thus <span
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3468 class="cmmi-12"><img
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3469 src="img/cmmi12-1a.png" alt="r" class="cmmi-12x-x-1a" align="middle" /> </span>is <br/>not constant for an incompressible fluid. <pb/>
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3470 </p><!--l. 3563--><p class="indent">
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3471
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3472 </p><!--l. 3564--><p class="noindent">with the given equation between <span
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3473 class="cmmi-12">p </span>and <span
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3474 class="cmmi-12"><img
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3475 src="img/cmmi12-1a.png" alt="r" class="cmmi-12x-x-1a" align="middle" /></span>, and the equation
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3476 </p>
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3477 <center class="par-math-display" >
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3478 <img
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3479 src="img/078_A_1916178x.png" alt=" dxa-dxb- gab ds ds = 1, " class="par-math-display" /></center>
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3480 <!--l. 3571--><p class="nopar">
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3481 </p><!--l. 3575--><p class="noindent">are sufficient, <span
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3482 class="cmmi-12">g</span><sub ><span
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3483 class="cmmi-8"><img
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3484 src="img/cmmi8-b.png" alt="a" class="8x-x-b" /><img
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3485 src="img/cmmi8-c.png" alt="b" class="cmmi-8x-x-c" align="middle" /></span></sub> being given, to define the six unknowns
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3486 </p>
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3487 <center class="par-math-display" >
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3488 <img
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3489 src="img/078_A_1916179x.png" alt="p, r, dx1, dx2-, dx3-, dx4-. ds ds ds ds " class="par-math-display" /></center>
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3490 <!--l. 3582--><p class="nopar">
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3491 </p><!--l. 3586--><p class="noindent">If the <span
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3492 class="cmmi-12">g</span><sub ><span
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3493 class="cmmi-8"><img
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3494 src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img
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3495 src="img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub> are also unknown, the equations (53) are <br/>brought in. These are eleven equations for defining the ten <br/>functions <span
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3496 class="cmmi-12">g</span><sub ><span
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3497 class="cmmi-8"><img
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3498 src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img
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3499 src="img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub>, so that these functions appear over-defined. <br/>We must remember, however, that the equations (57a) are <br/>already contained in the equations (53), so that the latter <br/>represent only seven independent equations. There is good <br/>reason for this lack of definition, in that the wide freedom of <br/>the choice of co-ordinates causes the problem to remain
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3500 <br/>mathematically undefined to such a degree that three of the <br/>functions of space may be chosen at will.<sup ><span
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3501 class="cmsy-8">*</span></sup>
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3502 </p>
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3503 <div class="center" >
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3504
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3505 <!--l. 3600--><p class="noindent">
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3506 </p><!--l. 3601--><p class="noindent"><span
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3507 class="cmbsy-10x-x-120">§ </span><span
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3508 class="cmbx-12">20. Maxwell&#8217;s Electromagnetic Field Equations for Free </span> <br/><span
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3509 class="cmbx-12">Space</span></p></div>
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3510 <!--l. 3606--><p class="indent"> Let <span
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3511 class="cmmi-12"><img
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3512 src="img/cmmi12-1e.png" alt="f" class="12x-x-1e" /></span><sub ><span
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3513 class="cmmi-8"><img
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3514 src="img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub> be the components of a covariant vector--the <br/>electromagnetic potential vector. From them we form, in <br/>accordance with (36), the components F<sub ><span
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3515 class="cmmi-8"><img
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3516 src="img/cmmi8-1a.png" alt="r" class="cmmi-8x-x-1a" align="middle" /><img
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3517 src="img/cmmi8-1b.png" alt="s" class="8x-x-1b" /></span></sub> of the covariant <br/>six-vector of the electromagnetic field, in accordance with <br/>the system of equations
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3518 </p>
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3519 <table width="100%"
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3520 class="equation"><tr><td><a
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3521 id="x1-78r59"></a>
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3522 <center class="math-display" >
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3523 <img
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3524 src="img/078_A_1916180x.png" alt=" @f @f Frs = ---r- ---s @xs @xr " class="math-display" /></center></td><td width="5%">(59)</td></tr></table>
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3525 <!--l. 3619--><p class="nopar">
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3526 </p><!--l. 3623--><p class="noindent">It follows from (59) that the system of equations
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3527 </p>
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3528 <table width="100%"
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3529 class="equation"><tr><td><a
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3530 id="x1-79r60"></a>
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3531 <center class="math-display" >
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3532 <img
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3533 src="img/078_A_1916181x.png" alt="@F @F @F ---rs-+ --st-+ --tr-= 0 @xt @xr @xs " class="math-display" /></center></td><td width="5%">(60)</td></tr></table>
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3534 <!--l. 3631--><p class="nopar">
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3535 </p><!--l. 3636--><p class="noindent">is satisfied, its left side being, by (37), an antisymmetrical <br/>tensor of the third rank. System (60) thus contains essenti-<br/>ally four equations which are written out as follows:--
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3536 </p><!--l. 3643--><p class="indent"> <sup ><span
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3537 class="cmsy-8">*</span></sup> On the abandonment of the choice of co-ordinates with <span
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3538 class="cmmi-12">g </span>= <span
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3539 class="cmsy-10x-x-120">-</span>1<span
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3540 class="cmmi-12">, </span>there
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3541 <br/>remain <span
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3542 class="cmti-12">four </span>functions of space with liberty of choice, corresponding to the four <br/>arbitrary functions at our disposal in the choice of co-ordinates.
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3543 <pb/>
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3544 </p><!--l. 3651--><p class="indent">
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3545
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3546 </p>
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3547 <table width="100%"
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3548 class="equation"><tr><td><a
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3549 id="x1-80r61"></a>
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3550 <center class="math-display" >
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3551 <img
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3552 src="img/078_A_1916182x.png" alt="@F23- + @F34-+ @F42-= 0 @x4 @x2 @x3 @F34- @F41- @F13- @x + @x + @x = 0 1 3 4 } @F41- + @F12-+ @F24-= 0 @x2 @x4 @x1 @F12 @F23 @F31 ----- + -----+ -----= 0 @x3 @x1 @x2 " class="math-display" /></center></td><td width="5%">(60a)</td></tr></table>
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3553 <!--l. 3665--><p class="nopar">
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3554 </p><!--l. 3669--><p class="indent"> This system corresponds to the second of Maxwell&#8217;s <br/>systems of equations. We recognize this at once by setting
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3555 </p>
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3556 <table width="100%"
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3557 class="equation"><tr><td><a
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3558 id="x1-81r61"></a>
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3559 <center class="math-display" >
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3560 <img
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3561 src="img/078_A_1916183x.png" alt="F = H , F = E 23 x 14 x} F31 = Hy, F24 = Ey F12 = Hz, F34 = Ez " class="math-display" /></center></td><td width="5%">(61)</td></tr></table>
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3562 <!--l. 3679--><p class="nopar">
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3563 </p><!--l. 3682--><p class="noindent">Then in place of (60a) we may set, in the usual notation of <br/>three-dimensional vector analysis,
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3564 </p>
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3565 <table width="100%"
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3566 class="equation"><tr><td><a
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3567 id="x1-82r62"></a>
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3568 <center class="math-display" >
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3569 <img
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3570 src="img/078_A_1916184x.png" alt="- @H--= curlE } @t divH = 0 " class="math-display" /></center></td><td width="5%">(60b)</td></tr></table>
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3571 <!--l. 3693--><p class="nopar">
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3572 </p><!--l. 3697--><p class="indent"> We obtain Maxwell&#8217;s first system by generalizing the <br/>form given by Minkowski. We introduce the contravariant <br/>six-vector associated with F<sup ><span
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3573 class="cmmi-8"><img
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3574 src="img/cmmi8-b.png" alt="a" class="8x-x-b" /><img
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3575 src="img/cmmi8-c.png" alt="b" class="cmmi-8x-x-c" align="middle" /></span></sup>
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3576 </p>
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3577 <table width="100%"
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3578 class="equation"><tr><td><a
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3579 id="x1-83r62"></a>
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3580 <center class="math-display" >
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3581 <img
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3582 src="img/078_A_1916185x.png" alt=" mn ma nb F = g g Fab " class="math-display" /></center></td><td width="5%">(62)</td></tr></table>
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3583 <!--l. 3705--><p class="nopar">
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3584 </p><!--l. 3709--><p class="noindent">and also the contravariant vector J<sup ><span
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3585 class="cmmi-8"><img
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3586 src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /></span></sup> of the density of the <br/>electric current. Then, taking (40) into consideration, the <br/>following equations will be invariant for any substitution <br/>whose invariant is unity (in agreement with the chosen co-<br/>ordinates):--
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3587 </p>
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3588 <table width="100%"
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3589 class="equation"><tr><td><a
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3590 id="x1-84r63"></a>
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3591
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3592 <center class="math-display" >
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3593 <img
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3594 src="img/078_A_1916186x.png" alt="-@--Fmn = Jm @xn " class="math-display" /></center></td><td width="5%">(63)</td></tr></table>
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3595 <!--l. 3718--><p class="nopar">
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3596 </p><!--l. 3721--><p class="noindent">Let
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3597 </p>
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3598 <table width="100%"
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3599 class="equation"><tr><td><a
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3600 id="x1-85r64"></a>
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3601 <center class="math-display" >
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3602 <img
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3603 src="img/078_A_1916187x.png" alt="F23 = H' , F14 = - E' x x F31 = H'y, F24 = -E'y} 12 ' 34 ' F = H z, F = -E z " class="math-display" /></center></td><td width="5%">(64)</td></tr></table>
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3604 <!--l. 3734--><p class="nopar">
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3605 </p><!--l. 3738--><p class="noindent">which quantities are equal to the quantities <span
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3606 class="cmmi-12">H</span><sub ><span
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3607 class="cmmi-8">x</span></sub> <span
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3608 class="cmmi-12">...</span> <span
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3609 class="cmmi-12">E</span><sub ><span
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3610 class="cmmi-8">z</span></sub> in <br/><pb/>
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3611 </p><!--l. 3742--><p class="indent">
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3612
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3613 </p><!--l. 3743--><p class="noindent">the special case of the restricted theory of relativity; and in <br/>addition
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3614 </p>
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3615 <center class="par-math-display" >
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3616 <img
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3617 src="img/078_A_1916188x.png" alt=" 1 2 3 4 J = jx, J = jy, J = jz, J = r, " class="par-math-display" /></center>
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3618 <!--l. 3750--><p class="nopar">
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3619 </p><!--l. 3754--><p class="noindent">we obtain in place of (63)
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3620 </p>
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3621 <table width="100%"
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3622 class="equation"><tr><td><a
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3623 id="x1-86r65"></a>
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3624 <center class="math-display" >
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3625 <img
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3626 src="img/078_A_1916189x.png" alt=" ' @E--+ j = curl H' @t } divE'= r " class="math-display" /></center></td><td width="5%">(63a)</td></tr></table>
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3627 <!--l. 3765--><p class="nopar">
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3628 </p><!--l. 3769--><p class="indent"> The equations (60), (62), and (63) thus form the generali-<br/>zation of Maxwell&#8217;s field equations for free space, with the <br/>convention which we have established with respect to the <br/>choice of co-ordinates.
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3629 </p><!--l. 3774--><p class="indent"> <span
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3630 class="cmti-12">The Energy-components of the Electromagnetic Field.</span>--<br/>We form the inner product
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3631 </p>
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3632 <table width="100%"
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3633 class="equation"><tr><td><a
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3634 id="x1-87r65"></a>
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3635 <center class="math-display" >
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3636 <img
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3637 src="img/078_A_1916190x.png" alt=" m ks = FsmJ " class="math-display" /></center></td><td width="5%">(65)</td></tr></table>
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3638 <!--l. 3780--><p class="nopar">
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3639
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3640 </p><!--l. 3784--><p class="noindent">By (61) its components, written in the three-dimensional <br/>manner, are
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3641 </p>
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3642 <table width="100%"
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3643 class="equation"><tr><td><a
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3644 id="x1-88r66"></a>
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3645 <center class="math-display" >
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3646 <img
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3647 src="img/078_A_1916191x.png" alt="k = rEx + [j .H]x 1 . . . . } . . . .
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3648 k4 = - (jE) " class="math-display" /></center></td><td width="5%">(65a)</td></tr></table>
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3649 <!--l. 3798--><p class="nopar">
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3650 </p><!--l. 3802--><p class="indent"> <span
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3651 class="cmmi-12"><img
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3652 src="img/cmmi12-14.png" alt="k" class="12x-x-14" /></span><sub ><span
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3653 class="cmmi-8"><img
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3654 src="img/cmmi8-1b.png" alt="s" class="8x-x-1b" /></span></sub> is a covariant vector the components of which are <br/>equal to the negative momentum, or, respectively, the energy, <br/>which is transferred from the electric masses to the electro-<br/>magnetic field per unit of time and volume. If the electric
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3655 <br/>masses are free, that is, under the sole influence of the <br/>electromagnetic field, the covariant vector <span
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3656 class="cmmi-12"><img
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3657 src="img/cmmi12-14.png" alt="k" class="12x-x-14" /></span><sub ><span
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3658 class="cmmi-8"><img
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3659 src="img/cmmi8-1b.png" alt="s" class="8x-x-1b" /></span></sub> will vanish.
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3660 </p><!--l. 3811--><p class="indent"> To obtain the energy-components <span
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3661 class="cmmi-12">T</span><sub><span
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3662 class="cmmi-8"><img
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3663 src="img/cmmi8-1b.png" alt="s" class="8x-x-1b" /></span></sub><sup><span
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3664 class="cmmi-8"><img
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3665 src="img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sup> of the electromagnetic <br/>field, we need only give to equation <span
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3666 class="cmmi-12"><img
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3667 src="img/cmmi12-14.png" alt="k" class="12x-x-14" /></span><sub ><span
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3668 class="cmmi-8"><img
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3669 src="img/cmmi8-1b.png" alt="s" class="8x-x-1b" /></span></sub> = 0 the form of <br/>equation (57). From (63) and (65) we have in the first place
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3670 </p>
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3671 <center class="par-math-display" >
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3672 <img
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3673 src="img/078_A_1916192x.png" alt=" @Fmn @ @F ks = Fsm ----- = ----(FsmFmn) - Fmr---sm. @xn @xn @xn " class="par-math-display" /></center>
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3674 <!--l. 3823--><p class="nopar">
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3675 </p><!--l. 3827--><p class="noindent">The second term of the right-hand side, by reason of (60), <br/>permits the transformation
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3676 </p>
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3677 <center class="par-math-display" >
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3678 <img
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3679 src="img/078_A_1916193x.png" alt=" @Fsm @Fmn @Fmn Fmn ----- = - 12Fmn -----= - 12gmagnbFab -----, @xn @xs @xs " class="par-math-display" /></center>
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3680 <!--l. 3839--><p class="nopar"> <pb/>
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3681
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3682 </p><!--l. 3846--><p class="indent">
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3683
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3684 </p><!--l. 3847--><p class="noindent">which latter expression may, for reasons of symmetry, also <br/>be written in the form
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3685 </p>
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3686 <center class="par-math-display" >
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3687 <img
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3688 src="img/078_A_1916194x.png" alt=" [ @F @F ] - 14 gmagnbFab ---mn+ gmagnb---abFmn . @xs @xs " class="par-math-display" /></center>
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3689 <!--l. 3857--><p class="nopar">
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3690 </p><!--l. 3860--><p class="noindent">But for this we may set
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3691 </p>
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3692 <center class="par-math-display" >
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3693 <img
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3694 src="img/078_A_1916195x.png" alt=" @ @ - 14----(gmagnbFabFmn) + 14FabFmn ----(gmagnb). @xs @xs " class="par-math-display" /></center>
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3695 <!--l. 3870--><p class="nopar">
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3696 </p><!--l. 3873--><p class="noindent">The first of these terms is written more briefly
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3697 </p>
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3698 <center class="par-math-display" >
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3699 <img
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3700 src="img/078_A_1916196x.png" alt=" @ - 14----(FmnFmn); @xs " class="par-math-display" /></center>
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3701 <!--l. 3880--><p class="nopar">
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3702 </p><!--l. 3884--><p class="noindent">the second, after the differentiation is carried out, and after <br/>some reduction, results in
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3703 </p>
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3704 <center class="par-math-display" >
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3705 <img
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3706 src="img/078_A_1916197x.png" alt=" @g - 1FmtFmngnr --st-. 2 @xs " class="par-math-display" /></center>
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3707 <!--l. 3893--><p class="nopar">
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3708 </p><!--l. 3896--><p class="noindent">Taking all three terms together we obtain the relation
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3709 </p>
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3710 <table width="100%"
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3711 class="equation"><tr><td><a
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3712 id="x1-89r66"></a>
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3713 <center class="math-display" >
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3714 <img
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3715 src="img/078_A_1916198x.png" alt=" @T n @g ks = ---s - 1gtm--mn-Tnt @xn 2 @xs " class="math-display" /></center></td><td width="5%">(66)</td></tr></table>
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3716 <!--l. 3904--><p class="nopar">
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3717 </p><!--l. 3908--><p class="noindent">where
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3718 </p>
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3719 <center class="par-math-display" >
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3720 <img
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3721 src="img/078_A_1916199x.png" alt=" n na 1 n ab Ts = - FsaF + 4dsFabF . " class="par-math-display" /></center>
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3722 <!--l. 3916--><p class="nopar">
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3723 </p><!--l. 3920--><p class="indent"> Equation (66), if <span
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3724 class="cmmi-12"><img
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3725 src="img/cmmi12-14.png" alt="k" class="12x-x-14" /></span><sub ><span
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3726 class="cmmi-8"><img
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3727 src="img/cmmi8-1b.png" alt="s" class="8x-x-1b" /></span></sub> vanishes, is, on account of (30), <br/>equivalent to (57) or (57<span
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3728 class="cmti-12">a</span>) respectively. Therefore the <span
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3729 class="cmmi-12">T</span><sub><span
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3730 class="cmmi-8"><img
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3731 src="img/cmmi8-1b.png" alt="s" class="8x-x-1b" /></span></sub><sup><span
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3732 class="cmmi-8"><img
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3733 src="img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sup> <br/>are the energy-components of the electromagnetic field. <br/>With the help of (61) and (64), it is easy to show that these <br/>energy-components of the electromagnetic field in the case <br/>of the special theory of relativity give the well-known Maxwell-<br/>Poynting expressions.
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3734 </p><!--l. 3929--><p class="indent"> We have now deduced the general laws which are satisfied <br/>by the gravitational field and matter, by consistently using a <br/>system of co-ordinates for which
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3735 <img
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3736 src="img/078_A_1916200x.png" alt=" V~ ---- -g" class="sqrt" /> = 1<span
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3737 class="cmmi-12">. </span>We have <br/>thereby achieved a considerable simplification of formulæ; <br/>and calculations, without failing to comply with the require-<br/>ment of general covariance; for we have drawn our equations <br/>from generally covariant equations by specializing the system <br/>of co-ordinates. <pb/>
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3738 </p><!--l. 3941--><p class="indent">
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3739
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3740 </p><!--l. 3942--><p class="indent"> Still the question is not without a formal interest, whether <br/>with a correspondingly generalized definition of the energy-<br/>components of gravitational field and matter, even without <br/>specializing the system of co-ordinates, it is possible to formu-<br/>late laws of conservation in the form of equation (56), and <br/>field equations of gravitation of the same nature as (52) or <br/>(52a), in such a manner that on the left we have a divergence <br/>(in the ordinary sense), and on the right the sum of the <br/>energy-components of matter and gravitation. I have found <br/>that in both cases this is actually so. But I do not think <br/>that the communication of my somewhat extensive reflexions <br/>on this subject would be worth while, because after all they <br/>do not give us anything that is materially new.
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3741 </p>
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3742 <div class="center" >
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3743
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3744 <!--l. 3959--><p class="noindent">
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3745 </p><!--l. 3960--><p class="noindent">E</p></div>
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3746 <div class="center" >
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3747
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3748 <!--l. 3964--><p class="noindent">
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3749 </p><!--l. 3965--><p class="noindent"><span
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3750 class="cmbsy-10x-x-120">§ </span><span
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3751 class="cmbx-12">21. Newton&#8217;s Theory as a First Approximation</span></p></div>
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3752 <!--l. 3969--><p class="indent"> As has already been mentioned more than once, the <br/>special theory of relativity as a special case of the general <br/>theory is characterized by the <span
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3753 class="cmmi-12">g</span><sub ><span
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3754 class="cmmi-8"><img
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3755 src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img
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3756 src="img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub>
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3757 having the constant values <br/>(4). From what has already been said, this means complete <br/>neglect of the effects of gravitation. We arrive at a closer
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3758 <br/>approximation to reality by considering the case where the <br/><span
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3759 class="cmmi-12">g</span><sub ><span
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3760 class="cmmi-8"><img
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3761 src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img
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3762 src="img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub> differ from the values of (4) by quantities which are small <br/>compared with 1, and neglecting small quantities of second <br/>and higher order. (First point of view of approximation.)
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3763 </p><!--l. 3981--><p class="indent"> It is further to be assumed that in the space-time territory <br/>under consideration the <span
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3764 class="cmmi-12">g</span><sub ><span
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3765 class="cmmi-8"><img
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3766 src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img
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3767 src="img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub> at spatial infinity, with a suitable <br/>choice of co-ordinates, tend toward the values (4); i.e. we are <br/>considering gravitational fields which may be regarded as <br/>generated exclusively by matter in the finite region.
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3768 </p><!--l. 3988--><p class="indent"> It might be thought that these approximations must lead <br/>us to Newton&#8217;s theory. But to that end we still need to ap-<br/>proximate the fundamental equations from a second point of <br/>view. We give our attention to the motion of a material
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3769 <br/>point in accordance with the equations (16). In the case of <br/>the special theory of relativity the components
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3770 </p>
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3771 <center class="par-math-display" >
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3772 <img
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3773 src="img/078_A_1916201x.png" alt="dx1 dx2 dx3 ----,----, ---- ds ds ds " class="par-math-display" /></center>
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3774
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3775 <!--l. 3999--><p class="nopar"> <pb/>
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3776 </p><!--l. 4006--><p class="indent">
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3777
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3778 </p><!--l. 4007--><p class="noindent">may take on any values. This signifies that any velocity
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3779 </p>
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3780 <center class="par-math-display" >
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3781 <img
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3782 src="img/078_A_1916202x.png" alt=" V~ -(----)2---(-----)2---(-----)2- dx1- dx2- dx3- v = dx4 + dx4 + dx4 " class="par-math-display" /></center>
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3783 <!--l. 4015--><p class="nopar">
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3784 </p><!--l. 4019--><p class="noindent">may occur, which is less than the velocity of light <span
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3785 class="cmti-12">in vacuo. </span> <br/>If we restrict ourselves to the case which almost exclusively <br/>offers itself to our experience, of <span
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3786 class="cmmi-12">v</span>
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3787 being small as compared <br/>with the velocity of light, this denotes that the components
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3788 </p>
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3789 <center class="par-math-display" >
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3790 <img
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3791 src="img/078_A_1916203x.png" alt="dx1-, dx2-, dx3 ds ds ds " class="par-math-display" /></center>
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3792 <!--l. 4028--><p class="nopar">
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3793 </p><!--l. 4032--><p class="noindent">are to be treated as small quantities, while <span
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3794 class="cmmi-12">dx</span><sub ><span
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3795 class="cmr-8">4</span></sub><span
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3796 class="cmmi-12">/ds, </span>to the <br/>second order of small quantities, is equal to one. (Second <br/>point of view of approximation.)
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3797 </p><!--l. 4036--><p class="indent"> Now we remark that from the first point of view of ap-<br/>proximation the magnitudes <img
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3798 src="img/cmr12-0.png" alt="G" class="12x-x-0" /><sub><span
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3799 class="cmmi-8"><img
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3800 src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img
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3801 src="img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub><sup> <span
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3802 class="cmmi-8"><img
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3803 src="img/cmmi8-1c.png" alt="t" class="8x-x-1c" /></span></sup> are all small magnitudes of <br/>at least the first order. A glance at (46) thus shows that in <br/>this equation, from the second point of view of approximation, <br/>we have to consider only terms for which <span
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3804 class="cmmi-12"><img
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3805 src="img/cmmi12-16.png" alt="m" class="cmmi-12x-x-16" align="middle" /> </span>= <span
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3806 class="cmmi-12"><img
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diff changeset
3807 src="img/cmmi12-17.png" alt="n" class="12x-x-17" /> </span>= 4<span
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diff changeset
3808 class="cmmi-12">. </span>Re-<br/>stricting ourselves to terms of lowest order we first obtain in <br/>place of (46) the equations
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3809 </p>
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3810 <center class="par-math-display" >
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3811 <img
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3812 src="img/078_A_1916204x.png" alt="d2xt- t dt2 = G 44 " class="par-math-display" /></center>
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3813 <!--l. 4048--><p class="nopar">
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3814 </p><!--l. 4052--><p class="noindent">where we have set <span
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3815 class="cmmi-12">ds </span>= <span
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3816 class="cmmi-12">dx</span><sub ><span
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3817 class="cmr-8">4</span></sub> = <span
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3818 class="cmmi-12">dt </span>; or with restriction to terms <br/>which from the first point of view of approximation are of <br/>first order:--
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3819 </p>
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3820 <center class="par-math-display" >
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3821 <img
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3822 src="img/078_A_1916205x.png" alt=" 2 d-xt-= [44, t] (t = 1, 2, 3) dt2 2 d-x4-= - [44, 4]. dt2 " class="par-math-display" /></center>
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3823 <!--l. 4063--><p class="nopar">
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3824 </p><!--l. 4066--><p class="noindent">If in addition we suppose the gravitational field to be a quasi-<br/>static field, by confining ourselves to the case where the <br/>motion of the matter generating the gravitational field is but <br/>slow (in comparison with the velocity of the propagation of <br/>light), we may neglect on the right-hand side differentiations <br/>with respect to the time in comparison with those with re-<br/>spect to the space co-ordinates, so that we have <pb/>
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3825 </p><!--l. 4078--><p class="indent">
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3826
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3827 </p>
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3828 <table width="100%"
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3829 class="equation"><tr><td><a
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3830 id="x1-90r67"></a>
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3831 <center class="math-display" >
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3832 <img
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3833 src="img/078_A_1916206x.png" alt=" 2 d-xt- = - 1 @g44- (t = 1, 2, 3) dt2 2 @xt " class="math-display" /></center></td><td width="5%">(67)</td></tr></table>
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3834 <!--l. 4084--><p class="nopar">
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3835 </p><!--l. 4087--><p class="noindent">This is the equation of motion of the material point accord-<br/>ing to Newton&#8217;s theory, in which <span
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3836 class="cmr-8">1</span>
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3837 <span
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3838 class="cmr-8">2</span><span
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3839 class="cmmi-12">g</span><sub ><span
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3840 class="cmr-8">44</span></sub> plays the part of the <br/>gravitational potential. What is remarkable in this result <br/>is that the component <span
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3841 class="cmmi-12">g</span><sub ><span
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3842 class="cmr-8">44</span></sub> of the fundamental tensor alone <br/>defines, to a first approximation, the motion of the material
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3843 <br/>point.
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3844 </p><!--l. 4096--><p class="indent"> We now turn to the field equations (53). Here we <br/>have to take into consideration that the energy-tensor of <br/>&#8220; matter &#8221; is almost exclusively defined by the density of <br/>matter in the narrower sense, i.e. by the second term of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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3845 <br/>right-hand side of (58) [or, respectively, (58a) or (58b)]. <br/>If we form the approximation in question, all the components <br/>vanish with the one exception of <span
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3846 class="cmmi-12">T</span><sub ><span
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3847 class="cmr-8">44</span> </sub> = <span
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3848 class="cmmi-12"><img
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3849 src="img/cmmi12-1a.png" alt="r" class="cmmi-12x-x-1a" align="middle" /> </span>= <span
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3850 class="cmmi-12">T. </span>On the left-<br/>hand side of (53) the second term is a small quantity of <br/>second order; the first yields, to the approximation in
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3851 <br/>question,
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3852 </p>
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3853 <center class="par-math-display" >
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3854 <img
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3855 src="img/078_A_1916207x.png" alt=" @ @ @ @ ----[mn, 1] + ----[mn, 2] +----[mn, 3]- ---- [mn, 4]. @x1 @x2 @x3 @x4 " class="par-math-display" /></center>
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3856 <!--l. 4114--><p class="nopar">
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3857 </p><!--l. 4118--><p class="noindent">For <span
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3858 class="cmmi-12"><img
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3859 src="img/cmmi12-16.png" alt="m" class="cmmi-12x-x-16" align="middle" /> </span>= <span
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3860 class="cmmi-12"><img
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3861 src="img/cmmi12-17.png" alt="n" class="12x-x-17" /> </span>= 4<span
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3862 class="cmmi-12">, </span>this gives, with the omission of terms differ-<br/>entiated with respect to time,
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3863 </p>
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3864 <center class="par-math-display" >
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3865 <img
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3866 src="img/078_A_1916208x.png" alt=" ( 2 2 2 ) - 1 @-g44+ @-g44 + @-g44 = - 1 \~/ 2g44. 2 @x21 @x22 @x23 2 " class="par-math-display" /></center>
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3867 <!--l. 4129--><p class="nopar">
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3868 </p><!--l. 4132--><p class="noindent">The last of equations (53) thus yields
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3869 </p>
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3870 <table width="100%"
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3871 class="equation"><tr><td><a
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3872 id="x1-91r68"></a>
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3873 <center class="math-display" >
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3874 <img
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3875 src="img/078_A_1916209x.png" alt=" \~/ 2g44 = kr " class="math-display" /></center></td><td width="5%">(68)</td></tr></table>
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3876 <!--l. 4139--><p class="nopar">
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3877 </p><!--l. 4143--><p class="noindent">The equations (67) and (68) together are equivalent to <br/>Newton&#8217;s law of gravitation.
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3878 </p><!--l. 4147--><p class="indent"> By (67) and (68) the expression for the gravitational <br/>potential becomes
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3879 </p>
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3880 <table width="100%"
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3881 class="equation"><tr><td><a
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3882 id="x1-92r69"></a>
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3883 <center class="math-display" >
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3884 <img
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3885 src="img/078_A_1916210x.png" alt=" k integral rdt - --- ---- 8p r " class="math-display" /></center></td><td width="5%">(68a)</td></tr></table>
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3886 <!--l. 4153--><p class="nopar">
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3887 </p><!--l. 4157--><p class="noindent">while Newton&#8217;s theory, with the unit of time which we have <br/>chosen, gives
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3888 </p>
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3889 <center class="par-math-display" >
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3890 <img
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3891 src="img/078_A_1916211x.png" alt=" K integral rdt - -2- ---- c r " class="par-math-display" /></center>
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3892 <!--l. 4163--><p class="nopar"> <pb/>
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3893 </p><!--l. 4170--><p class="indent">
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3894
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3895 </p><!--l. 4171--><p class="noindent">in which K denotes the constant 6<span
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diff changeset
3896 class="cmmi-12">.</span>7 <span
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diff changeset
3897 class="cmsy-10x-x-120">× </span>10<sup ><span
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diff changeset
3898 class="cmsy-8">- </span><span
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diff changeset
3899 class="cmr-8">8</span></sup><span
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3900 class="cmmi-12">, </span>usually called <br/>the constant of gravitation. By comparison we obtain
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3901 </p>
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3902 <table width="100%"
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3903 class="equation"><tr><td><a
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3904 id="x1-93r69"></a>
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3905 <center class="math-display" >
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3906 <img
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diff changeset
3907 src="img/078_A_1916212x.png" alt=" 8pK-- -27 k = c2 = 1.87 × 10 " class="math-display" /></center></td><td width="5%">(69)</td></tr></table>
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3908 <!--l. 4178--><p class="nopar">
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3909 </p>
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diff changeset
3910 <div class="center" >
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3911
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3912 <!--l. 4182--><p class="noindent">
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3913 </p><!--l. 4183--><p class="noindent"><span
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diff changeset
3914 class="cmbsy-10x-x-120">§ </span><span
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diff changeset
3915 class="cmbx-12">22. Behaviour of Rods and Clocks in the Static Gravi- </span> <br/><span
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diff changeset
3916 class="cmbx-12">tational</span>
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diff changeset
3917 <span
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3918 class="cmbx-12">Field. Bending of Light-rays. Motion of </span> <br/><span
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diff changeset
3919 class="cmbx-12">the Perihelion of a</span>
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diff changeset
3920 <span
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3921 class="cmbx-12">Planetary Orbit</span></p></div>
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3922 <!--l. 4189--><p class="indent"> To arrive at Newton&#8217;s theory as a first approximation we <br/>had to calculate only one component, <span
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diff changeset
3923 class="cmmi-12">g</span><sub ><span
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diff changeset
3924 class="cmr-8">44</span></sub>, of the ten <span
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diff changeset
3925 class="cmmi-12">g</span><sub ><span
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diff changeset
3926 class="cmmi-8"><img
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diff changeset
3927 src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img
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diff changeset
3928 src="img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub> of the <br/>gravitational field, since this component alone enters into the <br/>first approximation, (67), of the equation for the motion of the <br/>material point in the gravitational field. From this, however, <br/>it is already apparent that other components of the <span
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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diff changeset
3929 class="cmmi-12">g</span><sub ><span
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diff changeset
3930 class="cmmi-8"><img
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diff changeset
3931 src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img
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diff changeset
3932 src="img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub> must <br/>differ from the values given in (4) by small quantities of the <br/>first order. This is required by the condition
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diff changeset
3933 <span
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3934 class="cmmi-12">g </span>= <span
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3935 class="cmsy-10x-x-120">- </span>1<span
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diff changeset
3936 class="cmmi-12">.</span>
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3937 </p><!--l. 4200--><p class="indent"> For a field-producing point mass at the origin of co-ordin-<br/>ates, we obtain, to the first approximation, the radially <br/>symmetrical solution
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3938 </p>
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3939 <table width="100%"
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3940 class="equation"><tr><td><a
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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3941 id="x1-94r70"></a>
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3942 <center class="math-display" >
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3943 <img
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3944 src="img/078_A_1916213x.png" alt=" xrxs grs = - drs- a---3-(r, s = 1, 2, 3) r } gr4 = g4r = 0 (r = 1, 2, 3) a- g44 = 1- r " class="math-display" /></center></td><td width="5%">(70)</td></tr></table>
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3945 <!--l. 4213--><p class="nopar">
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3946 </p><!--l. 4217--><p class="noindent">where <span
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diff changeset
3947 class="cmmi-12"><img
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diff changeset
3948 src="img/cmmi12-e.png" alt="d" class="12x-x-e" /></span><sub ><span
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diff changeset
3949 class="cmmi-8"><img
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diff changeset
3950 src="img/cmmi8-1a.png" alt="r" class="cmmi-8x-x-1a" align="middle" /><img
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diff changeset
3951 src="img/cmmi8-1b.png" alt="s" class="8x-x-1b" /></span></sub> is 1 or 0, respectively, accordingly as <span
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3952 class="cmmi-12"><img
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3953 src="img/cmmi12-1a.png" alt="r" class="cmmi-12x-x-1a" align="middle" /> </span>= <span
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diff changeset
3954 class="cmmi-12"><img
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diff changeset
3955 src="img/cmmi12-1b.png" alt="s" class="12x-x-1b" /> </span>or <span
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diff changeset
3956 class="cmmi-12"><img
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diff changeset
3957 src="img/cmmi12-1a.png" alt="r" class="cmmi-12x-x-1a" align="middle" /></span><img
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3958 src="img/078_A_1916214x.png" alt="/=" class="neq" align="middle" /><span
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3959 class="cmmi-12"><img
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3960 src="img/cmmi12-1b.png" alt="s" class="12x-x-1b" />, </span> <br/>and <span
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diff changeset
3961 class="cmmi-12">r </span>is the quantity +<img
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3962 src="img/078_A_1916215x.png" alt=" V~ -x2-+-x2-+-x2- 1 2 3" class="sqrt" /> On account of (68a)
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3963 </p>
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diff changeset
3964 <table width="100%"
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diff changeset
3965 class="equation"><tr><td><a
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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diff changeset
3966 id="x1-95r71"></a>
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3967 <center class="math-display" >
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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3968 <img
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3969 src="img/078_A_1916216x.png" alt=" kM a = ----, 4p " class="math-display" /></center></td><td width="5%">(70a)</td></tr></table>
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3970 <!--l. 4225--><p class="nopar">
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3971 </p><!--l. 4229--><p class="noindent">if M denotes the field-producing mass. It is easy to verify <br/>that the field equations (outside the mass) are satisfied to the <br/>first order of small quantities.
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3972 </p><!--l. 4233--><p class="indent"> We now examine the influence exerted by the field of the <br/>mass M upon the metrical properties of space. The relation
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3973 </p>
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3974 <center class="par-math-display" >
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3975 <img
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3976 src="img/078_A_1916217x.png" alt="ds2 = gmndxmdxn. " class="par-math-display" /></center>
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3977 <!--l. 4240--><p class="nopar"></p><!--l. 4244--><p class="noindent">always holds between the &#8220; locally &#8221; (<span
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3978 class="cmsy-10x-x-120">§ </span>4) measured lengths <br/>and times <span
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3979 class="cmti-12">ds </span>on the one hand, and the differences of co-ordin-<br/>ates <span
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diff changeset
3980 class="cmmi-12">dx</span><sub ><span
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3981 class="cmmi-8"><img
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diff changeset
3982 src="img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub> on the other hand.
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diff changeset
3983 <pb/>
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3984 </p><!--l. 4251--><p class="indent">
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3985
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3986 </p><!--l. 4252--><p class="indent"> For a unit-measure of length laid &#8220; parallel &#8221; to the axis <br/>of <span
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3987 class="cmmi-12">x</span>, for example, we should have to set <span
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3988 class="cmmi-12">ds</span><sup ><span
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3989 class="cmr-8">2</span></sup> = <span
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3990 class="cmsy-10x-x-120">-</span>1; <span
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3991 class="cmmi-12">dx</span><sub >
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3992 <span
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3993 class="cmr-8">2</span></sub> = <span
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3994 class="cmmi-12">dx</span><sub ><span
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3995 class="cmr-8">3</span></sub> <br/>= <span
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3996 class="cmmi-12">dx</span><sub ><span
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3997 class="cmr-8">4</span></sub> = 0<span
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3998 class="cmmi-12">. </span>Therefore <span
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3999 class="cmsy-10x-x-120">-</span>1 = <span
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4000 class="cmmi-12">g</span><sub ><span
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4001 class="cmr-8">11</span></sub><span
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4002 class="cmmi-12">dx</span><sub><span
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4003 class="cmr-8">1</span></sub><sup><span
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4004 class="cmr-8">2</span></sup><span
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4005 class="cmmi-12">. </span>If, in addition, the <br/>unit-measure lies on the axis of <span
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4006 class="cmmi-12">x</span>, the first of equations (70)
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4007 <br/>gives
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4008 </p>
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4009 <center class="par-math-display" >
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4010 <img
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4011 src="img/078_A_1916218x.png" alt=" ( ) g = - 1 + a- . 11 r " class="par-math-display" /></center>
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4012 <!--l. 4261--><p class="nopar">
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4013 </p><!--l. 4264--><p class="noindent">From these two relations it follows that, correct to a first <br/>order of small quantities,
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4014 </p>
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4015 <table width="100%"
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4016 class="equation"><tr><td><a
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4017 id="x1-96r71"></a>
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4018 <center class="math-display" >
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4019 <img
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4020 src="img/078_A_1916219x.png" alt=" a-- dx = 1 - 2r " class="math-display" /></center></td><td width="5%">(71)</td></tr></table>
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4021 <!--l. 4271--><p class="nopar">
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4022 </p><!--l. 4275--><p class="noindent">The unit measuring-rod thus appears a little shortened in <br/>relation to the system of co-ordinates by the presence of the <br/>gravitational field, if the rod is laid along a radius.
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4023 </p><!--l. 4281--><p class="indent"> In an analogous manner we obtain the length of co-<br/>ordinates in tangential direction if, for example, we set
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4024 </p>
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4025 <center class="par-math-display" >
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4026 <img
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4027 src="img/078_A_1916220x.png" alt="ds2 = - 1; dx = dx = dx = 0 ; x = r, x = x = 0. 1 3 4 1 2 3 " class="par-math-display" /></center>
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4028 <!--l. 4287--><p class="nopar">
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4029 </p><!--l. 4291--><p class="noindent">The result is
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4030 </p>
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4031 <table width="100%"
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4032 class="equation"><tr><td><a
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4033 id="x1-97r72"></a>
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4034
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4035 <center class="math-display" >
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4036 <img
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4037 src="img/078_A_1916221x.png" alt="- 1 = g22dx22 = - dx22 " class="math-display" /></center></td><td width="5%">(71a)</td></tr></table>
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4038 <!--l. 4297--><p class="nopar">
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4039 </p><!--l. 4301--><p class="noindent">With the tangential position, therefore, the gravitational <br/>field of the point of mass has no influence on the length of a <br/>rod.
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4040 </p><!--l. 4306--><p class="indent"> Thus Euclidean geometry does not hold even to a first ap-<br/>proximation in the gravitational field, if we wish to take one <br/>and the same rod, independently of its place and orientation, <br/>as a realization of the same interval; although, to be sure, a <br/>glance at (70a) and (69) shows that the deviations to be ex-<br/>pected are much too slight to be noticeable in measurements <br/>of the earth&#8217;s surface.
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4041 </p><!--l. 4315--><p class="indent"> Further, let us examine the rate of a unit clock, which is <br/>arranged to be at rest in a static gravitational field. Here we <br/>have for a clock period
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4042 <span
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4043 class="cmmi-12">ds </span>= 1; <span
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4044 class="cmmi-12">dx</span><sub ><span
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4045 class="cmr-8">1</span></sub> = <span
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4046 class="cmmi-12">dx</span><sub ><span
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4047 class="cmr-8">2</span></sub> = <span
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4048 class="cmmi-12">dx</span><sub ><span
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4049 class="cmr-8">3</span></sub> = 0
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4050 </p><!--l. 4320--><p class="noindent">Therefore
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4051 </p>
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4052 <center class="par-math-display" >
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4053 <img
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4054 src="img/078_A_1916222x.png" alt=" 2 1 = g44dx 4; 1 1 1 dx4 = V~ -g--= V~ --------------- = 1 - 2(g44- 1) 44 (1 + (g44 - 1)) " class="par-math-display" /></center>
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4055 <!--l. 4331--><p class="nopar"> <pb/>
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4056 </p><!--l. 4338--><p class="indent">
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4057
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4058 </p><!--l. 4339--><p class="noindent">or
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4059 </p>
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4060 <table width="100%"
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4061 class="equation"><tr><td><a
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4062 id="x1-98r72"></a>
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4063 <center class="math-display" >
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4064 <img
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4065 src="img/078_A_1916223x.png" alt=" k integral dt dx4 = 1 + --- r--- 8p r " class="math-display" /></center></td><td width="5%">(72)</td></tr></table>
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4066 <!--l. 4345--><p class="nopar">
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4067 </p><!--l. 4348--><p class="noindent">Thus the clock goes more slowly if set up in the neighbour-<br/>hood of ponderable masses. From this it follows that the <br/>spectral lines of light reaching us from the surface of large <br/>stars must appear displaced towards the red end of the
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4068 <br/>spectrum.<sup ><span
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4069 class="cmsy-8">*</span></sup>
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4070 </p><!--l. 4355--><p class="indent"> We now examine the course of light-rays in the static <br/>gravitational field. By the special theory of relativity the <br/>velocity of light is given by the equation
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4071 </p>
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4072 <center class="par-math-display" >
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4073 <img
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4074 src="img/078_A_1916224x.png" alt="- dx21 - dx2 - dx23 + dx24 = 0 " class="par-math-display" /></center>
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4075 <!--l. 4362--><p class="nopar">
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4076 </p><!--l. 4366--><p class="noindent">and therefore by the general theory of relativity by the <br/>equation
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4077 </p>
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4078 <table width="100%"
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4079 class="equation"><tr><td><a
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4080 id="x1-99r73"></a>
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4081 <center class="math-display" >
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4082 <img
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4083 src="img/078_A_1916225x.png" alt=" 2 ds = gmndxmdxn = 0 " class="math-display" /></center></td><td width="5%">(73)</td></tr></table>
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4084 <!--l. 4372--><p class="nopar">
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4085 </p><!--l. 4375--><p class="noindent">If the direction, i.e. the ratio, <span
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4086 class="cmmi-12">dx</span><sub ><span
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4087 class="cmr-8">1</span></sub> : <span
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4088 class="cmmi-12">dx</span><sub ><span
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4089 class="cmr-8">2</span></sub> : <span
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4090 class="cmmi-12">dx</span><sub ><span
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4091 class="cmr-8">3</span></sub> is given, equation <br/>(73) gives the quantities
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4092 </p>
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4093 <center class="par-math-display" >
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4094 <img
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4095 src="img/078_A_1916226x.png" alt="dx1 dx2 dx3 dx--,dx--, dx-- 4 4 4 " class="par-math-display" /></center>
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4096 <!--l. 4381--><p class="nopar">
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4097 </p><!--l. 4385--><p class="noindent">and accordingly the velocity
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4098 </p>
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4099 <center class="par-math-display" >
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4100 <img
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4101 src="img/078_A_1916227x.png" alt=" V~ (----)2---(-----)2---(----)2-- dx1- dx2- dx3- dx4 + dx4 + dx4 = g " class="par-math-display" /></center>
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4102 <!--l. 4393--><p class="nopar">
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4103 </p><!--l. 4397--><p class="noindent">defined in the sense of Euclidean geometry. We easily <br/>recognize that the course of the light-rays must be bent with <br/>regard to the system of co-ordinates, if the <span
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4104 class="cmmi-12">g</span><sub ><span
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4105 class="cmmi-8"><img
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4106 src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img
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4107 src="img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub>
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4108 are not con-<br/>stant. If <span
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4109 class="cmmi-12">n </span>is a direction perpendicular to the propagation of <br/>light, the Huyghens principle shows that the light-ray, en-<br/>visaged in the plane (<span
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4110 class="cmmi-12"><img
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4111 src="img/cmmi12-d.png" alt="g" class="12x-x-d" />, n</span>)<span
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4112 class="cmmi-12">, </span>has the curvature <span
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4113 class="cmsy-10x-x-120">-</span><span
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4114 class="cmmi-12"><img
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4115 src="img/cmmi12-40.png" alt="@" class="12x-x-40" /><img
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4116 src="img/cmmi12-d.png" alt="g" class="12x-x-d" />/<img
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4117 src="img/cmmi12-40.png" alt="@" class="12x-x-40" />n.</span>
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4118 </p><!--l. 4406--><p class="indent"> We examine the curvature undergone by a ray of light <br/>passing by a mass M at the distance <img
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4119 src="img/cmr12-1.png" alt="D" class="12x-x-1" />. If we choose the <br/>system of co-ordinates in agreement with the accompanying <br/>diagram, the total bending of the ray (calculated positively if
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4120 <br/>
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4121 </p><!--l. 4413--><p class="indent"> <sup ><span
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4122 class="cmsy-8">*</span></sup> According to E. Freundlich, spectroscopical observations on fixed stars of
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4123 <br/>certain types indicate the existence of an effect of this kind, but a crucial <br/>test of this consequence has not yet been made. <pb/>
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4124 </p><!--l. 4420--><p class="indent">
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4125
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4126 </p><!--l. 4421--><p class="noindent">concave towards the origin) is given in sufficient approxi-<br/>mation by
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4127 </p>
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4128 <center class="par-math-display" >
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4129 <img
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4130 src="img/078_A_1916228x.png" alt=" integral + oo @g B = ---- dx2, - oo @x1 " class="par-math-display" /></center>
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4131 <!--l. 4428--><p class="nopar">
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4132 </p><!--l. 4432--><p class="noindent">while (73) and (70) give
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4133 </p>
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4134 <center class="par-math-display" >
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4135 <img
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4136 src="img/078_A_1916229x.png" alt=" V~ --------- ( g44 ) a ( x2) g = - --- = 1 - --- 1 + -22- . g22 2r r " class="par-math-display" /></center>
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4137 <!--l. 4439--><p class="nopar">
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4138 </p><!--l. 4443--><p class="noindent">Carrying out the calculation, this gives
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4139 </p>
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4140 <table width="100%"
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4141 class="equation"><tr><td><a
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4142 id="x1-100r74"></a>
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4143 <center class="math-display" >
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4144 <img
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4145 src="img/078_A_1916230x.png" alt=" 2a- kM--- B = /_\ = 2p /_\ . " class="math-display" /></center></td><td width="5%">(74)</td></tr></table>
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4146 <!--l. 4450--><p class="nopar">
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4147
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4148 <img
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4149 src="img/078_A_1916231x.png" alt="PIC" class="graphics" width="247.53888pt" height="240.4235pt" /><!--tex4ht:graphics
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4150 name="img/078_A_1916231x.png" src="078_A_1916_001.EPS"
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4151 -->
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4152 </p><!--l. 4459--><p class="noindent">According to this, a ray of light going past the sun under-<br/>goes a deflexion of 1<span
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4153 class="cmmi-12">.</span>7<span
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4154 class="cmsy-10x-x-120">''</span> ; and a ray going past the planet <br/>Jupiter a deflexion of about
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4155 <span
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4156 class="cmmi-12">.</span>02<span
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4157 class="cmsy-10x-x-120">''</span>.
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4158 </p><!--l. 4464--><p class="indent"> If we calculate the gravitational field to a higher degree <br/>of approximation, and likewise with corresponding accuracy <br/>the orbital motion of a material point of relatively infinitely <br/>small mass, we find a deviation of the following kind from <br/>the Kepler-Newton laws of planetary motion. The orbital <br/>ellipse of a planet undergoes a slow rotation, in the direction <br/>of motion, of amount
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4159 </p>
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4160 <table width="100%"
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4161 class="equation"><tr><td><a
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4162 id="x1-101r75"></a>
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4163 <center class="math-display" >
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4164 <img
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4165 src="img/078_A_1916232x.png" alt=" a2 e = 24p3--2-2------2- T e (1- e) " class="math-display" /></center></td><td width="5%">(75)</td></tr></table>
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4166 <!--l. 4476--><p class="nopar">
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4167 <pb/>
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4168 </p><!--l. 4483--><p class="indent">
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4169
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4170 </p><!--l. 4484--><p class="noindent">per revolution. In this formula <span
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4171 class="cmmi-12">a </span>denotes the major semi-<br/>axis, <span
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4172 class="cmmi-12">c </span>the velocity of light in the usual measurement, <span
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4173 class="cmmi-12">e </span>the <br/>eccentricity, T the time of revolution in seconds.<sup ><span
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4174 class="cmsy-8">*</span></sup>
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4175 </p><!--l. 4489--><p class="indent"> Calculation gives for the planet Mercury a rotation of the <br/>orbit of 43<span
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4176 class="cmsy-10x-x-120">''</span> per century, corresponding exactly to astronomical <br/>observation (Leverrier); for the astronomers have discovered <br/>in the motion of the perihelion of this planet, after allowing <br/>for disturbances by other planets, an inexplicable remainder <br/>of this magnitude.
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4177 </p><!--l. 4498--><p class="indent"> <sup ><span
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4178 class="cmsy-8">*</span></sup> For the calculation I refer to the original papers: A. Einstein, <br/>Sitzungsber. d. Preuss. Akad. d. Wiss., 1915, p. 831; K. Schwarzschild, <br/><span
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4179 class="cmti-12">ibid.</span>, 1916, p. 189.
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4180 </p>
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4181
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4182 </body></html>
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4183
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4184
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4185