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| author | Klaus Thoden <kthoden@mpiwg-berlin.mpg.de> |
|---|---|
| date | Thu, 02 May 2013 11:38:23 +0200 |
| parents | 22d6a63640c6 |
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<?xml version="1.0" encoding="utf-8" ?> <!--http://www.w3.org/TR/xhtml1/DTD/xhtml1-transitional.dtd--> <html> <head><title></title> <meta http-equiv="Content-Type" content="text/html; charset=iso-8859-1" /> <meta name="generator" content="TeX4ht (http://www.cse.ohio-state.edu/~gurari/TeX4ht/mn.html)" /> <meta name="originator" content="TeX4ht (http://www.cse.ohio-state.edu/~gurari/TeX4ht/mn.html)" /> <!-- xhtml,html --> <meta name="src" content="078_A_1916.tex" /> <meta name="date" content="2005-03-10 19:24:00" /> <link rel="stylesheet" type="text/css" href="078_A_1916.css" /> </head><body > <!--l. 16--><p class="noindent"><pb/></p> <div class="center" > <!--l. 17--><p class="noindent"> </p><!--l. 18--><p class="noindent"><span class="cmbx-12x-x-120">THE FOUNDATION OF THE GENERAL </span> <br/><span class="cmbx-12x-x-120">THEORY</span> <span class="cmbx-12x-x-120">OF RELATIVITY</span></p></div> <div class="center" > <!--l. 23--><p class="noindent"> </p><!--l. 24--><p class="noindent"><span class="cmcsc-10x-x-144"><small class="small-caps">b</small><small class="small-caps">y</small></span></p></div> <div class="center" > <!--l. 28--><p class="noindent"> </p><!--l. 29--><p class="noindent"><span class="cmbx-12x-x-120">A. EINSTEIN</span></p></div> <div class="center" > <!--l. 32--><p class="noindent"> </p><!--l. 33--><p class="noindent"><span class="cmti-12">Translated from “Die Grundlage der allgemeinen Rela- </span> <br/><span class="cmti-12">tivit</span><span class="cmti-12">ätstheorie,”</span> <span class="cmti-12">Annalen der Physik, </span>49, 1916.</p></div> <!--l. 36--><p class="noindent"><pb/> </p><!--l. 40--><p class="indent"> </p> <div class="center" > <!--l. 41--><p class="noindent"> </p><!--l. 42--><p class="noindent">THE FOUNDATION OF THE GENERAL THEORY <br/>OF RELATIVITY</p></div> <div class="center" > <!--l. 45--><p class="noindent"> </p><!--l. 46--><p class="noindent">B<span class="cmcsc-10x-x-120"><small class="small-caps">y</small> </span>A. EINSTEIN</p></div> <div class="center" > <!--l. 49--><p class="noindent"> </p><!--l. 50--><p class="noindent">A. F<span class="cmcsc-10x-x-120"><small class="small-caps">u</small><small class="small-caps">n</small><small class="small-caps">d</small><small class="small-caps">a</small><small class="small-caps">m</small><small class="small-caps">e</small><small class="small-caps">n</small><small class="small-caps">t</small><small class="small-caps">a</small><small class="small-caps">l</small> </span>C<span class="cmcsc-10x-x-120"><small class="small-caps">o</small><small class="small-caps">n</small><small class="small-caps">s</small><small class="small-caps">i</small><small class="small-caps">d</small><small class="small-caps">e</small><small class="small-caps">r</small><small class="small-caps">a</small><small class="small-caps">t</small><small class="small-caps">i</small><small class="small-caps">o</small><small class="small-caps">n</small><small class="small-caps">s</small> <small class="small-caps">o</small><small class="small-caps">n</small> <small class="small-caps">t</small><small class="small-caps">h</small><small class="small-caps">e</small> </span>P<span class="cmcsc-10x-x-120"><small class="small-caps">o</small><small class="small-caps">s</small><small class="small-caps">t</small><small class="small-caps">u</small><small class="small-caps">l</small><small class="small-caps">a</small><small class="small-caps">t</small><small class="small-caps">e</small> <small class="small-caps">o</small><small class="small-caps">f</small> </span> <br/>R<span class="cmcsc-10x-x-120"><small class="small-caps">e</small><small class="small-caps">l</small><small class="small-caps">a</small><small class="small-caps">t</small><small class="small-caps">i</small><small class="small-caps">v</small><small class="small-caps">i</small><small class="small-caps">t</small><small class="small-caps">y</small></span> </p></div> <div class="center" > <!--l. 55--><p class="noindent"> </p><!--l. 56--><p class="noindent"><span class="cmsy-10x-x-120">§ </span><span class="cmbx-12">1. Observations on the Special Theory of Relativity</span></p></div> <!--l. 61--><p class="indent"> <sub ><span 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. </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 class="cmti-12">same </span>laws also hold good in relation to any other system <br/>of co-ordinates <span class="cmmi-12">K</span><span class="cmsy-10x-x-120">' </span>moving in uniform translation relatively <br/>to K. This postulate we call the “ special principle of <br/>relativity. ” The word “ special ” is meant to intimate <br/>that the principle is restricted to the case when <span class="cmmi-12">K</span><span class="cmsy-10x-x-120">' </span>has a <br/>motion of uniform translation relatively to K, but that the <br/>equivalence of <span class="cmmi-12">K</span><span class="cmsy-10x-x-120">' </span>and K does not extend to the case of non-<br/>uniform motion of <span class="cmmi-12">K</span><span class="cmsy-10x-x-120">' </span>relatively to K. </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 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. </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/> </p><!--l. 94--><p class="indent"> </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 <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 <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. </p> <div class="center" > <!--l. 115--><p class="noindent"> </p><!--l. 116--><p class="noindent"><span class="cmsy-10x-x-120">§ </span><span class="cmbx-12">2. The Need for an Extension of the Postulate of </span> <br/><span class="cmbx-12">Relativity</span></p></div> <!--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 class="cmr-8">1</span></sub> prove to be a sphere, and that of S<sub ><span 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 <br/><pb/> </p><!--l. 146--><p class="indent"> </p><!--l. 147--><p class="noindent">be admitted as epistemologically satisfactory,<sup ><span class="cmsy-8">*</span></sup> unless the <br/>reason given is an <span 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 class="cmti-12">observable facts </span>ultimately <br/>appear as causes and effects. </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 class="cmr-8">1</span></sub>, in respect to which the body <br/>S<sub ><span class="cmr-8">1</span></sub> is at rest, but not to the space R<sub ><span class="cmr-8">2</span></sub>, in respect to which the <br/>body S<sub ><span class="cmr-8">2</span></sub> is at rest. But the privileged space R<sub ><span class="cmr-8">1</span></sub> of Galileo, <br/>thus introduced, is a merely <span class="cmti-12">factitious </span>cause, and not a thing <br/>that can be observed. It is therefore clear that Newton’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 class="cmr-8">1</span></sub> responsible for the ob-<br/>servable difference in the bodies S<sub ><span class="cmr-8">1</span></sub> and S<sub ><span class="cmr-8">2</span></sub>. </p><!--l. 168--><p class="indent"> The only satisfactory answer must be that the physical <br/>system consisting of S<sub ><span class="cmr-8">1</span></sub> and S<sub ><span class="cmr-8">2</span></sub> reveals within itself no imagin-<br/>able cause to which the differing behaviour of S<sub ><span class="cmr-8">1</span></sub> and S<sub ><span class="cmr-8">2</span></sub> can <br/>be referred. The cause must therefore lie <span class="cmti-12">outside</span> 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 class="cmr-8">1</span></sub> and S<sub ><span class="cmr-8">2</span></sub>, must be such <br/>that the mechanical behaviour of S<sub ><span class="cmr-8">1</span></sub> and S<sub ><span 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 class="cmr-8">1</span></sub> and <br/>S<sub ><span 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 class="cmr-8">1</span></sub> and S<sub ><span class="cmr-8">2</span></sub>. They take over the rôle of the <br/>factitious cause R<sub ><span class="cmr-8">1</span></sub>. Of all imaginable spaces R<sub ><span class="cmr-8">1</span></sub>, R<sub ><span class="cmr-8">2</span></sub>, etc., in <br/>any kind of motion relatively to one another, there is none <br/>which we may look upon as privileged a <span class="cmti-12">priori </span>without re-<br/>viving the above-mentioned epistemological objection. <span class="cmti-12">The </span> <br/><span class="cmti-12">laws of physics must be</span> <span class="cmti-12">of such a nature that they apply to </span> <br/><span class="cmti-12">systems of reference in any kind of</span> <span class="cmti-12">motion. </span>Along this road <br/>we arrive at an extension of the postulate of relativity. </p><!--l. 193--><p class="indent"> In addition to this weighty argument from the theory of <br/> </p><!--l. 196--><p class="indent"> <sup ><span 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. <pb/> </p><!--l. 203--><p class="indent"> </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 class="cmmi-12">K</span><span class="cmsy-10x-x-120">' </span>be a second system of <br/>reference which is moving relatively to K in <span class="cmti-12">uniformly </span> <br/><span class="cmti-12">accelerated </span>translation. Then, relatively to <span class="cmmi-12">K</span><span 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. </p><!--l. 219--><p class="indent"> Does this permit an observer at rest relatively to <span class="cmmi-12">K</span><span class="cmsy-10x-x-120">' </span>to <br/>infer that he is on a “ really ” accelerated system of reference? <br/>The answer is in the negative; for the above-mentioned <br/>relation of freely movable masses to <span class="cmmi-12">K</span><span class="cmsy-10x-x-120">' </span>may be interpreted <br/>equally well in the following way. The system of reference <br/><span class="cmmi-12">K</span><span 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 <span class="cmmi-12">K</span><span class="cmsy-10x-x-120">'</span>. </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 class="cmsy-8">*</span></sup> The <br/>mechanical behaviour of bodies relatively to <span class="cmmi-12">K</span><span 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 “ stationary ” or as “ privileged.” <br/>Therefore, from the physical standpoint, the assumption <br/>readily suggests itself that the systems K and <span class="cmmi-12">K</span><span class="cmsy-10x-x-120">' </span>may both <br/>with equal right be looked upon as “ stationary,” that is to <br/>say, they have an equal title as systems of reference for the <br/>physical description of phenomena. </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 “ produce ” 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 class="cmti-12">in</span> <span class="cmti-12">vacuo </span>must be modified, since we easily <br/> </p><!--l. 255--><p class="indent"> <sup ><span class="cmsy-8">*</span></sup> Eötvös has proved experimentally that the gravitational field has this <br/>property in great accuracy. <pb/> </p><!--l. 261--><p class="indent"> </p><!--l. 262--><p class="noindent">recognize that the path of a ray of light with respect to <span class="cmmi-12">K</span><span 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. </p> <div class="center" > <!--l. 268--><p class="noindent"> </p><!--l. 269--><p class="noindent"><span class="cmsy-10x-x-120">§ </span><span class="cmbx-12">3. The Space-Time Continuum. Requirement of General</span> <br/><span class="cmbx-12">Co-Variance for the Equations Expressing General </span> <br/><span class="cmbx-12">Laws of Nature</span></p></div> <!--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 class="cmr-8">1</span></sub> co-<br/>ordinate <span class="cmmi-12">x</span><sub ><span class="cmr-8">1</span></sub> means that the projection of the point-event on the <br/>axis of X<sub ><span 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 class="cmmi-12">x</span><sub ><span class="cmr-8">1</span></sub> times from the origin of co-<br/>ordinates along the axis of X<sub ><span class="cmr-8">1</span></sub>. To say that a point-event <br/>has the X<sub ><span class="cmr-8">4</span></sub> co-ordinate <span class="cmmi-12">x</span><sub ><span class="cmr-8">4</span></sub> = <span class="cmmi-12">t</span>, means that a standard clock, <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 class="cmsy-8">*</span></sup> will have <br/>measured off <span class="cmmi-12">x</span><sub > <span class="cmr-8">4</span></sub> = <span class="cmmi-12">t </span>periods at the occurrence of the event. </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 <br/>reader’s reflexions on the preceding paragraph (<span 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 <br/>relativity applies to the special case of the absence of a gravi-<br/>tational field. </p><!--l. 305--><p class="indent"> In a space which is free of gravitational fields we introduce <br/>a Galilean system of reference <span class="cmmi-12">K</span>(<span class="cmmi-12">x, y, z, t</span>)<span class="cmmi-12">, </span>and also a system <br/>of co-ordinates <span class="cmmi-12">K</span><span class="cmsy-10x-x-120">' </span>(<span class="cmmi-12">x</span><span class="cmsy-10x-x-120">'</span><span class="cmmi-12">, y</span><span class="cmsy-10x-x-120">'</span><span class="cmmi-12">, z</span><span class="cmsy-10x-x-120">'</span><span class="cmmi-12">, t</span><span 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/> </p><!--l. 312--><p class="indent"> <sup ><span class="cmsy-8">*</span></sup> We assume the possibility of verifying “ simultaneity ” for events im-<br/>mediately proximate in space, or--to speak more precisely--for immediate <br/>proximity or coincidence in space-time, without giving a definition of this <br/>fundamental concept. <pb/> </p><!--l. 320--><p class="indent"> </p><!--l. 321--><p class="noindent">of Z, permanently coincide. We shall show that for a space-<br/>time measurement in the system <span class="cmmi-12">K</span><span 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 class="cmmi-12">K</span><span class="cmsy-10x-x-120">'</span><span class="cmmi-12">,</span> Y’ plane of <span class="cmmi-12">K</span><span 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 class="cmmi-12"><img src="img/cmmi12-19.png" alt="p" class="12x-x-19" /></span>. With a measuring-rod at rest relatively <br/>to <span class="cmmi-12">K</span><span class="cmsy-10x-x-120">'</span>, the quotient would be greater than <span class="cmmi-12"><img 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 “ stationary ” 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 <span class="cmmi-12">K</span><span 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 class="cmmi-12">K</span><span class="cmsy-10x-x-120">'</span>. So, too, we are unable <br/>to introduce a time corresponding to physical requirements <br/>in <span class="cmmi-12">K</span><span class="cmsy-10x-x-120">'</span>, indicated by clocks at rest relatively to <span class="cmmi-12">K</span><span class="cmsy-10x-x-120">'</span>. To <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 “ stationary ” 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 <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 <br/>showing that the clock at the circumference “ really ” 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. <pb/> </p><!--l. 370--><p class="indent"> </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. </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 <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:-- </p><!--l. 389--><p class="indent"> <span class="cmti-12">The general laws of nature are to be expressed by equations </span> <br/><span class="cmti-12">which hold good for</span> <span class="cmti-12">all systems of co-ordinates, that is, are </span> <br/><span class="cmti-12">co-variant with respect to any substitutions</span> <span class="cmti-12">whatever (generally </span> <br/><span class="cmti-12">co-variant)</span>. </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 class="cmti-12">all</span> 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 <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 <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 <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. </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 class="cmmi-12">x</span><sub ><span class="cmr-8">1</span></sub> <span class="cmmi-12">, x</span><sub ><span class="cmr-8">2</span></sub> <span class="cmmi-12">, x</span><sub ><span class="cmr-8">3</span></sub> <span class="cmmi-12">, x</span><sub ><span class="cmr-8">4</span></sub> in such a way that for every point-event <br/><pb/> </p><!--l. 425--><p class="indent"> </p><!--l. 426--><p class="noindent">there is a corresponding system of values of the variables <br/><span class="cmmi-12">x</span><sub ><span class="cmr-8">1</span></sub> <span class="cmmi-12">...</span><span class="cmmi-12">x</span><sub ><span class="cmr-8">4</span></sub>. To two coincident point-events there corre-<br/>sponds one system of values of the variables <span class="cmmi-12">x</span><sub ><span class="cmr-8">1</span></sub> <span class="cmmi-12">...</span><span class="cmmi-12">x</span><sub ><span 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 class="cmmi-12">x</span><sub ><span class="cmr-8">1</span></sub> <span class="cmmi-12">...</span><span class="cmmi-12">x</span><sub ><span class="cmr-8">4</span></sub>, we introduce functions <br/>of them, <span class="cmmi-12">x</span><span class="cmsy-10x-x-120">'</span><sub ><span class="cmr-8">1</span></sub><span class="cmmi-12">, x</span><span class="cmsy-10x-x-120">'</span><sub ><span class="cmr-8">2</span></sub><span class="cmmi-12">, x</span><span class="cmsy-10x-x-120">'</span><sub ><span class="cmr-8">3</span></sub><span class="cmmi-12">, x</span><span class="cmsy-10x-x-120">'</span><sub ><span class="cmr-8">4</span></sub><span class="cmmi-12">,</span> 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. </p> <div class="center" > <!--l. 446--><p class="noindent"> </p><!--l. 447--><p class="noindent"><span class="cmbsy-10x-x-120">§ </span><span class="cmbx-12">4. The Relation of the Four Co-ordinates to Measure- </span> <br/><span class="cmbx-12">ment in</span> <span class="cmbx-12">Space and Time</span></p></div> <!--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:-- </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. </p><!--l. 466--><p class="indent"> For this purpose we must choose the acceleration of the <br/>infinitely small (“ local ”) system of co-ordinates so that no <br/>gravitational field occurs; this is possible for an infinitely <br/>small region. Let X<sub ><span class="cmr-8">1</span></sub> <span class="cmmi-12">, X</span><sub ><span class="cmr-8">2</span></sub> <span class="cmmi-12">, X</span><sub ><span class="cmr-8">3</span></sub>, be the co-ordinates of space, <br/>and <span class="cmmi-12">X</span><sub ><span class="cmr-8">4</span></sub> the appertaining co-ordinate of time measured in the <br/>appropriate unit.<sup ><span 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/> </p><!--l. 478--><p class="indent"> <sup ><span class="cmsy-8">*</span></sup> The unit of time is to be chosen so that the velocity of light <span class="cmti-12">in vacuo </span>as <br/>measured in the “ local ” system of co-ordinates is to be equal to unity. <pb/> </p><!--l. 485--><p class="indent"> </p><!--l. 486--><p class="noindent">in the sense of the special theory of relativity. By the <br/>special theory of relativity the expression </p> <table width="100%" class="equation"><tr><td><a id="x1-2r1"></a> <center class="math-display" > <img 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> <!--l. 492--><p class="nopar"> </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 <br/>continuum in infinite proximity, we call <span class="cmti-12">ds</span>. If the <span class="cmti-12">ds </span>belong-<br/>ing to the element <span class="cmmi-12">dX</span><sub ><span class="cmr-8">1</span></sub> <span class="cmmi-12">...</span><span class="cmmi-12">dX</span><sub ><span 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. </p><!--l. 506--><p class="indent"> To the “ linear element ” in question, or to the two infin-<br/>itely proximate point-events, there will also correspond <br/>definite differentials <span class="cmmi-12">dx</span><sub ><span class="cmr-8">1</span></sub> <span class="cmmi-12">...</span><span class="cmmi-12">dx</span><sub ><span 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 “ local ” system, is given for the region <br/>under consideration, the <span class="cmmi-12">dX</span><sub ><span class="cmmi-8"><img src="img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub> will allow themselves to be <br/>represented here by definite linear homogeneous expressions <br/>of the <span class="cmmi-12">dx</span><sub ><span class="cmmi-8"><img src="img/cmmi8-1b.png" alt="s" class="8x-x-1b" /></span></sub>:-- </p> <table width="100%" class="equation"><tr><td><a id="x1-3r2"></a> <center class="math-display" > <img src="img/078_A_19161x.png" alt="dXn = S ansdxs s " class="math-display" /></center></td><td width="5%">(2)</td></tr></table> <!--l. 520--><p class="nopar"> </p><!--l. 523--><p class="noindent">Inserting these expressions in (1), we obtain </p> <table width="100%" class="equation"><tr><td><a id="x1-4r3"></a> <center class="math-display" > <img src="img/078_A_19162x.png" alt=" 2 ds = Stsgstdxsdxt , " class="math-display" /></center></td><td width="5%">(3)</td></tr></table> <!--l. 530--><p class="nopar"> </p><!--l. 535--><p class="noindent">where the <span class="cmmi-12">g</span><sub ><span class="cmmi-8"><img src="img/cmmi8-1b.png" alt="s" class="8x-x-1b" /><img src="img/cmmi8-1c.png" alt="t" class="8x-x-1c" /></span></sub> will be functions of the <span class="cmmi-12">x</span><sub ><span class="cmmi-8"><img 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 “ local ” system of co-ordinates, for <span class="cmmi-12">ds</span><sup ><span 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 class="cmmi-12">g</span><sub ><span class="cmmi-8"><img src="img/cmmi8-1b.png" alt="s" class="8x-x-1b" /><img src="img/cmmi8-1c.png" alt="t" class="8x-x-1c" /></span></sub> <br/>are to be chosen here so that <span class="cmmi-12">g</span><sub ><span class="cmmi-8"><img src="img/cmmi8-1b.png" alt="s" class="8x-x-1b" /><img src="img/cmmi8-1c.png" alt="t" class="8x-x-1c" /></span></sub> = <span class="cmmi-12">g</span><sub ><span class="cmmi-8"><img src="img/cmmi8-1c.png" alt="t" class="8x-x-1c" /><img src="img/cmmi8-1b.png" alt="s" class="8x-x-1b" /></span></sub> ; the summation is <br/>to extend over all values of <span class="cmmi-12"><img src="img/cmmi12-1b.png" alt="s" class="12x-x-1b" /> </span>and <span class="cmmi-12"><img src="img/cmmi12-1c.png" alt="t" class="12x-x-1c" /></span>, so that the sum consists <br/>of 4 <span class="cmsy-10x-x-120">× </span>4 terms, of which twelve are equal in pairs. </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 class="cmmi-12">g</span><sub ><span class="cmmi-8"><img src="img/cmmi8-1b.png" alt="s" class="8x-x-1b" /><img 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 class="cmmi-12">g</span><sub ><span class="cmmi-8"><img src="img/cmmi8-1b.png" alt="s" class="8x-x-1b" /><img src="img/cmmi8-1c.png" alt="t" class="8x-x-1c" /></span></sub> assume the constant values <pb/> </p><!--l. 557--><p class="indent"> </p> <table width="100%" class="equation"><tr><td><a id="x1-5r4"></a> <center class="math-display" > <img 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> <!--l. 566--><p class="nopar"> </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. </p><!--l. 573--><p class="indent"> From the considerations of <span class="cmsy-10x-x-120">§ </span>2 and <span class="cmsy-10x-x-120">§ </span>3 it follows that <br/>the quantities <span class="cmmi-12">g</span><sub ><span class="cmmi-8"><img src="img/cmmi8-1c.png" alt="t" class="8x-x-1c" /><img 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 class="cmmi-12">g</span><sub ><span class="cmmi-8"><img src="img/cmmi8-1b.png" alt="s" class="8x-x-1b" /><img 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 <br/>uniform motion in a straight line. Then if we introduce new <br/>space-time co-ordinates <span class="cmmi-12">x</span><sub ><span class="cmr-8">1</span></sub><span class="cmmi-12">, x</span><sub ><span class="cmr-8">2</span></sub><span class="cmmi-12">, x</span><sub ><span class="cmr-8">3</span></sub><span class="cmmi-12">, x</span><sub ><span class="cmr-8">4</span></sub><span class="cmmi-12">, </span>by means of any substi-<br/>tution we choose, the <span class="cmmi-12">g</span><sup ><span class="cmmi-8"><img src="img/cmmi8-1b.png" alt="s" class="8x-x-1b" /><img 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 class="cmmi-12">g</span><sub ><span class="cmmi-8"><img 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 <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 class="cmmi-12">g</span><sub ><span class="cmmi-8"><img src="img/cmmi8-1b.png" alt="s" class="8x-x-1b" /><img src="img/cmmi8-1c.png" alt="t" class="8x-x-1c" /></span></sub> <br/>describe the gravitational field. </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 <br/>define the metrical properties of the space measured. </p> <div class="center" > <!--l. 609--><p class="noindent"> </p><!--l. 610--><p class="noindent">B. <span class="cmcsc-10x-x-120">M<small class="small-caps">a</small><small class="small-caps">t</small><small class="small-caps">h</small><small class="small-caps">e</small><small class="small-caps">m</small><small class="small-caps">a</small><small class="small-caps">t</small><small class="small-caps">i</small><small class="small-caps">c</small><small class="small-caps">a</small><small class="small-caps">l</small> A<small class="small-caps">i</small><small class="small-caps">d</small><small class="small-caps">s</small> <small class="small-caps">t</small><small class="small-caps">o</small> <small class="small-caps">t</small><small class="small-caps">h</small><small class="small-caps">e</small> F<small class="small-caps">o</small><small class="small-caps">r</small><small class="small-caps">m</small><small class="small-caps">u</small><small class="small-caps">l</small><small class="small-caps">a</small><small class="small-caps">t</small><small class="small-caps">i</small><small class="small-caps">o</small><small class="small-caps">n</small> <small class="small-caps">o</small><small class="small-caps">f</small> </span> <br/><span class="cmcsc-10x-x-120">G<small class="small-caps">e</small><small class="small-caps">n</small><small class="small-caps">e</small><small class="small-caps">r</small><small class="small-caps">a</small><small class="small-caps">l</small><small class="small-caps">l</small><small class="small-caps">y</small></span> <span class="cmcsc-10x-x-120">C<small class="small-caps">o</small><small class="small-caps">v</small><small class="small-caps">a</small><small class="small-caps">r</small><small class="small-caps">i</small><small class="small-caps">a</small><small class="small-caps">n</small><small class="small-caps">t</small> E<small class="small-caps">q</small><small class="small-caps">u</small><small class="small-caps">a</small><small class="small-caps">t</small><small class="small-caps">i</small><small class="small-caps">o</small><small class="small-caps">n</small><small class="small-caps">s</small></span></p></div> <!--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/> </p><!--l. 621--><p class="indent"> </p><!--l. 622--><p class="noindent">physics shall be covariant in the face of any substitution of <br/>the co-ordinates <span class="cmmi-12">x</span><sub ><span class="cmr-8">1</span></sub> <span class="cmmi-12">...</span><span class="cmmi-12">x</span><sub ><span 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 <br/>solution a fundamental rôle is played by the invariant <span class="cmti-12">ds </span> <br/>given in equation (3), which, borrowing from Gauss’s theory <br/>of surfaces, we have called the “ linear element. ” </p><!--l. 632--><p class="indent"> The fundamental idea of this general theory of covariants <br/>is the following:--Let certain things (“ tensors ”) be defined <br/>with respect to any system of co-ordinates by a number of <br/>functions of the co-ordinates, called the “ components ” 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 <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. </p> <div class="center" > <!--l. 653--><p class="noindent"> </p><!--l. 654--><p class="noindent"><span class="cmbsy-10x-x-120">§ </span><span class="cmbx-12">5. Contravariant and Covariant Four-vectors</span></p></div> <!--l. 658--><p class="indent"> <span class="cmti-12">Contravariant Four-vectors.</span>--The linear element is de-<br/>fined by the four “ components ” <span class="cmmi-12">dx</span><sub ><span class="cmmi-8"><img 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 </p> <table width="100%" class="equation"><tr><td><a id="x1-6r5"></a> <center class="math-display" > <img 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> <!--l. 668--><p class="nopar"> </p><!--l. 671--><p class="noindent">The <span class="cmmi-12">dx</span><span class="cmsy-10x-x-120">'</span><sub ><span class="cmmi-8"><img src="img/cmmi8-1b.png" alt="s" class="8x-x-1b" /></span></sub> are expressed as linear and homogeneous functions <br/>of the <span class="cmmi-12">dx</span><sub ><span class="cmmi-8"><img 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 “ tensor ” 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 class="cmmi-12">A</span><sup ><span class="cmmi-8"><img src="img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sup>, and which is transformed by the same law </p> <table width="100%" class="equation"><tr><td><a id="x1-7r6"></a> <center class="math-display" > <img 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> <!--l. 685--><p class="nopar"> <pb/> </p><!--l. 692--><p class="indent"> </p><!--l. 693--><p class="noindent">we also call a contravariant four-vector. From (5a) it <br/>follows at once that the sums <span class="cmmi-12">A</span><sup ><span class="cmmi-8"><img src="img/cmmi8-1b.png" alt="s" class="8x-x-1b" /></span></sup> <span class="cmsy-10x-x-120">± </span><span class="cmmi-12">B</span><sup ><span class="cmmi-8"><img src="img/cmmi8-1b.png" alt="s" class="8x-x-1b" /></span></sup> are also components <br/>of a four-vector, if <span class="cmmi-12">A</span><sup ><span class="cmmi-8"><img src="img/cmmi8-1b.png" alt="s" class="8x-x-1b" /></span></sup> and <span class="cmmi-12">B</span><sup ><span class="cmmi-8"><img src="img/cmmi8-1b.png" alt="s" class="8x-x-1b" /></span></sup> are such. Corresponding rela-<br/>tions hold for all “ tensors ” subsequently to be introduced. <br/>(Rule for the addition and subtraction of tensors.) </p><!--l. 701--><p class="indent"> <span class="cmti-12">Covariant Four-vectors.</span>--We call four quantities <span class="cmmi-12">A</span><sub ><span class="cmmi-8"><img 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 <span class="cmmi-12">B</span><sup ><span class="cmmi-8"><img src="img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sup> </p> <table width="100%" class="equation"><tr><td><a id="x1-8r6"></a> <center class="math-display" > <img src="img/078_A_19166x.png" alt=" n Sn AnB = Invariant " class="math-display" /></center></td><td width="5%">(6)</td></tr></table> <!--l. 710--><p class="nopar"> </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 class="cmmi-12">B</span><sup ><span class="cmmi-8"><img src="img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sup> on the right-hand <br/>side of the equation </p> <center class="par-math-display" > <img src="img/078_A_19167x.png" alt="S A'sB's = S AnBn s n " class="par-math-display" /></center> <!--l. 722--><p class="nopar"> </p><!--l. 726--><p class="noindent">by the expression resulting from the inversion of (5a), </p> <center class="par-math-display" > <img src="img/078_A_19168x.png" alt="S @xn-B's , s @x's " class="par-math-display" /></center> <!--l. 732--><p class="nopar"> </p><!--l. 736--><p class="noindent">we obtain </p> <center class="par-math-display" > <img src="img/078_A_19169x.png" alt=" @x S B's S ---n' An = S B'sA's . s n@x s s " class="par-math-display" /></center> <!--l. 745--><p class="nopar"> </p><!--l. 748--><p class="noindent">Since this equation is true for arbitrary values of the <span class="cmmi-12">B</span><span class="cmsy-10x-x-120">'</span><sup ><span class="cmmi-8"><img src="img/cmmi8-1b.png" alt="s" class="8x-x-1b" /></span></sup>, it <br/>follows that the law of transformation is </p> <table width="100%" class="equation"><tr><td><a id="x1-9r7"></a> <center class="math-display" > <img 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> <!--l. 756--><p class="nopar"> </p><!--l. 760--><p class="indent"> <span 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 class="cmmi-12"><img src="img/cmmi12-17.png" alt="n" class="12x-x-17" /> </span>in <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. </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 <br/><pb/> </p><!--l. 778--><p class="indent"> </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. </p> <div class="center" > <!--l. 784--><p class="noindent"> </p><!--l. 785--><p class="noindent"><span class="cmbsy-10x-x-120">§ </span><span class="cmbx-12">6. Tensors of the Second and Higher Ranks</span></p></div> <!--l. 789--><p class="indent"> <span class="cmti-12">Contravariant Tensors</span>.--If we form all the sixteen pro-<br/>ducts <span class="cmmi-12">A</span><sup ><span class="cmmi-8"><img src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img src="img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sup> of the components <span class="cmmi-12">A</span><sup ><span class="cmmi-8"><img src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /></span></sup> and <span class="cmmi-12">B</span><sup ><span class="cmmi-8"><img src="img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sup> of two contravariant <br/>four-vectors </p> <table width="100%" class="equation"><tr><td><a id="x1-10r8"></a> <center class="math-display" > <img 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> <!--l. 797--><p class="nopar"> </p><!--l. 801--><p class="noindent">then by (8) and (5a) <span class="cmmi-12">A</span><sup ><span class="cmmi-8"><img src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img src="img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sup> satisfies the law of transformation </p> <table width="100%" class="equation"><tr><td><a id="x1-11r9"></a> <center class="math-display" > <img 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> <!--l. 809--><p class="nopar"> </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 given sixteen <span class="cmmi-12">A</span><sup ><span class="cmmi-8"><img src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img src="img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sup> can be represented as the sums of the <br/><span class="cmmi-12">A</span><sup ><span class="cmmi-8"><img src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /></span></sup><span class="cmmi-12">B</span><sup ><span class="cmmi-8"><img 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). </p><!--l. 826--><p class="indent"> <span 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 class="cmr-8">3</span></sup> components, and so <br/>on. In the same way it follows from (8) and (9) that the <br/>contravariant four-vector may be taken in this sense as a <br/>contravariant tensor of the first rank. </p><!--l. 834--><p class="indent"> <span class="cmti-12">Covariant Tensors</span>.--On the other hand, if we take the <br/>sixteen products <span class="cmmi-12">A</span><sub ><span class="cmmi-8"><img src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img src="img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub> of two covariant four-vectors <span class="cmmi-12">A</span><sub ><span class="cmmi-8"><img src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /></span></sub> and <span class="cmmi-12">B</span><sub ><span class="cmmi-8"><img src="img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub>, </p> <table width="100%" class="equation"><tr><td><a id="x1-12r10"></a> <center class="math-display" > <img src="img/078_A_191613x.png" alt="Amn = AmBn, " class="math-display" /></center></td><td width="5%">(10)</td></tr></table> <!--l. 841--><p class="nopar"> </p><!--l. 845--><p class="noindent">the law of transformation for these is </p> <table width="100%" class="equation"><tr><td><a id="x1-13r11"></a> <center class="math-display" > <img 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> <!--l. 853--><p class="nopar"> </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. <pb/> </p><!--l. 864--><p class="indent"> </p><!--l. 865--><p class="indent"> <span class="cmcsc-10x-x-120">N<small class="small-caps">o</small><small class="small-caps">t</small><small 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. </p><!--l. 869--><p class="indent"> <span class="cmti-12">Mixed Tensors</span>.--We may also define a tensor of the <br/>second rank of the type </p> <table width="100%" class="equation"><tr><td><a id="x1-14r12"></a> <center class="math-display" > <img src="img/078_A_191615x.png" alt="An = AmBn m " class="math-display" /></center></td><td width="5%">(12)</td></tr></table> <!--l. 875--><p class="nopar"> </p><!--l. 879--><p class="noindent">which is covariant with respect to the index <span class="cmmi-12"><img 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 class="cmmi-12"><img src="img/cmmi12-17.png" alt="n" class="12x-x-17" /></span>. Its law of transforma-<br/>tion is </p> <table width="100%" class="equation"><tr><td><a id="x1-15r13"></a> <center class="math-display" > <img 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> <!--l. 889--><p class="nopar"> </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. </p><!--l. 899--><p class="indent"> <span 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 class="cmmi-12">A</span><sup ><span class="cmmi-8"><img src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img src="img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sup>, <br/>or the tensor <span class="cmmi-12">A</span><sub ><span class="cmmi-8"><img src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img 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 class="cmmi-12"><img src="img/cmmi12-16.png" alt="m" class="cmmi-12x-x-16" align="middle" /></span>, <span class="cmmi-12"><img src="img/cmmi12-17.png" alt="n" class="12x-x-17" /></span>, </p> <table width="100%" class="equation"><tr><td><a id="x1-16r14"></a> <center class="math-display" > <img src="img/078_A_191617x.png" alt=" mn nm A = A , " class="math-display" /></center></td><td width="5%">(14)</td></tr></table> <!--l. 910--><p class="nopar"> </p><!--l. 914--><p class="noindent">or respectively,</p> <table width="100%" class="equation"><tr><td><a id="x1-17r15"></a> <center class="math-display" > <img src="img/078_A_191618x.png" alt="Amn = Anm. " class="math-display" /></center></td><td width="5%">(14a)</td></tr></table> <!--l. 920--><p class="nopar"> </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> <center class="par-math-display" > <img 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> <!--l. 939--><p class="nopar"> </p><!--l. 942--><p class="noindent">The last equation but one depends upon the interchange of <br/>the summation indices <span class="cmmi-12"><img src="img/cmmi12-16.png" alt="m" class="cmmi-12x-x-16" align="middle" /> </span>and <span class="cmmi-12"><img src="img/cmmi12-17.png" alt="n" class="12x-x-17" /></span>, i.e. merely on a change of <br/>notation. </p><!--l. 946--><p class="indent"> <span 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 class="cmmi-12">A</span><sup ><span class="cmmi-8"><img src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img src="img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sup>, or the tensor <span class="cmmi-12">A</span><sub > <span class="cmmi-8"><img src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img src="img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub>, is <br/>therefore antisymmetrical, if always <pb/> </p><!--l. 956--><p class="indent"> </p> <table width="100%" class="equation"><tr><td><a id="x1-18r15"></a> <center class="math-display" > <img src="img/078_A_191620x.png" alt="Amn = - Anm, " class="math-display" /></center></td><td width="5%">(15)</td></tr></table> <!--l. 959--><p class="nopar"> </p><!--l. 963--><p class="noindent">or respectively,</p> <table width="100%" class="equation"><tr><td><a id="x1-19r16"></a> <center class="math-display" > <img src="img/078_A_191621x.png" alt="Amn = - Anm " class="math-display" /></center></td><td width="5%">(15a)</td></tr></table> <!--l. 968--><p class="nopar"> </p><!--l. 972--><p class="indent"> Of the sixteen components <span class="cmmi-12">A</span><sup ><span class="cmmi-8"><img src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img src="img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sup>, the four components <span class="cmmi-12">A</span><sup ><span class="cmmi-8"><img src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img 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 class="cmmi-12">A</span><sup ><span class="cmmi-8"><img src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img src="img/cmmi8-17.png" alt="n" class="8x-x-17" /><img src="img/cmmi8-1b.png" alt="s" class="8x-x-1b" /></span></sup> has only four numerically different <br/>components, while the antisymmetrical tensor <span class="cmmi-12">A</span><sup ><span class="cmmi-8"><img src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img src="img/cmmi8-17.png" alt="n" class="8x-x-17" /><img src="img/cmmi8-1b.png" alt="s" class="8x-x-1b" /><img 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. </p> <div class="center" > <!--l. 984--><p class="noindent"> </p><!--l. 985--><p class="noindent"><span class="cmbsy-10x-x-120">§ </span><span class="cmbx-12">7. Multiplication of Tensors</span></p></div> <!--l. 989--><p class="indent"> <span class="cmti-12">Outer Multiplication of Tensors</span>.--We obtain from the <br/>components of a tensor of rank <span class="cmmi-12">n </span>and of a tensor of rank <span class="cmmi-12">m </span> <br/>the components of a tensor of rank <span class="cmmi-12">n </span>+ <span 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, </p> <center class="par-math-display" > <img src="img/078_A_191622x.png" alt="Tmns = AmnBs, Tmnst = AmnBst , st sn Tmn = AmnB . " class="par-math-display" /></center> <!--l. 1005--><p class="nopar"> </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 <br/>first rank. </p><!--l. 1015--><p class="indent"> “ <span class="cmti-12">Contraction </span>” <span 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 (“ con-<br/>traction ”). Thus, for example, from the mixed tensor of the <br/>fourth rank <span class="cmmi-12">A</span><sub><span class="cmmi-8"><img src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img src="img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub><sup><span class="cmmi-8"><img src="img/cmmi8-1b.png" alt="s" class="8x-x-1b" /><img src="img/cmmi8-1c.png" alt="t" class="8x-x-1c" /></span></sup>, we obtain the mixed tensor of the second <br/>rank, </p> <center class="par-math-display" > <img src="img/078_A_191623x.png" alt=" t mt mt A n = A mn (= Sm A mn), " class="par-math-display" /></center> <!--l. 1030--><p class="nopar"> </p><!--l. 1034--><p class="noindent">and from this, by a second contraction, the tensor of zero <br/>rank, </p> <center class="par-math-display" > <img src="img/078_A_191624x.png" alt=" n mn A = A n = A mn . " class="par-math-display" /></center> <!--l. 1040--><p class="nopar"> <pb/> </p><!--l. 1047--><p class="indent"> </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). </p><!--l. 1054--><p class="indent"> <span class="cmti-12">Inner and Mixed Multiplication of Tensors</span>.--These consist <br/>in a combination of outer multiplication with contraction. </p><!--l. 1058--><p class="indent"> <span class="cmti-12">Examples</span>.--From the covariant tensor of the second rank <br/><span class="cmmi-12">A</span><sub ><span class="cmmi-8"><img src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img src="img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub> and the contravariant tensor of the first rank <span class="cmmi-12">B</span><sup ><span class="cmmi-8"><img src="img/cmmi8-1b.png" alt="s" class="8x-x-1b" /></span></sup> we form <br/>by outer multiplication the mixed tensor </p> <center class="par-math-display" > <img src="img/078_A_191625x.png" alt="Dsmn = AmnBs . " class="par-math-display" /></center> <!--l. 1066--><p class="nopar"> </p><!--l. 1069--><p class="noindent">On contraction with respect to the indices <span class="cmmi-12"><img src="img/cmmi12-17.png" alt="n" class="12x-x-17" /> </span>and <span class="cmmi-12"><img src="img/cmmi12-1b.png" alt="s" class="12x-x-1b" /></span>, we obtain <br/>the covariant four-vector </p> <center class="par-math-display" > <img src="img/078_A_191626x.png" alt="Dm = Dnmn = AmnBn . " class="par-math-display" /></center> <!--l. 1076--><p class="nopar"> </p><!--l. 1080--><p class="noindent">This we call the inner product of the tensors <span class="cmmi-12">A</span><sub ><span class="cmmi-8"><img src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img src="img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub> and <span class="cmmi-12">B</span><sup ><span class="cmmi-8"><img src="img/cmmi8-1b.png" alt="s" class="8x-x-1b" /></span></sup>. <br/>Analogously we form from the tensors <span class="cmmi-12">A</span><sub ><span class="cmmi-8"><img src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img src="img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub> and <span class="cmmi-12">B</span><sup ><span class="cmmi-8"><img src="img/cmmi8-1b.png" alt="s" class="8x-x-1b" /><img 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 class="cmmi-12">A</span><sub ><span class="cmmi-8"><img src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img src="img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub><span class="cmmi-12">B</span><sup ><span class="cmmi-8"><img src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img src="img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sup>. By outer multiplication and one contraction, we <br/>obtain from <span class="cmmi-12">A</span><sub ><span class="cmmi-8"><img src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img src="img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub> and <span class="cmmi-12">B</span><sup ><span class="cmmi-8"><img src="img/cmmi8-1b.png" alt="s" class="8x-x-1b" /><img src="img/cmmi8-1c.png" alt="t" class="8x-x-1c" /></span></sup> the mixed tensor of the second rank <br/>D<sub> <span class="cmmi-8"><img src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /></span></sub><sup><span class="cmmi-8"><img src="img/cmmi8-1c.png" alt="t" class="8x-x-1c" /></span></sup> = <span class="cmmi-12">A</span><sub > <span class="cmmi-8"><img src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img src="img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub><span class="cmmi-12">B</span><sup ><span class="cmmi-8"><img src="img/cmmi8-17.png" alt="n" class="8x-x-17" /><img 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 “ outer ” with respect to the indices <span class="cmmi-12"><img src="img/cmmi12-16.png" alt="m" class="cmmi-12x-x-16" align="middle" /> </span> <br/>and <span class="cmmi-12"><img src="img/cmmi12-1c.png" alt="t" class="12x-x-1c" /></span>, and “ inner ” with respect to the indices <span class="cmmi-12"><img src="img/cmmi12-17.png" alt="n" class="12x-x-17" /> </span>and <span class="cmmi-12"><img src="img/cmmi12-1b.png" alt="s" class="12x-x-1b" /></span>. </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 class="cmmi-12">A</span><sub ><span class="cmmi-8"><img src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img src="img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub><span class="cmmi-12">B</span><sup ><span class="cmmi-8"><img src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img src="img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sup> is a scalar if <span class="cmmi-12">A</span><sub > <span class="cmmi-8"><img src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img src="img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub> and <span class="cmmi-12">B</span><sup ><span class="cmmi-8"><img src="img/cmmi8-1b.png" alt="s" class="8x-x-1b" /><img src="img/cmmi8-1c.png" alt="t" class="8x-x-1c" /></span></sup> are tensors. But <br/>we may also make the following assertion: If <span class="cmmi-12">A</span><sub ><span class="cmmi-8"><img src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img src="img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub><span class="cmmi-12">B</span><sup ><span class="cmmi-8"><img src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img src="img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sup> is <br/>a scalar <span class="cmti-12">for any choice of the tensor </span><span class="cmmi-12">B</span><sup ><span class="cmmi-8"><img src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img src="img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sup>, then <span class="cmmi-12">A</span><sub > <span class="cmmi-8"><img src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img src="img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub> has tensor <br/>character. For, by hypothesis, for any substitution, </p> <center class="par-math-display" > <img src="img/078_A_191627x.png" alt="A' B'st = A Bmn . st mn " class="par-math-display" /></center> <!--l. 1108--><p class="nopar"> </p><!--l. 1111--><p class="noindent">But by an inversion of (9) </p> <center class="par-math-display" > <img src="img/078_A_191628x.png" alt="Bmn = @xm-@xn-B'st . @x's@x't " class="par-math-display" /></center> <!--l. 1118--><p class="nopar"> </p><!--l. 1121--><p class="noindent">This, inserted in the above equation, gives </p> <center class="par-math-display" > <img 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> <!--l. 1129--><p class="nopar"> </p><!--l. 1132--><p class="noindent">This can only be satisfied for arbitrary values of <span class="cmmi-12">B</span><span class="cmsy-10x-x-120">'</span><sup ><span class="cmmi-8"><img src="img/cmmi8-1b.png" alt="s" class="8x-x-1b" /><img src="img/cmmi8-1c.png" alt="t" class="8x-x-1c" /></span></sup> if the <br/><pb/> </p><!--l. 1136--><p class="indent"> </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. </p><!--l. 1142--><p class="indent"> The rule may also be demonstrated in this form: If <span class="cmmi-12">B</span><sup ><span class="cmmi-8"><img src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /></span></sup> <br/>and C<sup ><span class="cmmi-8"><img 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 class="cmmi-12">A</span><sub ><span class="cmmi-8"><img src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img src="img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub><span class="cmmi-12">B</span><sup ><span class="cmmi-8"><img src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /></span></sup>C<sup ><span class="cmmi-8"><img src="img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sup> is a scalar, then <span class="cmmi-12">A</span><sub > <span class="cmmi-8"><img src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img 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 class="cmmi-12">B</span><sup ><span class="cmmi-8"><img src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /></span></sup> the inner product <span class="cmmi-12">A</span><sub ><span class="cmmi-8"><img src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img src="img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub><span class="cmmi-12">B</span><sup ><span class="cmmi-8"><img src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /></span></sup><span class="cmmi-12">B</span><sup ><span class="cmmi-8"><img 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 class="cmmi-12">A</span><sub > <span class="cmmi-8"><img src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img src="img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub> satisfies the condition of <br/>symmetry <span class="cmmi-12">A</span><sub ><span class="cmmi-8"><img src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img src="img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub> = <span class="cmmi-12">A</span><sub ><span class="cmmi-8"><img src="img/cmmi8-17.png" alt="n" class="8x-x-17" /><img 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 class="cmmi-12">A</span><sub ><span class="cmmi-8"><img src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img src="img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub> + <span class="cmmi-12">A</span><sub ><span class="cmmi-8"><img src="img/cmmi8-17.png" alt="n" class="8x-x-17" /><img src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /></span></sub>), and from this the <br/>tensor character of <span class="cmmi-12">A</span><sub ><span class="cmmi-8"><img src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img 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. </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 class="cmmi-12">B</span><sup ><span class="cmmi-8"><img src="img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sup> the quantities <span class="cmmi-12">A</span><sub ><span class="cmmi-8"><img src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img src="img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub><span class="cmmi-12">B</span><sup ><span class="cmmi-8"><img src="img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sup> form a <br/>tensor of the first rank, then <span class="cmmi-12">A</span><sub > <span class="cmmi-8"><img src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img 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 class="cmmi-8"><img 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 <br/>tensor character of <span class="cmmi-12">A</span><sub ><span class="cmmi-8"><img src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img src="img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub><span class="cmmi-12">B</span><sup ><span class="cmmi-8"><img src="img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sup>, the inner product <span class="cmmi-12">A</span><sub > <span class="cmmi-8"><img src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img src="img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub><span class="cmmi-12">B</span><sup ><span class="cmmi-8"><img src="img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sup>C<sup ><span class="cmmi-8"><img 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 class="cmmi-12">B</span><sup ><span class="cmmi-8"><img src="img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sup> and C<sup ><span class="cmmi-8"><img src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /></span></sup>. From <br/>which the proposition follows. </p> <div class="center" > <!--l. 1173--><p class="noindent"> </p><!--l. 1174--><p class="noindent"><span class="cmbsy-10x-x-120">§ </span><span class="cmbx-12">8. Some Aspects of the Fundamental Tensor </span><span class="cmmi-12">g</span><sub ><span class="cmmi-8"><img src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img src="img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub></p></div> <!--l. 1179--><p class="indent"> <span class="cmti-12">The Covariant Fundamental Tensor</span>.--In the invariant <br/>expression for the square of the linear element, </p> <center class="par-math-display" > <img src="img/078_A_191630x.png" alt=" 2 ds = gmndxmdxn , " class="par-math-display" /></center> <!--l. 1186--><p class="nopar"> </p><!--l. 1190--><p class="noindent">the part played by the <span class="cmmi-12">dx</span><sub ><span class="cmmi-8"><img 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 class="cmmi-12">g</span><sub ><span class="cmmi-8"><img src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img src="img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub> = <span class="cmmi-12">g</span><sub ><span class="cmmi-8"><img src="img/cmmi8-17.png" alt="n" class="8x-x-17" /><img 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 class="cmmi-12">g</span><sub ><span class="cmmi-8"><img src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img 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 “ fundamental tensor.” 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 <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. <pb/> </p><!--l. 1206--><p class="indent"> </p><!--l. 1207--><p class="indent"> <span class="cmti-12">The Contravariant Fundamental Tensor.</span>--If in the deter-<br/>minant formed by the elements <span class="cmmi-12">g</span><sub ><span class="cmmi-8"><img src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img src="img/cmmi8-17.png" alt="n" class="8x-x-17" />,</span></sub> we take the co-factor of <br/>each of the <span class="cmmi-12">g</span><sub ><span class="cmmi-8"><img src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img src="img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub> and divide it by the determinant <span class="cmmi-12">g </span>= <img src="img/078_A_191631x.png" alt="|gmn |" class="left" align="middle" /><span class="cmmi-12">, </span> <br/>we obtain certain quantities <span class="cmmi-12">g</span><sup ><span class="cmmi-8"><img src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img src="img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sup><img src="img/078_A_191632x.png" alt="(= gnm)" class="left" align="middle" /> which, as we shall <br/>demonstrate, form a contravariant tensor. </p><!--l. 1215--><p class="indent"> By a known property of determinants </p> <table width="100%" class="equation"><tr><td><a id="x1-20r16"></a> <center class="math-display" > <img src="img/078_A_191633x.png" alt=" ns n gmsg = dm " class="math-display" /></center></td><td width="5%">(16)</td></tr></table> <!--l. 1221--><p class="nopar"> </p><!--l. 1224--><p class="noindent">where the symbol <span class="cmmi-12"><img src="img/cmmi12-e.png" alt="d" class="12x-x-e" /></span><sub><span class="cmmi-8"><img src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /></span></sub><sup><span class="cmmi-8"><img src="img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sup> denotes 1 or 0, according as <span class="cmmi-12"><img src="img/cmmi12-16.png" alt="m" class="cmmi-12x-x-16" align="middle" /> </span>= <span class="cmmi-12"><img src="img/cmmi12-17.png" alt="n" class="12x-x-17" /> </span>or <br/><span class="cmmi-12"><img src="img/cmmi12-16.png" alt="m" class="cmmi-12x-x-16" align="middle" /></span><img src="img/078_A_191634x.png" alt="/=" class="neq" align="middle" /><span class="cmmi-12"><img src="img/cmmi12-17.png" alt="n" class="12x-x-17" />.</span> </p><!--l. 1227--><p class="indent"> Instead of the above expression for <span class="cmti-12">ds</span><sup ><span class="cmr-8">2</span></sup> we may thus write </p> <center class="par-math-display" > <img src="img/078_A_191635x.png" alt="gmsdsndxmdxn " class="par-math-display" /></center> <!--l. 1233--><p class="nopar"> </p><!--l. 1237--><p class="noindent">or, by (16) </p> <center class="par-math-display" > <img src="img/078_A_191636x.png" alt=" st gmsgntg dxmdxn. " class="par-math-display" /></center> <!--l. 1243--><p class="nopar"> </p><!--l. 1246--><p class="noindent">But, by the multiplication rules of the preceding paragraphs, <br/>the quantities </p> <center class="par-math-display" > <img src="img/078_A_191637x.png" alt="dqs = gmsdxm " class="par-math-display" /></center> <!--l. 1253--><p class="nopar"> </p><!--l. 1257--><p class="noindent">form a covariant four-vector, and in fact an arbitrary vector, <br/>since the <span class="cmmi-12">dx</span><sub ><span class="cmmi-8"><img 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 </p> <center class="par-math-display" > <img src="img/078_A_191638x.png" alt="ds2 = gstdqsdqt . " class="par-math-display" /></center> <!--l. 1264--><p class="nopar"> </p><!--l. 1267--><p class="noindent">Since this, with the arbitrary choice of the vector <span class="cmmi-12">d<img src="img/cmmi12-18.png" alt="q" class="cmmi-12x-x-18" align="middle" /></span><sub ><span class="cmmi-8"><img src="img/cmmi8-1b.png" alt="s" class="8x-x-1b" /></span></sub><span class="cmmi-12">, </span>is a <br/>scalar, and <span class="cmmi-12">g</span><sup ><span class="cmmi-8"><img src="img/cmmi8-1b.png" alt="s" class="8x-x-1b" /><img src="img/cmmi8-1c.png" alt="t" class="8x-x-1c" /></span></sup> by its definition is symmetrical in the indices <br/><span class="cmmi-12"><img src="img/cmmi12-1b.png" alt="s" class="12x-x-1b" /> </span>and <span class="cmmi-12"><img src="img/cmmi12-1c.png" alt="t" class="12x-x-1c" />, </span>it follows from the results of the preceding paragraph <br/>that <span class="cmmi-12">g</span><sup ><span class="cmmi-8"><img src="img/cmmi8-1b.png" alt="s" class="8x-x-1b" /><img src="img/cmmi8-1c.png" alt="t" class="8x-x-1c" /></span></sup> is a contravariant tensor. </p><!--l. 1275--><p class="indent"> It further follows from (16) that <span class="cmmi-12"><img src="img/cmmi12-e.png" alt="d" class="12x-x-e" /></span><sub ><span class="cmmi-8"><img 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. </p><!--l. 1278--><p class="indent"> <span class="cmti-12">The Determinant of the Fundamental Tensor.</span>--By the <br/>rule for the multiplication of determinants </p> <center class="par-math-display" > <img src="img/078_A_191639x.png" alt="|g gan |= |g |× |gan |. ma ma " class="par-math-display" /></center> <!--l. 1286--><p class="nopar"> </p><!--l. 1289--><p class="noindent">On the other hand </p> <center class="par-math-display" > <img src="img/078_A_191640x.png" alt=" | | |gmagan |= |dn |= 1. m " class="par-math-display" /></center> <!--l. 1297--><p class="nopar"> </p><!--l. 1300--><p class="noindent">It therefore follows that </p> <table width="100%" class="equation"><tr><td><a id="x1-21r17"></a> <center class="math-display" > <img src="img/078_A_191641x.png" alt=" mn |gmn |× |g |= 1 " class="math-display" /></center></td><td width="5%">(17)</td></tr></table> <!--l. 1308--><p class="nopar"> </p><!--l. 1311--><p class="indent"> <span class="cmti-12">The Volume Scalar.</span>--We seek first the law of transfor-<br/><pb/> </p><!--l. 1315--><p class="indent"> </p><!--l. 1316--><p class="noindent">mation of the determinant <span class="cmmi-12">g </span>= <img src="img/078_A_191642x.png" alt="|gmn |" class="left" align="middle" /><span class="cmmi-12">. </span>In accordance with <br/>(11) </p> <center class="par-math-display" > <img src="img/078_A_191643x.png" alt=" | | ' ||@xm--@x- || g = |@x' @x' gmn |. s t " class="par-math-display" /></center> <!--l. 1324--><p class="nopar"> </p><!--l. 1327--><p class="noindent">Hence, by a double application of the rule for the multipli-<br/>cation of determinants, it follows that </p> <center class="par-math-display" > <img 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> <!--l. 1338--><p class="nopar"> </p><!--l. 1342--><p class="noindent">or </p> <center class="par-math-display" > <img src="img/078_A_191645x.png" alt=" | | V~ -g'= ||@xm-|| V~ g. |@x's | " class="par-math-display" /></center> <!--l. 1349--><p class="nopar"> </p><!--l. 1352--><p class="noindent">On the other hand, the law of transformation of the element <br/>of volume </p> <center class="par-math-display" > <img src="img/078_A_191646x.png" alt=" integral dt = dx1dx2dx3dx4 " class="par-math-display" /></center> <!--l. 1359--><p class="nopar"> </p><!--l. 1363--><p class="noindent">is, in accordance with the theorem of Jacobi, </p> <center class="par-math-display" > <img src="img/078_A_191647x.png" alt=" | | ' |@x's | dt = ||-@x- ||dt. m " class="par-math-display" /></center> <!--l. 1370--><p class="nopar"> </p><!--l. 1373--><p class="noindent">By multiplication of the last two equations, we obtain </p> <table width="100%" class="equation"><tr><td><a id="x1-22r18"></a> <center class="math-display" > <img 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> <!--l. 1379--><p class="nopar"> </p><!--l. 1382--><p class="noindent">Instead of <img src="img/078_A_191649x.png" alt=" V~ -- g" class="sqrt" /><span class="cmmi-12">, </span>we introduce in what follows the quantity <br/><img src="img/078_A_191650x.png" alt=" V~ ---- - g" class="sqrt" /><span class="cmmi-12">, </span>which is always real on account of the hyperbolic <br/>character of the space-time continuum. Theinvariant <img src="img/078_A_191651x.png" alt=" V~ ---- - g" class="sqrt" /><span class="cmmi-12">d<img src="img/cmmi12-1c.png" alt="t" class="12x-x-1c" /> </span> <br/>is equal to the magnitude of the four-dimensional element <br/>of volume in the “ local ” system of reference, as measured <br/>with rigid rods and clocks in the sense of the special theory <br/>of relativity. </p><!--l. 1391--><p class="indent"> <span 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 class="cmti-12">ds</span><sup ><span class="cmr-8">2</span></sup> can <br/>always be expressed in accordance with (1) by means of real <br/>quantities <span class="cmmi-12">dX</span><sub ><span class="cmr-8">1</span></sub> <span class="cmmi-12">...</span><span class="cmmi-12">dX</span><sub ><span class="cmr-8">4</span></sub><span class="cmmi-12">. </span>If we denote by <span class="cmmi-12">d<img src="img/cmmi12-1c.png" alt="t" class="12x-x-1c" /></span><sub ><span class="cmr-8">0</span></sub> the “ natural ” <br/>element of volume <span class="cmmi-12">dX</span><sub ><span class="cmr-8">1</span></sub><span class="cmmi-12">, dX</span><sub ><span class="cmr-8">2</span></sub><span class="cmmi-12">, dX</span><sub ><span class="cmr-8">3</span></sub><span class="cmmi-12">, dX</span><sub ><span class="cmr-8">4</span></sub><span class="cmmi-12">, </span>then </p> <table width="100%" class="equation"><tr><td><a id="x1-23r19"></a> <center class="math-display" > <img src="img/078_A_191652x.png" alt=" V~ ---- dt0 = -gdt " class="math-display" /></center></td><td width="5%">(18a)</td></tr></table> <!--l. 1404--><p class="nopar"> <pb/> </p><!--l. 1411--><p class="indent"> </p><!--l. 1412--><p class="indent"> If <img 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/>“ natural ” volume would correspond to a finite volume in <br/>the co-ordinates. Let us assume that this is never the case. <br/>Then <span class="cmmi-12">g </span>cannot change sign. We will assume that, in the <br/>sense of the special theory of relativity, <span 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. </p><!--l. 1423--><p class="indent"> But if <span class="cmsy-10x-x-120">-</span><span class="cmmi-12">g </span>is always finite and positive, it is natural to settle <br/>the choice of co-ordinates <span 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. </p><!--l. 1430--><p class="indent"> In place of (18), we then have simply <span class="cmmi-12">d<img src="img/cmmi12-1c.png" alt="t" class="12x-x-1c" /></span><span class="cmsy-10x-x-120">' </span>= <span class="cmmi-12">d<img src="img/cmmi12-1c.png" alt="t" class="12x-x-1c" />, </span>from <br/>which, in view of Jacobi’s theorem, it follows that </p> <table width="100%" class="equation"><tr><td><a id="x1-24r19"></a> <center class="math-display" > <img src="img/078_A_191654x.png" alt="| | |@x' | ||---s|| = 1 @xm " class="math-display" /></center></td><td width="5%">(19)</td></tr></table> <!--l. 1437--><p class="nopar"> </p><!--l. 1440--><p class="noindent">Thus, with this choice of co-ordinates, only substitutions for <br/>which the determinant is unity are permissible. </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 “ What are the laws of nature which are co-<br/>variant in face of all substitutions for which the determinant <br/>is unity? ” but our question is “ What are the generally co-<br/>variant laws of nature?” It is not until we have formulated <br/>these that we simplify their expression by a particular choice <br/>of the system of reference. </p><!--l. 1455--><p class="indent"> <span class="cmti-12">The Formation of New Tensors by Means of the Funda- </span> <br/><span 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, </p> <center class="par-math-display" > <img src="img/078_A_191655x.png" alt=" m ms A = g As, A = gmnAmn. " class="par-math-display" /></center> <!--l. 1468--><p class="nopar"> </p><!--l. 1471--><p class="noindent">The following forms may be specially noted:-- </p> <center class="par-math-display" > <img src="img/078_A_191656x.png" alt="Amn = gmagnbAab, ab Amn = gmagnbA " class="par-math-display" /></center> <!--l. 1482--><p class="nopar"> <pb/> </p><!--l. 1489--><p class="indent"> </p><!--l. 1490--><p class="noindent">(the “complements” of covariant and contravariant tensors <br/>respectively), and </p> <center class="par-math-display" > <img src="img/078_A_191657x.png" alt="B = g gabA . mn mn ab " class="par-math-display" /></center> <!--l. 1497--><p class="nopar"> </p><!--l. 1500--><p class="noindent">We call <span class="cmmi-12">B</span><sub ><span class="cmmi-8"><img src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img src="img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub> the reduced tensor associated with <span class="cmmi-12">A</span><sub ><span class="cmmi-8"><img src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img src="img/cmmi8-17.png" alt="n" class="8x-x-17" /></span><span class="cmsy-8"><sup class="htf"><strong>.</strong></sup></span></sub> Similarly, </p> <center class="par-math-display" > <img src="img/078_A_191658x.png" alt="Bmn = gmng Aab . ab " class="par-math-display" /></center> <!--l. 1509--><p class="nopar"> </p><!--l. 1512--><p class="noindent">It may be noted that <span class="cmmi-12">g</span><sup ><span class="cmmi-8"><img src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img src="img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sup> is nothing more than the comple-<br/>ment of <span class="cmmi-12">g</span><sub > <span class="cmmi-8"><img src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img src="img/cmmi8-17.png" alt="n" class="8x-x-17" />,</span></sub> since </p> <center class="par-math-display" > <img src="img/078_A_191659x.png" alt="gmagnbg = gmadn = gmn. ab a " class="par-math-display" /></center> <!--l. 1520--><p class="nopar"> </p> <div class="center" > <!--l. 1525--><p class="noindent"> </p><!--l. 1526--><p class="noindent"><span class="cmsy-10x-x-120">§ </span>9 <span class="cmbx-12">. The Equation of the Geodetic Line. The Motion of a </span> <br/><span class="cmbx-12">Particle</span></p></div> <!--l. 1531--><p class="indent"> As the linear element <span 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 class="cmmi-12">P</span><span class="cmsy-10x-x-120">' </span>of the four-dimensional continuum in such a way that <br/><span class="cmsy-10x-x-120"><img src="img/cmsy10-c-73.png" alt=" integral " class="10-120x-x-73" /></span> <span 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 </p> <table width="100%" class="equation"><tr><td><a id="x1-25r20"></a> <center class="math-display" > <img 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> <!--l. 1541--><p class="nopar"> </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 class="cmmi-12"><img src="img/cmmi12-15.png" alt="c" class="12x-x-15" /> </span>be a function of the co-ordinates <span class="cmmi-12">x</span><sub ><span class="cmmi-8"><img src="img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub><span 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 class="cmmi-12">P</span><span class="cmsy-10x-x-120">'</span><span class="cmmi-12">. </span> <br/>Any such line may then be supposed to be given by expres-<br/>sing its co-ordinates <span class="cmmi-12">x</span><sub ><span class="cmmi-8"><img src="img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub> as functions of <span class="cmmi-12"><img src="img/cmmi12-15.png" alt="c" class="12x-x-15" />. </span>Let the symbol <span class="cmmi-12"><img src="img/cmmi12-e.png" alt="d" class="12x-x-e" /></span> <br/>indicate the transition from a point of the required geodetic <br/>to the point corresponding to the same <span class="cmmi-12"><img src="img/cmmi12-15.png" alt="c" class="12x-x-15" /> </span>on a neighbouring <br/>line. Then for (20) we may substitute </p> <table width="100%" class="equation"><tr><td><a id="x1-26r21"></a> <center class="math-display" > <img 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> <!--l. 1567--><p class="nopar"> </p><!--l. 1570--><p class="noindent">But since <pb/> </p><!--l. 1575--><p class="indent"> </p> <center class="par-math-display" > <img 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> <!--l. 1583--><p class="nopar"> </p><!--l. 1587--><p class="noindent">and </p> <center class="par-math-display" > <img src="img/078_A_191663x.png" alt=" ( ) dxn- -d- d dc = dc (dxn) , " class="par-math-display" /></center> <!--l. 1595--><p class="nopar"> </p><!--l. 1599--><p class="noindent">we obtain from (20a), after a partial integration, </p> <center class="par-math-display" > <img src="img/078_A_191664x.png" alt=" integral c 3 c ksdxsdc = 0, 1 " class="par-math-display" /></center> <!--l. 1607--><p class="nopar"> </p><!--l. 1611--><p class="noindent">where </p> <table width="100%" class="equation"><tr><td><a id="x1-27r21"></a> <center class="math-display" > <img 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> <!--l. 1622--><p class="nopar"> </p><!--l. 1625--><p class="noindent">Since the values of <span class="cmmi-12"><img src="img/cmmi12-e.png" alt="d" class="12x-x-e" />x</span><sub ><span class="cmmi-8"><img src="img/cmmi8-1b.png" alt="s" class="8x-x-1b" /></span></sub> are arbitrary, it follows from this that </p> <table width="100%" class="equation"><tr><td><a id="x1-28r21"></a> <center class="math-display" > <img src="img/078_A_191666x.png" alt="ks = 0 " class="math-display" /></center></td><td width="5%">(20c)</td></tr></table> <!--l. 1633--><p class="nopar"> </p><!--l. 1637--><p class="noindent">are the equations of the geodetic line. </p><!--l. 1640--><p class="indent"> If <span class="cmti-12">ds </span>does not vanish along the geodetic line we may <br/>choose the “ length of the arc ” <span class="cmmi-12">s</span>, measured along the geodetic <br/>line, for the parameter <span class="cmmi-12"><img src="img/cmmi12-15.png" alt="c" class="12x-x-15" /></span>. Then <span class="cmmi-12">w </span>= 1<span class="cmmi-12">, </span>and in place of (20c) <br/>we obtain </p> <center class="par-math-display" > <img 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> <!--l. 1652--><p class="nopar"> </p><!--l. 1656--><p class="noindent">or, by a mere change of notation, </p> <table width="100%" class="equation"><tr><td><a id="x1-29r21"></a> <center class="math-display" > <img 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> <!--l. 1664--><p class="nopar"> </p><!--l. 1668--><p class="noindent">where, following Christoffel, we have written </p> <table width="100%" class="equation"><tr><td><a id="x1-30r21"></a> <center class="math-display" > <img 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> <!--l. 1678--><p class="nopar"> </p><!--l. 1682--><p class="noindent">Finally, if we multiply (20d) by <span class="cmmi-12">g</span><sup ><span class="cmmi-8"><img src="img/cmmi8-1b.png" alt="s" class="8x-x-1b" /><img src="img/cmmi8-1c.png" alt="t" class="8x-x-1c" /></span></sup> (outer multiplication with <br/>respect to <span class="cmmi-12"><img src="img/cmmi12-1c.png" alt="t" class="12x-x-1c" />, </span>inner with respect to <span class="cmmi-12"><img src="img/cmmi12-1b.png" alt="s" class="12x-x-1b" /></span>), we obtain the equations <br/>of the geodetic line in the form </p> <table width="100%" class="equation"><tr><td><a id="x1-31r22"></a> <center class="math-display" > <img 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> <!--l. 1692--><p class="nopar"> </p><!--l. 1696--><p class="noindent">where, following Christoffel, we have set </p> <table width="100%" class="equation"><tr><td><a id="x1-32r23"></a> <center class="math-display" > <img src="img/078_A_191671x.png" alt="{mn, t}= gta [mn, a] " class="math-display" /></center></td><td width="5%">(23)</td></tr></table> <!--l. 1704--><p class="nopar"> <pb/> </p><!--l. 1711--><p class="indent"> </p> <div class="center" > <!--l. 1712--><p class="noindent"> </p><!--l. 1713--><p class="noindent"><span class="cmsy-10x-x-120">§ </span>10 <span class="cmbx-12">. The Formation of Tensors by Differentiation</span></p></div> <!--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:-- </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 class="cmmi-12">s </span>measured from a fixed <br/>point on the curve, and if, further, <span class="cmmi-12"><img src="img/cmmi12-1e.png" alt="f" class="12x-x-1e" /> </span>is an invariant function <br/>of space, then <span class="cmmi-12">d<img src="img/cmmi12-1e.png" alt="f" class="12x-x-1e" /></span><img src="img/078_A_191672x.png" alt="/" class="left" align="middle" /> <span class="cmmi-12">ds</span> is also an invariant. The proof lies in <br/>this, that <span class="cmti-12">ds </span>is an invariant as well as <span class="cmmi-12">d<img src="img/cmmi12-1e.png" alt="f" class="12x-x-1e" />.</span> </p><!--l. 1734--><p class="indent"> As </p> <center class="par-math-display" > <img src="img/078_A_191673x.png" alt="df- -@f-dxm- ds = @xm ds " class="par-math-display" /></center> <!--l. 1741--><p class="nopar"> </p><!--l. 1745--><p class="noindent">therefore </p> <center class="par-math-display" > <img src="img/078_A_191674x.png" alt=" @f dx y = ------m- dxm ds " class="par-math-display" /></center> <!--l. 1751--><p class="nopar"> </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 class="cmmi-12">dx</span><sub ><span class="cmmi-8"><img src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /></span></sub>. Hence it immediately follows that </p> <table width="100%" class="equation"><tr><td><a id="x1-33r24"></a> <center class="math-display" > <img src="img/078_A_191675x.png" alt=" -@f- Am = @xm " class="math-display" /></center></td><td width="5%">(24)</td></tr></table> <!--l. 1763--><p class="nopar"> </p><!--l. 1767--><p class="noindent">is a covariant four-vector--the “ gradient ” of <span class="cmmi-12"><img src="img/cmmi12-1e.png" alt="f" class="12x-x-1e" />.</span> </p><!--l. 1770--><p class="indent"> According to our rule, the differential quotient </p> <center class="par-math-display" > <img src="img/078_A_191676x.png" alt=" dy x = --- ds " class="par-math-display" /></center> <!--l. 1775--><p class="nopar"> </p><!--l. 1779--><p class="noindent">taken on a curve, is similarly an invariant. Inserting the <br/>value of <span class="cmmi-12"><img src="img/cmmi12-20.png" alt="y" class="12x-x-20" />, </span>we obtain in the first place </p> <center class="par-math-display" > <img 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> <!--l. 1787--><p class="nopar"> </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 class="cmmi-12">d</span><sup ><span class="cmr-8">2</span></sup><span class="cmmi-12">x</span><sub > <span class="cmmi-8"><img src="img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub><img src="img/078_A_191678x.png" alt="/" class="left" align="middle" /> <span class="cmmi-12">ds</span><sup ><span class="cmr-8">2</span></sup> from (22), </p> <center class="par-math-display" > <img src="img/078_A_191679x.png" alt=" ( ) @2f @f dxm dxn x = --------- {mn, t} ---- --------. @xm@xn @xt ds ds " class="par-math-display" /></center> <!--l. 1802--><p class="nopar"> </p><!--l. 1806--><p class="noindent">Since we may interchange the order of the differentiations, <br/><pb/> </p><!--l. 1811--><p class="indent"> </p><!--l. 1812--><p class="noindent">and since by (23) and (21) <img src="img/078_A_191680x.png" alt="{mn, t}" class="left" align="middle" /> is symmetrical in <span class="cmmi-12"><img src="img/cmmi12-16.png" alt="m" class="cmmi-12x-x-16" align="middle" /> </span>and <span class="cmmi-12"><img src="img/cmmi12-17.png" alt="n" class="12x-x-17" /></span>, <br/>it follows that the expression in brackets is symmetrical in <span class="cmmi-12"><img src="img/cmmi12-16.png" alt="m" class="cmmi-12x-x-16" align="middle" /> </span> <br/>and <span class="cmmi-12"><img 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 class="cmmi-12">dx</span><sub ><span class="cmmi-8"><img src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /></span></sub><span 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 class="cmsy-10x-x-120">§ </span>7 that </p> <table width="100%" class="equation"><tr><td><a id="x1-34r25"></a> <center class="math-display" > <img 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> <!--l. 1827--><p class="nopar"> </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 </p> <center class="par-math-display" > <img src="img/078_A_191682x.png" alt="Am = -@f- @xm " class="par-math-display" /></center> <!--l. 1839--><p class="nopar"> </p><!--l. 1843--><p class="noindent">we can, by differentiation, form a covariant tensor of the <br/>second rank </p> <table width="100%" class="equation"><tr><td><a id="x1-35r26"></a> <center class="math-display" > <img 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> <!--l. 1851--><p class="nopar"> </p><!--l. 1854--><p class="noindent">We call the tensor <span class="cmmi-12">A</span><sub ><span class="cmmi-8"><img src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img src="img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub> the “ extension ” (covariant derivative) <br/>of the tensor <span class="cmmi-12">A</span><sub ><span class="cmmi-8"><img 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 class="cmmi-12">A</span><sub ><span class="cmmi-8"><img 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 <br/>that </p> <center class="par-math-display" > <img src="img/078_A_191684x.png" alt=" @f y ---- @xm " class="par-math-display" /></center> <!--l. 1864--><p class="nopar"> </p><!--l. 1868--><p class="noindent">is a covariant vector, if <span class="cmmi-12"><img src="img/cmmi12-20.png" alt="y" class="12x-x-20" /> </span>and <span class="cmmi-12"><img src="img/cmmi12-1e.png" alt="f" class="12x-x-1e" /> </span>are scalars. The sum of <br/>four such terms </p> <center class="par-math-display" > <img src="img/078_A_191685x.png" alt=" (1)f@(1) (4)@f(4) Sm = y -----+ .+ .+ y -----, @xm @xm " class="par-math-display" /></center> <!--l. 1877--><p class="nopar"> </p><!--l. 1881--><p class="noindent">is also a covariant vector, if <span class="cmmi-12"><img src="img/cmmi12-20.png" alt="y" class="12x-x-20" /></span><sup ><img src="img/078_A_191686x.png" alt="(1)" class="left" align="middle" /></sup><span class="cmmi-12">, <img src="img/cmmi12-1e.png" alt="f" class="12x-x-1e" /></span><sup ><img src="img/078_A_191687x.png" alt="(1)" class="left" align="middle" /></sup> <img src="img/078_A_191688x.png" alt=" ..." class="@cdots" /><span class="cmmi-12"><img src="img/cmmi12-20.png" alt="y" class="12x-x-20" /></span><sup ><img src="img/078_A_191689x.png" alt="(4)" class="left" align="middle" /></sup><span class="cmmi-12">, <img src="img/cmmi12-1e.png" alt="f" class="12x-x-1e" /></span><sup ><img 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 class="cmmi-8"><img src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /></span></sub>. For, if <span class="cmmi-12">A</span><sub ><span class="cmmi-8"><img 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 class="cmmi-12">x</span><sub ><span class="cmmi-8"><img 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) </p> <center class="par-math-display" > <img 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> <!--l. 1903--><p class="nopar"> </p><!--l. 1907--><p class="noindent">in order to ensure that S<sub ><span class="cmmi-8"><img src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /></span></sub> shall be equal to <span class="cmmi-12">A</span><sub ><span class="cmmi-8"><img src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /></span></sub>. <pb/> </p><!--l. 1912--><p class="indent"> </p><!--l. 1913--><p class="indent"> Therefore, in order to demonstrate that <span class="cmmi-12">A</span><sub ><span class="cmmi-8"><img src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img src="img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub> is a tensor if <br/><span class="cmti-12">any </span>covariant vector is inserted on the right-hand side for <span class="cmmi-12">A</span><sub ><span class="cmmi-8"><img 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 class="cmmi-8"><img 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 </p> <center class="par-math-display" > <img src="img/078_A_191692x.png" alt="A = y @f-. m @xm " class="par-math-display" /></center> <!--l. 1925--><p class="nopar"> </p><!--l. 1928--><p class="noindent">Now the right-hand side of (25) multiplied by <span class="cmmi-12"><img src="img/cmmi12-20.png" alt="y" class="12x-x-20" /></span>, </p> <center class="par-math-display" > <img src="img/078_A_191693x.png" alt=" 2 --@-f--- @f-- y@xm@xn - {mn,t}y @xt " class="par-math-display" /></center> <!--l. 1935--><p class="nopar"> </p><!--l. 1939--><p class="noindent">is a tensor. Similarly </p> <center class="par-math-display" > <img src="img/078_A_191694x.png" alt="-@y--@f- @xm @xn " class="par-math-display" /></center> <!--l. 1947--><p class="nopar"> </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 </p> <center class="par-math-display" > <img src="img/078_A_191695x.png" alt=" ( ) ( ) -@-- y @f-- - {mn, t } y-@f- . @xn @xm @xt " class="par-math-display" /></center> <!--l. 1960--><p class="nopar"> </p><!--l. 1964--><p class="noindent">As a glance at (26) will show, this completes the demon-<br/>stration for the vector</p> <center class="par-math-display" > <img src="img/078_A_191696x.png" alt=" @f y ---- @xm " class="par-math-display" /></center> <!--l. 1971--><p class="nopar"> </p><!--l. 1975--><p class="noindent">and consequently, from what has already been proved, for any <br/>vector <span class="cmmi-12">A</span><sub ><span class="cmmi-8"><img src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /></span></sub>. </p><!--l. 1978--><p class="indent"> By means of the extension of the vector, we may easily <br/>define the “ extension ” 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 <br/>formation. </p><!--l. 1985--><p class="indent"> As has already been observed, any covariant tensor of the <br/>second rank can be represented <sup ><span class="cmsy-8">*</span></sup> as the sum of tensors of the <br/> </p><!--l. 1989--><p class="indent"> <sup ><span class="cmsy-8">*</span></sup>By outer multiplication of the vector with arbitrary components A<sub > <span class="cmr-8">11</span></sub>, A<sub ><span class="cmr-8">12</span></sub>, <br/>A<sub ><span class="cmr-8">13</span></sub>, A<sub ><span class="cmr-8">14</span></sub> by the vector with components 1, 0, 0, 0, we produce a tensor with <br/>components </p> <center class="par-math-display" > <img src="img/078_A_191697x.png" alt="A11 A12 A13 A14 0 0 0 0 0 0 0 0 0 0 0 0. " class="par-math-display" /></center> <!--l. 2003--><p class="nopar"> </p><!--l. 2006--><p class="noindent">By the addition of four tensors of this type, we obtain the tensor <span class="cmmi-12">A</span><sub ><span class="cmmi-8"><img src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img src="img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub> with any <br/>ssigned components. <pb/> </p><!--l. 2012--><p class="indent"> </p><!--l. 2013--><p class="noindent">type <span class="cmmi-12">A</span><sub ><span class="cmmi-8"><img src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /></span></sub><span class="cmmi-12">B</span><sub ><span class="cmmi-8"><img src="img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub><span 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 </p> <center class="par-math-display" > <img src="img/078_A_191698x.png" alt="@Am-- @x - {sm, t}At , s @Bn- @x - {sn, t}Bt , s " class="par-math-display" /></center> <!--l. 2027--><p class="nopar"> </p><!--l. 2031--><p class="noindent">are tensors. On outer multiplication of the first by <span class="cmmi-12">B</span><sub ><span class="cmmi-8"><img src="img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub>, and <br/>of the second by <span class="cmmi-12">A</span><sub ><span class="cmmi-8"><img 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 </p> <table width="100%" class="equation"><tr><td><a id="x1-36r27"></a> <center class="math-display" > <img 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> <!--l. 2042--><p class="nopar"> </p><!--l. 2046--><p class="noindent">where we have put <span class="cmmi-12">A</span><sub ><span class="cmmi-8"><img src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img src="img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub> = <span class="cmmi-12">A</span><sub ><span class="cmmi-8"><img src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /></span></sub><span class="cmmi-12">B</span><sub ><span class="cmmi-8"><img src="img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub><span class="cmmi-12">. </span>As the right-hand side <br/>of (27) is linear and homogeneous in the <span class="cmmi-12">A</span><sub ><span class="cmmi-8"><img src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img 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 class="cmmi-12">A</span><sub ><span class="cmmi-8"><img src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /></span></sub><span class="cmmi-12">B</span><sub ><span class="cmmi-8"><img 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 class="cmmi-12">A</span><sub ><span class="cmmi-8"><img src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img src="img/cmmi8-17.png" alt="n" class="8x-x-17" /><img src="img/cmmi8-1b.png" alt="s" class="8x-x-1b" /></span></sub> the extension of the <br/>tensor <span class="cmmi-12">A</span><sub ><span class="cmmi-8"><img src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img src="img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub><span class="cmmi-12">.</span> </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). </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. </p> <div class="center" > <!--l. 2065--><p class="noindent"> </p><!--l. 2066--><p class="noindent"><span class="cmbsy-10x-x-120">§ </span><span class="cmbx-12">11. Some Cases of Special Importance</span></p></div> <!--l. 2070--><p class="indent"> <span 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 </p> <table width="100%" class="equation"><tr><td><a id="x1-37r28"></a> <center class="math-display" > <img 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> <!--l. 2078--><p class="nopar"> </p><!--l. 2081--><p class="noindent">The last member is obtained from the last but one, if we bear <br/>in mind that <span class="cmmi-12">g</span><sub ><span class="cmmi-8"><img src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img src="img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub><span class="cmmi-12">g</span><sup ><sub ><span class="cmmi-6"><img src="img/cmmi6-16.png" alt="m" class="cmmi-6x-x-16" align="middle" /></span><span class="cmsy-6">'</span><span class="cmmi-6"><img src="img/cmmi6-17.png" alt="n" class="6x-x-17" /></span></sub></sup> = <span class="cmmi-12"><img src="img/cmmi12-e.png" alt="d" class="12x-x-e" /></span><sub><span class="cmmi-8"><img src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /></span></sub><sup><span class="cmmi-8"><img src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /></span><span class="cmsy-8">'</span></sup><span class="cmmi-12">, </span>so that <span class="cmmi-12">g</span><sub ><span class="cmmi-8"><img src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img src="img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub><span class="cmmi-12">g</span><sup ><span class="cmmi-8"><img src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img src="img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sup> = 4<span class="cmmi-12">, </span>and conse-<br/>quently </p> <center class="par-math-display" > <img src="img/078_A_1916101x.png" alt="gmndgmn + gmndg = 0. mn " class="par-math-display" /></center> <!--l. 2090--><p class="nopar"> <pb/> </p><!--l. 2097--><p class="indent"> </p><!--l. 2098--><p class="noindent">From (28), it follows that </p> <table width="100%" class="equation"><tr><td><a id="x1-38r29"></a> <center class="math-display" > <img 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> <!--l. 2110--><p class="nopar"> </p><!--l. 2113--><p class="noindent">Further, from <span class="cmmi-12">g</span><sub ><span class="cmmi-8"><img src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img src="img/cmmi8-1b.png" alt="s" class="8x-x-1b" /></span></sub><span class="cmmi-12">g</span><sup ><span class="cmmi-8"><img src="img/cmmi8-17.png" alt="n" class="8x-x-17" /><img src="img/cmmi8-1b.png" alt="s" class="8x-x-1b" /></span></sup> = <span class="cmmi-12"><img src="img/cmmi12-e.png" alt="d" class="12x-x-e" /></span><sub> <span class="cmmi-8"><img src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /></span></sub><sup><span class="cmmi-8"><img src="img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sup><span class="cmmi-12">, </span>it follows on differentiation that </p> <table width="100%" class="equation"><tr><td><a id="x1-39r30"></a> <center class="math-display" > <img 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> <!--l. 2127--><p class="nopar"> </p><!--l. 2130--><p class="noindent">From these, by mixed multiplication by <span class="cmmi-12">g</span><sup ><span class="cmmi-8"><img src="img/cmmi8-1b.png" alt="s" class="8x-x-1b" /><img src="img/cmmi8-1c.png" alt="t" class="8x-x-1c" /></span></sup> and <span class="cmmi-12">g</span><sub > <span class="cmmi-8"><img src="img/cmmi8-17.png" alt="n" class="8x-x-17" /><img 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 </p> <table width="100%" class="equation"><tr><td><a id="x1-40r31"></a> <center class="math-display" > <img 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> <!--l. 2144--><p class="nopar"> </p><!--l. 2148--><p class="noindent">and</p> <table width="100%" class="equation"><tr><td><a id="x1-41r32"></a> <center class="math-display" > <img 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> <!--l. 2162--><p class="nopar"> </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) </p> <table width="100%" class="equation"><tr><td><a id="x1-42r33"></a> <center class="math-display" > <img 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> <!--l. 2176--><p class="nopar"> </p><!--l. 2180--><p class="noindent">Inserting this in the second formula of (31), we obtain, in <br/>view of (23) </p> <table width="100%" class="equation"><tr><td><a id="x1-43r34"></a> <center class="math-display" > <img 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> <!--l. 2190--><p class="nopar"> </p><!--l. 2194--><p class="noindent">Substituting the right-hand side of (34) in (29), we have </p> <table width="100%" class="equation"><tr><td><a id="x1-44r35"></a> <center class="math-display" > <img 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> <!--l. 2202--><p class="nopar"> </p><!--l. 2206--><p class="indent"> <span class="cmti-12">The “Divergence” of a Contravariant Vector.</span>--If we <br/>take the inner product of (26) by the contravariant funda-<br/>mental tensor <span class="cmmi-12">g</span><sup ><span class="cmmi-8"><img src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img src="img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sup><span class="cmmi-12">, </span>the right-hand side, after a transformation <br/>of the first term, assumes the form</p> <center class="par-math-display" > <img 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> <!--l. 2222--><p class="nopar"> <pb/> </p><!--l. 2229--><p class="indent"> </p><!--l. 2230--><p class="noindent">In accordance with (31) and (29), the last term of this ex-<br/>pression may be written </p> <center class="par-math-display" > <img 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> <!--l. 2243--><p class="nopar"> </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 class="cmmi-12">g</span><sup ><span class="cmmi-8"><img src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img src="img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sup><span class="cmmi-12">A</span><sub > <span class="cmmi-8"><img src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /></span></sub> = <span class="cmmi-12">A</span><sup ><span class="cmmi-8"><img src="img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sup><span class="cmmi-12">, </span>so that <span class="cmmi-12">A</span><sup ><span class="cmmi-8"><img src="img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sup> like <span class="cmmi-12">A</span><sub > <span class="cmmi-8"><img src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /></span></sub> <br/>is an arbitrary vector, we finally obtain </p> <table width="100%" class="equation"><tr><td><a id="x1-45r35"></a> <center class="math-display" > <img 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> <!--l. 2258--><p class="nopar"> </p><!--l. 2261--><p class="noindent">This scalar is the <span class="cmti-12">divergence </span>of the contravariant vector <span class="cmmi-12">A</span><sup ><span class="cmmi-8"><img src="img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sup>. </p><!--l. 2265--><p class="indent"> <span class="cmti-12">The “Curl” of a Covariant Vector.</span>--The second term in <br/>(26) is symmetrical in the indices <span class="cmmi-12"><img src="img/cmmi12-16.png" alt="m" class="cmmi-12x-x-16" align="middle" /> </span>and <span class="cmmi-12"><img src="img/cmmi12-17.png" alt="n" class="12x-x-17" /></span>. Therefore <br/><span class="cmmi-12">A</span><sub ><span class="cmmi-8"><img src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img src="img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub> <span class="cmsy-10x-x-120">- </span><span class="cmmi-12">A</span><sub ><span class="cmmi-8"><img src="img/cmmi8-17.png" alt="n" class="8x-x-17" /><img 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 </p> <table width="100%" class="equation"><tr><td><a id="x1-46r36"></a> <center class="math-display" > <img 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> <!--l. 2276--><p class="nopar"> </p><!--l. 2280--><p class="indent"> <span class="cmti-12">Antisymmetrical Extension of a Six-vector.</span>--Applying <br/>(27) to an antisymmetrical tensor of the second rank <span class="cmmi-12">A</span><sub ><span class="cmmi-8"><img src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img src="img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub><span 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 </p> <table width="100%" class="equation"><tr><td><a id="x1-47r37"></a> <center class="math-display" > <img 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> <!--l. 2294--><p class="nopar"> </p><!--l. 2298--><p class="noindent">which it is easy to prove is antisymmetrical. </p><!--l. 2301--><p class="indent"> <span class="cmti-12">The Divergence of a Six-vector.</span>--Taking the mixed pro-<br/>duct of (27) by <span class="cmmi-12">g</span><sup ><span class="cmmi-8"><img src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img src="img/cmmi8-b.png" alt="a" class="8x-x-b" /></span></sup><span class="cmmi-12">g</span><sup ><span class="cmmi-8"><img src="img/cmmi8-17.png" alt="n" class="8x-x-17" /><img src="img/cmmi8-c.png" alt="b" class="cmmi-8x-x-c" align="middle" /></span></sup><span 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 </p> <center class="par-math-display" > <img src="img/078_A_1916114x.png" alt=" nb ma -@-(gmagnbAmn) - gma@g---Amn - gnb@g---Amn. @xs @xs @xs " class="par-math-display" /></center> <!--l. 2314--><p class="nopar"> </p><!--l. 2317--><p class="noindent">If we write <span class="cmmi-12">A</span><sub><span class="cmmi-8"><img src="img/cmmi8-1b.png" alt="s" class="8x-x-1b" /></span></sub><sup><span class="cmmi-8"><img src="img/cmmi8-b.png" alt="a" class="8x-x-b" /><img src="img/cmmi8-c.png" alt="b" class="cmmi-8x-x-c" align="middle" /></span></sup> for <span class="cmmi-12">g</span><sup ><span class="cmmi-8"><img src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img src="img/cmmi8-b.png" alt="a" class="8x-x-b" /></span></sup><span class="cmmi-12">g</span><sup ><span class="cmmi-8"><img src="img/cmmi8-17.png" alt="n" class="8x-x-17" /><img src="img/cmmi8-c.png" alt="b" class="cmmi-8x-x-c" align="middle" /></span></sup><span class="cmmi-12">A</span><sub > <span class="cmmi-8"><img src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img src="img/cmmi8-17.png" alt="n" class="8x-x-17" /><img src="img/cmmi8-1b.png" alt="s" class="8x-x-1b" /></span></sub> and <span class="cmmi-12">A</span><sup ><span class="cmmi-8"><img src="img/cmmi8-b.png" alt="a" class="8x-x-b" /><img src="img/cmmi8-c.png" alt="b" class="cmmi-8x-x-c" align="middle" /></span></sup> for <span class="cmmi-12">g</span><sup ><span class="cmmi-8"><img src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img src="img/cmmi8-b.png" alt="a" class="8x-x-b" /></span></sup><span class="cmmi-12">g</span><sup ><span class="cmmi-8"><img src="img/cmmi8-17.png" alt="n" class="8x-x-17" /><img src="img/cmmi8-c.png" alt="b" class="cmmi-8x-x-c" align="middle" /></span></sup><span class="cmmi-12">A</span><sub > <span class="cmmi-8"><img src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img src="img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub><span class="cmmi-12">, </span>and in <br/>the transformed first term replace </p> <center class="par-math-display" > <img src="img/078_A_1916115x.png" alt="@gnb @gma -----and ----- @xs @xs " class="par-math-display" /></center> <!--l. 2328--><p class="nopar"> <pb/> </p><!--l. 2335--><p class="indent"> </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 </p> <table width="100%" class="equation"><tr><td><a id="x1-48r38"></a> <center class="math-display" > <img 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> <!--l. 2346--><p class="nopar"> </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. </p><!--l. 2357--><p class="indent"> We note that in an analogous way we may also form the <br/>extension of a mixed tensor:-- </p> <table width="100%" class="equation"><tr><td><a id="x1-49r39"></a> <center class="math-display" > <img 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> <!--l. 2365--><p class="nopar"> </p><!--l. 2369--><p class="indent"> On contracting (38) with respect to the indices <span class="cmmi-12"><img src="img/cmmi12-c.png" alt="b" class="cmmi-12x-x-c" align="middle" /> </span>and <span class="cmmi-12"><img src="img/cmmi12-1b.png" alt="s" class="12x-x-1b" /> </span> <br/>(inner multiplication by <span class="cmmi-12"><img src="img/cmmi12-e.png" alt="d" class="12x-x-e" /></span><sub><span class="cmmi-8"><img src="img/cmmi8-c.png" alt="b" class="cmmi-8x-x-c" align="middle" /></span></sub><sup><span class="cmmi-8"><img src="img/cmmi8-1b.png" alt="s" class="8x-x-1b" /></span></sup>), we obtain the vector </p> <center class="par-math-display" > <img src="img/078_A_1916118x.png" alt=" ab Aa = @A----+ {bg, b}Aag + {bg, a}Agb. @xb " class="par-math-display" /></center> <!--l. 2379--><p class="nopar"> </p><!--l. 2382--><p class="noindent">On account of the symmetry of <img src="img/078_A_1916119x.png" alt="{bg, a}" class="left" align="middle" /> with respect to the in-<br/>dices <span class="cmmi-12"><img src="img/cmmi12-c.png" alt="b" class="cmmi-12x-x-c" align="middle" /> </span>and <span class="cmmi-12"><img src="img/cmmi12-d.png" alt="g" class="12x-x-d" /></span>, the third term on the right-hand side vanishes, <br/>if <span class="cmmi-12">A</span><sup ><span class="cmmi-8"><img src="img/cmmi8-b.png" alt="a" class="8x-x-b" /><img 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 </p> <table width="100%" class="equation"><tr><td><a id="x1-50r40"></a> <center class="math-display" > <img 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> <!--l. 2396--><p class="nopar"> </p><!--l. 2400--><p class="noindent">This is the expression for the divergence of a contravariant <br/>six-vector. </p><!--l. 2404--><p class="indent"> <span class="cmti-12">The Divergence of a Mixed Tensor of the Second Rank.</span>--<br/>Contracting (39) with respect to the indices <span class="cmmi-12"><img src="img/cmmi12-b.png" alt="a" class="12x-x-b" /> </span>and <span class="cmmi-12"><img src="img/cmmi12-1b.png" alt="s" class="12x-x-1b" /></span>, and <br/>taking (29a) into consideration, we obtain </p> <table width="100%" class="equation"><tr><td><a id="x1-51r41"></a> <center class="math-display" > <img 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> <!--l. 2414--><p class="nopar"> </p><!--l. 2417--><p class="noindent">If we introduce the contravariant tensor <span class="cmmi-12">A</span><sup ><span class="cmmi-8"><img src="img/cmmi8-1a.png" alt="r" class="cmmi-8x-x-1a" align="middle" /><img src="img/cmmi8-1b.png" alt="s" class="8x-x-1b" /></span></sup> = <span class="cmmi-12">g</span><sup ><span class="cmmi-8"><img src="img/cmmi8-1a.png" alt="r" class="cmmi-8x-x-1a" align="middle" /><img src="img/cmmi8-1c.png" alt="t" class="8x-x-1c" /></span></sup><span class="cmmi-12">A</span><sub> <span class="cmmi-8"><img src="img/cmmi8-1c.png" alt="t" class="8x-x-1c" /></span></sub><sup><span class="cmmi-8"><img src="img/cmmi8-1b.png" alt="s" class="8x-x-1b" /></span></sup> in the <br/>last term, it assumes the form </p> <center class="par-math-display" > <img src="img/078_A_1916122x.png" alt="- [sm,r] V~ --gArs. " class="par-math-display" /></center> <!--l. 2425--><p class="nopar"> <pb/> </p><!--l. 2432--><p class="indent"> </p><!--l. 2433--><p class="noindent">If, further, the tensor <span class="cmmi-12">A</span><sup ><span class="cmmi-8"><img src="img/cmmi8-1a.png" alt="r" class="cmmi-8x-x-1a" align="middle" /><img src="img/cmmi8-1b.png" alt="s" class="8x-x-1b" /></span></sup> is symmetrical, this reduces to </p> <center class="par-math-display" > <img src="img/078_A_1916123x.png" alt=" 1 V~ ----@grs- rs - 2 -g @x A . m " class="par-math-display" /></center> <!--l. 2442--><p class="nopar"> </p><!--l. 2445--><p class="noindent">Had we introduced, instead of <span class="cmmi-12">A</span><sup ><span class="cmmi-8"><img src="img/cmmi8-1a.png" alt="r" class="cmmi-8x-x-1a" align="middle" /><img src="img/cmmi8-1b.png" alt="s" class="8x-x-1b" /></span></sup>, the covariant tensor <br/><span class="cmmi-12">A</span><sub > <span class="cmmi-8"><img src="img/cmmi8-1a.png" alt="r" class="cmmi-8x-x-1a" align="middle" /><img src="img/cmmi8-1b.png" alt="s" class="8x-x-1b" /></span></sub> = <span class="cmmi-12">g</span><sub ><span class="cmmi-8"><img src="img/cmmi8-1a.png" alt="r" class="cmmi-8x-x-1a" align="middle" /><img src="img/cmmi8-b.png" alt="a" class="8x-x-b" /></span></sub><span class="cmmi-12">g</span><sub ><span class="cmmi-8"><img src="img/cmmi8-1b.png" alt="s" class="8x-x-1b" /><img src="img/cmmi8-c.png" alt="b" class="cmmi-8x-x-c" align="middle" /></span></sub><span class="cmmi-12">A</span><sup ><span class="cmmi-8"><img src="img/cmmi8-b.png" alt="a" class="8x-x-b" /><img src="img/cmmi8-c.png" alt="b" class="cmmi-8x-x-c" align="middle" /></span></sup><span class="cmmi-12">, </span>which is also symmetrical, the last term, by <br/>virtue of (31), would assume the form </p> <center class="par-math-display" > <img src="img/078_A_1916124x.png" alt=" V~ ----@grs - 12 -g -----Ars. @xm " class="par-math-display" /></center> <!--l. 2456--><p class="nopar"> </p><!--l. 2459--><p class="noindent">In the case of symmetry in question, (41) may therefore be <br/>replaced by the two forms </p> <table width="100%" class="equation"><tr><td><a id="x1-52r42"></a> <center class="math-display" > <img 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> <!--l. 2469--><p class="nopar"> </p> <table width="100%" class="equation"><tr><td><a id="x1-53r42"></a> <center class="math-display" > <img 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> <!--l. 2478--><p class="nopar"> </p><!--l. 2482--><p class="noindent">which we have to employ later on. </p> <div class="center" > <!--l. 2487--><p class="noindent"> </p><!--l. 2488--><p class="noindent"><span class="cmsy-10x-x-120">§ </span><span class="cmbx-12">12. The Riemann-Christoffel Tensor</span></p></div> <!--l. 2492--><p class="indent"> We now seek the tensor which can be obtained from the <br/>fundamental tensor <span class="cmti-12">alone, </span>by differentiation. At first sight <br/>the solution seems obvious. We place the fundamental <br/>tensor of the <span class="cmmi-12">g</span><sub ><span class="cmmi-8"><img src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img src="img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub> in (27) instead of any given tensor <span class="cmmi-12">A</span><sub ><span class="cmmi-8"><img src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img 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 </p> <center class="par-math-display" > <img src="img/078_A_1916127x.png" alt=" @A Amn = ---m- {mn, r}Ar, @xn " class="par-math-display" /></center> <!--l. 2507--><p class="nopar"> </p><!--l. 2511--><p class="noindent">i.e. the extension of the four-vector A<sub ><span class="cmmi-8"><img 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> <center class="par-math-display" > <img 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> <!--l. 2529--><p class="nopar"> <pb/> </p><!--l. 2536--><p class="indent"> </p><!--l. 2537--><p class="noindent">This expression suggests forming the tensor <span class="cmmi-12">A</span><sub ><span class="cmmi-8"><img src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img src="img/cmmi8-1b.png" alt="s" class="8x-x-1b" /><img src="img/cmmi8-1c.png" alt="t" class="8x-x-1c" /></span></sub> <span class="cmsy-10x-x-120">- </span><span class="cmmi-12">A</span><sub ><span class="cmmi-8"><img src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img src="img/cmmi8-1c.png" alt="t" class="8x-x-1c" /><img 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 class="cmmi-12">A</span><sub ><span class="cmmi-8"><img src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img src="img/cmmi8-1b.png" alt="s" class="8x-x-1b" /><img src="img/cmmi8-1c.png" alt="t" class="8x-x-1c" /></span></sub> cancel those of <span class="cmmi-12">A</span><sub ><span class="cmmi-8"><img src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img src="img/cmmi8-1c.png" alt="t" class="8x-x-1c" /><img 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; <br/>because all these are symmetrical in <span class="cmmi-12"><img src="img/cmmi12-1b.png" alt="s" class="12x-x-1b" /> </span>and <span class="cmmi-12"><img 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 </p> <table width="100%" class="equation"><tr><td><a id="x1-54r42"></a> <center class="math-display" > <img src="img/078_A_1916129x.png" alt=" r Amst - Amts = B mstAr " class="math-display" /></center></td><td width="5%">(42)</td></tr></table> <!--l. 2551--><p class="nopar"> </p><!--l. 2555--><p class="noindent">where </p> <center class="par-math-display" > <img 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> <!--l. 2564--><p class="nopar"> </p> <table width="100%" class="equation"><tr><td><a id="x1-55r43"></a> <center class="math-display" > <img src="img/078_A_1916131x.png" alt="+{mt, a}{as, r} " class="math-display" /></center></td><td width="5%">(43)</td></tr></table> <!--l. 2571--><p class="nopar"> </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 class="cmmi-8"><img 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 <span class="cmmi-12">A</span><sub ><span class="cmmi-8"><img src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img src="img/cmmi8-1b.png" alt="s" class="8x-x-1b" /><img src="img/cmmi8-1c.png" alt="t" class="8x-x-1c" /></span></sub> <span class="cmsy-10x-x-120">- </span><span class="cmmi-12">A</span><sub ><span class="cmmi-8"><img src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img src="img/cmmi8-1c.png" alt="t" class="8x-x-1c" /><img src="img/cmmi8-1b.png" alt="s" class="8x-x-1b" /></span></sub> in conjunction with the fact <br/>that A<sub ><span class="cmmi-8"><img 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 class="cmsy-10x-x-120">§ </span>7, <br/>that <span class="cmmi-12">B</span><sub><span class="cmmi-8"><img src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img src="img/cmmi8-1b.png" alt="s" class="8x-x-1b" /><img src="img/cmmi8-1c.png" alt="t" class="8x-x-1c" /></span></sub><sup><span class="cmmi-8"><img src="img/cmmi8-1a.png" alt="r" class="cmmi-8x-x-1a" align="middle" /></span></sup> is a tensor (the Riemann-Christoffel tensor). </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 class="cmmi-12">g</span><sub ><span class="cmmi-8"><img src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img src="img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub> are constants, then <br/>all the <span class="cmmi-12">B</span><sub><span class="cmmi-8"><img src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img src="img/cmmi8-1b.png" alt="s" class="8x-x-1b" /><img src="img/cmmi8-1c.png" alt="t" class="8x-x-1c" /></span></sub><sup><span class="cmmi-8"><img 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 class="cmmi-12">g</span><sub ><span class="cmmi-8"><img src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img 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 class="cmmi-12">B</span><sub><span class="cmmi-8"><img src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img src="img/cmmi8-1b.png" alt="s" class="8x-x-1b" /><img src="img/cmmi8-1c.png" alt="t" class="8x-x-1c" /></span></sub><sup><span class="cmmi-8"><img 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 class="cmmi-12">g</span><sub ><span class="cmmi-8"><img src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img 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 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 class="cmti-12">finite </span>region of the <br/>continuum. </p><!--l. 2602--><p class="indent"> Contracting (43) with respect to the indices <span class="cmmi-12"><img src="img/cmmi12-1c.png" alt="t" class="12x-x-1c" /> </span>and <span class="cmmi-12"><img src="img/cmmi12-1a.png" alt="r" class="cmmi-12x-x-1a" align="middle" /> </span>we <br/>obtain the covariant tensor of second rank </p><!--l. 2607--><p class="indent"> <sup ><span class="cmsy-8">*</span></sup> The mathematicians have proved that this is also a <span class="cmti-12">sufficient </span>condition. <pb/> </p><!--l. 2612--><p class="indent"> </p> <table width="100%" class="equation"><tr><td><a id="x1-56r44"></a> <center class="math-display" > <img src="img/078_A_1916132x.png" alt=" r Gmn = B mnr = Rmn + Smn where 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> <!--l. 2626--><p class="nopar"> </p><!--l. 2630--><p class="indent"> <span class="cmti-12">Note on the Choice of Co-ordinates.</span>--It has already been <br/>observed in <span 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 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 class="cmmi-8"><img src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img 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 class="cmmi-8"><img src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img src="img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub>, so that the tensor G<sub ><span class="cmmi-8"><img src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img src="img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub> reduces to R<sub ><span class="cmmi-8"><img src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img src="img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub>. </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 class="cmti-12">generally </span>covariant equations, if this seems <br/>desirable in a special case. </p> <div class="center" > <!--l. 2650--><p class="noindent"> </p><!--l. 2651--><p class="noindent"><span class="cmcsc-10x-x-120">C. T<small class="small-caps">h</small><small class="small-caps">e</small><small class="small-caps">o</small><small class="small-caps">r</small><small class="small-caps">y</small> <small class="small-caps">o</small><small class="small-caps">f</small> <small class="small-caps">t</small><small class="small-caps">h</small><small class="small-caps">e</small> G<small class="small-caps">r</small><small class="small-caps">a</small><small class="small-caps">v</small><small class="small-caps">i</small><small class="small-caps">t</small><small class="small-caps">a</small><small class="small-caps">t</small><small class="small-caps">i</small><small class="small-caps">o</small><small class="small-caps">n</small><small class="small-caps">a</small><small class="small-caps">l</small> F<small class="small-caps">i</small><small class="small-caps">e</small><small class="small-caps">l</small><small class="small-caps">d</small></span></p></div> <div class="center" > <!--l. 2655--><p class="noindent"> </p><!--l. 2656--><p class="noindent"><span class="cmsy-10x-x-120">§ </span><span class="cmbx-12">13. Equations of Motion of a Material Point in the </span> <br/><span class="cmbx-12">Gravitational</span> <span class="cmbx-12">Field. Expression for the Field-com- </span> <br/><span class="cmbx-12">ponents of Gravitation</span></p></div> <!--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 class="cmr-8">0</span></sub>, may <br/>be, and is, so chosen that they have the special constant <br/>values given in (4). </p><!--l. 2670--><p class="indent"> If we consider precisely this movement from any chosen <br/>system of co-ordinates K<sub ><span class="cmr-8">1</span></sub>, the body, observed from K<sub ><span class="cmr-8">1</span></sub>, moves, <br/>according to the considerations in <span class="cmsy-10x-x-120">§ </span>2, in a gravitational field. <br/>The law of motion with respect to K<sub ><span class="cmr-8">1</span></sub> results without diffi-<br/><pb/> </p><!--l. 2677--><p class="indent"> </p><!--l. 2678--><p class="noindent">culty from the following consideration. With respect to K<sub ><span 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 class="cmr-8">1</span></sub>. If we set </p> <table width="100%" class="equation"><tr><td><a id="x1-57r45"></a> <center class="math-display" > <img src="img/078_A_1916134x.png" alt=" t G mn = -{mn, t } " class="math-display" /></center></td><td width="5%">(45)</td></tr></table> <!--l. 2689--><p class="nopar"> </p><!--l. 2693--><p class="noindent">the equation of the motion of the point with respect to K<sub ><span class="cmr-8">1</span></sub>, <br/>becomes </p> <table width="100%" class="equation"><tr><td><a id="x1-58r46"></a> <center class="math-display" > <img 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> <!--l. 2701--><p class="nopar"> </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 class="cmr-8">0</span></sub>, with respect to which the <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 class="cmti-12">first </span>derivatives of the <span class="cmmi-12">g</span><sub ><span class="cmmi-8"><img src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img 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 class="cmr-8">0</span></sub>, no relations sub-<br/>sist.<sup ><span class="cmsy-8">*</span></sup> </p><!--l. 2717--><p class="indent"> If the <img src="img/cmr12-0.png" alt="G" class="12x-x-0" /><sub><span class="cmmi-8"><img src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img src="img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub><sup><span class="cmmi-8"><img 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. </p> <div class="center" > <!--l. 2725--><p class="noindent"> </p><!--l. 2726--><p class="noindent"><span class="cmsy-10x-x-120">§ </span><span class="cmbx-12">14. The Field Equations of Gravitation in the Absence </span> <br/><span class="cmbx-12">of Matter</span></p></div> <!--l. 2731--><p class="indent"> We make a distinction hereafter between “ gravitational <br/>field ” and “ matter ” in this way, that we denote everything <br/>but the gravitational field as “ matter. ” Our use of the word <br/>therefore includes not only matter in the ordinary sense, but <br/>the electromagnetic field as well. </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/> </p><!--l. 2742--><p class="indent"> <sup ><span class="cmsy-8">*</span></sup> It is only between the second (and first) derivatives that, by <span class="cmsy-10x-x-120">§ </span>12, the <br/>relations <span class="cmmi-12">B</span><sub><span class="cmmi-8"><img src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img src="img/cmmi8-1b.png" alt="s" class="8x-x-1b" /><img src="img/cmmi8-1c.png" alt="t" class="8x-x-1c" /></span></sub><sup><span class="cmmi-8"><img src="img/cmmi8-1a.png" alt="r" class="cmmi-8x-x-1a" align="middle" /></span></sup> = 0 subsist. <pb/> </p><!--l. 2748--><p class="indent"> </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 class="cmmi-12">g</span><sub ><span class="cmmi-8"><img src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img src="img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub> have <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 class="cmr-8">0</span></sub>. <br/>Relatively to this system all the components of the Riemann <br/>tensor <span class="cmmi-12">B</span><sub><span class="cmmi-8"><img src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img src="img/cmmi8-1b.png" alt="s" class="8x-x-1b" /><img src="img/cmmi8-1c.png" alt="t" class="8x-x-1c" /></span></sub><sup> <span class="cmmi-8"><img 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 <br/>co-ordinates. </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 class="cmmi-12">B</span><sub><span class="cmmi-8"><img src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img src="img/cmmi8-1b.png" alt="s" class="8x-x-1b" /><img src="img/cmmi8-1c.png" alt="t" class="8x-x-1c" /></span></sub><sup> <span class="cmmi-8"><img 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 “ transformed away ” by any choice <br/>of the system of co-ordinates, i.e. it cannot be transformed to <br/>the case of constant <span class="cmmi-12">g</span><sub ><span class="cmmi-8"><img src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img src="img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub>. </p><!--l. 2772--><p class="indent"> This prompts us to require for the matter-free gravitational <br/>field that the symmetrical tensor <span class="cmmi-12">g</span><sub ><span class="cmmi-8"><img src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img src="img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub>, derived from the tensor <br/><span class="cmmi-12">B</span><sub><span class="cmmi-8"><img src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img src="img/cmmi8-17.png" alt="n" class="8x-x-17" /><img src="img/cmmi8-1c.png" alt="t" class="8x-x-1c" /></span></sub><sup> <span class="cmmi-8"><img 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 class="cmmi-12">g</span><sub ><span class="cmmi-8"><img src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img 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 class="cmmi-12">B</span><sub><span class="cmmi-8"><img src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img src="img/cmmi8-17.png" alt="n" class="8x-x-17" /><img src="img/cmmi8-1c.png" alt="t" class="8x-x-1c" /></span></sub><sup> <span class="cmmi-8"><img 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 </p> <table width="100%" class="equation"><tr><td><a id="x1-59r47"></a> <center class="math-display" > <img 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> <!--l. 2791--><p class="nopar"> </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 class="cmmi-8"><img src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img 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 class="cmmi-12">g</span><sub ><span class="cmmi-8"><img src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img src="img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub> and its derivatives, contains no derivations higher than <br/>second, and is linear in these derivatives.<sup ><span class="cmsy-8">*</span></sup> </p><!--l. 2802--><p class="indent"> These equations, which proceed, by the method of pure </p><!--l. 2806--><p class="indent"> <sup ><span class="cmsy-8">*</span></sup> Properly speaking, this can be affirmed only of the tensor </p> <center class="par-math-display" > <img src="img/078_A_1916137x.png" alt="Gmn + cgmngabGab, " class="par-math-display" /></center> <!--l. 2813--><p class="nopar"></p><!--l. 2816--><p class="noindent"><span class="cmr-10x-x-109">where </span><span class="cmmi-10x-x-109"><img src="img/cmmi10-15.png" alt="c" class="10-109x-x-15" /> </span><span class="cmr-10x-x-109">is a constant. If, however, we set this</span> tensor = 0<span class="cmr-10x-x-109">, we come back again </span> <br/><span class="cmr-10x-x-109">to the</span> <span class="cmr-10x-x-109">equations</span> G<sub ><span class="cmmi-8"><img src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img src="img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub> = 0<span class="cmr-10x-x-109">. </span><pb/> </p><!--l. 2823--><p class="indent"> </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’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. </p> <div class="center" > <!--l. 2836--><p class="noindent"> </p><!--l. 2837--><p class="noindent"><span class="cmbsy-10x-x-120">§ </span><span class="cmbx-12">15. The Hamiltonian Function for the Gravitational </span> <br/><span class="cmbx-12">Field. Laws</span> <span class="cmbx-12">of Momentum and Energy</span></p></div> <!--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:-- </p> <table width="100%" class="equation"><tr><td><a id="x1-60r48"></a> <center class="math-display" > <img src="img/078_A_1916138x.png" alt=" integral d Hdt = 0 } H = gmnGa Gb V~ ---- mb na - g = 1 " class="math-display" /></center></td><td width="5%">(47a)</td></tr></table> <!--l. 2855--><p class="nopar"> </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. </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 class="cmmi-12">g</span><sub ><span class="cmmi-8"><img src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img src="img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub> and the <span class="cmmi-12">g</span><sub><span class="cmmi-8"><img src="img/cmmi8-1b.png" alt="s" class="8x-x-1b" /></span></sub><sup><span class="cmmi-8"><img src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img src="img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sup>(= <span class="cmmi-12"><img src="img/cmmi12-40.png" alt="@" class="12x-x-40" />g</span><sup ><span class="cmmi-8"><img src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img src="img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sup><img src="img/078_A_1916139x.png" alt="/" class="left" align="middle" /> <span class="cmmi-12"><img src="img/cmmi12-40.png" alt="@" class="12x-x-40" />x</span><sub > <span class="cmmi-8"><img src="img/cmmi8-1b.png" alt="s" class="8x-x-1b" /></span></sub>). </p><!--l. 2870--><p class="noindent">Then in the first place </p> <center class="par-math-display" > <img 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> <!--l. 2885--><p class="nopar"> </p><!--l. 2888--><p class="noindent">But </p> <center class="par-math-display" > <img 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> <!--l. 2899--><p class="nopar"> </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 class="cmmi-12"><img src="img/cmmi12-16.png" alt="m" class="cmmi-12x-x-16" align="middle" /> </span>and <span class="cmmi-12"><img 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 class="cmmi-12"><img src="img/cmmi12-e.png" alt="d" class="12x-x-e" /></span>H, because they are multiplied by the <br/><pb/> </p><!--l. 2911--><p class="indent"> </p><!--l. 2912--><p class="noindent">quantity <img src="img/cmr12-0.png" alt="G" class="12x-x-0" /><sub><span class="cmmi-8"><img src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img src="img/cmmi8-c.png" alt="b" class="cmmi-8x-x-c" align="middle" /></span></sub><sup><span class="cmmi-8"><img src="img/cmmi8-b.png" alt="a" class="8x-x-b" /></span></sup>, which is symmetrical with respect to the in-<br/>dices <span class="cmmi-12"><img src="img/cmmi12-16.png" alt="m" class="cmmi-12x-x-16" align="middle" /> </span>and <span class="cmmi-12"><img 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 </p> <center class="par-math-display" > <img src="img/078_A_1916142x.png" alt=" a b mn a mb dH = - GmbG nadg + Gmbdga . " class="par-math-display" /></center> <!--l. 2924--><p class="nopar"> </p><!--l. 2927--><p class="noindent">Thus </p> <table width="100%" class="equation"><tr><td><a id="x1-61r48"></a> <center class="math-display" > <img 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> <!--l. 2939--><p class="nopar"> </p><!--l. 2942--><p class="noindent">Carrying out the variation in (47a), we get in the first place </p> <table width="100%" class="equation"><tr><td><a id="x1-62r49"></a> <center class="math-display" > <img 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> <!--l. 2950--><p class="nopar"> </p><!--l. 2954--><p class="noindent">which, on account of (48), agrees with (47), as was to be <br/>proved. </p><!--l. 2957--><p class="indent"> If we multiply (47b) by <span class="cmmi-12">g</span><sub><span class="cmmi-8"><img src="img/cmmi8-1b.png" alt="s" class="8x-x-1b" /></span></sub><sup><span class="cmmi-8"><img src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img src="img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sup>, then because </p> <center class="par-math-display" > <img src="img/078_A_1916145x.png" alt=" mn mn @gs--= @ga-- @xa @xs " class="par-math-display" /></center> <!--l. 2963--><p class="nopar"> </p><!--l. 2967--><p class="noindent">and, consequently, </p> <center class="par-math-display" > <img 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> <!--l. 2978--><p class="nopar"> </p><!--l. 2982--><p class="noindent">we obtain the equation </p> <center class="par-math-display" > <img src="img/078_A_1916147x.png" alt=" @ ( @H ) @H ---- gmsn---mn - ---- = 0 @xa @ga @xs " class="par-math-display" /></center> <!--l. 2990--><p class="nopar"> </p><!--l. 2994--><p class="noindent">or <sup ><span class="cmsy-8">*</span></sup> </p> <table width="100%" class="equation"><tr><td><a id="x1-63r49"></a> <center class="math-display" > <img 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> <!--l. 3004--><p class="nopar"> </p><!--l. 3008--><p class="noindent">where, on account of (48), the second equation of (47), and <br/>(34) </p> <table width="100%" class="equation"><tr><td><a id="x1-64r50"></a> <center class="math-display" > <img 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> <!--l. 3017--><p class="nopar"> </p><!--l. 3021--><p class="indent"> <sup ><span class="cmsy-8">*</span></sup> The reason for the introduction of the factor <span class="cmsy-10x-x-120">-</span>2<span class="cmmi-12"><img src="img/cmmi12-14.png" alt="k" class="12x-x-14" /> </span>will be apparent later. <pb/> </p><!--l. 3026--><p class="indent"> </p><!--l. 3027--><p class="indent"> It is to be noticed that <span class="cmmi-12">t</span><sub><span class="cmmi-8"><img src="img/cmmi8-1b.png" alt="s" class="8x-x-1b" /></span></sub><sup><span class="cmmi-8"><img 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 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 </p> <table width="100%" class="equation"><tr><td><a id="x1-65r51"></a> <center class="math-display" > <img src="img/078_A_1916151x.png" alt=" d integral integral ---- t4sdV = (lt1s + mt2s + nt3s)dS dx4 " class="math-display" /></center></td><td width="5%">(49a)</td></tr></table> <!--l. 3040--><p class="nopar"> </p><!--l. 3044--><p class="noindent">where <span class="cmmi-12">l, m, n </span>denote the direction-cosines of direction of the <br/>in ward drawn normal at the element <span 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 class="cmmi-12">t</span><sub><span class="cmmi-8"><img src="img/cmmi8-1b.png" alt="s" class="8x-x-1b" /></span></sub><sup><span class="cmmi-8"><img src="img/cmmi8-b.png" alt="a" class="8x-x-b" /></span></sup> we call the “ energy components ” <br/>of the gravitational field. </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 class="cmmi-12">g</span><sup ><span class="cmmi-8"><img src="img/cmmi8-17.png" alt="n" class="8x-x-17" /><img src="img/cmmi8-1b.png" alt="s" class="8x-x-1b" /></span></sup> these are ob-<br/>tained in the “ mixed ” form. Note that </p> <center class="par-math-display" > <img 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> <!--l. 3065--><p class="nopar"> </p><!--l. 3069--><p class="noindent">which quantity, by reason of (34), is equal to </p> <center class="par-math-display" > <img src="img/078_A_1916153x.png" alt=" @ ( ) ---- gnsGamn - gnbGsabGamn - gsbGnbaGamn, @xa " class="par-math-display" /></center> <!--l. 3079--><p class="nopar"> </p><!--l. 3083--><p class="noindent">or (with different symbols for the summation indices) </p> <center class="par-math-display" > <img 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> <!--l. 3093--><p class="nopar"> </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 </p> <center class="par-math-display" > <img src="img/078_A_1916155x.png" alt="k(tsm- 12dsmt), " class="par-math-display" /></center> <!--l. 3105--><p class="nopar"> </p><!--l. 3109--><p class="noindent">where <span class="cmmi-12">t </span>= <span class="cmmi-12">t</span><sub><span class="cmmi-8"><img src="img/cmmi8-b.png" alt="a" class="8x-x-b" /></span></sub><sup><span class="cmmi-8"><img src="img/cmmi8-b.png" alt="a" class="8x-x-b" /></span></sup>. Thus instead of equations (47) we obtain </p> <table width="100%" class="equation"><tr><td><a id="x1-66r51"></a> <center class="math-display" > <img 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> <!--l. 3120--><p class="nopar"> <pb/> </p><!--l. 3127--><p class="indent"> </p> <div class="center" > <!--l. 3128--><p class="noindent"> </p><!--l. 3129--><p class="noindent"><span class="cmsy-10x-x-120">§ </span><span class="cmbx-12">16. The General Form of the Field Equations of </span> <br/><span class="cmbx-12">Gravitation</span></p></div> <!--l. 3134--><p class="indent"> The field equations for matter-free space formulated in <br/><span class="cmsy-10x-x-120">§ </span>15 are to be compared with the field equation </p> <center class="par-math-display" > <img src="img/078_A_1916157x.png" alt=" 2 \~/ f = 0 " class="par-math-display" /></center> <!--l. 3140--><p class="nopar"> </p><!--l. 3144--><p class="noindent">of Newton’s theory. We require the equation corresponding <br/>to Poisson’s equation </p> <center class="par-math-display" > <img src="img/078_A_1916158x.png" alt=" 2 \~/ f = 4pkr, " class="par-math-display" /></center> <!--l. 3150--><p class="nopar"> </p><!--l. 3154--><p class="noindent">where <span class="cmmi-12"><img src="img/cmmi12-1a.png" alt="r" class="cmmi-12x-x-1a" align="middle" /> </span>denotes the density of matter. </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 <br/>general theory of relativity we must introduce a correspond-<br/>ing energy-tensor of matter T<sub><span class="cmmi-8"><img src="img/cmmi8-1b.png" alt="s" class="8x-x-1b" /></span></sub><sup><span class="cmmi-8"><img src="img/cmmi8-b.png" alt="a" class="8x-x-b" /></span></sup>, which, like the energy-com-<br/>ponents <span class="cmmi-12">t</span><sub > <span class="cmmi-8"><img 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 class="cmsy-8">*</span></sup> </p><!--l. 3168--><p class="indent"> The system of equation (51) shows how this energy-tensor <br/>(corresponding to the density <span class="cmmi-12"><img src="img/cmmi12-1a.png" alt="r" class="cmmi-12x-x-1a" align="middle" /> </span>in Poisson’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 class="cmmi-12">t</span><sub><span class="cmmi-8"><img src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /></span></sub><sup><span class="cmmi-8"><img src="img/cmmi8-1b.png" alt="s" class="8x-x-1b" /></span></sup> + <span class="cmmi-12">T</span><sub> <span class="cmmi-8"><img src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /></span></sub><sup><span class="cmmi-8"><img 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 </p> <table width="100%" class="equation"><tr><td><a id="x1-67r52"></a> <center class="math-display" > <img 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> <!--l. 3191--><p class="nopar"> </p><!--l. 3195--><p class="noindent">where we have set <span class="cmmi-12">T </span>= <span class="cmmi-12">T</span><sub><span class="cmmi-8"><img src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /></span></sub><sup><span class="cmmi-8"><img src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /></span></sup> (Laue’s scalar). These are the <br/> </p><!--l. 3198--><p class="indent"> <sup ><span class="cmsy-8">*</span></sup> <span class="cmmi-12">g</span><sub > <span class="cmmi-8"><img src="img/cmmi8-b.png" alt="a" class="8x-x-b" /><img src="img/cmmi8-1c.png" alt="t" class="8x-x-1c" /></span></sub><span class="cmmi-12">T</span><sub><span class="cmmi-8"><img src="img/cmmi8-1b.png" alt="s" class="8x-x-1b" /></span></sub><sup><span class="cmmi-8"><img src="img/cmmi8-b.png" alt="a" class="8x-x-b" /></span></sup> = <span class="cmmi-12">T</span><sub > <span class="cmmi-8"><img src="img/cmmi8-b.png" alt="a" class="8x-x-b" /><img src="img/cmmi8-1c.png" alt="t" class="8x-x-1c" /></span></sub> and <span class="cmmi-12">g</span><sup ><span class="cmmi-8"><img src="img/cmmi8-1b.png" alt="s" class="8x-x-1b" /><img src="img/cmmi8-c.png" alt="b" class="cmmi-8x-x-c" align="middle" /></span></sup><span class="cmmi-12">T</span><sub> <span class="cmmi-8"><img src="img/cmmi8-1b.png" alt="s" class="8x-x-1b" /></span></sub><sup><span class="cmmi-8"><img src="img/cmmi8-b.png" alt="a" class="8x-x-b" /></span></sup> = <span class="cmmi-12">T</span><sup ><span class="cmmi-8"><img src="img/cmmi8-b.png" alt="a" class="8x-x-b" /><img src="img/cmmi8-c.png" alt="b" class="cmmi-8x-x-c" align="middle" /></span></sup> are to be symmetrical tensors. <pb/> </p><!--l. 3204--><p class="indent"> </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) </p> <table width="100%" class="equation"><tr><td><a id="x1-68r53"></a> <center class="math-display" > <img 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> <!--l. 3218--><p class="nopar"> </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 class="cmsy-10x-x-120">§ </span>17. </p> <div class="center" > <!--l. 3236--><p class="noindent"> </p><!--l. 3237--><p class="noindent"><span class="cmsy-10x-x-120">§ </span><span class="cmbx-12">17. The Laws of Conservation in the General Case</span></p></div> <!--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 class="cmmi-12"><img src="img/cmmi12-16.png" alt="m" class="cmmi-12x-x-16" align="middle" /> </span>and <span class="cmmi-12"><img src="img/cmmi12-1b.png" alt="s" class="12x-x-1b" /></span>, and after multiplying the <br/>resulting equation by <span class="cmr-8">1</span> <span class="cmr-8">2</span><span class="cmmi-12"><img src="img/cmmi12-e.png" alt="d" class="12x-x-e" /></span><sub><span class="cmmi-8"><img src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /></span></sub><sup><span class="cmmi-8"><img src="img/cmmi8-1b.png" alt="s" class="8x-x-1b" /></span></sup>, subtract it from equation (52). <br/>This gives </p> <table width="100%" class="equation"><tr><td><a id="x1-69r54"></a> <center class="math-display" > <img 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> <!--l. 3254--><p class="nopar"> </p><!--l. 3257--><p class="noindent">On this equation we perform the operation <span class="cmmi-12"><img src="img/cmmi12-40.png" alt="@" class="12x-x-40" /></span><img src="img/078_A_1916162x.png" alt="/" class="left" align="middle" /> <span class="cmmi-12"><img src="img/cmmi12-40.png" alt="@" class="12x-x-40" />x</span><sub ><span class="cmmi-8"><img src="img/cmmi8-1b.png" alt="s" class="8x-x-1b" /></span></sub><span class="cmmi-12">. </span>We have</p> <center class="par-math-display" > <img 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> <!--l. 3273--><p class="nopar"> </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 class="cmmi-12"><img src="img/cmmi12-b.png" alt="a" class="12x-x-b" /> </span>and <span class="cmmi-12"><img src="img/cmmi12-1b.png" alt="s" class="12x-x-1b" /> </span>on the one hand, and <span class="cmmi-12"><img src="img/cmmi12-c.png" alt="b" class="cmmi-12x-x-c" align="middle" /> </span>and <span class="cmmi-12"><img src="img/cmmi12-15.png" alt="c" class="12x-x-15" /></span> <br/>on the other. The second term may be re-modelled by (31), <br/>so that we have </p> <table width="100%" class="equation"><tr><td><a id="x1-70r54"></a> <center class="math-display" > <img 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> <!--l. 3291--><p class="nopar"> </p><!--l. 3294--><p class="noindent">The second term on the left-hand side of (52a) yields in the <br/><pb/> </p><!--l. 3299--><p class="indent"> </p><!--l. 3300--><p class="noindent">first place </p> <center class="par-math-display" > <img src="img/078_A_1916165x.png" alt=" 1 @2 ( cb a ) - 2 -------- g Gcb @xa@xm " class="par-math-display" /></center> <!--l. 3308--><p class="nopar"> </p><!--l. 3312--><p class="noindent">or </p> <center class="par-math-display" > <img src="img/078_A_1916166x.png" alt=" 2 [ ( )] 1---@---- gcbgad @gdc-+ @gdb-- @gcb- . 4@xa@xm @xb @xc @xd " class="par-math-display" /></center> <!--l. 3324--><p class="nopar"> </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 </p> <center class="par-math-display" > <img src="img/078_A_1916167x.png" alt=" 1 --@3gab----- - 2 @x @x @x , a b m " class="par-math-display" /></center> <!--l. 3338--><p class="nopar"> </p><!--l. 3342--><p class="noindent">so that in consideration of (54), we have the identity </p> <table width="100%" class="equation"><tr><td><a id="x1-71r55"></a> <center class="math-display" > <img 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> <!--l. 3352--><p class="nopar"> </p><!--l. 3355--><p class="noindent">From (55) and (52a), it follows that </p> <table width="100%" class="equation"><tr><td><a id="x1-72r56"></a> <center class="math-display" > <img 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> <!--l. 3363--><p class="nopar"> </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 class="cmmi-12">t</span><sup ><span class="cmmi-8"><img 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. </p> <div class="center" > <!--l. 3377--><p class="noindent"> </p><!--l. 3378--><p class="noindent"><span class="cmbsy-10x-x-120">§ </span><span class="cmbx-12">18. The Laws of Momentum and Energy for Matter, as </span> <br/><span class="cmbx-12">a</span> <span class="cmbx-12">Consequence of the Field Equations</span></p></div> <!--l. 3383--><p class="indent"> Multiplying (53) by <span class="cmmi-12"><img src="img/cmmi12-40.png" alt="@" class="12x-x-40" />g</span><sup ><span class="cmmi-8"><img src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img src="img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sup><img src="img/078_A_1916170x.png" alt="/" class="left" align="middle" /> <span class="cmmi-12"><img src="img/cmmi12-40.png" alt="@" class="12x-x-40" />x</span><sub > <span class="cmmi-8"><img src="img/cmmi8-1b.png" alt="s" class="8x-x-1b" /></span></sub><span class="cmmi-12">, </span>we obtain, by the method <br/>adopted in <span class="cmsy-10x-x-120">§ </span>15, in view of the vanishing of </p> <center class="par-math-display" > <img src="img/078_A_1916171x.png" alt=" @gmn- gmn@xs , " class="par-math-display" /></center> <!--l. 3392--><p class="nopar"> </p><!--l. 3396--><p class="noindent">the equation </p> <center class="par-math-display" > <img src="img/078_A_1916172x.png" alt="@tas- 1@gmn- @x + 2@x Tmn = 0, a s " class="par-math-display" /></center> <!--l. 3404--><p class="nopar"> <pb/> </p><!--l. 3411--><p class="indent"> </p><!--l. 3412--><p class="noindent">or, in view of (56), </p> <table width="100%" class="equation"><tr><td><a id="x1-73r57"></a> <center class="math-display" > <img 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> <!--l. 3420--><p class="nopar"> </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 class="cmmi-12">g</span><sup ><span class="cmmi-8"><img src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img 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 </p> <table width="100%" class="equation"><tr><td><a id="x1-74r58"></a> <center class="math-display" > <img src="img/078_A_1916174x.png" alt="@T a ---s-= - GbasTba @xa " class="math-display" /></center></td><td width="5%">(57a)</td></tr></table> <!--l. 3443--><p class="nopar"> </p><!--l. 3446--><p class="noindent">The right side expresses the energetic effect of the gravitational <br/>field on matter. </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 class="cmsy-8">*</span></sup> </p> <div class="center" > <!--l. 3459--><p class="noindent"> </p><!--l. 3460--><p class="noindent">D. <span class="cmcsc-10x-x-120">M<small class="small-caps">a</small><small class="small-caps">t</small><small class="small-caps">e</small><small class="small-caps">r</small><small class="small-caps">i</small><small class="small-caps">a</small><small class="small-caps">l</small> P<small class="small-caps">h</small><small class="small-caps">e</small><small class="small-caps">n</small><small class="small-caps">o</small><small class="small-caps">m</small><small class="small-caps">e</small><small class="small-caps">n</small><small class="small-caps">a</small></span></p></div> <!--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’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, <br/> </p><!--l. 3475--><p class="indent"> <sup ><span class="cmsy-8">*</span></sup> On this question of. H. Hilbert, Nachr. d. K. Gesellsch. d. Wiss. zu <br/>Göttingen, Math.-phys. Klasse, 1915, p. 3. <pb/> </p><!--l. 3481--><p class="indent"> </p><!--l. 3482--><p class="noindent">without our having to introduce any new hypothesis what-<br/>ever. </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 <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. </p> <div class="center" > <!--l. 3500--><p class="noindent"> </p><!--l. 3501--><p class="noindent"><span class="cmbsy-10x-x-120">§ </span><span class="cmbx-12">19. Euler’s Equations for a Frictionless Adiabatic Fluid</span></p></div> <!--l. 3506--><p class="indent"> Let <span class="cmmi-12">p </span>and <span class="cmmi-12"><img 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 “ pressure,” the latter the “ density ” of a fluid; and let <br/>an equation subsist between them. Let the contravariant <br/>symmetrical tensor </p> <table width="100%" class="equation"><tr><td><a id="x1-75r58"></a> <center class="math-display" > <img 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> <!--l. 3517--><p class="nopar"> </p><!--l. 3521--><p class="noindent">be the contravariant energy-tensor of the fluid. To it belongs <br/>the covariant tensor </p> <table width="100%" class="equation"><tr><td><a id="x1-76r59"></a> <center class="math-display" > <img 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> <!--l. 3529--><p class="nopar"> </p><!--l. 3533--><p class="noindent">as well as the mixed tensor <sup ><span class="cmsy-8">*</span></sup> </p> <table width="100%" class="equation"><tr><td><a id="x1-77r59"></a> <center class="math-display" > <img 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> <!--l. 3540--><p class="nopar"> </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/> </p><!--l. 3554--><p class="indent"> <sup ><span 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 class="cmmi-12">T</span><sub><span class="cmr-8">4</span></sub><sup><span class="cmr-8">4</span></sup> equals <span class="cmmi-12"><img src="img/cmmi12-1a.png" alt="r" class="cmmi-12x-x-1a" align="middle" /> </span><span class="cmsy-10x-x-120">- </span><span class="cmmi-12">p</span>. This gives the definition of <span class="cmmi-12"><img src="img/cmmi12-1a.png" alt="r" class="cmmi-12x-x-1a" align="middle" /></span>. Thus <span class="cmmi-12"><img src="img/cmmi12-1a.png" alt="r" class="cmmi-12x-x-1a" align="middle" /> </span>is <br/>not constant for an incompressible fluid. <pb/> </p><!--l. 3563--><p class="indent"> </p><!--l. 3564--><p class="noindent">with the given equation between <span class="cmmi-12">p </span>and <span class="cmmi-12"><img src="img/cmmi12-1a.png" alt="r" class="cmmi-12x-x-1a" align="middle" /></span>, and the equation </p> <center class="par-math-display" > <img src="img/078_A_1916178x.png" alt=" dxa-dxb- gab ds ds = 1, " class="par-math-display" /></center> <!--l. 3571--><p class="nopar"> </p><!--l. 3575--><p class="noindent">are sufficient, <span class="cmmi-12">g</span><sub ><span class="cmmi-8"><img src="img/cmmi8-b.png" alt="a" class="8x-x-b" /><img src="img/cmmi8-c.png" alt="b" class="cmmi-8x-x-c" align="middle" /></span></sub> being given, to define the six unknowns </p> <center class="par-math-display" > <img src="img/078_A_1916179x.png" alt="p, r, dx1, dx2-, dx3-, dx4-. ds ds ds ds " class="par-math-display" /></center> <!--l. 3582--><p class="nopar"> </p><!--l. 3586--><p class="noindent">If the <span class="cmmi-12">g</span><sub ><span class="cmmi-8"><img src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img 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 class="cmmi-12">g</span><sub ><span class="cmmi-8"><img src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img 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 <br/>mathematically undefined to such a degree that three of the <br/>functions of space may be chosen at will.<sup ><span class="cmsy-8">*</span></sup> </p> <div class="center" > <!--l. 3600--><p class="noindent"> </p><!--l. 3601--><p class="noindent"><span class="cmbsy-10x-x-120">§ </span><span class="cmbx-12">20. Maxwell’s Electromagnetic Field Equations for Free </span> <br/><span class="cmbx-12">Space</span></p></div> <!--l. 3606--><p class="indent"> Let <span class="cmmi-12"><img src="img/cmmi12-1e.png" alt="f" class="12x-x-1e" /></span><sub ><span class="cmmi-8"><img 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 class="cmmi-8"><img src="img/cmmi8-1a.png" alt="r" class="cmmi-8x-x-1a" align="middle" /><img 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 </p> <table width="100%" class="equation"><tr><td><a id="x1-78r59"></a> <center class="math-display" > <img 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> <!--l. 3619--><p class="nopar"> </p><!--l. 3623--><p class="noindent">It follows from (59) that the system of equations </p> <table width="100%" class="equation"><tr><td><a id="x1-79r60"></a> <center class="math-display" > <img 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> <!--l. 3631--><p class="nopar"> </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:-- </p><!--l. 3643--><p class="indent"> <sup ><span class="cmsy-8">*</span></sup> On the abandonment of the choice of co-ordinates with <span class="cmmi-12">g </span>= <span class="cmsy-10x-x-120">-</span>1<span class="cmmi-12">, </span>there <br/>remain <span 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. <pb/> </p><!--l. 3651--><p class="indent"> </p> <table width="100%" class="equation"><tr><td><a id="x1-80r61"></a> <center class="math-display" > <img 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> <!--l. 3665--><p class="nopar"> </p><!--l. 3669--><p class="indent"> This system corresponds to the second of Maxwell’s <br/>systems of equations. We recognize this at once by setting </p> <table width="100%" class="equation"><tr><td><a id="x1-81r61"></a> <center class="math-display" > <img 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> <!--l. 3679--><p class="nopar"> </p><!--l. 3682--><p class="noindent">Then in place of (60a) we may set, in the usual notation of <br/>three-dimensional vector analysis, </p> <table width="100%" class="equation"><tr><td><a id="x1-82r62"></a> <center class="math-display" > <img src="img/078_A_1916184x.png" alt="- @H--= curlE } @t divH = 0 " class="math-display" /></center></td><td width="5%">(60b)</td></tr></table> <!--l. 3693--><p class="nopar"> </p><!--l. 3697--><p class="indent"> We obtain Maxwell’s first system by generalizing the <br/>form given by Minkowski. We introduce the contravariant <br/>six-vector associated with F<sup ><span class="cmmi-8"><img src="img/cmmi8-b.png" alt="a" class="8x-x-b" /><img src="img/cmmi8-c.png" alt="b" class="cmmi-8x-x-c" align="middle" /></span></sup> </p> <table width="100%" class="equation"><tr><td><a id="x1-83r62"></a> <center class="math-display" > <img 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> <!--l. 3705--><p class="nopar"> </p><!--l. 3709--><p class="noindent">and also the contravariant vector J<sup ><span class="cmmi-8"><img 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):-- </p> <table width="100%" class="equation"><tr><td><a id="x1-84r63"></a> <center class="math-display" > <img src="img/078_A_1916186x.png" alt="-@--Fmn = Jm @xn " class="math-display" /></center></td><td width="5%">(63)</td></tr></table> <!--l. 3718--><p class="nopar"> </p><!--l. 3721--><p class="noindent">Let </p> <table width="100%" class="equation"><tr><td><a id="x1-85r64"></a> <center class="math-display" > <img 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> <!--l. 3734--><p class="nopar"> </p><!--l. 3738--><p class="noindent">which quantities are equal to the quantities <span class="cmmi-12">H</span><sub ><span class="cmmi-8">x</span></sub> <span class="cmmi-12">...</span> <span class="cmmi-12">E</span><sub ><span class="cmmi-8">z</span></sub> in <br/><pb/> </p><!--l. 3742--><p class="indent"> </p><!--l. 3743--><p class="noindent">the special case of the restricted theory of relativity; and in <br/>addition </p> <center class="par-math-display" > <img src="img/078_A_1916188x.png" alt=" 1 2 3 4 J = jx, J = jy, J = jz, J = r, " class="par-math-display" /></center> <!--l. 3750--><p class="nopar"> </p><!--l. 3754--><p class="noindent">we obtain in place of (63) </p> <table width="100%" class="equation"><tr><td><a id="x1-86r65"></a> <center class="math-display" > <img 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> <!--l. 3765--><p class="nopar"> </p><!--l. 3769--><p class="indent"> The equations (60), (62), and (63) thus form the generali-<br/>zation of Maxwell’s field equations for free space, with the <br/>convention which we have established with respect to the <br/>choice of co-ordinates. </p><!--l. 3774--><p class="indent"> <span class="cmti-12">The Energy-components of the Electromagnetic Field.</span>--<br/>We form the inner product </p> <table width="100%" class="equation"><tr><td><a id="x1-87r65"></a> <center class="math-display" > <img src="img/078_A_1916190x.png" alt=" m ks = FsmJ " class="math-display" /></center></td><td width="5%">(65)</td></tr></table> <!--l. 3780--><p class="nopar"> </p><!--l. 3784--><p class="noindent">By (61) its components, written in the three-dimensional <br/>manner, are </p> <table width="100%" class="equation"><tr><td><a id="x1-88r66"></a> <center class="math-display" > <img src="img/078_A_1916191x.png" alt="k = rEx + [j .H]x 1 . . . . } . . . . k4 = - (jE) " class="math-display" /></center></td><td width="5%">(65a)</td></tr></table> <!--l. 3798--><p class="nopar"> </p><!--l. 3802--><p class="indent"> <span class="cmmi-12"><img src="img/cmmi12-14.png" alt="k" class="12x-x-14" /></span><sub ><span class="cmmi-8"><img 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 <br/>masses are free, that is, under the sole influence of the <br/>electromagnetic field, the covariant vector <span class="cmmi-12"><img src="img/cmmi12-14.png" alt="k" class="12x-x-14" /></span><sub ><span class="cmmi-8"><img src="img/cmmi8-1b.png" alt="s" class="8x-x-1b" /></span></sub> will vanish. </p><!--l. 3811--><p class="indent"> To obtain the energy-components <span class="cmmi-12">T</span><sub><span class="cmmi-8"><img src="img/cmmi8-1b.png" alt="s" class="8x-x-1b" /></span></sub><sup><span class="cmmi-8"><img 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 class="cmmi-12"><img src="img/cmmi12-14.png" alt="k" class="12x-x-14" /></span><sub ><span class="cmmi-8"><img 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 </p> <center class="par-math-display" > <img src="img/078_A_1916192x.png" alt=" @Fmn @ @F ks = Fsm ----- = ----(FsmFmn) - Fmr---sm. @xn @xn @xn " class="par-math-display" /></center> <!--l. 3823--><p class="nopar"> </p><!--l. 3827--><p class="noindent">The second term of the right-hand side, by reason of (60), <br/>permits the transformation </p> <center class="par-math-display" > <img src="img/078_A_1916193x.png" alt=" @Fsm @Fmn @Fmn Fmn ----- = - 12Fmn -----= - 12gmagnbFab -----, @xn @xs @xs " class="par-math-display" /></center> <!--l. 3839--><p class="nopar"> <pb/> </p><!--l. 3846--><p class="indent"> </p><!--l. 3847--><p class="noindent">which latter expression may, for reasons of symmetry, also <br/>be written in the form </p> <center class="par-math-display" > <img src="img/078_A_1916194x.png" alt=" [ @F @F ] - 14 gmagnbFab ---mn+ gmagnb---abFmn . @xs @xs " class="par-math-display" /></center> <!--l. 3857--><p class="nopar"> </p><!--l. 3860--><p class="noindent">But for this we may set </p> <center class="par-math-display" > <img src="img/078_A_1916195x.png" alt=" @ @ - 14----(gmagnbFabFmn) + 14FabFmn ----(gmagnb). @xs @xs " class="par-math-display" /></center> <!--l. 3870--><p class="nopar"> </p><!--l. 3873--><p class="noindent">The first of these terms is written more briefly </p> <center class="par-math-display" > <img src="img/078_A_1916196x.png" alt=" @ - 14----(FmnFmn); @xs " class="par-math-display" /></center> <!--l. 3880--><p class="nopar"> </p><!--l. 3884--><p class="noindent">the second, after the differentiation is carried out, and after <br/>some reduction, results in </p> <center class="par-math-display" > <img src="img/078_A_1916197x.png" alt=" @g - 1FmtFmngnr --st-. 2 @xs " class="par-math-display" /></center> <!--l. 3893--><p class="nopar"> </p><!--l. 3896--><p class="noindent">Taking all three terms together we obtain the relation </p> <table width="100%" class="equation"><tr><td><a id="x1-89r66"></a> <center class="math-display" > <img 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> <!--l. 3904--><p class="nopar"> </p><!--l. 3908--><p class="noindent">where </p> <center class="par-math-display" > <img src="img/078_A_1916199x.png" alt=" n na 1 n ab Ts = - FsaF + 4dsFabF . " class="par-math-display" /></center> <!--l. 3916--><p class="nopar"> </p><!--l. 3920--><p class="indent"> Equation (66), if <span class="cmmi-12"><img src="img/cmmi12-14.png" alt="k" class="12x-x-14" /></span><sub ><span class="cmmi-8"><img 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 class="cmti-12">a</span>) respectively. Therefore the <span class="cmmi-12">T</span><sub><span class="cmmi-8"><img src="img/cmmi8-1b.png" alt="s" class="8x-x-1b" /></span></sub><sup><span class="cmmi-8"><img 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. </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 <img src="img/078_A_1916200x.png" alt=" V~ ---- -g" class="sqrt" /> = 1<span 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/> </p><!--l. 3941--><p class="indent"> </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. </p> <div class="center" > <!--l. 3959--><p class="noindent"> </p><!--l. 3960--><p class="noindent">E</p></div> <div class="center" > <!--l. 3964--><p class="noindent"> </p><!--l. 3965--><p class="noindent"><span class="cmbsy-10x-x-120">§ </span><span class="cmbx-12">21. Newton’s Theory as a First Approximation</span></p></div> <!--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 class="cmmi-12">g</span><sub ><span class="cmmi-8"><img src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img src="img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub> 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 <br/>approximation to reality by considering the case where the <br/><span class="cmmi-12">g</span><sub ><span class="cmmi-8"><img src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img 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.) </p><!--l. 3981--><p class="indent"> It is further to be assumed that in the space-time territory <br/>under consideration the <span class="cmmi-12">g</span><sub ><span class="cmmi-8"><img src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img 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. </p><!--l. 3988--><p class="indent"> It might be thought that these approximations must lead <br/>us to Newton’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 <br/>point in accordance with the equations (16). In the case of <br/>the special theory of relativity the components </p> <center class="par-math-display" > <img src="img/078_A_1916201x.png" alt="dx1 dx2 dx3 ----,----, ---- ds ds ds " class="par-math-display" /></center> <!--l. 3999--><p class="nopar"> <pb/> </p><!--l. 4006--><p class="indent"> </p><!--l. 4007--><p class="noindent">may take on any values. This signifies that any velocity </p> <center class="par-math-display" > <img src="img/078_A_1916202x.png" alt=" V~ -(----)2---(-----)2---(-----)2- dx1- dx2- dx3- v = dx4 + dx4 + dx4 " class="par-math-display" /></center> <!--l. 4015--><p class="nopar"> </p><!--l. 4019--><p class="noindent">may occur, which is less than the velocity of light <span 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 class="cmmi-12">v</span> being small as compared <br/>with the velocity of light, this denotes that the components </p> <center class="par-math-display" > <img src="img/078_A_1916203x.png" alt="dx1-, dx2-, dx3 ds ds ds " class="par-math-display" /></center> <!--l. 4028--><p class="nopar"> </p><!--l. 4032--><p class="noindent">are to be treated as small quantities, while <span class="cmmi-12">dx</span><sub ><span class="cmr-8">4</span></sub><span class="cmmi-12">/ds, </span>to the <br/>second order of small quantities, is equal to one. (Second <br/>point of view of approximation.) </p><!--l. 4036--><p class="indent"> Now we remark that from the first point of view of ap-<br/>proximation the magnitudes <img src="img/cmr12-0.png" alt="G" class="12x-x-0" /><sub><span class="cmmi-8"><img src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img src="img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub><sup> <span class="cmmi-8"><img 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 class="cmmi-12"><img src="img/cmmi12-16.png" alt="m" class="cmmi-12x-x-16" align="middle" /> </span>= <span class="cmmi-12"><img src="img/cmmi12-17.png" alt="n" class="12x-x-17" /> </span>= 4<span class="cmmi-12">. </span>Re-<br/>stricting ourselves to terms of lowest order we first obtain in <br/>place of (46) the equations </p> <center class="par-math-display" > <img src="img/078_A_1916204x.png" alt="d2xt- t dt2 = G 44 " class="par-math-display" /></center> <!--l. 4048--><p class="nopar"> </p><!--l. 4052--><p class="noindent">where we have set <span class="cmmi-12">ds </span>= <span class="cmmi-12">dx</span><sub ><span class="cmr-8">4</span></sub> = <span 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:-- </p> <center class="par-math-display" > <img 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> <!--l. 4063--><p class="nopar"> </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/> </p><!--l. 4078--><p class="indent"> </p> <table width="100%" class="equation"><tr><td><a id="x1-90r67"></a> <center class="math-display" > <img 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> <!--l. 4084--><p class="nopar"> </p><!--l. 4087--><p class="noindent">This is the equation of motion of the material point accord-<br/>ing to Newton’s theory, in which <span class="cmr-8">1</span> <span class="cmr-8">2</span><span class="cmmi-12">g</span><sub ><span 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 class="cmmi-12">g</span><sub ><span class="cmr-8">44</span></sub> of the fundamental tensor alone <br/>defines, to a first approximation, the motion of the material <br/>point. </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/>“ matter ” is almost exclusively defined by the density of <br/>matter in the narrower sense, i.e. by the second term of the <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 class="cmmi-12">T</span><sub ><span class="cmr-8">44</span> </sub> = <span class="cmmi-12"><img src="img/cmmi12-1a.png" alt="r" class="cmmi-12x-x-1a" align="middle" /> </span>= <span 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 <br/>question, </p> <center class="par-math-display" > <img src="img/078_A_1916207x.png" alt=" @ @ @ @ ----[mn, 1] + ----[mn, 2] +----[mn, 3]- ---- [mn, 4]. @x1 @x2 @x3 @x4 " class="par-math-display" /></center> <!--l. 4114--><p class="nopar"> </p><!--l. 4118--><p class="noindent">For <span class="cmmi-12"><img src="img/cmmi12-16.png" alt="m" class="cmmi-12x-x-16" align="middle" /> </span>= <span class="cmmi-12"><img src="img/cmmi12-17.png" alt="n" class="12x-x-17" /> </span>= 4<span class="cmmi-12">, </span>this gives, with the omission of terms differ-<br/>entiated with respect to time, </p> <center class="par-math-display" > <img 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> <!--l. 4129--><p class="nopar"> </p><!--l. 4132--><p class="noindent">The last of equations (53) thus yields </p> <table width="100%" class="equation"><tr><td><a id="x1-91r68"></a> <center class="math-display" > <img src="img/078_A_1916209x.png" alt=" \~/ 2g44 = kr " class="math-display" /></center></td><td width="5%">(68)</td></tr></table> <!--l. 4139--><p class="nopar"> </p><!--l. 4143--><p class="noindent">The equations (67) and (68) together are equivalent to <br/>Newton’s law of gravitation. </p><!--l. 4147--><p class="indent"> By (67) and (68) the expression for the gravitational <br/>potential becomes </p> <table width="100%" class="equation"><tr><td><a id="x1-92r69"></a> <center class="math-display" > <img src="img/078_A_1916210x.png" alt=" k integral rdt - --- ---- 8p r " class="math-display" /></center></td><td width="5%">(68a)</td></tr></table> <!--l. 4153--><p class="nopar"> </p><!--l. 4157--><p class="noindent">while Newton’s theory, with the unit of time which we have <br/>chosen, gives </p> <center class="par-math-display" > <img src="img/078_A_1916211x.png" alt=" K integral rdt - -2- ---- c r " class="par-math-display" /></center> <!--l. 4163--><p class="nopar"> <pb/> </p><!--l. 4170--><p class="indent"> </p><!--l. 4171--><p class="noindent">in which K denotes the constant 6<span class="cmmi-12">.</span>7 <span class="cmsy-10x-x-120">× </span>10<sup ><span class="cmsy-8">- </span><span class="cmr-8">8</span></sup><span class="cmmi-12">, </span>usually called <br/>the constant of gravitation. By comparison we obtain </p> <table width="100%" class="equation"><tr><td><a id="x1-93r69"></a> <center class="math-display" > <img 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> <!--l. 4178--><p class="nopar"> </p> <div class="center" > <!--l. 4182--><p class="noindent"> </p><!--l. 4183--><p class="noindent"><span class="cmbsy-10x-x-120">§ </span><span class="cmbx-12">22. Behaviour of Rods and Clocks in the Static Gravi- </span> <br/><span class="cmbx-12">tational</span> <span class="cmbx-12">Field. Bending of Light-rays. Motion of </span> <br/><span class="cmbx-12">the Perihelion of a</span> <span class="cmbx-12">Planetary Orbit</span></p></div> <!--l. 4189--><p class="indent"> To arrive at Newton’s theory as a first approximation we <br/>had to calculate only one component, <span class="cmmi-12">g</span><sub ><span class="cmr-8">44</span></sub>, of the ten <span class="cmmi-12">g</span><sub ><span class="cmmi-8"><img src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img 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 class="cmmi-12">g</span><sub ><span class="cmmi-8"><img src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img 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 <span class="cmmi-12">g </span>= <span class="cmsy-10x-x-120">- </span>1<span class="cmmi-12">.</span> </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 </p> <table width="100%" class="equation"><tr><td><a id="x1-94r70"></a> <center class="math-display" > <img 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> <!--l. 4213--><p class="nopar"> </p><!--l. 4217--><p class="noindent">where <span class="cmmi-12"><img src="img/cmmi12-e.png" alt="d" class="12x-x-e" /></span><sub ><span class="cmmi-8"><img src="img/cmmi8-1a.png" alt="r" class="cmmi-8x-x-1a" align="middle" /><img src="img/cmmi8-1b.png" alt="s" class="8x-x-1b" /></span></sub> is 1 or 0, respectively, accordingly as <span class="cmmi-12"><img src="img/cmmi12-1a.png" alt="r" class="cmmi-12x-x-1a" align="middle" /> </span>= <span class="cmmi-12"><img src="img/cmmi12-1b.png" alt="s" class="12x-x-1b" /> </span>or <span class="cmmi-12"><img src="img/cmmi12-1a.png" alt="r" class="cmmi-12x-x-1a" align="middle" /></span><img src="img/078_A_1916214x.png" alt="/=" class="neq" align="middle" /><span class="cmmi-12"><img src="img/cmmi12-1b.png" alt="s" class="12x-x-1b" />, </span> <br/>and <span class="cmmi-12">r </span>is the quantity +<img src="img/078_A_1916215x.png" alt=" V~ -x2-+-x2-+-x2- 1 2 3" class="sqrt" /> On account of (68a) </p> <table width="100%" class="equation"><tr><td><a id="x1-95r71"></a> <center class="math-display" > <img src="img/078_A_1916216x.png" alt=" kM a = ----, 4p " class="math-display" /></center></td><td width="5%">(70a)</td></tr></table> <!--l. 4225--><p class="nopar"> </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. </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 </p> <center class="par-math-display" > <img src="img/078_A_1916217x.png" alt="ds2 = gmndxmdxn. " class="par-math-display" /></center> <!--l. 4240--><p class="nopar"></p><!--l. 4244--><p class="noindent">always holds between the “ locally ” (<span class="cmsy-10x-x-120">§ </span>4) measured lengths <br/>and times <span class="cmti-12">ds </span>on the one hand, and the differences of co-ordin-<br/>ates <span class="cmmi-12">dx</span><sub ><span class="cmmi-8"><img src="img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub> on the other hand. <pb/> </p><!--l. 4251--><p class="indent"> </p><!--l. 4252--><p class="indent"> For a unit-measure of length laid “ parallel ” to the axis <br/>of <span class="cmmi-12">x</span>, for example, we should have to set <span class="cmmi-12">ds</span><sup ><span class="cmr-8">2</span></sup> = <span class="cmsy-10x-x-120">-</span>1; <span class="cmmi-12">dx</span><sub > <span class="cmr-8">2</span></sub> = <span class="cmmi-12">dx</span><sub ><span class="cmr-8">3</span></sub> <br/>= <span class="cmmi-12">dx</span><sub ><span class="cmr-8">4</span></sub> = 0<span class="cmmi-12">. </span>Therefore <span class="cmsy-10x-x-120">-</span>1 = <span class="cmmi-12">g</span><sub ><span class="cmr-8">11</span></sub><span class="cmmi-12">dx</span><sub><span class="cmr-8">1</span></sub><sup><span class="cmr-8">2</span></sup><span class="cmmi-12">. </span>If, in addition, the <br/>unit-measure lies on the axis of <span class="cmmi-12">x</span>, the first of equations (70) <br/>gives </p> <center class="par-math-display" > <img src="img/078_A_1916218x.png" alt=" ( ) g = - 1 + a- . 11 r " class="par-math-display" /></center> <!--l. 4261--><p class="nopar"> </p><!--l. 4264--><p class="noindent">From these two relations it follows that, correct to a first <br/>order of small quantities, </p> <table width="100%" class="equation"><tr><td><a id="x1-96r71"></a> <center class="math-display" > <img src="img/078_A_1916219x.png" alt=" a-- dx = 1 - 2r " class="math-display" /></center></td><td width="5%">(71)</td></tr></table> <!--l. 4271--><p class="nopar"> </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. </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 </p> <center class="par-math-display" > <img 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> <!--l. 4287--><p class="nopar"> </p><!--l. 4291--><p class="noindent">The result is </p> <table width="100%" class="equation"><tr><td><a id="x1-97r72"></a> <center class="math-display" > <img src="img/078_A_1916221x.png" alt="- 1 = g22dx22 = - dx22 " class="math-display" /></center></td><td width="5%">(71a)</td></tr></table> <!--l. 4297--><p class="nopar"> </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. </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’s surface. </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 <span class="cmmi-12">ds </span>= 1; <span class="cmmi-12">dx</span><sub ><span class="cmr-8">1</span></sub> = <span class="cmmi-12">dx</span><sub ><span class="cmr-8">2</span></sub> = <span class="cmmi-12">dx</span><sub ><span class="cmr-8">3</span></sub> = 0 </p><!--l. 4320--><p class="noindent">Therefore </p> <center class="par-math-display" > <img 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> <!--l. 4331--><p class="nopar"> <pb/> </p><!--l. 4338--><p class="indent"> </p><!--l. 4339--><p class="noindent">or </p> <table width="100%" class="equation"><tr><td><a id="x1-98r72"></a> <center class="math-display" > <img 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> <!--l. 4345--><p class="nopar"> </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 <br/>spectrum.<sup ><span class="cmsy-8">*</span></sup> </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 </p> <center class="par-math-display" > <img src="img/078_A_1916224x.png" alt="- dx21 - dx2 - dx23 + dx24 = 0 " class="par-math-display" /></center> <!--l. 4362--><p class="nopar"> </p><!--l. 4366--><p class="noindent">and therefore by the general theory of relativity by the <br/>equation </p> <table width="100%" class="equation"><tr><td><a id="x1-99r73"></a> <center class="math-display" > <img src="img/078_A_1916225x.png" alt=" 2 ds = gmndxmdxn = 0 " class="math-display" /></center></td><td width="5%">(73)</td></tr></table> <!--l. 4372--><p class="nopar"> </p><!--l. 4375--><p class="noindent">If the direction, i.e. the ratio, <span class="cmmi-12">dx</span><sub ><span class="cmr-8">1</span></sub> : <span class="cmmi-12">dx</span><sub ><span class="cmr-8">2</span></sub> : <span class="cmmi-12">dx</span><sub ><span class="cmr-8">3</span></sub> is given, equation <br/>(73) gives the quantities </p> <center class="par-math-display" > <img src="img/078_A_1916226x.png" alt="dx1 dx2 dx3 dx--,dx--, dx-- 4 4 4 " class="par-math-display" /></center> <!--l. 4381--><p class="nopar"> </p><!--l. 4385--><p class="noindent">and accordingly the velocity </p> <center class="par-math-display" > <img src="img/078_A_1916227x.png" alt=" V~ (----)2---(-----)2---(----)2-- dx1- dx2- dx3- dx4 + dx4 + dx4 = g " class="par-math-display" /></center> <!--l. 4393--><p class="nopar"> </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 class="cmmi-12">g</span><sub ><span class="cmmi-8"><img src="img/cmmi8-16.png" alt="m" class="cmmi-8x-x-16" align="middle" /><img src="img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub> are not con-<br/>stant. If <span 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 class="cmmi-12"><img src="img/cmmi12-d.png" alt="g" class="12x-x-d" />, n</span>)<span class="cmmi-12">, </span>has the curvature <span class="cmsy-10x-x-120">-</span><span class="cmmi-12"><img src="img/cmmi12-40.png" alt="@" class="12x-x-40" /><img src="img/cmmi12-d.png" alt="g" class="12x-x-d" />/<img src="img/cmmi12-40.png" alt="@" class="12x-x-40" />n.</span> </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 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 <br/> </p><!--l. 4413--><p class="indent"> <sup ><span class="cmsy-8">*</span></sup> According to E. Freundlich, spectroscopical observations on fixed stars of <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/> </p><!--l. 4420--><p class="indent"> </p><!--l. 4421--><p class="noindent">concave towards the origin) is given in sufficient approxi-<br/>mation by </p> <center class="par-math-display" > <img src="img/078_A_1916228x.png" alt=" integral + oo @g B = ---- dx2, - oo @x1 " class="par-math-display" /></center> <!--l. 4428--><p class="nopar"> </p><!--l. 4432--><p class="noindent">while (73) and (70) give </p> <center class="par-math-display" > <img src="img/078_A_1916229x.png" alt=" V~ --------- ( g44 ) a ( x2) g = - --- = 1 - --- 1 + -22- . g22 2r r " class="par-math-display" /></center> <!--l. 4439--><p class="nopar"> </p><!--l. 4443--><p class="noindent">Carrying out the calculation, this gives </p> <table width="100%" class="equation"><tr><td><a id="x1-100r74"></a> <center class="math-display" > <img src="img/078_A_1916230x.png" alt=" 2a- kM--- B = /_\ = 2p /_\ . " class="math-display" /></center></td><td width="5%">(74)</td></tr></table> <!--l. 4450--><p class="nopar"> <img src="img/078_A_1916231x.png" alt="PIC" class="graphics" width="247.53888pt" height="240.4235pt" /><!--tex4ht:graphics name="img/078_A_1916231x.png" src="078_A_1916_001.EPS" --> </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 class="cmmi-12">.</span>7<span class="cmsy-10x-x-120">''</span> ; and a ray going past the planet <br/>Jupiter a deflexion of about <span class="cmmi-12">.</span>02<span class="cmsy-10x-x-120">''</span>. </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 </p> <table width="100%" class="equation"><tr><td><a id="x1-101r75"></a> <center class="math-display" > <img 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> <!--l. 4476--><p class="nopar"> <pb/> </p><!--l. 4483--><p class="indent"> </p><!--l. 4484--><p class="noindent">per revolution. In this formula <span class="cmmi-12">a </span>denotes the major semi-<br/>axis, <span class="cmmi-12">c </span>the velocity of light in the usual measurement, <span class="cmmi-12">e </span>the <br/>eccentricity, T the time of revolution in seconds.<sup ><span class="cmsy-8">*</span></sup> </p><!--l. 4489--><p class="indent"> Calculation gives for the planet Mercury a rotation of the <br/>orbit of 43<span 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. </p><!--l. 4498--><p class="indent"> <sup ><span 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 class="cmti-12">ibid.</span>, 1916, p. 189. </p> </body></html>