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On the Motion of Test Particles in General Relativity

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On the Motion of Test Particles in General Relativity 408 L. I NFELD AND A. SCH ILD finds for the coordinates before, a series of inertia planes U', V', 8" ., each perpendicular to its companion, every pair defined by t= tp(n'Chs —P') cosy nshs+r sinpp nP(Chs —1) +r a parameter s and effectuating an acceleration or x=tp nshs cospp Chs sinpp PShs +r +r rotation dP=f(s)ds and dx=g(s)ds respectively. The y=tp nP(1 C—hs) r—cospp PShs+r sinip (n' —P'Chs) succession of elements U(s) and U'(s) would then Spa generate a collection of time tracks or space curves One verifies, that t' —x' —y' —s'=tp' —r' —sp' and that permitting to locate, for every s, a collection of points for P=O the formulae reduce to those of the rectilinear preserving rigid distances. A few examples have been hyperbolic acceleration found above. indicated by Herglotz. ' Putting tp —0 one finds the time tracks of a rigid One may take a constant separation plane U(= OI'Z) body occupying the instantaneous present space at the and a constant acceleration plane (OXT) and make the origin of time. This motion has not been described by uniform revolutions defined by P=o.u and y= ~u. One Herglotz. will have the superposition of a rotation and an acceler- ation along the axis of rotation Another conception of a rigid body has been put t = x pshaw, x= xpChg, =r cosy, s= r sinx, P/n = x/~ = u forward in early discussions, in which it was defined as y a collection of points, not time tracks, preserving This kind of motions has been called by Herglotz the constant distances among them. The vectors connecting loxodrorrti c group. pairs of points which show the constant distance Again take a constant separation plane U(=OVZ) according to this conception need not be normal to the and a translation in it, the latter (Xf) being the limit time tracks showing the motions of the points. of a rotation about an acceleration plane U' at an Any displacement of such a rigid body of the second infinite distance. The resulting group of motions is kind is an example of a complex revolution in (1+3)- defined by dimensional time-space, and in general the axis of such ' t= xpshg, x= xpChg, y= yp+Xlt' s= sp a complex revolution consists of a pair of planes, a 2-dimensional plane in space, and a (1+1)-dimensional and has been called by Herglotz the hyperbolic group plane in time-space (a separation plane and an inertia Finally, one may take a stationary duration in time plane, to speak with Robb's terminology') which are as the limit of an acceleration, and put it together mutually perpendicular. A continued revolution about with a circular displacement, y=cot. This leads to a separation plane means an accelerated motion. A x= xp y= r cos~t, s= 'f sin~t. These circular mo- revolution about an inertia plane is an instantaneous tions were called by Herglotz the elliptic group. Obvi- cyclic circular displacement. ously for res~1 the velocity in the time tracks tends to Now it is obvious that one can conceive, as com- the velocity of light. For larger r one has no longer panions of the separation planes U, V, W considered time tracks, but spatial curves, and so the particles of this rotating rigid of the second kind cannot extend 4A. Robb, A Theory of Time and Space (1914); The Absolute . Relations of Time and Space (Cambridge University Press, 1941). beyond r=+ '. REVIEWS OF MODERN PH YSI CS VOLUM E 21, NUM B ER 3 J UL Y, 1949 On txe .V. .'otion oI': .. es1. .. ari:ic..es in Genera. .Xe..a1:ivity L. INPKLD AND A. SCHILD* Urtinersity of Torolto, Cartada l. INTRODUCTION ' gravitation as well as in the Maxwell-Lorentz theory 'N this paper we give a simple derivation of the of electromagnetism the physical laws fall naturally - - geodesic motion of test particles from Einstein s into two independent classes. The first class consists of gravitational equations for empty space. The history the partial differential equations which (with suitable of this problem is connected with the development of boundary conditions at infinity) determine the 6eld in basic physical concepts. terms of the distribution and motion of the matter Classical physics is dominated by a characteristic which "generates" it. The second class consists of the duality of field and matter. In Newton's theory of dynamical equations governing the motion of matter under the forces "exerted" the field. *Now at Carnegie lnstitgte of Technology, Pittsburgh, by The complete Peg n.a, independence of the dynamical laws from the field MOT ION OF TEST PA RT I CLES 409 equations is a direct consequence of the linearity of the field equations. Early in the development of Einstein's general theory. of relativity, it was suspected that the non-linear character of the gravitational equations made unneces- sary the separate assumption of a dynamical law, such as the geodesic postulate for test particles. This im- portant problem was approached by two distinct lines different of matter. of attack, using concepts I'io. 1. One line of attack is based on the existence of an energy momentum tensor and on the equation Rp. 2gp.R—Tpv- (1.01) Here matter is represented by local concentrations of the energy momentum tensor T„„.A compact particle, whose macroscopic interaction with other particles is purely gravitational (i.e., an electrically neutral parti- cle), must be pictured as a narrow world tube of time- like direction, T„, being essentially diferent from zero in the interior of the tube and zero in the exterior. momentum tensor and which is based on a particle A limiting process may be considered in which the world tube shrinks to a world line while at the same picture of matter. Matter is regarded as consisting of each is a singu- time an integral, involving T„„and representing the point particles; particle represented by of the field time-like world muss of the particle, tends to zero. In this limit a world . larity gravitational along a of line is obtained, embedded in a continuous gravitational line. The advantage this model is twofold. Firstly, held; this world line represents a test particle. By virtue the gravitational field, satisfying the covariant of (1.01) and the Bianchi identities, Rpp 2gppR Op (1.02) divergence of T„„must be zero everywhere. It is a simple consequence of these conservation equations that is alone sufFicient to represent neutral matter. Secondly, we know convergence difhculties) that solu- the worid lime Of a test particle, obtained by the limiting (ignoring process indicated above, is not arbitrary but must be u tions of (1.02) do exist which represent systems of geodesic of the continuous metric field obtained by the particles. This was shown by Einstein, Infeld, and same limiting process. This important argument was Hoffmann' who actually obtained such solutions by an developed by %eyl, ' Eddington, ' Robertson' and others. approximation method based on the idea of quasi- The idea of representing matter by a continuous stationary fields. In the same papers it was shown that energy momentum tensor goes back, in part, to Mie's4 the motion of the particle singularities is completely work on field theory. In its post-relativistic form, determined by the gravitational field equations (1.02) Mie's program demands the existence of continuous of empty space. non-gravitational fields which can represent the ele- The restricted problem of the geodesic motion of a mentary particles of nature; T„„must then be regarded test particle was attacked' long before the more general as an explicit function of these new fields. However, m-body problem was solved. It might appear that, this procedure is unsatisfactory on two counts. Firstly, since the complicated problem of e particles is now there is at present no consistent theory which represents- solved, the geodesic motion of a test particle should matter by fields such that uQ the 6eld variables are follow as a simple corollary. However, this is not the free of singularities. Secondly, the division of fields case, as was drawn to our attention by J. A. Wheeler. into gravitational components and into components- The reason is as follows: Einstein, Infeld and Ho8manns with non-gravitational sources is artificial. Ke should use a new approximation procedure, which is well suited either have a unitary theory of all fields, or else, to the case of slowly varying fields, but which is restricting ourselves to gravitational phenomena only, inadequate for our present problem: the motion of a consider the gravitational equations without any energy particle of small mass in an arbitrarily strong external momentum tensor. 6eld. %e shall use a diferent approximation method The preceding remarks lead us to an altern'ative proceeding by powers of the mass of the particle. procedure which avoids the introduction of an energy Let us now ask what exactly we understand by the H. Weyl, Xaum-Zeit-Materie (Verlag. Julius Springer, Berlin, ~ A. Einstein, L. Infeld, and B. Hoffmann, Ann. Math. 39, 65 1923), fifth edition, $38. (193g); A. Einstein and L. Infeld, Ann. Math. 41, 455 (1940); a ' A. S. Eddington, The Mathematical Theory of Relativity new and improved treatment of the theory will appear shortly in (Cambridge University Press, London, 1923), $56. the Canadian J. Math, (1, No. 3). ' H. P. Robertson, Proc. Edinb. Math. Soc. 2, 5, 63 (1936). 'A. Einstein and J. Grommer, Sitz Preuss. Akad. Wiss, 1 ' Q.
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