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Equations of Motion in General Relativity

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Equations of Motion in General Relativity Gravitation and Cosmology Lecture 27: Equations of motion in general relativity Equations of motion in general relativity Reading: Adler, Bazin & Schiffer, Introduction to General Relativity, Chapter 10. How many equations? We recall that in the theory of stellar structure, the equations for the gravitational field were -B’’ B’ æB’ A’ö 1 B’ Rtt = + ç + ÷ - = -4pG (3p + r) B (25.7t) 2A 4A è B A ø r A -B’’ B’ æB’ A’ö 1 A’ Rrr = + ç + ÷ + = 4pG (r - p) A (25.7r) 2B 4B è B A ø r A r æB’ A’ö 1 2 Rqq = -1 + ç - ÷ + = -4pG (r - p) r (25.7q) 2A è B A ø A 2 Rjj = sin q Rqq (25.7j) We also had an equation of hydrodynamic equilibrium dp B’ + (p + r) = 0 . (25.12) dr 2B and an equation of state ær(r)ö p(r) = rc Fç ÷ . è rc ø Now, Eq. 25.7j is redundant with Eq. 25.7q because of rotational invariance. But it might seem as though we had 4 unknowns (A(r), B(r), r(r) and p(r)) and 5 equations to determine them. Are these unknowns overdetermined? Is it possible to solve them at all? Clearly, the equation of hydrodynamic equilibrium is an identity that follows from mn 1 mn mn æR - g Rö = T = 0 (27.1) è 2 ø ; n ; n and so is not independent from it. Einstein was the first to remark on the rather striking fact that the equations of motion describing the motion of particles under gravitational forces follow from the field equations----rather than being distinct from them. The subject was first elaborated by Einstein, Infeld and Hofmann† and then simplified and extended by Einstein and Infeld‡. In this lecture I can only give you some of the flavor of the theory, rather than the whole subject. † Einstein, Infeld and Hofmann, Ann. of Math. 39 (1938) 66. 127 Gravitation and Cosmology Motion of particles Motion of particles mn The basic equation is T ; n = 0 . Isolate a piece of matter in a world-tube D: Clearly, mn mn ì m ü sn Ö`g T ; n = ¶n (Ö`g T ) + Ö`g í ý T = 0 (27.2) îs nþ Thus, 4 mn 4 ì m ü sn d x ¶n (`Ög T ) + d x Ö`g í ý T = 0 (27.3) ò ò s n D D î þ mn Now, since the matter is inside D, T is zero on S2 , hence Diagram of the world-tube D 4 mn 4 mn 3 mn ò d x ¶n (`Ög T ) = ò d x ¶n (Ö`g T ) = ò d x `Ög T nn (27.4) D S1 + S3 S1 + S3 where on S the unit normal is n = æ-1, 0, 0, 0ö and on S it is n = (1, 0, 0, 0) . Now, suppose we 1 n è ø 3 n employ locally freely-falling coordinates to describe the barycenter of the particles, for which dt Ö`g = 1 . (27.5) dt Then if we ignore pressure (that is, Tmn refers to isolated, non-interacting particles----dust) Tmn = r Um Un (27.6) and ï dt ï ï dtï d3x Ö`g Tmn n = d3x r Um - d3x r Um ò n ò ïdtï ò ïdtï ï ït=b ï ït=a S1 + S3 t=b d2xm º t ò d m 2 (27.7) t=a dt df where m = ò d3x r . ‡ Einstein and Infeld, Can. J. Math. 1 (1949) 209. 128 Gravitation and Cosmology Lecture 27: Equations of motion in general relativity Similarly we use Eq. 27.5 to write t=b s n 4 ì m ü sn 3 ì m ü s n ì m ü dx dx ò d x Ö`g í ý T = ò d x dt í ý r U U » ò dt í ý m (27.8) s n s n t=a s n dt dt D î þ D î þ î þ so that t=b éd2xm ì m ü dxs dxnù ò dt m ê 2 + í ý ú = 0 . (27.9) t=a ë dt îs nþ dt dt û That is, we recover the equation of motion for a test particle in a gravitational field, d2xm ì m ü dxs dxn 2 + í ý = 0 dt îs nþ dt dt since the range of integration over t may be as restricted as much as we like. We have therefore shown that the conservation of the energy-momentum tensor in the presence of a gravitational field (as represented by the geometry of space-time), which follows from the Bianchi identity mn 1 mn æR - g Rö = 0 , è 2 ø ; n is equivalent to Newton’s Second Law with gravitation treated as an external force. It is this possibility that has led J.A. Wheeler to coin the term ‘‘geometrodynamics’’----and that led Einstein in the last years of his life to seek a geometrical description of all of physics. 129 Gravitation and Cosmology Motion of particles 130.
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