Lorentz-Invariant Equations of Motion of Point Masses in the General Theory of Relativity

Abstract

After a general discussion of the problem of motion in the general theory of relativity a simple derivation of the law of motion is given for single poles of the gravitational field, which is based on a method originally developed by Mathisson. This law follows from the covariant conservation law for the matter energy-momentum tensor alone, without reference to any field equations, and takes the form of a geodesic of the (unknown) metric. Expanding this metric in terms of a power series in a parameter $\ensuremath{\lambda}$ and using the Minkowski proper time to parametrize the world lines of the particles, the (Lorentz-invariant) form of the approximate laws of motion follows. A method is developed to obtain the equations of motion (including the explicit form of the metric in terms of the particle variables) from Einstein's field equations. A systematic linearization procedure leads to a series of second-order differential equations for the metric; the $n\mathrm{th}$ order approximation of the equations of motion, as well as the explicit form of the matter tensor in $(n+1)\mathrm{st}$ order, is obtained as an integrability condition on the $(n+1)\mathrm{st}$ order approximation for the metric. No coordinate conditions are required to obtain the general form of the equations of motion; they are needed only to reduce the approximation equations to wave equations and thus to allow their explicit integration in terms of retarded or symmetric potentials. In developing the approximation method it is shown that consistency requires that any set of approximate equations is solved "up to" rather than "in" $n\mathrm{th}$ order; this implies that the form of the lower order metric be maintained, but with the motion corresponding to the $n\mathrm{th}$ order solutions rather than to lower order ones. In particular, the equations for the first-order metric imply zero-order equations of motion which restrict the particles to zero acceleration; the equations for the second-order metric imply first-order equation of motion involving the first-order metric, but without the previous restriction. In the retarded case the equations of motion contain retarded interactions and radiation reaction terms of the form familiar from electrodynamics; no such terms appear in the symmetric case. The equations of the symmetric case are derivable from a Fokker-type variational principle. The relation of the results obtained to work on Lorentz-invariant equations by other authors is discussed. In Appendix I a discussion of alternative derivations is presented; Appendix II contains remarks on Wheeler-Feynman type considerations for general relativistic equations of motion.