ON THE DERIVATION OF THE EQUATIONS OF MOTION

Abstract

General Relativity is unique, among the class of field theories, in the treatment of the equations of motion. The equations of motion of massive particles are completely determined by the field equation 1-5. Einstein's field equations, as well as most field equations in gravity theory, have a specific analytical form: They are linear in the second order derivatives and quadratic in the first order, with coefficients that depend on the variables. We propose for the N-body problem of equations that are Lorentz invariant a novel algorithm for the derivation of the equations of motion from the field equations. It is: (1) Compute a static, spherically symmetric solution of the field equation. It will be singular at the origin. This will be taken to be the field generated by a single particle. (2) Move the solution on a trajectory P and apply the instantaneous Lorentz transformation based on instantaneous velocity. (3) Take, as first approximation, the field generated by N particles to be the superposition of the fields generated by the single particles. (4) Compute the leading part of the equation. Hopefully, only terms that involve the acceleration will be dominant. This is the "inertial" part. (5) Compute the quadratic part of the equation. This is the agent of the "force". (6) Equate for each singularity, the highest order terms of the singularities that came from the linear part and the quadratic parts, respectively. This is an equation between the inertial part and the force. The algorithm was applied to Einstein's equations. The approximate evolution of the scalar curvature leads, in turn, to an invariant scalar equation. The algorithm for it did produce Newton's law of gravitation. This is, also, the starting point for the embedding of the trajectories in a common field.