Classical mechanics of special relativity in a Riemannian space-time

Abstract

It is shown that the two-body problem with action-at-a-distance interaction potential in the framework of a canonical manifestly covariant mechanics, for which there exists a consistent and well-developed quantum version, can be embedded in a Riemannian manifold (with conformal metric) where the curvature replaces the explicit appearance of the potential. The geodesic motion in the conformal space coincides with the solutions of the Hamilton equations of motion in the corresponding flat space problem. The reparametrization invariant form of the action principle which leads to the geodesic equation is shown to be equivalent to the underlying canonical mechanics with a particular choice of parameter. This program realizes the possibility of describing a class of nongravitational interactions in space-time by essentially geometrical means. A relation is proposed between the Einstein tensor in the conformal space and the (nonconserved) energy momentum tensor associated with relative motion, with the help of a Brans–Dicke type field. The structure of the result suggests a generalization of the principle of equivalence, for dealing, at least, with the two-body problem, which treats a manifold of relative space-time as the globally defined coordinates for which there exists, at every point, a local frame in which the metric is conformal (in the absence of nongravitational forces, this procedure reduces to the usual one). In this way, one can treat gravitational and a certain class of nongravitational forces in a unified way.