THE EQUIVALENCE PRINCIPLE IN KALUZA–KLEIN GRAVITY

Abstract

In four-dimensional general relativity the space–time outside of an isolated spherical star is described by a unique line element, which is the Schwarzschild metric. As a consequence, the "gravitational" mass and the "inertial" mass of a star are equal to each other. However, theories that envision our world as being embedded in a larger universe, with more than four dimensions, permit a number of possible non-Schwarzschild 4D exteriors, which typically lead to different masses, violating the weak equivalence principle of ordinary general relativity. Therefore, the question arises as to whether the violation of this principle, i.e. the equality of gravitational and inertial mass, is a necessary consequence of the existence of extra dimensions. In this paper, in the context of Kaluza–Klein gravity in 5D, we show that the answer to this question is negative. We find a one-parameter family of asymptotically flat non-Schwarzschild static exteriors for which the inertial and gravitational masses are equal to each other, and equal to the Deser–Soldate mass. This family is consistent with the Newtonian weak field limit as well as with the general-relativistic Schwarzschild limit. Thus, we conclude that the existence of an extra dimension, and the corresponding non-Schwarzschild exterior, does not necessarily require different masses. However, to an observer in 4D, it does affect the motion of test particles in 4D, which is a consequence of the departure from the usual (4D) law of geodesic motion.