The equivalence principle in Kaluza-Klein gravity

TL;DR

This paper demonstrates that in 5D Kaluza-Klein gravity, it is possible to have non-Schwarzschild exteriors where inertial and gravitational masses are equal, challenging the notion that extra dimensions necessarily violate the equivalence principle.

Contribution

The authors construct a family of asymptotically flat, non-Schwarzschild solutions in 5D that preserve the equivalence of inertial and gravitational masses, extending understanding of extra-dimensional effects.

Findings

Existence of non-Schwarzschild exteriors with equal masses in 5D

These solutions are consistent with Newtonian and Schwarzschild limits

Extra dimensions can preserve the equivalence principle under certain conditions

Abstract

In four-dimensional general relativity the spacetime 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 embedded in a larger universe, with more than four dimensions, permit a number of possible non-Schwarzschild 4D exteriors, which typically lead to {\it different} masses, violating the weak equivalence principle of ordinary general relativity. Therefore, the question arises of 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.