Hilbert's forgotten equation, the equivalence principle and velocity dependence of free fall

TL;DR

This paper explores how the acceleration of free-falling objects in general relativity depends on velocity, revealing a historical equation by Hilbert and discussing coordinate transformations that remove this dependence, with suggestions for experimental tests.

Contribution

It uncovers a lesser-known relativistic result from 1917 regarding velocity-dependent acceleration and analyzes how coordinate choices affect this dependence in general relativity.

Findings

Relativistic correction to acceleration differs by a factor of two from flat space estimates.

Velocity dependence can be eliminated through appropriate coordinate transformations.

Proposes experiments to test velocity dependence in gravitational free fall.

Abstract

Referring to the behavior of accelerating objects in special relativity, and applying the principle of equivalence, one expects that the coordinate acceleration of point masses under gravity will be velocity dependent. Then, using the Schwarzschild solution, we analyze the similar case of masses moving on timelike geodesics, which reproduces a little known result by Hilbert from 1917, describing this dependence. We find that the relativistic correction term for the acceleration based on general relativity differs by a factor of two from the simpler acceleration arguments in flat space. As we might expect from the general theory, the velocity dependence can be removed by a suitable coordinate transformation, such as the Painlev{\'e}-Gullstrand coordinate system. The validity of this approach is supported by previous authors who have demonstrated vacuum solutions to general relativity producing true flat space metrics for uniform gravitational fields. We suggest explicit experiments could be undertaken to test the property of velocity dependence.