The force of gravity in Schwarzschild and Gullstrand-Painlev\'e coordinates

TL;DR

This paper derives simplified equations of motion for test masses in Schwarzschild and Gullstrand-Painlevé coordinates, interpreting gravity as a flux of superluminal particles and analyzing relativistic corrections.

Contribution

It provides exact Newtonian-form equations of motion in these coordinates, eliminating the affine parameter, and offers a novel interpretation of gravity as a graviton flux with relativistic corrections.

Findings

Simplified equations of motion in specific coordinates

Gravity interpreted as superluminal graviton flux

First-order relativistic correction from two graviton interaction

Abstract

We derive the exact equations of motion (in Newtonian, F=ma, form) for test masses in Schwarzschild and Gullstrand-Painlev\'e coordinates. These equations of motion are simpler than the usual geodesic equations obtained from Christoffel tensors in that the affine parameter is eliminated. The various terms can be compared against tests of gravity. In force form, gravity can be interpreted as resulting from a flux of superluminal particles (gravitons). We show that the first order relativistic correction to Newton's gravity results from a two graviton interaction.