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

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.