The Force of Gravity in Schwarzschild and Gullstrand-painlevé Coordinates * 1 Gullstrand-painlevé Coordinates

We derive the exact equations of motion (in Newtonian, F = ma, form) for test masses in Schwarzschild and Gullstrand-Painlevé 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. 1 Gullstrand-Painlevé Coordinates In general relativity, the Schwarzschild solution for a spherically symmetric (non rotating) black hole has been known since 1915. The usual choice of coordinates is the one Karl Schwarzschild used in its discovery, Schwarzschild coordinates, which are characterized as keeping the metric diagonal, but have a coordinate singularity at r = 2M : (dτ) = (1− 2M/r) dt − dr/(1− 2M/r)− r(dθ + sin(θ) dφ), (1) where we have chosen coordinates with G = c = 1. Note that if we multiply r and t by M , the metric will end up with an overall multiple of M which we can cancel. For convenience, we will do this both with the Schwarzschild and GP coordinates. The reader can reinsert M by making the reverse substitution. Gullstrand-Painlevé (GP) coordinates were discovered by Allvar Gullstrand [1] and Paul Painlevé [2] in 1921/1922: dτ = (1− 2M/r)dt − 2 √ 2M/r dt dr − dr − r(dθ + sin(θ) dφ). (2) This essay received an “honorable mention” in the 2009 Essay Competition of the Gravity Research Foundation. Gullstrand had a primary role in denying Einstein a Nobel prize for relativity.