Particle paths in a Schwarzschild spacetime via the Regge calculus

As a test of the Regge calculus Williams and Ellis (1984) computed, among other quantities, the precession of the perihelia of Mercury. Unfortunately they did not obtain anywhere near the correct values. Their results varied between -2.74 and 0.42 radians per orbit. Their best result was 9.1*10 ^-4 radians per orbit whereas the correct analytic value is 5.0*10^-7 radians per orbit. A continuous-time version of their equations is presented. It is shown numerically that, for a sufficiently fine discretization, the global error between the Regge and Schwarzschild geodesics varies linearly with the typical length scale for the Regge simplices. Some simple modifications to the continuous-time equations are then presented. It is shown both numerically and analytically that the modified equations yield paths that converge quadratically to the Schwarzschild geodesics. The modified Regge equations are then applied to the problem of computing the precession of the perihelia of Mercury. The result is a precession of 5.0*10^-7 radians per orbit.