Compact calculation of the Perihelion Precession of Mercury in General Relativity, the Cosmological Constant and Jacobi's Inversion problem

Abstract

The geodesic equations resulting from the Schwarzschild gravitational metric element are solved exactly including the contribution from the Cosmological constant. The exact solution is given by genus 2 Siegelsche modular forms. For zero cosmological constant the hyperelliptic curve degenerates into an elliptic curve and the resulting geodesic is solved by the Weierstra$\ss$ Jacobi modular form. The solution is applied to the precise calculation of the perihelion precession of the orbit of planet Mercury around the Sun.