Orbits of massless particles in the Schwarzschild metric: Exact solutions

Abstract

The problem of finding all the orbits of test particles in the exterior Schwarzschild black hole metric has an exact solution in terms of elliptic functions. In this paper, we develop in detail the case of massless particles, including a derivation of an exact formula for the deflection of light. It is shown that the mass of the black hole can be determined from a simple relationship between the angular deflection and the time delay of neighboring light rays.