The Application of Weierstrass elliptic functions to Schwarzschild Null Geodesics

Abstract

In this paper we focus on analytical calculations involving null geodesics in some spherically symmetric spacetimes. We use Weierstrass elliptic functions to fully describe null geodesics in Schwarzschild spacetime and to derive analytical formulae connecting the values of radial distance at different points along the geodesic. We then study the properties of light triangles in Schwarzschild spacetime and give the expansion of the deflection angle to the second order in both $M/r_0$ and $M/b$ where $M$ is the mass of the black hole, $r_0$ the distance of closest approach of the light ray and $b$ the impact parameter. We also use the Weierstrass function formalism to analyze other more exotic cases such as Reissner-Nordstr\om null geodesics and Schwarzschild null geodesics in 4 and 6 spatial dimensions. Finally we apply Weierstrass functions to describe the null geodesics in the Ellis wormhole spacetime and give an analytic expansion of the deflection angle in $M/b$.