The Application of Weierstrass elliptic functions to Schwarzschild Null Geodesics

TL;DR

This paper employs Weierstrass elliptic functions to analytically describe null geodesics in Schwarzschild and other spacetimes, deriving formulas for light deflection and exploring exotic geometries like wormholes.

Contribution

It introduces a novel application of Weierstrass elliptic functions to derive explicit analytical solutions for null geodesics in various spherically symmetric spacetimes, including higher dimensions and wormholes.

Findings

Derived analytical formulas for null geodesics in Schwarzschild spacetime.

Expanded the deflection angle to second order in key parameters.

Extended the formalism to exotic spacetimes like Reissner-Nordström and Ellis wormholes.

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 $M/b$.