Applications of the Gauss-Bonnet theorem to gravitational lensing

TL;DR

This paper introduces a novel geometrical approach to gravitational lensing using the Gauss-Bonnet theorem, revealing light focusing as a topological effect and providing a new method to compute deflection angles.

Contribution

It applies the Gauss-Bonnet theorem to optical metrics in gravitational lensing, offering a new topological perspective and calculation method for deflection angles in various lens models.

Findings

Light focusing is a topological effect in gravitational lensing.

A new method to calculate deflection angles from Gaussian curvature.

Application to Schwarzschild, Plummer, and isothermal sphere lenses.

Abstract

In this geometrical approach to gravitational lensing theory, we apply the Gauss-Bonnet theorem to the optical metric of a lens, modelled as a static, spherically symmetric, perfect non-relativistic fluid, in the weak deflection limit. We find that the focusing of the light rays emerges here as a topological effect, and we introduce a new method to calculate the deflection angle from the Gaussian curvature of the optical metric. As examples, the Schwarzschild lens, the Plummer sphere and the singular isothermal sphere are discussed within this framework.