# Equivalence of Gibbons-Werner method to geodesics method in the study of gravitational lensing

## TL;DR

This paper demonstrates that the Gibbons-Werner method, which uses the Gauss-Bonnet theorem, is equivalent to the traditional geodesics method for calculating gravitational lensing, confirmed through examples in Kerr-Newman spacetime.

## Contribution

It proves the equivalence of the Gibbons-Werner and geodesics methods for asymptotically flat spacetimes, providing a unified understanding of gravitational deflection calculations.

## Key findings

- Gibbons-Werner method can derive the geodesics method.
- The gravitational deflection angle depends on geodesic curvature.
- Equivalence shown through Kerr-Newman spacetime example.

## Abstract

The Gibbons-Werner method where the Gauss-Bonnet theorem is applied to study the gravitational deflection angle has received much attention recently. In this paper, we study the equivalence of the Gibbons-Werner method to the standard geodesics method, and it is shown that the geodesics method can be derived with the Gibbons-Werner method, for asymptotically flat case. In the geodesics method, the gravitational deflection angle of particle depends entirely on the geodesic curvature of the particle ray in the Euclidean space. The gravitational deflection of light in Kerr-Newman spacetime is calculated by different technologies under the Gibbons-Werner framework, as an intuitive example to show the equivalence.