Gravitational lensing using Werner’s method in Cartesian-like coordinates

The Gibbons-Werner method, which employs the Gauss-Bonnet theorem and the optical/Jacobi metric to compute deflection angles in the weak-field regime, has become widely popular in recent years. Werner extended this method to stationary spacetimes, where the optical/Jacobi metric takes the form of a Finsler metric of Randers type, by adopting an osculating Riemannian metric. Werner's method is particularly valuable as it provides a concise expression for the deflection angle, holds potential for application to gravitational lensing in Finsler spaces beyond the Randers type, and may stimulate the broader application of Finsler geometry across diverse fields. However, it has not been widely adopted due to the cumbersome calculations involved in the conventional coordinates $(r,\ensuremath{\phi})$. This paper aims to reduce the computational burden of Werner's method, thereby making it a more practical and accessible approach. To this end, we introduce Cartesian-like coordinates $(X,Y)$ to construct the osculating Riemannian metric and calculate the deflection angle using the Gauss-Bonnet theorem. We demonstrate the effectiveness of our method by computing the deflection of massive particles in Kerr spacetime, rotating Bardeen (Hayward) regular spacetime and Teo rotating wormhole spacetime. Future research aims to extend Werner's method to the study of gravitational lensing in Finsler spaces beyond the Randers type. Although this paper focuses on Randers metrics, it provides a foundation for such an extension.