Gravitational Lensing Using Werner's Method in Cartesian-like Coordinates

The Gibbons-Werner method for calculating deflection angles using the Gauss-Bonnet theorem and optical/Jacobi metric 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 significant as it provides a concise expression for the deflection angle, retains applicability for gravitational lensing in Finsler geometry beyond the Randers type, and has the potential to stimulate widespread application of Finsler geometry across diverse fields. However, because of the cumbersome calculations required in Werner's method using conventional coordinates $(r,\phi)$, it has not been widely adopted. The aim of this paper is to alleviate the computational burden associated with Werner's method. 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 illustrate the current method with examples of the deflection of massive particles in Kerr spacetime, rotating Bardeen (Hayward) regular spacetime, and Teo rotating wormhole spacetime, respectively.