Gravitational bending angle of light for finite distance and the Gauss-Bonnet theorem

TL;DR

This paper extends the calculation of light bending angles to non-asymptotically flat spacetimes using the Gauss-Bonnet theorem, accounting for finite distances in gravitational lensing.

Contribution

It introduces a method linking the bending angle to the surface integral of Gaussian curvature, enabling finite-distance corrections in non-flat spacetimes.

Findings

Finite-distance corrections are significant in Kottler and Weyl conformal gravity spacetimes.

The proposed method accurately computes bending angles for non-asymptotically flat cases.

Application demonstrates the method's effectiveness in specific gravitational models.

Abstract

We discuss a possible extension of calculations of the bending angle of light in a static, spherically symmetric and asymptotically flat spacetime to a non-asymptotically flat case. We examine a relation between the bending angle of light and the Gauss-Bonnet theorem by using the optical metric. A correspondence between the deflection angle of light and the surface integral of the Gaussian curvature may allow us to take account of the finite distance from a lens object to a light source and a receiver. Using this relation, we propose a method for calculating the bending angle of light for such cases. Finally, this method is applied to two examples of the non-asymptotically flat spacetimes to suggest finite-distance corrections: Kottler (Schwarzschild-de Sitter) solution to the Einstein equation and an exact solution in Weyl conformal gravity.