Application of the Gauss-Bonnet theorem to lensing in the NUT metric

TL;DR

This paper applies the Gauss-Bonnet theorem to analyze light deflection in the NUT spacetime by projecting lightlike geodesics onto a two-dimensional optical metric, revealing new insights into gravitational lensing in non-spherical geometries.

Contribution

It extends the use of the Gauss-Bonnet theorem to the NUT metric, a non-static, non-spherically symmetric spacetime, demonstrating how light deflection can be understood through optical geometry.

Findings

Deflection angle is related to an area integral of Gaussian curvature.

Optical metrics on cones with different opening angles are locally isometric.

The approach generalizes known results from static, spherically symmetric spacetimes.

Abstract

We show with the help of Fermat's principle that every lightlike geodesic in the NUT metric projects to a geodesic of a two-dimensional Riemannian metric which we call the optical metric. The optical metric is defined on a (coordinate) cone whose opening angle is determined by the impact parameter of the lightlike geodesic. We show that, surprisingly, the optical metrics on cones with different opening angles are locally (but not globally) isometric. With the help of the Gauss-Bonnet theorem we demonstrate that the deflection angle of a lightlike geodesic is determined by an area integral over the Gaussian curvature of the optical metric. A similar result is known to be true for static and spherically symmetric spacetimes. The generalisation to the NUT spacetime, which is neither static nor spherically symmetric (at least not in the usual sense), is rather non-trivial.