Application of the Gauss\u2013Bonnet theorem to lensing in the NUT metric

Mourad Halla,Volker Perlick

doi:10.1007/s10714-020-02766-z

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.

Highlights

The NUT metric is a solution to the vacuum Einstein equation that was found by Newman, Unti and Tamburino (NUT) in 1963 [1]
We illustrate the optical metric by way of an embedding diagram, we demonstrate that it has negative Gaussian curvature and we show that the optical metrics on cones with different opening angles are locally isometric
When Gibbons and Werner [6] applied the Gauss–Bonnet theorem to the spatial paths of lightlike geodesics in static and spherically symmetric spacetimes, many readers found this idea attractive because it related the lensing features in such spacetimes to geometric quantities such as the Gaussian curvature of a two-dimensional Riemannian metric

Summary

Introduction

The NUT metric is a solution to the vacuum Einstein equation that was found by Newman, Unti and Tamburino (NUT) in 1963 [1] It describes the vacuum spacetime around a source that is characterised by two parameters, called m and l in the following. As to the influence on light rays, i.e., to the lensing features of a NUT source, there is a paper by Nouri-Zonoz and Lynden Bell [5]. 7 we use the Gauss–Bonnet theorem to rewrite the deflection angle as an integral over the Gaussian curvature of the optical metric As the latter is negative, this result demonstrates that all light rays are deflected towards the centre and that the deflection angle increases with decreasing impact parameter. We use Einstein’s summation convention with greek indices for the four spacetime coordinates and with latin indices for the two coordinates of the two-dimensional manifold on which the optical metric lives

The NUT metric

Geodesics in the NUT metric

The deflection angle

Optical metric and Gaussian curvature

Gravitational lensing and Gauss–Bonnet theorem

Discussion and conclusions