Gravitational lensing in Brill spacetimes

Abstract

We consider the Brill metric which is an electrovacuum solution to Einstein's field equation. It depends on three parameters, a mass parameter $m$, a NUT parameter $l$ and a charge parameter $e$. If the charge parameter is small, the metric describes a black hole; if it is sufficiently big, it describes a wormhole. We determine the relevant lensing features both in the black-hole and in the wormhole case. In particular, we give formulas for the photon spheres, for the angular radius of the shadow and for the deflection angle. We illustrate the lensing features with the help of an effective potential and in terms of embedding diagrams. To that end we make use of the fact that each lightlike geodesic is contained in a (coordinate) cone and that it is a geodesic of a Riemannian optical metric on this cone. By the Gauss-Bonnet theorem, the sign of the Gaussian curvature of the optical metric determines the sign of the deflection angle. In the wormhole case the deflection angle may be negative which means that light rays are repelled from the center.