Gravitational lensing in the charged NUT–de Sitter spacetime

Abstract

It is a long standing open question if a gravitomagnetic charge, the gravitational analogon to a hypothetical magnetic charge in electrodynamics, exists in nature. It naturally occurs in certain exact solutions to Einstein's electrovacuum field equations with cosmological constant. The charged NUT-de Sitter metric is such a solution. It describes a black hole with electric and gravitomagnetic charges and a cosmological constant. In this paper we will address the question how we can observe the gravitomagnetic charge using gravitational lensing. For this purpose we first solve the equations of motion for lightlike geodesics using Legendre's canonical forms of the elliptic integrals and Jacobi's elliptic functions. We fix a stationary observer in the domain of outer communication and introduce an orthonormal tetrad. The orthonormal tetrad relates the direction under which the observer detects a light ray to its latitude-longitude coordinates on the observer's celestial sphere. In this parameterisation we rederive the angular radius of the shadow, formulate a lens map, discuss the redshift and the travel time. We also discuss relevant differences with respect to spherically symmetric and static spacetimes and how we can use them to determine if an astrophysical black hole has a gravitomagnetic charge.