Gravitational lensing in stationary and axisymmetric black hole spacetimes: acceleration and gravitomagnetic charge

In this thesis we investigate gravitational lensing in two different families of black hole spacetimes that are exact solutions of Einstein’s electrovacuum field equations with a cosmological constant. The first family of spacetimes are the charged C-de Sitter metrics. They are axisymmetric and static. The charged C-de Sitter metric is usually interpreted as an accelerating black hole with an electric charge and a cosmological constant. The second family of spacetimes are the charged NUT-de Sitter metrics. They are axisymmetric and stationary. The charged NUT-de Sitter metric is usually interpreted as a black hole with electric charge, cosmological constant and a gravitomagnetic charge. Both families of spacetimes are generalisations of the Reissner-Nordström-de Sitter metrics, however, they are of rather exotic nature. In Boyer-Lindquist-like coordinates both spacetimes contain conical singularities on the axes and, in the case of the charged NUT-de Sitter metrics, closed timelike curves, which violate causality. Due to these mathematical peculiarities the physical relevance of both families of spacetimes is rather unclear. Although we can be rather certain that conical singularities and closed timelike curves are unlikely to exist in nature both spacetimes may still serve as first highly mathematically idealised approximations for the spacetimes of a linearly accelerating black hole or a black hole with gravitomagnetic charge.
In this thesis we will address the question how we can use gravitational lensing for identifying if an astrophysical black hole can be described by one of the charged C-de Sitter metrics or one of the charged NUT-de Sitter metrics. We will first analytically solve the equations of motion using elementary as well as Jacobi’s elliptic functions and Legendre’s elliptic integrals. Then we will rederive the angular radius of the shadow, formulate an exact analytic lens equation and derive the redshift and the travel time of the light rays.