Lightlike Geodesics and Gravitational Lensing in the Spacetime of an Accelerating Black Hole

Abstract

The C-metric is a solution to Einstein's vacuum field equation that describes an accelerating black hole. In this paper we discuss the propagation of light rays and the resulting lensing features in this metric. We first solve the lightlike geodesic equation using elliptic integrals and Jacobi elliptic functions. Then we fix a static observer in the region of outer communication of the C-metric and introduce an orthonormal tetrad to parameterise the directions of the light rays ending at the position of the observer using latitude-longitude coordinates on the observer's celestial sphere. In this parameterisation we rederive the angular radius of the shadow, we formulate a lens equation, and we derive the redshift and the travel time of light rays. We discuss the relevance of our theoretical results for detecting accelerating black holes described by the C-metric and for distinguishing them from non-accelerating black holes.