Acceleration plays an important role in gravitational physics. When a black hole moves towards the centre of a galaxy it is slowly accelerated. Due to the galactic environment the whole situation is rather complex and thus we cannot describe this scenario in a mathematically sound way. However, in general relativity there are in fact exact solutions to Einstein's vacuum field equations that describe an accelerating black hole. In this talk I will focus on the non-spinning variant, the so-called C-metric. It describes the spacetime of a linearly accelerating black hole. While in this spacetime the acceleration is caused by rather exotic objects, namely a string and a strut, it is the best first guess for describing an accelerating black hole in space.
The most important question is now how we can detect such an object and distinguish it from other compact objects, in particular non-accelerating black holes in space. In this talk I will show how we can approach this problem using gravitational lensing. For this purpose I will first give a brief introduction of the C-metric and its mathematical peculiarities. Then I will demonstrate how to analytically solve the equations of motion using elementary and Jacobi's elliptic functions and Legendre's elliptic integrals. In the last part of my talk I will then discuss gravitational lensing in the C-metric. For this purpose I will fix a static observer in the domain of outer communication outside the photon sphere. Next I will demonstrate how we can relate the constants of motion of the light rays to latitude-longitude coordinates on the observer's celestial sphere. Finally, I will derive the shadow of the black hole, write down a lens equation, and calculate the redshift and the travel time of the light rays.