Strong-field gravitational lensing by black holes

In this thesis we study aspects of strong-field gravitational lensing by black holes in general relativity, with a particular focus on the role of integrability and chaos in geodesic motion. We first investigate binary black hole shadows using the Majumdar–Papapetrou static binary black hole (or di-hole) solution. It is shown that the propagation of null geodesics on this spacetime background is a natural example of chaotic scattering. We demonstrate that the binary black hole shadows exhibit a self-similar fractal structure akin to the Cantor set. Next, we use techniques from the field of non-linear dynamics to quantify these fractal structures in binary black hole shadows. Using a recently developed numerical algorithm, called the merging method, we demonstrate that parts of the Majumdar–Papapetrou di-hole shadow may possess the Wada property. We then study the existence, stability and phenomenology of circular photon orbits in stationary axisymmetric four-dimensional spacetimes. We employ a Hamiltonian formalism to describe the null geodesics of the Weyl–Lewis–Papapetrou geometry. Using the Einstein–Maxwell equations, we demonstrate that generic stable photon orbits are forbidden in pure vacuum, but may arise in electrovacuum. Finally, we apply a higher-order geometric optics formalism to describe the propagation of electromagnetic waves on Kerr spacetime. Using the symmetries of Kerr spacetime, we construct a complex null tetrad which is parallel-propagated along null geodesics; we introduce a system of transport equations to calculate certain Newman–Penrose quantities along rays; we derive generalised power series solutions to these transport equations through sub-leading order in the neighbourhood of caustic points; and we introduce a practical method to evolve the transport equations beyond caustic points.