Black Holes and Wave Mechanics

This is a short introductory course on wave mechanics on black hole space-times. Its main focus is on the the classical Klein-Gordon field on the Schwarzschild spacetime. As well as being of interest in its own right, the scalar field serves as a simple ‘toy model ’ which offers insight into the dynamics of gravitational radiation. As an example of a quantum-mechanical field, we also consider the Dirac equation. The course covers the scattering, absorption and emission of radiation; the self-force problem for black hole inspirals; and acoustic black holes. The course begins with a brief introduction to General Relativity. The interval, the metric, geodesics and Killing vectors are introduced. Next we define the simplest black hole solution: the Schwarzschild space time. We discuss geodesics, coordinate singularities, and the relative merits of various coordinate systems. The scalar wave equation is introduced along with its conserved current. Appropriate physical boundary conditions are discussed, and the behaviour of the field at the origin, the horizon and infinity is scrutinised. We examine the field dynamics by evolving the 1 + 1 PDEs in the time domain. We show that bombarding the black hole with Gaussian wavepackets results in the excitation o