This is a course on wave mechanics on black hole space-times. The main focus is on the the scalar 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.

The course consists of five one-hour lectures, outlined below.

The course begins with a brief introduction to General Relativity. Basic concepts are defined: the interval; covariance; geodesics; congruences; Killing vectors; hypersurfaces; action principles, etc. Einstein's field equations are introduced. I will demonstrate the use of the "GRTensorII" package for Maple. Next we look at the Schwarzschild space time. We examine the geodesics, and derive weak-field and strong-field approximations for the bending of light. We look at radial infall, redshift, and the stability of circular orbits. Coordinate singularities are discussed, along with the relative merits of alternative coordinate systems.

In this talk, I briefly introduce the Kerr spacetime, which describes a rotating black hole. We will look at two effects of rotation: frame-dragging and the ergosphere. We introduce the "no-hair theorem"" and the laws of classical black hole mechanics. Next, we consider small perturbations of black hole spacetimes. The scalar wave equation is introduced along with a 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 of "quasi-normal modes" (QNMs). These modes have specific frequencies and decay rates that may provide a signature for the underlying black-hole space-time. Two methods for calculating QNMs are discussed.

In the third lecture, we look at time-independent scattering. We consider a monochromatic, long-lasting planar wave impinging on a black hole. In the long-wavelength regime $\lambda \gg r_s$, a perturbative approach may be used. We show how to expand the scattering amplitude in a Born series, and compute the lowest-order amplitude in two coordinate systems. For higher couplings $\lambda \gtrsim r_s$, a partial wave approach is more appropriate. A range of interesting diffraction effects are seen in the scattered signal, including a "glory" peak. We discuss methods for solving the radial equation.

Black holes are not completely black! In fact, they emit radiation with a (nearly) thermal spectrum, and a temperature inversely proportional to black hole mass. Hawking's discovery means we may associate an entropy with the area of a black hole's horizon. It has inspired many attempts to reconcile GR and quantum mechanics. In this lecture we will look at Quantum Field Theory in curved spacetime, and I will give a heuristic explanation for the Hawking effect.

In this lecture, I will discuss the possibility that systems with some of the properties of black holes may be created in the laboratory. In 1981, Unruh showed that under certain assumptions (inviscidity; irrotationality; barotropy) the Navier-Stokes equations describing perturbations to steady fluid flow are formally equivalent to the Klein-Gordon equation on an effective metric. Further, if the fluid flows faster than the speed of sound in the fluid, then a analogue horizon may be created. I will introduce two models for acoustic holes, and discuss similarities with, and differences to, astrophysical black holes.