The vacuum exterior of a spherically symmetric body is represented by the Schwarzschild space-time. If such a body collapses, it could form a black hole as described in Section 8.2. However, most astrophysically significant bodies are rotating. And, if a rotating body collapses, the rate of rotation will speed up, maintaining constant angular momentum. This is a most important process to model, the details of which are extremely complicated. Nevertheless, the end result could be expected to be a stationary rotating black hole. The field representing such a situation will be described in this chapter.

The Schwarzschild solution is the unique vacuum spherically symmetric space-time. It is also asymptotically flat, static, non-radiating (of algebraic type D), and includes an event horizon. The simplest rotating generalisation of this would be expected to be a stationary, axially symmetric and asymptotically flat space-time. To describe a rotating black hole, it should also include an event horizon. In fact, there is only one solution (Carter, 1971b) that satisfies all these properties, and that is the one obtained by Kerr (1963). The main geometrical properties of this very important type D solution will be described here. Other stationary, axially symmetric space times that do not have horizons but could well represent the exteriors of some rotating sources will be described in Chapter 13.

The Kerr solution

Although it was not originally discovered in this form, it is convenient to present this solution here in terms of the spheroidal-like coordinates of Boyer and Lindquist (1967).