To present the Kerr metric in intuitive terms, let us picture a distant, spherically symmetric star rotating in space about a vertical axis through its center. To describe this situation, it is natural to use spherical coordinates r, theta, phi on R3 and a time coordinate t on R, interpreting r as distance to the center of the star, theta as colatitude, and phi as longitude--the star rotating in the positive phi direction. Only the gravitational field of the star is to be modelled, not the star itself. (The word "star" should be interpreted liberally, as also including the black hole possibility.)
These coordinates could be used for a Newtonian description, where it is irrelevant whether the star is rotating or not, and they serve also (initially, at least) for the two relativistic cases: Schwarzschild spacetime, where the star does not rotate, and Kerr spacetime, where it does. When the Kerr metric is presented in terms of these familiar coordinates, they are called Boyer-Lindquist coordinates.
A major purpose of this chapter is to extend the domain on which the Kerr metric is defined. This is readily done in the case of an ordinary star, since only its exterior is modelled. Black holes are another story. With the star's bulk gone, the spacetime must extend from the distant vantage point at which our description begins, on into the physically unobservable regions at its core, and beyond. If the black hole rotates rapidly enough, this spacetime is readily constructed. However, in the principal case, where the rotation is not too rapid, the extension comes in stages. To know how to construct these we need to understand some of the geometry of the initial Boyer- Lindquist domain.
The basic geometric invariant, Riemannian curvature, can already be computed at this level. Also we notice some special submanifolds, notably the horizons, that will grow as the initial domain is extended. The next chapter will complete the construction of the maximal Kerr black holes.