This chapter gives a concise exposition of most of the material needed for the study of Kerr spacetime in the chapters that follow. The subjects discussed include: manifold theory, tensor calculus, differential geometry, general relativity, and differential forms. The amount of detail varies considerably from topic to topic. Roughly speaking, brevity is possible when the topic is covered in a variety of readily available sources. The section on manifold theory, for example, does little more than record basic definitions and fix notation. By contrast, the special topic of extensions of analytic manifolds requires more attention because it is usually dealt with informally but is crucial to the construction of Kerr spacetime.
Tensor calculus (Secs. 1 and 2) is a generally accepted common language for differential geometry and relativistic physics, with the fundamentals expressed invariantly as well as in coordinate terms.
The Cartan calculus of differential forms (Sec. 1.8 and App. B) is at least close to general acceptance, and it is the most efficient way to compute the curvature of Kerr spacetime--and manifolds in general.
Perhaps less widely known is the Newman-
Penrose formalism (Ch.5), which is particularly well-suited to
analysis of the relation between the curvature and geodesics of
spacetimes.