This book is an account of the global geometry of Kerr spacetime, the relativistic model of the gravitational field of a rotating central mass. Actually, the Kerr exact solution is a family of spacetimes depending on parameters m (mass) and a (angular momentum per unit mass). Schwarzschild spacetime, where there is no rotation, is the limiting case a=0.
Only three of the book's five chapter deal directly with Kerr spacetime. Chapter 1 supplies general background material and Chapter 5 is an exposition of Petrov types and null congruences, with the Kerr case used throughout as the main example. Its goal also is to give some perspective on the place of Kerr spacetime within all spacetimes. Chapter 2 establishes the basic geometry of the Kerr metric; Chapter 3 constructs maximal analytic extensions and examines their global structure; and Chapter 4 (the longest) is a detailed study of Kerr geodesics. The introduction to each chapter (linked to Contents) gives a fuller summary of its contents.
As the title of the book indicates, it is Kerr geometry that is studied--curvature, geodesics, isometries, totally geodesic submanifolds, topological structure, and so on. Of course, the physical interpretations of these geometrical invariants are emphasized--but there is no attempt to "do physics." Here Kerr spacetime is viewed only as the arena in which physics and related disciplines operate--from observational astronomy to quantum gravity theory.
However, as the title may also suggest, we are interested not just in the exterior of a rotating star but in the entire extent of maximally extended Kerr spacetimes. The two main Kerr subfamilies, slowly rotating (a2 < m2) and rapidly rotating (a2 > m2), are separated by the extreme case (a2=m2). The fast case is the least interesting, both physically and geometrically, and is largely subsumed in the other two. The extreme case, though exceptional, deserves attention at least as a stepping stone to the more intricate slowly rotating caseÊa2 < m2. Here the maximal Kerr black hole is a vast, symmetrical spacetime, whose construction pattern can be seen, for example, in Figure 3.3 of Chapter 3.
Mathematicians interested Lorentz geometry will find many of its novelties--particularly its differences from positive-definite Riemannian geometry--illustrated by Kerr spacetime, for example:
To make the book accessible to readers with varying interests, prerequisites have been kept to a minimum. (In particular this book is not a sequel to my book Semi-Riemannian Geometry.)
A major obstacle in the vast literature on Kerr spacetime is the great variety of notational languages or formalisms that are used. In spite of temptations there are only three in this book: elementary tensor calculus, differential forms, and (in Chapter 5) the Newman- Penrose formalism. The specifically Kerr notation is also quite conventional.
To my best knowledge, many results in the book are new, particularly the determination of the isometry groups (Secs. 3.7 and 3.8), topological structure (Sec. 3.9), and the qualitative classification of the global trajectories of geodesics (Secs 4.7 through 4.10).