The Geometry of Kerr Black Holes

Barrett O'Neill

• Publisher: Dover Publications
• Year: 2014
• ISBN:
  9780486783116
• Topic: Physical Sciences
• Subtopic: Astronomy & Astrophysics

This unique monograph by a noted UCLA professor examines in detail the mathematics of Kerr black holes, which possess the properties of mass and angular momentum but carry no electrical charge. Suitable for advanced undergraduates and graduate students of mathematics, physics, and astronomy as well as professional physicists, the self-contained treatment constitutes an introduction to modern techniques in differential geometry.
The text begins with a substantial chapter offering background on the mathematics needed for the rest of the book. Subsequent chapters emphasize physical interpretations of geometric properties such as curvature, geodesics, isometries, totally geodesic submanifolds, and topological structure. Further investigations cover relativistic concepts such as causality, Petrov type, optical scalars, and the Goldberg-Sachs theorem. Four helpful appendixes supplement the text.

CHAPTER

ONE

BACKGROUND

This chapter gives a concise exposition of most of the material needed for the study of Kerr black holes in the chapters that follow. The topics 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 (Sections 1.2 and 1.3) 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 (Section 1.8 and Appendix 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 (Chapter 5) which is particularly well-suited to analysis of the relation between the curvature and geodesics of spacetimes.

1.1 Manifolds

Roughly speaking, a manifold is a topological space whose local equivalence to Euclidean space Rⁿ permits calculus to be globally established on it. Accordingly, we assume a familiarity with the basic calculus of Rⁿ. If ϕ = (f₁, …, f_n) is a mapping from an open set

of R^m into Rⁿ we say that ϕ is smooth (or infinitely differentiable or C^∞) if each of the real-valued functions f_j (1

j

n) has continuous partial derivatives of all orders.

MANIFOLDS AND COORDINATE SYSTEMS

A coordinate system of dimension n in a topological space S is a homeomorphism ξ from an open set

of S onto an open set

of Rⁿ. The open set

is a coordinate neighborhood of ξ, and the real-valued functions x¹, …, xⁿ on

such that ξ = (x¹, …, xⁿ) are called its coordinate functions.

A smooth manifold M of dimension n is a Hausdorff space furnished with a collection

(called a smooth atlas) of n-dimensional coordinate systems such that

1. The coordinate neighborhoods for all

cover all of M, and

2. Any two

overlap smoothly;...

Table of contents

• Cover
• Title Page
• Copyright Page
• Dedication
• Contents
• Preface
• Introduction
• Chapter 1 Background
• Chapter 2 Beginning Kerr Spacetime
• Chapter 3 Maximal Extensions
• Chapter 4 Kerr Geodesics
• Chapter 5 Petrov Types
• Appendix A Units
• Appendix B Differential Forms
• Appendix C Carter Constant
• Appendix D Exterior Products
• Index of Notations
• Bibliography
• Index