Dynamical black holes

In this chapter we review what is known about dynamical black hole-solutions of Einstein equations. We discuss the Robinson–Trautman black holes, with or without a cosmological constant. We review the Cauchy-data approach to the construction of black-hole spacetimes. We propose some alternative approaches to a meaningful definition of black hole in a dynamical spacetime, and we review the nonlinear stability results for black-hole solutions of vacuum Einstein equations.

Projection diagrams

In this chapter we show that one can usefully represent classes of non-spherically symmetric geometries in terms of two-dimensional diagrams, called projection diagrams, using an auxiliary two-dimensional metric constructed out of the spacetime metric. Whenever such a construction can be carried out, the issues such as stable causality, global hyperbolicity, the existence of event or Cauchy horizons, the causal nature of boundaries, and the existence of conformally smooth infinities become evident by inspection of the diagrams.

Further selected solutions

In previous chapters we presented the key notions associated with stationary black-hole spacetimes, as well as the minimal set of metrics needed to illustrate the basic features of the world of black holes. In this chapter we present some further black holes, selected because of their physical and mathematical interest. We start, in Section 5.1, with the Kerr–de Sitter/anti-de Sitter metrics, the cosmological counterparts of the Kerr metrics. Section 5.2 contains a description of the Kerr–Newman–de Sitter/anti-de Sitter metrics, which are the charged relatives of the metrics presented in Section 5.1. In Section 5.3 we analyse in detail the global structure of the Emparan–Reall ‘black rings’: these are five-dimensional black-hole spacetimes with R × S 1 × S 2-horizon topology. The Rasheed metrics of Section 5.4 provide an example of black holes arising in Kaluza–Klein theories. The Birmingham family of metrics, presented in Section 5.5, forms the most general class known of explicit static vacuum metrics with cosmological constant in all dimensions, with a wide range of horizon topologies.

Basic notions

The aim of this chapter is to set the ground for the remainder of this work. We present our conventions and notations in Section 1.1. We review some basic facts about the topology of manifolds in Section 1.2. Lorentzian manifolds and spacetimes are introduced in Section 1.3. Elementary facts concerning the Levi-Civita connection and its curvature are reviewed in Section 1.4. Some properties of geodesics, as needed in the remainder of this book, are presented in Section 1.5. The formalism of moving frames is outlined in Section 1.6, where it is used to calculate the curvature of metrics of interest.

Extensions, conformal diagrams

In this chapter we present a systematic approach to extensions of spacetimes, as needed to construct black-hole spacetimes. On the way we introduce the conformal diagrams, which are a useful tool for visualizing the geometry of the extensions. We focus on extensions of metrics containing a two-by-two stationary Lorentzian block.We discuss causality for such metrics in Section 6.1; the possible building blocs are described in Section 6.2; these building blocs are put together in Section 6.3. The general rules governing the construction are explained in Section 6.4, with the causal aspects of the construction highlighted in the short Section 6.5. The method is applied to those Birmingham metrics which have not been analysed previously in Section 6.6.

An introduction to black holes

In this chapter the basics of the geometry of stationary black-hole spacetimes are presented. We start in Section 4.1 with a brief review of astrophysical black holes. We continue in Section 4.2 with the presentation of the flagship black hole, the Schwarzschild solution: we construct there its various extensions, and analyse some of its properties. The general notions arising in the context of black-hole geometries are presented in Section 4.3. A systematic discussion of extensions of spacetimes is carried out in Section 4.4. The charged counterparts of the Schwarzchild metric, namely the Reissner–Nordström metrics, are analysed in Section 4.5. The Kerr metric, expected to describe the most general vacuum, stationary, and rotating black holes, is presented in Section 4.6. The electrovacuum Majumdar–Papapetrou spacetimes, containing two or more disconnected black-hole regions, are described in Section 4.7.

Some applications

The aim of this chapter is to present key applications of causality theory, as relevant to black-hole spacetimes. For this we need to introduce the concept of conformal completions, which is done in Section 3.1. We continue, in Section 3.2, with a review of the null splitting theorem of Galloway. Section 3.3 contains complete proofs of a few versions of the topological censorship theorems, which are otherwise scattered across the literature, and which play a basic role in understanding the topology of black holes. In Section 3.4 we review some key incompleteness theorems, also known under the name of singularity theorems. Section 3.5 is devoted to the presentation of a few versions of the area theorem, which is a cornerstones of ‘black-hole thermodynamics’. We close this chapter with a short discussion of the role played by causality theory when studying the wave equation.

Elements of causality

A standard part of studies of black holes, and in fact of mathematical general relativity, is causality theory, which is the study of causal relations on Lorentzian manifolds. An essential issue here is understanding the influence of energy conditions on the causality relations. The highlights of such studies include the incompleteness theorems, known also as singularity theorems, of Penrose, Hawking and Geroch, the area theorem of Hawking, and the topology theorems of Hawking and others. The aim of this chapter is to provide an introduction to the subject, with a complete exposition of those topics which are needed for the global treatment of the uniqueness theory of black holes. In particular we provide a coherent introduction to causality theory for metrics which are twice differentiable.