Naked singularity

This graduate-level text covers in detail the technical aspects of recent work on global aspects of general relativity theory and the existence of space-time singularities.

Since the explosion of work on those topics some 20 years ago, progress has been relatively slow, but some major topics remain unresolved. Pankaj S. Joshi provides a workmanlike introduction to these themes, and then a detailed analysis of some specific topics of interest. Inevitably a considerable part of the book (for example those pages covering basic features of differential geometry, causal structure, and details of focussing effects and consequent singularity theorems) is fairly repetitive of what appears in other texts, for example The Large-scale Structure of Space-Time (by Stephen Hawking and George Ellis) and General Relativity (by Bob Wald). However this book attains uniqueness in its specific approach, for example through an emphasis on reflecting space-times, which exclude some undesirable pathological features that otherwise might occur, and by exploring light-cone cuts of infinity in asymptotically flat space-times, determining the metric from asymptotic properties there. It also achieves a specific flavour and usefulness through tackling some specific topics in depth, notably the issue of cosmic censorship. This is the question of whether gravitational collapse leads to a naked singularity that can be seen from far away and can influence what happens there, or to a black hole with an event horizon that hides the final stages of collapse and the ensuing space-time singularity from the outside world.

As the author notes, it has not been possible to establish the validity of cosmic censorship despite several serious attempts, and this remains one of the most outstanding open issues in classical gravitational theory today. He discusses this issue in depth, showing how some spherically symmetric examples with inflowing radiation lead to censorship violation, as do some self-similar examples; however these space-times have special symmetries that could be the reason a naked singularity occurs, and so are not regarded as serious counter-examples.

He explores further models (the Vaidya solutions and Tolman-Bondi models) where naked singularities occur, looking at general constraints on naked singularity formation, with the aim of helping to formulate a rigorous statement of cosmic censorship (restricting the allowed forms of matter, for example) that might indeed be obeyed. We do not yet have such a statement.

Other sections look at the way upper limits on some cosmologically interesting parameters (the age of the universe, particle masses, and the cosmological constant) can be obtained by use of these global methods, a use of these techniques that is valuable in terms of relating them to observational quantities; and give a brief introduction to quantum effects near the space-time singularity by quantising the conformal factor and relating this to an operator approach.

This shows that on the basis of general quantum principles the singularity is no longer an inevitability, and indeed nonclassical, nonsingular states occur with finite probability. Thus black holes without a singularity might occur once quantum effects are taken into account.

The book is readable and covers interesting material, some of it new and not covered so fully in earlier texts. It can be recommended for the serious student who wishes to study these matters in some detail, particularly the material on cosmic censorship. However it is presented at a technical level; the reader who wants a more informal approach should look elsewhere, for example to Kip Thorne's excellent presentation in his recent book Black Holes and Time Warps, which covers much of the same material in a highly readable and less technical way. That text in turn does not present the detailed mathematical methods and results that justify the statements and diagrams presented there. The present book does so in a pleasant way.

George Ellis is professor of applied mathematics, University of Cape Town.