Book review

In the late sixtieth Penrose and Hawking pointed out that a space-time could contain incomplete geodesics ending in a singularity, even it is not highly symmetric as is the case with the Schwarzschild space-time. Many singularities are only apparent, being related to a poor choice of coordinates, such that we are naturally lead to the problem of the extension of a given spacetime through such an "apparent singularity". Actually, this the first aim of this book: to give conditions in which a space-time can be extended and to extend it, when possible. After the extension was performed, it is still possible for the space-time to contain non-removable singularities and these are likely to have a physical significance.

The contents of the book look as follows. The first three chapters have a rather introductive nature. They are dealing with the definitions of singularities, the Riemann tensor and different ways to attach a boundary to a space-time (boundary that is empty every time when the spacetime is singularity-free). The chapter four develops the mathematical techniques needed to face the problem (first of all the Sobolev spaces, energy inequalities a.o.) and there are discussed different results on the existence and the differentiability of the solutions Einstein fields equations). Chapter five is concerned with the problem of the analytic extension, while the sixth one discusses some attributes of the singularities (strength, global hyperbolicity, past simplicity, past hyperbolicity a.o.). In the seventh chapter there are considered various situations when is possible to extend the space-time. Finally, the chapter eight presents results about the cosmic censorship conjecture (asserting that in a space-time with a "physically realistic stress-energy tensor" there cannot exist naked singularities (which are not hidden inside an event horizon)).

This is, I think, the first monograph about singularities which is really new, after the book of Hawking and Ellis (The large scale structure of space-time, Cambridge, 1973). The reader should know that here one cannot find the proofs of the classical theorems, but there are developed, in more details, and are used to discuss the singularities, the techniques of Sobolev spaces rather than those of differential topology.

A great part of this book is based on the works of the author and his collaborators. Many of the results are for the first time presented in a book. It is written with sure hands and it is very clear. As prerequisites, it is assumed that the reader is familiar with the theory of space-time, as developed by Hawking and Penrose. The topics touched being very special, I think that the book is addressed to advanced graduate students and researchers in the field.

The author is a well known expert. He is also the author of Elementary General Relativity (Cambridge, 1979) and coauthor (with F. de Felice) of Relativity on Curved Manifolds (Cambridge, 1991), both very successful. The book is in good technical conditions, has a list of references and an index, too.