##
**The geometry of Kerr black holes.**
*(English)*
Zbl 0828.53078

Wellesley, MA: A. K. Peters. xvii, 381 p. (1995).

The Kerr metric, depending on two parameters \(m\) and \(a\), is one of the most famous solutions of Einstein’s vacuum field equations. It was discovered in 1964 by R. Kerr and describes the vacuum region around some rotating object with mass \(m\) and angular momentum \(ma\). For \(m > 0\) and \(a^2 \leq m^2\), the Kerr metric is of particular relevance since it gives the unique mathematical model of a stationarily rotating uncharged black hole. (The generalization to the charged case leads to the Kerr- Newman metric which is not treated in the book under review). Such a Kerr black hole features a ring-shaped singularity hidden behind two horizons. In the limiting case \(a^2 = m^2\) (“extreme Kerr”) the two horizons coincide, whereas in the limiting case \(a = 0\) (Schwarzschild) one of the two horizons is swallowed by the singularity. A Kerr black hole has a highly non-trivial causal structure; e.g., there is causality violation inside the inner horizon. All these features can be studied by explicit calculations, i.e., not just by qualitative or indirect arguments. In particular, the geodesic equation on a Kerr spacetime is completely integrable, which gives explicit information on the null geodesics and thus on the causal structure. For all these reasons, a detailed investigation of the Kerr metric is one of the most instructive and most fruitful exercises in general relativity.

Many textbooks on general relativity contain sections on the Kerr metric. Moreover, there is a celebrated book by S. Chandrasekhar [“The mathematical theory of black holes”, Oxford Univ. Press (1983; Zbl 0511.53076)] that can be viewed, to a great extent, as a monograph on the Kerr metric. Nonetheless, the book under review fills a gap in the literature. It is the first self-contained and comprehensive exposition of the Kerr metric from the view-point of global Lorentzian geometry. (This is a view-point quite different from Chandrasekhar’s.)

The book is organized as follows. It begins with a brief introduction on the history (and pre-history) of how the Kerr metric was discovered and investigated. Chapter 1 is entitled “Background” and introduces basic notions from differential geometry and from general relativity. For the most part, this chapter is a concise presentation of elementary textbook material; however, this chapter also contains a section on “gluing” semi-Riemannian manifolds which is not standard. The results of this section are extensively used later when maximal Kerr spacetimes are constructed.

In Chapter 2 the Kerr metric is introduced. Here the author does not bother to give anything like a “derivation” of the Kerr metric; he just writes it down in Boyer-Lindquist coordinates. The main part of Chapter 2 is devoted to studying the causal features of what the author calls “Boyer-Lindquist blocks”, i.e., of maximal connected domains covered by a single Boyer-Lindquist chart. This already gives the opportunity to study the stationary limit surface, the ring singularity, and causality violations. In the fast rotating case, \(m^2 < a^2\), which is usually considered unphysical, there is only one Boyer-Lindquist block and this is a maximal, i.e. inextendible, spacetime. In the extreme case \(m^2 = a^2\), there are two Boyer-Lindquist blocks and in the slowly rotating case \(m^2 > a^2\), there are three. None of them is a maximal spacetime. This makes it possible to glue Boyer-Lindquist blocks together along an interface which becomes a horizon in the resulting spacetime.

In Chapter 3 this gluing procedure is used to produce maximal Kerr spacetimes in the extreme and in the slow case. The result is well known and can be found in several other books. Nevertheless, the procedure is presented here in a more detailed manner than usual and attention is given to several subtleties which are usually disregarded by other authors. The remainder of Chapter 3 is devoted to an analysis of maximal Kerr spacetimes that goes well beyond standard results. This includes topological and bundle theoretical investigations and, in particular, a thorough discussion of global isometries.

In Chapter 4, which is the longest chapter of the book, the geodesics in Kerr spacetimes are studied in detail. In agreement with the general style of the book, the author concentrates on those results which can be phrased as mathematical theorems. To give an idea, by way of example, of what kind of results are presented in this chapter, there is a theorem presenting a criterion by which a non-equatorial geodesic hits the ring singularity. Many results of Chapter 4 seem to be new, and the presentation gives a good idea of how additional results might be found.

Chapter 5 deals with the Petrov classification and with the Goldberg- Sachs theorem which historically played a major role in the discovery of the Kerr metric. Here this chapter appears somewhat disconnected, and the same is true of the four short appendices. The book ends with a rather selective list of references. Unfortunately, there are very few hints to the literature in the text.

This is definitely a book on the mathematics and not on the physics of Kerr spacetimes. As a matter of fact, there are only a few remarks on physical interpretation and some of them are highly misleading. In particular, the Kerr metric is advertised as “the standard relativistic model of the gravitational field of a rotating star”. This produces the completely wrong impression that the Kerr metric could be easily matched to an interior solution of Einstein’s field equation with reasonable matter tensor. To mention another example, the ring singularity is characterized as a “circle of infinite gravitational forces”. This is meaningless since the Newtonian concept of a gravitational force has no invariant counterpart in general relativity. It is the tidal force that goes to infinity at a curvature singularity.

Still, as far as the mathematical aspects of the Kerr metric are concerned, this is a very good book. It is clearly and carefully written and contains only a few minor misprints. The terminology is precise, sometimes even at the risk of being pedantic, which is to be viewed as a necessity for the matter at issue. By and large, the book is to be recommended to all mathematicians and physicists who are interested in the mathematical formalism of general relativity.

Many textbooks on general relativity contain sections on the Kerr metric. Moreover, there is a celebrated book by S. Chandrasekhar [“The mathematical theory of black holes”, Oxford Univ. Press (1983; Zbl 0511.53076)] that can be viewed, to a great extent, as a monograph on the Kerr metric. Nonetheless, the book under review fills a gap in the literature. It is the first self-contained and comprehensive exposition of the Kerr metric from the view-point of global Lorentzian geometry. (This is a view-point quite different from Chandrasekhar’s.)

The book is organized as follows. It begins with a brief introduction on the history (and pre-history) of how the Kerr metric was discovered and investigated. Chapter 1 is entitled “Background” and introduces basic notions from differential geometry and from general relativity. For the most part, this chapter is a concise presentation of elementary textbook material; however, this chapter also contains a section on “gluing” semi-Riemannian manifolds which is not standard. The results of this section are extensively used later when maximal Kerr spacetimes are constructed.

In Chapter 2 the Kerr metric is introduced. Here the author does not bother to give anything like a “derivation” of the Kerr metric; he just writes it down in Boyer-Lindquist coordinates. The main part of Chapter 2 is devoted to studying the causal features of what the author calls “Boyer-Lindquist blocks”, i.e., of maximal connected domains covered by a single Boyer-Lindquist chart. This already gives the opportunity to study the stationary limit surface, the ring singularity, and causality violations. In the fast rotating case, \(m^2 < a^2\), which is usually considered unphysical, there is only one Boyer-Lindquist block and this is a maximal, i.e. inextendible, spacetime. In the extreme case \(m^2 = a^2\), there are two Boyer-Lindquist blocks and in the slowly rotating case \(m^2 > a^2\), there are three. None of them is a maximal spacetime. This makes it possible to glue Boyer-Lindquist blocks together along an interface which becomes a horizon in the resulting spacetime.

In Chapter 3 this gluing procedure is used to produce maximal Kerr spacetimes in the extreme and in the slow case. The result is well known and can be found in several other books. Nevertheless, the procedure is presented here in a more detailed manner than usual and attention is given to several subtleties which are usually disregarded by other authors. The remainder of Chapter 3 is devoted to an analysis of maximal Kerr spacetimes that goes well beyond standard results. This includes topological and bundle theoretical investigations and, in particular, a thorough discussion of global isometries.

In Chapter 4, which is the longest chapter of the book, the geodesics in Kerr spacetimes are studied in detail. In agreement with the general style of the book, the author concentrates on those results which can be phrased as mathematical theorems. To give an idea, by way of example, of what kind of results are presented in this chapter, there is a theorem presenting a criterion by which a non-equatorial geodesic hits the ring singularity. Many results of Chapter 4 seem to be new, and the presentation gives a good idea of how additional results might be found.

Chapter 5 deals with the Petrov classification and with the Goldberg- Sachs theorem which historically played a major role in the discovery of the Kerr metric. Here this chapter appears somewhat disconnected, and the same is true of the four short appendices. The book ends with a rather selective list of references. Unfortunately, there are very few hints to the literature in the text.

This is definitely a book on the mathematics and not on the physics of Kerr spacetimes. As a matter of fact, there are only a few remarks on physical interpretation and some of them are highly misleading. In particular, the Kerr metric is advertised as “the standard relativistic model of the gravitational field of a rotating star”. This produces the completely wrong impression that the Kerr metric could be easily matched to an interior solution of Einstein’s field equation with reasonable matter tensor. To mention another example, the ring singularity is characterized as a “circle of infinite gravitational forces”. This is meaningless since the Newtonian concept of a gravitational force has no invariant counterpart in general relativity. It is the tidal force that goes to infinity at a curvature singularity.

Still, as far as the mathematical aspects of the Kerr metric are concerned, this is a very good book. It is clearly and carefully written and contains only a few minor misprints. The terminology is precise, sometimes even at the risk of being pedantic, which is to be viewed as a necessity for the matter at issue. By and large, the book is to be recommended to all mathematicians and physicists who are interested in the mathematical formalism of general relativity.

Reviewer: V.Perlick (Berlin)

### MSC:

53Z05 | Applications of differential geometry to physics |

83C57 | Black holes |

83-02 | Research exposition (monographs, survey articles) pertaining to relativity and gravitational theory |