##
**A relativist’s toolkit. The mathematics of black-hole mechanics.**
*(English)*
Zbl 1058.83002

Cambridge: Cambridge University Press (ISBN 0-521-83091-5/hbk). xvi, 233 p. (2004).

There already exists a vast array of available books on the theory of gravitation in a curved space-time, or general relativity. Space-time is represented as a four-dimensional differential manifold with metric, a generalization of flat Minkowski space-time [R. Adler, M. Bazin, and M. Schiffer, General Relativity. McGraw-Hill, New York (1965; Zbl 0144.47604); B. F. Schutz, A First Course in General Relativity. Cambridge University Press, Cambridge (1985), reprint (1986; Zbl 0604.53029); C. W. Misner, K. S. Thorne, and J. A. Wheeler, Gravitation. Freeman, New York (1973); R. M. Wald, General Relativity. The University of Chicago Press, Chicago (1984; Zbl 0549.53001); C. M. Will, Theory and Experiment in Gravitational Physics. Second edition, Cambridge University Press, Cambridge (1993; Zbl 0785.53068); N. Straumann, General Relativity: With Applications to Astrophysics. Springer, Berlin (2004; Zbl 1059.83001)]. The most successful application of general relativity is the mathematical theory of black holes [S. W. Hawking and G. F. R. Ellis, The Large Scale Structure of Space-Time. Cambridge University Press, Cambridge (1973; Zbl 0265.53054); S. Chandrasekhar, The Mathematical Theory of Black Holes. Oxford University Press, Oxford (1983; Zbl 0511.53076); P. M. Schwarz and J. H. Schwarz, Special Relativity: From Einstein to Strings. Cambridge University Press, Cambridge (2004; Zbl 1104.83008)].

The book under review which should be considered as an updated but less complete companion to Wald’s monograph (loc. cit.) covers the mathematical theory of black holes in the final Chapter 5. It offers a thorough review of the solutions to the Einstein field equations that describe isolated black holes, a description of the fundamental properties of black holes that are independent of the details of any particular solution, and an introduction to the four laws of black-hole mechanics. The most important aspect of black-hole space-times is that they contain an event horizon, a null hypersurface that marks the boundary of the black hole and shields external observers from events going inside.

To set the stage, Chapter 1 of the book is devoted to a brisk review of the fundamentals of differential geometry. The collection of topics is standard. Tensors are defined in the old-fashioned way in terms of how their components transform under a coordinate transformation, and not in terms of multilinear mappings of vectors and dual vectors. The languages of vector bundles and Lie group theory are avoided because the author thinks that the old approach has the advantage of economy [B. F. Schutz, Geometrical Methods of Mathematical Physics. Cambridge University Press, Cambridge (1980; Zbl 0462.58001); W. A. Poor, Differential Geometric Structures. McGraw-Hill, New York (1981; Zbl 0493.53027); B. Felsager, Geometry, Particles, and Fields. Springer-Verlag, New York (1998; Zbl 0897.53001)].

Chapter 2 develops the relevant techniques to understand the behaviour of the event horizon as a whole. The gain of mass of an accreting black hole is explained by integration over the event horizon.

The integration requires techniques that are introduced in Chapter 3. Other topics covered in Chapter 3 include the initial-value problem of general relativity and the Darmois-Lanczos-Israel-Barrabès formalism for junction conditions and thin shells.

Chapter 4 is devoted to a systematic treatment of the Lagrangian and Hamiltonian formulations of general relativity with the goal of arriving at correct definitions of black-hole mass and angular momentum. The most compelling definitions come from the gravitational Hamiltonian, whose value for a given solution to the Einstein field equations depends on a specifiable vector field. If this vector corresponds to a time translation at spatial infinity, then the Hamiltonian gives the total mass of the space-time. If, on the other hand, the vector corresponds to an asymptotic rotation about an axis, then the Hamiltonian gives the total angular momentum of space-time in the direction of this axis. This insight into black-hole mechanics is both deep and beautiful, and in the book under review it represents the central point for defining black-hole mass and angular momentum.

What sets its exposition apart from what can be found in other books on gravitational physics is that the text pays careful attention to the boundary terms that must be included into the gravitational action to produce a well-posed variational principle [D. Lovelock and H. Rund, Tensors, Differential Forms, and Variational Principles. Wiley, New York (1975; Zbl 0308.53008)]. These boundary terms have been around for a long time, but it is only recently that their importance has been fully recognized. In particular, they are directly involved in defining the mass and angular momentum of an asymptotically flat space-time. The proofs provided by the book are informal. For a more rigorous and complete exposition the reader is referred to the text by Wald (loc. cit.). Instead of mathematical rigour, emphasis is laid on practicality to help the reader acquire advanced skills of intuition and become a competent researcher in the fields of relativity and gravitational physics. The aspects of quantum cosmology seem not to activate these advanced skills.

The book under review which should be considered as an updated but less complete companion to Wald’s monograph (loc. cit.) covers the mathematical theory of black holes in the final Chapter 5. It offers a thorough review of the solutions to the Einstein field equations that describe isolated black holes, a description of the fundamental properties of black holes that are independent of the details of any particular solution, and an introduction to the four laws of black-hole mechanics. The most important aspect of black-hole space-times is that they contain an event horizon, a null hypersurface that marks the boundary of the black hole and shields external observers from events going inside.

To set the stage, Chapter 1 of the book is devoted to a brisk review of the fundamentals of differential geometry. The collection of topics is standard. Tensors are defined in the old-fashioned way in terms of how their components transform under a coordinate transformation, and not in terms of multilinear mappings of vectors and dual vectors. The languages of vector bundles and Lie group theory are avoided because the author thinks that the old approach has the advantage of economy [B. F. Schutz, Geometrical Methods of Mathematical Physics. Cambridge University Press, Cambridge (1980; Zbl 0462.58001); W. A. Poor, Differential Geometric Structures. McGraw-Hill, New York (1981; Zbl 0493.53027); B. Felsager, Geometry, Particles, and Fields. Springer-Verlag, New York (1998; Zbl 0897.53001)].

Chapter 2 develops the relevant techniques to understand the behaviour of the event horizon as a whole. The gain of mass of an accreting black hole is explained by integration over the event horizon.

The integration requires techniques that are introduced in Chapter 3. Other topics covered in Chapter 3 include the initial-value problem of general relativity and the Darmois-Lanczos-Israel-Barrabès formalism for junction conditions and thin shells.

Chapter 4 is devoted to a systematic treatment of the Lagrangian and Hamiltonian formulations of general relativity with the goal of arriving at correct definitions of black-hole mass and angular momentum. The most compelling definitions come from the gravitational Hamiltonian, whose value for a given solution to the Einstein field equations depends on a specifiable vector field. If this vector corresponds to a time translation at spatial infinity, then the Hamiltonian gives the total mass of the space-time. If, on the other hand, the vector corresponds to an asymptotic rotation about an axis, then the Hamiltonian gives the total angular momentum of space-time in the direction of this axis. This insight into black-hole mechanics is both deep and beautiful, and in the book under review it represents the central point for defining black-hole mass and angular momentum.

What sets its exposition apart from what can be found in other books on gravitational physics is that the text pays careful attention to the boundary terms that must be included into the gravitational action to produce a well-posed variational principle [D. Lovelock and H. Rund, Tensors, Differential Forms, and Variational Principles. Wiley, New York (1975; Zbl 0308.53008)]. These boundary terms have been around for a long time, but it is only recently that their importance has been fully recognized. In particular, they are directly involved in defining the mass and angular momentum of an asymptotically flat space-time. The proofs provided by the book are informal. For a more rigorous and complete exposition the reader is referred to the text by Wald (loc. cit.). Instead of mathematical rigour, emphasis is laid on practicality to help the reader acquire advanced skills of intuition and become a competent researcher in the fields of relativity and gravitational physics. The aspects of quantum cosmology seem not to activate these advanced skills.

Reviewer: Walter Schempp (Siegen)

### MSC:

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

83C05 | Einstein’s equations (general structure, canonical formalism, Cauchy problems) |

83C57 | Black holes |

83F05 | Relativistic cosmology |

53B50 | Applications of local differential geometry to the sciences |

53C80 | Applications of global differential geometry to the sciences |