Exact solutions of Einstein’s field equations.
2nd ed.

*(English)*Zbl 1057.83004
Cambridge Monographs on Mathematical Physics. Cambridge: Cambridge University Press (ISBN 0-521-46136-7/hbk; 978-0-511-53518-5/ebook). xxix, 701 p. (2003).

This book is a second edition of a previous version (1980; Zbl 0449.53018) with the same title, the fourth author of this edition being new. It took five years to complete the first edition (over 2000 publications have been used). Since that time, because of the highly inflationary nature of the number of publications on the topic treated, a new edition has been decided (over 4000 new publications have been used). Several new chapters have been added and old ones updated. This work collects and classifies exact (i.e. analytic) solutions to Einstein’s field equations which relate gravitation and matter on a Lorentzian manifold. The authors tell us that their main preoccupation is mathematical; many solutions do not have a physical interpretation. Still, they recall that some exact solutions have a physical meaning and exact solutions can be used for checking numerical computations. This book will certainly be the best tool for researchers to locate work already published in this area.

The field of exact solutions to Einstein’s equations owes its nontriviality to the constraints imposed on the energy-momentum tensor (since without constraints, the metric allows to define this tensor) and these constraints may constitute a door of entry of physics in this area. The book mirrors the development of the discipline, from the first “classical” solutions discovered, to new ones mainly because of the use of new techniques (involving Lie group theory, representation theory, algebraic geometry). The coordinate free differential geometric methods underline this evolution.

The volume contains four parts. Part I (chapters one to ten) presents the mathematical language and methods needed afterwards. The authors first introduce to differential and Riemannian geometry. Other chapters develop the classification criteria: the spectral ones for Weyl’s tensor in the decomposition of the curvature tensor, which are called “the Petrov types” and the spectral ones for the Ricci tensor, which are called “the Segre types”. The authors deal also with a classification based on the energy-momentum tensor. Here, the following cases are considered: vacuum, electromagnetic fields, pure radiation (null fields), dust and perfect fluids. But situations combining such fields are not treated in the book. Chapter seven on the Newman-Penrose formalism announces Part IV devoted to special methods. Chapter eight deals with Lie group theoretic methods fundamental in this work; the authors focus on isometry groups and homothety groups. Chapter nine is devoted to the invariant characterization of metrics using especially Cartan invariants. Chapter ten introduces to Lie point symmetries, prolongation, Bäcklund transformations which will be used in part IV too.

Part II presents symmetry methods for looking for exact solutions to Einstein’s equation (“generation technique”), while part III is devoted to Petrov types. In part II, all chapters consider spacelike or timelike orbits, except the last one which considers null orbits. Special attention is devoted to isotropy.

Part III studies algebraically special solutions. Einstein’s equation of specific Petrov types are handled using a repeated principal null direction. It can be diverging and shear free and the nondiverging case is examined. A chapter looks at perfect fluid solutions without using symmetries. So, symmetries are still appearing in this part but not all the time.

A final part is devoted to special methods typically, “generation methods”. A last section of the volume provides a synthetic perspective, showing interconnections between the main classification schemes with tables.

The book ends with a considerable number of references (the reviewer has never seen a book with so many references) and a useful index. The authors tell us that publications which rediscover previous results are not cited and that in many cases, they checked the proofs of the results of the publications they used; publications with mistakes are not cited.

The authors cannot be praised enough for the considerable amount of time and energy which has been put in this very useful work. While writing and publishing is professionally rewarded in the academic arena, leading to a plethora of texts of various quality, we are fortunate that some sacrifice so much for providing the right pointers to the important advances as well as a synthetic view, so that reading which is the dark side of the moon, gives some deeper value to the previous endeavour. This work arising from this dark side, will be a lighthouse for those navigating in the ever expanding ocean of exact solutions to Einstein’s equations.

The field of exact solutions to Einstein’s equations owes its nontriviality to the constraints imposed on the energy-momentum tensor (since without constraints, the metric allows to define this tensor) and these constraints may constitute a door of entry of physics in this area. The book mirrors the development of the discipline, from the first “classical” solutions discovered, to new ones mainly because of the use of new techniques (involving Lie group theory, representation theory, algebraic geometry). The coordinate free differential geometric methods underline this evolution.

The volume contains four parts. Part I (chapters one to ten) presents the mathematical language and methods needed afterwards. The authors first introduce to differential and Riemannian geometry. Other chapters develop the classification criteria: the spectral ones for Weyl’s tensor in the decomposition of the curvature tensor, which are called “the Petrov types” and the spectral ones for the Ricci tensor, which are called “the Segre types”. The authors deal also with a classification based on the energy-momentum tensor. Here, the following cases are considered: vacuum, electromagnetic fields, pure radiation (null fields), dust and perfect fluids. But situations combining such fields are not treated in the book. Chapter seven on the Newman-Penrose formalism announces Part IV devoted to special methods. Chapter eight deals with Lie group theoretic methods fundamental in this work; the authors focus on isometry groups and homothety groups. Chapter nine is devoted to the invariant characterization of metrics using especially Cartan invariants. Chapter ten introduces to Lie point symmetries, prolongation, Bäcklund transformations which will be used in part IV too.

Part II presents symmetry methods for looking for exact solutions to Einstein’s equation (“generation technique”), while part III is devoted to Petrov types. In part II, all chapters consider spacelike or timelike orbits, except the last one which considers null orbits. Special attention is devoted to isotropy.

Part III studies algebraically special solutions. Einstein’s equation of specific Petrov types are handled using a repeated principal null direction. It can be diverging and shear free and the nondiverging case is examined. A chapter looks at perfect fluid solutions without using symmetries. So, symmetries are still appearing in this part but not all the time.

A final part is devoted to special methods typically, “generation methods”. A last section of the volume provides a synthetic perspective, showing interconnections between the main classification schemes with tables.

The book ends with a considerable number of references (the reviewer has never seen a book with so many references) and a useful index. The authors tell us that publications which rediscover previous results are not cited and that in many cases, they checked the proofs of the results of the publications they used; publications with mistakes are not cited.

The authors cannot be praised enough for the considerable amount of time and energy which has been put in this very useful work. While writing and publishing is professionally rewarded in the academic arena, leading to a plethora of texts of various quality, we are fortunate that some sacrifice so much for providing the right pointers to the important advances as well as a synthetic view, so that reading which is the dark side of the moon, gives some deeper value to the previous endeavour. This work arising from this dark side, will be a lighthouse for those navigating in the ever expanding ocean of exact solutions to Einstein’s equations.

Reviewer: A. Akutowicz (Berlin)

##### MSC:

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

83C15 | Exact solutions to problems in general relativity and gravitational theory |

35Q75 | PDEs in connection with relativity and gravitational theory |

58J72 | Correspondences and other transformation methods (e.g., Lie-Bäcklund) for PDEs on manifolds |

58J70 | Invariance and symmetry properties for PDEs on manifolds |

34C14 | Symmetries, invariants of ordinary differential equations |

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