*(English)*Zbl 1234.83001

In general relativity, there are solutions of the field equations (like the Schwarzschild and the Kerr metrics) which represent strong field configurations having the feature that the curvature manifests itself in unusual global properties of space-time. Since the time when J. A. Wheeler and others gave them the name ‘black holes’, more and more relativists considered them as astrophysical objects. Black holes were assumed to be possible end stages of the star development and thus became an object of astrophysical considerations. Wheeler and colleagues had still strong criteria to be imposed on possible physical matter, i.e., on its energy-momentum tensor. These limitations and the fact that the known black hole solutions are singular led them to conclude that there is no such solution of the classical (i.e., not quantized) gravitation field equations. In the hope to arrive at regular classical black hole (and cosmological) solutions, at that time many authors tried to modify Einstein’s gravitational field equations. Now, the situation has changed: mostly, without having developed a quantized general relativity (‘Quantum Gravity’), one relies on that a future Quantum Gravity will solve the singularity problem such that one can consider the black hole solutions to Einstein’s classical equations as space-times physically interpretable up to the Planck regime. This is also the standpoint of the authors of the present book. They are of the opinion that after 40 years of black hole discussion by relativists and astrophysical discoveries supporting the belief in the existence of black holes (the observation of pulsars is considered to be the beginning of such discoveries), “we cannot imagine our world without black holes”.

This is the second comprehensive book by V. P. Frolov on black holes, now co-authored by A. Zelnikov, while its 1998 forerunner was written together with I. D. Novikov [*V.P. Frolov* and *I.D. Novikov*, Black hole physics. Basic concepts and new developments. Dordrecht: Kluwer Academic Publishers (1998; Zbl 0978.83001)]. The book is based on lectures that Frolov has given at different universities during 30 years. The main part (presented in Chapters 1-8) provides one with material sufficient for a one-term self-consistent course on black holes. After giving a survey over this fast developing area of research, general relativity is prepared by chapters on physics in accelerated reference frames in Minkowski space, Riemannian geometry, and particle motions in curved space-time. Chapter 5 introducing Einstein’s equations and discussing gravitational radiation and gravity in higher-dimensions is followed by three chapters devoted to spherically symmetric and rotating black holes. Besides very informative appendices, in Chapters 9 and 10, the book also contains additional material, e.g., concerning classical and quantum fields near black holes, black holes and higher dimensions, wormholes, and ‘time machines’.

This book is a very good basis for teaching graduate courses on black holes. Moreover, in view of the diversity of considered topics reaching from mathematics over special and general relativity theory to astrophysics, this book deserves a great readership going beyond readers who are particularly interested in black hole physics.

##### MSC:

83-01 | Textbooks (relativity) |

83C57 | Black holes |

83C15 | Closed form solutions of equations in general relativity |

83F05 | Relativistic cosmology |

83C35 | Gravitational waves |

83A05 | Special relativity |

83C10 | Equations of motion |

85A40 | Cosmology (astronomy and astrophysics) |

83C47 | Methods of quantum field theory in general relativity |

83C75 | Space-time singularities, cosmic censorship, etc. |

85A15 | Galactic and stellar structure |

83D05 | Relativistic gravitational theories other than Einstein’s |