## Black Hole Uniqueness TheoremsThis timely review provides a self-contained introduction to the mathematical theory of stationary black holes and a self-consistent exposition of the corresponding uniqueness theorems. The opening chapters examine the general properties of space-times admitting Killing fields and derive the Kerr-Newman metric. Heusler emphasizes the general features of stationary black holes, the laws of black hole mechanics, and the geometrical concepts behind them. Tracing the steps toward the proof of the "no-hair" theorem, he illustrates the methods used by Israel, the divergence formulas derived by Carter, Robinson and others, and finally the sigma model identities and the positive mass theorem. The book also includes an extension of the electro-vacuum uniqueness theorem to self-gravitating scalar fields and harmonic mappings. A rigorous textbook for graduate students in physics and mathematics, this volume offers an invaluable, up-to-date reference for researchers in mathematical physics, general relativity and astrophysics. |

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### Contents

1 Preliminaries | 1 |

2 Spacetimes admitting Killing fields | 6 |

3 Circular spacetimes | 31 |

4 The Kerr metric | 42 |

5 Electrovac spacetimes with Killing fields | 56 |

6 Stationary black holes | 84 |

7 The four laws of black hole physics | 102 |

8 Integrability and divergence identities | 122 |

### Common terms and phrases

2—dimensional 2—surfaces arbitrary asymptotically ﬂat axial Killing ﬁeld axisymmetric spacetime Bekenstein black hole solutions boundary Carter chapter Chrusciel conclude conﬁgurations constant coordinates Corollary curvature deﬁned deﬁnition denote domain of outer dominant energy condition Einstein’s equations electric electromagnetic ﬁeld electrovac spacetimes Ernst equations Ernst potential establish event horizon expression fact ﬁeld equations ﬁnally ﬁnd formula fulﬁl gravitational harmonic mappings Hence Heusler Hodge dual hypersurface implies inﬁnity integrability conditions invariant Kerr metric Kerr—Newman Killing horizon Komar Lagrangian Laplacian law of black magnetic Masood—ul—Alam Math non—negative nonrotating null obtain orthogonal outer communications Phys proposition Quantum Grav respect Ricci tensor Ricci—circularity Riemannian manifold rotation scalar ﬁelds Schwarzschild metric self—gravitating sigma model Skyrme spacelike spherically symmetric static spacetime stationary and axisymmetric stationary spacetime Straumann stress—energy tensor surface gravity target manifold timelike Killing ﬁeld tion uniqueness theorem vacuum vanishes variation vector ﬁeld Wald yields zeroth law