Black Hole Uniqueness Theorems ( Cambridge Lecture Notes in Physics )

Publication series ：Cambridge Lecture Notes in Physics

Author： Markus Heusler  

Publisher： Cambridge University Press‎

Publication year： 1996

E-ISBN: 9780511893223

P-ISBN(Paperback): 9780521567350

Subject： P145.8 collapsed star (black hole)

Keyword： 物理学

Language： ENG

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Black Hole Uniqueness Theorems

Description

This 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. Strong emphasis is given to the geometrical concepts. The general features of stationary black holes and the laws of black hole mechanics are then reviewed. Critical steps towards the proof of the 'no-hair' theorem are then discussed, including the methods used by Israel, the divergence formulae derived by Carter, Robinson and others, and finally the sigma model identities and the positive mass theorem. The book is rounded off with an extension of the electro-vacuum uniqueness theorem to self-gravitating scalar fields and harmonic mappings. This volume provides a rigorous textbook for graduate students in physics and mathematics. It also offers an invaluable, up-to-date reference for researchers in mathematical physics, general relativity and astrophysics.

Chapter

Cover

Title

Copyright

Contents

Preface

1 Preliminaries

1.1 Conventions

1.2 Differential forms

2 Spacetimes admitting Killing fields

2.1 Killing fields

2.2 Basic identities

2.3 The vacuum variational principle

2.4 Asymptotic flatness and stationarity

2.5 Static spacetimes

2.6 Foliations of static spacetimes

2.7 Stationary and axisymmetric spacetimes

3 Circular spacetimes

3.1 The metric

3.2 The orbit manifold

3.3 The orthogonal manifold

3.4 Weyl coordinates and the Papapetrou metric

4 The Kerr metric

4.1 The vacuum Ernst equations

4.2 Conjugate solutions

4.3 The Kerr solution

5 Electrovac spacetimes with Killing fields

5.1 The stress-energy tensor

5.2 Maxwell's equations with symmetries

5.3 The electrovac variational principle

5.4 The electrovac circularity theorem

5.5 The circular Einstein-Maxwell equations

5.6 The Kerr-Newman solution

6 Stationary black holes

6.1 Basic definitions

6.2 The strong rigidity theorem

6.3 The weak rigidity theorem

6.4 Properties of Killing horizons

6.5 The topology of the horizon

7 The four laws of black hole physics

7.1 The zeroth law

7.2 The first law

7.3 The second law

7.4 The generalized entropy

8 Integrability and divergence identities

8.1 The circularity theorem

8.2 The staticity theorem

8.3 Divergence identities

8.4 Static identities and further applications

9 Uniqueness theorems for nonrotating holes

9.1 The Israel theorem

9.2 Uniqueness of the Schwarzschild metric

9.3 Uniqueness of the Reissner-Nordstrom metric

9.4 Uniqueness of the magnetically charged Reissner-Nordstrom metric

9.5 Multi black hole solutions

10 Uniqueness theorems for rotating holes

10.1 Outline of the reasoning

10.2 The Ernst system and the Kinnersley group

10.3 The uniqueness proof

10.4 The Robinson identity

10.5 Appendix: The sigma model lagrangian

11 Scalar mappings

11.1 Mappings between manifolds

11.2 Harmonic mappings

11.3 Skyrme mappings

11.4 The SU(2) Skyrme model

11.5 Conformal scalar fields

12 Self-gravitating harmonic mappings

12.1 Staticity and circularity

12.2 Nonexistence of soliton solutions

12.3 Uniqueness of spherically symmetric black holes

12.4 Divergence identities

12.5 The uniqueness theorem for nonrotating black holes

12.6 The uniqueness theorem for rotating black holes

References

Index

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