Fundamental Theories of Physics

Transcription

1 Black Hole Physics

2 Fundamental Theories of Physics An International Book Series on The Fundamental Theories 0/ Physics: Their Clarification, Development and Application Editor: ALWYN VAN DER MERWE, University o/denver, U.S.A. Editoral Advisory Board: LAWRENCE P. HORWITZ, Tel-Aviv University, Israel BRIAN D. JOSEPHSON, University o/cambridge, U.K. CLIVE KILMISTER, University 0/ London, U.K. PEKKA J. LAHTI, University o/turku, Finland GUNTER LUDWIG, Philipps-Universitiil, Marburg, Germany ASHER PERES, Israel Institute o/technology, Israel EDUARD PRUGOVECKI, University o/toronto, Canada MENDEL SACHS, State University 0/ New York at Buffalo, U.S.A. ABDUS SALAM, International Centre/or Theoretical Physics, Trieste, Italy HANS-JURGEN TREDER, Zentralinstitutfiir Astrophysik der Akademie der Wissenschaften, Germany Volume 96

3 Black Hale Physics Basic Concepts and New Developments by Valeri P. Frolov Department of Physics, University of Alberta, Edmonton, Alberta, Canada and Igor D. Novikov Theoretical Astrophysics Center, University ofcopenhagen, Copenhagen, Denmark SPRINGER SCIENCE+BUSINESS MEDIA, B.V.

4 A C.I.P. Catalogue record for this book is avaiiable from the Library of Congress. ISBN ISBN (ebook) DOI / Printed on acid-free paper All Rights Springer Science+Business Media Dordrecht Originally published by Kluwer Academic Publishers in 1998 Softcover reprint ofthe hardcover Ist edition 1998 No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording or by any information storage and retrieval system, without written permission from the copyright owner

5 To our parents

6 Contents Preface xix I Basic Concepts 1 1 Introduction: Brief History of Black Hole Physics 3 2 Spherically Symmetric Black Holes 2.1 Spherically Symmetric Gravitational Field Spherically Symmetric Gravitational Field in Vacuum Schwarzschild metric Schwarzschild reference frame Radial Motion of Test Particles in the Schwarzschild Field Radial motion of light Radial motion of particles Spacetime Within the Schwarzschild Sphere Lemaitre reference frame R- and T-regions Eddington-Finkelstein coordinates. 2.5 Contracting and Expanding T-Regions Formation of a Black Hole in a Gravitational Collapse Gravitational collapse White holes Eternal Black and White Holes Incompleteness of the Lemaitre frame Complete empty spherically symmetric spacetime Kruskal coordinates Celestial Mechanics in the Gravitational Field of the Black Hole Equations of motion of a free test particle Circular motion Motion of an ultra-relativistic particle Non-inertial circular motion

7 V III CONTENTS 2.9 Gravitational Capture The Motion of Particles Corrected for Gravitational Radiation Tidal Interaction of Extended Bodies with a Black Hole Equations of motion Tidal forces Rotating Black Holes Formation of a Rotating Black Hole Gravitational Field of a Rotating Black Hole Kerr metric The (3 + I)-split of the spacetime outside the black hole Reference Frames. Event Horizon Chronometric reference frame Ergosphere. Event horizon Reference frame of locally non-rotating observers Celestial Mechanics Near a Rotating Black Hole Equations of motion. First integrals General properties Motion in the equatorial plane Motion off the equatorial plane Gravitational capture Spacetime of a Rotating Black Hole Charged Rotating Black Holes Kerr-Newman geometry Motion of test particles Problem of "visualization" of Black Holes and the Membrane Paradigm 84 4 Black hole Perturbations (Written jointly with N. Andersson) Introduction Weak Fields in the Schwarzschild Metric Scalar field near a spherically symmetric black hole A useful set of basic solutions Weak fields of higher spins Gravitational perturbations of a Schwarzschild black hole Solutions in the high- and low-frequency limits Evolution of Wave Fields Around a Black Hole Early "numerical relativity" A Green's function analysis Quasinormal Modes Simple approximations The complete spectrum of black hole modes Quasinormal modes for charged and rotating black holes 105

8 CONTENTS IX Quasinormal-mode contribution to the radiation Power-Law Tails Late-time behavior Analyzing the Green's function Gravitational Radiation from a Test Particle in the Field of a Black Hole Particles plunging into the black hole Scattering of Waves by Black Holes The scattering problem Approximate results Wave scattering The complex angular momentum paradigm Wave Fields around a Rotating Black Hole The Teukolsky equation Superradiant scattering Radiation from a test particle moving in the Kerr background Scatt.ering of waves by Kerr black holes St.ability of Black Holes Gravitational Waves from Binary Systems The in spiraling phase Black hole collisions General Properties of Black Holes Asymptotically Flat Spacetirnes Asymptotic properties of Minkowski spacet.ime Definition and properties of asymptotically fiat spacetime Penrose-Carter conformal diagrams Bondi-Metzner-Sachs group of asymptotic symmetries Massless fields in asymptotically fiat. spacetime Event Horizon and its Properties Event horizon Penrose theorem Surface topology of black holes Ehlers-Sachs Theorem Light wavefront and light rays Optical scalars. Ehlers-Sachs theorem Focusing theorem Hawking's Area Theorem Trapped Surfaces. Apparent Horizon "Teleological nature" of the event horizon Trapped surfaces R- and T-regions. Apparent horizon Dynamics of apparent horizons. Numerical results 181

9 x CONTENTS 5.6 Theorems on Singularities Inside Black Holes Singularities in general relativity Theorems on singularities Cosmic Censorship Conjecture. Critical Behavior in Black Hole Formation Cosmic censorship conjecture Hoop conjecture Critical behavior in black hole formation Can One See What Happens "Inside a Black Hole"? Formulation of the problem Wormholes in the Schwarzschild geometry Causal structure of spacetime with a wormhole Energy and information extraction Stationary Black Holes "Black Holes Have No Hair" General Properties of Stationary Black Holes Stationary spacetime with a black hole Static black holes Penrose process Stationary nonstatic black holes are axisymmetric Killing Horizon Definition and properties of the Killing horizon Surface gravity Constancy of surface gravity on H Angular velocity Electric field potential Uniqueness Theorem for Static Black Holes Uniqueness Theorem for Stationary Black Holes Analytic Continuation of the Kerr-Newman Metric Inside the Event Horizon Generalization of the Uniqueness Theorems Physical Effects in the Gravitational Field of a Black Hole Extraction of Energy from a Black Hole Irreducible mass Black hole as an amplifier Electromagnetic Field of a Test Charge Electric field of a point-like charge in the black hole exterior Electromagnetic field in the black hole interior Comparison with the field of a uniformly accelerated charge The shift in the self-energy

10 CONTENTS XI 7.3 Mutual Transformation of Electromagnetic and Gravitational Waves Geometrical optics approximation Mutual transformation of photons and gravitons Interaction of Cosmic Strings with a Black Hole Gravitational capture of cosmic strings by a black hole Stationary string near a rotating black hole Separation of variables Uniqueness theorem 7.5 Black Hole in an External Field Black hole in an external field Perturbation theory Deformed black holes Interaction Between Black Holes Interaction of two non-relativistic black holes Interaction of relativistically moving black holes Momentarily static black holes configurations 7.7 Black Hole Collisions Head-on collision. Numerical Results Event horizon structure for colliding black holes Black Hole Electrodynamics Introduction Maxwell's Equations Black hole electrodynamics and membrane paradigm Maxwell's equations in (3+1)-form Stationary Axisymmetric Electrodynamics Invariant variables Fields in the plasma surrounding a black hole Magneto-hydrodynamic approximation Singular surfaces Membrane Interpretation and "Stretched" Horizon Boundary conditions at the event horizon Slow change of black hole parameters Electromagnetic Fields in Vacuum Near a Black Hole Electric field of a point-like charge Black hole in a homogeneous magnetic field Magnetosphere of a Black Hole Magnetospheric models Efficiency ofthe power-generation process near a rotating, magnetized black hole Black hole as a unipolar inductor

11 XII CONTENTS 9 Astrophysics of Black Holes 9.1 Introduction The Origin of Stellar Black Holes Stellar Black Holes in the Interstellar Medium 9.4 Disk Accretion onto Black Holes Evidence for Black Holes in Stellar Binary Systems 9.6 Supermassive Black Holes in Galactic Centers Dynamical Evidence for Black Holes in Galactic Nuclei 9.8 Primordial Black Holes Black Holes and Gravitational Wave Astronomy (Written jointly with N. Andersson) II Further Developments Quantum Particle Creation by Black Holes 10.1 Quantum Effects in Black Holes Introduction Particle creation Black hole evaporation Vacuum polarization Quantum fluctuations of the metric 10.2 Particle Creation by Black Holes General theory Modes and bases Bogoliubov transformations and S-matrix Rotating black holes and higher spins Density Matrix and Generating Functional Density matrix Generating functional Particular Cases Hawking effect Stimulated radiation Scattering of coherent waves Probability distribution Black hole in a "thermal bath" 10.5 Energy, Angular Momentum, and Entropy Emission Loss of energy and angular momentum Entropy of black hole radiation Radiation of a charged rotating black hole

12 CONTENTS XliI 11 Quantum Physics of Black Holes 11.1 Vacuum Polarization near Black Holes Semi-classical approximation Wald's axioms Point-splitting method Conformal trace anomaly Choice of State and Boundary Conditions for Green's Functions Unruh vacuum Hartle-Hawking vacuum Boulware vacuum Mode expansion for Hadamard's functions in the black hole exterior (T;:)ren and (rj;2)ren in the Spacetime of a Black Hole Christensen-Fulling representation Asymptotic values of (T,;L)rCn and (rj;2)rcn at the horiwn and at infinity Numerical results Thermal atmosphere of black holes 1l.3.5 Analytical approximations Exact results Vacuum polarization of massive fields 11.4 Quantum Mechanics of Black Holes Introduction Euclidean approach The no-boundary wavefunction of a black hole Creation of black hole pairs by an external field Thermodynamics of Black Holes 12.1 Black Holes and Thermodynamics Mass Formulas Integral mass formula Differential mass formula Four Laws of Black Hole Physics Four laws of black hole thermodynamics Generalized second law Entropy as the Noether charge Black Hole as a Thermodynamic System Equilibrium of a black hole and thermal radiation in a box Heat capacity. Thermal fluctuations Euclidean Approach to Black Hole Thermodynamics Euclidean formulation Boundary conditions

13 xiv CONTENTS Calculation of the Euclidean action Thermodynamical parameters Conical singularity method Statistical-Mechanical Foundations of Black Hole Thermodynamics Introduction Black hole entropy Sakharov's induced gravity and black hole entropy Black hole entropy in superstring theory Black Holes in Unified Theories 13.1 Non-Einsteinian Black Holes Introduction Low-energy effective action in string theory 13.2 Four-Dimensional Black Holes Dilaton black holes Black holes with non-abelian hair Quantum hair Lower-Dimensional Black Holes Three-dimensional black holes Two-dimensional black holes Multi-Dimensional Black Holes Einsteinian multi-dimensional black holes D-dimensional black holes in string theory Kaluza-Klein black holes The Interior of a Black Hole 14.1 Introduction Physical Fields Inside a Schwarzschild Black Hole Instability of Cauchy Horizons Inside a Black Hole Interior of a charged spherical black hole Linear instability of a Cauchy horizon of a charged spherical black hole Instability of the Cauchy horizon of a rotating black hole 14.4 Structure of a Classical Black Hole Interior Formulation of the problem and overview Spherical symmetric charged black hole. Vaidya solution Mass inflation More realistic models of the classical black hole interior General structure of a classical black hole interior Quantum-Electrodynamical Instability of Cauchy Horizons 14.6 Complete picture? New Worlds Inside a Black Hole?

14 CONTENTS xv 15 Ultimate Fate of Black and White Holes Role of Planck Scales in Black Hole Physics White Hole Instability Classical instability Quantum instability What is Left After the Quantum Decay of a Black Hole Possible outcomes of black hole evaporation Elementary black holes: maximon, friedmon, and so on Virtual black holes Information Loss Puzzle Black Holes, Wormholes, and Time Machines Topological and Causal Structure of Spacetime Locally Static Multiply Connected Spacetimes Non-potential gravitational fields Clock synchronization Spacetimes with Closed Timelike Curves Chronology horizon Possible obstacles to creation of a time machine Classical Physics in the Presence of Closed Timelike Curves Chronology Protection Conjecture Quantum Theory and Time Machines Conclusion 617 Appendices 618 A Mathematical Formulas 619 A.1 Differential Manifold. Tensors 619 A.2 Metric. Space and Time Intervals 620 A.3 Causal Structure A.4 Covariant Derivative 622 A.5 Geodesic Lines A.6 Curvature A.7 Lie- and Fermi-Transport. 625 A.8 Symmetries and Conservation Laws 626 A.9 Geometry of Congruence of Lines 627 A.10 Stationary Congruences A.10.1 Killing congruence A.10.2 Congruence of locally non-rotating observers 630 A.11 Local Reference Frames. 631 A.12 Geometry of Subspaces

15 XVI A.13 Integration in Curved Space A.14 Conformal Transformations A.15 Einstein Equations... CONTENTS B Spherically Symmetric Spacetimes B.1 Spherically Symmetric Geometry B.2 Reduced Action.... B.3 Generalized Birkhoff's Theorem.. B.4 Spherically Symmetric Vacuum Solutions B.4.1 Schwarzschild metric B.4.2 Scaling properties B.5 Kruskal Metric B.5.1 Derivation of Kruskal metric. 643 B.5.2 Relation between Kruskal and Schwarzschild metrics 645 B.5.3 Kruskal spacetime as maximal analytical continuation of the Schwarzschild metric. 646 B.5.4 Einstein-Rosen bridge B.6 Tolman Solution C Rindler Frame in Minkowski Spacetime C.1 Uniformly Accelerated Motion. C.2 Rindler Frame.... C.3 Light and Particle Propagation.... C.4 Maximal Analytical Extension of Rindler Spacetime D Kerr-Newman Geometry D.1 Kerr-Newman Metric D.2 Christoffel Symbol... D.3 Symmetries.... D.4 Motion of Test Particles D.4.1 Integrals of motion D.4.2 Hamilton-Jacobi method D.5 Stationary Congruences in the Kerr-Newman Geometry. D.5.1 Killing congruence.... D.5.2 Congruence of locally non-rotating observers D.6 Algebraic Properties D.7 Analytic Extension... E Newman-Penrose Formalism E.1 Complex Null Tetrad. Spin Coefficients. E.2 Covariant Derivatives. Ricci and Weyl Tensor E.3 Newman-Penrose Equations E.4 Bianchi Identities

16 CONTENTS XVII F Wave Fields in a Curved Spacetime 674 F.l Scalar Field F.2 Electromagnetic Field F.3 Gravitational Perturbations 677 G Wave Fields in the Kerr Metric 678 G.1 Teukolsky Equation G.2 Separation of Variables. Spin-Weighted Spheroidal Harmonics. 680 G.3 Radial Equation G.4 Massless Scalar Field G.5 Electromagnetic Field G.6 Gravitational Perturbations 689 H Quantum Fields in Kerr Spacetime 692 H.1 Quantum Theory in an External Field 692 H.2 Vacuum. Many-Particle States H.3 S-Matrix H.4 Massless Fields Quantization in Kerr Spacetime 698 H.4.1 IN-, UP-, OUT-, and DOWN-modes H.4.2 DN-modes H.5 Wald's Bases, Bogoliubov Transformation, and S-matrix 704 H.6 Averaging over "Non-observable" States I Quantum Oscillator 1.1 Action.... I.2 Quantization and Representations... I.3 Quantum Oscillator at Finite Temperature 1.4 Two Mode Coupled Oscillators Bibliography Index

17 Preface It is not an exaggeration to say that one of the most exciting predictions of Einstein's theory of gravitation is that there may exist "black holes": putative objects whose gravitational fields are so strong that no physical bodies or signals can break free of their pull and escape. The proof that black holes do exist, and an analysis of their properties, would have a significance going far beyond astrophysics. Indeed, what is involved is not just the discovery of yet another even if extremely remarkable, astrophysical object, but a test of the correctness of our understanding of the properties of space and time in extremely strong gravitational fields. Theoretical research into the properties of black holes, and into the possible corollaries of the hypothesis that they exist, has been carried out with special vigor since the beginning of the 1970's. In addition to those specific features of black holes that are important for the interpretation of their possible astrophysical manifestations, the theory has revealed a number of unexpected characteristics of physical interactions involving black holes. By the middle of the 1980's a fairly detailed understanding had been achieved of the properties of the black holes, their possible astrophysical manifestations, and the specifics of the various physical processes involved. Even though a completely reliable detection of a black hole had not yet been made at that time, several objects among those scrutinized by astrophysicists were considered as strong candidates to be confirmed as being black holes. Furthermore, profound links were found between black hole theory and such seemingly very distant fields as thermodynamics, information theory, and quantum theory. The branch of physics that is now referred to as black hole physics was born and actually took shape as a full-blooded scientific discipline during the past two decades at the junction of the theory of gravitation, astrophysics, and classical and quantum field theories. In 1986 we published in Russian a book "Physics of Black Holes" devoted to this relatively young and rapidly developing branch of physics. In 1989 the book was translated and published in English. During the years that have passed since then, the physics of black holes enjoyed a period of rapid growth. First of all, great progress was achieved in the observational astrophysics of black holes. Ten years ago there were only a few black hole candidates in binary systems in which the companion was a normal star. In spite of enormous efforts, no unequivocal signature of black holes had been found in these systems. Today after great improvements in the quantity and quality of the observational material, our confidence that we observe the manifestaxix

18 xx PREFACE tion of black holes in at least a few binaries is almost a hundred per cent. Moreover, impressive progress in optical, radio, and X-ray astronomy greatly bolstered the evidence for supermassive black holes (up to several billion solar mass) in the centers of galaxies. During the same period of time, great progress was made also in the development of the theoretical and mathematical aspects of black hole physics. Black hole collisions might be one of the most powerful sources of gravitational radiation in the Universe. The gravitational wave observations that might become possible after construction of the LIGO and other gravitational antennas of the new generation required developments in numerical relativity and a n a 1 yfor the i analysis ~ s of the gravitational radiation generated by black holes. More than five years ago we were requested to prepare a second edition of the book to reflect the enormous progress in our knowledge concerning black holes. We started work on this project, and quite soon we realized that it is virtually impossible to keep the book unchanged and restrict ourselves to tiny "cosmetic" improvements. As a result, in order to update the book, we practically rewrote it though we preserved its original structure and certainly used much of the old material. This monograph is the result of our attempts to understand the modern status of black hole physics. This new book is twice longer than the old one; it contains four new chapters. We also added a lot of new material in the form of appendices which cover more mathematical subject. This book is written to introduce the reader to the physics of black holes and the methods employed in it, and to review the main results of this branch of physics. But attention is focused primarily on questions that were answered relatively recently, and thus could not be adequately reflected in earlier textbooks and reviews. Those aspects that are relatively familiar are presented concisely (but as clearly as possible). We have tried to make the representation lucid not only for a specialist, but also for a broad spectrum of physicists and astrophysicists who do not have a special knowledge of black hole physics. An attempt is made to explain, first and foremost, the physical essentials of the phenomena, and only after this do we pass on to the mathematical means of describing them. These objectives determined both the spectrum of selected topics and the style of presentation. A conscious attempt has been made to avoid excessive rigor in the formulations and proofs of theorems on black holes. Quite often, only the principal idea is given instead of a complete proof; then the successive stages of the proof are outlined, and references to the original papers are supplied where the reader will find the required details. This approach was chosen not only because the excellent monographs of Penrose (1968), Hawking and Ellis (1973), Chandrasekhar (1983), and Thorne, Price and Macdonald (1986) cover most of the material omitted in this book, but also because we are of the opinion that excessive rigor stands in the way of understanding the physical ideas that are at the foundation of the specific properties of black holes. The material in the book is partly based on the courses of lectures which the authors have presented during a number of years at the University of Copenhagen

19 PREFACE xxi and University of Alberta. One of the authors (V.F.) is grateful to the Killam trust for its financial support during the work on this book. The authors also would like to thank the Danish National Research Foundation for the support of this project through its establishment of the Theoretical Astrophysics Center, and the Natural Sciences and Engineering Research Council of Canada for its financial support. While working on this monograph, we enjoyed the unfailing support of our colleagues from the P.N. Lebedev Physical Institute (Moscow), University Observatory, NORDITA, and Theoretical Astrophysics Center (Copenhagen), and the University of Alberta (Edmonton). We wish to express our sincere thanks to Marek Abramowicz, Roger Blandford, Bernard Carr, Vitaly Ginzburg, Pavel Ivanov, David Kirzhnitz, Hans-Peter Kiinzle, Vladimir Lipunov, Draza Markovic, Michael Nowak, Don Page, Christopher Pethick, Martin Rees, Kip Thorne, and John Wheeler for helpful discussions of the problems of black holes. We are much indebted to Werner Israel for numerous stimulating discussions and for his enormous work in reading the draft of the book. His advice concerning the contents and style were very important for us in our work. We want also to mention particularly Nils Andersson, who is a co-author of Chapter 4 and Section 9.9. We also appreciate the help of Vasily Beskin in preparing the new version of Chapter 8. Their help was invaluable. Additional thanks go to Patrick Sutton for his help with proofreading. This book would never have appeared if we had not received support and encouragements from our friends and colleagues. We owe the deepest debt of gratitude to our wives Raya Frolova and Eleonora Kotok. They were not only patient, but contributed countless hours helping us in the preparation of the electronic version of the book. We would also like to thank Andrei Frolov. His skills as the computer guru helped us to overcome numerous problems which are inevitable in any project of such a scale. Copenhagen and Edmonton September 1998 Valeri Frolov and Igor Novikov