Oscillations and Waves: in Strong Gravitational and Electromagnetic Fields

371 Pages·1991·9.546 MB·English

by  Professor Dr. Nail R. Sibgatullin (auth.)

Texts and
Monographs
in Physics
W. Beiglbock
J.L. Birman
R.P. Geroch
E.H. Lieb
T. Regge
W. Thirring
Series Editors
Nail R.Sibgatullin
Oscillations
and Waves
in Strong Gravitational
and Electromagnetic Fields
With 26 Figures
Spri nger-Verlag
Berlin Heidelberg New York
London Paris Tokyo
Hong Kong Barcelona
Professor Dr. Nail R. Sibgatullin
Faculty of Mechanics and Mathematics, Moscow State University
SU-119899 Moscow, USSR
Translator:
Dr. Nathaniel M. Queen
Department of Mathematics, The University of Birmingham
P.O. Box 363, Birmingham B15 2IT, United Kingdom
Editors
Wolf Beiglb6ck Elliott H. Lieb
Institut fiir Angewandte Mathematik Department of Physics
Universitiit Heidelberg Joseph Henry Laboratories
1m Neuenheimer Feld 294 Princeton University
D-6900 Heidelberg 1 Princeton,NJ08540, USA
Fed. Rep. of Germany
Joseph L. Birman Tullio Regge
Department of Physics, The City College Istituto di Fisca Teorica
of the City University of New York Universita di Torino, C. so M. d'Azeglio, 46
New York, NY 10031, USA 1-10125 Torino, Italy
Robert P. Geroch Walter Thirring
Enrico Fermi Institute Institut fiirTheoretische Physik
University of Chicago der UniversitiitWien, Boltzmanngasse 5
5640 EllisAve. A-1090Wien, Austria
Chicago, IL 60637, USA
Title of the original Russian edition:
Kolebaniya i volny v silnykh gravitatsionnykh i elektromagnitnykh polyakh
© Nauka, Moscow 1984
ISBN-13: 978-3-642-83529-2 e-ISBN-13: 978-3-642-83527-8
DOl: 10.1007/978-3-642-83527-8
Library of Congress Cataloging·in·Publication Data. Sib~atullin, N. R. (Nail' Rakhimovich) [KolebaniG i
volny v sil 'nykh gravitatsionnykh i elektromagnitnykh pohakh. English 1O scillations and waves in strong gra
vitational and electromagnetic fields / Nail R. Sibgatullin. p. cm.-(Texts and monographs in physics) Trans
lation of: Kolebani2. i volny sil 'nykh gravitatsionnykh i elektromagnitnykh polGkh. Includes bibliographi
cal references. ISBN 0-387-19461-4 (U.S.) 1. Gravitational fields. 2. Electromagnetic fields. 3. Perturbation
(Quantum dynamics) I. Title. II. Series. QB283.S5513 1990 530.1 '4-dc20 89-11542
This work is subject to copyright. All rights are reserved, whether the whole or part of the material is
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<{;J Springer-Verlag Berlin Heidelberg 1991
Softcover reprint ofthe hardcover 1st edition 1991
The use of registered names, trademarks, etc. in this publication does not imply, even in the absence
of a specific statement. that such names are exempt from the relevant protective laws and regulations
and therefore free for general use.
57/3140-543210 - Printed on acid-free paper
Dedicated to the memory of my father,
Rakhim Kharrulovich Sibgatullin
Preface to the English Edition
This book is an updated and modified translation of the Russian edition of 1984.
In the present edition, certain sections have been abridged (in particular, Sects. 6.1
and 8.3) and the bibliography has been expanded. There are more detailed discus
sions of the group properties of integrable systems of equations of mathematical
physics (Sect. 3.4) and of the Riemannian problem in the context of the infinite
dimensional internal symmetry groups of these systems of equations. There is an
extended discussion of the reasons for the acceleration and retardation of pulsars
in connection with more recent achievements of X-ray astronomy. Part of the
material of Chap. 8 of the Russian edition has been included in Chap. 7; thus the
number of chapters has been reduced to seven.
S. Chandrasekhar set for me an example of brilliant analytical penetration into
the essence of physical problems, and my book touches on his work in many in
stances. The results of modem quantum theories of strong fields are not presented,
but they can be found in the fundamental monographs Quantwn Electrodynamics
of Strong Fields by W. Greiner, B. Muller, J. Rafelski (Sprioger-Verlag, Berlin,
Heidelberg, New York 1985) and Quantwn Effects in Intense External Fields [in
Russian] by A. Grib, S. Mamaev, W. Mostepanenko (Energoatomizdat, Moscow
1988).
This book was translated by Dr. N. M. Queen; I am very grateful to him.
I thank sincerely H. Latta, C.-D. Bachem, V. Rehman, S. von Kalckreuth for
preparing of the english manuscript.
Moscow, August 1990 Nail R. Sibgatullin
Preface
"The current of life which flows day and night in my veins flows in the
universe and dances a measured dance."
RabindraNJ( Tagor
This book is devoted to the investigation of waves and oscillations in the presence
of strong gravitational and electromagnetic fields.
A detailed study is made of the propagation of gravitational and electromag
netic waves in the pseudo-Riemannian manifolds of the general theory of rela
tivity. Much attention is given to the classical problems of the theory of black
holes and waves in their vicinity. Rigorous results of the mathematical theory
of black holes as well as of stationary axially symmetric fields are expounded,
and the properties of electromagnetic and gravitational waves propagating in the
gravitational fields of charged and neutral black holes are analyzed in detail.
An account is given of the principles of relativistic hydrodynamics, magne
tohydrodynamics, and the acoustics of a relativistic gas. Scale-invariant motions
of an ultrarelativistic gas are analyzed in detail in the framework of the spe
cial and general theories of relativity. An outline is given of the theory of the
equations of state of an ideal gas under strong compression, and also at high
temperatures. The development of nonhomogeneities in models of the Universe
with a cosmological magnetic field is investigated. The reader who is interested
only in problems of relativistic hydrodynamics can confine himself to Chaps. 5
and 6 of the book, where a brief introduction to cosmology is given at the same
time. Chapter 2 represents an introduction to the classical theory of black holes.
Chapters 1 and 4 contain an account of the principles of the wave dynamics
of gravitational and electromagnetic fields in general relativity. Some acoustic
phenomena in strong gravitational fields and manifestations of weak nonlinearity
for oscillations and waves in restricted systems in external electromagnetic and
gravitational fields are considered in Chap. 7.
As a guide for the reader, each chapter is prefaced with an elementary intro
duction to the physical problems.
A knowledge of the elements of tensor analysis is sufficient for reading the
book. From the material, the novice reader can, if he wishes, master the various
mathematical methods of the theory of waves in Newtonian mechanics and in
the special and general theories of relativity, and can gain an idea of the new
results in this field.
The author has made use of material from lectures given by him in the
Faculty of Mechanics and Mathematics at Moscow University. A bibliography
X Preface
is provided to enable the reader to make a detailed study of problems related to
the subjects of the book but treated here with insufficient completeness.
The author is deeply grateful to Academician L. I. Sedov, who suggested
that this book be written, for numerous fruitful discussions of the problems
considered, and to Professors V.I. Arnol'd, A. A. Starobinsky and R. A. Sunyaev
for valuable remarks on the manuscript.
The author expresses sincere gratitude to the editor, V. V. Rozantseva, for
her work on improving the manuscript, and to Drs. G. A. Alekseev and Alberto
Garsia for a number of helpful remarks.
Moscow, December 1983 Nail R. Sibgatullin
Contents
1. Gravitational Waves in Strong Gravitational Fields . . . . . . . . . . 1
1.1 Formalism of Complex Null Tetrads and Petrov Classification
of Algebraic Types of the Weyl Tensor .......•.......... 1
1.2 Gravitational Waves and Generalized Solutions
of the Equations of the Electrovacuum ................... 17
1.2.1 Properties of Families of Isotropic Geodesics
in General Relativity ........................... 20
1.2.2 Propagation of Breaks in the Gravitational Field
and Their Algebraic Classification ................ 23
1.2.3 Decay of an Arbitrary Break
in the Vacuum Gravitational Field ................ 30
1.2.4 The Interaction of Short Gravitational
and Electromagnetic Waves in Arbitrary External
Electromagnetic Fields ..... . . . . . . . . . . . . . . . . . . . . 32
1.2.5 Algebraic Structure of Perturbations
of the Weyl Tensor
in the Case of High-Frequency Waves ............. 39
1.2.6 Behavior of Short-Wave Perturbations
of the Gravitational Field Near Caustic Surfaces 40
1.3 Interaction of Gravitational and Electromagnetic Waves 45
1.3.1 Curvature of Space-Tune in a Plane
Electromagnetic Wave .......................... 45
1.3.2 Nonlinear Interaction of Plane Waves ............. 48
1.3.3 Propagation of Weak Electromagnetic
and Gravitational Waves in the Field
of a Strong Electromagnetic Wave .•.............. 52
1.3.4 Oscillatory Character of Solutions Near a Singularity 56
1.4 Conditions on Surfaces with Strong Breaks in Theories
of the Gravitational Field . . . . . . . . . . • . . . . . . . . . . . . . . . . . . 59
2. The Classical Theory of Black Holes . . . . . . . . . . . . . . . . . . . . . . . 65
2.1 Asymptotically Flat Gravitational Fields . . . • . . . . . . . . . . • . . 7(]
2.2 Basic Elements of the Theory of Lie Groups and Exterior Forms 7f.
2.2.1 The Concept of Lie Groups ..................... 7f.
2.2.2 The Concept of Skew and Differential Forms ....... 78
2.2.3 Frobenius's Theorem •...•...................... 8(]
2.3 Stationary Gravitational Fields ...•..........•......•... 82
XII Contents
2.4 Energetics of Black Holes ............................. 103
2.4.1 Temperature of a Black Hole .................... 103
2.4.2 Electrostatic Potential of a Black Hole . . . . . . . . . . . . 104
2.4.3 Formula for the Mass of a Black Hole ... . . . . . . . . . 105
2.4.4 "Thermodynamics" of Black Holes ............... 107
3. Stationary Axially Symmetric Fields in General Relativity 113
3.1 Canonical Equations of Massless Fields Admitting Abelian
Two-Parameter Groups of Motions ...................... 114
3.2 Infinite-Dimensional Algebra and Lie Group of the Equations
for the Neutrino Electrovacuum ........................ 125
3.3 General Solution of the Einstein-Maxwell Equations
for Ernst Data Regular Locally on the Symmetry Axis ...... 137
3.4 Lie-Backlund Groups of Integrable Systems
of Mathematical Physics .............................. 157
4. Propagation of Waves in the Gravitational Fields of Black Holes 168
4.1 Propagation of Short Waves in the Field
of a Charged Black Hole .............................. 170
4.1.1 Short Waves in the Nordstrom-Reissner Field ....... 170
4.1.2 Short Waves in the Neighborhood
of a Rotating Charged Black Hole ................ 174
4.2 Asymptotic Theory of Scattering of Wave Packets
in the Gravitational Field of a Black Hole ................ 176
4.3 Wave Fields Outside a Collapsing Star .................. 184
4.3.1 Derivation of the Basis Equations ................ 185
4.3.2 Boundary Conditions and General Properties
of the Reflection and Transmission Coefficients
of Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189
4.3.3 Properties of Radiation Emitted by a Collapsing Body
Near the Horiwn .............................. 192
4.3.4 Behavior of the Transmission Coefficient for Small w 193
4.3.5 Laws of Attenuation of the "Tails"
of the Multipole Radiation ...................... 199
5. Relativistic Hydrodynamics .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202
5.1 Relativistic Dynamics of a Point and Gas of Free Particles .. 203
5.2 Thermodynamic Equilibrium in an Ideal Gas ............. 205
5.3 Relativistic Dynamics and Acoustics of an Ideal Gas ....... 212
5.3.1 Shock Waves and Vortex Motions of an Ideal Gas ... 214
5.3.2 Potential Motions .............................. 215
5.3.3 Acoustic Waves in a Relativistic Gas .............. 216
5.3.4 Nonlinear Acoustics of an Expanding Universe:
Relativistic Theory ............................ 218
5.4 Relativistic Magnetohydrodynamics ..................... 221
xm
Contents
5.4.1 Shock Waves in Magnetohydrodynamics
and the Hugoniot Adiabat ...................... . 223
5.4.2 Properties of MHO Breaks ..................... . 225
5.4.3 Relative Positions of the Poisson and Hugoniot Adiabats 227
5.5 Hydrodynamical Flow Resulting from Production
of Ultrarelativistic Particles in the Field of a Black Hole .... 229
5.6 Self-Similar Motions of an Ultrarelativistic Gas
with Spherical or Cylindrical Symmetry ................ . 231
5.6.1 Qualitative Investigation of (5.6.5) ............... . 233
5.6.2 Ejection of Matter from a Singular Point (Axis)
=
at the Instant t 0 ............................ . 235
5.6.3 Solution of the Cauchy Problem 236
6. Some Problems of the Dynamics of Waves
in Relativistic Cosmology ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239
6.1 Development of Inhomogeneities in Models of the Universe
with a Cosmological Magnetic Field .................... 243
6.1.1 Unperturbed Solution ........... . . . . . . . . . . . . . . . 244
6.1.2 Notation for Small Perturbations
and Coordinate Restrictions .... . . . . . . . . . . . . . . . . . 244
6.1.3 Equations of Conservation of Energy-Momentum
and of the Magnetic Induction ................... 246
6.1.4 A Closed System for the Odd Perturbations ........ 247
6.1.5 A Closed System for the Even Perturbations ........ 248
6.1.6 Exact Solutions of the Linearized Equations
Corresponding to Perturbations
of the Coordinate System ....................... 248
6.1.7 Closed System of Equations for the Even Perturbations 250
6.1.8 Analysis of the Closed System of Equations
for Odd Perturbations .......................... 251
6.1.9 Analysis of the Closed System for Even Short-Wave
Perturbations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251
6.1.10 Evolution of Perturbations of Arbitrary Finite Scales
Near a "Pancake" Singularity .................... 253
6.2 Self-Similar Motions of a Photon Gas
in the Friedman-Lemaitre Model .. . . . . . . . . . . . . . . . . . . . . . 255
6.2.1 Derivation of a Closed System of Ordinary Differential
Equations and Conditions on Shock Waves ......... 257
6.2.2 Friedman Solution and Qualitative Investigation
of the System of Equations for x(O and V(O ...... 259
6.2.3 Discussion of the Results ....................... 264
7. Acoustic Phenomena in Strong Gravitational and Magnetic Fields 269
7.1 Propagation of Nonlinear Short Acoustic Waves ........... 271
7.1.1 Derivation of the Model Equations ....... . . . . . . . . 271