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Texts and Monographs in Physics W. Beiglbock J.L. Birman R.P. Geroch 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 1Oscillations 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 concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. Duplication of this publication or parts thereof is only permitted under the provisions ofthe German Copyright Law of September 9,1965, in its current version, and a copyright fee must always be paid. Violations fall under the prosecution act of the German Copyright Law. <{;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 .....................
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