Table Of Content

Magnetohydrodynamics: Waves and Shock Waves in Curved Space-Time MATHEMATICAL PHYSICS STUDIES Series Editor: M. PLATO, Universite de Bourgogne, Dijon, France VOLUME 14 Magnetohydrodynamics: Waves and Shock Waves in Curved Space-Time by Andre Lichnerowicz Chaire de Physique Mathematique, College de France, Paris, France SPRINGER-SCIENCE+BUSINESS MEDIA, B.V. Library of Congress Cataloging-in-Publication Data Lichnerowicz, Andre, 1915- Magnetohydrodynamics : waves and shock waves in curved space-time by Andre Lichnerowicz. p. em. -- <Mathematical physics studies ; v. 14J ISBN 978-90-481-4390-0 ISBN 978-94-017-2126-4 (eBook) DOI 10.1007/978-94-017-2126-4 1. Magnetohydrodynamics. 2. Shock waves. 3. Calculus of tensors. 4. Mathe~atical physics. I. Title. II. Series. OC718.5.M36L53 1994 538' . 6--dc20 94-7970 CIP ISBN 978-90-481-4390-0 Printed on acid-free paper All Rights Reserved © 1994 Springer Science+B usiness Media Dordrecht Originally published by Kluwer Academic Publishers in 1994 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. Table of Contents Preface. XI Chapter - Tensor Distributions. I. 1. - Tensor distributions over a pseudo-Riemannian manifold .......................................................................... .. 1.2. - Covariant derivative of a tensor distribution.................. 4 6 1.3. - The distributions y+, y- and corresponding to a hypersurface........................... ... .. .. . ... .. .... .... .. .. .. .. .. .. ... ... .. 6 1.4.- Tensor discontinuities relative to a hypersurface.......... 10 1.5.- Generalization of V'T0 .................................................... 12 I. 6. - The case of continuous tensors......................................... 1 4 Chapter II • Maxwell's equations and electromagnetic waves over a curved space-time. II. 1. - Space-time....................................................................... 18 11.2. - Singular 2-form.............................................................. 19 11.3. - Maxwell's equations and the formal Cauchy problem...... 21 11.4. - Maxwell's equations and potential-vector....................... 25 11.5. - Ordinary electromagnetic waves........ ............................. 28 v vi TABLE OF CONTENTS Chapter Ill - Einstein's equations and the leray theorem. Ill. 1. - The system of Einstein equations...................................... 33 111.2. Harmonic quantities......................................................... 38 II I .3. - Einstein's equations and harmonic cherts........................ 41 111.4.- Strictly hyperbolic matrices........................................... 44 111.5.- Leray systems.................................................................. 47 11 I .6. - The Einstein equations in the exterior case..................... 49 II I .7. - The case cf pure matter.................................................... 53 Chapter IV - Gravitational and electromagnetic shock waves. IV. 1. Gravitational shock waves............................................... 60 IV.2. Singular double 2-form....... .................................... ...... . 64 IV.3.- Einstein's equations in the weak sense. Conditions of shock................................................................................. 68 IV.4. A geometric consequence................................................... 76 IV.S. Propagation of discontinuities.......................................... 77 IV.6. Consequences..................................................................... 81 IV.7. Electromagnetic shock waves ............................................ 84 IV.8.- Propagation of discontinuities for an electromagnetic shock wave ......................................................................... 87 IV.9. Gravitational shock wave in an electromagnetic field...... 91 IV. 10.- Gravitational and electromagnetic shock waves (with Maxwell tensor)................................................................ 94 TABLE OF CONTENTS vii Chapter V - Relativistic hydrodynamics. Y. 1. - Equations of relativistic hydrodynamics........................... 98 Y.2. - Characteristic hypersurfaces of the hydrodynamic system ................................................................................. 102 Y.3.- A Leray system and the Cauchy problem........................... 107 Y.4. Sonic waves........................................................................ 1 14 Y.5. Main property of the rays in hydrodynamics .................... 116 Y.6.- Thermodynamic hypothesis ................................................ 12 1 Chapter VI - The equations of magnetohydrodynamics. Yl.1. Electromagnetic field with induction................................ 124 Yl.2. Maxwell equations and energy tensor of Minkowski ......... 128 Yl.3. The case of the magnetohydrodynamics............................. 129 Yl.4.- The main equations of relativistic magnetohydrodynamics ...................................................... 131 Vl.5. - The equation of continuity and the system of the linesofflow ....................................................................... 133 Yl.6. Magnetosonic waves and Al fven waves............................... 13 5 Yl.7. Ma gnetoson i c speeds and Al fven speeds.............................. 14 0 Yl.8. Representations of the wave cones in !R3 ......................... 1 44 Yl.9.- Main property of the rays associated to the magnetosonic waves ............................................................ 150 VI. 10.- Property of the rays associated to the Alfven waves ......... 156 viii TABLE OF CONTENTS Chapter VII • Magnetohydrodynamic shock waves. VII. 1. - The main system of the shock waves................................. 159 VII. 2. - Analysis of the shock equations......................................... 162 Vll.3.- Newformoftheshockequationsfor ~«~«~0 ............... 168 Vll.4.- Alfven shocks.................................................................... 170 Chapter VIII • Hugoniot's function and applications. VIII. 1. - Relativistic Hugoniot's relation for ~ « ~ « ~ 0 .... .. .. .. .. .. .. 17 4 VIII.2. Differential of Hugoniot's function .................................... 177 VIII.3. Differential of S along a straight line (~« ~ « ~ 0) ......... 179 = Vlll.4.- The case where ~« ~ Cll 0 is impossible....................... 182 VIII .5. - Orientation of shock wave fronts in the time................... 185 Vll1.6. Thermodynamic inequalities for the shocks..................... 186 VIII.7. I sen tropic curves and Hugoniot's curve........................... 191 VIII.8. Speeds of shock waves....................................................... 195 Vlll.9.- Isentropic curves and weak shocks................................... 197 VI ll. 10 . - The case 1: ~ > 0.......... .. .. .. .. .. . .. .. .. .. .. .. .. . .. .. .. .. .. .. .. .. .. .. .. .. .. .. . 19 9 VIII. 1 1. - The case 1:~ < 0 ................................................................. 202 VIII. 12. - Conclusions ........................................................................ 210 Annex • Shock waves and alfven waves. AI. 1. - Singular shocks ................................................................. 211 Al.2.- Compatibility between shock waves andAlfven waves...... ... .......... ...... ........ .. .. ...... .. ...... .. ........ .... .... ........... 220 TABLE OF CONTENTS ix Annex II - Magnetosonic rays. All. 1 . - Directions of the rays....................................................... 226 All.2.- Action of 6 on the direction of the ray............................ 229 Annex Ill - Classical approximations of. the relativistic shock equations. Alii. 1.- The frame connected with the shock ......... ,....................... 232 Alll.2.- Classical approximation................................................... 232 Bibliography................................................................................... 235 Note - Approach to a quantum theory of fields for a curved space-time. I • Tensor propagators............................................................. 237 Nl.l. Orientations over a space-time........................................ 237 Nl.2. Global hyperbolicity ......................................................... 239 Nl.3. Bi tensors and Di rae bitensors.......................................... 2 40 Nl.4. Linear differential-tensor operators associated with g .. 242 Nl.5. Elementary Kernels and propagators............................... 244 Nl.6.- Tensor propagators relative to the space-time of Minkowski ......................................................................... 249 Nl.7.- Propagators relative to the operator (D. + IJ)................. 252