Magnetohydrodynamics: Waves and Shock Waves in Curved Space-Time

292 Pages·1994·5.548 MB·English

by  André Lichnerowicz (auth.)

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