Post-Newtonian Hydrodynamics: Theory and Applications

412 Pages·2022·2.98 MB·English

by  Gilberto Medeiros Kremer

Post-Newtonian
Hydrodynamics

Post-Newtonian
Hydrodynamics
:
Theory and Applications
By
Gilberto Medeiros Kremer
Post-Newtonian Hydrodynamics: Theory and Applications

By Gilberto Medeiros Kremer
Departamento de Física
Universidade Federal do Paraná
Curitiba, Brazil
E-mail: [email protected]

This book first published 2022

Cambridge Scholars Publishing

Lady Stephenson Library, Newcastle upon Tyne, NE6 2PA, UK

British Library Cataloguing in Publication Data
A catalogue record for this book is available from the British Library

Copyright © 2022 by Gilberto Medeiros Kremer

All rights for this book reserved. No part of this book may be reproduced,
stored in a retrieval system, or transmitted, in any form or by any means,
electronic, mechanical, photocopying, recording or otherwise, without
the prior permission of the copyright owner.

ISBN (10): 1-5275-7969-7
ISBN (13): 978-1-5275-7969-9
To Maria Rachel

Contents
PREFACE xiii
1 THE BOLTZMANN EQUATION:
AN OVERVIEW 1
1.1 Non-relativistic Boltzmann equation . . . . . . . 2
1.2 Boltzmann equation in special relativity . . . . . 7
1.3 Boltzmann equation in
gravitational fields . . . . . . . . . . . . . . . . . 14
2 FIRST POST-NEWTONIAN
APPROXIMATION 27
2.1 Preliminaries . . . . . . . . . . . . . . . . . . . . 28
2.2 The first post-Newtonian approximation . . . . . 31
2.3 The solution of Einstein’s field
equations . . . . . . . . . . . . . . . . . . . . . . 37
2.3.1 The Weinberg method . . . . . . . . . . . 37
2.3.2 Explicit expressions for the components . 42
vii
2.3.3 The Chandrasekhar method . . . . . . . . 47
2.4 Hydrodynamic equations for an
Eulerian fluid . . . . . . . . . . . . . . . . . . . . 52
2.5 Brans-Dicke post-Newtonian
approximation . . . . . . . . . . . . . . . . . . . 58
2.5.1 Brans-Dicke theory . . . . . . . . . . . . . 58
2.5.2 Post-Newtonian Brans-Dicke theory . . . 66
2.5.3 Hydrodynamic equations for an
Eulerian fluid . . . . . . . . . . . . . . . 75
2.6 Non-perfect fluid hydrodynamic
equations . . . . . . . . . . . . . . . . . . . . . . 78
2.7 The gravitational potentials . . . . . . . . . . . . 86
2.8 The conservation laws . . . . . . . . . . . . . . . 92
2.9 The post-Newtonian virial
theorem . . . . . . . . . . . . . . . . . . . . . . . 104
3 SECOND POST-NEWTONIAN APPROXIMA-
TION 111
3.1 Preliminaries . . . . . . . . . . . . . . . . . . . . 112
3.2 Equation for determination Ψ . . . . . . . . . . 115
ij
3.3 Equation for determination Ψ . . . . . . . . . . 120
0i
3.4 Equation for determination Ψ . . . . . . . . . . 122
00
3.5 A gauge choice . . . . . . . . . . . . . . . . . . . 124
3.6 Hydrodynamic equations for an
Eulerian fluid . . . . . . . . . . . . . . . . . . . . 125
3.7 Conservation laws in general relativity . . . . . . 138
3.7.1 Energy-momentum pseudo-tensor of the
gravitational field . . . . . . . . . . . . . . 140
viii
3.7.2 The total linear momentum density con-
servation . . . . . . . . . . . . . . . . . . 145
3.7.3 The total energy density conservation . . 148
4 POST-NEWTONIAN KINETIC THEORY 163
4.1 First post-Newtonian approximation . . . . . . . 164
4.1.1 Post-Newtonian Boltzmann equation . . . 164
4.1.2 Post-Newtonian Maxwell-Ju¨ttner
distribution function . . . . . . . . . . . . 167
4.1.3 Post-Newtonian macroscopic fields . . . . 169
4.1.4 Post-Newtonian transfer and Eulerian
hydrodynamic equations . . . . . . . . . 173
4.2 Second post-Newtonian
approximation . . . . . . . . . . . . . . . . . . . 176
4.2.1 Post-Newtonian Boltzmann equation . . . 177
4.2.2 Post-Newtonian Maxwell-Ju¨ttner
distribution function . . . . . . . . . . . . 179
4.2.3 Post-Newtonian macroscopic fields . . . . 181
4.2.4 Post-Newtonian hydrodynamic
equations . . . . . . . . . . . . . . . . . . 185
4.3 Post-Newtonian Jeans equations . . . . . . . . . 190
4.3.1 Stationary and spherically symmetrical
self-gravitating systems . . . . . . . . . . 191
4.3.2 Stationary and axisymmetrical
self-gravitating systems . . . . . . . . . . 194
5 STELLAR STRUCTURE MODELS 203
5.1 The polytropic equation of state . . . . . . . . . 204
5.2 Stellar mean molecular weight . . . . . . . . . . . 207
ix