Inhomogeneous Cosmological Models

334 Pages·2006·4.01 MB·English

by  Andrzej Krasiński

This unique volume provides a comprehensive survey of our understanding of
the Universe based on the exact solutions of the theory of relativity. More pre-
cisely, it describes those models that fit with astronomical observations of
galaxy clusters, cosmic voids and other key features of our Universe.
This authoritative account achieves two important goals. Firstly, it collects
together all independently derived cosmological solutions from the birth of
relativity in 1915 to the present day, and clearly shows how they are inter-
related. Secondly, it presents a coherent overview of the physical properties of
these inhomogeneous models. It demonstrates, for instance, that the forma-
tion of voids and the interaction of the cosmic microwave background radia-
tion with matter in the Universe can be explained by exact solutions of the
Einstein equations, without the need for approximations.
This book will be of particular interest to graduates and researchers in
gravity, relativity and theoretical cosmology, as well as historians of science.

INHOMOGENEOUS COSMOLOGICAL MODELS

Inhomogeneous
Cosmological
Models
ANDRZEJ KRASINSKI
N. Copernicus Astronomical Center and School of Sciences,
Polish Academy of Sciences, Warsaw
I CAMBRIDGE
UNIVERSITY PRESS
CAMBRIDGE UNIVERSITY PRESS
Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, Sao Paulo
Cambridge University Press
The Edinburgh Building, Cambridge CB2 2RU, UK
Published in the United States of America by Cambridge University Press, New York
www.cambridge.org
Information on this title: www.cambridge.org/9780521481809
© Cambridge University Press 1997
This publication is in copyright. Subject to statutory exception
and to the provisions of relevant collective licensing agreements,
no reproduction of any part may take place without
the written permission of Cambridge University Press.
First published 1997
This digitally printed first paperback version (with corrections) 2006
A catalogue record for this publication is available from the British Library
Library of Congress Cataloguing in Publication data
Krasifiski, Andrzej.
Inhomogeneous cosmological models / Andrzej Krasinski.
p. cm.
Includes bibliographical references.
ISBN 0 521 48180 5 (hardcover)
1. Cosmology - Mathematical models. 2. Relativity (Physics) - Mathematical models. I. Title.
QB981.K76 1997
523.1'01'5118-dc20 95-23549 CIP
ISBN-13 978-0-521 -48180-9 hardback
ISBN-10 0-521-48180-5 hardback
ISBN-13 978-0-521-03017-5 paperback
ISBN-10 0-521 -03017-X paperback
Contents
List of illustrations page xi
Preface xiii
Acknowledgements xv
1 Preliminaries 1
1.1 The scope of this review 1
1.2 The sources in the Einstein equations 5
1.3 The FLRW and Kantowski-Sachs limits of solutions 9
1.3.1 Limits of spacetimes 9
1.3.2 The invariant definitions of the FLRW and Kantowski-
Sachs spacetimes 10
1.3.3 Criteria for a FLRW limit 13
1.3.4 Different representations of the FLRW metrics 14
1.4 Spherically symmetric perfect fluid spacetimes 16
2 The Szekeres-Szafron family of solutions 19
2.1 The formulae and general properties 19
2.1.1 The p=0 subfamily 20
2.1.2 The jS'^O subfamily 21
2.2 Common properties of the two subfamilies 22
2.3 The invariant definitions of the family 23
2.4 Properties of the Szekeres solutions 24
2.4.1 The p'=0 subfamily 24
2.4.2 The jS'^O subfamily 26
2.5 Common characteristics and comparisons of the two subfamilies
of Szekeres solutions 31
2.6 The spherically symmetric and plane symmetric Szafron spacetimes
with/3'=0 40
2.7 The Szafron spacetimes of embedding class 1 45
2.8 Other subcases of the Szafron pf =0 spacetime 48
2.9 Stephani's (1987) solutions 52
2.10 Other generalizations of various subcases of the P'=0 subfamily 55
2.10.1 Solutions with heat-flow 55
2.10.2 Solutions with electromagnetic field 58
2.10.3 Solutions with viscosity 60
viii Contents
2.10.4 Other solutions 60
2.11 Solutions with Kantowski-Sachs geometry and their other
generalizations 61
2.11.1 The Kompaneets-Chernov (1964) spacetimes and their
subcases 61
2.11.2 The van den Bergh and Wils (1985) solution 62
2.11.3 Generalizations of the linear barotropic case, p=(y-\)e 63
2.11.4 Generalizations of the "stiff fluid" case 63
2.11.5 Generalizations of the case e=3p 64
2.11.6 Generalizations of the proper K-S model, p=0 64
2.11.7 Other papers 64
2.12 Solutions of the /3' ^0 subfamily with a G /S symmetry 64
3 2
2.13 Other subcases of the Szafron /3' ^0 spacetime 75
2.14 Electromagnetic generalizations of the solutions of the p'^O
subfamily 84
2.15 Generalizations of the solutions of the /3' =£0 subfamily for
viscosity 94
2.16 Other generalizations of the j8' i= 0 subfamily 97
3 Physics and cosmology in an inhomogeneous Universe 100
3.1 Formation of voids in the Universe 100
3.2 Formation of other structures in the Universe 108
3.3 Influence of cosmic expansion on planetary orbits 112
3.4 Formation of black holes in the evolving Universe 114
3.5 Collapse in electromagnetic field 116
3.6 Singularities and cosmic censorship 118
3.7 Influence of inhomogeneities in matter distribution on the cosmic
microwave background radiation 124
3.8 Other papers discussing the L-T model 127
3.9 General theorems and considerations 137
4 The Stephani-Barnes family of solutions 142
4.1 Definition and general properties 142
4.2 The type D solutions 145
4.2.1 The hyperbolically symmetric Barnes models 145
4.2.2 The plane symmetric Barnes models 146
4.2.3 The Kustaanheimo-Qvist (spherically symmetric Barnes)
models 147
4.3 Rediscoveries of the Kustaanheimo-Qvist (K-Q) class of
metrics 149
4.4 Invariantly defined subcases of the K-Q class 152
4.5 The McVittie (1967) class 156
4.6 Explicit K-Q solutions corresponding to f(u)=(au2+bu+c)~5/2 157
Contents ix
4.7 Physical and geometrical considerations based on the McVittie
(1933) solution 164
4.8 Other solutions of the K-Q class 167
4.9 The conformally flat solutions 168
4.10 The spherically symmetric Stephani solution 171
4.11 Other subcases of the Stephani (1967a) solution 175
4.12 Electromagnetic generalizations of the K-Q spacetimes 179
4.13 Other solutions with charged fluid, related to the Stephani-Barnes
family 187
4.14 Generalizations of the Stephani-Barnes models with heat-flow 189
5 Solutions with null radiation 195
5.1 General remarks 195
5.2 The energy-momentum tensor of null radiation 195
5.3 The Kerr, Kerr-Newman and Demianski (1972) solutions 197
5.4 The FLRW metrics in null coordinates 200
5.5 Superpositions of the special Demianski (1972) solution with
the FLRW backgrounds 201
5.6 Superpositions of the Kerr solution with the FLRW models and
other conformally nonflat composites 204
5.7 Conformally flat solutions with null radiation 211
6 Solutions with a "stiff fluid'Vscalar field source 215
6.1 How to generate a "stiff fluid" solution from a vacuum solution 215
6.2 The allowed coordinate transformations 217
6.3 The Einstein equations with a stiff fluid source in an A -symmetric
2
spacetime 218
6.4 Subcases of the Tomita (1978) class 221
6.5 Electromagnetic generalizations of the stiff fluid spacetimes 224
6.6 Two- and many-fluid generalizations of the stiff fluid spacetimes 226
6.7 Explicit solutions with a stiff perfect fluid source 227
7 Other solutions 238
7.1 Other A -symmetric perfect fluid solutions 238
2
7.2 A -symmetric solutions with nonperfect fluid sources 246
2
7.3 Oleson's Petrov type N solutions 247
7.4 Martin and Senovilla's (1986a,b) and Senovilla and Sopuerta's
(1994) solutions 250
7.5 Spherically symmetric solutions with expansion, shear and
acceleration 252
7.6 Thakurta's (1981) metric 255
7.7 Wainwright's (1974) spacetimes and Martin-Pascual and
Senovilla's (1988) solutions 255