Colliding plane waves in general relativity

254 Pages·1991·1.657 MB·English

by  J. B. Griffiths

Colliding plane waves
in general relativity
J. B. GRIFFITHS
Department of Mathematical Sciences
Loughborough University of Technology
OXFORD UNIVERSITY PRESS
(1991)
PREFACE
For many years after Einstein proposed his general theory of relativity,
only a few exact solutions were known. Today the situation is completely
different, and we now have a vast number of such solutions. However,
very few are well understood in the sense that they can be clearly inter-
preted as the fields of real physical sources. The obvious exceptions are
the Schwarzschild and Kerr solutions. These have been very thoroughly
analysed, and clearly describe the gravitational fields surrounding static
and rotating black holes respectively.
In practice, one of the great difficulties of relating the particular fea-
tures of general relativity to real physical problems, arises from the high
degree of non-linearity of the field equations. Although the linearized the-
ory has been used in some applications, its use is severely limited. Many
of the most interesting properties of space-time, such as the occurrence
of singularities, are consequences of the non-linearity of the equations.
In this book we will be considering one of the most obvious situa-
tions in which the effects of the non-linearity of Einstein’s equations will
be manifest. We will be considering the interaction between two waves.
By restricting our attention to somewhat idealized situations, it will be
possible to describe some types of wave interaction in terms of exact so-
lutions. Moreover, these solutions have a clear physical interpretation in
terms of combinations of gravitational or electromagnetic waves and their
interaction.
Much attention has been focused on these problems in recent years.
An initial approach to the subject was pioneered by Szekeres (1970, 1972)
and Khan and Penrose (1971). More recently, an alternative approach us-
ing an analogy with stationary axisymmetric solutions has been exploited
by Chandrasekhar and Ferrari (1984) and their co-workers.
After spherically symmetric situations, the most studied and best
understood space-times are those that are stationary and have axial sym-
metry. In these situations the field equations can be reduced to a single
equation involving a complex potential – the Ernst equation. It is now
known that, with this, all possible stationary axisymmetric solutions can
be generated in a finite number of steps using standard techniques. In
their 1984 paper, Chandrasekhar and Ferrari showed that the main field
equations for colliding plane waves can also be written as the same Ernst
equation. In fact, it is then found that most of the techniques that have
been developed for stationary axisymmetric space-times can also be ap-
plied to colliding plane waves. This has introduced considerable mathe-
matical interest in the subject in recent years.
viii Preface
In fact colliding plane wave space-times have been found to have
a surprisingly rich structure. Initially, it was widely believed that the
collision of plane waves would necessarily produce a future space-like cur-
vature singularity. This seemed to be implied by the focusing properties
of plane waves. However, numerous counterexamples have subsequently
been produced in which the curvature singularity is replaced by a Killing–
Cauchy horizon. Extensions of the space-time through this horizon may,
or may not, contain a space-like curvature singularity, or even a time-like
curvature singularity which could be avoided by an observer travelling on
a time-like world line. Recent research has clarified the singularity struc-
ture of most colliding plane wave space-times. These have been found to
have a surprisingly rich variation.
In view of the recent advances, it is clearly time to present a com-
prehensive and unified review of the now vast literature on this topic.
The purpose of this book is to provide such a review. Interesting lectures
on this topic have been presented by Chandrasekhar (1986) and Ferrari
(1989). However, in view of the considerable interest in the subject, a
more thorough review is now required.
The first eight chapters of this book cover the background to the sub-
ject, presenting the field equations and a discussion of some qualitative
aspects of their solution. A detailed discussion of the Khan–Penrose solu-
tion is included in this part, since it is the simplest solution and exhibits
the general character of most colliding plane wave solutions. Further
exact solutions for colliding plane gravitational waves are obtained and
describedinChapters9to14. Thecollisionandinteractionofelectromag-
netic waves is then considered in Chapters 15 to 19. The final chapters
contain an attempt to summarize all related results for the collision of
plane waves of different types and in non-flat backgrounds. A few general
conclusions and some outstanding problems that still require attention
are also indicated.
In the preparation of this book, I have been greatly assisted by a
number of colleagues. I am extremely grateful to Chris Clarke, John
Stewart and Sean Hayward for providing me with most helpful comments
on the first draft of this work. This final version has been substantially
expanded and seems to bear little resemblence to that initial draft. I am
also very grateful to Parvinder Singh for reading through a late version,
and to Roger Penrose and the American Physical Society for permission
to copy Figure 4.1. Finally, I must record my debt to most of the authors
of papers on colliding plane waves for regularly sending me preprints of
their work prior to publication.
I have also benefited greatly from numerous discussions on colliding
wave problems with many colleagues including Professor Chandrasekhar,
Preface ix
Alex Feinstein, Chris Clarke, Sean Hayward, Valeria Ferrari and Basilis
Xanthopoulos. The views expressed in the book, however, are my own
and I take full responsibility for any errors that it contains.
Loughborough J. B. G.
October 1990
CONTENTS
1 INTRODUCTION 1
1.1 Why consider wave interactions? 1
1.2 Simplifying assumptions 2
2 ELEMENTS OF GENERAL RELATIVITY 4
2.1 Basic notation 4
2.2 Components of the curvature tensor 5
2.3 Spin coefficients 7
2.4 Einstein–Maxwell fields 10
3 COLLIDING IMPULSIVE GRAVITATIONAL WAVES 11
3.1 The approaching waves 11
3.2 The solution describing the interaction 13
3.3 The structure of the solution 15
4 PLANE WAVES 17
4.1 The class of pp-waves 17
4.2 The class of plane waves 18
4.3 Particular cases 21
4.4 Global properties 22
5 GEOMETRICAL CONSIDERATIONS 26
5.1 The focusing of congruences 26
5.2 General theorems 29
5.3 Colliding waves 31
6 THE FIELD EQUATIONS 34
6.1 The coordinate system 34
6.2 The derivation of the field equations 36
6.3 The Einstein and Einstein–Maxwell equations 39
6.4 Integrating the field equations 40
7 BOUNDARY CONDITIONS 42
7.1 General discussion 42
7.2 Junction conditions for colliding plane waves 43
xii Contents
8 SINGULARITY STRUCTURE 48
8.1 Singularities 48
8.2 The singularity in region IV 50
8.3 The Khan–Penrose solution 53
8.4 The structure of other solutions 58
9 THE SZEKERES CLASS OF VACUUM SOLUTIONS 63
9.1 The solution in region IV 63
9.2 The approaching waves 66
9.3 The singularity structure 67
10 OTHER VACUUM SOLUTIONS WITH ALIGNED
POLARIZATION 69
10.1 A general method 69
10.2 The non-singular ‘solution’ of Stoyanov 74
10.3 The solution of Ferrari and Iban˜ez and Griffiths 76
10.4 The soliton solution of Ferrari and Iban˜ez 79
10.5 The degenerate Ferrari–Iban˜ez solutions 81
10.6 An odd order solution 85
10.7 The second Yurtsever and the Feinstein–Iban˜ez solutions 86
10.8 The first Yurtsever solutions 90
10.9 Further explicit solutions 92
11 ERNST’S EQUATION FOR COLLIDING GRAVITATIONAL
WAVES 95
11.1 A derivation of the Ernst equation 95
11.2 Boundary conditions 98
11.3 Colinear solutions 100
12 SOLUTION-GENERATING TECHNIQUES 103
12.1 The colinear case 103
12.2 Rotations and Ehlers transformations 105
12.3 Geroch transformations 108
12.4 The Neugebauer–Kramer involution 110
12.5 A combined transformation 113
12.6 Other methods 114
Contents xiii
13 VACUUM SOLUTIONS WITH NON-ALIGNED
POLARIZATION 118
13.1 The Nutku–Halil solution 118
13.2 The Panov solution 120
13.3 The Chandrasekhar–Xanthopoulos solution 121
13.4 Other solutions 127
14 THE INITIAL VALUE PROBLEM 131
14.1 The initial data 131
14.2 The colinear case 132
14.3 The non-colinear case 135
15 COLLIDING ELECTROMAGNETIC WAVES:
THE BELL–SZEKERES SOLUTION 137
15.1 The Bell–Szekeres solution 137
15.2 The structure of the solution 139
15.3 Extensions of the solution 142
15.4 A non-colinear collision 143
16 ERNST’S EQUATION FOR COLLIDING
ELECTROMAGNETIC WAVES 146
16.1 The field equations 146
16.2 A simple class of solutions 151
16.3 The Bell–Szekeres solution 153
17 COLLIDING ELECTROMAGNETIC WAVES:
EXACT SOLUTIONS 156
17.1 A technique of Chandrasekhar and Xanthopoulos 156
17.2 Two particular examples 158
17.3 Another type D solution 160
17.4 A technique of Halilsoy 163
17.5 Other solutions 165
18 COLLIDING ELECTROMAGNETIC WAVES:
DIAGONAL SOLUTIONS 168
18.1 The generation technique of Panov 168
18.2 An alternative approach 169
18.3 Electromagnetic Gowdy cosmologies 173
18.4 Other methods 176
xiv Contents
19 ELECTROMAGNETIC WAVES COLLIDING WITH
GRAVITATIONAL WAVES 177
19.1 A simple example 177
19.2 General initial data 178
19.3 A general class of solutions 180
20 OTHER SOURCES 184
20.1 Scalar fields 184
20.2 Perfect fluid solutions 186
20.3 Null fluids and the uniqueness problem 189
20.4 Plane shells of matter 192
20.5 Neutrino fields 194
20.6 Other null fields 199
21 RELATED RESULTS 201
21.1 Other wave interactions 201
21.2 Collisions in non-flat backgrounds 202
21.3 Solitons 203
21.4 Alternative gravitational theories 205
21.5 Numerical techniques 206
22 CONCLUSIONS AND PROSPECTS 208
22.1 General conclusions 208
22.2 Prospects for further work 210
APPENDIX: COORDINATE SYSTEMS 214
REFERENCES 216
INDEX 228
1
INTRODUCTION
The subject to be discussed in this book is the collision and interaction of
gravitational and electromagnetic waves. This is a particularly important
topic in general relativity since the theory predicts that there will be a
non-linear interaction between such waves. The effect of the non-linearity,
however, is unclear. It is appropriate therefore to look in some detail at
the simplest possible situation in which the effect of the non-linearity will
be manifest: namely the interaction between colliding plane waves.
1.1 Why consider wave interactions?
In classical theory, Maxwell’s equations are linear. An immediate conse-
quenceofthisisthatsolutionscanbesimplysuperposed. Thisleadstothe
prediction that electromagnetic waves pass through each other without
any interaction. This prediction is very thoroughly confirmed by observa-
tions. Radio waves are transmitted at many different frequencies, yet it is
possible for a receiver to select any one particular station and to receive
that signal almost exactly as it was transmitted. The only interference
that is detected arises from other transmitters using the same frequency,
and from the difficulty of isolating just one frequency within the receiver.
After many years’ experience, no interaction has ever been detected
betweenpropagatingelectromagneticwaves. Thisappliesnotjusttoradio
waves, but to all types of electromagnetic radiation, including light. That
we can see clearly through a vacuum, even though light is also passing
through it in other directions undetected, is one of the best established
of scientific observations. More remarkably, this applies not just to local
phenomena, but to the vast regions of space. The light that reaches us
from distant galaxies arrives without any apparent interaction with the
light that must have crossed its path during the millions of years that
it takes to reach us. The apparent linearity of the field equations for
electromagnetic waves is thus one of the best established scientific facts.
However, this is not the entire story. Einstein’s equations which
describe gravitational fields are highly non-linear. It follows that gravi-
tational waves, if they exist, cannot pass through each other without a
significant interaction. In this book we will be using the standard gen-
eral theory of relativity. In this theory gravitational waves are predicted,
thoughtheirmagnitudesaresosmallthatthepossibilityofdetectingthem
1
2 Introduction
is only just coming within the scope of the most sophisticated modern ap-
paratus. The purpose of this book is to contribute to an understanding
of the character of the interaction that is theoretically predicted between
gravitational waves.
In Einstein’s theory, gravitational waves are considered as perturba-
tions of space-time curvature that propagate with the speed of light. As
these waves pass through each other, theoretically there will be a non-
linear interaction through the gravitational field equations. It will be
necessary to consider the general character of these interactions, and how
the propagating waves are modified by them.
Consider again electromagnetic waves. According to Einstein’s the-
ory, all forms of energy have an associated gravitational field. Electro-
magnetic waves must therefore be coupled to an associated perturba-
tion in the space-time curvature. In the full Einstein–Maxwell theory,
Maxwell’sequationsdescribingtheelectromagneticfieldremainlinear, in-
dicatingthatthereisnodirectelectromagneticinteractionbetweenwaves.
However, Einstein’s equations, which apply to the gravitational field, are
highly non-linear. Thus, as two electromagnetic waves pass through each
other, there will be a non-linear interaction between them due to their
associated gravitational fields.
This non-linear interaction between electromagnetic waves that is
predicted by Einstein’s theory must necessarily be very small in order
to be consistent with the fact that such interactions have not yet been
detected. An interaction, however, is predicted, though its magnitude is
likely to be similar to that between gravitational waves.
Since the interactions we will be considering are so weak, it may be
considered appropriate initially to use approximation techniques. A num-
ber of authors have considered this approach. The modern techniques of
numericalrelativityhavealsoproducedsomeinterestingresults. However,
these approaches will not be used in this book. The method adopted here
will be to concentrate on exact solutions of the Einstein–Maxwell field
equations. This has the advantage of being able to clarify something of
the global structure of wave interactions. This turns out to be one of
their most remarkable features. It leaves us, however, with the problem
of finding exact solutions, and these are only possible in a very limited
number of situations.
1.2 Simplifying assumptions
The problem that is to be considered in this book is the interaction be-
tween two waves. A simple case in which the waves propagate in the same
direction has been analysed by Bonnor (1969) and Aichelburg (1971).