The Einstein Equations and the Large Scale Behavior of Gravitational Fields: 50 Years of the Cauchy Problem in General Relativity

486 Pages·13.312 MB·English

by  Robert Bartnik, Jim Isenberg (auth.), Piotr T. Chruściel, Helmut Friedrich (eds.)

Piotr T. Chrusciel
Helmut Friedrich
The Einstein Equations
and the
Large Scale Behavior
of Gravitational Fields
50 Years of the Cauchy Problem
in General Relativity
Springer Basel AG
Editors:
Piotr T. Chrusciel Helmut Friedrich
Laboratoire de MathematiQues Max.planck~nstitut fur Gravitationsphysik
et de PhysiQue TheoriQue Am Muhlenberg 1
CNRS UMR 6083 14467 Golm
Avenue Grammont Germany
37200 Tours e-mail: [email protected]
France
e-mail: [email protected]
2000 Mathematical Subject Classification 35J60, 35L45, 58J05, 58J45, 53C50, 83C05
A CIP catalogue record for this book is available from the
Library of Congress, Washington D.C., USA
Bibliographic information published by Die Deutsche Bibliothek
Die Deutsche Bibliothek lists this publication in the Deutsche Nationalbibliografie;
detailed bibliographic data is available in the Internet at <http://dnb.ddb.de>.
ISBN 978-3-0348-9634-4 ISBN 978-3-0348-7953-8 (eBook)
DOI 10.1007/978-3-0348-7953-8
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© 2004 Springer Basel AG
Originally published by Birkhăuser Verlag ,Basel, Switzerland in 2004
Softcover reprint of the hardcover 1s t edition 2004
Printed on acid-free paper produced of chlorine-free pulp. TCF
(J)
ISBN 978-3-0348-9634-4
987654321 www.birkhauser-science.com
Contents
Preface .................................................................. Vll
R. Bartnik and J. Isenberg
The Constraint Equations .................... . . . . . . . . . . . . . . . . . . . . . . . 1
H.L. Bray and P. T. Chrusciel
The Penrose Inequality ............................................. 39
L. Andersson
The Global Existence Problem in General Relativity 71
H. Friedrich
Smoothness at Null Infinity and the Structure of Initial Data. . . . . . . . 121
L. Lehner and O. Reula
Status Quo and Open Problems in the Numerical Construction
of Spacetimes ....................................................... 205
A.D. Rendall
The Einstein-Vlasov System 231
Y. Choquet-Bruhat
Future Complete U(1) Symmetric Einsteinian Spacetimes,
the Unpolarized Case ............................................... 251
L. Andersson and V. Moncrief
Future Complete Vacuum Spacetimes 299
J.L. Friedman
The Cauchy Problem on Spacetimes That Are Not
Globally Hyperbolic ................................................ 331
M. T. Anderson
Cheeger-Gromov Theory and Applications to General Relativity 347
vi Contents
G.J. Galloway
Null Geometry and the Einstein Equations 379
T. Barbot and A. Zeghib
Group Actions on Lorentz Spaces, Mathematical Aspects:
A Survey ........................................................... 401
R. Geroch
Gauge, Diffeomorphisms, Initial-Value Formulation, Etc. 441
Index.................................................................... 479
On the cover the reader will find the general form of a toroidal Gowdy metric and
the vacuum Einstein equations for that metric. The mountain landscape is a result of a
numerical simulation of the time evolution of the function P appearing in a closely related
metric. We are very grateful to Beverly Berger for making the results of her numerical
simulations available to us.
The Einstein Equations and the
Large Scale Behavior of Gravitational Fields
50 years of the Cauchy problem in general relativity
Preface
The essence of the theory of general relativity is encoded in the specific features of
individual solutions and also in the structure of the set of all solutions to Einstein's
equations. Extracting this information from the field equations has been a major
effort in the field since the birth of general relativity and still much has to be done.
At early stages, constructing explicit solutions played a dominant role. The
analysis and interpretation of some of these solutions revealed unexpected features
of general relativity. Guided by explicit solutions, approximation methods were
invented to describe weak gravitational fields. Using highly sophisticated versions
of these techniques to interpret recent observations, one was able to establish the
first indirect proof of the reality of gravitational radiation (see [1, 22]).
Strong gravitational fields and highly dynamical situations are not accessible
by those methods. Techniques of abstract analysis are required to establish the
general qualitative picture, to clarify conceptional questions, and to supply struc
tural insights for numerical work. Numerical computer codes are needed to provide
quantitative information for comparison with observational data.
Due to the intricacies of Einstein's equations, and the lack of mathematical
tools, the initial investigation of the solution manifold proceeded at a slow pace
(cf. [6, 20, 21, 16] for early work). A fundamental breakthrough in the field was
achieved by Yvonne Choquet-Bruhat's article Theoreme d'existence pour certains
systemes d'equations aux derivees partielles non lineaires [8], published in 1952,
where she proves local existence of solutions to the Cauchy problem for Einstein's
field equations.
Independent strands of development, which emphasized the geometrical con
tent of the theory, began in the late 1950's. Attempts to understand gravitational
radiation and gravitational collapse brought new insights, highlighting in particular
the important role of the global structure of space-time.
viii The Einstein Equations
The concept of asymptotic flatness led to a general notion of gravitational
radiation field which does not require mathematical approximations [2, 18] (com
pare [23]). These studies relied on the analysis of formal expansions and assump
tions on the fall-off behavior of solutions at null infinity. The singularity theo
rems showed that geodesic incompleteness is a stable feature of Einstein's theory
[19,9,10]. The proofs exploited, however, only certain general features of the field
equations and supplied little information on the solutions.
Because the asymptotic behavior and the singularity structure of its solutions
express the individuality ofa quasi-linear hyperbolic equation in a most subtle and
precise way, it is quite remarkable that these results could be obtained at all. In
fact, if Einstein's equations are looked at purely from the PDE point of view,
their general covariance poses even further difficulties by the need to invent and
control gauge conditions and to determine the underlying manifold as part of the
solution. The geometrical nature of the equations supplies, however, arguments of
remarkable strength which help in various considerations.
The developments indicated above raised questions about the large scale
structure of solutions to the field equations. The underlying assumptions needed
justification, one had to face the possible failure of predictability or the occurrence
of naked singularities. Attempts to resolve these questions by the original meth
ods failed. Combined with physical insight, the traditional calculational tools and
stability arguments of the physicist can be surprisingly powerful, but they cannot
supply definitive, general, and detailed answers to global problems. This requires
control of the long time behavior of solutions to Einstein's equations in terms of
suitably prescribed Cauchy or other boundary data.
The existence results up to the early 1980's were local in time (see [3, 7]
for a survey), but the situation changed drastically in the last twenty-five years.
The analysis of the Cauchy problem evolved into a tool for performing large scale
studies of classes of solutions with distinguished qualitative features. The questions
indicated above played guiding roles in most of the new work.
In view of this progress it seemed appropriate to mark the fiftieth birthday of
the local existence theorem by organizing a summer school in which younger par
ticipants would learn from leading experts about recent progress in the field, while
their older colleagues would have an opportunity to meet, exchange ideas, and
present their work. This is how the Cargese Summer School "50 years of the Cauchy
problem in general relativity" was born. The reader will find the audio, video, and
supporting material from several lectures held there on the enclosed DVD.
After the school had taken place it became clear to us that a volume describ
ing the state of the art in mathematical general relativity would be useful. This
was not planned as a proceedings of the school: not all topics which we felt should
be represented were covered during the school; moreover, the contents of some of
the talks was already adequately covered elsewhere. This is how the idea of this
book was conceived.
The reader will find here an overview of the current knowledge on global
properties of solutions of Einstein equations, and on techniques which in recent
The Einstein Equations ix
years brought progress to the understanding of the subject. It should be recog
nized that there are three notable gaps: Firstly, the techniques developed in the
the proof of the non-linear stability of Minkowski space-time are not represented
at all. However, those techniques have already been presented in detail in mono
graphs ([5, 11]) as well as in introductory reviews ([4, 12]), which have recently
been complemented by [13]. Secondly, the theory oflow-regularity solutions of the
Cauchy problem developed in [14, 15] is not presented. This aspect of the field is
rapidly evolving, and it is useful to wait a few years before attempting to present
a synthetic overview. Finally, an important event in the field, which took place
after the structure of this book had been finalized, is a new proof of the stabil
ity of Minkowski space-time based on wave coordinates [17]. While the new proof
provides less information about the solutions than the previous ones, it leads to
dramatic simplifications of the global existence argument.
There does not appear to be a natural ordering of the articles in this book
except the most uninspired one, namely that by alphabet. The current ordering
arose from a mixture oflogical and pedagogical arguments, together with the desire
to guide a newcomer to the field through its physical and mathematical, theoretical
and technical aspects. In any case, the readers should feel strongly encouraged to
start reading wherever they like and to look at whatever raises their interest, and
try to discover the intricate web of relations between the different contributions.
We are pleased that mathematical relativity is thriving, and that young peo
ple contribute new deep results, or solve long standing problems. We hope that
this volume will help keep the field alive and inspire new developments.
Both this book, and the Cargese Summer School, would not have been pos
sible without the financial support of the European Community (under the Euro
Summer Schools Program), the Centre National de Recherche Scientifique, the
Clay Institute (Boston), the Tomalla Foundation (Zurich), la Collectivite Terri
toriale Corse, the Centre International de Mathematiques Pures et Appliquees
(Nice), the International Association of Mathematical Physics, Ie Laboratoire de
Mathematique Physique et Theorique de Tours, and Compaq. We wish to express
our warmest thanks to all those Institutions.
Throughout her whole career Yvonne Choquet-Bruhat has kept contributing
key results to a field which she helped grow and evolve. Her work has been a
constant inspiration for scientists entering the field. It is a pleasure to dedicate
this volume to her.
Piotr T. Chrusciel, Helmut Friedrich, Tours and Colm, April 2004.
x The Einstein Equations
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The Einstein Equations xi
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