Table Of ContentINTRODUCTION TO GENERAL RELATIVITY, BLACK HOLES, AND COSMOLOGY
Introduction to General Relativity,
Black Holes, and Cosmology
Yvonne Choquet-Bruhat
French Academy of Sciences, Paris,
and the American Academy of Arts and Science
With a foreword by
Thibault Damour
Permanent Professor, Institut des Hautes E´tudes Scientifique
Member of the French Academy of Sciences, Paris
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Foreword
YvonneChoquet-Bruhathasmademanydeepandlastingcontributions
to mathematical and physical aspects of General Relativity, starting
with her epoch-making 1952 proof of the well-posedness of the Cauchy
problem for Einstein’s equations. We are all very fortunate that she
has undertaken to present, in terms accessible to all, a comprehensive
account of all the aspects of General Relativity. Indeed, this beautiful
bookisquiteuniquebothintherangeoftopicsitcoversandintheway
each topic is treated.
First, the range of topics presented by Yvonne Choquet-Bruhat is
truly remarkable. She covers successively the basics of Riemannian
geometry (Chapter I) and of Special Relativity (Chapter II); the math-
ematical and physical definition of General Relativity (Chapter III);
the main properties and consequences of Einstein’s field equations
(Chapter IV); the mathematics and physics of the Schwarzschild
spacetime (Chapter V); a mathematically rigourous discussion of black
holes (Chapter VI); a brief, but up to date, discussion of relativistic
cosmology (Chapter VII); a thorough presentation of the Cauchy
problem (Chapter VIII); and last, but not least, a detailed discussion
of two of the most important phenomenological matter models, namely
relativistic fluids (Chapter IX) and the relativistic kinetic theory of
N-particle systems (Chapter X).
But,mostimportantly,thetreatmentofeachoneofthesebroadtopics
isbothverycomprehensiveandremarkablyconcise.Shehassucceededin
reachingaLandau-andLifshitz-likeidealofcoveringallthecrucialissues
in the most concise way, while expounding each topic in a mathematic-
ally rigourous way. This rare combination of qualities makes this book
particularly valuable. For instance, her discussion of the Schwarzschild
spacetimecovers,inonego:(i)aderivationofthesolutionthatincludes
aproofofBirkhoff’stheorem;(ii)theformofthesolutioninfivedifferent
types of coordinates; (iii) a preliminary discussion of the event horizon
(which is developed in the following chapter); (iv) the motion of planets
andoflight,andtheircomparisonwiththemostrecentobservations;(v)
the stability of circular orbits; (vi) a presentation of Fermat’s principle
that includes its little-known generalization to arbitrary spacetimes due
to G. Ferrarese; (vii) the redshift and time-delay effects; (viii) a dis-
cussion of spherically symmetric interior solutions that includes several
theoremsabouttheirgeneralproperties;(ix)theReissner-Nordstro¨mso-
lution;(x)theSchwarzschildsolutioninanydimension;and(xi)aprecise
but concise account of the results of Gu Chao Hao, Christodoulou, and
vi Foreword
others on spherically symmetric gravitational collapse. The chapter is
then nicely capped by some problems, followed by their solutions.
Iam surethat this remarkably concise and complete book by Yvonne
Choquet-Bruhat will allow many readers to achieve a deep understand-
ing of General Relativity through her unique mathematico-physical
knowledge of one of the fundamental pillars of modern physics. Thank
you, Yvonne for sharing with us the deep wisdom you have acquired
during your lifelong exploration of the beautiful theoretical landscape
opened, nearly a century ago, by Albert Einstein.
Thibault Damour
Institut des Hautes E´tudes Scientifiques
Preface
Special Relativity (1905) and General Relativity (1915), both due to
Einstein’s genius, are fundamental pillars of modern physics. They have
revolutionized the scientific concepts of space and time, formerly due
to everyday experience at a human scale, and also to previous scien-
tific work and experiments made with clocks, very precise and reliable
at the considered scale. These concepts of absolute space and absolute
time had become ingrained in the minds of common folk as well as
scientists, and were difficult to overthrow for a long time.1 Nowadays, 1For instance, the fact that two twins
the world explored by humanity, at microscopic as well as cosmological who live different lives age differently
wascalledthe‘twinparadox’,although
scales, has become very much greater, and old concepts have had to be
infactthereisnoparadoxthere,except
replaced by more general ones. Spectacular improvements in technology
inthehuman-builtdefinitionoftime.
have changedthelives ofalargepartof humanity, andnewinformation
processes have permitted easier access to scientific knowledge and the
acceptance of new concepts.
Modern physical theories have a mathematical formulation, often
geometrical, with consequences deduced from mathematical theorems.
The validity of a physical theory results from the verification of its
consequences by observations or experiments.
General Relativity is a beautiful geometric theory, simple in its gen-
eral mathematical formulation, which has numerous consequences with
strikingphysicalinterpretations:gravitationalwaves,blackholes,cosmo-
logical models. Several of these consequences have been verified with a
greataccuracy.TheEinsteinequationspresentawidevarietyofnewand
interestingmathematicalproblemswithpossiblephysicalinterpretation.
Theaimofthisbookistopresentwithprecision,butassimplyaspos-
sible, the foundations and main consequences of General Relativity. It
is written for an audience of mathematics students interested in physics
and physics students interested in exact mathematical formulations—
or indeed for anyone with a scientific mind and curious to know more of
the world we live in. The mathematical level of the first seven chapters
is that of undergraduates specializing in mathematics or physics; these
chapters could be the basis for a course on General Relativity. The next
three chapters are more advanced, though not requiring very sophisti-
cated mathematics; they are aimed at graduate students, lecturers, and
researchers.Noapriorispecializedphysicsknowledgeisrequired.These
chapters could serve as the text for a course for graduate students.
I wish to everyone a good trip in this strange but fascinating world.
Yvonne Choquet-Bruhat
viii Preface
Acknowledgements
My gratitude goes to IHES, its scientific director Jean Pierre Bour-
guignon, and its permanent professor Thibault Damour, for inducing
PierreCartier,whokindlyaccepted,togivemeadeskinhisoffice.Ihave
thus benefited for the last ten years from a pleasant environment and
working facilities together with instructive seminars and conversations
that have been intellectually stimulating as well as friendly.
I am very fortunate that I have been able to meet regularly at IHES
Thibault Damour, who has always been ready to answer my questions
andallowmetobenefitfromhisgreatandwideknowledgeofrelativistic
physics and cosmology, as well as of the varied elaborate mathematical
toolsusedinthisfield—fromtheoldesttothemostrecent.Hewaskind
enough to read the whole of a preliminary version of this book. The
present version has been very much improved through his information
and his constructive criticism. Thank you Thibault—this book would
never have been completed without your help and encouragement.
IalsothankRichardKerner,whospenttimereadingmytext,pointing
out useful corrections.
I thank Franc¸ois Bachelier for his pertinent help with tex.
Cover painting: Cosmos Birth by Genevi`eve Choquet-Kastylevsky.
Notation
The sign ≡ denotes an identity and the sign := a definition, although
sometimes when there is no confusion possible I denote by the simple
sign = one or the other of these.
In the text, I usually denote by a single letter a geometric ob-
ject, for example X or T for a vector or tensor. In equations, I
prefer to use indices, since this makes operations simpler to write
and understand—without speaking of possible numerical or algebraic
computing applications.
Thespacetimemetricsignatureis−++...+.IntheLorentziancase,
the spacetime indices are Greek letter. When specified, a time index is
denoted by zero, and the space indices by Latin letters from the middle
of the alphabet, i,j,....
Definitions of new notions are in bold characters and are assembled
in the index.
Note on references
I have not tried to find for each result an original source. This would
havebeenadifficultandsometimescontroversialtask.Inthesidenotes,
Igivereferences,assembledattheendofthebook,topapersIhaveused,
where the interested reader can find relevant details, complements, and
references to previous papers about the subjects treated.
• The reference YCB-OUP2009 is to Yvonne Choquet-Bruhat General
RelativityandtheEinsteinEquations,publishedbyOxfordUniversity
Press in 2009.
• The references CB-DMI and II are to Y. Choquet-Bruhat and
C. DeWitt-Morette, Analysis, Manifolds and Physics, Parts I and II,
published by North-Holland in 1982 and 2000, respectively.
