INTRODUCTION 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 3 3 GreatClarendonStreet,Oxford,OX26DP, UnitedKingdom OxfordUniversityPressisadepartmentoftheUniversityofOxford. ItfurtherstheUniversity’sobjectiveofexcellenceinresearch,scholarship, andeducationbypublishingworldwide.Oxfordisaregisteredtrademarkof OxfordUniversityPressintheUKandincertainothercountries (cid:2)c YvonneChoquet-Bruhat2015 Themoralrightsoftheauthorhavebeenasserted FirstEditionpublishedin2015 Impression:1 Allrightsreserved.Nopartofthispublicationmaybereproduced,storedin aretrievalsystem,ortransmitted,inanyformorbyanymeans,withoutthe priorpermissioninwritingofOxfordUniversityPress,orasexpresslypermitted bylaw,bylicenceorundertermsagreedwiththeappropriatereprographics rightsorganization.Enquiriesconcerningreproductionoutsidethescopeofthe aboveshouldbesenttotheRightsDepartment,OxfordUniversityPress,atthe addressabove Youmustnotcirculatethisworkinanyotherform andyoumustimposethissameconditiononanyacquirer PublishedintheUnitedStatesofAmericabyOxfordUniversityPress 198MadisonAvenue,NewYork,NY10016,UnitedStatesofAmerica BritishLibraryCataloguinginPublicationData Dataavailable LibraryofCongressControlNumber:2014934913 ISBN978–0–19–966645–4(hbk.) ISBN978–0–19–966646–1(pbk.) Printedandboundby CPIGroup(UK)Ltd,Croydon,CR04YY LinkstothirdpartywebsitesareprovidedbyOxfordingoodfaithand forinformationonly.Oxforddisclaimsanyresponsibilityforthematerials containedinanythirdpartywebsitereferencedinthiswork. 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.