AFirstCourseinLoopQuantumGravityRodolfo Gambini and Jorge Pullin13Great Clarendon Street, Oxfordox26dpOxford University Press is a department of the University of Oxford.It furthers the Universitys objective of excellence in research, scholarship,and education by publishing worldwide inOxfordNew YorkAucklandCape TownDar es SalaamHong KongKarachiKuala Lumpur MadridMelbourneMexico CityNairobiNew Delhi Shanghai Taipei TorontoWith oces inArgentinaAustriaBrazil ChileCzech RepublicFranceGreeceGuatemalaHungaryItalyJapanPolandPortugal SingaporeSouth KoreaSwitzerlandThailandTurkeyUkraineVietnamOxford is a registered trade mark of Oxford University Pressin the UK and in certain other countriesPublished in the United Statesby Oxford University Press Inc., New YorkcR. Gambini and J. Pullin 2011The moral rights of the authors have been assertedDatabase right Oxford University Press (maker)First published 2011All rights reserved. No part of this publication may be reproduced,stored in a retrieval system, or transmitted, in any form or by any means,without the prior permission in writing of Oxford University Press,or as expressly permitted by law, or under terms agreed with the appropriatereprographics rights organization. Enquiries concerning reproductionoutside the scope of the above should be sent to the Rights Department,Oxford University Press, at the address aboveYou must not circulate this book in any other binding or coverand you must impose the same condition on any acquirerBritish Library Cataloguing in Publication DataData availableLibrary of Congress Cataloging in Publication DataData availableTypeset by SPI Publisher Services, Pondicherry, IndiaPrinted and bound byCPI Group (UK) Ltd, Croydon, CR0 4YYISBN978019959075913579108642PrefaceLoopquantumgravityhas emergedas apossible avenue towards thequantization of general relativity. Looking at the top 50 cited papers of alltime of the arXiv.org:gr-qc preprint repository (which includes papers ongravityasa whole,notonlyquantum gravity),according tothe SPIRESdatabaseinSLACinitslatestedition(2006)onendsthat13papersare on loop quantum gravity. Although a smaller eld than string theory,theothermainapproachtoquantumgravitypursuedtoday,thenumberof researchers working inloopquantumgravityis signicant. At themoment both string theory and loop quantum gravity remain incompleteparadigms andas aconsequence controversies about whichis amorepromising approach naturally arise. We will address some of these in thisbook.Manypeople, includingof coursephysicsundergraduates, areinter-ested in learning about loop quantum gravity. There are indeed excellentrecenttextbooksbyRovelli (2007)andThiemann(2008)forthosewhowanttopursuethetopicindepth, roughlyspeakingatthelevel of anadvancedUSgraduateschool course. This typeof treatment however,is largely inadequate for physics undergraduates and for others whowant to get some minimal graspof the subject ina relatively shortperiod of time and without the depth expected for someone that isto do research in the eld. An additional complication is that thesegraduate-level treatments of the subject have knowledge of generalrelativityas aprerequisite. This becomes abarrier for manyreaders.Although recent books by Hartle (2003) and Schutz (2009) make teachinggeneral relativity at an undergraduate level possible, most undergraduatestudentshavecurriculathatplacegeneral relativityattheveryendoftheir careers(if their institutionoersthecourseat all), whichmakessuchcourses unsuitable as prerequisites for acourse inloopquantumgravity. Something that is not always appreciated is that undergraduates,particularlyinthe last semester of their career, are busyindividuals.Most are taking several courses, perhaps conducting some undergraduateresearchwork, preparingforthegraduaterecordexaminations(GRE),and applying to graduate schools, sometimes on top of holding a job. Thetimeavailabletoaspeciccourseisverylimited.iv PrefaceInthese notes we will attempt tointroduce loopquantumgravitywithout assuming previous knowledge of general relativity. The onlybackgroundwewillassumeisknowledgeofMaxwellselectromagnetism,a minimal knowledge of LagrangianandHamiltonianmechanics, andspecialrelativityandquantummechanics.Thiswillinevitablyimplywewill be taking many shortcuts in the coverage of loop quantum gravity, butthis will be the price we pay to introduce the topic in a way that is widelyaccessibletoundergraduatesatmostphysicsprogramsintheUSwithintheconnesofaone-semesterthree-creditcourse.Someundergraduatesandother readers mayfeel slightlydisappointedinnot getting moredetails andamorecompletepicture, but webelievethemajoritywillwelcome abookthat is light andnimble as agoodintroductiontoasubjectthatmayotherwiseappearintimidating. Someexpertsmayfeelweareshort-changingreaders byoversimplifyingseveral issues for thesakeof expediency. Wewill trytobecareful towarnreaderswhenwearedoingso. Anothergoal wehadinmindwastocreateashort book.Given that we are only introducing people to the topics, being deliberatelysupercial, and not attempting a full discussion, it is not worth trying tobeexhaustiveanddiscussingallissuesinfulldetail.Longbookstendtobe intimidating to the reader and we think we will serve a larger audiencewithacompactbook.Theorganizationof thisbookisasfollows: inChapter1weaddressthequestionof whyoneshouldquantizegravity. Chapter2will reviewMaxwells electromagnetism and in particular its relativistic formulation.Chapter 3will introduce some minimal elements of general relativity.Chapter4willdealwiththeHamiltonianformulationofmechanicsandeldtheories, includingconstraints. Chapter 5will discuss YangMillstheories.Chapter6willcoverquantummechanicsandsomeelementsofquantumeldtheory.Chapter7introducesAshtekarsnewvariablesforgeneralrelativity.Chapter8developsthelooprepresentationforgeneralrelativity. Chapter 9 presents quantumcosmology as an application.Chapter10discussesseveral miscellaneousapplicationsincludingblackhole entropy, the master constraint program and uniform discretizations,spinfoams, possibleexperimental signatures, andtheproblemof time.The bookends witha chapter onthe controversies surrounding loopquantumgravity.AcknowledgementsWe have beneted enormously from detailed comments on a draft versionof the bookbyseveral people, whoinmanycases clearlyspent alotof time tryingtohelpus: FernandoBarbero, MartinBojowald, SteveCarlip, Jonathan Engle, Kristina Giesel, Gaurav Khanna, Kirill Krasnov,DanieleOriti,CarloRovelli,ParampreetSingh,MadhavanVaradarajan,and Richard Woodard. To them we will always be grateful. This work wassupportedinpartbytheGravitationalPhysicsProgramoftheNationalScience Foundation, The Foundational Questions Institute (fqxi.org), theHorace Hearne Jr. Institute for Theoretical Physics, CCT-LSU and PDT(Uruguay).AGabrielayMarthaContents1 Whyquantizegravity? 12 Specialrelativityandelectromagnetism 82.1 Spaceandspace-time 92.2 Relativisticmechanics 142.3 Maxwelltheory 183 Someelementsofgeneralrelativity 233.1 Introduction 233.2 Generalcoordinatesandvectors 253.3 Curvature 293.4 TheEinsteinequationsandsomeoftheirsolutions 333.5 Dieomorphisms 373.6 The3 + 1decomposition 403.7 Triads 424 Hamiltonian mechanics including constraints and elds 474.1 UsualmechanicsinHamiltonianform 474.2 Constraints 484.3 Fieldtheories 514.4 Totallyconstrainedsystems 575 YangMillstheories 625.1 Kinematicalarenaanddynamics 625.2 Holonomies 666 Quantummechanicsandelementsofquantumeldtheory 716.1 Quantization 716.2 Elementsofquantumeldtheory 766.3 Interactingquantumeldtheoriesanddivergences 816.4 Renormalizability 867 GeneralrelativityintermsofAshtekarsnewvariables 917.1 Canonicalgravity 917.2 Ashtekarsvariables:classicaltheory 927.3 Couplingtomatter 977.4 Quantization 98viii Contents8 Looprepresentationforgeneralrelativity 1048.1 Thelooptransformandspinnetworks 1048.2 Geometricoperators 1108.3 TheHamiltonianconstraint 1179 Anapplication:loopquantumcosmology 1249.1 Theclassicaltheory 1249.2 TraditionalWheelerDeWittquantization 1279.3 Loopquantumcosmology 1289.4 TheHamiltonianconstraint 1309.5 Semiclassicaltheory 13110 Furtherdevelopments 13410.1Blackholeentropy 13410.2Themasterconstraintanduniformdiscretizations 14510.3Spinfoams 14910.4Possibleobservationaleects? 15610.5Theproblemoftime 16311 Openissuesandcontroversies 168References 174Index 1811Whyquantizegravity?Inourcurrentunderstanding, thereexistfourfundamental interactionsinnature: electromagnetism, weakinteractions, stronginteractions, andgravity. Everyoneisfamiliar withelectromagnetism. Weakinteractionsare involved in the decay of nuclei. Strong interactions keep nucleitogether. The rules of quantummechanics have beenappliedtoelec-tromagnetism, theweakandstronginteractions. Itissortofnatural toapply the rules of quantum mechanics to such interactions since they playkey roles in the dynamics of atoms and nuclei and one knows that at suchscalesclassicalmechanicsdoesnotgivecorrectpredictions.Therulesofquantummechanics havenot beenappliedtogravityinasatisfactorymanneruptonow.Loopquantumgravityisanattempttodoso,butitisanincompletetheory.Before continuingwe shouldclarifythat fromnowongravityismeanttobedescribednotbythetheoryof Newtonthatonelearnsinelementary physics courses, but by Einsteins general theory of relativity.Numerous experimental tests of high accuracy agree with the predictionsof general relativity (Will 2005). In such description, gravity is not reallyaninteractionbut rather adeformationof space-time. Thelatter isnotatandthereforeobjectsdonotfollownaturallystraighttrajecto-riesinspace-time. Thisaccountsforwhatonenormallyperceivesasagravitational force, which in reality does not exist as such. In everydayparlance, we reinterpret the curved space-time around us as generating agravitationalforce.Aswewill discussinthisbook, thefact that gravityisnot aforcebutadeformationof space-timewill makeitsquantizationharder. Thestandardcomputationaltechniquesofquantumeldtheoriesallassumeoneworks onagivenbackgroundspace-time, eventhoughintheendthegoalistocreateanS-matrixassumingthatspace-timeisatintheasymptotic past, future, and spatial innity. But in gravity space-time is aeld and therefore the object to be quantized without the presence of any2 Whyquantize gravity?backgroundspace-time. Thelackof abackgroundstructuretechnicallytranslates itself intothetheorybeingnaturallyinvariant under dieo-morphisms1of the space-time points, since there is nothing to distinguishonepointfromanother. Thereislittleexperienceinapplyingquantumeldtheorytechniquestodieomorphisminvarianttheories, exceptforcertaintopological theorieswithnolocal degreesoffreedom. Moreover,gravity is not a very important force in the microscopic realm, where oneexpects quantumeects tobedominant. Tounderstandthis, considerthe ratio of gravitational to electromagnetic forces between, say, a protonandanelectron; gravityis about 1040times weaker. This is therootcauseofwhyeventodaywedonothaveasingleexperimentthatclearlyrequiresquantumgravityforitsexplanation.Itisperhapsthersttimeinthehistoryof physics that oneis tryingtobuildatheorywithoutexperimental guidance. Ifgravityishardertoquantize, andindomainswhere it really is important quantum eects are expected to be small, whybother quantizing it at all? Could we not keep it as a classical interaction?This is not a moot question: attempts to quantize gravity have been madesincethe1930s. If innearly80yearswehavenotsucceeded, whykeeptrying?Tobeginwith, eventhoughthere are nocalls fromexperimentallyaccessible situations where one needs to quantize gravity for their descrip-tion, therearemanyphysical processesonecanimaginethatrequireaquantum theory of gravity for their description. A simple example wouldbetostudythecollisionoftwoparticlesatenergiessohighthatgravitybecomes relevant. Another example we will discuss later in the book wouldbetoconsiderablackholethatevaporatesviaHawkingradiationuntilits mass is comparable to Plancks mass (105g). Or another point we willcoverlaterinthebook: whathappenedtotheuniverseclosetotheBigBang?Thereisalsotheissueofconceptual clarityandunityinphysicsthatsuggeststhatquantizingthreeofthefourfundamentalinteractionswhile keeping gravity classical is unsatisfactory. It is to be noted that thepointof viewof unicationof theories(ormorepreciselyunicationofframeworks underlying theories) has been highly successful in history. Forinstance, puttingNewtons mechanicsandMaxwells electromagnetismonthesamefootingledtospecial relativity. Incorporatinggravityinto1A dieomorphism is a map that assigns to each point of the manifold another pointand that it is dierentiable. It essentially moves the points of the manifold around.Whyquantize gravity? 3thisframeworkledtogeneral relativity. Incorporatingspecial relativityintoquantummechanicsledtoquantumeldtheory.Similarly,unifyingelectromagnetismandtheweakinteractionsledtotherstsatisfactorytheory of the latter. In all instances putting theories on the same footinghas led to the prediction of new physics, some of which have had dramaticimplications. For instance, incorporating gravity, an apparently weakforce, into special relativity leads to the notion of black holes and the BigBang. Quantum eld theory led to the notion of particles and antiparticlesbeingcreatedallthetimeinthevacuum.Moreover, since we do not have a complete theory of quantum gravityitis hard to argue preciselythat there are no experimental consequencesof suchapotential theory. Therecertainlyareunexplainedphenomenarelated to gravity out there, for instance those associated with dark energyanddarkmatterincosmology, thatsomeconjecturemayeventuallyrequireamodicationofthetheoryofgravity.Thatquantumgravityisinanywayresponsibleforthoseeectsremainstobeseen.Butif weignoretheoretical considerations, isthereapractical needto quantize gravity? As we argued before, there are no outstandingexperimentsthatweknowof thatrequireatheoryof quantumgravityfor their explanation. Thereis noconclusiveanswer tothis point, butit canbe arguedthat it will be hardto have a consistent theoryofclassical gravityinteractingwithquantumelds. Onequicklyrunsintoproblemswiththeuncertaintyprinciple.EppleyandHannah(1977)andPage and Geilker (1981) have devised thought experiments that illustratethis point (see however Mattingly(2006) for criticisms). For instance,consider a quantumobject ina state withvery small uncertainty inmomentumandthereforealargeuncertaintyinposition. Wemeasureitspositionwithgreataccuracyusingaverysharppackageof(classical)gravitational waves. That will require asuperpositioninvolvingwavesof very highfrequency, which, however, inclassical gravity canhavearbitrarilysmall momentum. Throughthemeasurementof thepositionwith great accuracy, the uncertainty in momentumin the quantumsystemhas suddenlybecomeverylarge. Wehavethereforepotentiallyproduced a large change in the momentum of the total system, suggestingconservationofmomentummaybeviolated.Such experiments are not conclusive proof, since they cannot be carriedoutinpractice(theaboveonehastheproblemthatgravitationalwavesare verydicult togenerate andcontrol due tothe weakness of the4 Whyquantize gravity?gravitational interaction). Carlip(2008)hasarguedthattoconsistentlycouple a classical gravitational eld to a quantum system will require non-linearmodicationsofquantummechanicsthatcouldbeexperimentallytestedinsomefuture.Inadditiontothis thetwomainparadigms of physics, general rel-ativity and quantumeld theories, have problems of their own. Ingeneral relativitypowerful mathematical theoremsprovedinthe1960sand1970s indicate that under generic conditions space-times becomesingular. Examples of such singularities are the Big Bang we believe to bepresent at the origin of the universe and the singularities that arise insideblackholes. Asingularitygenericallyisassociatedwithadivergenceinquantitiesthat indicatesthetheoryhasbeenpushedbeyonditsrealmof applicability. Near suchsingularities one usuallyencounters energydensitiesthatarenotcompatiblewithacompletelyclassical treatmentanymore. Theexpectationisthereforethatatheoryunifyingquantumeldtheorywithgeneral relativitycouldoer anewperspective, andperhapseliminatethesingularitiesaltogether.Similarly, thequantumtheoryof elds has theproblemthat manyquantumoperators are in the mathematical sense not functions butdistributions, like the Dirac delta. Whenone studies interactions onehas to consider products of these operators and such products are usuallynotwell dened. Someof thedivergencesinquantumeldtheoriescanbe eliminated, redening the coupling constants througha procedureknownas renormalization, andone canuse the theories toformulatephysicalpredictions.Inspiteofthis,asfreestandingmathematicaltheo-ries quantumeldtheories areusuallypoorlydened, andhavetobetreatedusing perturbative series that are not reallyconvergent (theyarewhat is technicallyknownas asymptoticexpansions). Singularitiesinquantumeldtheoryarise due to the distributional nature of theelds and operators. It is clear that if one changes the underlying pictureof spaceandtimethedistributional natureof eldsandoperatorsmaychange.Thismayopenthepossibilityofeliminatingthesingularitiesofeldtheory.Summarizingthelasttwopoints: thecurrentphysical paradigmsforgravityandfor eldtheories are incomplete andinclude singularities.Thereappearsthedistinctpossibilitythatbyunifyingtheseparadigms,singularities could be eliminated. We will take the point of view that aes-thetics, the previously mentioned thought experiments, and the attractiveWhyquantize gravity? 5possibility that unication may cure the problems of the standaloneparadigmsof gravityandquantumeldtheory, areenoughmotivationtosaythatgravityneedstobequantized.Thepreviousdiscussionalsohighlightssomeoftheopenproblemsofthe eld that people are attempting to address in contemporary research.IsthereasingularityattheBigBangordidourcurrentuniverseevolvefrom a previous universe? If so, are there remnants of information comingfrom the previous universe? Does such a potential modication of the BigBanginuencetherestofthecosmological evolution, inparticularhowinationdeveloped, nucleosynthesis, andtheformationof structureinthe universe? What happens inside a black hole when curvatures becomelarge?Does one againtravel intoanother space-time region?We nowknow that black holes radiate like black bodies and could eventually evap-orate. How is such evaporation described in detail? Since the nal productof the evaporation is purely thermal radiation with no distinctive featuresapart from its temperature, what happened to the information included inthe matter that formed the black holes? Are there any phenomenologicalconsequences of quantizing gravity that we could observe? We will touchuponall thesetopicsintheapplicationschaptertowardstheendofthebook.Letusnowturntoabitofhistoryoftheeld. Wewill notattempta detailedhistory here, just give some minimal background. Agoodconcisetreatmentof thehistoryof quantumgravityisinthearticlebyRovelli (2002). Althoughonealreadyencounters mentionof thequan-tization of gravity in papers by Einstein in 1916 and Rosenfeld andBronsteinthenwrote the rst detailedpapers onthe subject inthe1930s, signicantattemptstoquantizegravitystartedonlyintheearly1960s.Threedierentapproachesemerged.Oneapproachwascanonicalquantization, whichwewill largelyfollowinthis book, sinceit is theone that resembles the elementary treatments of quantummechanicsofundergraduatetextbooksthatstudentsmaybefamiliarwith. Inthisapproachonehastoseparatespace-timeintospaceandtimeandaswewillsee,thiswilladdcomplications.Ittookquiteawhiletounderstandhowtoworkout theHamiltonianformulationof theories liketheonethatdescribesgravity. Wewill getaavorinthisbookof whyittooksomeeort. Anotherapproachwastostudythetheoryperturbatively,byassumingthatspace-timeisatplussmall perturbations. Suchper-turbative approaches workedwell for electromagnetismandthe weak6 Whyquantize gravity?andstronginteractions(inthelattercaseincertainparticularregimes).Ingravitythisapproachranintotrouble. Inelectromagnetismandtheweak and strong interactions one can formulate the theory in termsof acouplingconstant that is dimensionless (inelectromagnetism, forinstance, it is the ne structure constant). Ingravity one cannot dothat. Havingacouplingconstant withdimensions implies that if onemakes expansions inpowers of the coupling constant, as one does inperturbationtheory, extrapowersof momentumhavetobeintroducedateachordertokeeptheexpressiondimensionallyuniform. Theextrapowersofmomentummaketheintegralsarisingintheinteractiontermsdivergent. One can correct such divergences by modifying the action, butit requires aninnitenumber of modications tocureall divergences.Tohavetospecifyaninnitenumberoftermsbyhandimpliesthatthetheorydoesnothavepredictivepower. Thisproblemisknownasnon-renormalizability. Stelle(1977)showedthatonecouldcuretheproblemby adding some higher order terms to the action, but the resulting theoryofgravityhasunphysical properties. AgoodreviewoftheperturbativeapproachisthatofWoodard(2009).Wewillpresentahighlysimplieddiscussioninthis book. Thethirdapproachthat was triedis theoneknownasFeynmanpathintegral. Suchanapproachrequiressummingprobabilityamplitudes overallclassicaltrajectories,whichinthe caseofgravityrequiressummingoverall possiblespace-times. Thishasprovedformidablydiculttodo.Remarkably,loopquantumgravitytechniquesarehelpingdenethepathintegral inarigorous way, inanapproachcalledspinfoams.Wewillcoverspinfoamsonlybrieyinthisbook.At the same time as these approaches were encountering diculties, aparallel lineof thought was beingpursued, namelythat of unicationof the elementary interactions into a single theory. Amotivation forthis comes fromthe weak interactions: it turns out that one cannotquantizetheweakinteractions bythemselves, but onlywhentheyareintegratedintoatheorythatuniesthemwithelectromagnetism.Couldthesituationbesimilaringravity?Coulditbethatintegratinggravitywith the other interactions into a single theory would help with itsquantization?Thispointofviewtendstobefavoredbymostphysicistswithbackgrounds inparticle physics. Over the years it has ledto aseries of theories that attempt to unify gravity with the other interactionsandatthesametimeprovideatheoryofquantumgravity. ThevariousapproachesincludedtheKaluzaKleintheories,supergravity,andlately,Whyquantize gravity? 7stringtheoryandM-theory. Wewill notattemptadiscussionof theseapproaches here. Atextbook for undergraduates on string theory isavailablebyZwiebach(2009).Inadditiontotheapproachesdescribedabove, thereareotherideasthatarepursuedbysmallergroupsofresearchers. Theseincludecausaldynamical triangulations, causal sets, matrix models, Regge calculus,twistors, noncommutatiave geometries, andthe asymptotic safetysce-nario. We will not discuss themin this book. Agood introductoryoverviewispresentedinthebookbySmolin(2002).In the mid 1980s, Ashtekar noted that one could rewrite the equationsof gravity in terms of variables that made the theory resemble the theoriesof particle physics. This raised hopes that techniques from particle physicscould be imported to the quantization of gravity. The resulting approachtoquantizinggravityiscalledloopquantumgravityandistheonewewill coverinthisbook. Itisanattempttounderstandthequantizationofgravitybyitself,withouttheneedtounifyitwithotherinteractions.Currently, both string theory and loop quantum gravity are incompletetheories. Somepeopleviewthemascompetingtheoriesandimplythatif oneends upbeingcorrect theother will not be. Our point of viewis more conservative. It could end up being that string theory andloopquantumgravitybothprovidequantumtheoriesof gravitycastindierent languageandhighlightingdierent aspects of theprobleminmorenaturalwaysineachapproach.Atthemomentitisstillunclearifthisisthecase.2Special relativityandelectromagnetismNewtonslawsofmechanicstaketheirsimplestformincertainreferenceframescalledinertialframes.Onecannotdistinguishapreferredinertialframe,theyareallequivalent,andinallofthemNewtonslawstakethesame form. This is the principle of Galilean relativity. However, when oneconsiders electromagnetism as formulated by Maxwell there do appear toexist preferred frames of reference. This caught the attention of Einstein,who found the situation unsatisfactory. He was particularly troubled thatinordertodescribeamagnetmovingclosetoawire, orawiremovingclosetoamagnet, oneneededtousetwodierent physical laws eventhough the end result, the production of a current in the wire, wasexactlythesameinbothcases.Thiswasonlyoneofthemanyapparentconictsthatarosewhentryingtounderstandelectromagneticbehaviorinmechanicalterms.Einsteinsobservationwasthatallphysicallawsshouldbesubjecttotheprincipleofrelativity,theyshouldtakethesameforminallinertialframes. Inparticular, if Maxwellsequationstakethesameforminallinertial frames, the speed of light should take the same value in allinertial frames. This last observation indicates that inertial frames cannotberelatedbyGalileantransformations, sincethelatterdonotkeepthespeedoflightconstant.Ifoneacceptsthispointofview,Galileantrans-formations have to be abandoned along with the concomitant hypothesesaboutspaceandtimethatwereusedtobuildNewtonstheory.Intheirplaceoneneeds tointroducenewtransformationlaws torelateeventsviewedfromdierentinertialreferenceframes.SuchtransformationsaretheLorentztransformations. Galileanrelativitywas built ondailylifeobservations that seemedrootedincommonsense, but inrealitywereonly observations made in a relatively narrow range of relative speeds. ItturnsoutthatGalileanrelativityisonlyaslowspeedapproximationofa more fundamental (and more importantly, physically correct) relativitySpace and space-time 9principle.Thelatterhasmanyimplicationsinvolvingourideasofspace,time, and simultaneity. In this chapter we will explore some of these ideasand in particular introduce the mathematical notation for it that we willlaterusetodescribegeneralrelativity.2.1 Spaceandspace-timeLet us start withsome elementaryvector notationinordinarythree-dimensional space. We start by setting up Cartesian coordinates in spacexiwithi = 1, 2, 3.Thedistancebetweentwopointsinspaceisgivenbyswith,s2=_x1_2+_x2_2+_x3_2=3

i=1_xi_2(2.1)with xithe separation of the two points along the i-th coordinate. OnecanchooseanewsetofCartesianaxes,likethoseshowninFig.2.1(forsimplicity we draw a two-dimensional example), and the distance betweentwopointsremainsunchanged,s2=3

i=1_xi

_2(2.2)sxxyxxyyyFig.2.1 TwosetsofCartesianaxes.10 Special relativityand electromagnetismwherewehavedenotedbyxi

thenewsetofaxis.Thisnotationofusinga prime in the index rather than in the name of the coordinate will proveuseful lateron. If therotationbetweentheaxesshowninthegureisgivenbyananglethenonecanrelatethevaluesofxiandxi

by,xi

= i

ixi3

i=1i

ixi, (2.3)andalsoxi

= i

ixi, (2.4)wherewehaveintroducedtheEinsteinsummationconvention,inwhichanyindexthat is repeatedis summedover fromonetothree. At themomenttheuseofsubscriptsorsuperscriptsinagivenquantitymakesnodierence,butitwilllateron.Thematrixisgivenbyi

i=___cos sin 0sin cos 00 0 1___. (2.5)Avector is acollectionof three numbers Aithat transformundercoordinatetransformationslikethecoordinatesthemselves,thatis,Ai

= i

iAi. (2.6)Youareprobablyfamiliarwithallthis,wearejustxingnotationhere.Somethingyouareperhaps not sofamiliar withis withthenotionoftensor. A tensor is a multi-index generalizationofa vector. The key ideaisthateachindexinatensortransformsasifitweresittingonavector,without any regard to what happens to the other indices. For instance, ifweconsideratensorwithtwoindicesSij,itwilltransformas,Si

j

= i

ij

jSij(2.7)whereagainweareassumingtwosummations,oneoniandoneonj.In ordinary Newtonian physics, if one uses Cartesian coordinates, theseareallthevectortransformationsoneneeds.Inparticularthetimevari-abletremainsunchanged.Inspecialrelativitythesituationisdierent.Special relativityreferstospace-timeandinvolvestransformationsthatmixspaceandtimetogether.Remarkably,wewillseethatthelanguageSpace and space-time 11forcoordinatetransformationsthatwehavedescribeduptonowappliesalmostunchangedtospecialrelativity.Let us now consider a space-time, that is, a four-dimensional space withcoordinatesxwith = 0, 1, 2, 3,withthe1, 2, 3componentscoincidingwithxi, andthezerothcomponent will begivenbyct wherecis thespeedof light. Thereasonweneedc is tohavethesameunits inallthecomponents. Intheoretical physicsitiscommontochooseunitsinwhichc = 1. Inthistextwewill choosethisconventionunlesswewanttoemphasizetheroleofthespeedoflight,inwhichcasewewillstateitexplicitly.Achoiceofc = 1requiresmeasuringtimeinunitsofdistance.Apointinspace-timeisanevent,itisanassignmentofapointinspaceandaninstantintime.Up to now there is nothing special being done from the point of view ofphysics. We could have set up a similar notation in Newtonian mechanics,bundlingspaceandtimeintofour-dimensionalvectors.Butitwouldnothave been very useful. Since in Newtonian mechanics time is unchanging,we wouldonlybe doingtransformations that involve the components1, 2, 3 of the four-vectors and leaving the zeroth unchanged. There wouldbenothinggained.For something physically new to happen we need to introduce the ideaofdistanceinspace-timethatisphysicallyusefulinspecialrelativity.Suchdistancebetweentwoeventsisgivenby,s2= (ct)2+ (x1)2+ (x2)2+ (x3)2. (2.8)Notice that this is not a true distance in that it can be positive, negative,orevenzero(evenforpointsthatdonotcoincide).ThisdistanceiskeptinvariantundertheLorentztransformations,thatis1,x

=

x,s2= (ct

)2+ (x1

)2+ (x2

)2+ (x3

)2. (2.9)Lorentztransformationscomeinvariouskinds. Firstof all, ordinaryspatial rotationsareincluded, forinstancetherotationweconsideredinFig.2.1yieldsthefollowingLorentztransformation,1When we are considering space-time quantities repeated indices are summed from0 to 3.12 Special relativityand electromagnetism

=_____1 0 0 00 cos sin 00 sin cos 00 0 0 1_____. (2.10)Wealsohaveboostswhichmeanthatwearechangingtoanewframeofreferencethatismovingwithrespecttotheoriginalone.Forinstanceifwechangetoacoordinatesystemthatismovingalongthex1directionwithvelocityvwithrespecttotheoriginalone,wehave,

=_____cosh sinh 0 0sinh cosh 0 00 0 1 00 0 0 1_____, (2.11)with the parameter (which ranges from to ) given by =tanh1(v) (remember wearesettingc = 1). If youexplicitlyworkoutthe transformation x

=

xwith the above matrix you will nd thatt

= t cosh xsinh , (2.12)x

= t sinh + xcosh . (2.13)Theseexpressionsmayappearunfamiliar, butabitof algebraandtrigidentitiesleadstothefamiliarexpressions,t

= (t vx) , (2.14)x

= (x vt) , (2.15)where = 1/1 v2. If the velocity vis small compared to the speed oflight, which in our units means v 1, then the above expressions can beapproximated by t

= t and x

= x vt, that is, a Galilean transformation(to see this it is good to restore the speed of light in the equations whichread t

= (t vx/c2), the second is unchanged and =_1 v2/c2andv c).So, just like ordinary rotations keep spatial distances unchanged,Lorentz transformations keep unchanged the space-time distance weintroduced in (2.8). Lorentz transformations mix space and time and it isthereforenaturaltothinkofspaceandtimeasafour-dimensionalsingleentity. That minus sign that appears in the space-time distance, however,hasrathersignicantimplications.Toseesomeofthose,itisveryusefulSpace and space-time 13xxttx = tx = tx = tx = tFig.2.2 EectofaLorentztransformationinspace-time.toconsidersomespace-timediagrams. Letussuppresstwoof thethreespatial dimensions for simplicity and consider the Lorentz transformation,specicallytheboostinthexdirection, graphically. InFig. 2.2weseesignicant dierences with respect to Fig. 2.1. This transformation is notarotation,butrathertheaxesseemtocloseinsymmetricallyaroundthe line x = t, which in turn remains invariant (as does the line x = t),thatisitcoincideswithx

= t

.Noticethatthelinex = tcorrespondsto the trajectory of an object moving (rightwards/leftwards) at the speedof light inspace-time. Weknowthat thespeedof light is invariant inspecial relativity and the invariance of the x = t lines reects that fact.Remember that we are suppressing two spatial directions, if not, the x = tline wouldcorrespondtoalight cone t2= (x1)2+ (x2)2+ (x3)2. Sincetheequationis quadratic, thereis afuturelight coneandapast lightcone with respect to the origin of the diagram (cone pointing upwards ordownwards).Sincetheoriginofthediagramwaschosenarbitrarily,thissituationcanapplytoanypointof space-time. Thatis, ateverypointinspace-time there is afuture andpast light cone. All points withinthefutureorpastlightconesof agivenpointaresaidtobetime-likeseparatedwithrespecttothepointinquestion.Thespace-timedistancebetweenthosepointsisnegative.Pointsoutsidethelightconeare space-likeseparated and their distance to the origin is positive, like an ordinaryspatialdistance.Finally,pointsalongthelightconearelight-likeornull14 Special relativityand electromagnetismseparatedandtheir distance tothe originis zero! Notice that inthispicturethenotionof simultaneityis not absolute. Consider theframet, x. There, pointsatt = 0areall simultaneous, theycorrespondtothehorizontal line at the origin. However, if we consider the same line but inthet

, x

frame,weseethatitdoesnotcorrespondtoasinglevalueof t

,thatis,thoseeventsarenotsimultaneousinthatreferenceframe.The non-universality of simultaneity can be very confusing to deal withat rst. This leads tomanyparadoxes.Thesimplest waytoresolvesuch paradoxes consists of drawing a space-time diagram. Then it usuallybecomesveryclearwhichpreconceptionfromspatialintuitionhavegonewrong. For instance a moving objects length appears to contract. This isknown as Lorentz contraction. Does that mean that one could trap theobjectinsideaboxintowhichitwouldnottatrestjustbyhavingitmove? This is the core of the pole in the barn paradox, which we leaveasoneoftheexercisesattheendofthechapter.Somebasicsaboutspace-timediagrams: aswementioned, lightraysmove alongthe light cone, inthe diagrams where one suppresses twodimensionsitmovesalongthelinesat45degreest = x(withc = 1).A particle at rest moves upwards along a vertical line. All moving particlesmoveat less thanthespeedof light, that is theydescribetrajectoriessteeper than 45 degrees with respect to the horizontal. Events that occurin the future light cone of a given event are causallyconnected to it, thatis, they can be the consequence of said event. Events outside the light coneof an event cannot be caused by it, it would require communication fasterthan light for that to happen. Uniform motion in space-time correspondsto straight lines, the slope giving the velocity. Accelerated trajectories alleventually asymptote to t = x since nothing can move faster than light,evenacceleratingobjects. Apointtobenotedisthatwhenonegoestoamovingframe,inadditiontotheclosinginoftheaxiswementionedabove, there is a rescaling of the axis, which can be signicant for speedsclosetothespeedof light. This needs tobetakenintoaccount whendrawingspace-timediagramstoclarifyparadoxes.2.2 RelativisticmechanicsJust likeinordinaryspace, onedenes vectors inspace-timeas acol-lectionof fournumbersAthattransformlikethecoordinates, thatisA

=

A. Asanexampleof afour-vector, consideratrajectoryinRelativisticmechanics 15space-time, that is, acurve x() where is aparameter. If we con-struct the tangent to the space-time curve associated with the trajectoryU dx/d,onecanquicklyseethatittransformsasthecoordinates,sincethetransformationmatrixdoesnotdependon.Ausefuloperationtohaveamongvectorsisthatofascalarproduct.Weknowthatthescalarproductofanordinaryvectorwithitselfgivesthelengthof thevector. Inordinaryspacewecoulddenethescalarproduct in the following fashion, for instance, starting from the notion ofdistance,s2= ijxixj(2.16)where ij is the identity matrix in three dimensions, that is it is one if i = jand zero otherwise. So we would say that s2is the scalar product of xiwithitself.Similarlyonecandeneascalarproductinfourdimensions,except that thematrixthat is involvedis not theidentitymatrixanymore,s2= xx(2.17)but is given by = diag(1, 1, 1, 1). This matrix is called the Minkowskimetric. The word metric denotes that it helps us measure distances. Butonecanusethemetrictodenethescalarproductof anyvectorwithanotherorwithitself.Noticethat,justlikethespace-timedistance,thescalarproductofafour-vectorwithitselfisnotpositivedenite.Ifitisnegative the vector is called time-like if positive space-like and if zero thevectoriscallednull (therearevariationsintheliteratureastowhichisdenedaspositiveandwhichasnegative).The transformationmatrixfor vectors keeps the Minkowski metricinvariant,=

, (2.18)and the notions of time-like, space-like, and null are coordinate invariant.Ifweconsideracurveinspace-timewecanmeasurethelengthalongthecurve,l =_ dxddxd d. (2.19)16 Special relativityand electromagnetismOne has to be careful since the argument of the square root could becomenegative. Theabovedenitioniscorrectaslongasitispositive. Thatmeans the curve is space-like. If the curve is time-like, that is, eachsuccessivepoint is withinthelight coneof theprevious one, thenthetangent vector to the curve is time-like and the scalarproduct inside thesquarerootisnegative,sowedene,=_ dxddxd d. (2.20) iscalledthepropertime. Thereasonforthisisthatif weconsideraspace-timetrajectorythatdoesnotmovespatially, thatisaparticleatrest, then= t, that is thetimemeasuredat rest withrespect totheparticle.Particletrajectoriesarealwaystime-likeor,ifoneconsidersmassless particles, null. In that case the length of the trajectory vanishes.We can use this concept to dene a physically relevant vector, the four-velocity, U dx/d. Sinced2= dxdxweimmediatelyseethatUU= 1,thatisthefour-velocityisautomaticallynormalized.Arelatedfour-vectorforaparticleisthefour-momentum,denedasp= mU. The quantity m is known as the rest mass of the particle. SuchquantityisaninvariantunderLorentztransformationsandcorresponds,intherestframe,tothezerothcomponentofp.Startingfromtherestframe, wherep= (m, 0, 0, 0), wecangotoaframewheretheparticleismovingwithvelocityvviaaLorentztransformation. Saywechoosetheboostinthexdirectionwehavebeendiscussing.Inthatcasep

=(m, vm, 0, 0), where as before = 1/1 v2. Suppose we choose asmall v, correspondingtoaparticlemovingslowlywithrespecttothespeed of light. If we Taylor expand 1 +v2/2 +. . .. So to leading orderin velocities we have p0= m + mv2/2 and p1= mv. We therefore see thatfor small velocities the spatial components of the four-momentumarejust theordinarymomentum. What about thezerothcomponent?Weidentifyinitthekineticenergymv2/2butwhataboutthetermm?Wecanidentifythat termas theenergyof amass inits rest frame. Thisleadstothefamousformula(restoringc)E= mc2. Thatis, thezerothcomponentofthefour-momentumistheenergyoftheparticleincludingacontributionmc2fromitsrestmassandwhatevercontributionscomefrom kinetic energy. So we see that this vector really captures the energyandmomentumcontentofaparticle.Relativisticmechanics 17Tocompletetheformulationofrelativisticmechanics,oneintroducesafour-dimensional versionofNewtonslaw. Forthisoneintroducestheacceleration four-vector a= dU/d, and one has that it is proportionaltothefour-force,f= ma. (2.21)Noticethatthefour-forceisnotanarbitraryvector, sinceithastobeorthogonal tothe four-velocity(this canbe easilyseendierentiatingUU= 1).Itthereforehasonlythreeindependentcomponents.Beforemovingontoelectromagnetism,weneedtointroduceanothersmall itemdealing with indices. In usual spatial vector algebra, onenormally does not care where indices are placed. In space-time it isusefultomakeadistinctionbetweensubindicesandsuperindices.Givena four-vector Awe dene the same four-vector with its index down asA A. Thetwofour-vectorsarenotthesame, oneof thecompo-nents diers by a sign. One of the reasons this is useful is because one cantake a vector with an index down, take another vector with the index up,and their (Lorentz invariant) scalar product is just obtained by summingtheproducts of their components, that is AB AB. Similarly,givenavectorwithitsindexdownwecanraiseitwithA= A.Thematrix= diag(1, 1, 1, 1)coincideswith.Thissoundslikealotoffusstodealwithasimpleminussign.Butlateroninthebookwewill wanttodiscussspace-timesthatarenotat. Inthiscasegetsreplacedbyamorecomplicated, genericallynon-diagonal, non-constantmatrix. In this case raising or lowering an index implies much more thanjust ipping a sign. If there is such a dierence, why do we call the objectwithanindexdownthesamenameas theobject withtheindexup?Thereasonisthatthephysical contentoftheobjectremainsthesame,itisjustrearrangedinadierentformoncethemetricisgiven.Somephysical quantities are naturally dened with their index down, some withtheirindexup. Forinstance, thefour-velocityisnaturallydenedwithitsindexup, evenincurvedspace-times. Bynaturallywemeanthatwe do not need involve the metric of space-time in its denition with theindexup,wedowiththeindexdown.Atthispointitisconvenienttomentionsomeadditionaljargonthatisusedwithtensors, theideaof symmetryandantisymmetry. Asetofindices (either all lower or all upper but never mixed) are saidtobe18 Special relativityand electromagnetismsymmetricifswappinganypairofthemproducesthesameobject. Thesimplestexampleiswhichsatises= sinceitisasymmetricmatrix. Buttheconceptextendstomanyindicesaswell. Atensor, incontrast, issaidtobeantisymmetricinasetof (eitherupperorlower)indices, such that ifone swaps a pairof them the tensor changes sign.Inthenextsectionwewillseeanexampleofantisymmetrictensorplayinganimportantrole.2.3 Maxwell theoryLet us turn our attention to electromagnetism. Maxwells theory ofelectromagnetismisatheoryoftwovectorelds,calledtheelectriceldEiandthe(pseudo-vector)magneticeldBi.Sinceweareinterestedinfundamental physics, we will only consider the theory in vacuum coupledto charges, that is we will not discuss material media. Maxwells equationsare,2writtenintraditionalvectornotation,

B

Et= 4

J, (2.22)

E= 4, (2.23)

E +

Bt= 0 (2.24)

B= 0, (2.25)where is the density of charge and

J is the current density. Letus attempt to rewrite these equations inspace-time notation. To dothiswerecall thedenitionof acrossproductof twovectorsinindexnotation,(A B)i= ijkAjBk, (2.26)wherethequantityijkisknownastheLeviCivitasymbolandisequalto one if i, j, kare an even permutation of 1, 2, 3 (that is 2, 3, 1 or 3, 1, 2),is zeroif twoindices arerepeatedandis 1otherwise. WecanwriteMaxwellsequationsas,2As is common in theoretical physics we choose natural units where c = 0 = 0 = 1.Maxwelltheory 19

ijkjBk0Ei= 4Ji, (2.27)iEi= 4J0, (2.28)

ijkjEk + 0Bi= 0, (2.29)iBi= 0. (2.30)Pleaserecall that for spatial indices it makes nodierenceif wewritethem as subscripts or superscripts. The notation imeans /xiand thenotation 0means /t. Notice that we have replaced the charge densitybyJ0sowearebuildingafour-vectorJ= (, J1, J2, J3).Wenowproceedtodeneaeldtensor,F=_____0 E1E2E3E10 B3B2E2B30 B1E3B2B10_____(2.31)and we note that the same objectwith its indices up is dened as FF.BothobjectsareantisymmetricF= F.Wecanidentifyvarious components of the electric and magnetic elds, for example, F0i=Ei,Fij= ijkBk(itisagoodexerciseforyoutocheckthese).WiththisnotationtherstfourMaxwellequationscanberewrittenas,jFij+ 0Fi0= 4Ji, (2.32)iF0i= 4J0, (2.33)or,remarkablyas,F= 4J. (2.34)If we now introduce the space-time LeviCivita symbol dened againas+1if, , , areanevenpermutationof0, 1, 2, 3(thatis1, 2, 3, 0or2, 3, 0, 1 or 3, 0, 1, 2), zero if two indices repeat and 1 otherwise, one canrewritetheotherfourMaxwellequationsas

F= 0. (2.35)Toseethis, for instanceif = 0, then, , andhavetobespatialfor theLeviCivitatobenon-zero, sowecanrewritetheequationas

ijkiFjk= ijki

jkmBmandthenusetheidentityijk

jkm= 2imtogetiBi= 0. Asimilarconstructioncanbedoneforthethreeothervalues20 Special relativityand electromagnetismof.ItisworthnoticingthattheLeviCivitasymbolisinvariantunderLorentztransformations(uptoanoverallsigngivenbythedeterminantof

).Let us pause andsee what has happenedhere. We have rewrittenMaxwells equations in space-time notation. The resulting equations havebeenexpressedentirelyintermsofvectorsandtensors. ThatmeansweknowhowtotransformtheseequationsunderLorentztransformationsandtheywill havethesameforminall referenceframes. SoMaxwellstheory was secretly Lorentz invariant, we were just not writing it in a waythat made that manifest. We now have. What do we mean by this? Well,supposethatyouaregivenanelectricandamagneticeldEi, Biandyouareaskedtotransformthemtoamovingframe.Physicallyweknowtheyarenotinvariant.Forinstance,considerachargeatrest.Ithasanelectriceldandnomagneticeld.Ifwegotoaframewherethechargeis moving, there will be a magnetic and an electric eld. Computing themin ordinary vector notation is not straightforward. However, in space-timenotationitiseasy,simplywritetheeldtensorFandthenapplytheLorentz transformation (remember, just treat every index as if the othersdidnt exist), that is F

=

Fand you will get the elds in thenew frame. The transformation will mix together Eiand Bias expected.We see here the space-time notation starting to bear fruits. And it reectsthespiritofEinsteinwealludedtoatthebeginningofthechapter.Toconclude the discussionof Maxwell theoryit is agoodideatointroducethenotionof potentials. Inelectromagnetismoneknowsthatthereisanelectrostaticpotentialandavectorpotential

Asuchthat,

E=

At , (2.36)

B=

A. (2.37)To rewrite these equations in space-time notation, we begin by introduc-ingaspace-timevectorpotential,A= (,

A), (2.38)intermsofwhichwehave,F= AA. (2.39)Problems 21Notice the simplicity of these expressions and howeasy they are toremember.Inparticularyounowknowhowtoquicklytransformpoten-tials tomovingreference frames. Notice, for instance, that if one hasa charge at rest one can choose

A = 0, but if one goes to a framewhere the charge is moving there will be a non-vanishing

Aas thecomponents of Aget mixed by Maxwells equations. If one assumes thatthe electromagnetic eld derives from a potential, as given by (2.39), thehomogeneous set of Maxwells equations (2.35) is automatically satised.NoticethatthedenitionofFintermsofAisleftunchangedifonerescalesA A + withanarbitraryfunction. Suchtransforma-tions are knownas gauge transformations andleave Maxwells theoryunchanged.FurtherreadingThe treatment of the material in this chapter is barely what is needed toworkwithspecialrelativity,tensornotation,andelectromagnetism.Thebook and online notes by Carroll (2003) follow a similar presentation andcanbeusedforamorein-depthstudy.ThebookbyRindler(1977)alsohasagoodintroductorypresentation.Problems1.The pole in the barn paradox. A pole vaulter runs very fast into a barnthatisshorterthantherunnerspole. SinceLorentztransformationsimplylengthsofmovingobjectscontract,anattendantstandingnearthedoorclaimsthepoleisshorterthanthebarnandclosesthedooroncetherunnerhasenteredit.Soitappearsthatapolelongerthanthe barn has been tted into it. Is this possible? Sort the situation outusingaspace-timediagram.2.Show that Dand Dare not invariant under Lorentz trans-formationsbut Dis.3.Showexplicitlythattheboost(2.11)indeedleavesthedistance(2.8)invariant.4.ShowthatF= 0isequivalenttohalf of theMaxwell equa-tions.5.Obtain the magnetic eld due to an current by considering the electriceldofaninnitechargedwire,andLorentzboostingitintoaframe22 Special relativityand electromagnetisminwhichthewireismovingalongitslength. Doesityieldtheusualresult?6.ShowthatthehomogeneousMaxwell equations(2.35)areautomati-callysatisedif theeldtensorisrelatedtoavectorpotential asin(2.39).7.WritetheLorentzforcelaw,

F= q

E + qv

Bincovariantform.3Someelementsofgeneral relativity3.1 IntroductionOneof themaincontributionsof Newtonianphysicswastoestablishasimplelawtodescribegravity. Thelatteristhemostuniversal forceinnature,allphysicalobjectsexperienceit.Newtonslaw,thatestablishesthatthegravitationalforcebetweentwopointmassesisproportionaltothemassesandinverselyproportional tothedistancesquared, success-fully explained the solar system, the motion of projectiles and of everydayobjects. However, it isclear that Newtonsgravitycannot bethenalpicturesinceitisclearlyincompatiblewithspecial relativity. Theforcebetween two masses acts instantaneously, there is no delay in propagation.Forexample,ifonehastwomassesandincreasesoneofthem,theotherwill experienceanincreasedforceimmediately, nomatterhowfarawayitis.Gravityalsohasanotherprivilegedcharacteristicunlikethatof anyotherforce.ThemassthatappearsinNewtonssecondlawrepresentingthe inertiaof aparticle tobe acceleratedis alsothe same mass thatappearsinNewtonslawofgravitationalattraction.Asaconsequenceofthat,givenagravitationaleld,allmassesexperiencethesameaccelera-tioninit.Thisissometimescalledtheequivalenceprinciple.Thiswouldnotbethecaseinanelectromagneticeld,wheregivenaeld,dierentcharges suer dierent accelerations. This fact allows us tomimic(atleastlocally)gravitationaleectsbygoingtoanacceleratedframe.Thisistheoriginof theEinsteinelevatorthoughtexperiment. If someoneisinasmallelevatorthatisfreefallingintheEarthsgravitationaleldoranelevatorthatisoatinginouterspace,nolocalexperimentcarriedout inside the elevator could decide which is which. Similarly an elevatorxedonthe groundonEarthandanelevator inouter space that isacceleratedwithanaccelerationof 9.8m/scannotbedistinguishedbylocal experiments. Theneedforlocalityisthat, if theelevatorislarge,one coulddroptwo objects andinanaccelerating frame theywould24 Some elements of general relativitydropinparallel(neglectingthegravitationalattractionbetweenthetwomasses)whereasontheEarththeywouldgraduallymovetogethersincethe gravitational eld points out radially. What we conclude with all thisis that if oneincludes acceleratingframes of reference, oneis defactoincludinggravity.Theequivalenceofinertialandgravitationalmasshasbeentestedwithenormous accuracy(1013) byusingtorsionbalancesandstudyingtheinuenceofthegravityoftheEarth,Sun,andgalacticcenters using dierent masses in the balance (Schlamminger et al. 2008).Thefactthatacceleratedframesarerelatedwithcurvedgeometriescan also be motivated by a thought experiment also considered by Ehren-fest.Supposeyouareinacarouselthatisnotmovingandyoumeasureitsdiameteranditscircumferenceusingaruler. Youwill getthattheyareproportional andtheproportionalityfactor is . Nowsupposethecarousel starts toaccelerateandit achieves speeds comparabletothespeedof light. If younowtrytomeasurethecircumference, therulerwill Lorentzcontractandyouwill measurealongercircumferencethanbefore.Whenyoumeasure thediameter,asthe ruleristransversetothemotion,itwillnotLorentzcontract.Youwillconcludethatthevalueofhas changed. Kaluzas explanation of this apparent paradox is that therelation between diameter and circumference is in at spaces. By goingto an accelerating frame you have made space look curved, at least in thissense. One can build similar thought experiments with clocks that suggestthat space-time is curved. Unfortunately, as we will see, these argumentsare naive. One cannot make a at space curved just by changing referenceframes. One of the major mathematical challenges in dealing with curvedspace-times is how to recognize true curvature from coordinate dependentnotions. Wewill deal withthis inSection3.3. TheEhrenfest carouselalso involves several other subtleties related to rotating frames that makeits interpretationdicult, agoodsummaryis present inthe relevantWikipediaarticle.Summarizing, general relativity will be a theory where space-time willbe generically curved and in which the physics will be invariant under anychangesofreferenceframe.Itwillalsobeatheoryofgravity,wherethelatterwill notberepresentedbyaforce, butbythecurvatureofspace-time.Assuchitisuniqueamongthefundamentalinteractions,sinceallothers aredescribedbyaforcelivinginabackgroundspace-time. Aswediscussedintheintroduction, wewill notattempttocoverindetailgeneralrelativity,justintroducesomebasicelements.General coordinates and vectors 253.2 General coordinatesandvectorsAswejuststated, general relativityisatheoryof gravity, butinsteadofdescribinggravityasaforce,itdescribesitasadeformationofspace-time. Todeal withthistheorywethereforeneedtolearnabitof howtohandlecurvedgeometries. Therstobservationisthatthemetricinacurvedspace-timeisnotgivenanymorebytheMinkowski metric.Generically, the distance betweentwoinnitesimallyclose points inacurvedspace-timeisgivenby,ds2= gdxdx, (3.1)withgasymmetricmatrixpossiblydependentonthecoordinates.Forinstance, in the vicinity of the Earth, the metric looks approximately like,ds2= (1 + 2(x)) dt2+ (1 2(x)) (dx2+ dy2+ dz2) (3.2)where (x) is the Newtonian potential at the point one is evaluating themetric.Inunitsinwhichc = 1thegravitationalpotentialatthesurfaceof the Earth is about 1010, which explains why we are usually justied inignoring space-time curvature. However, that tiny deviation from atnessisresponsibleforallobjectsfallingtowardstheEarthat9.8 m/s2!Animmediatecomplicationthat arises whenonedeals withcurvedgeometries is that there is no concept of Cartesian coordinates (or gener-ically, there exist nopreferredset of coordinates). Infact there is noconceptofstraightline.Oneisforcedtousewhatonenormallyreferstoas curvilinear coordinates. But this adds several complications. First ofall, itmakesithardertotell ifaspace-timeisatornot. Forinstance,considerthemetric,ds2= dt2+ dr2+ r2_d2+ sin2d2_. (3.3)We would all recognize that (apart from the dt2bit) this is just the usualexpression for distance in spherical coordinates. This space-time is indeedat. But the metric is not , in fact it has a spatial dependence. Worse,there even are points ( = 0, ) where one of the metric coecients seemsto vanish! So this is the rst challenge that curved geometries pose to us:howdoweknowifageometryiscurved?Onecouldarguethatinaatgeometryonecanalwayschoosecoordinatessuchthatthemetricis,andtherefore, ifyouarehandedametricyouproveitisatifyoucanchangecoordinatestothoseinwhichitissimply.Butthatgenerally26 Some elements of general relativityturns out to be an incredibly challenging task! To explicitly construct thecoordinatetransformationrequiresthe solvingofasetofcoupledpartialdierentialequations,whichgenericallyonedoesnotknowhowtosolve.Thatis,ifoneisgivenasetofcoordinatesxinwhichthemetricisgtondcoordinatesx

inwhichthemetricis

oneneedstosolve,gdxdx=

dx

dx

(3.4)andbythis it is meant tosolve x

viewedas afunctionof x. Theresulting equations are non-linear and generically with non-constantcoecients, thatiswhythisissuchahardproblem(historicallyitwasgiventhenameof theequivalenceproblemandhauntedGauss andmanyotherfamousmathematicians). Thebottomlineis: oneneedstodevelopabetterframeworktotellifametriciscurved.The use of curvilinear coordinates which are forced upon us in curvedspace-time creates an additional complication: how to do vector analysis.First of all we need to reconsider our denition of vectors. A curved space-timebehaveslocallyateverypointlikeavectorspace.ThepicturethatemergesisshowninFig. 3.1. Soateachpointinthespaceoneindeedhasavectorspace.Onecantakeitselements,addthem,computetheirscalar products, and multiply them times a scalar, just like in any vectorspace. But the rst warning is that one is not supposed to mix vectors atdierentpointsinthespacesincetheybelongindierentvectorspaces.In ordinary vector calculus we are accustomed to sliding vectors aroundtoaddthem,etc.Thisisonlypossibleinatspaces.IncurvedoneswePFig.3.1 AtagivenpointP,curvedspace-timesbehavelikevectorspaces.General coordinates and vectors 27will have tobe more careful. Animmediate problemcrops upinthedenitionof aderivativeof avector. Thenotionof derivativeinvolvescomparingquantitiesinnearbypoints.Butwejustsaidthatwecannotmixvectorsatdierentpoints. Thishasapractical implication. Inatspace-time in Cartesian coordinates we had that a vector transformed asthecoordinatesA

=

Awiththematricesbeingonlydependenton velocities. In curvilinear coordinates the relationship between theoldandnewcoordinatesisingeneral acomplicatednon-linearfunctionx

= x

(x).However,thetransformationlawforthedierentialofthecoordinatesislinear,dx

=

dxwith

=x

x . (3.5)Since, as we argued, one can only think of vectors at a point, the dieren-tial of the coordinates is a more natural thing to use as a guideline (sinceits value is local) than the coordinates themselves. We will therefore deneavectorincurvilinearcoordinatesasasetofnumbersthattransformasthe coordinate dierentials. Vectors with lower indices transform with theinversematrix,thatis,A =

Awith

=xx

. (3.6)Let us return to the issue of how to take derivatives of a vector. Just bytaking derivatives with respect to the coordinates of the above expression,and for the moment assuming the matrices are coordinate independent,wehave,A = (

A) =

A. (3.7)The left and right members look funny because we are mixing a quantityinthenewcoordinatesystemx

withderivativesintheoldcoordinatesystemx. This can be corrected by noting that since, in Cartesiancoordinates,x

=

xthendx

=

dxor,xx

=

, (3.8)andthereforeapplyingthechainruleto(3.7)weget, A =

A. (3.9)28 Some elements of general relativityThatis,thederivativeofavectortransformsasatensor.Crucialinthisderivationisthatthederivativesailedthroughthematrixinthelastidentity in (3.7) since does not depend on the coordinates. This is true inCartesian coordinates. When one is dealing with curvilinear coordinates,this is not true anymore. The matrices implementing coordinate transfor-mations generically depend on the coordinates. The partial derivative of avector fails to be a tensor, one picks inhomogeneous terms, depending onthe transformation law, that do not vanish and are coordinate dependent.Beingabletotakederivativesiscrucial forsettingupphysical theories,sothisneedstobecorrected.Thewaytocorrectitisbydeninganewtypeof derivativecalledacovariantderivative.Thecovariantderivativediersfromtheordinarypartialderivativeinatermthatislinearinthevectorwearediscussing(otherwise the derivative would fail to be a linear operator). It is denotedby ,A= A+ A. (3.10)Whatarethethreeindexedobjects?Theyarecalledaconnection,because they allow us to connect neighboring points on the space-timeinordertocomputederivatives.So where does one get the connection? The connection in general is anadditional elementthatneedstobeprovidedbywhomeverisprovidingyouwiththe curvedspace-time, just like theyprovidedyouwiththemetric. One cannot dovector analysis if onlysuppliedwithametric,one needs this additional element: the connection. It turns out thatif one demands that the connectionbe symmetric inits lower indices= andif one demands that the connectionbe suchthat thecovariant derivative of the metric is zero, g= 0, then the connectioniscompletelyanduniquelydeterminedbythemetric1. Notethattheseare choices, generally it is not determined by the metric. But it turns outthatthegeometriesthatareuseful ingeneral relativitydosatisfythoserequirements, theyare calledRiemanniangeometries. It just happensthat nature works that way with gravity, but it could have been dierent.People triedtoexploit more general geometries tounifygravitywith1In the technical jargon the symmetry of the connection is called torsion free andthe annihilation of the metric by the covariant derivative is called the connection ismetric-compatible.Curvature 29electromagnetismmanyyearsago, butsucheortsprovedunsuccessfulintheend(Goenner2004).Theexpressionfor theconnectioninterms of themetricinaRie-manniangeometryis,=12g(g + gg). (3.11)If youarecurious about wherethis comes from, onetakes g= 0,writes it explicitly, and cycles its indices and sums. This formula isknown as Christoel formula (or archaically Christoel symbol). Noticethat this formulagives automaticallyvanishingconnectioncoecientsif the metric is Minkowski and one is in Cartesian coordinates. Butit will give non-trivial coecients if the metric is at space, say, inspherical coordinates. These extra terms are related to those extra termsthatyouhavealreadyencounteredinpastcourseswhenyoudiscussedgradient, curl, anddivergenceincurvilinearcoordinates, forinstanceinelectromagnetism. Notice that the connection is not a tensor: it can vanishinonecoordinatesystemanddoes not vanishinanother. This makessense: we could not have xed the problem that the derivative of a vectoris not a tensor by adding a bit to its denition that is a tensor. Inevitablywhatweaddedhadtofailtobeatensoritself.3.3 CurvatureWestill donothaveasatisfactorydenitionof curvature. True, if theconnectioncoecientsvanishidenticallyeverywherethenthemetricisat. But otherwisewedonot know. Curvature, infact, is not alocalconcept. Itturnsoutonecanalwayschooselocally(atonepoint)coor-dinates wherethemetricis Minkowski andtheconnectioncoecientsvanish. ThisiswhatoneoperationallydidintheEinsteinelevatorneartheEarthwhenoneallowedittofreefall:locallyonecouldnotseeanyeectsof gravity. Duetothispropertywecansetuplocal coordinateswherethingslookMinkowskian.Sohowdoweknowweareinacurvedspace?Theanswerisnecessarilynon-local, thatisyouneedtotakeawalkaroundyourspace(orininnitesimal calculations, oneneedstoinvolve at least second derivatives). Consider that you and a friend try toestablish if the Earth is at or not. To do so, you grab a straight piece ofmetalandwalkaroundtheEarthwhiletryingtokeepthestraightpieceofmetalasparalleltoitselfasyoucanasyoumove(andtangenttothe30 Some elements of general relativityabFig.3.2 Onecandetermineifasurfaceiscurvedbyattemptingtotransportastraightline around it ina closed circuit and seeing if it comes back pointing inthe samedirection. HerestartingintheNorthPole, gotopoint aontheequator, thentopointb,andthenreturntotheNorthPoletondthestraightlinepointsatadierentangle.Theangleisbiggerthebiggertheareaenclosedbythecircuit.Earth, we are trying to ascertain the curvature of the surface of the Earthasatwo-dimensionalspaceandyouandyourfriendashavingnegligibleheight, sinceotherwisewewouldbemakingmeasurementsoutsidethespaceofinterest).Suppose,asinFig.3.2,thatyouandyourfriendstartattheNorthPoleandwalkdowntowardstheequator. Yourmetal rodwill pointsouthwardattheequatoratpointa. Supposeyounowgotopointb, againkeepingtherodasparallel toitself aspossibleandyouthenbacktracktotheNorthPole. WhenyouarriveattheNorthPoleyoull realizethattherodisatananglewithrespecttotheorientationyoupickedat departure. Theanglewill bebigger thebigger theareaenclosedbyyourtrajectory(if yougodownandupagainonthesamemeridian,forinstance,theangleisunchangedasyouenclosezeroarea).This notion of curvature is a bit dierent from the colloquial usage of theword. Consideracircle. Ifonehadone-dimensional livingorganismsonthecircle, andtheytriedtheabovementionedmethodtodetermineiftheyarelivinginaatorcurvedspaceandattemptedtocarryametalrodaroundthecircle, theywouldconcludethatitisat. Thatis, themetal rod will depart and arrive at a point pointing in the same directionaftergoingaroundthecircle.Thesameistrueforacylinder.Soarewemissingsomethingwiththisnotionofcurvature? Theanswerisyes.Thenotion of curvature we have introduced is the best that can be done fromwithinaspace. Itiscalledintrinsiccurvature. ThereisnomechanismCurvature 31thatonecandeviseforthepoorcreatureslivingonacirclethatwilltellthemthatitisnotatusingmeasurementstheycanmakewithinthecircle.Thereexistsanothernotionofcurvaturewhichpertainstohowaspacerelatestoahigherdimensional spaceandobserversexternal toit(like us observing the creatures living on a circle). Such a notion is calledextrinsiccurvatureandwewillreturntoitlateroninthischapter.Goingbacktotheintrinsiccurvature,weneedtotranslatetheabovedescribed non-local notion of the curvature to something useful for dier-ential calculus purposes. One possibility is to consider moving around aninnitesimally small loop. The change in the direction of the vector wouldbesmalltoo.IfwebuildaninnitesimalclosedloopliketheoneshowninFig.3.3wecouldconsidermovingfromatobviapointcorviapointd. In one case we would advance rst through dx, and then through dx,andintheothertheorderwouldbereversed. Thisleadsustoconsiderthedoublederivative(onceinthexdirection,onceinthexdirection)todisplacefromatob. Intheother casetheorder of thederivativesshouldbe reversed. The expressionwe want toconsider, therefore, is() V.Noticethatoneisforcedtoinvolveanobjectwithat least one index like a vector V, since covariant derivatives on a scalarare just ordinary partial derivatives andtherefore commute (at leastwhentheconnectionis symmetricas wewill assume). Now, obviouslytherateofchangeinthedirectionof thevectorshouldbeproportionalto the vector itself. And as we see from the previous expression, it shouldproduceanobjectwiththreeindices, , and. ThatmeansthatthedxdxadbcFig.3.3 Theinnitesimal versionoftakingacircuitaroundacurvedspace.32 Some elements of general relativityproportionalityfactoroughttobeanobjectwithfourindices. SuchanobjectisusuallycalledtheRiemanntensororcurvaturetensor,( ) V= RV. (3.12)Anobject withfour indices mayappear intimidating, but it is rathernatural inthis context: we needtoproduce as aresult avector (oneindex) but the result will dependonhowthe innitesimal element ofareaisoriented(twoindices)andonthevectorwestartfrom(anotherindex).Ifthecurvaturetensoriszerothederivativescommute.Sincewecanconstructaniteclosedcircuitbycombininginnitesimal ones(atleastifnonon-trivialtopologicalfeaturesareinvolved),ifthecurvaturetensorvanishesatall points, thenweareassuredthespaceisat. Wehavethereforesucceededincharacterizinginanunambiguouswaytheideaof curvature. Itdoesnotmatterhowcrazythecoordinatesoneisworkingwith, this denitionwill workbecauseatensor that vanishesinonecoordinatesystemvanishesinallcoordinatesystems.Noticethattheconnectioncoecientsingeneral will benon-zeroandthecovariantderivativeswill becomplicated, butneverthelessif thespaceisatthecurvaturewill bezero. Youarecapableofworkingout, ifyouwish, theexplicit expressionof the curvature tensor interms of the connectioncoecients. Itjustinvolveswritingouttheexpressionforthecovariantderivativesandplayingaroundabit. Sincewewill notusethisformulainthebook,weomitit.Atthispointitisgoodtointroduceanadditional tensoroperationwewill nduseful. Whenonehas atensor withapair of indices oneupandone down, one cansumover themandthe result is atensorwith two indices less (you saw a particular case in Problem 2.2). Such anoperation is called contraction and is the generalization to this contextoftheideaofthetraceoneusesinmatrices. Ifoneviewsamatrixasatensor withanindex up andanindex down, summing overthe indices isnothing more thansumming the diagonalelements ofthe matrix,thatistaking the trace. If one wants to contract two indices that are both up ordown, onemustraiseoneorloweroneusingthemetricaswediscussedbefore. When a tensor has many indices there are several possible traces(contractions) that one can introduce. In the case of the Riemann tensor,commononesaretheRiccitensor,R R(3.13)The Einstein equations and some of their solutions 33andthecurvaturescalar,R = Rg. (3.14)3.4 TheEinsteinequationsandsomeoftheirsolutionsUptonowall thishasbeenamathematical construction. TheEinsteinequations are the physical equations that determine which geometryoccurs in nature. They state that a linear combination of the componentsof theRicci tensoraredeterminedbytheamountof energyandstresspresentinmatter.Specically,R12gR = 8GT. (3.15)InthisequationTisconstructedfrommattereldsandisknownasastress-energytensor or energy-momentumtensor andGis Newtonsconstant. This equationis similar inspirit totheequationthat deter-mines the Newtonian potential 2 = 4G, where is the gravitationalpotential and is the density of matter. Just like the left-hand side of thatequationhassecondderivativesof , theleft-handsideof theEinsteinequationshassecondderivativesof themetric, whichplaystheroleofgravitational potential. Thereare, however, important dierences. TheNewtonian equation only has second spatial derivatives. That means thatanychangeintimeontheright-handsideinstantaneouslypropagatestotheleft-handside. Newtonstheorythereforeclearlyviolatesspecialrelativity. The Einstein equations contain both space and time derivatives.This means that changes in the matter properties do not propagateinstantaneously to the metric. This respects the spirit of relativity, whereoneknowsthatnothingcanpropagateinstantaneously.At this point it may be useful to consider some examples of solutions oftheEinsteinequations.Ofcourse,atspaceisasolutionofthevacuumequations (whenT= 0). But there exist other solutions invacuum.Animportantsolutionisthegeneral solutioninvacuumwithsphericalsymmetry,knownastheSchwarzschildsolution,ds2= _1 2GMc2r_dt2+_1 2GMc2r_1dr2+ r2_d2+ sin2d2_.(3.16)34 Some elements of general relativityThis solution plays an analogous role to the potential = GM/rinNewtons theory. Just like that potential is asolutionof 2 = 0,the Schwarzschildsolutionis a solutioninvacuum. Unfortunatelyitssimpleappearanceis quitedeceptive. Thesolutionwas foundbyKarlSchwarzschildin1916whileghtingontheRussianfrontofWorldWarI and also while gravely ill (he died a few months later). So the discoveryof thesolutionwasextremelymeritorious(Einsteinhadconsideredhisequations toohardtosolveinclosedform, preferringanapproximatesolution, andwas surprisedwhenhe heardof Schwarzschilds result).Theproper interpretationof thespace-timesolutionit represents wasonly found many years later by Kruskal (1960)and Szekeres (1960). It isquite remarkable that the best minds in physics of the twentieth century,Einstein,Weyl,Eddington,andothers,werebaedbythesolutionanddiedwithoutknowingitstruemeaning.Thereasontheywereallbaedwas they managed to confuse what are coordinate dependent eects withtrue physical eects. We do not need to concern ourselves here with otherpeoples confusions(whichinthemindofmanyamateurphysicistsstillpersist today!), sowewill just stateour modernunderstandingof themetric. Adetaileddiscussionof all thepropertiesof thespace-timethesolutionrepresents is beyondthescopeof this text, but wewill list afew. Forvaluesof rmuchlargerthanM(inunitsinwhichc = G = 1)space-timeisapproximatelyatandordinarylooking. However, thingschange dramatically close to r = 2M. First of all, at that point the metrichas asingularity, whichit turns out is not atruesingularity, onecanactuallyndcoordinates where the metric is regular at r = 2M, andthatiswhatKruskalandSzekeresdid.Carefulstudiesindicatethatthesurfacer = 2Misactuallynot atime-likesurfaceat all but it isnull.Thatisanobjectat r = 2Mismovingatthe speedoflighteventhoughitremainsatr = 2M! Theconsequenceof thisisthatanyobjectthatpenetrates closer tor = 0thanr = 2Mcannot get backout unless itmoves faster thanlight. There exists what is usuallycalledatrappedregion in the space-time. This is what is normally called a black hole. Thesurfaceatr = 2Miscalledtheevent horizon. Intheregioninteriortor = 2Mthesignof thedt2anddr2coecients change. This indicatesthatthevariablerbehaveslikeatime.Justlikeoutsider = 2Moneinevitablymarchesintothefutureast increases, insider = 2Moneinevitably marches towards r = 0, which is not a point at all but actuallyaspace-likesurfacethat lies tothefutureof all points interior totheThe Einstein equations and some of their solutions 35blackhole.Thatis,ifyoufallintoablackholeyouinevitablyendupatr = 0. At that point there is a genuine singularity that cannot be removedbychangingcoordinates. Closetothesingularitythevaluesof all non-vanishingscalarsyoucanbuildbytakingcontractionsof thecurvaturetensor diverge. An object approaching that region would be ripped apartby gravitational tidal forces. Black holes are expected to form when starsexhaustthenuclearfuelthat,whenburned,producesradiationpressurethatkeepstheirsurfacebuoyantagainstthegravitationalself-attractionofthematterconstitutingthestar. Whenthefuel isexhaustedthereisno pressure and the surface starts to collapse. Eventually it will cross ther = 2Msurface and all the matter in the star will be trapped in the blackhole.Another solution of interest is the FriedmannRobertsonWalker(FRW)cosmologyds2= dt2+a(t)2_dx2+ dy2+ dz2_, (3.17)where a(t) is determinedbythe Einsteinequations dependingonthematter content. This solutionrepresents aspace-timethat has spatialsections that arehomogeneous (thereis nodependenceonx, y, z) andisotropic(all threedirectionsaretreatedequally). Webelievethatouruniverse today is very close to homogeneous on large scales. This is calledtheCopernicanprinciple, whichstatesthatourpositionintheuniverseisnotprivilegedandthereforeitshouldlookthesamewayatall otherpoints at the same cosmic time. From our vantage point we observe that atlarge scales the universe is very approximately isotropic. For instance, thecosmic microwave radiation background has a temperature of 2.72Kelvinthatisisotropicuptoonepartin105.Sothismetriccouldapproximateour current universereasonablywell. Weallowfor spatial distancestochangewithtime.Itisnotasolutionofthevacuumequations(unlessaisaconstant, inwhichcaseonehasaatspace-time)butitsolvestheequations coupled to various forms of matter depending on the functionalformof a(t). For instance, if Thas as onlycomponent T00= (t) itrepresents agas of particles of densitythat arenot moving. This isactually not a bad approximation to the matter in our universe today. Inagasof movingparticlestherewouldbeothercomponentsof Tthatarenon-vanishingthat areproportional tothevelocities, but sinceweareusingunitswherec = 1thevelocities,sayofgalaxies,areverysmallnumbers. Inthatcasea(t) = t2/3. Weobservethatatt = 0thereseems36 Some elements of general relativityto be a problem. Indeed there is a singularity that cannot be removed bychangingcoordinates. Againcurvaturesblowupasoneapproachesthesingularity. ThisistheBigBangthatmostmoderncosmologytheoriespredict as the originof our universe. We will see that loopquantumgravity, whenappliedtocosmologies, indeedeliminatesthissingularity,but nevertheless thereis aregionof largecurvatureinthespace-timeclose to where classical general relativity predicts the Big Bang occurred.Thosehighcurvaturesimplyanymatterpresentisathightemperature.The radiationemanating frommatterinthat eraiswhatconstitutes thecosmic microwave background today. The low temperature we observe forit is due to the fact that between the moment of emission of the radiationandtodaytheuniversehasexpandedandtheradiationcooled.A couple of extra remarks about FRW. First of all, what we presentedhere is what is known as the at model. It is clear that a spacebeing homogeneous does not necessarily mean that its spatial sections arenecessarily at. For instance a sphere is clearly homogeneous and curved.A hyperboloid is another example of a homogeneous space. One can easilygeneralizethemetric(3.17)tohomogeneousbutcurvedspatialsections,but we will not need it in this book. A second remark is that in the cosmo-logical context a dierent kind of matter is sometimes considered calledacosmological constant. ForsuchmatterT= g/(8G)withaconstant. The origin of such matter is a historical accident. When Einsteinrsttriedtosolvehisequationsforthehomogeneousandisotropiccasehe readily noted, as we did above, that the universe expands or contracts.However, this was in 1917 and at the time it was widely believed that theuniverse was static. Einstein noted that if instead of using as the left-handside of the eld equations R gR/2, one used RgR/2 + g,the theory was still consistent but admitted static universes as solutions.When it was later realized that the universe expands, Einstein called theintroduction of the cosmological constant the biggest blunder of my life(Gamow 1970). In modern cosmology, observations of distant supernovaesuggestthatthecosmological constantisnon-zero. Ontheotherhand,theenergymomentumtensorforthevacuumstateof aquantumeldtheory has exactly the form of a cosmological constant, which leads sometobelievethatitcouldbeitsorigin. Unfortunately, anaivecalculationshowsthatthevaluepredictedbyquantumeldtheoryis120ordersofmagnitudetoobig.ItisexpectedthataquantumtheoryofgravitymayDieomorphisms 37shedlightonthisproblem.Someproposalsexist,butnonehasachievedwide consensus amongtheorists to date (foranalternativeviewpointseeBianchiandRovelli(2010)).3.5 DieomorphismsAtthispointitisagoodideatointroducetheconceptof amanifold.Amanifoldisatopological set(asetinwhichonecandeneneighbor-hoods of elements andtakelimits) that is mappable(at least inlocalpatches)toRn, thatis, onecansetupcoordinatesonittodistinguishitselements. Forinstance, acollectionof chairsisnotamanifoldsincethereisnonotionof neighborhood. Butatabletopis. Atabletopwithanail stickingout of it is not (the tabletopmaybe mappable toR2andthe nail to Rbut the neighborhoodof the point where the nailentersthetabletopcannotbemappedtoRn). Theequationsofgeneralrelativityareinvariantundergeneral coordinatetransformations. Thereare two ways to deal with coordinate transformations. One of them, mostcommonlyconsideredinphysics, is tokeepthepoints of themanifoldxed and change the mapping into Rn. This is a change of coordinates.Butthereisanotherpossibleperspective, andthatistokeepthemaptoRnxedandmovearoundthepointsinthemanifold. Thewaywemove aroundpoints is througha type of mathematical mapcalleddieomorphisms. A dieomorphism is a one-to-one map that given a pointpinthemanifold, mapsittoanewpoint(p), anddoessorespectingtopologicalnotionsofproximity,soifonewasabletotakederivativesof quantities at point pbeforethedieomorphism, onecanstill doitafterit. Thislatterperspectiveoncoordinatetransformationsiscalledtheactiveviewandthemorecommononeinphysicscontextsiscalledthe passive view. The active viewis more natural inthe context of atheory of gravity like general relativity. The latter is a theory of a metricthat lives ona manifold. Onsucha manifoldthere are no preferredpointsuntil oneintroducesthemetric. Eachpointhasthesamestatusas any other. Therefore it ought to be possible to move the pointsaround without anyconsequence for the theory. This, ina nutshell,is thereasonwhyanytheoryof gravityof geometricoriginhas tobeinvariantunderdieomorphisms(orif youwant, undergeneral coordi-natetransformations). It just reectsthat thetheoryhasnopreferredbackgroundstructuresthatcantellapartonepointfromanotheronthe38 Some elements of general relativitymanifold.Thisisaconceptthattakessomeworktogetusedtobecauseall otherphysicsonehasnormallydealtwithdonothavethistypeofinvariance.Dieomorphismsmovearoundnotonlythepointsofthemanifoldbuteverythingthatissittingonit.Tounderstandthispointbetteritisgood to discuss rst the more general concept of a map between sets andhowmapsmovearoundobjects.SupposeyouhavetwodierentsetsMandNandyouhaveamapthatassignstoeachpointin MapointinN. Notice that Mand Ndont even have to have the same dimensionalityinprinciple,wewillassumetheydojustforsimplicity.SoifpisapointinMthenq= (p)isapointinN. Wewill call thisoperationapushforwardofthepointpintoN.Now,supposeyouhaveafunctionfthatassignsareal numbertoeachpointinN. Suchafunctioninprincipledoes not knowhowtoact inM. But wecanuseour maptoteachithowtoactinM. Wedeneanewfunction(f)inthefollowingway(f)(p) f((p)) = f(q) and it is clear that such a function acts on M.Usingthemapinthiswayiscalledthepull backofthefunctionffromNtoM.Whataboutvectorsandtensors, canwepushforwardandpull backthose too? A vector can be thought of as an operator acting on functions.How? Givenavector V, one candene the derivative of afunctionf along suchvector Vf andthe result is a function. So say onehasavectoractingonM. OnecandeneanoperatorinNactingonfunctionsbyconsideringthepushforwardofthevectordenedlikethisV((f)).Thatis,thevectoractsonthepullbackofthefunction.This sounds a bit abstract. Can we nd out the components of such avector?Wecan.Consider,(V )f= V((f)) = V(f ). (3.18)Tocompute this explicitlywe needtointroduce coordinates. SaythecoordinatesinMarecalledxandthecoordinatesinNarecalledy.Thenthelastmemberofthepreviousidentitycanbewrittenas,V(f )x= Vyxfy. (3.19)To write the above expression we need to assume that the map is smooth,since we are computing derivatives, and we will assume it is invertible soone cangobackandforthbetweenxandy. Suchamapis calledaDieomorphisms 39dieomorphism. Suddenly, theabstract has becomeveryconcrete: theeect of the dieomorphismon the vector is to multiply its compo-nents times thematrixy/x. But that is thematrixof coordinatetransformations! So if we had assumed that the two manifolds werethe same, the practical eect of the dieomorphismis the same as acoordinatetransformationinthemanifold,aswewereanticipatingfromthebeginning.Itshouldbeclearfromtheabovediscussionthatcertainobjectscanbepushedforwardandcertainothers pulledbackbut theconverseisgenericallynottrue. Unlessthemapisinvertible. Thenonecanrepeatthe above construction swapping Nand Mjust by considering the inverseof themap, anddenepushforwardsandpull backsforall quantities.Sincedieomorphismsareinvertiblemapswewill beabletopull backandpushforwardviadieomorphismsanyobjectwelike.Onelastthingweneedtoconsideristheconceptofaone-parameterfamilyof dieomorphisms t. This is afamilyof maps suchthat fort = 0onehastheidentitymapandthemapsmovethepointsfartherawayforbiggervaluesoftheparametert. Startingfromagivenpointp, the continuous family of dieomorphisms t(p) generates a curveonthemanifold. Wecancomputethetangent vector tosuchacurveV= dx/dt.Thenotionof dieomorphismcanactuallybeusedtodeneanewderivativeof vectors andtensors. It will havethepropertyof beingavector or tensor without the need to introduce extra structures. It is calledtheLiederivative.GivenaoneparameterfamilyofdieomorphismstheLiederivativeofatensorTalongavectorisgivenbyLVT(p) = limt0t(T(t(p))) T(p)t. (3.20)So we push forward the point p via a dieomorphism along a curve whosetangentvectorisV,evaluatethetensoratthenewpointandthenpullback the whole thing to p. That way, we can subtract from it T(p), sinceweareallowedtoaddandsubtractvectorsandtensorsprovidedwedoso at a given point. It is clear that the result of this operation is a tensorofthesamerankastheonewestartedwith.Wewill not explicitlyderiveheretheformof theLiederivativeonanarbitrarytensor.Inprinciplebygeneralizingabittheresultsforthevectors we showed above you could work it out, we just quote the formulaandrefertothebookbyCarroll(2003)forfurtherdetails,40 Some elements of general relativityLVT12k12m= VT12k12m(V1) T2k12m(3.21)(V2) T1k12m +_1V_T12k2m+_2V_T12k1m +Liederivativeswill proveuseful whendiscussingsymmetriessinceasymmetryusuallymeansthatcertainthingsremaininvariantwhenoneperforms adieomorphism. For instance, aspace-timeinvariant undertranslationswillbegivenbyametricthathaszeroLiederivativealongaparticular vector eldwhichis tangent toastraight line (it is notheretical tomentionastraight line here since we are talkingabout aparticular family of space-times for which straight lines exist!). The sameholdsforinvarianceunderrotationsaroundanaxis, themetricinthatcase will have zero Lie derivative with respect to a vector that is tangenttoacircle.You may wonder why, if the Lie derivative exists without havingto introduce a connection, didwe bother withdening the covariantderivative?Thelatter provedcrucial todenethenotionof curvatureinacoordinateindependentway. OnecannotusetheLiederivativetodothat.3.6 The3 +1decompositionWe will eventually be discussing a Hamiltonian formulation of general rel-ativity.IfyourecalltheHamiltonianformulationofordinarymechanics,it is given in terms of a set of canonical variables q, p at a given instant oftime t. If one is dealing with elds rather than a mechanical system thenthe canonical variables are functions of position (x) and their canonicalmomentaareas well (x), bothgivenat aninstant of time. Generalrelativity, aswehavediscussedituptonow, treatsspaceandtimeonthesamefooting.ThisisnotwhatisdoneinHamiltonianformulations.Sowewill needtobreakthatequal footinginordertodiscussgeneralrelativity in a Hamiltonian fashion. Some people consider this a signicantdrawbackof theHamiltonianformulationsinceit imposes anarticialdistinctiononthevariables of thetheory. It is actuallyslightlyworsethan that. Although we have not emphasized it, the space-times of generalrelativity cannot only have arbitrary metrics but also arbitrary topologies.Inthe Hamiltonianformulation, since one will single out a time-likeThe3 + 1decomposition 41directioninwhichthingsevolve,thetopologywillcomeoutnaturallytobe R where is a three-dimensional manifold (of arbitrary topology)and R is the real line, and it cannot change upon evolution. Some peopleconsiderthistoorestrictive, thoughitiscrucial inordertohaveawellposed initial value problem. We will see that the Hamiltonian formulationis smart in the sense that the framework ends up noticing the arbitrarydistinctionweintroducedandactuallycompensates for it, makingthenal result insensitive to our choice of a time-like direction. However, therestrictions ontopologyremain. But this will havetowait till furtherchapters.Herewewill thereforeassumethatspace-timehasa Rtopologyand that there is a time-like direction characterized by a vector twhoseorbits2are acurve parametrizedbyaparameter t andsuchthat t =constant surfaces are spatial slices . We introduce a vector eldnnormal to . One can then dene a positive denite spatial metric on ,qab gab + nanb. (3.22)Intheaboveexpressiontheindices a, bgofrom1to3. Onedoes notneedtoidentifycoordinatesinsuchaway, butitmakesthingseasierifwe just view the coordinates 1, 2, 3 as coordinatizing . Otherwise this isagreatplacetousetheframeworkwesetupwhenwediscusseddieo-morphisms, setting a map between the three-dimensional surface and thefour-dimensional space-timeandpullingbackthings appropriately. Wecandecomposethevectortincomponentsnormalandtangentialtoas,ta= Nna+Na.Thisexpressionistrickysincetheindexontareallycould go from 0 to 3, and so does the index on na, but not the one in Na.Soifyouwanttoevaluatethezerothcomponentofthatexpressiononlythersttermcontributes. ThequantitiesNandNaareusuallyknownasthelapseandtheshift vector. Theycanbeviewedasascalarandavectorlivingin. Intermsofthesethefour-dimensional metriccanbewrittenas,ds2=_N2+ NaNa_dt2+ 2Nadtdxa+ qabdxadxb. (3.23)One can dene the extrinsiccurvature, a notion we mentioned before, as,Kab=12Lnqab, (3.24)2The orbit of a vector is a curve such that its tangent is the vector of interest.42 Some elements of general relativityNnataNa3Fig.3.4 Thevectorsinvolvedinthe3+1decompositionofspace-time.anditiscloselyrelatedtothetimederivativeofthemetric, qab Ltqab= 2NKab +L

Nqab. (3.25)These are the variables that we will use to construct a Hamiltonianformulationfor gravity. The congurationvariable will be the three-dimensional metricqabanditsconjugatemomentumwill berelatedtotheextrinsiccurvatureKab. Justliketheusual momentumisrelatedtothetimederivativeofthecongurationvariable,soisthemomentumofthemetric. Thesevariables will evolveintime, helpingus build, withthelapseandtheshift, thespace-timemetric. Thelapseandshiftwillendupbeingarbitraryfunctions, reectingthefreedomtochoosefourcomponentsofthemetrictotakewhatevervaluewewantbychoosingacoordinatesystem.3.7 TriadsWhen one works in loop quantum gravity one uses a description of gravitythat is slightly dierent from the one we introduced above. Since one willwork in the Hamiltonian picture when one is interested in the spatial partof themetricas wediscussedabove, let us consider three-dimensionalspaceratherthanspace-timeandintroduceasetof threevectoreldsEai ,i = 1, 2, 3thatareorthogonal.Wecanwrite,qab= Eai Ebjij, (3.26)so on the right-hand side we have at space in coordinates i, jand on theleft-handsidewehavecurvedspacewithcoordinatesa, b. WethereforeTriads 43seethattheEaiareallweneedtoconstructthemetric.Suchquantitiesare called triads. So we now have two dierent types of indices, spaceindices a, b, c that behave like regular vector indices in a curved space, andinternal indices i, j, k. Both sets of indices range from 1 to 3. However,ifonewishestoraiseandlowerinternalindicesoneshoulddothatwiththeatmetricij. Suchindicescanbecontractedamongthemselvesjustlikeordinaryindices.If oneis goingtooperateonobjects that haveinternal indices oneneeds tointroduce anappropriate derivative. Sayyouhave anobjectwithjustaninternalindexGi.Wedeneaderivativeonitbasedonourexperiencedeningthecovariantderivativeas,DaGi= aGi+ aijGj(3.27)and just like in the covariant derivative the components of the spinconnection aijhave to be specied externally. In order for the derivativeofascalar(forexampleGiGi)tobetheordinarypartial derivativeonemusthavethatDaGi= aGiajiGj(3.28)andif oneneedstotakethederivativeof anobjectwithmixedindicesoneusesbothconnections,thatis,DaEbi= aEbi ajiEbj+ bacEci. (3.29)If one demands the convenient property that the derivative annihilate thetriads DaEbi= 0, thenthespinconnectiongets completelydeterminedbytheconnectionweintroducedwiththecovariantderivatives. Italsoannihilates the spatial metric. The explicit formula is not very importantatthispoint.Onecandeneacurvaturefor thederivativeweintroduced, prettymuchasinthecaseofthecovariantderivative,(DaDbDbDa) Gi= abijGj(3.30)wherewecallabijthecurvatureoftheconnection,anditcanberelatedtotheRiemanntensorbut, again, theexplicitformulawill notbeusedsoforreasonsofbrevityweomitit.Thereisonelastpointof curvedspace-timesthatweneedtotouchupon. This has to do with howto performintegrations on curved44 Some elements of general relativitymanifolds. Suppose one wishes to compute a volume integral in at spaceinCartesiancoordinates.TheintegralisgivenbyV=_ _ _dxdydz. (3.31)So the volume is just given by the integral of the coordinate dierentials int