GeneralRelativityandCosmology 1LecturesonGeneralRelativityandCosmologyPedroG.FerreiraAstrophysicsUniversity ofOxford,DWBuilding, KebleRoadOxford OX13RH,UKlastmodiedFebruary27,2012PreambleThefollowingsetoflectureswasdesignedfortheOxfordUndergraduatecourseandaregivenduringthe3rdyearoftheBAorMPhyscourse. Ihavetriedtokeepthemathematicaljargontoaminimumandgroundmostoftheexplanationswithphysical examplesandapplications.Yetyouwill seethat, althoughthereisverylittleemphasisondierential geometryyoustillhavetolearnwhattensorsorcovariantderivativesare. Sobeit.Thesenotesarenotveryoriginal andarebasedonanumber of booksandlectures. InparticularIhaveusedGravitationandCosmology, StevenWeinberg(Wiley,1972)Gravity: AnIntroductiontoEinsteins General Relativity, James B. Hartle (AddisonWesley,2003)PartIIGeneral Relativity,GWGibbons(canbefoundonlineonhttp://www.damtp.cam.ac.uk/research/gr/members/gibbons/partiipublic-2006.pdf)General Relativity: Anintroductionfor Physicists, M.PHobson, G.P.EfstathiouandA.N.Lasenby(CUP,2006)GeneralRelativity,AlanHeavens(notpubliclyavailable).Theories of GravityandCosmologyT. Clifton, P.G. FerreiraandC. Skordis (PhysicsReports,2012)CosmologicalPhysicsJohnPeacock(CUP,1998)Buttherearemanytextsouttherethatyoucanconsult.For astart, I will assume that the time coordinate, t andthe spatial coordinate, x=(x1, x2, x3)canbeorganizedintoa4-vector (x0, x1, x2, x3) =(ct, x)wherecisthespeedoflight. Throughout theselecturenotes I will usethe(, +, +, +)conventionforthemetric.ThismeansthattheMinkowskimetricisamatrixoftheform=_____1 0 0 00 1 0 00 0 1 00 0 0 1_____(1)GeneralRelativityandCosmology 2You willnote that thisis theoppositeconventiontothe oneusedwhenyouwererst learningspecial relativity. Infactyouwill ndthat, ingeneral (but notalways), booksonGeneralRelativity will use the convention we use here, while books on particle physics or quantum eldtheorywillusetheoppositeconvention.Wewill beusingtheconventionthatRomanlabels(likei, j, etc)span1to3andlabelspatialvectorswhileGreeklabels(suchas,,etc)span0to3andlabelspace-timevectors.Wewill alsobeusingtheEinsteinsummationconvention. Thismeansthat, wheneverwehaveapairofindiceswhicharethesame, wemustaddoverthem. Soforexample, whenwewriteds2= dxdxwemeands2=3

=03

=0dxdxNote that thepairedindicesalwaysappear withoneas asuperscript(i.e. up) andthe otheroneasasubscript(i.e. down).Finally, I amextremelygrateful toTessaBaker, TimClifton, RosannaHardwick, AlanHeavens,Anthony Lewis,Ed Macaulay and Patrick Timoneyfor their help in putting togetherthelecturenotes,exercisesandspottingerrors.1 WhyGeneral Relativity?Throughoutyourdegree, youhavelearnthowalmostall thelawsof physicsareinvariantifyoutransformbetweeninertial referenceframes. WithSpecial RelativityyoucannowwriteNewtons 2ndlaw, the conservation of energy and momentum, Maxwells equations, etc in a waythattheyareunchangedundertheLorentztransformation. Oneforcestandsapart: gravity.Newtonslawofgravity, theinversesquarelaw, ismanifestlynotinvariantundertheLorentztransformation.Youcannowaskyourselfthequestion: canwewritedownthelawsofphysicssotheyareinvariant under any transformation?Not only betweenreferenceframes with constant velocitybutalsobetweenacceleratingreferenceframes. Itturnsoutthatwecanandindoingso, weincorporategravityintothemix. ThatiswhattheGeneralTheoryofRelativityisabout.2 NewtonianGravityIntheselectureswewill bestudyingthemodernviewofgravity. Einsteinstheoryofspace-time is one of the crowning achievementsof modern physicsand transformed the way we thinkabout the fundamental laws of nature. It superseded a spectacularly successful theory, Newtonstheory of gravity. If we are to understand the importance and consequences of Einsteins theory,weneedtolearn(orrevise)themaincharacteristicsofNewtonstheory.GeneralRelativityandCosmology 3NewtonsLawofUniversal GravitationfortwobodiesAandBwithmassesmAandmBataseparationrcanbestatedinasimpliedformas:F= GmAmBr2whereG = 6.672 1011m3kg1s2. TheresultinggravitationalaccelerationfeltbymassmAisg= GmBr2Wecanrewritetheexpressionintermsofvectorsr = (x1, x2, x3).F = GmAmBrBrAr3istheforceexertedonthemassmAandthegravitationalaccelerationF = mAgConsidernowafewsimpleapplications. WecanusetheaboveexpressiontoworkoutthegravitationalaccelerationonthesurfaceoftheEarth. IfweTaylorexpandaroundtheradiusoftheEarth,Rwendg= GMr2= GM(R + h)2 GMR2(1 2hR)wherehistheheightabovethesurfaceoftheEarthandMisthemassoftheEarth. WithM= 5.974 1027gandR= 6.378 108cmwendafamiliarvalue: g 9.8ms2.ThepowerofNewtonstheoryisthatitallowsustodescribe, withtremendousaccuracy,theevolutionoftheSolar System. Todoso,weneedtosolvethetwobodyproblem. WehavethattheequationsofmotionaremirA= GmAmBrBrAr3withr= |r|= |rA rB|. Weareinterestedinhowrevolvesandwecannditsevolutionbysimplifyingthissystem. LetusrstdenethetotalmassM= mA + mBandthereducedmass =mAmBMIgnoringthemotion ofthe centreof mass,wehavethat the Lagrangian for thissystemwillbeL =12 r r + GMrGeneralRelativityandCosmology 4Itismoreconvenienttotransformtosphericalcoordinatessox1= r cos() sin()x2= r sin() sin()x3= r cos()thentheLagrangianbecomesL =12( r2+ r2 2+ r2sin2() 2) + GMrTheangularpartsoftheEuler-Lagrangeequationsare ddt(r2 ) = r2sin() cos() 2ddt_r2sin2() _= 0Wecanintegratethesecondequationtogiver2sin2() = Jand,multiplyingtherstequationby2r2 ,wecanintegratetond2r4 2= J2 J2sin2()We can now choose a coordinate system such that the orbit lies on the equatorial plane ( = /2)andJ= J = Jwhichcorrespondsto = 0. Weareleftwithtwocoordinates,rand.Replacingthetheangularmomentumconservationequationintheradialequationofmo-tion,wehave r J2r3+ GMr2= 0whichwecanintegratetogiveusanexpressionfortheconservedenergyE=12 r2+12J22r2 GMr(2)NotethatwecandeneaneectivepotentialenergywhichisgivenbyVeff(r) =12J2r2 GMrAgain,usingconservationofenergy,wecanuseJdt = r2dtoreexpresstimederivativesintermsofangularderivativesddt=Jr2ddandd2dt2=Jr2dd(Jr2dd)GeneralRelativityandCosmology 5TheEulerLagrangeequationthenbecomesJr2dd_Jr2drd_J2r3= GMr2Ifwechangetou = 1/rwehaved2ud2+ u =J2GM (3)This is the simple harmonic oscillator equation with a constant driving force. We can solve thisequationtondu =1r=G(M)J2__1 +_1 +2EJ2(GM)2 cos( 0)__whereEistheconservedenergyof thesystem(andanintegrationconstant) and0istheturningpointoftheorbit(andthesecondintegrationconstant). Youshouldrecognizethisastheequationforaconicwithellipticityegivenbye =_1 +2EJ2(GM)2Inotherwords, thetrajectoriesof thetwobodyproblemcorrespondtoclosedorbitsintheformofellipses.The Solar System is almost perfectly described in terms of these trajectories. In fact almostalltheplanetshavequasi-circularorbitswitheccentricitiese 1 10%. Mercurystandsout,withe 20%. Furthermore,Mercurysorbitdoesnotactuallycloseinonitselfbutprecessesatarateofabout5600arcsecondspercentury. Thisprimarilyduetotheeectoftheotherplanets slowly nudging it around and, again, can be explained with Newtonian mechanics. Butsincethemid19thcentury, ithasbeenknownthatthereisstill anunaccountedamountofprecession,about43arcsecondspercentury. Weshallnditsorigininlaterlectures.Finally, it is convenient to dene the Newtonianpotential or gravitational potential in termsofthepotentialenergy,V ,ofthesystem:V mThegravitationalforcewillthenbeF = Vandthegravitationalaccelerationisgivenbyg = ThegravitationalpotentialsatisestheNewtonPoissonequation2 = 4GGeneralRelativityandCosmology 63 TheEquivalencePrincipleInthelastsection,weworkedwithtwoequations: NewtonsLawofUniversalAttractionandNewtons 2ndLaw. In both of theseequations a massappears and weassumedthat theyareoneandthesame. Butletusnowrewritethemandbeexplicitaboutthedierenttypesofmasscomingin,theinertialmassmIandthegravitationalmass,mG:mIa = FV = GmGMGrWeassumedthatmI= mGbut are they? Their equivalence has beentestedtosurprisingprecisionwithLunar LaserRangingobservations. Thisinvolvesbouncingalaserpulseoreectingmirrorssittingonthesurface of the moon and measuringits orbit as it sits in the joint gravitational eld of both theEarth and the Sun. The distance between the Earth and the Moon is 384,401 km and has beenmeasuredwithaprecisionofunderacentimetreThe Lunar Ranging experiment can be seen as a type of E otvos Experiment where the EarthandMoonarethetestmassessittinginthegravitationaleldoftheSun. AlaboratorybasedE otvos Experiment canbeconstructedinthefollowingway. Consider twomasses madeofdierentmaterialattachedtoeitherendofarod. Therodissuspendedfromastringonthesurfaceof theEarth. Eachof themasses will besubjectedtotwoforces: thegravitationalpull tothecentreandthecentrifugal force. Hencetherodwill hangatananglerelativetothevertical direction. Therodis freetorotateif thereis adierence inthegravitationalacceleration between the masses. This can only happen if there is a dierence between mGandmI.We can look at the numbers here. Consider the two masses, which have gravitational massesmG1andmG2andinertial massesmI1andmI2. Denotethecomponentof thegravitationalforceinthedirectionthatmakes therodtwist tobegtandtheaccelerationof eachof themassestobeat1andat2. WethenhavethatmI1at1= mG1gtmI2at2= mG2gtIf the ratios of inertial to gravitational mass are the same for both bodies, then the accelerationwill bethesameforboth. Anydierenceinthegravitationalandinertialmasseswill leadtoatwistinthependulumwhichcanbecharacterizedintermsofadimensionlessparamater:=at1at2at1 + at2=_mG1mI1mG2mI2__mG1mI1+mG2mI2_Usingberylliumandtitanium,thecurrentbestconstraintonis=at1at2at1 + at2= (0.3 1.8) 1013GeneralRelativityandCosmology 7Twisting direction Fiber Earths rotation

! T

mI ! a

mG! g Figure1: ThependulumfortheEotvosExperimentwhichisafactorof4betterthantheoriginalEotvosexperimentof1922. Todate,oneofthemost successfultests is to usethe Earth-Moon systemin the gravitational eldof the Sun as agiantEotvosexperiment. Thedierence,withregardstothelabbasedEotvosexperiments,isthat themassesofthetestbodies(i.e. theEarth andMoon) are notnegligibleanymore. Thetest can be done by using lasers reected o mirrors left on the Moon by the Apollo 11 missionin1969and,asmentionedearlier,theconstraintis = (1.0 1.4) 1013Thisisone ofthe mostaccuratelytestedprinciplesofphysicsandweexpectthisconstrainttoimproveby5ordersofmagnitudewhenspacebasedtestscanbeperformed.If thegravitational andinertial masses arethesame, thenwehavethat aparticleinagravitationaleldwillobeyr =mGmIg = gandthiswill betrueof anyparticle. Thishasaprofoundconsequence-it meansthatI canalwaysndatimedependentcoordinatetransformation(fromrtoR),r =R+b(t)suchthatR =g b(t) = 0In other words, it is always possible to pick an accelerated reference frame such that the observerdoesnt feel the gravitational eld at all. For example, consider a particle at rest near the surfaceof theEarth. Itwill feel agravitational pull of g= 9.8ms2. Nowplaceitinareferenceframesuchthatb=12gt2(suchasafreelyfallingelevator). Thentheparticleatrestinthisreferenceframewontfeelthegravitationalpull.GeneralRelativityandCosmology 8 A B t = 0 first pulseemitted by A t = t1 first pulsereceived by B A B A B t = !"Asecond pulseemitted by A t = t1+!"B second pulsereceived by B A B z t Figure2: ReferenceframesforthegravitationalredshiftEinstein had an epiphany when he realized this. As he said ... for an observer falling freelyfromtheroof ofahousethereexists-atleastinhisimmediatesurroundings-nogravitationaleld. ItledhimtoformulatetheEquivalencePrinciple ofwhichtherearethreeversionsthatwewillstatehere.WeakEquivalencePrinciple(WEP): All uncharged, freely falling test particles follow thesametrajectories,onceaninitialpositionandvelocityhavebeenprescribed.StrongEquivalencePrinciple(SEP): TheWEPis valid, andfurthermore inall freelyfallingframes one recovers(locally, and upto tidal gravitational forces)the samelawsofspecialrelativisticphysics,independentofpositionorvelocity.EinsteinEquivalencePrinciple(EEP): TheWEPisvalidfor massivegravitatingobjectsaswell astestparticles, andinall freelyfallingframesonerecovers(locally, anduptotidal gravitational forces) the same special relativistic physics,independent of position orvelocity.You will note that these three equivalent principles have dierent remits. While the rst makes asimplestatementabout thetrajectoriesoffreelyfallingbodies,thesecondonesayssomethingaboutthelaws of physics obeyedbythefreelyfallingbodies andthethirdaddresses goingbeyondthesimplied, point-liketestmassapproximation. Wewill useWEP, SEPandEEPinterchangeablythroughouttheselecturesalthoughtheycanbetestedindistinctways.Letusbrieystudyoneof theconsequences of theEquivalencePrinciple. Consider twoobservers,oneat positionAwhichisat aheighthabovethesurfaceoftheEarthand anotherat a position Bon the surface of the Earth. ObserverA emits a pulse every tAto be receivedby observerBeverytB. What is the relation betweentAand tBand how is it aected bythegravitationaleld?GeneralRelativityandCosmology 9Given what we have seen above, we can think of this as two observers moving upwards withanaccelerationg. WehavethatthepositionsofAandBaregivenbyzA(t) =12gt2+ hzB(t) =12gt2Nowassume the rst pulse is emittedat t =0byAandis receivedat time t1byB. AsubsequentpulseisthenemittedattimetAbyAandthenreceivedattimet1 + tB. WehavethatzA(0) zB(t1) = h 12gt21= ct1zA(tA) zB(t1 + tB) = h +12gt2A12g(t1 + tB)2 h 12gt21gt1tB= c(t1 + tBtA)where we have discarded higher order terms in t. Combining the two equations, and assumingt1 h/cwehavethattA (1 +ghc2 )tBInotherwords, thereisgravitational timedilationduetothedierenceinthegravitationalpotentialsatthetwopointsBA= ghsoRateReceived =_1 BAc2_RateEmittedThisisknownasthegravitational redshiftoflight.4 GeodesicsIn the previous section, we saw how important accelerated reference frames could be. From thevariousequivalenceprinciples, itseemsthatanacceleratingreferenceframewill beindistin-guishablefromaframeinagravitationaleld. Letusthenstudyhowparticlesandobserverstravelthroughspace.Inatspace,aparticlefollowsstraightlinesgivenbysolutionstothekinematicequationsofmotiond2xd2= 0Theparticlewilltraceoutapathinspace-timex()andthesolutionisoftheformx() = x(i) + u( i)wherex(i)anduareintegrationconstants.GeneralRelativityandCosmology 10In the presence of a gravitational eld, the path taken by a particle will be curved. Alterna-tively,in an accelerating reference frame, the same will happen. We call Geodesics the shortestpaths between two points in space-time. We have just written down the geodesic in a at, forcefreespace. If wewishtodosointhepresenceof agravitational eld, weneedtosolvetheGeodesicequation. AsEinsteinargued,accordingtotheprincipleofequivalence, thereshouldbeafreelyfallingcoordinatesystemyinwhichparticlesmoveinastraightlineandthereforesatisfyd2yd2= 0whereisthepropertimeoftheparticleandhencec2d2= dydyNowchooseadierentcoordinatesystem,x;itcan beat rest,accelerating,rotating, etc. Wecanreexpresstheysintermsofthex,y(x). Usingthechainrulewehave0 =dd_yxdxd_=yxd2xd2+2yxxdxddxdMultiplyingthroughbytheinverseJacobianx/yweendupwiththegeodesicequationwiththeformd2xd2+ dxddxd= 0 (4)wherewehavedenedtheaneconnection=xy2yxxWecanalsoexpressthepropertimeinthesenew(orarbitrarycoordinates)asc2d2= yxdxyxdx gdxdxwherewehavedenedthemetric:g= yxyxThesituationisslightlydierentforamasslessparticle. Neutrinosorphotonsfollownullpathssod= 0. Insteadofusingwecanusesomeotherparameter. Wethenhaved2yd2= 0dyddyd= 0GeneralRelativityandCosmology 11neOnecanrepeatthesamederivationasabovetondd2xd2+ dxddxd= 0gdyddyd= 0So,givenanandg,wecanworkouttheequationsinagivenreferenceframe.It turnsout that wecan simplifythe calculationevenfurther: can be found from g.Takingthepartialderivativeofthemetric,wehavethatgx=2yxxyx +2yxxyxFromthedenitiionofwecanreplaceitintheaboveexpressiongx= yxyx + yxyxwhichcanbewrittenasgx= g + gWecannowpermuteindicesandaddthemtosolveandnd=12g_gx+gx gx_Hence,as advertised,givenametric,g,wecan ndthe connectioncoecentsand thensolvethegeodesicequation.It often useful to nd the geodesic equations in terms of a variational principle. In fact, it isalsoaconvenientmethodfor,givenametric,calculatingtheconnectioncoecients. Considera path in spacetime x(). We can dene the proper time elapsedbetweentwopoints on thatcurve,AandB,tobecAB=_BALd =_BAd_ x xThederivativesare takenwithregardsto. It is nowpossibletodenean action for thepathx():S= mc2andwecanminimizethisactiontondthepathwhichtakesthemostamountofpropertimebetween points A and B. This path will be the geodesic. For example, if we choose x0= c = ctwehavethatS= mc_dt_c2_dxdt_2GeneralRelativityandCosmology 12Moregenerally,wecanusetheEulerLagrangeequations:dd_ L x_ =LxGiven that L is independent of x(and d= Ld) we have that the equation of motion becomesL x= 121Ldxd= 12dxdTheresultingequationsthenbecomed2xd2= 0Althoughtheaboveactionisreparametrizationinvariant(i.e. wecanchangeourdenitionofandtheactionisunaected), thesquarerootisdiculttoworkwith. ItiseasiertoworkwithS= m_ddxddxd= m_dL2TheEuler-LagrangeequationthenbecomesL2x dd_L2 x_ = 2dLdL xAgain, ifwechoosetheaneparametertobelinearin wehavethattherighthandsidebecomesdLd=dd_cdd_ = 0So, for a massiveparticle it makes senseto choose = . Finally we,can rederivethe geodesicequationfromtheactionprincipleasabove, butinthepresenceofagravitational eld. WenowhaveS= m_dgdxddxd= m_dL2and can apply the Euler-Lagrange equation to obtain the Geodesic equation in a general frame.5 CoordinateTransformationsandMetricsWe have seenthat coordinatetransformations toacceleratedreference frames leadtonon-trivial geodesic equations. We have consideredone suchtransformationwhenderivingthegravitationalredshift. Letusnowlookatcoordinatetransformationsinmoredetail.Youhaveextensivelystudiedcoordinatetransformationsbetweeninertialreferenceframes(in the absence of gravitational elds) in Special Relativity. For example, consider a coordinateGeneralRelativityandCosmology 13transformationalongthez-axisfromastationaryframetoonemovingatavelocityv. Wehavethatthecoordinatetransformationcanbeexpressedinamatrixformas_____ x0 x1 x2 x3_____=_____ 0 0 0 1 0 00 0 1 0 0 0 __________x0x1x2x3_____where= v/cand= 1/1 2. Thiscoordinatetransformationisoftheform x= x(x)andgiventhatitislinear,wehavethattheJacobianissimply: xx=_____ 0 0 0 1 0 00 0 1 0 0 0 _____Now,inspecialrelativitywehavethatthespacetimeintervalds2= c2dt2+ (dx1)2+ (dx2)2+ (dx3)2isinvariantundercoordinatetransformations. Wecanrestatethisasds2= dxdx= d xd xinotherwords,thetransformationleavestheactualformofthespacetimeintervalinvariant.What happens if we nowconsider acoordinatetransformationtoanacceleratingrefer-enceframe? Letusconsiderthesimplestcase, anacceleratingreferenceframe, withaccelera-tiong, alongthex3direction. Wehavethatthetransformationbetweentheoldcoordinates,(x0, x1, x2, x3)andthenewcoordinates,( x0, x1, x2, x3),isgivenbyx0=c2g eg x3c2sinh( gc2 x0)x1= x1x2= x2x3=c2g eg x3c2cosh( gc2 x0)Wecantransformtheexpressionforthespace-timeintervaltothenewcoordinatesystem:ds2= dxdx= (dx0)2+ (dx1)2+ (dx2)2+ (dx3)2= e2g x3c2(d x0)2+ (d x1)2+ (d x2)2+ e2g x3c2(d x3)2Note that the equivalent gravitational potential has the form = g x3and so the interval takestheformds2= e2c2(d x0)2+ (d x1)2+ (d x2)2+ e2c2(d x3)2(5)GeneralRelativityandCosmology 14Infact, thisexpressionisvalidinamoregeneral settingthanjust aconstant gravitationalacceleration. Aweak,staticgravitationaleld,isequivalenttoametricofthisform.Fortheremainder of thissectionlet usfamiliarizeourselves abit morewithcoordinatetransformationsandhowtheyaectthemetric. Withageneral coordinatetransformation, x= x(x)wendthatthespacetimeintervalchangesasds2= dxdx= x xx xd xd x gd xd xInotherwords, underageneral coordinatetransformation, wehavethat g. Wecalltheobjectgthemetric.The metric contains informationabout the geometryof space (andspace-time) we areconsidering. Itisinstructivetoworkthroughafewexamples. Forexample, considera2-Dsheetplane,withcoordinates(x, y). Theintervalonthatplaneisgivenbyds2= dx2+ dy2andhencethemetricisverysimple: itisadiagonalmatrixwithentriesgij=_1 00 1_Wecantransformtopolarcoordinatesx = r cos y= r sin tondds2= dr2+ r2d2Notethatnowthemetricismorecomplicated:gij=_1 00 r2_yetitstill describesaplane. Wecouldhaveconsideredadierentsurface, aspherewithunitradius. It is a two dimensional surface and hence needs two coordinates, (, ). The innitesimalintervalisdenedasds2= d2+ sin2()d2withmetricgij=_1 00 sin2()_Thegeometryofthesurfaceofasphereisobviouslyverydierenttothegeometryofaplaneanditisinthemetricthatthisinformationisencoded.GeneralRelativityandCosmology 15Weareofcourse, interestedinthegeometryof3-Dspaceandof4-Dspacetime. So, forexample, theinterval (andmetric)forEuclidean(orat)3-DspaceinCartesiancoordinatesareds2= (dx1)2+ (dx2)2+ (dx3)2andgij=___1 0 00 1 00 0 1___Insphericalcoordinatesx1= r sin() cos()x2= r sin() sin()x3= r cos()wehavethattheinterval(andmetric)areds2= dr2+ r2d2+ r2sin2()d2andgij=___1 0 00 r200 0 r2sin2()___Again, thesetwometrics(inCartesianandspherical coordinates)describeexactlythesamespace.Wehavealreadyseenexamplesof space-timemetricsabove. TheMinkowski metricandthemetricofanacceleratedobserver(knownasaRindlermetric).Let us now consider two important metric. The rst one is that of a Euclidean, homogeneousandisotropicspacetime. Wehavethatds2= c2dt2+ a2(t)_(dx1)2+ (dx2)2+ (dx3)2_(6)whichhasametricg=_____1 0 0 00 a2(t) 0 00 0 a2(t) 00 0 0 a2(t)_____Aparticularlyimportantmetricisastatic(i.e. timeindependent), sphericallysymmetricSchwarzschildmetric:ds2= _1 2GMc2r_c2dt2+_1 2GMc2r_1dr2+ r2d2+ r2sin2()d2(7)GeneralRelativityandCosmology 16whichhasametricg=________1 2GMc2r_0 0 00_1 2GMc2r_10 00 0 r200 0 0 r2sin2()_______Thismetricisofparticularimportance. Itcorrespondstothespacetimeofapointlikemassandcanbeusedtodescribethespacetimearoundstars,planetsandblackholes.Tonish, letuscalculatethegeodesicsforthehomogeneousmetrictondoutwhattheconnectioncoecientsare. Theactionis:L2= c2t2+ a2(t)

( xi)2wheref=dfd. TheEuler-Lagrangeequationsare: x0+acdadt

( xi)2= 0 xi+ 21acdadt x0 xi= 0Wecannowreadotheconnectioncoecients (andbecareful not toover count withthefactorof2inthesecondexpression):0ij=1cadadtiji0j=1acdadtij6 TheNewtonianLimitandtheGravitational RedshiftRevisitedInterestinglyenough,withwhat wehavedone wecanalready start relatingthe geodesicequa-tion with the Newtonian regime of gravity. Let us look at the case where the gravitational eldisextremelyweakandstationaryandparticlesaremovingatnon-relativisticspeedssov c.Letusstartwiththegeodesicequation:d2xd2+ dxddxd= 0Thenon-relativisticapproximationmeansthatdxi/dd(ct)/d sothegeodesicequationsimpliestod2xd2+ 00_dx0d_2 0Ifthegravitationaleldisstationarywehavethatg/t = 0andwehave00= 12gg00xGeneralRelativityandCosmology 17Nowconsideraweakeldg= + hwhereweassumethat |h| 1. Wecanexpand00torstorderinhtond00= 12h00xOnlythespatialpartsofsurvive(whichare1). Hencewehave00= 12h00xThegeodesicequationthenbecomesd2xd2=12_dx0d_2h00xwhichinvectorialnotationbecomesd2rd2=12_dx0d_2h00Forsmallspeedsdt/d 1andcomparingwiththeNewtonianresultd2rdt2= weseethath00= 2c2Henceintheweak-eldlimitg00= _1 +2c2_Notethat, ifwetakethemetricthatwefoundinequation5andexpandtolinearorder(i.e.taketheweakeldlimit),wegetexactlythesameexpression.Wecan rederiveour expressionfor thegravitational redshiftdirectly from themetric. Con-siderthepropertimeagainds2= c2d2= gdxdxandpickastationarysystemsodxi= 0. Wethenhaved=g00dt.Intheweakeldcasewehaved _1 +c2_dtGeneralRelativityandCosmology 18Notethattand onlycoincideif=0soclocksrunslowlyinpotential wells. Wecannowcomparetherateofchangeattwopoints,AandB,togetdAdB=_g00(A)g00(B)whichintheweakeldlimitgivestheratiooffrequenciesBA=dAdB 1 BAc2whichisequivalenttotheexpressionwefoundinSection3. Wecandenethegravitationalredshiftzgrav ABA=BAc2Thisisaverysmalleect-fortheSunitis 106.Onewaytolookforthiseectisbystudyingthespectrallinesemittedfromatomswhichareveryclosetoamassivebodyandhencedeepintoagravitational potential. ThiseecthasbeenobservedintheSunandwhitedwarfsbuttheseobservationsarenotveryaccurate.Aclassictest of thegravitational redshift was undertakenbyPoundandRebkain1960atHarvard. Theyuseda22.5metrehightower wheretheyplacedanunstablenucleus, Fe57atthetopandthebottom. Thenucleus(atthetop)wouldemitgamma-rayswithacertainfrequencyrelatedtotheirenergy. Theserayswouldfall tothebottomandinteractwiththeFe57there. Ifthegammaraysoftheobserverwerethesameastheemitter, theFe57atthebottomoftheshaftwouldreact. Butbecauseofthegravitationalredshift,thefrequencywasshiftedandtheabsorptionwaslessecient. Bychangingthevelocityofthesourceatthetopof the tower, the experimenterscould compensate for the gravitational eect and measure it towithin1%.Better measurements of the gravitational redshifting of light can be obtained on (or near) theEarth where,eventhough the gravitational eldismuch,muchweaker,thereis thepossibilitytomakeveryprecisemeasurements. Onewaytodothisistosendarocketupintoorbitwitha hydrogen-maser clock and emitting pulses to a ground station. At an altitude of 104Km, thechangein gravitational potentialwill be gh/c2 1010. Note that thiseect isminute,almost5ordersofmagnitudesmallerthanthesimpleDopplereectduetothemotionoftherocket.Yetitisstillpossibletoconstraintheeecttowithin0.002%.7 OrbitsI: thePerihelionofMercuryIt is nowtime torevisit thetwobodyproblem. Wehave alreadyworkedthis out for theNewtonian case but we can now see what happens if we consider the more general case. Strictlyspeaking we will be studying the motion of mass in a central potential sourced by a mass M.We canuse the Schwarzschildmetric that we introducedinequation7of the previoussection. ThetotalmassisMandthereducedmassis. Theactionforthegeodesicequationfor(t(), r(), (), ()) inthismetricisL2=_1 2GMrc2_c2t2 r21 2GMrc2r2(2+ sin2 2)GeneralRelativityandCosmology 19TheangularEuler-LagrangeequationsareexactlyasintheNewtoniantwobodyproblemwepreviouslysolveddd(2r2 ) = 2r2sin cos 2dd(2r2sin2 ) = 0andwecansolvetheminthesameway, placingtheorbitontheequatorial plane, choosingintegrationconstantssuchthat = 0sothatr2 =JThetimelikecomponentofthegeodesicobeysdd_c2_1 2GMrc2_ t_= 0whichcanbeintegratedtogive_1 2GMrc2_ t = kForamassiveparticlewehaveL2= c2c2=c2k2 r2_1 2GMc2r_ J22r2whichcanberewritten r2+_1 2GMc2r_J22r2= c2k2c2+2GMrRearrangingwend r2+J22r2 2GMr2GMJ22r3c2= constant (8)WecancomparethisexpressionwiththeonewefoundintheNewtoniancaseinequation2: r2+h2r2 2GMr= constantThereisanextratermintheGeneral Relativisticcase. Furthermore, intheNewtoniancasewearetakingderivativeswithregardstotwhileintheGeneralRelativisticcaseweareusingtheaneparameter.Wewouldnowliketondtheorbitsofmotioninthissystem. AsintheNewtoniancase,usingconservation of angular momentumwe can change the independentvariable from to .Furthermore,wecantransformtou = 1/rsothat r =drddd= 1u2dudJu2= JdudGeneralRelativityandCosmology 20DividingthroughbyJ2/2,equation8becomes_dud_2+ u22GM2J2u 2GMc2u3= constantDierentiatingbyanddividingby2dudwendd2ud2+ u =Gm2J2+3GMc2u2Youwill seethatthisequationhasaverysimilarformtoequation3withanextrabit. Itisuseful to rescale u so as to assess how important the correction is. Dene U=J2GM2u. We thenhaved2Ud2+ U= 1 + U2(9)with 3G2M22/J2c2. Wewill seethatinthecaseofMercury, isoforder107, soverysmall. We therefore assume that Ucan be split into a Newtonian part, U0and a small, generalrelativisticcorrection, U1. WehavethatU0=1 + e cos()whichwecanplugintoequation9tond(tolowestorder):d2U1d2+ U1= U20= [1 + 2e cos() + e2cos2()] = [1 +e22+ 2e cos() +e22cos(2)]ThecomplementaryfunctionisasbeforebuttheparticularintegraltakestheformU1= __1 +e22_+ e sin() e26cos(2)_Withtime, thedominanttermwill beproportional tosothat, addingthecomplementaryfunctionandtheparticularintegralwehaveU 1 + e cos() + e sin()ThiscorrespondstotheTaylorexpansionofU 1 + e cos[(1 )]I.e. theperiodoftheorbitisnow2/(1 )andnot2. Theorbitdoesnotcloseinonitself.WecanworkoutwhatthiscorrectionisforMercury. TakingM 2 1030kg, theorbitalperiodT=88daysandthemeanorbital radiusr=5.8 1010m, wend 107sothatprecessionrateisapproximately43arcsecondspercentury. Thiseect, rstdetectedbyLeVerrierin the mid19thcenturyisobscuredbyanumberofother eects. Theprecessionof theequinoxesof thecoordinatesystemscontributestoabout5025percenturywhiletheotherplanets contribute about 531per century. The Sunalsohas aquadropolemoment whichcontributes afurther 0.025percentury. Takingall theseeects intoaccount still leaves aprecessionof forwhichthecurrentbestestimateis = 42.969 0.0052percenturyThepredictionfromGeneralRelativityis 42.98percenturyGeneralRelativityandCosmology 218 OrbitsII:GravitationalLensing andtheShapiro TimeDelayWe now want to study what happens to a light ray propagating in a gravitational eld. We rstworkout what happensin aNewtonian universe;weare going to modela photon as a massiveparticle travelling at the speed of light, c with angular momentum per unit mass h = cR. Recallthatu = 1/rsatisesd2ud2+ u =GMh2=GMc2R2Wehavesolveditbeforeforclosedorbitsbutwenowwanttopickintegrationconstantsthatleadtounboundedorbits:u =sin R+GMc2R2whereRisthedistanceofclosestapproachintheabsenceofgravity. Considertheasymptoticbehaviour: whenr wehaveu 0whichgivesus twosolutionsfor : = GM/(c2R)and+ = + GM/(c2R). ThetotaldeectionisN=2GMc2RWecannowrepeatthecalculationforintherelativisticcase. AsinSection7, wehavethatthe geodesic equation for a light ray must be parametrized in terms of an ane parameter andnotpropertime. Wecanusesomeoftheresultsfoundin7(and,onceagain,taking = /2)r2 = h_1 2GMrc2_ t = kForamasslessparticlewehaveL2= 00 =_1 2GMc2r_c2t2 r2_1 2GMc2r_ h2r2whichcanberewrittenintermsofuash2_dud_2= c2k2h2u2+2GMc2h2u3which,whendierentiatedgivesd2ud2+ u =3GMc2u2Again,wecantreattherighthandsideasasmallperturbationtotheorbitu0 =sin R(10)GeneralRelativityandCosmology 22Thisequationcorrespondstoastraightline. Therstorderequationisd2u1d2+ u1=3GMc2R2sin2 =3GM2c2R2(1 cos 2)whichcombinedwithu0leadstothecomplete,rstorder,solutionu =sin R+3GM2c2R2_1 +13 cos 2_At large distances, u0andassuming sin we have twopossible solutions for : = 2GMc2Rand+ = +2GMc2R. ThetotaldeectionisthenGR=4GMc2RWendthatGR= 2N.For alight raygrazingthelimbof theSun, thedeectionwill be 1.75, famouslymeasuredbyArthurEddingtonduringhisEclipseexpeditionin1919. Thetightestobserva-tional constraint come from observations due to Shapiro, David, Lebach and Gregory who usedaround2500daysworthofobservationtakenover20years-theyused87VLBIsitesand541radiosourcesyieldingmorethan1.7 106observationsandandobtainedaconstrainton: = (0.99992 0.00023) 1.75whichis3ordersofmagnitudebetterthanEddingtonsoriginalobservations.AnotherrelativisticeectinvolvinglightraysistheShapirotimedelay, rstproposedin1964. Again, take the Schwarzschild metric in the equatorial plane and apply to electromagneticwave(radar)propagatingatthespeedoflight. Wehavethatds2= 0so0 =_1 2GMc2r_c2dt2dr2_1 2GMc2r_ r2d2Taketheunperturbedsolution,giveninequation10. Wehavedrr2=cos()Rdwhichcanbeusedtondr2d2= dr2tan2 =R2dr2r2R2Themetriccanthenberewrittenasc2dt2= dr2__1 2GMc2r_2+_1 2GMc2r_1R2r2R2_WecannowexpandtorstorderinGM/(rc2)tondcdt = rdrr2R2_1 +2GMrc2GMR2r3c2_GeneralRelativityandCosmology 23ThisexpressioncanbeeasilyintegratebetweenpointsAandBtogivect = __r2R2+2GMc2ln__r2R2 1 +rR__GMc21 R2r2__BA(11)Wecannowapplythistoaplanetarysystem. LetustakeAtobetheEarthandBtobeVenus. The expressionin equation 11 for t is the coordinate elapsed,not the time elapsedontheEarth,. Torelatethesetwotimeintervalsrecallthatds2= c2d2whered isthepropertimeelapsedatontheEarth. Wecanassumeacircularorbitsothatdr = d = 0butclearlywehaved = 0. Wearethenleftwithc2d2=_1 2GMc2rE_c2dt2r2d2whererEistheEarth-Sundistance. Wecansimplifytod=_1 2GMc2rEr2c2_ddt_2dtOnacircularorbitwecanuseKeplerslaw_ddt_2=GMr3sowend=1 2GMc2rEGMc2rEt =1 3GMc2rEt _1 3GM2c2rE_tTo understand what one should expect, take r R and look at the expression to see whichtermdominatesasR 0. Thelogarithmictermwilldivergesothat,whenalightraypassesclosetothesource, thereisalargetimedelay. So, bymonitoringaregularpulseof lightasitpassesbehindamassivebody,oneshouldseealargeincludeintheperiod,acharacteristicspikeduringthetransit. Shapiroproposedanexperiment whereonewouldsendlight rays(orradarsignals)fromtheEarthwhichwouldthenbereectedoVenusandback. If theEarthandVenusarealigned,theSuninducesaneectontheorderofmicroseconds. Infact,whenreceivingsignalsfrom distantsatellitessuchVoyager and Pioneer,one hastoincludetheShapirotimedelayeectinprocessingtheirsignals. ThebestconstraintsareduetoBertotti,IessandTortorausingradiolinkswithCassiniin2002whichgiveust = (1.00001 0.00001)tGRwhere tGRisthepredictionfromGeneralRelativity.GeneralRelativityandCosmology 249 TheEquivalencePrincipleandGeneral CovarianceTheEquivalencePrinciplehasledustomoveawayfrompreferredframesorevenpreferredcoordinatesystems. Amoderntheoryofgravitymusttakethatintoaccount,i.e. itshouldbepossibletowritethelawsofphysicsinaformwhichistrueinanycoordinatesystem. ThisisknownasthePrincipleofGeneral Covariance.ToimplementGeneralCovariancewehavetolearnalittlebitmore about geometry. For astart let us recall how we transform between dierent coordinate systems. Consider a coordinatetransformation x= x(x). TheJacobianmatrixofthetransformationisdenedtobe xxGivenanothercoordinatetransformation x= x( x),wecanapplythechainruletoget: xx= x x xxandx x xx= Howdo dierenttypesoffunctions ofxtransform undercoordinate transformations. Thesimplestcaseisascalareld,(x)-itremainsunchangedunderacoordinatetransformation.A simple example of a scalar is d, which we used in Section 4 to construct the invariant actionforthegeodesic.Thenexttypeoffunctionsarevectorselds. Consideracurveinspacetime,parametrizedbysox= x(). ThetangentvectoreldisgivenbyT=dxdSupposewenowchangecoordinatesto x. WenowhavethatthetangentvectorinthesenewcoordinatesisT=d xd. (12)Usingthechainrulewehaved xd= xxdxdSothetangentvectoreldtransformsasT= xxTAvectoreldwiththeindicesup(andwhichthereforetransformsinthisway)isknownasacontravariantvectoreld.GeneralRelativityandCosmology 25There is a dierent type of vector eld, with an index down which is known as a covariantvectoreld. AnexampleisF=fxi.e. thegradientofafunctionf(x). Thechainrulegivesfx= xxf xSodeniningF=f xwehaveF=x xFNotehowthetransformationmatrixistheinverseoftheoneforcontravarianttensors. Thisof course, meansthatif youcontractsomethingwithanupindexwithsomethingwithadownindexyouhaveT F= xxx xTF= TF= TFi.e. theresultingobjectisascalarandunchangedbyacoordinatetransformation.Itshouldbeobviousthatwecangeneralizethistoobjectswitharbitrarynumberofupanddownindices. Theseobjectsareknownastensors. Wehavealreadyhadtodeal withoneofthem,themetric. Themetricisa2ndranktensorandtransformsas g=x xx xgWecanalsodenetheinverseofthemetricwhichissimplythecontravariantversionofthemetric,gandsatises:gg= Wecangeneralizetoanarbitrarytensor. Forexamplea2ndrankcontravarianttensorwillberepresentedasM, a2ndrankcovarianttensorwill beof theformNanda2ndrankmixedtensorwill havetheformO. Wecanhavehigherordertensors(andwewill comeacrossonelateron). Forexamplea4thordermixedranktensorwillbeoftheformR.Wehavenowconstructedthisarrayof objectsandwishtodooperationsonthem. Themost important operation weneedto dois dierentiation. Let us seewhywecant usenormalderivatives. Weve alreadyseenthat F=fxtransforms inthecorrect way. Let us nowconsiderthe2ndderivativeofF:FxGeneralRelativityandCosmology 26Againletusconsiderachangeofcoordinates:Fx=2fxx= xx x_ xxf x_= xx xx2f x x+2 xxxf xIfwewereworkingwithacovarianttensorwewouldnthavetheextraterm. Sothenormal,2ndderivativeofascalar(i.e. theHessian)isnota2ndcovarianttensor.Wecandenethecovariantderivative, whichobeysthefollowingproperties f=fxItobeystheLiebnitzrule(MN) = (M)N+ M(N)foranytwotensorsMandN(wehavehiddentheindices). commuteswithcontractionsbetweenindices.Wecanconstructsuchanoperatorasfollows. Itisanormalderivativewhenappliedtoascalar. AppliedtocontravariantorcovariantvectoreldsitactsasV= V+ VU= UUThe objects that wehave denoted by are exactly the connection coecientsthat we cameacrosswhenconstructingthegeodesicequations. Wecanusethemtoconstructthecovariantderivativesof2ndranktensorstoo. So,forexamplewehaveM= M M MN= N+ N+ NO= O+ OOFinally,thecovariantderivativeconstructedinthiswaysatisesthemetricitycondition:g= 0Now let us revisit our curve on space-time x= x(). We can dene the absolute derivativeofavectorValongthatpathtobeDVD TVwherethetangentvectorTisdenedinequation12. WesaythatthevectorVisparallelytransportedalongthatpathifDVD= 0GeneralRelativityandCosmology 27ThisisadierentialequationforV. Wecanstartatapointx( = 0)andintegratetondthevalueVat,forexample, thepointx( = 1). Theresultispathdependent. TheparalleltransportequationappliedtoTisDTD= 0canberewrittenasd2xd2+ dxddxd= 0Notethatthisisnothingmorethanthegeodesicequationwefoundinequation4.Wenowhavethetoolstoconstructlawsof physicswhichareinvariantundercoordinatetransformations. Weneedonlyapplythefollowingtworules:1. WhereverweseetheMinkowskimetric,,replacebyageneralmetric,g.2. Whereverweseeapartialderivative,x,replacebyacovariantderivative, .LetusapplythisprescriptiontoNewtons2ndlawappliedinspecialrelativityd(mV)d= FIfitistobecoordinateinvariant,wehaveD(mV)D= Fwherethetotalderivativeisdenedintermsofthecovariantderivativeabove.10 TheCurvatureof Space-Time: RiemannCurvatureTensorWe have seen in previous lectures that by performing a coordinate transformation, it is possibletoremovetheeectofgravitylocally. SuchasetofcoordinatesisknownastheLocalInertialFrame(LIF).Theyare,for example,thexedcoordinatesdenedrelativetoanobjectinfreefall. But given that we can always transform to a LIF, how can we tell if we are in the presenceofagravitationaleld?WhenwetransformtoaLIF, wendacoordinatesystemsuchthatg andtheconnectioncoecients, (whicharebuiltofrstderivativesofg)vanish. Hence, thegrav-itational eldmustarisethroughsecond-derivativesof g, i.e. neighbouringpointswill feeldierent accelerations because the connection coecientsdier. We can schematicallythink of(x) g(x)andifweTaylorexpandaroundapointxwehave(x + x) g(x) + g(x)xHencetheforces(whichcomeintothegeodesicequationviatheconnectioncoecients)willbe dierent if g(x) = 0. This is a 4thrank object although not necessarilya tensor (noteGeneralRelativityandCosmology 28 !a !b ! V Figure3: ParalleltransportofvectorVthatitisbuiltoutofnormalderivatives,notcovariantones). Weneedsomethingofthisformwhich is a tensor to relate to the gravitational eld and we can nd it if we revisit our equationforparalleltransport.RecallthatourequationforparalleltransportisDVD= 0whichcanbeexplicitelywrittenasdVd= dxdVWehavethenthatthechangearoundaninnitesimallength,xisV= VxLetusnowparallelytransportthevectorVaroundaparallelogramwithsidesaandb. Wehavethatthetotalchangeisgivenbyaddingupthefourcontributions:V= (x)V(x)a(x + a)V(x + a)b+(x + b)V(x + b)a+ (x)V(x)bNotethatwearenotprogressingsequentiallyaroundeachcornerofthesquare. WecannowtaketheTaylorexpansionofthemiddletwoterms:V= (V)xab+(V)xabIfwenowrelabelthe2ndterm, ,then andusingtheproductrule,wehave:V= (V+ VVV)abWecannowusetheparalleltransportequationabovetoreplaceVandVandwendV= RVabGeneralRelativityandCosmology 29whereR + (13)We have that Ris the Riemann curvature tensor and it quanties the curvature of a surface(in this case space-time). If there was no curvature, the parallel transport around a closed loopwouldbringavectorbackontoitself. Risindeeda4thranktensoranddependsonthesecondderivative. Itshouldthereforebeuseful forteasingouttheeectsof agravitationaleld. In fact wecan denethe Riemanncurvature tensorin termsofcovariant derivativesandvectorsthrough:()V= RVTheRiemanntensorsatisesanumberofsymmetryproperties:R= RR= RR= RR + R + R= 0wheretherstindexhasbeenloweredusingthemetric: R=gR. WealsohavetheBianchi identity:R +R +R= 0,Finally,wecandenetheRiccitensorandscalar:R RR gR11 BuildingtheEinsteinFieldEquationsWe now want to progress to the equations that tell us how the gravitational eld is sourced. WemayndahintofhowtoconstructtheeldequationfromNewtoniangravity. InNewtoniangravitywehavethePoissonequation:2 = 4GConsider nowtwo neighbouring particles, xiand xi=xi+Niinthis gravitational eld.Newtons2ndLawgivesusd2xidt2= i(x)d2 xidt2= i(x +N)GeneralRelativityandCosmology 30IfwetaketheTaylorexpansionofthe2ndequationwehave,tolowestorder:d2Nidt2= jiNjWecandenethetidaltensor:Eij jisothatd2Nidt2+ EijNj= 0(letusnotworryaboutthefactthatupanddownindicesdontmatchjustthisonce).WecancallthisthegeodesicdeviationequationforNewtoniangravity. NowletusrevisitthePoissonequation;wehavethatitcanberewrittenasEii= 4GWe can now use this link between the geodesic deviation equation and the Poisson equationtoconstructtheappropriateeldequationsforGeneralRelativity. Inconstructinggeodesics,wesawthatthemetric, gplayedtheroleofgravitationalpotentialsandwenowneedasetofequationswhichareinvariantundercoordinatetransformations. Considernowafamilyofgeodesicsx(, ). Wemovealongageodesicbyxingandvarying. Wecanmovefromonegeodesictothenextonebyxingandvarying. Wehavethevectorwhichistangenttoagivengeodesicissimply:T=dxd |whilethevectorwhichisorthogonalornormaltoageodesicatapointisN=dxd |We have that normal derivatives commute so T= N. We can think of as a time coordi-nate and as a spatial coordinate, which means we can pick a coordinate system: (, , x2, x3).Wethenhavethatthetangentandnormalvectorstakeaparticularlysimpleform:T= 0N= 1Again, from the commutation of the normal derivativeswe have NTTN= 0 whichremainstrueifwereplacethenormalderivativesbycovariantderivatives:NTTN= 0Take the equation which relates the Riemann curvature tensor with the commutator of covariantderivatives:( )T= RTGeneralRelativityandCosmology 31NowcontractitwithNandT:NT(TT) = RTTN EN(14)WenowhavethatD2ND2= T(TN)whichwecanusetheabovecommutationrelationtorewriteasD2ND2= T(NT)UsingLeibnitzrulewehaveD2ND2= TNT+ TNTIfweaddthisequationtoequation14(andusethegeodesicequationforT)wend:D2ND2+ EN= TNT+ TNT= TNT+ N(TT) NTT= TNTTNT= 0We have then that the geodesic deviation equation in general relativity has a similar form tothat in Newtonian gravity but with a tidal tensor dened in equation 14. Clearlythe Riemanncurvature(orsomereducedversionofit)mustplaytherolethat 2playsintheNewtoniangravity. Infact,intheNewtonianlimitwehaveEii= Ri0i0sotheNewtonPoissonequationisoftheformRTT 4GTheformoftheequationisschematicallyGeometry Energy, inotherwords, thematterwill source thegeometryinsomeway. This is givingus ahint that thegeneral relativisticequationshouldbeoftheformR GTwhereTisatensorwhichmustbedeterminedbythematterdistribution.12 TheEnergy-MomentumTensorWe now need toconstruct the object,Twhichwill be usedfor the General RelativisticFieldequations. Clearyitmustinvolve. FromSpecialRelativityweknowthat,ifwechangetoaGeneralRelativityandCosmology 32movingframe, withvelocityvandboostfactor,,wehavethat 2. Whichmeansthattransformslikeacomponentofa2ndranktensorandhencetsnicelyinanobjectsuchasT. ArstguesswouldbeT= UUwhereUisthe4-velocityoftheuid, U(c, v). Ifwetakethedivergenceofthistensor,weobtainaconservationequation:T= 0Setting= 1wehavetwofamiliarconservationequations. Firstofall,conservationofmass:t+ i(vi) = 0andmomentumt(vi) + k(vivk) = 0ThelatterequationcanbereexpressedastheEulerequationforapressurelessuid.Wewanttorepresentamoregeneral, perfectuid, onethatincludepressure, Pandhasaformwhichisinvariantundercoordinatetransformation-i.e. apropertensor. ThiscanbeachievedwithT= ( +Pc2)UU+ Pgwherethe4-velocityoftheuidsatisesUU= c2. Inthelocalrestframeoftheuid, wehaveU= (c, 0). Theenergy-momentumconservationequationnowbecomesT= 0Wecanconstructtheenergy-momentumtensorforjustaboutanything: vectorelds(liketheelectricandmagneticelds), gasesof particlesdescribedbydistributionfunctions, fermions,scalarelds,etc.13 TheEinsteinequationsandtheNewtonianLimitLet us now attempt to construct the eld equations which are tensorial and which have at most2ndderivatives. Theonlytensorsatourdisposal are, R, Tandg. WealsohavetwoscalarsT gTand theRicci scalar,R;it turnsout that one ofthesewillbe redundantinwhatfollowssowewilldiscardT. ThemostgeneralequationisR= AT+ Bg+ CRg(15)whereA,BandCareconstants. Fromenergy-momentumconservationwehavethatT= 0GeneralRelativityandCosmology 33andthemetricityconditiongivesusg= 0IfwetaketheBianchiidentityR +R +R= 0,contractwithandmultiplybygwend2RR = 0 (16)WecanreplacethisexpressioninEquation15tondC=12. ThisallowsustodenetheEinsteintensor:G R12gRWeareleftwithtwoconstantsAandBsothatG= AT+ BgWhataretheseconstants? LetusrstfocusonA. Wecangetanideaof wheretheycomefrombycomparingtheeldequationswiththeNewtonPoissonequation. Todosowehavetomaketwoapproximations. Firstofallweneedtoconsidertheweakeldlimitofgravitysothatwecanexpandthemetricaroundaatspace,Minkowski spacetime:g= + hwhere |h| 1. Second, wewill consideratimeindependentsourcewithlowspeeds. TheenergymomentumtensorthenbecomesT00= c2Tij 0Torstorderinhtheaneconnectionsare 12(h+ hh)andtheRiemanntensorbecomes(notethatwecandiscardproductsofsbecausetheyare2ndorder)R=12(hhh + h)TheRiccitensoristhenR= R=12(hhh + h)GeneralRelativityandCosmology 34TondA wewill focuson the G00componentofthe eldequations. ThismeansweneedR00.Wehavechoseastaticsourcesothatall derivativeswithregardsto0vanish. ThisleavesuswithR00= 12h00 122h00Nowtakingg00 (1 + 2c2)wehaveR00=1c22FortheRicci scalarwecandothesame. Thereisatrickwecanuse: assuming |Tij| 0(asdeclaredabove),andsettingB= 0wehave |Gij| 0andsoRij 12gijR =12ijRTheRicciscalarisR = gR R= R00 + Rii= R00 +32R. (17)So 2R00= Rand G00= R0012g00R R00 +R00= 2R00We can nowreplace it all in the eldequations:2c22 = Ac2IfthisistoagreewiththePoissonequationwemusthaveA =8Gc4Finally,itisaconventionthatwehaveB= andwecallthecosmological constant. WethenhavethattheEinsteineldequationsare:G=8Gc4Tg(18)Wehavecompletedourquestforanewtheoryofgravity. ThesetofEinsteinFieldEqua-tionsgiveninequations18andthegeodesicequations, giveninequations4replaceNewtonsUniversal law of gravitation and Newtons 2ndlaw. In Einsteins theory, the picture is dierent-astheAmericanphysicistJohnArchibaldWheelersaid: Spacetellsmatterhowtomoveandmattertellsspacehowtocurve.14 BlackHolesWenowknowhowto, givenadistributionof mass, derivethecorrespondingself-consistentmetric. TheEinsteinFieldEquationsare, of course, atangledmessof non-linearequationswith10unknownfunctionsofspacetime. Ithelpstoconsidersymmetriccongurations. WeGeneralRelativityandCosmology 35nowlookatonesuchconguration, thatofasphericallysymmetric, staticmetricinvacuum.Ifwewritesuchametricinsphericalpolarcoordinateswehaveds2= c2f(r)dt2+ g(r)dr2e(r)dtdr + h(r)(d2+ sin2d2)Withajudiciousredenitionofrandtwecaneliminatee(r)andh(r)andwecanthenworkwithds2= c2f(r)dt2+ g(r)dr2+ r2(d2+ sin2d2)Wewill solvetheEinsteinFieldEquationsinemptyspace, whereT=0. TheequationscanberewrittenasR=0andsothechallengeistoconstructtheRicci tensorforsuchaspacetimewithg00= f(r)grr= g(r)g= r2g= r2sin2Thereare9non-zeroconnectioncoecientsandtheseare:r00= 12grrrg00=12fg00r=12g00rg00= 12ffrrr=12grrrgrr=12ggr= 12grrrg= rgr= 12grrrg = r sin2gr=12grg=1r= 12gg = sin cos r=12grg=1r= 12gg =cos sin WenowneedexpressionsforRwhichareR00=12fg+14fffg14fgg2+12rfgRrr=12ff+14fff214fgfg+12rggR=1g 1 +r2g_ffgg_R= sin2RGeneralRelativityandCosmology 36Therearenoo-diagonaltermsandthereareonlythreeequations. WecannowtakethersttwotondgfR00 + Rrr= 1r_ff+gg_ = 0Thiseasilysolvedtogivefg= whereisaconstant. ReplacinggintheR= 0wendf+ rf = whichintegratedgivesusf= (1 +r)Matchingthis expressionto the weak eld limit (i.e. the Newtonian regime) we nd = 1 and = 2GM/c2rsothatthenalsolutionistheSchwarzschildmetric:ds2= _1 2GMc2r_c2dt2+_1 2GMc2r_1dr2+ r2d2+ r2sin2()d2whichwehaveusedextensivelythroughouttheselecturenotes.Wehavefocusedontheweakeldregionofthisspacetimebutletustryanunderstandabit more about its peculiarities. For astart, somethingodd seemsto happenat rS= 2GM/c2,knownastheSchwarzschildradius: themetricseemstoblowup. NeverthelessweknowthattheRiccitensoris0andifwecalculatetheRiemanntensorwendthatRR= 12r2Sr4i.e. itisalsonite. ItturnsoutthatatrSwedonthaveagenuinespacetimesingularitybuta coordinate singularity. This doesnt mean that odd things dont happen at that, or near thatpoint.Let us considergeodesics in this spacetime. Usingthe geodesicsthat wederivedabove,wehavethattheradialequationis r = GMr2+h2r3_1 3GMc2_Wecan ndthestableminimaofthe lefthand side(towhichwecan associatecircularorbits)tondthattheyareatr =2h2_4h412r2Sc2h22rSc2Thereisclearlyalimitof hbelowwhichthereisnosolutionanditisgivenbyh2=3r2Sc2.ThiscorrespondstotheInnermostStableCircularOrbit: rISCO=3rs. Therearenocircularorbitswithsmallerradii, all orbitsareinspirallingtowardsrS. Interestinglyenough, outsidethisorbit,circularorbitsdosatisfyKeplerslaw: (d/dt)2= GM/r3.GeneralRelativityandCosmology 37Letusnowstudywhathappenstoaninfallingparticle. Wecanseth = 0tond r = rSc22r2Taking r = 0atr wendanintegralofmotion r2=rSc22rIntegratingthisequation,taking= 0atr = r0anddening= r/rSwendcrS=23(3/203/2)Takingxi = 0wendthatittakeaniteamountofpropertimeforaparticletoreachr = 0fromanyradius(withinorwithouttheSchwarazschildradius). If, however, wewishtondtheamountoftimeelapsedforanobserveratinnity,weneedtointegratedrdt= _rsc2r_1/2 _1 rSr_whichgivesctrS=23(3/203/2) + 2(1/201/2) + ln(1/2+ 1)(1/201)(1/21)(1/20+ 1)IfwetheendpointtobetheSchwarzschildradius(i.e. = 1),wehavethatct . Inotherwords,fromanexternalobserverittakesaninniteamountoftimefortheparticletofallin.Wecansolvethegeodesicequationsforradiallightraysbylookingdirectlyatthemetric.Wethenhave_1 rSr_1/2cdt = _1 rSr_1/2drwhichcanberewrittenascdtdr= (1 rSr)1andintegratedtogivect = r rs ln|rrS1| + r0wherer0is aconstant ofintegration. With rSwe havetheusual light coneswith whichwearefamiliarinMinkowski space. Thisisalsoapproximatelytrueforr/rS 1. Butforr/rS,thelightconetipsoversothatall forwardmovingparticlesnecessarilymoveinwardstowardsr=0. ItisnotpossibletocausallyexittheSchwarzschildradius. TheSchwarzschildradiusworksasahorizonbeyondwhichwecantseeanything. Itisaneventhorizon.GeneralRelativityandCosmology 38Finally,suchstrongeectsmaystillbeatplayeveniftheobjectisnotablackhole. Verydenseobjectslikeneutronsstarscanbeagoodlaboratoryfortestinggravity. Andindeed,Binarypulsarsareincrediblyuseful astronomicalobjectsthatcanbeusedtoplaceverytightconstraints on General Relativity. Pulsars are rapidly rotating neutrons stars that emit a beamof electromagneticradiation, and wererst observedin 1967. When thesebeams passover theEarth, asthestarrotates, wedetectregularpulsesofradiation. TherstpulsarobservedinabinarysystemwasPSRB1913+16in1974, byHulseandTaylor. Therotational periodofthispulsarisabout59msasitorbitsaroundanotherneutronstar. Binarypulsarsarehighlyrelativistic. ForexampletheHulse-Taylorpulsarprecesses relativisticallymorethan30000timesfasterthantheMercury-Sunsystem. Theyarealsoasourceofgravitational radiation,i.e. wavesinspacetimethatpropagateawayfromthesystemandtakeenergyaway. Wecanpredicthowmuchenergyingravitational radiationabinarysystemwill emit, inthecontextofGeneral Relativity-itagreesalmostperfectlywiththeangularmomentumdecayobservedintheHulse-Taylor pulsar. IndeedtheHulse-Taylor pulsar is anincrediblyrichlaboratoryfor General Relativity. At least 5General Relativistic eects have beenmeasured: orbitalprecession(alsoknownasperiastronadvance), therateof changeof theorbital period, thegravitationalredshiftandtwoversionsoftheShapirotime-delayeects.15 HomogeneousandIsotropicSpace-TimesThe EinsteinField Equations are a tangled mess of ten nonlinear partial dierential equations.Theyareincrediblyhardtosolveandforalmostacenturytherehavebeenmanyattemptsatndingsolutionswhichmightdescribereal worldphenomena. Fortheremainderof theselectureswearegoingtofocusononesetofsolutionswhichapplyinaveryparticularregime.We will solve the FieldEquations for the whole Universe under the assumptionthat it ishomogeneousandisotropic.Formanycenturies wehavegrowntobelievethatwedontliveinaspecial place, thatwearenotatthecenteroftheUniverse. And, oddlyenough, thispointofviewallowsustomakesomefarreachingassumptions. Soforexample,ifweareinsignicantand,furthermore,everywhere is insignicant, thenwecanassumethat atanygiventime, theUniverselooksthesameeverywhere. Infactwecantakethatstatementtoanextremeandassumethatatanygiventime, theUniverselooksexactlythesameateverysinglepointinspace. Suchaspace-timeisdubbedtobehomogeneous.Thereis another assumptionthat takes into account the extreme regularity of the Universeandthatisthefactthat,atanygivenpointinspace, theUniverselooksverymuchthesameinwhateverdirectionwelook. Againsuchanassumptioncanbetakentoanextremesothatatanypoint, theUniverselookexactlythesame, whateverdirectiononelooks. Suchaspacetimeisdubbedtobeisotropic.Homogeneityandisotropyaredistinctyetinter-relatedconcepts. Forexampleauniversewhichis isotropic will be homogeneous while auniverse that is homogeneous may not beisotropic. A universewhichisonlyisotropicaround onepoint isnothomogeneous. A universethat is bothhomogeneous andisotropicis saidtosatisfytheCosmological Principle. It isbelievedthatourUniversesatisestheCosmologicalPrinciple.HomogeneityseverelyrestrictthemetricsthatweareallowedtoworkwithintheEinsteinGeneralRelativityandCosmology 39eldequation. Forastart, theymustbeindependentofspace, andsolelyfunctionsof time.Furthermore, we must restrict ourselves to spaces of constant curvature of which there are onlythree: aateuclideanspace,apositivelycurvedspaceandanegativelycurvedspace. Wewilllookatcurvedspacesinalaterlectureandwillrestrictourselvestoaatgeometryhere.ThemetricforaatUniversetakesthefollowingform:ds2= c2dt2+ a2(t)[(dx1)2+ (dx2)2+ (dx3)2]Wecalla(t)thescalefactorandtisnormallycalledcosmictimeor physicaltime. Theenergymomentumtensormustalsosatisfyhomogeneityandisotropy. Ifweconsideraperfectuid,werestrictourselvestoT= ( +Pc2)UU+ Pgwith U= (c, 0, 0, 0) and and Pare simplyfunctions of time. Note that both the metric andtheenergy-momentumtensorarediagonal. Sog00= 1 gij= a2(t)ijT00= c2Tij= a2PijAs we shall see, with this metric and energy-momentum tensor, the Einstein eld equations aregreatlysimplied. Wemustrstcalculatetheconnectioncoecients. Wehavethattheonlynon-vanishingelementsare(andfromnowonewewilluse.=ddti.e. nottobeconfusedwithddthatwehaveusedpreviouslyforthegeodesicequations):0ij=1ca aiji0j=1c aaijandtheresultingRiccitensorisR00= 3c2 aaR0i= 0Rij=1c2(a a + 2 a2)ijAgain,theRiccitensorisdiagonal. WecancalculatetheRicciscalar:R = R00 +1a2Rii=1c2_6 aa+ 6_ aa_2_tondthetwoEinsteinFieldequations:G00= R0012Rg00=8Gc4T00 3_ aa_2= 8GGij= Rij12Rgij=8Gc4Tij2a a a2=8Gc2a2PGeneralRelativityandCosmology 40Wecanusetherstequationtosimplifythe2ndequationto3 aa= 4G( + 3Pc2)Thesetwoequationscanbesolvedtondhowthescalefactor, a(t), evolvesasafunctionoftime. TherstequationisoftenknownastheFriedman-Robertson-WalkerequationorFRWequation and the metric is one of the three FRW metrics. The latter equation in a is known astheRaychauduriequation.BothoftheevolutionequationswehavefoundaresourcedbyandP. ThesequantitiessatisfyaconservationequationthatarisesfromT= 0andinthehomogeneousandisotropiccasebecomes + 3 aa( +Pc2) = 0It turns out that theFRWequation, theRaychauduri equationandtheenergy-momentumconservationequationarenotindependent. Itisastraightforwardexercisetoshowthatyoucan obtain one from the other two. We are therefore left with two equations for three unknowns.Onehastodecidewhatkindof energyweareconsideringandinalaterlecturewewillconsideravarietyofpossibilities. Butfornow, wecanhintatasubstantial simplication. Ifwe assumethat the systemsatisesan equation of state,so P= P() and, furthermore that itisapolytropicuidwehavethatP= wc2(19)wherewisaconstant,theequationofstateofthesystem.16 PropertiesofaFriedmanUniverseIWecannowexplorethepropertiesoftheseevolvingUniverses. Letusrstdosomethingverysimple. Letuspicktwoobjects(galaxiesforexample)thatlieatagivendistancefromeachother. Attimet1theyareatadistancer1whileatatimet2, theyareatadistancer2. Wehavethatduringthattimeinterval,thechangebetweenr1andr2isgivenbyr2r1=a(t2)a(t1)and, because of thecosmological principle, this is truewhatevertwopoints wewouldhavechosen. Itthenmakestosensetoparametrizethedistancebetweenthetwopointsasr(t) = a(t)xwherexiscompletelyindependentof t. Wecanseethatwehavealreadystumbleduponxwhenwewrotedownthemetricforahomogeneousandisotropicspacetime. ItisthesetofGeneralRelativityandCosmology 41coordinates (x1, x2, x3) that remain unchanged during the evolution of the Universe. We knownthat thereal,physical coordinatesaremultipliedbya(t)but(x1,x2, x3)are timeindependentandareknownasconformal coordinates. Wecanworkouthowquicklythetwoobjectsweconsideredaremovingawayfromeachother. Wehavethattheirrelativevelocityisgivenbyv= r = ax = aaax = aar HrIn other words, the recession speed between two objects is proportional to the distance betweenthem. Thisequalityappliedtoday(att0)isv= H0rand isknownas HubblesLawwhereH0istheHubbleconstantand isgivenbyH0= 100h kms1Mpc1andhisadimensionlessconstantwhichisapproximatelyh 0.7.Howcanwemeasurevelocitiesinanexpandinguniverse? Consideraphotonwithwave-lengthbeingemittedat onepoint andobservedat some other point. Wehave that theDopplershiftisgivenby (1 +vc)Wecanrewriteitinadierentialformddvc= aadrc= aadt =daaandintegrate tond a. We therefore have that wave lengths are stretchedwiththeexpansionof theUniverse. It isconvenient todenethefactorbywhichthewavelengthisstretchedbyz=ree1 + z a0awherea0isthescalefactortoday(throughouttheselecturenoteswewillchooseaconventioninwhicha0= 1). Wecallztheredshift.Forexample, if youlookatFigure4youcanseethespectrameasuredfromagalaxy; afewlinesareclearlyvisibleandidentiable. MeasuredinthelaboratoryonEarth(toppanel),theselineswill haveaspecicsetof wavelengthsbutmeasuredinaspecic, distant, galaxy(bottompanel)thelineswill beshiftedtolongerwavelengths. Henceameasurementof theredshift(orblueshift), i.e. ameasurementoftheDopplershift, will beadirectmeasurementofthevelocityofthegalaxy.TheAmericanastronomer, EdwinHubblemeasuredthedistancestoanumberof distantgalaxiesandmeasuredtheirrecessionvelocities. Thedatahehadwaspatchy, asyoucanseefromFigure5, buthewasabletodiscernaapattern: mostofthegalaxiesaremovingawayfromusandthefurtherawaytheyare, thefastertheyaremoving. Withmoremoderndata,thisphenomenonisstriking,asyoucanseeintheFigure5. Thedataisneatlytbyalawoftheformv= H0rwhereH0isaconstant(knownasHubblesconstant). CurrentmeasurementsofthisconstantgiveusH0= 67kms1Mpc1.GeneralRelativityandCosmology 42Figure4: Asetof spectrameasuredinlaboratory(toppanel)onadistantgalaxy(bottompanel)Figure5: Therecessionvelocityofgalaxies,Hubblesdatacirca1929(left)andSNdatacirca1995(right)17 Energy, PressureandtheHistoryoftheUniverseWecannowsolvetheFRWequations for arangeof dierent behaviours. Inthenal fewlectures we will look, insome detail, at the nature of matter andenergyinanexpandingUniversebutfornow, wewill restrictourselvestodescribingthemintermsoftheirequationofstateintheformgiveninequation19,P= wc2.Let us start o with the case of non-relativistic matter. A notable example is that of massiveparticles whose energy is dominated by the rest energy of each individual particle. This kind ofmatter is sometimes simply called matter or dust. We can guess what the evolution of the massdensityshouldbe. TheenergyinavolumeV isgivenbyE=Mc2soc2=E/V whereisthe mass density. But in an evolvingUniversewe have V a3so 1/a3. Alternatively,noteGeneralRelativityandCosmology 43thatP nkBT nMc2 c2soP 0. Hence,usingtheconservationofenergyequations: + 3 aa =1a3ddt(a3) = 0andsolvingthis equationwe nd a3. We cannowsolve the FRWequation(taking(a = 1)0):_ aa_2=8G30a3a1/2 a =_8G03_1/2tonda t2/3. If a(t0) =1, where t0is thetimetoday, wehavea=(t/t0)2/3Youwillnoticeafewthings. Firstof all, att =0wehavea=0i.e. thereisaninitial singularityknownastheBigBang. Furthermorewehavethatv= aar=23t0r. i.e. bymeasuringHubbleslawwemeasuretheageoftheUniverse. Andnallywehavethat a 1. For w< 1/3the expansionrateisaccelerating, notdecelerating. For thespecialcaseofw = 1/3wehavea t.Finally, weshouldconsider theveryspecial caseof aCosmological Constant. SuchoddsituationarisesintheextremecaseofP= c2. Youmayndthatsuchanequationofstateisobeyedbyvaccumuctuationsof matter. Suchtypeof mattercanbedescribedbythewefoundinequation18. Thesolutionsarestraightforward: isconstant, aaisconstantanda exp(Ht).Throughout this section, we have considered one type of matter at a time but it would makemoresensetoconsideramix. ForexampleweknowthattherearephotonsandprotonsintheUniversesointheveryleastweneedtoincludebothtypesofenergydensityintheFRWequations:_ aa_2=8G3_M0a3+R0a4_In fact, the current picture of the universe involves all three types of matter/energy we con-sideredin thissectionand, dependingon theirevolutionas a functionof a,they willdominatethedynamicsof theUniverseatdierenttimes. InFigure6weplottheenergydensitiesasafunctionofscalefactorandwecanclearlyseethethreestagesintheUniversesevolution:aradiationera, followedbyamatter eraendingupwithacosmological constant eramorecommonlyknownasaera.18 GeometryandDestinyUntilnowwehaverestrictedourselvestoaatUniversewithEuclideangeometry. Beforewemoveawayfromsuchspacesletusrevisitthemetric. Wehaveds2= c2dt2+ a2(t)(dx2+ dy2+ dz2)Letustransformtosphericalpolarcoordinatesx = r cos sin GeneralRelativityandCosmology 45y = r sin sin z = r cos andrewritethemetricds2= c2dt2+ a2(t)(dr2+ r2d2+ r2sin2d2)We could in principle work out the FRW and Raychauduri equations in this coordinate system.Letusnowconsidera3dimensionalsurfacethatispositivelycurved. Inotherwords,itisthe surface of a 3 dimensional hypersphere in a ctitious space with 4 dimensions. The equationforthesurfaceofasphereinthis4dimensionalspace,withcoordinates(X, Y, Z, W), isX2+ Y2+ Z2+ W2= R2Nowinthesamewaythatwecanconstructsphericalcoordinatesinthreedimensions,wecanbuildhypersphericalcoordinatesin4dimensions:X = Rsin sin cos Y = Rsin sin sin Z = Rsin cos W = Rcos Wecannowworkoutthelineelementonthesurfaceofthishyper-sphereds2= dX2+ dY2+ dZ2+ dW2= R2_d2+ sin2(d2+ sin2d2)_Note how dierentit is from the at geometry. Ifwe transform Rsin into rfor it all to agreewehavethatd2=dr2R2r2We can now repeat this exercise for 3-D surface with negative curvature- a hyper-hyperboloidesotospeak. In our ctitious4-D space(not to beconfusedwithspacetime),wehavethat thesurfaceisdenedbyX2+ Y2+ Z2W2= R2Letusnowchangetoagood coordinatesystemforthatsurface:X = Rsinh sin cos Y = Rsinh sin sin Z = Rsinh cos W = Rcosh Thelineelementonthatsurfacewillnowbeds2= dX2+ dY2+ dZ2+ dW2= R2_d2+ sinh2(d2+ sin2d2)_GeneralRelativityandCosmology 46WecanreplaceRsinh byrtogetd2=dr2R2+ r2We can clearly write all three space time metrics (at, hyperspherical,hyper-hyperbolic) inauniedway. Ifwetaker=Rsin forthepositivelycurvedspaceandr=Rsinh forthenegativelycurvedspacewehaveds2= c2dt2+ a2(t)_d2r1 kr2+ r2(d2+ sin2d2)_(20)where k is positive, zero or negative for spherical, at or hyperbolic geometries, and |k| = 1/R2.We cannowrepeat thecalculationwe undertookfor aat geometryandndthe con-nectioncoecients, Ricci tensor andscalar andthe evolutionequations. Take the metricg= diag(1,a21kr2, a2r2, a2r2sin 2)andnotethatforthischoiceofcoordinates,theiandjlabelsnowrunoverr,and. Wendthattheconnectioncoecientsare:0ij=1ca a giji0j=1c aaijijk=ijkwhere gijandarethemetricandconnectioncoecientsoftheconformal3-space(thatisofthe3-spacewiththeconformalfactor,a,dividedout):rrr=kr1 kr2r= r(1 kr2)r= (1 kr2)r sin2()r=1r=sin(2)2r=1r=1tan()TheRiccitensorandscalarcanbecombinedtoformtheEinsteintensorG00= 3 a2+ kc2c2a2Gij= 2a a + a2+ kc2c2 gij(21)GeneralRelativityandCosmology 47whiletheenergy-momentumtensorisT00= c2Tij= a2P gijCombiningthemgivesustheFriedmanequation_ aa_2=8G3 kc2a2whiletheRaychauduriequationsremainsas3 aa= 4G3( + 3Pc2)LetusnowexploretheconsequencesoftheoverallgeometryoftheUniverse, i.e. thetermproportionaltokintheFRWequations: Forsimplicity, letusconsideradustlleduniverse.We cansee that the termproportional tokwill onlybeimportant at late times, whenitdominatesovertheenergydensityof dust. Inotherwords, intheuniversewecansaythatcurvaturedominates at late times. Let us now consider the twopossibilities. First of all, let ustakek< 0. Wethenhavethat_ aa_2=8G3 + |k|c2a2Whenthecurvaturedominateswehavethat_ aa_2= |k|c2a2soa t. Inthiscase,thescalefactor growsatthespeedoflight. Wecanalsoconsiderk> 0.FromtheFRWequationsweseethatthereisapoint, when8G3=kc2a2andtherefore a=0whentheUniversestopsexpanding. AtthispointtheUniversestartscontractingandevolvestoaBigCrunch. Clearlygeometryisintimatelytiedtodestiny. IfweknowthegeometryoftheUniverseweknowitsfuture.ThereisanotherwaywecanfathomthefutureoftheUniverse. Ifk=0, thereisastrictrelationshipbetweenH= aaand. IndeedfromtheFRWequationwehaveH2=_ aa_2=8G3 = c 3H28GWecallcthecritical density. Itisafunctionofa. IfwetakeH0=100hKms1Mpc1, wehavethatc= 1.9 1026h2kgm3which corresponds to a few atoms of Hydrogen per cubic meter. Compare this with the densityofwaterwhichis103kgm3. NowletustakeanotherlookattheFRWequationandrewriteitas12 a24G3a2= 12kc2GeneralRelativityandCosmology 48whichhastheformEtot= U+ KandweequateEtotto kc2sothatKisthekineticenergy,Uisthegravitationalenergy. Weseethatif=c, itcorrespondstothetotalenergyofthesystembeing0, i.e. kineticandgravitationalenergybalancethemselvesoutperfectly. Letuslookatthecaseofnonzerok.k< 0 0 > candthetotalenergyisnegative,gravitationalenergywinsoutandtheUniverserecollapses.Werecoveranimportantunderlyingprinciplebehindall this, thegeometryisrelatedtotheenergydensity.It is convenientto dene a more compact notation. The fractional energy density or densityparameter. Wedene cItwill beafunctionofaandwenormallyexpressitsvaluetodayas0. Iftherearevariouscontributions to the energydensity,we can denethe fractional energy densitiesof each one ofthesecontributions. For exampleR RcM Mc Itisconvenienttodenetwoadditionals:3H20k kc2H201a2 k0a2andwehave: = R + M+ Wenowhave < 1: < c,k < 0,Universeisopen(hyperbolic) = 1: = c,k = 0,Universeisat(Euclidean) > 1: > c,k > 0,Universeisclosed(spherical)IfwedividetheFRWequationthroughbycwendthatitcanberewrittenasH2(a) = H20_M0a3+R0a4+K0a2+ _(22)wherethesubscript0indicatesthatthesequantitiesareevaluatedatt0. Wewill normallydropthesubscriptwhenreferringtothevarioussevaluatedtoday. Whenwerefertothesat dierent times, we will explicitely say so or add an argument (for example M(a) or M(z)).GeneralRelativityandCosmology 49Howdoesevolve? Withoutlossofgenerality, letusconsideraUniversewithdust, taketheFRWequationsanddividebyH2toobtain 1 =kc2a2H2 kt2/3I.e., if = 1, it is unstable and driven away from 1. The same is true in a radiation dominateduniverseandforanydeceleratingUniverse: =1isanunstablexedpointand, aswesawabove,curvaturedominatesatlatetimes.19 PropertiesofaFriedmanUniverseILetusrevisitthepropertiesofaFRWuniverse, nowthatweknowabitmoreaboutthetheevolution of the scale factor. Distances play an important role if we are to map out its behaviourindetail. WehavealreadybeenexposedtoHubbleslawv= H0dfromwhichwecanextractHubblesconstant. From HubblesconstantwecandeneaHubbletimetH=1H0= 9.78 109h1yrandtheHubbledistanceDH=cH0= 3000h1MpcThese quantitiesset the scale of the Universeand give us a rough idea of how old it is and howfarwecansee. Theyareonlyroughestimatesandtogetarmerideaofdistancesandages,weneedtoworkwiththemetricandFRWequationsmorecarefully.Toactuallygureouthowfarwecansee, weneedtoworkouthowfaralightraytravelsoveragivenperiodoftime. Tobespecic,whatisthedistance,DMtoagalaxythatemitteda light ray at time t, which reachesus today?Let us look at the expressionfor the metric usedinequation20foralightray. Wehavethatdr21 kr2=c2dt2a2(t)(23)Thetimeintegralgivesusthecomovingdistance:DC= c_t0tcdta(t)Fromequation22wehavethat k=k/D2H. Performingtheradial integral (andassumingtheobserverisatr = 0wehave_DM0dr1 kr2=___DHksinh1[kDM/DH] for k> 0DMfor k= 0DH|k|sin1[_|k|DM/DH] for k< 0GeneralRelativityandCosmology 50so we nd an expression for the proper motion distance (also known as the transversecomovingdistance,DMintermsofthecomovingdistance)DM=___DHksinh[kDC/DH] for k> 0DCfor k= 0DH|k|sin[_|k|DC/DH] for k< 0Supposenowwewelook at an objectofanitesizewhichistransversetoour lineofsightand liesat a certain distance from us. If we dividethe physicaltransversesizeof the object bythe angle that object subtends in the sky (the angular size of the object) we obtain the angulardiameterdistance:DA=DM1 + zHence,if we know that size of an object and its redshift we can work out, for a given Universe,DA.Alternatively, wemayknowthebrightnessorluminosityofanobjectatagivendistance.WeknowthattheuxofthatobjectatadistanceDLisgivenbyF=L4D2LDLisaptlyknownastheluminositydistanceandisrelatedtootherdistancesthrough:DL = (1 + z)DM= (1 + z)2DAIt turns out that, in astronomy, one often works with a logarithmic scale, i.e. with magnitudes.Onecandenethedistancemodulus:DM 5 log_DL10pc_anditcanbemeasuredfromtheapparentmagnitudem(relatedtotheuxattheobserver)and the absolutemagnitude M(what it wouldbe if the observerwasat 10 pc from the source)throughm = M+ DMWenow haveaplethoraofdistanceswhichcanbe deployedina rangeof dierentobserva-tions. They clearly depend on the universe we are considering, i.e. on the values of H0, and thevarious s. While kwill dictate the geometry, Dcwill depend on how the Universe evolves. Itis useful to rewrite Dcin a few dierent ways. It is useful to use the FRW in the form presentedinequation22. WecantransformthetimeintegralinDctoanintegralina:DC=_t0tcdta(t)= c_1adaa2H(a)= DH_1adaa2_M/a3+ R/a4+ k/a2+ GeneralRelativityandCosmology 51Aninterestingquestionishowfarhaslighttravelled, fromthebigbanguntil now? Thisisknownastheparticlehorizon,rPandanaiveestimatewouldberP ct0butthatdoesnttakeintoaccounttheexpansionofspacetime. ThecorrectexpressionisgivenaboveanditisrP= DM(0)wheretheargumentimpliesthatitisevaluatedfromt=0tot=t0. Applyingitnowtothesimplecaseofadustlled,atUniverse. WehavethatrP= 3ct0Unsurprisingly,theexpansionleadstoanextrafactor.We could ask a dierent question: how far can light travel from now until the innite future,i.e. how much will we ever see of the current Universe. Known as the eventhorizon it is by theintegralofequation23fromt0until . ForexampleinaatUniversewehaverE=_t0cdta(t)For a dust or radiation dominated universe we have that rE= but this is not so for a universedominatedbyacosmologicalconstant.Wehavebeenfocusingondistances but wecanalsoimproveour estimateof ages. WedenedtheHubbletimeaboveandthatisaroughestimateoftheageoftheUniverse. TodobetterweneedtoresorttotheFRWequationsagain,asabovewehavethat a = aHsodt =daaH _t00dt =_10daaH= t0which,combinedwithequation22givesust0=1H0_10daa_M/a3+ R/a4+ k/a2+ We canuse the above equationquite easily. For aat, dust dominatedUniverse we ndt0= 2/(3H0). Ifwenowincludeacosmologicalconstantaswell,wendt0= H10_10daa_M/a3+ At= 0wesimplyretrievethematterdominatedresult,butthelargeris,theoldertheUniverse. Tounderstandwhy,recalltheRaychauduriequationforthisUniverse: aa= 4G3 +3DividebyH20andwehavethatthedecelerationparameterq0 a(t0) a(t0) a2(t0)=12M GeneralRelativityandCosmology 52If M += 1 then q0=32M1. If M0whichbothhavethesameexpansionratetoday. Thelatterisacceleratingwhichmeansitwasexpandingmoreslowlyinthepastthantheformer. Thismeansitmusthavetakenlongertoreachitscurrent speed and hence is older. Furthermore we can see that our inference about the Universedependsonourknowledgeofthevariouss. Inotherwords,ifwewanttomeasuretheageoftheUniversewemustalsomeasurethedensityinitsvariouscomponents.Finally,letusrevisitHubbleslaw. Weworkedouttherelationshipbetweenvelocitiesanddistancefortwoobjectswhichwereveryclosetoeachother. If wewanttoconsiderobjectswhichare further apart (not too distant galaxies) we can Taylor expand the scale factor today,wendthata(t) = a(t0) + a(t0)[t t0] +12 a(t0)[t t0]2+ Assumethatthedistancetotheemitterattimetisroughlygivenbyd=c(t0 t)wecanrewriteitas(1 + z)1= 1 H0dc q0H202_dc_2+ Forq0= 0andsmallzwerecovertheHubblelaw,cz= H0d. Aswegotohigherredshift,thisismanifestlynotgoodenough.20 TheCosmological DistanceladderGivenour model of arange of possible universes, we wouldlike topindownwhichset ofcosmological parameters (like t0, H0, M, ) correspond to our Universe. We can ask questionslike: whatistheageoftheUniverse, isitacceleratingordecelerating,whatisitsdensityandgeometry? Interestinglyenough, all thesequestionsmustbeansweredtogetherandtodosoweneedtogoout,observeandmeasure.TherststepistomapouttheUniverseandmeasuredistancesandredshiftsaccurately.Byfar theeasiestquantitytomeasureistheredshift. Bylookingat theshiftin thespectraofknownelementsitispossibletoinfertherecessionvelocityofthegalaxydirectly. Measuringdistancesismuchharder. Themostdirectmethodistouseparallaxtomeasurethedistanceto a star. Let us remember what you do here. Imagine that you look at an object in the sky. Itcan be described in terms of twoangles. It has a position on the celestial sphere. Now imaginethatwemoveadistance2dfrom wherewewere. Theobjectmaymoveananglefrom whereit was. The angle that it has moved will be related to the distance D and displacementd. If wesay = 2 then we have tan =dDIf is small then we can use the small angle approximationtoget =dDThe motion of the earth around the sun gives us a very good baseline with which to measuredistance. Thedistancefromtheearthtothesunis1AUsowehavethatD=1whereisGeneralRelativityandCosmology 53SunEarth (spring)Earth (autumn)starvery distantstarsFigure 7: The motion of the Earth around the Sun supplies us with a long baseline for parallaxmeasurments.measuredinarcseconds. Disthengiveninparsecs. Oneparseccorrespondsto206,265AUor3.09 1013km. Thisisatremendousdistance, 1pc 3.26lightyears. Allstarshaveparallaxangles less than one arcsecond. The closeststar, ProximaCentauri,has a distance of 1.3pc. In1989asatellitewaslaunchedcalledHipparcostomeasurethedistancesto118,000starswithanaccuracyof0.001arcseconds. Thiscorrespondstodistancesofhundredsofparsecs. Thismayseemfarbutitisnt. Thesunis8kpcawayfromthecentreofthegalaxy.Wewouldliketobeabletolookfurther. Thebasictoolfordoingthisistotakeanobjectofknownbrightnessandseehowbrightitlooks. TakeastarwithagivenluminosityL. Theluminosityistheamountoflightit pumpsoutpersecond. Howbrightwillit lookfrom wherewestand? Wecanthinkofstandingonapointof asphereof radiusDcentredonthestar.The brightness will be B=L4D2The further away it is the dimmer it will look. If we know theluminosityofastarandwemeasureitsbrightness,thenwewillknowhowfarawayitis.Howcanwedothatinpractice? Starshavevaryingluminositiesandareverydierent.Isthereanywayinwhichwecanuseinformationaboutastarsstructuretoworkoutitsluminosity? Letusstartbylookingatthecoloursofstars. Dierentstarswill emitdierentspectra. Somewill lookredder, othersmoreyellow, whileotherswill beblue. Theircolours(or spectra) areintimatelytiedtotheir temperature. Remember ablackbodywhat blackbodylookslike. Itsspectrumpeaksatacertainvaluewhichisgivenbyitstemperature. Forexample, theSunisyellow-white, hasatemperatureof5800K.ThestarBellatraisblueandhasatemperatureof 21,500K. Betelgeuseisredandhasatemperatureof 3500K. Nowwemightthinkthatwehaveitmade.The luminositymust be relatedtothe temperature somehow. If we assume that it isblack body, the energy ux is F =T4where is the Stefan-Boltzmannconstant =5.6108Wm2K4. So luminosity is simply the surface of the star times its ux L = 4R2T4.Thereisindeedaverytightconnectionbutstarscanhavedierentradii. Forexamplemainsequencestarshaveone type ofradius whilered giants havemuchlarger radii. We can look atthe H.R. diagram and nd stars with the same temperature which have very dierent luminosi-ties. Howeverifwecanidentifywhattypeofstarstheyarethenwecan, giventheircolours,readotheirluminosities.Supposewelookatthespectraoftwostars,AandB,andweidentifysomespectrallines.Thesecorrespondtothesameabsorption/emissionlinesbutinAtheyrenarrowerthaninB.What leadstothethickness of thelines? If therearerandomvelocities, theywill Dopplershifttheline. Thelargerthespreadinvelocities, themoreshiftstherewill be. Butclearlyfortheretobealargerspread, theyhavetobeclosertothecoreof thestari.e. theradiusGeneralRelativityandCosmology 54Figure8: TheluminosityofCepheidstarsvariesperiodicallyovertime.hastobesmaller. I.e. broaderlinesimplysmallerR. Sobyreadingothethicknessof thelines we can pinpoint what type of stars they are and then from their colours we can infer theirluminosity. Forexample: SunhasT 5800K. Itisamainsequencestarwithaluminosityof1 L. Aldebaranisagiantstarwhich, eventhoughitiscooler, T 4000K,hasaluminosityof370 L. Thismethod,knownasspectroscopicparallaxcanbeusedtogooutto10kpc.Howcanwe move out beyond10kpc? There are some stars whichhave averyusefulproperty. Their brightness varies withtimeandthelonger their variation, thelarger theirluminosity. ThesestarsknownasCepheidstarsareinterestingbecausetheyhavea)periodsofdays(whichmeanstheirvariationscanbeeasilyobserved)andb)areveryluminouswithluminositesofabout100 1000 Landthereforetheycanbeseenatgreatdistances. Itwasfoundthattheirperiodofoscillationisdirectlyrelatedtotheirintrinsicluminosity.These stars pulsate because their surface oscillates up and down like a spring. The gas of thestar heats up and then cools down, and the interplay of pressure and gravity keeps it pulsating.Howdoweknowtheintrinsicluminosityofthesestars? Wepickoutglobularclusters(verybrightaglomerationsintheGalaxywithabout106stars)andweusespectroscopicparallaxtomeasuretheirdistances. Thenwelookforthevaryingstars, measuretheirbrightnessandperiod and buildup aplot. Thereis another classof star called RRLyrae whichalsooscillate.Theyhavemuchshorterperiods, andarelessluminous(100 L)buthaveamuchtighterrelationshipbetweenperiodandluminosity. Wecanusethesestarstogooutto30Mpc(i.e.30millionparsecs). Ifwewanttogofurther,weneedsomethingwhichisevenbrighter.Themethodofchoiceformeasuringverylargedistancesistolookfordistantsupernovae.Asyouknow,supernovaearetheendpointofstellarevolution,massiveexplosionsthatpumpout an incredible amount of energy. Indeed supernovae can be as luminous as the galaxies whichhost them with luminosities of around 109L. So we can see distant supernovae,measure theirbrightness and if we know their luminosities, use the inverse square law to measure the distance.A certaintypeof supernova(supernovaeIa) seemtohaveverysimilar behaviours. Theydontall have the same luminosities but the rate at which they fade after explosion is intimately tiedGeneralRelativityandCosmology 55Figure9: There is atight relationshipbetweenthe period(x-axis) andthe luminosity(ormagnitudeinthey-axis)forCepheidandRRLyraestar.to the luminosity at the moment of the explosion. So by following the ramp up to the explosionandthesubsequentdecayitispossibletorecalibrateasupernovaexplosionsothatweknowitsluminosity.SupernovaeIaarisewhenawhitedwarf whichisjustmarginallyheavierthantheChan-drasekhar massgobblesupenoughmaterial tobecomeunstableandcollapse. Theelectrondegeneracypressureisunabletoholditupanditcollapsesinaeryexplosion. Supernovaecanbeusedtomeasuredistancesouttoadistanceof about1000Mpc. Theyareextremelyrare, onepergalaxyperhundredyears, sowehavetobeluckytondthem. Howeverthereare 109galaxies to look at sothe current practiceis to stare at large concentrationsof galaxiesandwaitforaneventtoerupt. Oforder500SNhavebeenmeasuredinthepastdecades.FinallyI want tomentionanother distanceindicatorwhichcanbeusedtomeasurethedistances out toabout 100Mpc. Whenwe lookat distant galaxies there is averyusefulspectral line to measure. It has a wavelength of 21cm and corresponds to the energy associatedwith the coupling of the spin of the nucleus (a proton) with the spin of an electron in a Hydrogenatom. Iftheyarealigned,theenergywillbehigherthaniftheyareanti-aligned. Onceagain,thislinewill haveacertainwidthduetotheDopplereectasaresultofinternal motionsinthegalaxy. InparticulartherotationofthehydrogenwillinduceaDopplereect. Thefastertherotation,thelargertheDopplereectandthewiderthespectralline.Weknow, fromNewtoniangravitythattherotationisintimatelytiedtothemassofthegalaxy, sothewider theline, thefaster thespeedof rotationandhence themoremassivethegalaxy. Butthemoremassivethegalaxy, themorestarsitshouldcontainandthereforethemoreluminousitshouldbe. Sobymeasuringthe21cmlineitispossibletomeasuretheluminosityofdistantgalaxies. ThisisknownastheTullyFisherrelation.WecannowusethesetechniquestopindownvariouspropertiesofourUniverse.GeneralRelativityandCosmology 56Figure 10: Toppanel: the light curves of Supernovae Ia. Bottompanel: the light curveshavebeenrecalibrated(orstretched)sothattheyallhavethesamedecayrate. Notethat,followingthisprocedure,allcurveshavethesameluminosityatthepeak.21 TheThermal historyoftheUniverse: EquilbriumWeshall nowlookathowthecontentsof theUniverseareaectedbyexpansion. Therstproperty which we must consideris that as the Universeexpands,its contentscool down. Howcanweseethat? LetusfocusontheradiationcontainedintheUniverse. Intheprevioussectionswefoundthattheenergydensityinradiationdecreasesas 1a4.Whatelsecanwesayaboutradiation? Letusmakeasimplifyingassumption, thatitisinthermalequilibriumandthereforebehaveslikeablackbody. Forthistobetrue,theradiationmust interact very eciently with itself to redistribute any uctuations in energy and occupy themaximumentropy state. You have studiedthe properties of radiation (or relativistic particles)in thermal equilibrium in statistical mechanics in the 2ndyear and found that it can be describedintermsofanoccupationnumberpermodegivenbyF() =2exphkBT 1whereisthefrequencyofthephoton. Thiscorrespondstoanenergydensitypermode()d=83dc3hexphkBT 1GeneralRelativityandCosmology 57Ifweintegrateoverallfrequencieswehavethattheenergydensityinphotonsis:=215(kBT)_kBT hc_3. (24)Wehavethereforethat T4. HenceifradiationisinthermalequilibriumwehavethatT 1aIs this the temperature of the Universe?Two ingredients are necessary. First of all, everythingelsehastofeel thattemperaturewhichmeanstheyhavetointeract(evenif onlyindirectly)with the photons. For examplethe scatteringo photons of electronsand positronsis throughtheemissionandabsorptionof photons. Andonceagain, atsucientlyhightemperatures,everythinginteractsquitestrongly.Anotheressentialingredientisthattheradiationmustdominateovertheremainingformsof matter intheUniverse. Wehavetobecareful withthisbecauseweknowthatdierenttypesof energywill evolveindierentwaysastheUniverseexpands. Forexamplewehavethattheenergydensityofdust(ornon-relativisticmatter)evolvesasNR a3ascomparedto a4soevenif wasdominantatearlytimesitmaybenegligibletoday. Howeverwealsoknowthatthenumberdensityof photons, n a3asdoesthenumber densityofnon-relativisticparticles,nNR a3. Ifweaddupallthenon-relativisiticparticleintheformofneutronsandprotons(whichwecallbaryons), wendthatnumberdensityofbaryons, nBisverysmallcomparedtothenumberdensityofphotons. Infactwecandenethebaryontoentropyratio,B:B=nBn 1010AswecanseetherearemanymorephotonsintheUniversethanparticleslikeprotonsandneutrons. SoitissafetosaythatthetemperatureofthephotonssetsthetemperatureoftheUniverse.Wecanthinkof theUniverse as agiganticheat bathwhichis coolingwithtime. Thetemperature decreases as the inverse of thescale factor. Tostudytheevolutionof matterintheUniversewemustnowusestatistical mechanicstofollowtheevolutionof thevariouscomponents as the temperature decreases. Let us start owithanideal gas of bosons orfermions. Itsoccupationnumberpermode(nowlabeledintermsofmomentump)isF(p) =gexp_EkBT_1wheregisthedegeneracyfactor, E=p2c2+ M2c4istheenergy, isthechemical poten-tial and+(-)correspondstotheFermi-Dirac(Bose-Einstein)distribution. Wecanusethisexpressiontocalculatesomemacroscopicquantitiessuchasthenumberdensityn =gh3_d3pexp_EkBT_1GeneralRelativityandCosmology 58theenergydensityc2=gh3_E(p)d3pexp_EkBT_1andthepressureP=gh3_p2c23Ed3pexp_ET_1Itisinstructivetoconsidertwolimits. Firstofallletustakethecasewherethetempera-tureoftheUniversecorrespondstoenergieswhicharemuchlargerthantherestmassoftheindividual particles, i.e. kBT Mc2andletustake 0. Wethenhavethatthenumberdensityobeysn =(3)2g_kBT hc_3(B.E.)n =3(3)42g_kBT hc_3(F.D.)where(3) 1.2comesfromdoingtheintegral. Theenergydensityisgivenbyc2= g230(kBT)_kBT hc_3(B.E.)c2=78g230(kBT)_kBT hc_3(F.D.)andpressuresatisesP=c2/3. Asyoucanseethesearethepropertiesof aradiation. Inother words, even massiveparticles will behave like radiation at sucientlyhigh temperatures.AtlowtemperatureswehavekBT Mc2andforbothfermionsandbosonsthemacroscopicquantitiesaregivenby:n = g_2h2_32(MkBT)3/2exp(Mc2kBT )c2= Mc2nP = nkBT Mc2n = .Thislastexpressiontellsusthatthepressureisnegligibleasitshouldbefornon-relativisticmatter.Thiscalculationhasalreadygivenusaninsightintohowmatterevolvesduringexpansion.At sucientlyearly times it all looks like radiation. As it cools down and the temperature fallsbelowmassthresholds, thenumberofparticlesbehavingrelativisticallydecreasesuntil whenwegettotoday,thereareeectivelyonlythreetypeofparticleswhichbehaverelativistically:thethreetypesofneutrinos. Wedenotetheeectivenumberofrelativisticdegreesoffreedombygandtheenergydensityinrelativisticdegreesoffreedomisgivenby = g230(kBT)_kBT hc_3GeneralRelativityandCosmology 5922 The Thermal historyof theUniverse: The CosmicMicrowaveBackgroundUntil nowwe have consideredthings evolving passively, subjectedtothe expansionof theUniverse. But we know that the interactions between dierent components of matter can be farmore complex. Let us focus on the realm of chemistry,in particular on the interaction betweenoneelectronandoneproton. Fromatomicphysicsandquantummechanicsyoualreadyknowthat an electron and a proton may bind together to form a Hydrogen atom. To tear the electronaway we need an energy of about 13.6eV . But imagine now that the universeis sucientlyhotthatthereareparticleszippingaroundthatcanknocktheelectronoutoftheatom. Wecanimagine that at high temperaturesit will be very dicultto keepelectrons and protons boundtogether. If the temperature of the Universe is such that T 13.6eVthen we can imagine thattherewillbeatransitionbetweenionizedandneutralhydrogen.We can work this out in more detail (although not completely accurately) if we assume thatthis transition occurs in thermal equilibrium throughout. Let us go through the steps that leadtotheSahaequation. Assumewehaveanequilibriumdistributionof protons, electronsandhydrogenatoms. Letnp, neandnHbetheirnumberdensities. Inthermal equilibrium(withT M)wehavethatthenumberdensitiesaregivenbyni= gi_2h2_32(MikBT)32exp iMic2kBTwherei = p, n, H. Inchemicalequlibriumwehavethatp + e= HsothatnH= gH_2h2_32(MHkBT)32expMHc2kBTexp (p + e)kBTWecanusetheexpressionsfornpandnetoeliminatethechemicalpotentialsandobtain:nH= nenpgHgpge_2h2_32(MHkBT)32(MpkBT)32(MekBT)32expMHc2+ Mpc2+ Mec2kBTThereareaseriesofsimplicationswecannowconsider: i)Mp MH,ii)thebindingenergyisB MHc2+Mpc2+Mec2= 13.6eV , iii)nB= np +nHiv)ne= npand nallygp= ge= 2andgH= 4. SoweendupwithnH= n2p(MekBT)32_2h2_32expBkBTWecangofurtheranddeneanionizationfractionX npnB. QuiteclearlywehaveXis1iftheUniverseiscompletelyionizedand0ifitisneutral. Usingthedenitionofthebaryontoentropyfractionwehave1 X= X2Bn_2h2_32(MekBT)32expBkBT(25)GeneralRelativityandCosmology 60Figure11: TheevolutionoftheionizationfractionasafunctionofredshiftFinally we havethat weare in thermal equilibriumsowehave an expressionfor nand weget1 XX2 3.8B_kBTMec2_32expBkBT(26)ThisistheSahaequation. Ittellsushowtheionizationfraction, Xevolvesasafunctionof time. Atsucientlyearlytimeswewill ndthatX=1, i.e. theUniverseiscompletelyionized. Asitcrossesacertainthreshold, electronsandprotonscombinetoformHydrogen.This happens when the temperature of the Universe is T 3570Kor 0.308eV , i.e. when it wasapproximately380, 000yearsold, ataredshiftofz 1100. Wewouldnaivelyexpectthistohappenat13.6eV buttheprefactorsinfrontoftheexponential playanimportantrole. Onewaytothinkaboutitisthat,atagiventemperaturetherewillalwaysbeafewphotonswithenergies larger than the average temperature. Thus energetic photons only become unimportantatsucientlylowtemperatures.Whatdoesthisradiationlookliketous? Atveryearlytimes, beforerecombination, thisradiationwillbeinthermalequilibriumandsatisfythePlanckspectrum:()d=8hc33dexp(h/kBT) 1After recombination, the electrons andprotons combine toformneutral hydrogenandthephotons will beleft topropagatefreely. Theonlyeect will betheredshiftingduetotheexpansion. The net eect is that the shape of the spectrum remains the same, the peak shiftingas T 1/a. So even though the photons are not in thermal equilibrium anymore, the spectrumwill still bethat of thermal equilibriumwiththetemperatureT0=3000o/1100Kelvin, i.e.T0= 2.75oKelvin.Thehistoryofeachindividual photoncanalsobeeasilydescribed. Letsworkbackwards.After recombination,a photondoes not interactwithanythingandsimplypropagates forwardat the speed of light. Its path will be a straight line starting o at the time of recombination andendingtoday. Beforerecombination,photonsare highlyinteractingwithaverydensemediumof charged particles, the protons and electrons. This means that they are constantly scatteringoparticles, performingsomethingakintoarandomwalkwithaverysmallsteplength. ForGeneralRelativityandCosmology 61allintentsandpurposes,theyaregluedtothespotunabletomove. Soonecanthinkofsuchaphotonshistoryasstartingostuckatsomepointinspaceand, atrecombination, beingreleasedtopropogateforwarduntilnow.We can take this even further. If we look from a specic observingpoint (such as the Earthorasatellite), wewill bereceivingphotonsfromall directionsthathavebeentravellinginastraightlinesincetheUniverserecombined. Allthesestraightlineswillhavestartedo at thesametimeandatthesamedistancefromus-i.e. theywillhavestartedofromthesurfaceofasphere. Thissurface,knownasthesurfaceoflastscatteringiswhatweseewhenwelookatthe relic radiation. It is very much like a photograph of the Universe when it was 380,000 yearsold.23 TheThermal historyof theUniverse: outof equili-biriumandBigBangNucleosynthesisWe have assumed that the Universe is in thermal equilibrium throughout this process. We havecomeupwithanexpressionfortheionizationfractionwhichisnotcompletelyaccuratebutqualitativelyhasthecorrectbehaviour. Thereisanothersituationwhereassumingthermalequilibriumw