1968ApJ. . .151. . 431M 103 9 10 l l 10 for theanisotropyparameters whileestimating(a)theinfluenceofanisotropy onthe Lagrangian formforthosefieldequationswhichcontrol theanisotropyparameters;(iv) most importantoftheseare(i)theuseparametrizations forthemetricwhichallow allows onetoignorethedetailsofsolutions thenon-linearequationsofmotion neous cosmologiesusingsomenewtechniquesdeveloped in§§II-IVofthispaper.The dominated expansionphasebetweenabout10°andKwheretheneutrinosare and (v)aderivationofanadiabaticinvariant for theanisotropyparameterswhich a definitionof“anisotropyenergy”(relatedtothe Hamiltonianoftheanisotropyequa- the stressesinducedincollisionlessradiationby theanisotropicexpansion;(iii)a in theneutrinoradiationmustbetakenintoaccount. collisionless andtheanisotropicstressesfrommomentumdistribution tions) whichcontributestotheexpansionratein samewayasotherformsofenergy; the expansionandanisotropytobetreatedin differentways;(ii)acomputationof This excludes,inthesemodelswhichwebelievetobetypical,anypossibilitythatpure various directionsisreducedbythisneutrinoviscositytothepointwhereanisotropy 0.03 percent(of3°K).Themostimportantmechanisminreducingtheanisotropyis than 6X10°K.Furtherreductionsinanisotropytakeplaceduringtheradiation- Thorne 1967ö)whichiscontrolledbytheaverageexpansionrateattemperaturesless anisotropy couldinfluencetheprimordialheliumproduction(HawkingandTayler1966; no longerseriouslyaffectstheaverageexpansionrateonceneutrinosarecollisionless. neutrino viscosityattemperaturesjustabove10Kwhenthefrequencytrfor for variousreasonsthatpartofthetemperatureanisotropyinmicrowaveradia- late thattheUniverse,frombeginning,wasremarkablysymmetric.Ishallattempt photon temperatureshouldbelessthan003percent,independentoftheamountinitialanisotropy, expansion rateBr{dR/dt)~t~.Anylargeinitialanisotropyintherates background radiation,thentheprecise(0.2percent)measurementofitsisotropy from collisionlessradiation,areincluded.Theseshowthatthepresentanisotropyofblack-body The AstrophysicalJournal,Vol.151,February1968 collisions betweenneutrinosandthermalelectronsorpositronsiscomparabletothe to providesomeexplanationherebyshowingthat,inalargeclassofhomogeneousbut cosmology. Assuchitsurelydeservesabetterexplanationthanisprovidedbythepostu- if theUniversehascooledtoitspresentstatefromtemperaturesaboveabout2XIO°K. expansion ratesaregiveninwhichtheeffectsofviscosityradiation,andanisotropicpressures tion atthepresentepochwhichisduetoprimordialanisotropiesshouldbelessthan anisotropic universessatisfyingEinstein’sequations,theanisotropydiesawaysorapidly (Partridge andWilkinson1967)becomesthemostaccurateobservationaldatumin © American Astronomical Society •Provided bytheNASA Astrophysics DataSystem The resultsstatedabovearebasedonastudyof the Einsteinequationsforhomoge- * Permanentaddress:Department ofPhysicsandAstronomy,UniversityMaryland,College Park If oneaccepts,asIshallhere,thecosmologicalinterpretationof3°Kmicrowave Solutions oftheEinsteinequationswithflathomogeneousspacelikehypersurfacesbutanisotropic THE ISOTROPYOFUNIVERSE I. INTRODUCTIONANDCONCLUSIONS Peterhouse, Cambridge,England Charles W.Misner* Received June22,1967 ABSTRACT a) Epitome 431 1968ApJ. . .151. . 431M 2 -2 10 proposal (Gamow1948;AlpherandHerman1948,1950),notonlyallowedforthe many decadesofexpansion. 432 CHARLESW.MISNERVol.151 interpretation (Dickeetal.1965)andpublicationofdataconfirmingitsexistence 0.07) X10~whichwewilltreatasanobservationalupperlimit.Thevariationswith variations withaquadrupoledistribution(12-hourperiod)thevalueAT/T—(0.16± ropy ofrelativisticcosmologicalmodelsthanthosederivedfromtheisotropy Thorne (1967a)hasdiscussedhowmeasurementsofthevariationtemperature but alsostimulatedanumberofnewapproachestoproblemsincosmology,astrophysics, radiation temperatureasthesequantitieschangebymanyordersofmagnitudeover observation. present limitbyimprovedobservations,itwouldimplysuchalowvelocityfortheSolar The isotropyobservationsofPartridgeandWilkinson(1967)giveforthetemperature of thebackgroundradiationwithdirectionputmuchmorestringentlimitsonanisot- native explanationisnotruledoutonenergeticgroundssincerecentburningofhydrogen Field andHitchcock1966;ThaddeusClauserHowellShakeshaft1966), expansion rate;(b)theanisotropyinand(c) The firstexampleofthisapproachwouldbeEinstein’s(1917)failuretofindstatic be devotedtocalculationswhichtry“predict”thepresentlyobservableUniverse. relation ofobservationaldatasufficienttodistinguishamongasmallnumbersimple System thatnotheoryofthe3°Kradiationispresentlypreparedtotoleratesuchan could notbeduetotheexpansionanisotropyeffectsunderdiscussioninthispaper,but extragalactic (Hubble)redshifts(KristianandSachs1966;Kantowski1966). thermalizing it.Thispaperpresumestheconventionalcosmologicalinterpretation. to heliuminstarscouldsupplythenecessaryenergyifmechanismsbefoundfor tion ofthis3°Kradiationisnotsomuchestablishedasitunchallenged,andanalter- and cosmic-rayphysics.AsFowler(1967)haspointedout,thecosmologicalinterpreta- present paperisconcernedwithaclassofsolutions whichgeneralizethestandard feasible approacheswouldattempttosurveymuch morelimitedclassesofsolutions between observationandtheoryonthispointisreviewed byNorth[1965]).Thedifficulty expansion whichisrequired(notmerelypermitted)ifoneassumesspatialisotropyin cosmological solutions,whichcanberegardedasalostopportunitytopredicttheHubble cosmological solutionsofEinstein’sequations,Isuggestthatsometheoreticaleffort a dipoledistribution(24-hourperiod)forwhichtheyfindAT/r=(0.03±0.07)X10 pressure whilemaintaining thehomogeneityofspace(translationinvariance). Wefind verse maybelargelyindependentoftheinitial conditionsadmittedforstudy.The lies inthetreatmentofinitialconditions.Ideally onemighttrytoshowthatalmost in usingrelativisticcosmologyforpredictiverather thanmerelydescriptivepurposes than takingtheuniqueproblemofrelativisticcosmologytobecollectionandcor- the large-scaleinhomogeneityofUniverse(SachsandWolfe1967;SciamaRees their measurementplaceslimitsonthevelocityofSolarSystem(Sciama1967)and (Penzias andWilson1965),ledtofurtherobservations(RollWilkinson1966; the original(A=0)Einsteinequations(see [1945];thehistoricalinteraction the Einsteinequationstoseewhethersomepresently observablepropertiesoftheUni- all solutionsoftheEinsteinequationswhichlead tostarformationalsohavemany the Universebyincludinganisotropybothin expansionrateandintheradiation radiation-filled k=0Robertson-Walker“hotbig-bang” modelsoftheearlyepochs other propertiescompatible(orincompatible!)with observation.Moremodestbutmore 1967), Iftheamplitudeofdipolevariationsweretobereducedone-third that allthesemodelsinwhich temperaturesabove10°Koccurredinthe pastbecome 1 1 © American Astronomical Society •Provided bytheNASA Astrophysics DataSystem Dicket conceptionofprimordialblack-bodyradiation,whichrevitalizedanearlier I wishtoapproachrelativisticcosmologyfromanunfamiliarpointofview.Rather R.H.Dicke(1964,unpublished); seeDicke,Peebles,Roll,andWilkinson(1965)Peebles (1966). b) Introduction 1968ApJ. . .151. . 431M 2a2ß 10 510 142aa21 2a we writethespatialmetric componentsintheformgij{i)—e()ijwhere 2/3¿yisa equations intosimpleforms,suchastheFriedmann equationfortheRobertson-Walker solutions inanalyticforms.Theeffortisdirected rather towardcastingthedifferential in thisdirectionthanothers.Thepresentpaper isprimarilydevotedtosolutionsof fluid) andcontributeamajorpartoftheenergydensity. Sinceneutrinostravelinginthe where Zio=Z/IOK.Moredetailsofthisstandardmodelforthe“hotbig-bang" highly isotropicbythetimetheycoolto3°K,sothislimitedstudyisnotinconsistent ignored. Pressureanisotropiesthatarisefromthe viscosity ofcollision-dominatedradia- direction ofmostrapidexpansionsufferthegreatest redshifts,thepressurewillbeweaker covering atleast10°toK,wheretheneutrinos arecollisionless(hencenota at anyepoch,andsothequalitativebehaviorof solutionsiseasytodescribe. cosmologies, sothattherelativeimportanceofvarious effectscanbeeasilyestimated tion arealsoconsidered.Idonot,however,aim particularlytowardobtainingexact the Einsteinequationsinwhichpressureanisotropies thatariseinthiswayarenot tensor isnotappropriatefordiscussinganisotropiccosmologiesinthetemperaturerange, Once oneacceptsthermalneutrinosfromearlyepochs,however,afluidstress-energy Universe canbefoundinWagoner,Fowler,andHoyle(1967)orZel'dovich(1966). light andbycollisions)toproducehomogeneity.Indeed,theseeffectshavebeenasevere nikov 1963)andbyHeckmannSchiicking(1962)forthecasewherestress- form byKasner(1921)foremptyspace-time(seealsoTaub1951;LifshitzandKhalat- most generalsolutionoftheEinsteinequationsforgij(t)hasbeengiveninanalytic be complementedbyaconsiderationoffiniteamplitudeirregularitiesalltypes,not ment ofgalaxiesfrommodestinitialperturbationswithamplitudesfargreaterthan restriction oneffortstoproduceminorirregularities(suchasgalaxies)frommorehomo- with theconjecturethat“Einstein'sequationspredictUniverseisisotropic." No. 2,1968ISOTROPYOFUNIVERSE433 as wellphotons,orfaattemperatureswherethermalelectron-positronpairsmust expands fromearlyhightemperaturesexceedingthep-mesonrestmass,appropriate and p=aTise«:twithTe~T~ifairGa/Sc).Forauniversewhich when onespecializesthemetricbyassumingtwoofthreespacedirectionsareequiva- which includesthe&=0Robertson-Walkerformasaspecialcasewithgijebij.The received nodetailedattention(Misner1967Z>). which couldbesignificantinsmoothingvariousmodeschaoticinitialconditionshave merely thosedesignedtoproducegalaxies(Misner1967a).Anumberofphenomena The possibilitythat“Einstein'sequationspredicttheUniverseishomogeneous" also beincluded.Inthislattercaseonehas energy densityconstantisnotabut^atoincludenon-degeneratep-and¿-neutrinos lent. ThesolutionintheRobertson-Walkercaseforpurelyaradiationfluidwithp=\p radiation fluidwithp—alsogivesolutionsinanalyticform(seeThorne1967a) stress-energy tensors,correspondingtoapressurelessfluidplusmagneticfield,or energy tensoristakenintheformforapressurelessperfectfluid.Moregeneral include Hawking1966c;Peebles1967;SachsandWolfe1967).Severalauthors(Peebles geneous initialconditions(Gamow1948;recentstudiescontainingfurtherreferences ever, largeinitialrandomvelocitieswhichmightbethoughttypicalofachaoticor does notcomeunderstudyhere,sincewesimplyassumehomogeneitythroughout.How- statistical fluctuationscouldproduce.Thisapproachneedsfurtherdiscussion,andshould “generic" initialsingularitydohaveatendency(limitedinitseffectsbythevelocityof 1965, 1967;Harrison1967a,b;SilkMichie1968)haveconsideredthedevelop- 2i1 © American Astronomical Society •Provided bytheNASA Astrophysics DataSystem To describetheanisotropy oftheUniverseindependentlyitsover-all expansion The metricformassumedthroughoutthepresentpaperisds=—dt+gij(t)dxdx\ c) AnisotropyParameters 2 t =(1.1sec)Zio“,(i‘U 1968ApJ. . .151. . 431M ß2ß 2ß 200 1i a A a 2/S01n 2 2 00 1/23a significant. Wedefinethe“rateofshear”matrix = — 2 tr [o-(ói3)(r] + | tr [(Dbe2ß)e~ß(re~ß] . In the last term here we will wish to integrate by parts in the action integral where the term eZab(\ tr a2) occurs. Neglecting total time derivatives then we have e3aô(J tr <72) = — 2e3a tr { Now in this last term we use, from equation (3.3) and e~ßeß = 1, the formula De~ß = -e-ß( With equations (3.15) and (3.16) now established it is trivial to see that the anisotropy equations (3.9d) indeed follow from varying ß in the Lagrangian (3.12). Note that the only property of used in establishing the Lagrangian (3.12) was equation (2.32) or equivalently (2.11). This property holds for more general forms of matter than collisionless null radiation. For instance, a homogeneous collisionless gas of particles of mass m would have a stress-energy tensor also given by equation (2.8) but 0 2 2 b with (k ) = m + qa(f (i)qb so equation (2.11) would again follow. [Note also that 5î(^) is arbitrary; it need not be a thermal distribution.] And for collision-dominated fluids equation (2.11) is just the law for adiabatic expansion work, dll = d(gl,2Tm) = —pdV = — \Tkkdg112 since T00 will depend only on a, not on ß. c) Anisotropy Energy Let us choose our stress-energy tensor so that = P6 + Py + P„(l + V) , (3.17) 3 4 a where p& = /xT , p7 = a7r , and p„ = avT*. Here T = 7V~ is the “central temperature” of the collisionless radiation (such as neutrinos below 1010 ° K) whose energy density is pv(l + F). As in § I we use p7 for the energy density of collision-dominated radiation (such as photons above 3000° K, but also including neutrinos above 1010 ° K), and p& for the energy density of pressure-free matter. The terms independent of ß do not contribute to the Euler-Lagrange equations in our variation principle for equations (3.9d), so we can take the Lagrangian to be given by Za 4 a SttZ, = ^(Tij(7ije — 87ra„r0 F(ß)e~ . (3.is) The corresponding Hamiltonian is then er a Stt/z = 2 4“ %TravToW(ß)e- . (3-i9) © American Astronomical Society • Provided by the NASA Astrophysics Data System 1968ApJ. . .151. . 431M 2 2 3a ß 2 448 CHARLESW.MISNERVol.151 Da, wedefinenothbut Because ofthewayitentersFriedmann-likeequation(3.9a)forexpansionrate so forward tosolvethisequationfora{t)ifeither(i)theanisotropyenergywerenegligible Since theentireßdependenceinthisequationoccurspßterm,itwouldbestraight- to bethe“anisotropyenergy.”Thenequation(3.9a)reads are discussedfurtherin§IVbelow. principal valuesoflnT.Thuswedefine ßij and(Tijmustvanish,withtheresultthatßwillthereafterremainzerobyequation constant differenceAH=definedinequation(1.2)wehavefrom(3.20) that pßalsogiveslimitsonthekinematicquantitiesanandßij.IntermsofrmsHubble to activegravitationalmassinproducingexpansionDaequation(3.21),whilehis or (ii)theanisotropyenergypßwereknownasafunctionofa.Thesetwopossibilities H). Aconvenientmeasureofthetemperatureanisotropyisrmsdifferencein By usingequation(3.21)forZT=Dawethenfindtherelatedestimate(1.6)(AH/ a Hamiltoniangoverningtheanisotropymotion,thispositivedefinitepropertymeans (3.9d). Althoughpß=he~isprimarilyadynamicalquantity,inthatitcorresponds gives ßijßij«-4-F<^(pß/pv)sowithAInT AT/T wededucefromequation(3.24) where Tjarethetemperaturesonprincipalaxesoftemperaturedistribution can assert When theanisotropyissmall,thisnotagood estimate,butthenequation(2.23) eDß, etc.,andinequations(3.3)wefind Returning to the adiabatic case where rj = 0 but possibly av 7^ 0, we see that the time dependence of the anisotropy energy is governed by the formula dh/dt = —dL/dt which, from equations (3.18) and (3.19), can be written Dh = —hDa — (ÜT^iDo^o-ijO-ije3* . (4-i7) During any expansion then, when Da > 0, this shows that D(hea) = D(pße^) < 0 . (4.is) Thus, quite generally pß = he~Za decreases more rapidly than e“4°, and therefore more rapidly than the energy of isotropic radiation such as p7 or pv. Since pß/pv is thus a monotonically decreasing function of /, we see from the estimate (3.25) that the oscilla- tions of the temperature anisotropy (A ln T) must also have a decreasing amplitude in an expanding universe of the type considered in this paper. The oscillation amplitude of the Hubble constant anisotropy kH/H must also decrease monotonically. e) Adiabatic Invariants The preceding rigorous limit showing that pß decreases faster than T4 is not a good estimate, since we found that pß & T* when av = 0, and our small anisotropy example of equation (4.41) with T oc t~l¡2 showed pß œ T5. If the ß oscillations were rapid com- pared to the expansion rate one could easily derive such estimates from equation (4.17). For the present we adopt this (false) hypothesis, which means that we are studying the anisotropy Lagrangian (3.18) while ignoring the expansion equation (3.21). The reader may regard this as an evolutionary step in the process of becoming acquainted with this Lagrangian: after the simplest case where a(/) is assumed constant (which we treated in §§ ITLd-e)) the case next at hand is that where a(t) is slowly changing. We write equation (4.17) for pß = he~Za in the form AflnCp^«.)] =2 (4.19) r where we have set (d/dt) = (Da) (d/da). The ratio of the values of p;je '“ at two different epochs is then obtained by integration: ( P0e6a) 2(ppV/pß)da . (4.20) (ßßC 6a ft-x: The value of this integral is, by definition, not changed if we replace the integrand by its average value over the interval. By this well-known argument, then we are led to study the equation obtained from equation (4.19) by replacing its right-hand side with a smoothly varying function of a which has nearly the same average value as 2(pvV/pß) over long intervals. © American Astronomical Society • Provided by the NASA Astrophysics Data System 1968ApJ. . .151. . 431M a a 6a _1 ZaSa 2 5a 2 l2 2 = ba 28104 2ha 6a 00 No. 2,1968ISOTROPYOFUNIVERSE453 fixed a.Usingthisestimateinequation(4.19)showsthatpß^isanadiabaticinvariant, that is,itisaquantitywhichapproachesconstancyinthelimitßoscilla- the virialtheorem,(pV)=%pßwhenaverageistakenoverßoscillationat sharp collisionswiththewallswhenpV/pßmayapproachbrieflyitsmaximumvalue,1. exponential behavior,thenthereisagainasimple,butdifferent,average,namely, tions arerapidcomparedtochangesina. of pß,sothatinthesmallanisotropycasewherepßisquadraticßandDßwehaveby more accuratelyconstantinthelimitofrapidlarge-amplitudeßoscillationscouldbe In thiscase,then,theright-handsideofequation(4.19)isnegligiblewhenaveraged walls spendsmostofitstimeinfreemotionwherepV/pß<^l,andverylittle difficult large-ßcaseisdeferredtothenextsection,butsincewehaveafullsolutionfor small ß(ifp&=0also),wemaystudythebehaviorofpß^inthisexample.Fromequa- obtained frommorepreciseestimatesofthesmallquantity(ppV/pß). over ßoscillations,andp^isanadiabaticinvariant.Adiabaticinvariantswhichare variable. (Infactitisconvenientphysically,as wellasmathematically,tothinkof rapid ßoscillationinaslowlychangingpotential, theexpansionequationfora(t)re- amplitude ßij(t)motionssufficienttojustifytheestimate(pV/pß) ßw ß+) Thus V/Vw — e~( ~ « 1 implies that the walls are moving out with a velocity 2 /2 1/2 8W' « 2. The system point (ß+ß~) which has velocity (/V + #_ ) ~ 2 can there- fore never catch up with the receding wall unless ß+' « 2 so ß- is near zero. We must therefore turn to the exceptional case where ßJ = 0. If ßJ = 0 while ßw and ß+ are large, equation (4.26c) gives (i/V)2 = 1 - V^V (4-28) and equation (4.26b) gives i/V = i - v^v. (4-29) If we further assume that V^V — pvV/pß<&l and choose ß+ positive we can then ß ßw ß+ compute ßw' — ß+ = —V-uTW = —e~^~ ^ which has the formal solution e ~ — ißw ß+) l (const. — Í2) showing that e~ ~ — Vw~ V cannot remain small compared to unity 1 1 for an Q interval as long as AQ = (VW~ V)~~ which, although large, is finite. Thus when initially ßJ = 0 and ß+' « 2, the system point (ß+,ß-) will eventually gain on the moving wall which also has ft/ « 2 initially, and approach it. But equations (4.28) l 2 and (4.29) are valid also when Vw~ V is not small, and show then that (|ft/) = (èft/) from which follows the strict inequality | ft/1 > ft/ for all values of ft/ except ß^' — 2 which we have just seen cannot persist, and ft/ = 0. Thus starting from the condition VvTW — e~(ßw~ß+) when in addition to (pv/p) <$C 1 and (p7/p) <3C 1 one further assumes \ß\ £!+ © American Astronomical Society • Provided by the NASA Astrophysics Data System 1968ApJ. . .151. . 431M a 3al2 456 Here itisclearthattheterm^e~negligiblesoonaftera=0(whenpp^),sowe .1945,TheMeaningofRelativity(2ded.;Princeton,NJ. :PrincetonUniversityPress),AppendixI. ———. 1950,Rev.Mod.Phys.,22,153. from theU.S.NationalScienceFoundation(SeniorPostdoctoralFellowship1966- Then onerequiresthesolutionofequation(4.31)withinitialconditionsß±=0.Itis radiation (e.g.,photons)firstbecomescollisionlessduringamatter-dominatedphase. simple result may equivalentlyomitthecorrespondingterminequation(4.30)tofind Flanders, H.1963,DifferentialFormswithApplicationsto the PhysicalSciences(NewYork:Academic Alpher, R.A.,andHerman,C.1948,Nature,162,774. ment andconclusion.MythanksalsogototheCambridgeUniversityDepartmentof This rapiddecreaseofHubbleconstantanisotropyisnotsharedbythetemperature which impliesdß±/daœe~.Thusinmatter-dominateduniversesweobtainthe Gamow, G.1948,Nature,162, 680. Field, G.B.,andHitchcock,J.L.1966,Phys.Rev.Letters, 16, 817. Applied MathematicsandTheoreticalPhysics(D.A.M.T.P.)fortheirhospitality, ideas andencouragementledmetobeginthesestudies,R.A.Matzner,who with particularappreciationtoJ.FaulknerandP.A.Strittmatter,whoseprovocative or equivalentlyequation(1.9). anisotropy sinceß±=const,isclearlyasolutionofequation(4.31).Thus(AT/T)oc Fowler, W.A.1967,paperpresented attheSecondInternat.Conf.HighEnergyPhysics andNuclear Dicke, R.H.,Peebles,P.J.E.,Roll,G.,andWilkinson, D. T.1965,ApJ.,142,414. Cartan, E.1946,LeçonssurlagéométriedesespacesdeRiemann(2ded.;Paris:Gauthier-Villars),chap.vii. Bahcall, J.N.1964,Phys.Rev.,136,B1164. to J.WeberoftheUniversityMaryland,andfromU.K.ScienceResearchCouncil. assistance ofJ.ElderD.A.M.T.P.Financialsupportismostgratefullyacknowledged ties providedbyagrantfromI.B.M.(U.K.)weremadeavailablethroughthegenerous to S.W.HawkingandM.J.Reesfortheirmanyusefulconversations.Computerfacili- carried outthecomputationsreportedinfiguresandcorrectedanerroneousargu- Saslaw, andP.A.G.Scheuerformanyinformativestimulatingconversations,but Einstein, A.1917,Sitzungsber.preuss.Akad.Wiss.,p.142, translatedinThePrincipleofRelativity,by Eckart, C.1940,Phys.Rev.,58,919. Chernikov, N.A.1963,ActaPhys.Polon.,23,629. Chapman, S.,andCowling,T.G.1939,TheMathematicalTheoryofNon-UniformGases(Cambridge: T° =const,isapossiblebehavior.Aparticularcaseofinterestwhensomeform Ehlers, J.1961,Ahh.Akad.Wiss.Mainz,No.11. Doroshkevich, A.G.,Zel’dovich,Ya.B.,andNovikov,I. D.1967,Zh.ET.F.Pis'ma,5,119(trans. Curtis, A.R.1950,Proc.Roy.Soc.,A,200,248. 1967), fromaU.S.NationalAeronauticsandSpaceAdministrationgrant(NsG-436) v © American Astronomical Society •Provided bytheNASA Astrophysics DataSystem Press). I wishtothankmycolleaguesD.W.Sciama,E.A.Spiegel,G.F.R.Ellis,C. A. Einstein,H.Lorentz,Weyl,andMinkowski (London: Methuen,1923;reprintedNew Structure, HebrewUniversity, Jerusalem,Israel,March1,1967. York: Dover). in J.E.T.P.Letters,5,96). Cambridge UniversityPress). 2/23a3 (AH/H) =|(iS+ßJ)cc-ocT.(4.32) +e Sal2 ß± =Kdß±/da)o(l—e~)(4.33) CHARLES W.MISNER 2 2 dß ,,dß +± da REFERENCES Vol. 151 (4-31) No. 2,1968 ISOTROPY OF UNIVERSE 457 Harrison, E. R. 1965, Ap. 142,1643. • . 1967ô, Phys. Rev. Letters, 18, 1011. . 1967Ô, preprint. Hawking, S. W. 1966a, Proc. Roy. Soc., London A, 294, 511. • . 1966&, ibid., 295, 490. ■ —. 1966c, Ap. /., 145, 544. Hawking, S. W., and Tayler, R. J. 1966, Nature, 209, 1278. 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