Ideal observational cosmology

PHYSICSREPORTS(Review Sectionof PhysicsLetters)124, Nos.5 & 6 (1985) 315—417. North-Holland, Amsterdam

IDEAL OBSERVATIONAL COSMOLOGY

G.F.R. ELLIS

DepartmentofAppliedMathematics,Universityof Cape Town,Rondebosch,CapeTown77(X), SouthAttica

S.D. NEL

Departmentof Mathematics,Universityof SanFrancisco,Cal~fornia94117, U.S.A.

R. MAARTENS

Depamnen:ofAppliedMathematics,Universityofthe Wftwatersrand,Jan SmutsAvenue,Johannesburg,SouthAfrica

W.R. STOEGERVaticanObservatory,I.C012() Cilia Del Vaticano,Europe

and

A.P. WHITMAN

VaticanObservatoryandPon:ifIca ljnjversidadaCattiuica deRio deJaneiro,PUC/RJ,RanMarquesde&io Vicente293,22.451RiodeJaneiro,Brazil

ReceivedOctober1984

Content~:

1. Introduction 317 4.3. Imagesize andshape 3314.4. Numbercountsandgalaxy clustering 333

Part I: Ideal CosmographicObservations (G.F.R. Ellis, 4.5. Identifications 334R.Maartens,S.D. Nel) 320 5. Observationalquantitiesandcosmography 334

2. Cosmologicalmodelsandobservations 320 5.1. Observationalquantities 3342.1. Space—time 320 5.2. The space—timeon thepastnull cone 3362.2. Matterandradiationdescription 320 5.3. Observationalrelationsandcosmography 3372.3. Thenatureof cosmologicalobservations 321 6. An example: Isotropic observationsand spherical sym-

3. Observationalcoordinates 324 metry 3413.1. Coordinatechoiceandnull geodesics 324 6.1. Isotropicobservations 3413.2. Themetric 326 6.2. Sphericallysymmetricspace—times 343

4. Galacticmeasurements 328 6.3. FRW universemodels 3454.1. Redshiftsand4-velocities 329 7. Conclusionto part I 3474.2. Imageintensities 331

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IDEAL OBSERVATiONAL COSMOLOGY

G.F.R.ELLIS

DepartmentofAppliedMathematics,Universityof CapeTown,Rondebosch,CapeTown7700, SouthAfrica

S.D.NEL

Departmentof Mathematics,Universityof SanFrancisco, California 94117, U.S.A.

R. MAARTENS

DepartmentofAppliedMathematics,Universityofthe Witwatersrand,Jan SmutsAvenue,Johannesburg,SouthAfrica

W.R. STOEGER

Vatican Observatory,1.00120Cittâ Del Vaticano,Europe

and

A.P. WHITMAN

Vatican ObservatoryandPontifIcaUniversidadaCatólicadeRiodeJaneiro,PUC/RJ,RuaMarquesdeSâoVicente293, 22.451Rio deJaneiro, Brazil

NORTH-HOLLAND-AMSTERDAM

G.F.R. Ellis eta!., Idealobservationalcosmology 317

Part II: Ideal Cosmological Data (S.D. Ne W. Stoeger, 12.2. D(wo, z*) astheminimal dataset 389G.F.R. Ellis, A.P. Whitman,R. Maartens) 348 13. Propagatingthesolution off C-(p) 390

8. Introductionto partII 348 13.1. Integrationoff C(p) in theanalyticcase 3919. Settingup theproblem 350 13.2. TheBondi’-Sachsmethod 393

9.1. TheNewman—Penroseformalism 350 13.3. Integrationto thefuture of C(p) 3939.2. Observationalcoordinates 351 14. Powerseriesapproximationsandtheangulardependence9.3. Thenull tetrad 353 of thefinal data 395

10. Thecentralconditions 357 15. Sphericallysymmetricspace—times:anexample 39610.1. Deriving thecentralconditions 357 15.1. Simplification of thefield equations 39710.2. Observationalangularcoordinates 366 15.2. Theobservationalintegrationscheme 40110.3. Central conditions on the fluid 4-velocity com- 15.3. Observationalintegrationwith FRWinitial data 403

ponents 368 16. Conclusionto partII 40711. Cosmologicallysignificantobservations 370 Acknowledgements

11.1. Observationsof distantgalaxies 370 References 40911.2. The maximaldataset 374 Appendix A: Central limits for the metric and 4-velocity11.3. Thebackgroundradiation 375 components 41111.4. Observationalspace-times 376 AppendixB: NP spincoefficients andcurvaturecomponents 414

12. Solvingthefield equationson thelight cone 377 Appendix C: Transformation properties of NP spin12.1. Themodelon thelight cone 377 coefficientsandcurvaturetensorcomponents 415

Abstract:Following Kristian andSach’sdirect observationalapproachto cosmology,this paperanalysesin detail theinformation that canbe obtained

from idealisedastronomicalobservations,firstly in thecosmographiccasewhen no gravitationalfield equationsareassumed,andsecondlyin thecosmologicalcasewhenEinstein’s field equationsof GeneralRelativityare taken to determinethe space—timestructure.It is shownthat if idealobservationsare available, in the cosmographiccasethey are insufficient to determinethe space—timestructureon the past light coneof theobserver;howeverin thecosmologicalcasetheyarepreciselynecessaryandsufficient to determinethespace—timegeometryon thelight coneandinits causalpast(atleastdownto wherecausticsorcuspsfirst occur).Therestrictedcaseof sphericallysymmetricspace—timesisanalysedin detail, andnecessaryandsufficientobservationalconditionsthatsuchaspace—timebe spatially homogeneousareproven.

A subsequentpaperwill examinethesituationof realisticobservationaldata.

1. Introduction

The aim of this series of papers is to characterise in detail the way in which cosmologicalobservationscan be used to directly determine the geometry of cosmologicalspace—time. In carryingout this programme, we are following the ideasof the beautiful paper by Kristian and Sachs[1], butdiffer from them in that they examinedin detail the way this could be done near our present space—timeposition (they obtained power-series solutions to the field equations and observational equations);whereas we will aim to determine what could be done, in principle, to determine the geometry atsubstantial distancesfrom our present space—timeposition. This difference in emphasisenablesus toconsider in a new way, the question of what is and what is not decidablein cosmologyon the basisofastronomicalobservations;and sowe can consider what the limits to verification in cosmologyare (Ellis[4]; and paper II [2]). Our investigation is geared toward bringing cosmologically interpretableastronomical observations into detailed confrontation with cosmologicaltheory without introducing apriori assumptionsabout its geometry, such as assumingthat space—timeis isotropic and homogeneous.The underlying theme, therefore, is that as far as possible,cosmologyshould be a directly observation-ally based subject. The philosophy of this approach has been described at length and justified inKristian and Sachs[1], Ellis [4, 20, 31, 51, 52]. A similar approach hasbeen used by Dautcourt [53,54];the methodsusedand the results obtained are similar but not identical to ours.

The present paper (paper I) considers what can be found out, in principle, on the basis of ideal

318 G.F.R.Ellis eta!., Idealobservationalcosmology

observationsaimedat directly revealingthe structureof distant regionsof the universe. In part I, weproceedwithout assumingwe know the dynamic laws determiningthe very large-scalestructureofspace—time.Thus it is assumedhere that all observationalquantitiescan be measuredto indefiniteaccuracy,andon thisbasiscarefulconsiderationis given to what can be deducedaboutthe space—timemetric and matter contentfrom theseobservationsalone (that is, with minimal further assumptions).This is done first in the context of general cosmologicalobservations,and then in the context of asituationwheresuchobservationsturnout to beisotropic. In part II, weexaminein detailthe way theseidealobservationscan beusedto directlydeterminethe space—timestructurewhenit is assumedthatthelarge-scalestructureof space—timeis governedby Einstein’sfield equations.Thisagainis donefirst for thecaseof generalobservations,andthenfor isotropicobservations.Thesecondpaperin theseries(paperII[2])considerspracticallimitationson whatonecanin factmeasure,andsomeimplicationstheserestrictionshavefor theaimsof observationalcosmology(someof theseissueshavingalsobeendiscussedin [4]).Thethird paper(paperIII [3]) looksattheseissuesin thecontextof “nearlyRobertson—Walker”space—times.

The limits on what can be determinedby direct observationareintimately tied in with the way theobservationsarecarriedout, andtheway the resultsof the observationsareanalysed;andthisquestionis consideredin a companionset of papers[5,6], which leadto someproposalson the bestway to carryout someof the proceduresin an observationalcosmologyprogramme.

The limitationson what onecan measureandverify in cosmologyhaveseriousimplicationsfor afullcosmological theory, that attempts to relate theoretical principles and astronomicalobservationstogetherinto asatisfactoryview of our understandingof the natureof the universe.We will not takeupthesebroaderissuesin this seriesof papers;nevertheless,we believethat this serieshasusefulinsightsto offer, which shouldeventuallybe incorporatedin suchan overall cosmologicaltheory (cf. [52]).

In this paper, then, we consider the kinds of cosmologicalobservationsthat can be made,andexamine in detail what featuresof the universe can in principle be determineddirectly by idealelectromagneticobservations(that is, by radio, infra-red, optical, ultra-violet, X-ray, or y-ray obser-vations)of distantobjects.We regard the information that is obtainedby otherkinds of observations(element abundances,agesof star clusters, backgroundradiation isotropy, and so on) as indirectevidenceabout the structureof the universe,for thesedo not directly restrict the detailedspace—timestructure,but ratherput integralrestraintson allowablecosmologicalmodels.The relationbetweentherole given to suchindirectevidence,andto the direct observationalevidenceaboutdistantplacesin theuniverse,is oneof thefeaturesthat needsexplorationin afull cosmologicaltheory (cf. thediscussionsin[4], [31], and[52]; it will not be pursuedhere).

In thesepapers,we assumethat locally valid physicallaws hold everywherein the universe— at allpoints of space—time— and that the universemay be modelledwith sufficient accuracyby the usual4-dimensionalmanifold-with-metric(M, g), with tensorfields or distributions representingits matterandradiationcontent.Among otherthings,thisinvolvesassumingthat the orderingof matterinto stars,star clusters,galaxies,clustersof galaxies,andintergalacticmatteras determinedby local astronomicalobservationsalsoholdsat largedistances,in such away that theconventional“fluid approximation”forthe matterandthe radiationcontentof the universeholdson a“suitably large” scale.This meansthat awell-definedaverage4-velocity Ua, propermassdensityp, andconservedstress-energytensorTabcan beusedto describeaccuratelythe matterandradiationcontentof the universeon ascaleon whichgalaxiesmaybe regardedas particlesof the cosmologicalfluid, with the U” definedby averagingoverclustersofgalaxies.

If de Vaucouleurs’further hierarchicalordering of the clustersof galaxies into superclusters,and

G.F.R.Ellis et aL, Idealobservationalcosmology 319

these,in turn, into super-superclustersis correct, it is possiblethat no suitable averagingscaleexists(see, e.g., de Vaucouleurs[16, 55]). Any attempt to constructhierarchical models using the fluidapproximationcould thereforebe fundamentallymisconceived.A kinetic theory approach(see,e.g.,Stewart [56]) would probably also run into major problemsin this case.In fact, whether any fullyhierarchicalmodelcouldbe constructedwithin the framework of generalrelativity would dependto alarge extenton whetherthe apparentlysevereproblemsconcerningthe definition of a stress-energytensorcould be overcome.

Commonto all but the most eccentricinterpretationsof the observationaldata is the implicitassumptionthat theuniverseis large— so large that on a cosmologicalscalewe are unableto moveoffour local galactic world line. The spatial distancesand time scales involved are so greatthat weeffectively view the universefrom a single space—timepoint — “hereandnow”. This restrictionimposesfundamentallimitations on what is empirically decidablein cosmology[20].

Finally, we assumethat thereare no singularities,causticsor other unboundedbehaviourwhichdirectly affect the largescalespace—timestructurein the regionwe areconsidering.As far asthe matterand radiationcontentof the universeis concerned,this is just an assumptionconsistentwith the fluidapproximation.But we also in particular restrict attention down our past light coneto the limitingredshiftbeyondwhich eitherwecannotperformastronomicalobservationsor therearenull caustics—

points wherethe null geodesicson the light cone crossor becometangent.Eventually we hope togeneraliseourtreatmentto beyondnull causticsthat mayoccuron ourpastlight cone;but that involvestechnically difficult unsolvedproblems.

In part I of thispaper,weshall find out that cosmography(that is, analysisof the observationaldatawithout the useof specific gravitationalfield equations)can give usrather little information abouttheuniverse,evenif werestrictourambitionsto understandingthe space—timestructureonly in the vicinityof ourpastlight cone.In particular,we areunableto determinecosmographically(by directobservations)the relationshipbetweenthe radial coordinatey down our light coneandthe distance,as measuredforexampleby theredshift z, to anyobjectweobserve.Further,we areunablewithoutatheory of gravity todeterminethe embeddingof our 3-dimensionalpast light cone in the 4-dimensionalspace—timemanifold.

As aparticularcaseof thisobservationalindeterminacy,wefind that we cannotprove on the basisofobservationsalonewhetheror not space—timeis sphericallysymmetricaboutourposition— evenif alltheseobservationsare isotropicaroundus. Evenif we go fartherandmakethe importantassumptionthat the universehas a Friedmann—Robertson—Walker(FRW) geometry,we cannotdetermineobser-vationallywhetherthe spatialsectionshavepositive,zeroor negativecurvature(Weinberg[41]).Nor doisotropicobservations,togetherwith the assumptionof sphericalsymmetry,proveFRWgeometry.

In part II, it will be madeclear how ideal observationscan give a rathergood descriptionof theuniverse,when we assumeEinstein’s field equationsto be valid, and makea “no-news” assumptiondeterminingthe behaviourof the universemodel outsideour pastlight cone(suchassumptionsareinevitablein any attempt to predictthe future, unlesswe live in a “small” spatiallycompactuniverse,seee.g.[20, 31, 52]); but furtherpapersin the seriesshowthat realisticlimits on what onecan in factobservedrasticallycurtail what one can hope to prove directly from observations.Neverthelessthediscussionof the caseof “ideal” observationsis useful in that it providesa sketchof the maximumwecould everhopeto attain, andso setsthe stagefor a discussionof how to utilise best the realisticallyattainablesets of observations,and provides the basis on which we can evaluatewhat is in factachieved.


Part I: IdealCosmographicObservations

G.F.R. Ellis, R. Maartens, S.D. Nel

2. Cosmologicalmodels andobservations

We assumethat ordinary local physicsholdseverywherein the universe,both becausewithout thisassumptionit is virtually impossibleto extract any useful information aboutdistant regionsof theuniversefrom observations,andbecausethereis someobservationalevidenceto supportthis view (seee.g. Dyson [7], Barrow [8]). In the first part of this paper,we considerthe situationwhenwe do notassumewe know the theory which determinesthe large-scaledynamicaldevelopmentof space—time,that is, we takea cosmographicview of the data(seee.g. Robertson[10],Rindler [11],Weinberg[121).We do soboth becauseof the manytheoriesof gravity now available,andbecauseof the legitimacy ofthe cautionaryviewsexpressedby Schücking[9] andothers(seee.g. Weinberg[12] chapter14). In thesecondpart, we shall take the further stepof assumingthat the theory which correctly predictsthisdevelopmentis known (andfor want of solid proof that anymore complextheory is needed,we willadoptthe conventionalcosmologicalviewpoint by takingGeneralRelativity to be that theory).

We will assumethat our presentunderstandingof the “local” astronomyof the universe— theorderingof matterinto stars,starclusters,galaxies,clustersof galaxies— is reasonablyaccurate,andthatessentially the same ordering holds at large distancesfrom us. This view is also supportedbyastronomicalobservations,andis the basisof mostcurrentinterpretationsof the cosmologicaldata.

On the basisof theseassumptions,the ingredientsof a cosmologicalmodel are (Ellis [13], Ehlersand Sachs[14]):

2.1. Space—time

Thespace—timeconsistsof a manifold M with metric g (seee.g. Hawking andEllis [15]for detaileddiscussionof these assumptions).Through the relationsds2= gq dxt dx’, the metric determinestheresultsof space—timemeasurements;its derivativesdetermine(in the usual way, see e.g. [12, 15]) theconnectionandcurvatureof space—time.We takethe normal form of g as diag(—1, +1, +1, +1).

2.2. Matterandradiation description

One must specify the material and energycontent of the space—time,representingthe matter—

orderedinto galaxies,clustersof galaxies,andintergalacticmatter— andradiation in the universe.Weassumethis is further ordered(see e.g. [13]; but note that this assumptioncould be wrong, cf. deVaucouleurs[16]) so as to determinea well-defined (cosmological)average4-velocity of matter U’~,

propermass-densityp, and stresstensorTa,. The latter implies the total energydensity ~s,isotropicpressurep, and possibleother stresscomponentsdue to matter and radiation; the descriptioniscompleted(for dynamicalpurposes)by equationsof state* limiting andrelatingthe variablesdescribingthe matter and radiation components.**The averagingscale involved in defining thesedynamicalvariables— andso the space-timemetric of cosmology— is that of clustersof galaxies.For the purposeof

* We include“energyconditions” [15]in this concept.** Theserelationswill usually imply theconservationequationsT~

5= 0, but may not alwaysdo so (e.g. in Hoyle’s theorytheserelationsare

modified).

G.F.R.Ellis etaL, Idealobsert,àtitinalcosmology 321

assessingobservationsonewill needin addition a descriptionof the luminouspropertiesof the matterdistribution:thesurfacebrightnessandspectraof sources,the numbersof sourcesof differentkinds andsizes,and the time-evolutionof thesequantities.

The usualdynamicaldescription[12,13] is to assumeT~hasthe “perfect fluid” form,

T~=/2u~ub+p(g~+u~ub),U”Ua=~l, (1)

with equationsof staterelatingthe quantitiesp, ~zandthe pressurep. More specifically, T~,is takentobe composedof a radiation component (for which p~= P~rad/

3� 0) and a pressure-freemattercomponent(Pmat = 0, !Lmat >0) which areseparatelyconservedatmost times; atlate timesin the usualpicture, only the latter will be dynamically important. We shall usethis “matter plus radiation”description,in particularassuming(1) is valid, in thesepapers.Use of this descriptionfor the radiationrelieson it beingnearlyisotropicabout the 4-velocity u”; this conditionappearsto be satisfiedat leastnearour presentspace—timeposition.

Given this information,the space—timeis filled with auniquefamily of flow lines, the integralcurvesof the velocity vector U”, representingthe world lines of “fundamentalobservers” (typical galacticobserversmoving with the averagemotion of the matter in the universe). The metric tensorgdeterminesproper time r alongtheseworld lines; if theyaregiven in termsof local coordinatesx’ by

= x’(r), thenthe 4-velocity is given

u~dxt/dr. (2)

2.3. Thenatureofcosmologicalobservations

It follows from theseinterpretationsthat the universeis large relativeto our galaxy; thereforewe areessentiallyrestricted,on a cosmologicalscale,to observationsmadefrom one point on oneworld line.Not only arespatialdistancesso largethat we areeffectively unableto moveoff theworld line C of thelocal clusterof galaxies,but the scaleof time variation of cosmologicalquantities(galacticredshifts,themicrowavebackgroundradiation temperature,the Hubble constant,and so on) is such that we areeffectivelyonly ableto observetheuniversefrom onepointq (“here andnow”) on theworld line C (thewhole recordedhistory of mankindcorrespondingto a spatialdistanceof only afew kiloparsecs).

The metric tensorg determinesthe local light cones which limit possible observationsin theuniverse.We can, at q, receiveinformation-bearingsignals from the universeonly on future-directedtimelike or null curvesfrom eventsin our pastto q (fig. 1; andcf. Heckmanand Schucking[17],Hoyle[181,Hawking andEllis [19],Ellis [13,20]). Onecan distinguishfour differenttypesof observationsthatcan be made:

(i) Local physicalexperiments.These observations(which determinethe local physical laws thatunderlie everyday life) certainly contain somecosmologicalinformation (e.g. they imply limits on thebackgroundradiation present; for example, local thermodynamicscan only take the form it doesbecauseof the dark night sky). They might conceivablyimply detailed informationaboutthe universe(cf. Harrison’sdiscussionof the “bootstrap”approachto cosmology[21],and[51]).Howeverwedo notknow how to decodesuch information, exceptfor that implied by measurementsof the backgroundradiation(which we include in (iii) and(iv) below).

322 G.F.R.Ellis etal., Idealobservationalcosmology

‘Ctime

~space

q

C(q) post light conegalaxyworld—lines

, ‘

Fig. 1. Theeventq (“here andnow”) is thepresenttime on theworldline of our clusterof galaxiesC. At q, we canreceivemessagestravellingonnull geodesicsgeneratingourpastlight coneC(q),or on timelikelinesoriginatinginsideC(q), from thepointof origin of themessageup to q.Thuswe canonly seeothergalaxiesat theinstantwheretheir worldlines crossC(q).

(ii) Local geophysicaland astronomicalobservations.These are distinguishedfrom (i) becausewecannotcarryout repeatableexperimentsrelatedto them; rather,we can merely observewhat is there(the earthhasone moon, the sun a particular chemical composition,andso on). Theseobservationscarry valuable information aboutour presentsituation, which placelimitations on the history of ourregion of the universe (the neighbourhoodof the world line C) very long ago. They thereforesetimportantconstraintsthat aviable cosmologicalmodelmustsatisfy; particularlysignificantsuchfeaturesare,

* local elementabundancesand particle abundances(implying in particular the values of “con-served”quantities:charge,leptonnumber,baryonnumberatrecenttimes,. .

* estimatesof the ages of local objects (stars, star clusters, meteorites,the Earth, radioactiveelements,etc.);

* estimatesof the local matterdensityin the universe.Howeversuchobservationsdo not, without additional assumptions,conveyinformation aboutdistantregionsof the universe(i.e. regionsfar away from the world line C; see [4] for furtherdiscussion).

(iii) Extragalactichigh-energymassiveparticles.Somecosmicrayscouldcomefrom distantregionsinthe universe,and so conveyusefulinformation aboutthesedistantregions. Howeverinterpretingtheobservationspresentsseriousproblems,for one hasto first establishthat the observedparticlesare ofextragalacticorigin (cf. e.g.BurbidgeandBrecher[22]),andthenrelatethem to specificsourcesdespitethe interactionsthat havetakenplacebetweenthem andgalacticandextragalacticfields andradiation.With the kinds of resolution that are likely to be obtained,suchobservationsare likely to say muchmoreaboutastrophysicalprocessesin the universethanaboutthe universeitself.They can, like (ii), beregardedas observationswhich do not directly give detailed information aboutdistant regions, butwhich setconstraintsto be satisfiedby a universemodel.

(iv) Astronomical observationsof distant objects and of background radiation. These are “cos-mological observations”proper; they might in the future include gravitational wave and neutrino

G.F.R.Ellis eta!., Idealob.~èrvationalcosmology 323

observations,but currentlywe maysafelyrestrictourattentionto electromagnetic-waveobservationsofdistantobjectson our past light cone, andof backgroundradiation.We neglectherepossiblewavebackscatteringeffects(seee.g.Harrison[23]),asothercurvatureeffectswould almostcertainlybecomenoticeablebefore theseeffectsbecameimportant; we will also, for the present,ignore the effectsofdispersion and absorption in the intergalactic medium [23], although this point will have to bereconsideredat a later stage.Thereforewe accept the usual assumptionthat the geometricopticsapproximation is adequatefor describing the processesinvolved in cosmologicalobservationsbyelectromagneticradiation(refs. [13,14], Ehlers[24]).

It is only through suchobservationson our pastlight conethat we currently,or in the foreseeablefuture,candirectly obtain high-resolutioninformationaboutdistantpartsof theuniverse.This seriesofpaperswill thereforeconcentrateon analysisof suchobservationsmadeon thepastlight coneof thespace—timepoint q; howeverfor somepurposesof discussionand presentationit will be interestingtoconsiderwhat couldbe determinedif wecould makeobservationson an openinterval I of ourworldline C which containstheeventq, and wewill sometimesconsiderthis casealso.

Given a cosmologicalmodel as outlinedabove,standardtheory showswhat observationscould bemade,andpredictswhat the resultsof suchobservationswould be (the measuredredshifts,observedangles,apparentmagnitudes,galacticnumbercounts,andsoon; seee.g.[1, 13, 14] for thegeneralcaseand Sandage[25],Weinberg [12]for the caseof Robertson—Walkeruniverses;and [18,5, 4, 6] fordiscussionof how the theory can be madeto correspondquite closely to the actual measurementsmade).Whatwewish to characterisenow, is the inversequestion:giventheresultsof suchastronomicalobservations,in whatwaydo theylimit the space—timeandits mattercontent?(fig. 2). Theotherkindsof

——-——————

I COSMOLOGICAL MODEL DIRECT OBSERVATIONS ISpace—time Observational mop redshifts

I ILMattsr content ! Inverse? ~ ~I INDIRECT OBSERVATIONS

constralnts o~undancesa’es IdensIties J

L — ——framework of assumed physical lows and astronomical conditions — —‘

Fig. 2. The aim of observationalcosmology:to determine thespace—timeand itsmatter content,from theastronomicalobservations(using indirectobservationsto confirm the model thus arrived at). This is the inverse of the usual procedure,where cosmologicalmodels areused to predictobservational relations, which arethen comparedwith actualobservations.

observations(see(iHiii) above)canbe regardedas settingadditional constraintson thecosmologicalmodel (cf. [2]),whenthis questionis answered.

We considerthis questionin detail in section5, after looking in turn at thecoordinatesto be used(section3) andthe informationwhich canbe obtainedfrom galacticobservations(section4).

324 G.F.R. Ellis eta!., Ideal observationalcosmology

3. ObservatIonalcoordinates

3.1. Coordinatechoice andnull geodesics

In order to carry out a systematic investigation, it is convenient to introduce observationalcoordinates(cf. Temple[26],Kristian and Sachs[1]).We basethesecoordinateson theworld line C(representingthehistoryof our galaxy).We assumethis is a geodesicat the time when theobservationsaremade(whenwe may assumep 0). The coordinates{x1 } = {w, y, 0, ~}aredefinedin the followingway:

(i) w usx°.Let thesurfaces{w = constant)be thepastlight conesof theeventson C. Normalisew bythecondition: w measurespropertime alongC, i.e. wic = Tic; andchoosew = w

0 to correspondto theeventq (“hereandnow”). Then w is completelydeterminedwhenw0 hasbeenchosen.It is a functionwhich is C°,andis C’ almost everywherewithin asimply convexnormal neighbourhoodof q; it is notC

1 on C (the verticesof the light cones),andwill be multiple-valuedaftercausticpointsoccuron thelight cones.

Thenull geodesicvectorfield generatingthe ruling geodesicsof theselight coneswill be written as

k = 8/öv’~’k’ = dx’/dv, where k1 us w1, k~k1= 0. (3)

This definition necessarilyimplies [27]that k is thehypersurface-orthogonal(k~1,= kb;a) null geodesic(ka;bkb = 0)vectorfield orthogonalto thenull surfaceson which w is constant(w,1k’ = 0), andwith v anaffine parameteralongthegeodesics.Further,this definition plus thecentralconditionimplies

k~u~=w,,u1 kaU”iCl. (4)

(Note that thederivativeimplied in this central limit is well-defined,eventhoughw is not C’ at C; andthat (4) showsthat k is past-directed,becauseu is future-directed.)This showsthat the affineparameterv is uniquelydefinedon thenull geodesics,if wespecify that v = 0 on theworld line C (sotheeventq isgivenby w = wo, v = 0).

cii) 9 us x2, ~ us x3. Let the geodesicsgeneratingthe null conebe given by {0, q~constant)in thesurface{w = constant}, so ~ = 0 = 4s,~k1.We normalise 9, 4’ by the centralcondition that they arestandardsphericalcoordinatesbasedon aparallelypropagatedorthonormaltetradea (u = e

0, u e~= 0,e~e,. = ô~~)alongC. Thenonehas,Vueaic= 0; and

lim{(ds2/v2) } = dir usdO2+ sin2 0 dqS2.

v—~.0 w~constantV= constant

The freedomin thesecoordinatescorrespondsto a rigid rotation of the triade~at one point q0 on C.

Thecoordinates0, 4’ arejust sphericalcoordinateson thecelestialspherewith respectto the(physicallynon-rotating)referenceframee~.

(iii) y usx’. The coordinatey measuresdistancedown the null geodesics,and so representsboth

G.F.R. Ellis ci at, Idealobservationalcosmology 325

spatial distancefrom C, and time differencefrom q. There are various choicesof y that might besuitablefor different purposes,for example,(1) y = v, the uniqueaffine parameterdown the null geodesicsthrough C determinedby the centralconditionson C (vic = 0, U’~’kaiC= 1);(2) y = r, areadistancedown thenull conesfrom C [1] (alsocalled“correctedluminosity distance”in[1] ,and“observerareadistance”in [13]).Thisquantityis definedby eq.(7) below,andis furtherdiscussedin section4.3;(3) y = z, galacticredshiftobservedfrom C;(4) y chosenasin oneof (1)—(3) on the initial null cone w = w0, and then specified thereafterto becomoving with the fluid: y.1u

1 = 0.When oneof thesespecific choiceshasbeenmade,y is uniquely definedon all the null cones.Whileany of thesechoicesmight be most suitablefor investigationof particularproblems,it seemsthat ingeneraluseful simplification is given by choice (4); and so unlessotherwisestated,we will usesuch achoice of coordinatey as a coordinatecomoving with the fluid, and determined by a uniquespecificationon the initial null conew = w

0.In all thesecases,wewill have:C is given by y = 0. In addition, the limiting behavioursof r and v

arethesameasthey go to zero(seeequations34 and 35 of [1]),so in cases(1) and (2), y —~v andy—* rasy-+O.

Becauseof eq. (3) andbecausew,,k’ = 0= 0,1k1 = qS,,k’, in termsof thecoordinatesx wehave

k = 8, k’ = dx’/dv = (11/3)8’~ (5)

for some function /3 >0; the significance of /3 follows from setting j = 1 in (5), showing that(11/3)= dy/dy= {rate of changeof coordinatey down the geodesicrelative to theaffine parameterv}.In particular,/3 is a constantif andonly if y is an affine parameter;andwhenwechoosey = r, /3-4 1 asy-+O.

Havingmadeaspecific choiceof coordinatey, eacheventis assignedcoordinates{x’} = {w, y, 0, 4’},were w is the time of observation;9, 4’ representsthedirectionof observation;andy is a measureofdistanceto the objectobserved(fig~3). Oneshouldnoteherethat thesecoordinatesdo notnecessarilycoverall the space—time,but that they do coverthat part which is observablefrom theworld line C.Theymay give a many-onecoveringof someof this region. This occurswhen thenull conedevelopscaustics,which can happenin two essentiallydifferent ways. Firstly the gravitational-lenseffect maycausedifferentneighbouringpathsof light to join thesameobject to theobserver(e.g.whentheselightrayspassnearmassiveobjects,seee.g. PressandGunn[28],Walsh, CarswellandWeymann[29]);andthen the samegalaxy, and perhapsthe samespace—timeevent,may be assignedtwo or more setsofcoordinates.In this case,oneor moreof the imagesmight appearto be anomalouslybright. Second,.asimilar effect may occurfor essentiallytopological reasons:if thespace-sections(or time-sections;butwe normallyexcludesuchcausalviolations)arecompact,we mayseethe samegalaxy in very differentdirections in the sky (e.g. Gott [30],Ellis [31]);in generalthe different imagesseen at one time w

0would correspondto light emittedfrom thegalaxy at differenteventsin its history;andonemight seethesamespace—timeeventin differentdirectionsin thesky(thenin generalthis would occurat differenttimes of observationw). This possibility arisesif thegravitationalfield causesthespace-sectionsto closeup, and the ageof the universe is large enough (as in the Eddington—Lemaitreuniverses,see e.g.Petrosianet al. [32]);or if the universehas“unnatural” compact(spatiallyfinite) topologies(seee.g.Ellis [33,52]).

326 G.F.R. Ellis ci aL, Ideal observationalcosmology

proper time4 ~

e ent I post null cone~time a~ww~\ u Clq): (w =constont

e~ null geodesic:(9,$ constant)

It

e~u

poroIlel~ observed event:propagated u (coordinates w0,y,8,~)

e~ q0 distance y

Fig. 3. Observationalcoordinates{w, y, 8, 4,} basedon theeventq on theworld line C. Here w is thetime of observation;8, 4’ thedirection ofobservation;and y the distancefrom q to the event observed,down the null geodesicjoining theseevents(variousquantities may be usedasmeasuresof this distance,seetext).

Thus an importantpart of the processof settingup andusing observationalcoordinates,is to see ifthere are any identificationswhich should be madeamongstthe observedevents or the observedgalaxies(cf. [29,30]) in orderto correctlyrepresenttheactualspace—timetopology.This maybe difficultto decide,for in generalit would bepossibleto observethesamegalaxy in differentpartsof thesky (inparticular,to seeour own galaxyat someremotetime in its pasthistory)without realisingthis; for onewould effectively seethegalaxiesfrom different directions,andat differentredshiftsandareadistances,and so with different selectioneffectsoperatingand at different stagesof their history. Neverthelesswithout thesepossibleidentificationshavingbeenconsideredandobservationallyverified or rejected,one cannotknowtheproperrelation of thecoordinatesystemusedto the space—timeand thegalaxiesin the space—time.Oneshouldnoticethat theexistenceof null-conecaustics(causedby thegravitationallenseffect) is likely to be thegenericsituationin theoriesof gravity suchasGeneralRelativity, in whichgravity is attractive.The crucial questionis not so muchwhetherthey exist or not, but ratherdo theyoccurnearenoughto affect our observationsof distantobjects.Thus an interestingquestionis, howtocharacteriseobservationalwaysof detectingcaustics.

Our coordinatesmay also give a one-manycovering,i.e., thesamecoordinatesmay designatedistinctevents.This could happenif r or z wereusedasthe y-coordinate.In general,thereis no guaranteethateither of thesequantitiesis a monotonic function of affine distancev down the null geodesics;forexample,in a Robertson—Walkeruniverse,theareadistanceis notmonotonic(Hawking andEllis [15]),andin asphericallysymmetricspace—time,the redshiftmaynot be monotonic(Ellis, MaartensandNel[42]).

3.2. The metric

Someof themetric tensorcomponentsin thecoordinatesystem{x’ } follow immediately.From (3),kaka = 0~w.1g”w,1 = g°°=0. Moreover k = g’k1, so (5) implies g’°=(1/13)8’,. Then g”gJk =

~ g°’gi~= 8°k~ glk = $80k.Thereforewecan write themetric tensorcomponentsin the form:

G.F.R. Ellis eta!., Idea!observationalcosmology 327

/3 V2 V3\ /0 1/fl 0 0

g• = ( /3 0 0 0 jk J1/13 8 O~2 O~3

~ v2 0 h,., h,.3 J’ g —‘ 0 U~ h33/h —h,.3Jh (6\ V3 0 h,.3 h33/ \ 0 o~ —h,.3/h h,.2/h

where h usdet(hM)= h~h33— (h,.~)2, 8 us —(a+ /3(V202 + V

3u3))//32, 02 us —(v

2h33— v3h,.3)/f3h, O~3us

—(v3h~—v2h,.3)/$h.It is sometimesconvenientto define quantitiesr, f’, by therelations(Sachs[27],d’Inverno andStachel[34])

r4 sin2 0 = h us det(h~), fLY us h~/r2 (~‘det(f,.,)= sin2 0) (7)

where here and subsequently,we use I, J as indices rangingover the values2, 3; then r (the areadistance,seesection4.3 below) andf

11 give an alternativerepresentationof thequantitieshL,.

The metric componentscanbe given direct geometricinterpretations.The form of g,1 andgW showthat k1 us w~is a (hypersurface-orthogonal)null geodesicvector field, with /3 related to the affineparameterv througheq. (5); that is, the resultsof theprevioussectionconcerningk also follow fromthemetricform (6). Thecurveson which (v, 9, 4’) areconstant,arethecurveswith tangentvector8/ow;their direction and the magnitudeof their tangent vector are characterisedby the scalarproductsg01 (a = (8/8w)~(8/Ow); /3 = (8/Ov).(8/ow); V2 = (8/a0)~(8/Ow); v3 = (8/34’). (8/Ow)). Thesequantitiesthereforerefer to metric characteristicsout of, or transverseto, thenull surfaces{w = constant},for thecurves transvect thesesurfaces.The intrinsic geometryof the null surfacesis representedby thecomponentsgKA (K, A = 1, 2, 3) of the metric, with non-zerocomponentsg,., = h,, (these3-surfacesaredegenerate).The quantity r

2 determinesthe areaof cross-sectionsof the null cone,while fu is itsconformaltwo-structure[34].Particularchoicesof y will simplify someof thecomponents;for example(cf. section3.1 (iii)), thechoicey = v implies ,8 = 1; or onecouldusethe freedomin y to setoneof v

2,v3 zeroeverywhere,and in additionto set theotherone to zeroon a 3-dimensionalsurface.

While themetric form (6) implies that thesurface{w = constant}arenull surfaces,it doesnot, as itstands,guaranteethat thesenull surfacesare thepast light conesof a regulargeodesicworld line C.This featureimposeslimits on thebehaviourof themetric tensorcomponentsnearC. Theselimits canbe found by an adaptationof the procedureusedby ManasseandMisner [35] to fit coordinatestospacelikegeodesicscentredon aworld line; thedetailsaregiven in appendixA (andcf. [26]).Whentheaffine parameterv or areadistancer is usedfor thecoordinatey, the essentialresult is:

lim a = —1, limfi = 1, lim(v,/y2)= 0, lim(h,., dx’ dx~/y2)=dir. (8)

y-~O y—O y-+O y-.O

If a generalcoordinatey is usedwhich is regularly relatedto v at the origin (which will be true forchoice (3) and (4) of section3.1 (iii)), the correspondingresultsare obtainedfrom (8) by making acoordinatetransformationy’ = y‘(w, y, 0, 4’), w’ = w, 9’ = 0, 4” = 4’ which is regularasy -+0; onefinds

lim a = —1, lim /3 = flo(w, x’), lim(v,) = 0, lim(h~dx’ dx’/y2) = flo2diP. (8’)y-.O y-.O y-.O

Togetherwith (6), theseconditions imply that the null surfacesw = constantaie the null conesof aregulartimelike geodesicC, andmoreoverthat thedirections0, 4’ arebasedon aparallelypropagatedtetradof vectors alongC, and that w measurespropertime alongC. That is, we haveobservational

328 G.F.R. Ellis ci aL, Ideal observationalcosmology

coordinates(as describedin section3.1), if andonly if the metric tensorcomponentsobeyeqs.(6) and (8).One might note here, that Minkowski space—timeis given in observationalcoordinates(with

y = v = r) by themetric form (6) with a= —1, /3 = 1, v, = 0, andh1~dx’ dx1 = r2 dIP.Thuseq. (8) just

says that the space—timeis like Minkowski space—time(given in observationalcoordinateswithy = v = r) nearC.

4. Galacticmeasurements

As explained in section2, we shall restrict our attention exclusively to observationsof distantgalaxiesandquasars,and of the radiation background.In both of thesecasesinformation reachestheobserveron thecentralworld line C by meansof electromagneticsignals.

Eachsingleobservationis madeover a finite time interval,andthusinvolvesa 1-parameterfamily ofpastlight cones,ratherthana singlelight cone.However,in thecharacteristicfinal valueproblem,dataneedsto be specified on just one past light coneC(q). [We are interested,for the time being, indeterminingthesolutiononly in theinterior of C(q).] This difficulty canbe overcomeby appealingtothe fluid approximationalreadyassumedandto the scaleof time variationof cosmologicalquantities.We construct an idealised final past light cone C(q). All the available cosmological informationobtainedfrom actual observationsmadeat different timesandin different directionsis simply assumedto havebeenobtainedby observingfrom a singlepoint q on C, and is usedasfinal dataon the singlepast light coneC(q). This is clearly a “smoothing approximation” of sorts,similar to that used inconventionalcosmologywhenonerefers to spatiallyhomogeneoushypersurfacesin the universe.Thejustification for this approximationis that the scale of time variation of cosmological quantities isexpectedto be extremelylong comparedwith the time intervalsover which observationsaremadeonC. In fact, it can be arguedthat suchan approximationis necessaryfor consistency,oncethe fluidapproximationhasbeen adopted.The averagingscale involved in the fluid approximationis that ofclustersof galaxies,while theentirerecordedhistory of mankindcorrespondsto a spatialdistanceof afew kiloparsecs.

Now, thestructureof the acgualpastlight conesof pointson C will, in general,be very complicated.In particular,causticsare boundto occurin thevicinity of massivestarsandblackholes.The existenceof suchcausticsmight causea numberof problemsthat needto be consideredcarefully whenthe actualobservationsareinterpreted.

As far asthe idealisedpast light coneC(q) is concerned,however,wemay assumethat it doesnotdevelopsuchcaustics,preciselybecausethe fluid approximationalreadyincludesthe assumptionthatlocal irregularitiesin thematterdistributionhavebeensmoothedout. This meansthat, for thepurposesof the characteristicfinal value problem,black holesmay be ignored. (Strictly speaking,by the verynatureof singularities,any such “hole-free” assumptionneedsto be formulatedin global terms; seeCollinsandEllis [81].)

Causticscausedby theglobal topologicalpropertiesof theuniversecannotbe smoothedout in thisway. In thesecases, the light conewraps aroundand intersectsitself, with geodesicsthat initiallycorrespondto widely different directionsof observationintersectingeachother. A good exampleisprovidedby theEinsteinstaticuniverse.Theconformalstructureof sucha universemaybe representedin theusualway (see,e.g., Hawking andEllis [15]) by a cylinder with topologyR x S3. The pastlightconeof an eventq on C wraps aroundthecylinder andrefocussesat a secondpoint q’ to thepast of,andconjugateto, q. Thus, if theuniversehascompactspace-sectionsandan age largeenoughso that


we havealreadyseenaroundit, theobserveron C will seedifferent images,in different directions,ofthesameobject. This is particularly interestingin thecaseof “small universes”wherethepresent-dayscaleof thesecompactspace-sectionsis a fraction of a Hubble radius(Ellis [52]).

We noted in section3 that thenull-coordinatesystem(w, V, xA) breaksdown at this point and thatone needsto determinewhat identificationsmustbe madeamongobservedobjects.In principle, thiscould be done,given a sufficiently detailedcatalogueof sourcecharacteristics.We could thendefine agroupG of identificationsby stating for eachobject observedon C(q) in the direction x~and atredshift z, which otherobjectsseenin different directionsxAa andat different redshiftsZa are in factthesameobject.

In practice,sucha procedurewould be exceedinglydifficult, if not impossible,to carryout. In orderto avoid this problem,we restrictourselvesin this paperto a regionof C(q),not too far from C, wherenull causticshavenot yet occurred.

4.1. Redshiftsand 4-velocities

Thecoordinatew measuresproper time alongtheobserver’sworld line C, while propertime alonggeneralgalacticworld linesis denotedby r (fig. 4). Accordingly, the time dilation observedfrom C foreventsalongagalactic world line is determinedby the ratio d w/d’r alongthat world line. In particular,applyingthistothelight emittedwith wavelengthA~andobservedatC, wherethewavelengthis observedtobeA0, we find that the observedredshiftz of the light is determinedby

1+ZusAo/Ae=dW/dr. (9)

This resultcan also be obtainedby using the relation [24,13],

ij —II a\ iii b\

I T Z — t,r~U )emitterI~f~~bU)observer,

andnoting eqs.(3), (4); from this form, we also notethat

1 + Z = U°

1ejmj~~~r,

showingthat thiscomponentof the4-velocityof thematteris directly observable,becausethe redshift

Fig. 4. A time interval dT at theobservedgalaxy is measuredasa time interval dw by theobserver;this affectsall observations,in particularobservationsof thefrequenciesof light waves(andhencetheobservedredshifts,seeeq. (9)).

330 G.F.R. Ellis eta!., Idealobservationalcosmology

is directly measurablefrom the observedsource spectrumas long as clearly identified emissionorabsorptionlines arepresent.

In fact, thematter4-velocity is given by eq. (2); combiningthis with (9), we find

= dw/dr= 1 + z; u2 = dO/dr= (1 + z)d9/dw; u3 = d4’/dr = (1 + z)dçb/dw. (10)

Thus we can hope to directly measureall thesecomponentsof the matter4-velocity u, u°followingdirectly from z,while u2 andu3 follow from observationsof thepropermotionsof galaxiesin theskywith respectto a local non-rotatingreferenceframe(the observedpositionof a galaxyat time w relativeto sucha frameis givenby 9, 4’ atthat time; so theobservedpropermotionsfollow from thederivativesof 9(w), 4’(w) if onecanobservefor long enoughto measurethesederivatives*).The final componentu1 of the4-velocity follows from thenormalisationof U” (see(1)), which implies

u’ = {—u’u~h1~— 2(1 + z)u’v1— 1— a(1+ z)

2}/2 (1 + z)f3. (11)

This relation gives u’ in termsof the observable4-velocitycomponentsu°,u’ and themetric tensorcomponentsh,,, v,, a, /3.

The velocity field u must includethe4-velocityof theworld line C, becausethis world line is oneofthe family of galactic world lines; so for coordinatesy satisfying the limiting conditions (8’), thecomponentsu of this field must include thevalues U’Ic = 8’o (they take this form at C becausey isconstanton C). Howeverthecoordinatesaresingularon C, and in fact thelimit relationsto besatisfiedare

urnu°= 1~ urn z = 0; lim u’ = 0, lim u’ = u’~(w,x’), (12)y-.O y-.O y—.O y-’~O

where u’o(w, x’) are arbitrary functions(seeappendixA for a derivationof this limit).Special coordinatechoicescan be used to simplify u. If we make the choice of a fluid comoving

coordinatey (which will be ourusualchoice, seesection3.1 (iii)), then

u’ = dy/dr = 0~a = —(1 + z)2{1+ 2u’vj(l + z)+ 2h,~u’u~}, (11’)

theequivalence(showingthat a is determinednow by U’, z, v, and h,1)following from (11).

The metric form andthe form of u’ also imply the form of the velocity componentsu1 = g~u’.Onecan write thesecomponentsas

uo {a(1+ z)2— 1+2h,

1u’u’}/2(1 + z); u, = /3(1+ z); Uj = (1 + Z)Vj + h~1u’. (13)

The limiting values of these quantitiesat the centre are, u0-+—1, u,-+/3o(w,x’), u,-+O when acoordinatey is usedsuchthat (8’) holds. We may notefrom (13) that the relationbetweenthe fluidvorticity and theobservedpropermotionsis not asimple one in general.Forexample,supposethat theobservedpropermotionsvanish(u’ = 0), andwe useacomovingcoordinatey to simplify the resultingform; thenwefind u0= —(1 + z)

1, u1 = /3(1+ z), u, = (1 + Z)Vj. As in generalz, /3 andVj can— aslong

* Onemight haveto measurepositionson anopenintervalof C ratherthanatonespace—timepointq on C; this dependson theaveragingscales

involved, we will discussthispoint lateron.

G.F.R.Ellis etaL, Idealobservationalcosmology 331

as they satisfy the central conditions— be any function of all the coordinatesx’, the vorticily vector~77”’”UbUc.d (seee.g. [36,13]) will in generalnot be zeroevenif the observedproper motionsare

zero.* Similarly the other kinematic quantitiesdescribingthe fluid motion [36,13] will in generalberelatedin a fairly complexway to theobserved4-velocitycomponentsu. * *

Finally we note that the derivativeof any function along the fluid flow lines is given by f,~u’=(1 + z) Of/Ow + U

1 Of/Oy+ (1 + z){(Of/39)(d9/dw)+ (Of/Oçb)(dçb/dw)},whereu’ is given by (11)or (11’).

4.2. Imageintensities

In a given cosmologicalmodel, the specific intensityof radiationreceivedfrom asourceof surfacebrightnessI~andspectrum5(v) dependsonly on thesequantitiesand the observedredshift of thesource;it is independentof otherspace—timeproperties,suchastheareadistanceof thesource(seee.g.[13,18, 14, 24]). In fact, essentiallybecauseof theconservationof photons(we areat presentignoringthe effects of absorptionbetweenthe source and the observer),the observedspecific intensity ofradiationI~at frequencyv is given by

Ip = Ig 5[v(1 + z)]/(1 + z)3, (14)

where‘g is thesurfacebrightnessof thesourcein the directionof observation;as one looks acrosstheimageof thesource,theobservedintensityvaries accordingto thevariationof surfacebrightnessat thecorrespondingpoints of the source. This intensity variation is directly measurable,down to somelimiting valueSL(v)of the intensitycorrespondingto thedetectorlimit [4,5, 6].

4.3. Imagesizeand shape

The shapeandsizeof the imagedependson thepathtakenthroughspace—timeby thenull geodesicswhich arethe light raysfrom thesourceto theobserver.In general,the space—timecurvaturewill bothintroducedistortion into the image (a sphericalgalaxymay look elliptical; an elliptical galaxymay lookmoreor lesselliptical, dependingon theorientationof theaxes;andsoon), andcausefocussing(sothatat a particularvalue v of the affine parameteralongthenull geodesics,theareadistancer will takeadifferent valuefrom thevalue v = r in flat space—time);seee.g. Sachs[37],Penrose[38],Kristian andSachs[1].In principle,onecan measureboth thesefeaturesdirectly if the intrinsic sizeandshapeof theobjectobservedis known.

In fact, they are both representedby the metric componentsh1,, which are, in principle, directly

measurable.For an objectof knownsizeandshapeobservedat time w0 and lying at distanceYo in thedirection 0~,4’~onehas(from (6))

d12 = hz,(wo,Yo, 00, 4,~)dx’ dx~ (15)

whered12 representsdistancesat theobjectperpendicularto the line of sight, which areknown if thesize, shapeand orientationof the object are known; and dx’ representthe correspondingangulardisplacementsat the image,which aredirectly measurable.Comparingtheangularmeasurementswith

• Note thatthis cannothappenat theorigin q (seebelow).

** In part II we will give thecomponentsof thesequantitiesrelativeto suitablychosentetrads.


the knowndimensions,one can deduceh~.Looking in all directionsat time w0, onecan in principle*determinethe functionsh1~(w0,y, 0, 4’), if y is chosento be ameasurablequantity (e.g. redshift z) onw = w0.

The areadistance r (also called “corrected luminosity distance” in [1]; called “observer areadistance”in [13])is definedby the relationdS0= r

2 dfl0 (fig. 5), wheredS0 is thecross-sectionalareaof

an objectperpendicularto the light ray, and d110 is the solid anglesubtendedby that object at theobserver.It follows directly from (15) and the relation of the angles0, 4’ to solid anglesat C (which istheusualrelation,asis implied by thelimiting behaviour(8)) that theareadistancer is givenby eq. (7).Alternatively, this resultcanbe foundfrom the fact [37]that the rate of changeof cross-sectionalareaof a bundleof null geodesicsis given by d(logdS)/dv = k”;a 0. Now (5), (6) show

k”;a = d(log h”2)/dv = (1//3)d(log h”2)/dy, (16)

where h andet(h,j); thus r2 = Ah1”2 wheredA/dv= 0, andevaluatingA asy -+0, by (8), again leadsto(7).

The focussingeffect is thereforerepresentedby the relation betweenthe areadistancer, which isessentiallygiven by h, and other measuresof distance(e.g. the affine parameterv). Becauseh,~ismeasurable,r is in principledirectly observable;for example,it canbefounddirectly by measuringthesolid angle subtendedby the imageof an object of known size. It can also be determined(basically,becauseof the inversesquarelaw for photonpropagationwhich follows from photonconservation,andis essentiallyequivalentto (14))by measuringthe observedluminosity of a sourceof known intrinsicluminosity, see[1, 13, 14]. Howeverboth waysof determiningr in practicedependcrucially on thedetectionlimit SL(P)which determinestheboundaryof theobservedimage,so detailedknowledgeofthe source brightnessdistribution and spectrumis neededbefore r can be estimatedeither frommagnitudeor angulardiametermeasurements(see [5, 4, 6]). Given this knowledge,r is a measurablequantity.

Any distortionin theimagewill bereflectedin the fact that fu h11/r

2 is unequalto themetric of theunit sphere,which is its limiting valueat C; thedifference(fil dx’ dx~— dIP) thereforeindicatessuchdistortion.

The sheard~of the null geodesicsgeneratingthe null cone is a measureof the rateof distortion

solidangledfl

0

observer

sourceFig. 5. In a curved space—time,light travelson null geodesics,and thearea distancer of the sourcefrom theobserveris definedby therelation:dS0 r

2dfl0.

The practicalityof this proposalis discussedin [2,41.

G.F.R. Ellis eta!., Ideal observationalcosmology 333

down thesegeodesics.It is defined(seee.g. [37,15]) by projectingthe covariantderivativeof k~downthenull geodesicsinto the “screenspace” orthogonalto k and the 4-velocity u of the fundamentalobservers,and thensubtractingoff the trace.As the size andshapeof theimageareindependentof themotion of theobserver[37],onemay carryout this calculationfor the particularobservermoving withzero proper motion and zero redshift; the corresponding 4-velocity will then be U” =

—&o + [(1+ a)/2Jo”1 and the projection h~ into the screen space, defined by h~=

g~— k~kb+ kaUb+ Uakb, hascomponentsh11 = 8,’Ofg~.The covariantderivativesof ka hascomponents

k;j = (1/2/3) 0g11/Oy— (11/3)/3.4~°~ (17)

so its projectioninto thescreenspaceis A.1 anh,tmhJ’km fl = (1/2/3)O~Of OgH/Oy;and its trace-freepart is

= (r~/2/3)o,’ Of Of11IOy. (18)

The magnitudeof the shearis ê,determinedby ê2= (~)~,gIK~j~g

From theseresults it follows (see section5 below) that while h1~,and so both r and fu, can in

principle bedirectlyobserved,thenull geodesicexpansion0 andshear(Tab canonly bedeterminedfrom

this knowledgeof h,~if themetric component/3 is known. This implies that 0 and d~2arenot directlyobservable(seesection5).

4.4. Numbercountsandgalaxyclustering

Supposeone countsgalaxies seenin a solid angled110 aroundthe direction of observation(9, 4’),

down to a distancey. An incrementfrom y to y + dy will result in including dN new galaxiesin thecount, wheredN is thenumberof galaxiesdetectedin thevolume dV correspondingto the range(y,y + dy) in the coordinatey; this volume is [1,13] dV= (r

2 dQ0)((U’~ka)dv). If the numberdensityof

galaxiesat theposition y is n, then (n dV) galaxieswill be includedin this volume;howevernot all ofthem will be counted,in general,for somewill be too faint to be detected,while othersmay not beselectedfor inclusion in thegalacticnumbercountbecauseof variousselectioneffects (e.g. they maybeconfusedwith starsif theirimagesarevery small). Wewill write dN in the form, dN = fn d V, wherefis the selectionfunction representingthe fraction of galaxiesin dV that are actually detectedandincluded in the number count; one can estimatef from a knowledgeof the galactic brightnessdistributionandspectrum,theareadistancer andredshiftz, and thedetectionlimit SL(v)(see [39,6]);in general,f will dependon w, y, 9 and4’. Froni the formulaefor dN andd V one finds, on using (5)and (9),

dN=fnr2dt2o(1+z)dy. (19)

The total numberof galaxiesN(y)seenup to a coordinatevaluey is then givenby N(y)= f~’dN. BothdN andN will in generaldependon w, 9, 4’.

NumbercountsN(y) ordN(y,dy) in somedirection9, 4’ atsometime wo aredirectlyobservableify is an observablecoordinate(e.g. the redshiftzor theareadistancer) on thenull conew = wo. If y isnot directly observable(e.g. if it is chosento be the affine parameterv) on w = wo, then N(y) ordN(y,dy) can only be foundfrom numbercountsin termsof someother distanceparameterwhich is

334 G.F.R. Ellis et a!., Idealobservationalcosmology

directly measurable.Othervariationsof numbercounts,suchasthe VI Vm~tests,will still be basedonthe relation(19).

As well asthegalaxynumbercounts,galaxyclusteringasdeducedfrom observedcorrelationsin thedistributionsof galaxiesin thesky hasbeenusedto estimatevarious cosmologicalparameters(seee.g.Fall [46],White [49],Peebles[48]for a review).We shall not pursuethis methodof investigationhere;what one is determiningin this way are the galaxy formation and evolution conditions,rather thancurrent space—timeconditions on the past light cone. Thus while theseinvestigationsgive valuableinformation on the evolutionof the universe,they do so by way of providing constraintson possibledevelopments,muchaselementabundanceobservationsdo, ratherthanby providing directinformationwhich canbeusedin a cosmographicprogramme,We regardthem asgiving indirect, ratherthandirect,evidence.

4.5. Identifications

Finally, although considerableproblems would have to be overcome, it might be possible todeterminein particularcasesthat onehadseen thesameobject in different directions in the sky (cf.section3.1 above).This would give a setof imageidentifications,recognisingdifferent imagesasin factbeingimagesof thesameobject.

5. Observationalquantities and cosmography

5.1. Observationalquantities

We areconcernedherewith observationsmadeby meansof electromagneticradiationreceivedfromour past light cone(see section 2.3 (iv)). We may regard this as coming from two different typesofsources.

(i) Backgroundradiation. Measurementsof cosmological“backgroundradiation”— that is, radiationwhich is believedto comefrom cosmologicaldistances,but cannotbe ascribedto any observationallyseparateddiscretesources— providesinformationon the integratedemissionand absorptiondue to allunresolvedmatterdown the line of sight. In particular,the total radiationreceivedlimits the integratedbrightnessof sources(“Olber’s limits”), while boundson theanisotropyof the radiationimply limits onthe anisotropyof thematterdistribution.However asthemeasurementsrefer to integratedemission(see e.g. [13,23] for the relevantformulae; in general the mattercontributingwill consistboth of adistributionof unresolveddiscretesources,and continuasuchasan intergalacticgasor the plasmaof aprimeval fireball), they do not directlygive detailedinformationon thedistantspace—timegeometryormatter distribution,* but rather, like local geophysicaland astrophysicalobservations,provide im-portantconstraintsthat any cosmologicalmodel hasto satisfy [4].On theother hand, they do directlygive two items of importanceabout the point of observationq, namely the energydensity /.~radlqofradiation,and the isotropyof this radiation,there.We shall denotethis information(thatis, the specificintensity I(i.’, 9, 4’) of thebackgroundradiationatq as a functionof frequencyii anddirection 0, 4’ in

As an example,local isotropy of themicrowavebackgroundradiation doesnotnecessarilyimply thatthespace—timeis isotropic, in general(see[42]).

G.F.R.Ellis et a!., Idealobservationalcosmology 335

the sky)by BR0, or “observedbackgroundradiation”; clearly this determinesin particularthequantity~Lrad

1q.

(ii) Discretesources.Supposeone could makeobservationsfrom C for all w (the time along C) insomeopen interval I: then through observationsof distant extendedsources(galaxies,radio sources,etc.) one could aim to determinedetailedcosmological information on the past null cones {w =

constant,w E I}. At sometime w E I, onecould measurefrom the imageandspectrumof a source,itsposition (9, 4’) in the skyandits redshiftz; soobservationof thesourcefor long enoughwould enableone to measureits apparentmotion (d9/dw, d4’/dw). Onecould measurethe image size, shapeandintensity distribution; and thus derive, from an assumedknowledgeof the intrinsic propertiesof theobject observed,thedistortion effect and areadistancer (seesection4.3). Now oneof theobservablequantitieszor r could be chosenasthecoordinatey, so that all fourcoordinateswere observationallydetermined.Carryingout this procedurefor sourcesat all 9, 4’ andall y down to * somevaluey~of y,one will know the average4-velocity componentsu” (a = 0, 2, 3), given by (10), and the intrinsicgeometryof the null cone(the metric componentsh

1.,, equivalentto r, fr,) for w E I, O<y � y* and 9,4’ ESz

In addition, as y hasbeen chosento be an observablequantity, the number countsdN when yincreasesto y + dy aredirectly measurableanddeterminethe quantity(fn r

2 (1+z)/3dy), see(19). Asz and r are known, one can estimatethe selectionfunction f (which dependson r, z, the galaxyproperties,and the observationallimits and selectioneffects), and so determinethe quantity (n13) intermsof known quantities.In principle* onecould thereforealso find (n/3) for w E I, 0 � y � y* and 9,4’ ES2.

Carrying out observationsat otherwavelengthswould enableone to obtain a firmer graspon thequantitiesjust discussed(n would be the total sourcedensityandf would in generalvary widely withthewavelengthof observation)but will not enableoneto determinenew kinds of quantities;andthesameis true for any possibleextensionto gravitationalwaveor neutrinoobservations.Observationsofdiscretecosmologicalsourceswhoseimagescannotbe resolved(“quasi-stellarobjects”) would enableone to obtain thesamekind of information— their areadistancecould be estimatedfrom theapparentmagnitude,on estimating the source magnitude— except that the distortion effect could not beinvestigated.Measurementstakenover a sufficiently long time period,if possible,would enableone todeterminethe time variationof all thesequantities,e.g. thechangein redshiftz of particularsources,orin their areadistancer, or in thenumbersof galaxiesout to somegivendistancey.

It is clear thenthat themostdetailedcosmologicalinformationonecould hope to obtain by directobservationof distantregionsfrom pointson C for w E I, is a knowledgeof: u”(w, y, 0, 4’) for a = 0, 2,3; hu(w, y, 0, 4’); and [nfl] (w, y, 0, 4,); all down to somevaluey* (which couldvary with w, 9, 4’). Wewill call this maximalattainabledatasetgiving direct information aboutdistantregions— i.e. the set offunctions{(u°,u2, u3 !z~~nfl): w E I, 0~ y ~ y”, 0, 4’ E S2}— the setD(I, y*). In practice(seepaperII)much less will be achievable;so certainly the setD(I, y*) representsthe mostdetailedcosmologicalinformation onecould obtain by astronomicalobservationsof distant objectsmadeat times w E I ontheworld line C.

Exactly whatmight be observedatan instantw = w0 (correspondingto thepoint q on C) will depend

on theaccuracyof measurements,and the time of observationavailableat the “point” q. The mostonemight reasonablyhope for, is to measurethe quantities D(wo, y*) anD(I, ~ plus the time

* Problemsarising in attemptingto do thisfor all y down to y on eachnull geodesicwill be consideredin [3] (andsee[4,52]).

336 G.F.R.Ellis el aL, Ideal observationalcosmology

derivativesof someof thesequantities(the prime candidateasa quantity whosetime derivativemightpossiblybe observed,is the redshiftz of distantgalaxies[45,43]). Denotingthesetof time derivativesof thequantitiesin D(I, y*) evaluatedat w w0, by D(wo,y*), the maximuminformation onecouldhope for at q is D(wo, y*) plus D(w0,yo). However,even attainmentof D(wo,y*) is fraught withdifficulty; for example,the time intervalsinvolved may not allow one to measurethe propermotionsneededto determineu

2, u3 [4,2]; and supposingonecould determinetime changesof someof thequantitiesin D(I, y*) at w = w

0, onewould haveto showthat this changewascosmologicalratherthanastrophysicalin its significance,which would be extremelydifficult to do. Thus even in the mostfavourablecase(thatof redshifts),definite measurementof quantitiesin D(wo, y*) is likely to remainelusive(considerfor example,the interpretationsput forward to explainredshiftchangesin SS433,seee.g. [44]).Thus we will take the view that (apart from possibleidentifications,see below) the setD(w0, y*) representsthe maximal attainable detailed information at time w = w0 about cosmologicalvariablesin distantregions (andin practicewe do notexpect to attainthat maximumof data).Should iteventuallyprove possible to determineunambiguoustime variations of quantitiesof cosmologicalsignificance,our approachwould be to usethis datato checkinformation obtainedfrom theprimarydataD(wo, y*). In addition, in principle onecould determinethe identificationsof different imagesofthe sameobject. Theseare specifiedby giving a groupG(wo,y*) of identification of objectsseen inC(q) down to y*; thus G states,for each object seen,say at (wo, y1, 01, 4,1), which other objectsobserved— say at (wo, y~,9~,& ) — arein factthesameobjectsseenin adifferent direction andlorat adifferent distance. In principle this is perfectly feasible; a sufficiently detailed catalogueof sourcecharacteristicswould enableone to determinewhenthesameobjecthasbeenseentwice.In practice,atany reasonabledistances,considerableobstaclesmight arisein making suchidentifications;howeverforthepurposesof this paper,wemay takeit that this groupcould be determined.

5.2. Thespace—timeon thepastnull cone

On the other hand,we will havecompletelycharacterisedthecosmologicalquantitieson the nullcone w = w0 down to y = y* if we have the following information: the metric tensor components

= {a, fi, v1, hu} (see(6)), thematter4-velocitycomponentsu, the total numberdensity n of sources,and the radiationdensity~d at eachpoint of this null conedown to y = y~ Assumingtheequationofstateof the matteris known from local astronomicalobservations,knowledgeof n will determinethelocal restmassdensityof matterPmatat eachpoint~thentheenergydensity/.Lmat and isotropicpressurePmat will bedetermined,andsothemattercontributionto the total stresstensorwill beknown (see(1)).At the presenttime the backgroundradiation is very isotropic, so it is reasonableto assume— fordynamicalpurposes— that it remainsisotropic for substantialdistancesbackinto the past,evenif thegeometryis relatively anisotropicthere.That is, wecanreasonablyusethe “perfect fluid” form (1) alsofor the radiation, and so determinethe radiation contribution to the total stress-energytensor, atrelatively recenttimes. (In very anisotropicuniverses,onewould needa more complexdescriptionofthe radiationat early times;we will not go into that questionhere.*)

We assumethecoordinatey hasbeenuniquely determined(seesection3.1(iii)); so the coordinates{xt } areessentiallyunique— the only remainingfreedomis a rigid rotationof the axesdeterminingtheangles0, 4’ down to C. Considera particulareventq (correspondingto w = wo) on a chosenworld line

If thereis a largeamount of “hidden matter”present,this estimationbecomeshighly problematic.

* We discounthere thepossibilitythat thereis atpresentalargeanisotropicflux of neutrinosorgravitationalwaves.While thereis no evidence

againstsuch anassumptionneither is thereany for it.

G.F.R. Ellis eta!., Idea! observatitinalcosmo!ogy 337

C in a cosmologicalspace—time.Knowledge of the local cosmological variable— that is, both thespace—timemetric and the matter and radiation distribution (as representedby the stress tensorTab)0fl the null conew = w0 down to y = y* will determinethedata setS(wo,y*), definedto bethequantities{(g,j; u’; n; grad); w = w0, 0 � y sy”, 0, 4, E S

2}. Becausethecoordinatesystemis unique,thesecomponentswill all be uniquelydetermined(upto the freedomcorrespondingto a rotationof thecoordinateaxes).On theotherhand,a null coneon which thedatasetS takesthat preciseform, will beimbeddablein that cosmologicalmodel asthepastlight coneof thepoint q on theworld line C, downto thedistancey = y”. Any variationin thedatasetS (exceptthat correspondingto a rotationof thecoordinateaxes)will correspondto a realdifferencein the space—timemetric or thematterdistributionon the null cone(the numberdensityn determiningthe densityand pressurep of the mattervia theequationof state),showing it could not be the light coneof that eventq in the space—time.Thus, thedatasetS(wo,y*) uniquelycharacterisesthe cosmologicalvariables,locally on the null conew = wo downto y = y~’,up to a rigid rotation of theangles0, 4’.

Given this detailedlocal structureon thenull coneC(q)an{w = wo}, oneneedsto know in additionthenon-localstructureof thenull cone,that is, the identificationsG that haveto bemadeamongsttheobservedobjects(seesection5.1).

As well as thedetaileddescriptionof thespace-timeimpliedby a knowledgeof its metric tensorandthe matter variableseverywhere,one is particularly interestedin certain quantitieswhich can bethoughtof as derivativesof the primaryvariables;in particular,onewishesto know the fluid “kineticquantities” [35,13]; the space—timecurvature tensor, and its resolution into Weyl and Ricci tensorparts;andthegradientof thematterdensity.Thus it is of interestto considerwhetheronehasthedatanecessaryto determinesuch (covariant) derivative quantities; in some cases this would requireknowledgeof S(w,y*) on an openintervalof w, but in othercasesarestrictedsetof datain S may besufficient. We will considerparticularcasesof interestin the subsequentsections.

5.3. Observationalrelationsand cosmography

The vertex q on the null conew = w0 is a uniquepoint on that null cone, and the observational

situationthereis quite different from that at otherpointson thenull cone.We shall first considerthesituation at generalpoints on the null cone(other than thevertex); andthen return to considerthesituationat thevertex.

(i) Generalpoints.At a generalpoint on thenull cone,theobservablequantities{BR0, D, G} and thedataneeded{S, G} are thosedescribedabove.It is clear that the observationaldata givesus no directinformationon thequantitiesa, fi, v1 consequently,the4-velocitycomponentu

1 and thenumberdensitynare not directly determinableeither.For the 4-velocitycomponentu1, this is becauseut is not directlyobservableandits determinationdependson the quantitiesa, fi, v

1 (see eqs.(11)). For the numberdensityn, it is becausethe numbercountsN(y) determinedirectly only thequantity [nfl], but fi is notdirectly observable*;becausen is not known,neitheris thematterdensitySUmat. Howevertheavailablefreedomof choice of thecoordinatey can be usedto assign a chosenvalue to at most two of thesequantities(the usualchoiceof y asa comovingcoordinatesetsu

1 = 0 and then a will be given in terms* Notethat $ would beknownif, insteadof choosingyasanobservablequantity(seesection5.1 (ii) above),it waschosenasanaffineparameter;

but thenN(y) would notbedirectlyobservable.Thusalthough$ would be known in thiscase,[‘~1 would notbeknown.Similarly,otherchoicesofy will not changetheessentialsituation:a cannotbedetermineddirectly from observations,in the cosinographiccontext (where thespace—timestructureis notknownapnon).

338 G.F.R. Ellis etal., Idealobservationalcosmology

of the otherunknownsby eq. (11’); and onecan still choosey on C(q) so asto simplify one of theotherunknowns).

For the radiationdensitygrad, thesituation is intriguing: we cannotdirectly measurethe radiationdensity at the point (wo, y, 0, 4’) on the null cone w = w0, but we do haveways of estimating itindirectly from observablequantities.This is becausewe can determine,Urad and the isotropyof theradiationat thevertexq of thecone.From the radiationintensityobservedin a direction 0, 4, at q, wecan estimatethe intensity of the radiation at a point a distancey along the null geodesicin thatdirection, which is travellingalongthe null direction k (for this is just the radiationwhich travelsto qand is measuredthere*). We cannotdirectly measureany of the other radiationcomponentsat y;howeverif we believethe radiationfield at y to be nearlyisotropic,we may estimatethe radiationinany direction to be nearly thesameas the radiationin thek-direction. At leastnearq, it is likely thatthe radiationfield is highly isotropic (becausethedominantcomponent— the microwaveradiation— isknown to be very isotropic atq, seee.g. [12,15]). Thus we may usethe “perfect fluid” idealisation(1),andcanobtain reasonableestimatesof ~ out to reasonabledistancesfrom q; in fact onemay assume/.Lrad cc T

4 whereT is the radiationtemperaturewhich variesasT = Tq(l + z) [12,13, 24]. This radiationtemperatureis well-definedbecausethedominantmicrowaveradiation hasa spectrumvery closeto thatof ablackbody. Consequently,although suchestimatesmay be less reliableat large distancesfrom q(where the radiation stresstensormay vary considerablyfrom a perfect fluid form), at reasonablecosmologicaldistancesone may— for dynamicalpurposes— take /.Lrad as a known quantity dependingonly on BR~andthe redshiftz, eventhoughit is not directly measurable.

Oncomparingthedatasets{BR0, D, G} and {S, G}, it is clear that an attemptat direct observational

determinationof all the required data S on the null cone w = w0 fails; one does not have anobservationalway of determiningthe “rigging” (in the notation of Schouten[40])of the null cone(representedby the metric componentsa, fi, VA); or thematter4-velocity componentu’ anu”; or thematternumberdensityn; as functionsof y, 0, 4,. The available coordinatefreedomcan reducethisuncertaintyby at most two functions.Furthermorethis indeterminatenesswould remain even if wecouldobservefor a finite time interval I on theworld line C; for this would thenenableus to determinethequantities{BR0, D(w, y*), G} for w EI, but would not extendtheobservationsin sucha way astomaketheunknownquantitiesin S(w,y*) measurable.

Thus, the cosmographicideal of determiningthe cosmologicalmodel on the past null cone w = w0directly, without useofsomedynamicaltheoryfor the space—timecurvature, is notattainableevenif theresults of ideal cosmologicalobservationsare available. The crucial feature here is that the metriccomponentsa, VA do not affect the null geodesicpropertiesdirectly; andwhile $ doesaffect the nullgeodesicsin that it determinesthe relationof the affine parameterv to thecoordinatey, we havenodirect observationalway of determiningthis relation (which is not apropertyof the intrinsic geometryof thenull cone,but ratherof its imbeddin.gin the space—time).

What is the situation if we try to measurethe various derivatives of u’, ~, k’ and g,1 that are ofinterest?Again,we considerageneralpoint on thenull conew = w0. If we know thedataD we havethe 0, 4’ variationsof thequantitiesobserved,andsocanmeasuretheir 0, 4, coordinatederivatives;forexample, 9h1~/a0,t9h~/o4’,öz/ô0, oz/a4’ would be known(where thesederivativeswould be derivativeswith w andy held constant).By consideringthey-derivativeswherey is an observablecoordinate(e.g.r or z), onecould determinethederivativesdown the null geodesics,e.g. ôr/Oz, 0h11/Oz. Observationsover a very extendedperiod(using a fixed referenceframein a local inertial systemasa basisfor these

* As remarkedbefore,we are at presentassumingtheuniverseis transparentto this radiation.

G.F.R.Ellis et aL, Ideal observationalcosmology 339

measurements)would be neededto determinethe w-derivatives.Supposingall this waspossible,onewould haveobtainedall the coordinatederivativesof thedataD on thenull cone.

Howevereventhen,muchof theinformationonewantedwould bemissing.Firstly, onecould clearlynot obtain derivativesof thequantitiesnot in D; thus for example,onecould notdirectly measurethederivativesof the matterdensity,asthis densityis not itself directly measurable.Secondly,assomeofthe metric componentsare not known from the data D, one would not be able to determineobservationallymanyof thecovariantderivativesonewants,evenif thepartialderivativeswereknown.As an example, one would like to know the quantity Ui;j which determinesthe fluid kinematicquantities(its expansion,shear,vorticity, etc.). Howeverthe componentsu~dependon the metriccomponentsa, fi, VA (see eq. (13)) which are not known, preventing one from determiningthesequantitiesor theirderivatives.Thus,onecannotmeasuredirectly the fluid kinematicquantities.tAs anexample,the fluid expansion0 is given by 0 = (j3ht”2)1 O(.$ht”2u1)/Dx’ in the observationalcoordinates;this cannotbe directly determinedbecauseof lack of knowledge of fi and of ~ Asanotherexample,onemight think that thecomponentu i;jk~k~of U;j would be measurablebecausethisis the component of Uj;j along the direction of sight k at y; however, uj;jk1k~= (Ujk’);jk~=

d(1+ z)/dv= (fl’) d(1 + z)/dy.This is not directlymeasurable,for while d(1+ z)/dycanbe determinedif y is measurable,the unknownfactor$ thenpreventsonedeterminingthe requiredquantity.Similarproblemsarise if one attemptsto determine,for example,ua;be’~e~’where eaanka/(u,,kt~)= eia—

(heree~”is a spatial unit vectorin a fundamentalobserver’srest frame,pointing in the directionofobservationk, seeeqs.(29,30) of [1]).

Similarproblemsalso ariseif onetries to determinethe covariantderivativesof k (e.g. theopticalscalars[37]).Its covariantderivativesdependon /3 (see (17)); so for example,although r and h~areknown, their covariantderivatives0 and~ (see (16) and(18))cannotbe directly determined.This inturn preventsonefrom determiningcertainof the Weyl tensorcomponentswhich could otherwisebe,determinedfrom ô~(seepart II for theequations).

(ii) The vertex. Thesituationat thevertexq (y = 0) is quite different, in two importantways.Firstly,at ageneralpoint on the null cone,one can (from the vertex q) only actually “look down” the nulldirectionk = O/t9y, whereasat thevertexonecanlook down all pastnull directions(the congruencek isdegenerateatC, andpointsin all pastnull directionsfrom this line). Secondly,thelimiting behaviourofthe metric tensorcomponentsthere is known (eqs. (8)) as is that of some of the matter4-velocitycomponents(eq. (12)). Consequently,the only quantities in the data set S needingobservationaldeterminationat this point are the radiationandmatterdensities,determiningthe quantities~ andp(this reflects the fact that, nearenoughto C, the metric is simply that of Minkowski space-timeinobservationalcoordinates),andthe limiting valuesU’1q. Thesequantitiesare, in principle,observable.In fact thecomponent/3 of the metric will havethelimiting value 1 at q if y is chosento be the areadistancer, which is observable.Consequentlyonecan determinein addition manyof the interestingcovariantderivativesat q, in termsof observationalquantities.

Thedetailsof theobservablerelationsat q andhow theycan determinethecovariantderivationsofinterestingquantitiesat that point, aregiven in thepaperby Kristian andSachs[1].Theirresultsshowthe following. From the (redshift,areadistance)relation,onecandetermine:from the linear termin r,the symmetric part of the derivative U:;j1q, which determinesthe fluid expansion,shear, and ac-

* Therearea few exceptionalcases:for thecaseof dust with zeropropermotionsanddistortion,onecanfind directly thevorticity andcertaincomponentsof theexpansiontensor(seepart IT), if theconservationequationsT”~.a=0hold.


celeration*atq (cf. [36,13]); and from thequadratictermin r, variousderivativesof thefluid kinematicquantitiesand Ricci tensorcomponentsat q. Theseincludethe derivative0. an (U’~a);bU” in the linearcombination(~0+ Ron), and [(R®+ R)— 3R11] (nosum), oncethe Weyl tensoris known (seeeqs.(36),(70—74)of [1]).Fromthequadratictermof the (distortioneffect, areadistance)relationonecandeterminetheWeyl tensorat q completely(seeeqs.(38)and (81,82)of [1]).Fromthe(numbercount,areadistance)relation,onecan determine:from the quadraticterm, thevalueof n at q; from the termcubic in r, thevalue of the gradientsn.,, q, and in particularof n anfl.aU” at q; and from the termquartic in r, thevalueof (n;~+ ~flRab)lq (seeeq. (39) of [1]).From the (apparentmotion, areadistance)relation, onecandetermine:from the termindependentof r, the fluid shearandvorticity; from the termlinear in r,combinationsof certainRicci tensorcomponents,Weyl tensorcomponents,and derivatives of thekinematicquantities(seeeq. (49) of [1]).The relevantRicci tensorcomponentsarethosewhich, in anorthonormalframeea with u = e0, would belabelledR01,R~,R03andR12,R~,R31. The first threeofthese Ricci tensor componentscannotbe determineddirectly becausethey are only known incombinationwith unknownspatialderivativesof theexpansionand vorticity, while thesecondthreecouldbe determinedif theWeyl tensorandthekinematicquantitiesaretakento beknown from otherobservationalrelations.One shouldnote particularly two things: the numbercountrelations will notenableone to determinethe Ricci tensorcomponents,becauseonecannotindependentlydeterminethequantitiesn ;abjq from otherobservations,and so cannotseparatethesetermsfrom the termsR~Iqbyobservationalmeans.Secondly,onecanmeasurethe time derivativeof n from theobservationsmadeatonetime of theangularvariationof the r

3 term in the numbercountrelation; that is, onewould nothave to observeover an extendedtimeperiod to obtain this derivative. (Essentiallythe sameremarkappliesto the time derivative9. of 0 if the accelerationvanishesin an open neighbourhoodof q; thenthe Raychaudhuriidentity relates 0 and Ron in terms of known quantities, enabling both to bedetermined.)

Accordingly, thesituationat the vertexq is that, if onehad ideal observationsmadefrom thepoint qatone’sdisposal,onecould (without makingany assumptionsaboutthe field equations)determinethenumberdensityn there,andits first spatialandtime derivatives;the fluid kinematicquantitiesthere,and someof their derivatives (including particularly the derivatives0’ and th” of the fluid expansionandvorticity, respectively,providedtheaccelerationvanishesin an openneighbourhoodof q); theWeyltensor; and, in general, three of the ten Ricci tensor components(four if z~= 0 in an openneighbourhoodof q). TheRicci tensorcomponentsonewould beunableto determinedirectlyarethosewhich, in an orthonormalframewith u = e

0,would be labeledthe “diagonal”componentsRon,R~,R22,R33, as well as the componentsR01, R~,R03 (Ron would be determinableif U” = 0 in an openneighbourhoodof q, andR01,R~,R03if thederivativesof 0 and tu” wereknown).However,onewouldbe ableto determinethecombinationsR11— R22,R~— R33, R33— R11.

Oneshouldnote here,that in making and interpretingtheseobservations,an assumptionis madeabouttheaveragingscaleinvolvedwhenone talks aboutobservationsmade“at a point” (cf. [1]),andthis scalemust be determinedin the context of settingup the “fluid” approximation,and finding anaveragedspace-timemetric which determinesthe averagegalaxy motion in this approximation.Thisscale is such that other possible ways of determiningdistances— e.g. by parallaxmeasure— will noteffectively enableoneto moveaway from thepoint q. If thesituation wasotherwise— e.g. if parallaxmeasurescouldbe extendedto considerabledistanceson acosmologicalscale— thenmuchmorecould

* The accelerationshouldvanish,as a resultof theassumptionthat C is a geodesic(section3.1). This would follow from thestatementthat

pressuregradientsarenegligible at q (section2 above),if oneassumesmomentumconservation.

G.F.R. Ellis ci aL, Idealobservationalcosmology 341

be determinedwithout the useof field equations(in particular,one might be able to determinethequantity /3 at significantdistancesdown thenull cone).If onecould observefor a long enoughtime, itwould be conceivablethat aswell asthequantitieslisted aboveasobservableatq, onemight determinesomeof their time derivatives.However,it is most unlikely that this could be achievedin practice.Infact in this sectionwe have,asremarkedbefore,erredgreatlyon theside of generosityin allowing forwhat might be proved by observation;in particular, by allowing for the possibility that one mightdeterminedatafrom observedpropermotionsof galaxies(a more realistic interpretation— seepaperII — would insist that one cannotin practicemeasuresuchpropermotions; in this case,what one candetermineby cosmographicmethodsis evenmore restrictedthan indicatedhere).

(iii) Summary.In summary,then, theconclusionof this sectionis that by ideal observationsdownthepastnull cone, onecould in principle determinethe groupG relatingthe coordinatesusedto theactualspace-timetopology, andthedataset {BR0, D}. However,this is not sufficientto determinethecosmologicalmodel on this light cone, for many significant aspectsof the dataset {S, G} remainundeterminedby such ideal observations.At the point q (“here and now”), one can in principledeterminemost of the interestingcosmologicalquantities; the major exception is that one cannotdeterminefour of theRicci tensorcomponentsdirectly by observation.

Whenthe field equationsareintroduced,this situationis quitedifferent, aswe shall seein thesecondpart of this paper.We concludeour cosmographicinvestigationby examining,asaparticularexample,thesituationwhereourobservationsturn out to be isotropic.

6. An example:Isotropicobservationsandsphericalsymmetry

To illustrate the situation we considerhere briefly the relationsbetweenspherical symmetry ofobservations,and sphericalsymmetryof the cosmology;and thedegreeto which cosmographycanbesuccessfulin sphericallysymmetricspace-times.

6.1. Isotropic observations

The cosmologicalobservationsmadeat time w will be isotropic aboutq if theobservablerelationsimpliedby thedatasetBRa,D andthe groupG areisotropic(i.e. areinvariantunder thegroupS0(3)acting in the obviousway with 0, 4’ as coordinates).Specifically, theobservationswill be isotropic attimes w whenthe following conditionshold:

[Si]: no distortioneffectis observed,implyingfu is independentof y (seeeq. (18)); for otherwisetherewould be observabledistortion axespicking out preferreddisplacementsdo, d4’ aboutsomedirections0, 4’ in thesky (cf. [1]).Accordingly,by (7), (8), isotropic observationsimply:

h,, dx’ dr’ = r2 dD2~ f,, dx’ dr’ = ando2+ sin20d4’2 (20i)

(here,r is the areadistance,cf. sections3.2 and4.3).[S2]:no proper motionsareobserve4for otherwisethesemotionswould definevector fields on the

celestialsphere.Accordingly,by (10),isotropic observationsimply

u’=04~dO/dw=0=d4’/dw. (2Oii)


[S3]:Relationsbetweenobservablequantitiesareisotropic. From (20i, ii) and thediscussionof thedataset D, the remainingobservablevariablesin D are u°(observablethrough the redshift z), the areadistancer (which is the remainingsignificant featureof thenull coneintrinsic metric), and thequantity[n/3] determinedby the numbercounts. The relationsbetweentheseobservablequantitiesmust beisotropic, i.e. they mustnot involve thecoordinates0, 4’. This will be true if the following two relationshold at the time w: firstly, the (areadistance,redshift)relationis isotropic, i.e.

r = r(z, w)~z = z(r, w). (20iii)

Secondly,the (numbercount,areadistance)relationis isotropic; through(20iii), this will also imply thatthe (numbercount,redshift)relation is isotropic.That is, from eq. (19), we musthaveaswell as(20iii),

[fn/3] = g(z,w), (2Oiv)

for somefunction g.[S4]:The backgroundradiation BR0 is isotropic, that is, the specific intensity I~of background

radiationIs independentof theangles0, 4’:

I~(w,0, 4’)= I~(w). (20v)

Theconditionsso far areall “local” conditions,that is they refer to propertiesof observationsin eachdirection in the sky. Finally, thereis a “non-local” condition(referringto the topologyof theuniverseratherthansimply its local metric structure).That is,

[S5]:The group ofidentificationsG is invariant underrotations. This implies that any further imagesof any galaxy or eventthat may be visible mustbe eitherantipodalto theoriginal image,or collinear(i.e. lying along the sameline of sight) with it. (An exampleof this situation is the Einstein staticuniverse,with orwithout the “antipodal” identificationin that universe.)

Therefore we have the caseof isotropic observations (1.0.) if the conditions [S1—S5]hold (andspecifically thedataD obeys(20i—iv), and thedataBR0 obeys(20v)).

Supposetheseconditionsarefulfilled~Thenthe remainingnon-trivial observabledatasetat thetimew0, is the data set: D1~,an{(u°, r, n/3):w = w0, y � y*}, ~ an{I~(wo)}, and ~ (a group ofidentificationsG satisfying [S5]).In the caseof isotropic observations,this is the maximal set ofnon-zerodatawe could hopeto obtain.

It is immediately clear that in this casetoo— assumingobservationsare isotropic and we havedeterminedthemaximal ideal setof observations— wedo not havesufficient observationsto completelydeterminethecosmologyon C(q).While wewould know thequantitiesu’, h~in the set S(w0, y*), wewould not know a, /3, v, andconsequentlyalso would not know u

1 or n, anduseof the freedomtochoosey would not completelyresolvethis indeterminacy.As a particularexamplewe notethat (whenparticularfield equationshavenot beenassumed)isotropic observationsabout C do not imply that thespace—timeis sphericallysymmetricabout C. For example,supposewe havetied down the coordinatefreedom by choosingy = r. Then while (fnf3) will be isotropic, f and /3 could both be non-trivialfunctionsof 0, 4’ (/3 is unknown,andf is affectedby suchfactorsasgalacticabsorptionwhich is knownto be patchy,see e.g. [47]).Then whenone looked in different directions, thesameaffine parametervalue (which is uniquely determinedgeometricallyby the centralcondition (4)) could correspondtodifferent valuesof r; atconstantvaluesof v — representingbeing atthesamedistancefrom q down the

G.F.R.Ellis ci aL, Idealobservationalcosmology 343

null conein different directions— thevalueof n would then also vary,but in suchawaythat (J~t/3),andso thenumbercountsobserved,were isotropic.The surfacesof constantn and r would waveup anddown the null conerelativeto theuniquely definedaffine parameterdistancev (fig. 6). This situationmight well be regardedasextremelyunlikely~but thepresentpoint is that theobservationswe areableto makecannotdisproveit. Secondly,the quantitiesv1 could dependarbitrarily on 0, 4’; correspond-ingly thevelocity componentsu1 can also dependon 0 and 4’; onecould therefore,for example,haveanon-zerovorticity of thematter,eventhough all the observationswere isotropic andin particulartheobservedpropermotionsvanished*(seefig. 6).

apparent directionof galaxy G Is constant1 C

kpast null geodesucs q

move around C c—cd

— - ~ ~ constant distance from q(measured by V)

surfaces of constant6 — - - number density

galaxy world - - - - - —

line movingaround C r

C

Fig.6. Evenif observationsaboutC areisotropic, therearevariouswaysthespace—timecan beanisotropicaboutC. (i) Surfacesof constantdensity(andconstantareadistance)can lie at different affine distancesfrom C (but still giveisotropic observationsif they movetogetherin theappropriateway).(ii) Galaxiescanmovearoundtheworld line C, aslong asthenull geodesicsfrom C alsomovearoundin sucha waythattheapparentpositionof thegalaxy 0 remainsconstant.

At thepoint of observationq, thesituation would, asbefore,be betterdeterminedby the possibleobservations.The vanishingdistortion would imply that the Weyl tensorwaszero at q; an isotropic(redshift, distance)relation would imply the fluid shearand accelerationvanishedat q, while zeropropermotionswould imply the fluid vorticity also vanishedthere(cf. [36,13]). Variousgradientsof thekinematicquantitiesat q could be determined,and thespatial gradientof thenumberdensity n wouldbe zero. However,while the Ricci tensorcomponentsR~(a� b) would all be zero, one would notobservationallyobtain directlimits on theotherRicci tensorcomponentsat q.

6.2 Sphericallysymmetricspace—times

Supposethat the space—timeis sphericallysymmetricabout theworld line C. Thenthe metric form(6) andfluid stresstensor(1) will be much restricted.Onecan show, by transformingfrom theusualcoordinatesin a spherically symmetric space—time(see e.g. appendixA of [15]) to observationalcoordinates,that, assumingthecoordinatey is chosento be sphericallysymmetriccoordinate(whichwill necessarilybe true if y is chosenasan observablequantitysuchas r, or if y is chosenasa function

* Although the space—timewill necessarilycontain irrotational flows in the caseof pressure-freematterwhich obeys the usualmomentum

conservationequations(seepartH); andthe flowswill necessarilybe irrotationalat q itself, if no propermotionsaremeasured(seenextparagraph).


of the uniqueaffine parameterv: y = y(v)), not only will the conditions [S1]—[S51hold but also themetric tensorcomponentswill takethe form

a=a(w,y), /3=f3(w,y), v1=O, r=r(w,y) (21i)

and theotherobservablequantitiesandphysical featureswill be sphericallysymmetricabouttheworldline C:

u°=u°(w,y), u1=u’(w,y), n=n(w,y),

(2lii)

~ =~(w,y), p=p(w,y), j(Lrad/.~rad(W,Y).

Thatis, onehasthe caseof a SphericallySymmetricSpace—Time(SSS)if, in additionto [S1]—[S5],therearecoordinatessuchthatequations(21) hold.

From this, it is clear that the data set Sjsot(wo,y*) neededto determinea sphericallysymmetriccosmologyat time w

0 (basedon observationalcoordinateswhich reflect this sphericalsymmetry)is thedataset ((a, /3, r; U°, U’; n; grad): w = w0, 0 � y � y*}, plus the groupof identifications G1~~(thequantities~mat andP~twill be determinedfrom n by thepresumedequationsof state).It is clear thatthemaximumobservationallyavailabledataset(D1~~,BR1~~,~ — evenif completelydetermined—would be insufficient to determinethe space—timegeometry, even if we knew a priori that thisspace—timewas isotropic;becausealthough u’ anda maybe known(e.g. througheq. (11’) if y is chosenasa fluid comovingcoordinate),/3 is not observationallydeterminable;andso onecannotdeterminendirectly from the observationseither. (The crucial feature, again, is that /3 = dy/dy, and we cannotobservationallydeterminethe affine parameterv as a function of any observablecoordinate.)Thus, ifwe do not assumedynamicalequationsfor the space—timemetric, not only are ideal sphericallysymmetricobservationsunableto prove that the space—timeis sphericallysymmetric,but in additiontheyare notsufficientto fully determinethespace—timegeometryormatterdistributionevenif wemaketheassumptionthat the space—timeis sphericallysymmetricaboutC. Thus thecosmographicaim cannotbecarriedout in thecaseof asphericallysymmetricuniverseviewedfrom thecentreof symmetry(whenthe observationsare necessarilyspherically symmetric) either; although the data set required todeterminethe space—timeis less than in the general case, the observabledata is neverthelessnotsufficient to completelydeterminethe space—timeandits contents.

This is true alsoatthevertexq. If weassumethespace—timeis sphericallysymmetricaboutC, thenitfollows that theRicci tensoris sphericallysymmetricatthevertexq of thenull cone,andconsequentlythe componentsR~(a� b) of the Ricci tensorin a local orthonormalframemustvanishthere; all theWeyl tensorcomponentswill also be zerothere.The observationalrelationsevaluatedat thevertexwillenableone to determinethe quantities0 U~’;a (the fluid expansion),0re 0,1u~,n and n. Howeverthey will still not enableoneto determinetheRicci tensorcomponentsRon (unlessu” = 0, when thiscan be found)andR11 (in this caseR1,= R22= R33) at q directly. Consequentlyin this casewecannotdetermineobservationallyany of the non-zeroRicci tensorcomponentsat q.

A consequenceof this situation is that, if theobservationsaresphericallysymmetricandwe assumethe space—timeto be sphericallysymmetric,we are unableto directly observationallyproveor disprovethat the universeis a Friedmann—Robertson—Walker(FRW) universe.That is, whenwe do not assumethat we know thefield equationsfor thespace—time,onecannotobservationallyprove(ordisprove)thata spherically symmetricuniverseis in addition spatially homogeneous,even if ideal observationsareavailablefor analysis.This arisesbecauseof our inability to observationallydetermine/3 and n.

G,F.R. Ellisci aL, Idealobservationalcosmology 345

6.3. FRWuniversemodels

Supposewe go onestagefurther andassumewe know thegeometryof the universeto be that of aFRW universe, that is, the universeis spatially homogeneousas well as beingisotropic aboutC (orequivalently[36],theuniverseis isotropic for everyfundamentalobserver).Thenwe arestill unabletoobservationallydeterminesignificant featuresof theuniversedirectly (Weinberg[12,41]).

We recall that a FRW universemaybe describedin termsof coordinates{y’} = (t, P, 0, 4’) wherethemetric tensorcomponents,matter4-velocitycomponents,andgalaxynumberdensityaregivenby

ds2= —dt2+ R2(t){dP+j2(F) d122}, u’ = &‘~, n = n(t) (22)

wheref(F) is sin i, i~or sinh F according to whethera normalisedconstantk (the sign of the spatialcurvature)is + 1, 0, or —1. The spatialdistancesbetweenthesameclustersof galaxiesat different times tscale with the function R(t); in addition, if k � 0 the magnitudeof the function R determinesthecurvatureof the3-spaces{t = constant}.

First considerthesituationat thevertex q. From thepowerseriesgiven in [1], it is clear that onecandirectly determine0~and 0’~at q, from the linear andquadratictermsin the (redshift, areadistance)relation; and thequantitiesno andn

0, from thequadraticandcubic termsin the (numbercount,areadistance) relation. The quartic term in that relation gives the combination (n’+(n/6)R~(2u”u~’+ g°~”))of a secondderivativeof n, andRicci tensorterms;but thesetwo termscannotbe separatedfrom eachother by observationalmeans.Thus while at the vertex one can determineobservationallythe expansion0~(the “Hubble constant”) and its derivative 0’o (essentially the“decelerationparameter”), plus the number density n0 and its time derivative n’0, one cannotdeterminethe Ricci tensorthere,andso cannotmeasurethe magnitudeR0of the scalefunctionR(t)there,oreventhe sign of k [41](k andR0 aregivenby thespatialRicci tensorcomponentsin the ratiok/R0

2).Thus direct observationin a FRW universecannotgiveus the sign or magnitudeof the spatialcurvatureat thepointof observation(the vertexq).

To considerthesituationat a generalpoint on thepastnull cone,wechangefrom coordinates(22)tocoordinates(~,F, 0, 4’) where

= J (1/Rfr))dr+ F. (23a)

Thenthe space—timemetric andthematteraredescribedby

dS2=R2(~_~{_d~2+2dti~dF+f2(fr)dD2},u’=(1/E(~—F))6-’o, n=ñ(t~—Y); (23b)

herethe functions1~(~—F), n(~— F) arethe samespace—timefunctionsasR(t), n(t) but expressedinnew coordinates,i.e. R(~— F) = R(t), n(i~— F) = n(t), with ,~given by (23a).Thesecoordinatesarecomoving coordinateswhich have the “observationalcoordinate”form of section3 (with a = /3 =

RØ~F)), except that ~ is not normalisedto proper time along C. The vectorfield k is given by*k~= RØi~)8,°,k’ = (R(*)/R2Ø~— F))8~

1anddi5’dv = (R(~)/R2(~?— F))—~.1/R(t~)at C. The observable

quantities(effectively, the data~ arenow given by the relations

* Here/c, ~ W,j wherek, w satisfythecentralconditions(4); that is, k is thesamevectorfield asin theprecedingsections.


1 + z= E(/1~ — F), r~= /2(i) 1~2(~)/(1+ z)2,

dN = fñ(~— F) r2 di20E(~)(1+ z)’ dF. (24)

The coordinateF is not directly observable,andreducesto (r/1~(i~))at thecentreC (ratherthan to theareadistancer).

To determinethe local structureof theuniverse,we wish to know the functionsR(t), n(t) and theconstantk; particularly,wewish to know n asn(~0)andR0asR(~0)where~‘o is the time of observation(unless k = 0, when R0 has no particular significance). Now supposeideal local observationsareavailable,i.e. we know at time ~ the observerareadistanceas a function of redshift, and numbercountsasa functioneitherof redshiftor of areadistance;say

w= t~o~r=g(1+z), dN/dr=h(r) (25)

whereg, h areobservationallydeterminedfunctions.Then(24), (25) show

— F) = ~~)/(1 + z), F = J’{(i + z)g(1+ z)/1~o)}, (26a)

n(~0— F) = {h(r)/fr2 dfl

0 (1 + z)}- (dr/dF) . (1~(~o))_1. (26b)

Clearly the redshift z directly determinestheexpansionratio (the value of R at theobjectobserved,relativeto the valueof R at theobserver).However,this doesnot uniquely determinethe space—timegeometry;on the contrary, we may arbitrarily choosea value for k (so determiningthe choice offunction f(r), and henceof the inverse functionf~

1{y}) and a value for R(~),andthen (26a) willdetermineF uniquely as a functionof z, andhence(by (25)) as a functionof r; in addition, (26b)willthen(becausedr/dF is nowuniquely specified)determinefl(~?

0— F) from h(r). As z(f) is known,onecanin principle thendeterminet(F), andhenceF(t), from the relation t0 — t = f~{R(~0)/(1+ z)(F)} dF, whichfollows from (23a);sofinally R(t) andn(t) will follow from theknown formsof P(~0— F) andn(~0— F).Thus thesituationis that, whateverfunctionalrelations(25) are observed,onecanfind universemodelsinwhich theseareexactlythepredictedrelations, no matterwhatvaluewe maychoosefor thepresentvalueR0 of R, and the sign k of the curvature of the homogeneousspatial sections. In particular, localobservationsof galacticredshifts,magnitudesandnumbercountscannotby themselvesdetermineif thespace-sectionsof a FRW universeareof positive,zero,or negativespatialcurvature[12].

The situationwill be differentif onecanobservationallydeterminea non-trivial groupof G of galaxyidentifications;for particularidentificationswill only becompatiblewith particularsignsk of thespatialcurvature.Howeverthis cannotnecessarilybe usedas a conclusivetestfor the sign k, becauseevenifthespatialsectionsarecompact,theremay nothavebeensufficient time sincetheorigin of theuniversefor light to travel roundtheuniverse;i.e. wemaynotbeableto makeobservationswhich can in principledeterminethegroup G. Thusadeterminationof 0 might besufficient to determinek; but in generalitwill not be possibleto determineG.

Finally, one shouldnote that we haveso far in this section,assumedthat n(t) is a function to bedeterminedby observation.However,it is possibleto proceeddifferently, assumingthat the numberofgalaxiesin a comovingvolume is conserved(galaxiesareneithercreatednor destroyedduring the timeof observation);then n(t) = no/R

3(t), andconsequentlyn’ can be expressedin termsof observation-ally determinablequantities. In this case, one can in principle determineboth the Ricci tensor


componentsat q (which givebothk andR0), and the relationdr/dF, from thenumbercounts(cf. (26b));andnow thereare sufficient observationsto checktheFRW natureof theuniverse,by comparingthisvalue of dildf with that implied by (26a)and theobservationallydeterminedvaluesof k and R0. Asusual,considerableproblemswould arise in practicein attemptingthis (cf. paperII); in addition,it isnot clearthat it is reasonableto assumethis featureof “galaxy conservation”evenin thestandardFRWcase,wherenormalphysicsis assumed(asgalaxiesandradiosourcesareknown to evolve significantlyon cosmictime-scales),let alonewhenothertheoriesof gravityareadopted,for non-conservationof thecomoving numberof galaxiesis a particular feature of various of the alternativetheories(e.g. theSteadyStatetheory). Thus it only really makessenseto adopt this assumptionin the context of aparticularchoice of field equations;it really belongs to the cosmologicalapproachto analysingtheobservations,where particular field equationsare assumed,ratherthan the cosmographicapproachwhentheseequationsarenot takento beknown.An attemptto checktheFRW natureof theuniversein this way is really a testof the “galaxy conservation”assumption,which is almostcertainly violatedfor largeenoughtime-scales.

7. ConclusIonto pert I

We haveexaminedin detail the informationabout theuniversethatcan be extracteddirectly fromobservationsof distant objects. The information one can extract directly from ideal cosmologicalobservations— whenwe assumeall the observationalproblemshavebeen overcome— is ratherrestric-ted.Assumingwe haveovercomeall theproblemsthatmakeit difficult to determinetheareadistancerof particularsources,andthenumbercountsdN asa functioneitherof r orof redshiftz,westill cannotdeterminedirectly major featuresof significanceon our past null cone.In fact the most we coulddeterminedirectly about the space—timeis the dataset D(wo, y*) as{(u°,u

2, U3 h1~n/3): w = wo,

o� y <ye, 0, 4’ E S2}, BR

0, and thegroupof identifications0, while thedatawe needto specify thecosmologyon the past light cone is the data set S(wo,y*)=_{(uo, u’, u

2, u3 a, /3, v1, h~n; ~

w = w0, 0� y � y*, 0, 4’ E S2} plusthegroup0. Theremainingcoordinatefreedomin y canbe usedto

fix u’ anda, butwe areleft unableto determineobservationally/3, v~,or thenumberdensityof sourcesn (the latter, basicallybecausewe areunableto observationallydeterminewhat variation in the radialcoordinatey correspondsto any particulardistanceattheobjectobserved).Details aregiven in section5. As a particular exampleof this indeterminacy,we cannotdirectly observationallyprove that thespace—timeis sphericallysymmetric about us or not, even if all our cosmologicalobservationsareisotropic about iii. Even if we makethemajorassumptionthat the universehas the geometryof aFriedmann—Robertson—Walkeruniverse,we cannotdeterminedirectly from observationswhetherthespatialsectionshavepositive, negativeor zerospatialcurvature(Weinberg[41]andsection6); andweare unable,even if observationsare isotropic aboutus and we assumethe space—timeis sphericallysymmetric,to provetheuniversehasa FRW geometry(section6).

Other evidenceis availablewhich gives other clues and boundson the natureof the universe;particularlybackgroundradiationprovidesintegratedinformationdown thelight cone,andastrophysi-cal information(galaxytypesandclustering,elementabundances,andso on)put physicallimitationsontheuniverse;howeverthesedo not directlygiveusdetailedinformationon thespace—timegeometryonourpastnull cone.Ratherthey provideadditionalinformation which canbe combinedwith the directobservationalinformation, in an overall cosmologicaltheory. Our aim hasbeento provide a precise


delineation of the featuresof the universe which can, in principle, be determined directly byobservation.

The implicationsof the resultswe havedetailedare, firstly, that direct observationsdetermineonlysomeof the informationwe needto fully characterisetheuniverseon ourpastlight cone;andsecondly,that thereforeone cannot usecosmologicalobservationsdirectly to test rival theories of gravity, fordifferent theoriesof gravity will in generalbe compatiblewith thesamesetof observations(theywillsimply correspondto differentspace—timegeometrieswhich give theobservedobservations).Thuswhencosmologicalresultsare usedto limit theoriesof gravity, it will be on the basisof detailedphysicalarguments,orassumptionswhich arebasedon a priori knowledgeof thegeometryof theuniverse(notdirectly testableby cosmologicalobservations).

It is clear that direct observationscannotdeterminethe space—timeor the mattervariablesoff thepastlight coneC(q).

The cosmographicinvestigationsof this paperplace ratherpreciselimits on what can be obser-vationally directly verified without useof a specific theoryof gravity. In thesecondpartof this paper,we examine how the use of Einstein’s field equationsenables the theoretically available idealobservations(examinedin this part) to determinethe space—timegeometry,on and off the past nullconeC(q). The analysiscarriedout so far is essentialto that next step,which usesthe observationalcoordinatesintroduced in this part; particularly important are the limiting values of the metriccomponents(derivedin appendixA), and the expressions(derived in section4) for observationallymeasurablequantities.Thus thecosmographicanalysisof part I providesthe foundationon which onecan basean examinationof the cosmologicaldeductionsonecan makefrom thedirect observations,given a specific theoryof gravity that determinesthe space—timestructure.

The secondpaperin this serieswill usethe foundationprovidedby this onein order to investigatethe possibledeductionsone can makewhen realistic limits on what one can observeare taken intoaccount.It is this setof deductionsthat arethebasictools availableto the theoreticalcosmologistwhenhe makesasynthesisof thedataavailablefrom direct cosmologicalobservations,indirectobservations,andphysicalarguments.

Part II: IdealCosmologicalDataS.D. Nel, W. Stoeger,G.F.R. Ellis, A.P. Whitman,R. Maartens

8. Introductionto part II

In part I the focuswascosmography.We consideredhowmuchcanbe,in principle, foundout aboutthe structureof the universeon the basisof ideal observations,without assuminga knowledgeof thedynamicsgoverningtheevolutionof that structure— Le., without assuminga theoryof gravity. This wasdonefirst for generalcosmologicalobservations,and thenin thecasewheresuchobservationshappento be isotropic. The resultsof that cosmographicanalysiswere that, evenafter all the observationalproblemshave been overcome, it is impossible in principle to determinemajor featuresof thespace—timestructureon our past light cone— not to mention off the light cone— on the basisofobservationsalone.

In this part, we turn from cosmographyto cosmology, and examinehow assumingand usingEinstein’s field equationsasthedynamicaltheorygoverningthe largescaledevelopmentof space—time

G.F.R.Ellis eta!., Idea!observationalcosmology 349

enablestheideal observationsdescribedin part Ito determinethespace-timegeometryboth on andoffour pastlight coneC(p).

In thework we presentin this part,we againavoid assumingany symmetryof space—time,exceptafter thegeneraltreatmentwhen in section15 we specialiseour results to the sphericallysymmetriccase,andthen further to the situation in which spatial homogeneityholds. This lack of a symmetryassumptionis crucial to ourobservationalapproach:following Kristian andSachs,we wish to establishthe structureof the universewith a minimum of a priori assumptionsandwith maximumweight oncosmologicallysignificantobservations.

We first setup the problemin section9, introducing the Newman—Penrose(NP) spinorformalism,which we shall useextensively,and discussingin detail the coordinatesand tetrad we shall employwithin that framework.Then, in section 10, assumingthat the galacticworld line of the observeris aregulargeodesic,we derive thecentral conditions— the limiting behaviourof all the spin coefficients,metric variablesandcurvaturecomponentsas one approachesthe central world line, the world linefrom which the observationsare made.That is, we calculatethe behaviourof all thesequantitiesasv -*0, wherev is the affine distancedown ourpast light cone.Thesecentralconditionsareessentialforproperlysolving the field equations.In section11, we summarisein this formalism thecosmologicallysignificant informationthat can be derivedfrom astronomicalobservations,andfrom it constitutethemaximal observabledata set D(wo, z*). (As previously mentioned,this paperassumesideal obser-vations. The next paper in this series considerswhat can be done in an observationalcosmologyprogrambasedon realistic observations.)

In section12, wepresentthemajorresultof thepaper,theproofthat D(w0, z*) determinesa uniquesolution to the field equationson our past light cone down each null geodesicgeneratorto v =

min(v*, v~)where v’~is the valueof theaffine distanceparameteron our light conecorrespondingtothe limiting redshift z” beyondwhich we are without observationaldata,and v~the distancewherecausticsfirst occur. To be moreexact,given D(wo, z*), we proveexistenceanduniquenessof a solutionto Einstein’sequationson any null generatorof C(p) down to the valuev~of v wherecausticsfirstoccur, ordown to vi’, if v~<v~.Further,wealso prove that D(wo, z*) is theminimal datasetrequiredto determinetheuniquesolution.

In section13, we discussbriefly and muchless rigorously thepropagationof the solutionoff C(p)and into its interior. It seemsreasonablyclear from the argumentswe presentthat D(w0, z*) willdeterminethe structure of space—timewithin the past Cauchy developmentof that part of C(p)determinedby D(wo, z*), at leastwhen we haveanalyticdata[seebelow]. In carryingout theseproofs,we considerwhat we may call the characteristicfinal value problem in cosmology.Astronomicalobservationsconstitutea datasetarrayedalongourpastlight coneC(p)backto somelimiting redshiftz~’.We want this characteristicdatato determinethe solutionof theequations— andthusthestructureof space—time— not so muchto the futureof C(j), but ratheron C(p) itself andbackinside,into thepastof C(p). Thus,thedatais, in thestrict sense,characteristicfinal valuedata.We could also hopetouseD(wo,z*) asinitial datato determinethe structureof space-timein someregionto the futureofC(p),but this is moreproblematic.We briefly considerhow thesolutionto the futureof C(j,) may bedeterminedwhen a suitable “no-news” or “no-interference”assumptionis made(e.g. by assuminganalyticity).

We indicate in section 14 how polynomial data determinesa unique solution in this context,describingthe detailedwork of Nel [57],which extendsthe importantclassicalwork of Kristian andSachs[1].Finally, in section15, we presentan importantresult,a testfor thespatialhomogeneityof aspherically symmetric, pressure-freecosmological model. We demonstratethat •such a universe is


spatially homogeneousif and only if the Observations— that is, the r(z) and the (M dv/dz)(z)relations— take precisely the FRW form. In arriving at this result, we make contact with moreconventionaltheoreticalapproaches,specializingourgeneralresultsto the casein which the space—timeis a priori sphericallysymmetricaboutoneworld line andthencarryingout an observationalanalysis.This work also answersthe question:What observationsin a universeisotropic aboutoneworld linewill imply that it is isotropic aboutall world lines?

In this part,weusenotationwhich is standardin relativistic cosmology(andcompatiblewith part I)for the most part. However,becausethe Newman—Penrose(NP) spin coefficientsareused,themetricsignaturechosenis (+— — —). The only two symbols that are slightly unusual are M, denoting themass-energydensity,and t, denotingpropertime. Theseareusedto avoid confusionwith particularNPspin coefficients.

9. Settingup theproblem

In this sectionwe review briefly the Newman—Penrose(NP) formalism, which we shall later useextensively, review the observationalcoordinatesintroduced in part I, specify the null tetrad weemploy,andwrite down the field equationsin theNP formalism.

9.1. TheNewman—Penroseformalism

TheNewman—Penroseformalism(NewmanandPenrose[58],Pirani [59])is basedon acomplexnull

tetradbasis

{ea}={n,k,m,ñi},

whereit andk arereal null vectorfields andm is a complexnull vectorwhoserealand imaginarypartsareboth spacelike.The non-zeroscalarproductsbetweenthesevectorsare

kn=—m•,n=1, (27a)

othersbeingzero.Thus, themetric tensorcanbe written

gab = 2{k(aflb) — m(ornb)). (27b)

The affine connectionis describedby 12 complex rotation coefficientsfor the basis vectors, usuallycalled “spin coefficients”,which aregiven in appendixB. We canrepresentthe irreduciblepartsof thecurvaturetensorR~d (the Weyl tensorcomponents~ the trace-freepart of theRicci tensorR~,andtheRicci scalar)in termsof componentsrelativeto this basis.Thesearealso given in appendixB.Finally, we write the tetradderivativesas

fl’2Va, D~k4V0, (5Em’

2Va. (28)

The irreduciblepartsof the curvaturetensormay now be expressedin termsof the spin coefficientsand their derivatives,andone can then give the Ricci, Bianchi and contractedBianchi identities in

G.F.R. Ellis ci a!., Idealobservationalcosmology 351

termsof thesevariables(Pirani [59]).Thesewill be given in section9.3 below [notethat therearetwomisprintsin theequationsgiven by Pirani], afterwehavediscussedthecoordinatesystemandthenulltetradwe shalluse.

9.2. Observationalcoordinates

In part I we introducedobservationalcoordinatesbasedon theworld line of our galaxy, which wedenoteby C, and on the family of past light conesC(q) of points q on C. We briefly review thosecoordinateshere,using thesignatureadoptedin this partof thepaper,as theyarecrucialto properlyunderstandingthe remainderof thepaper.

The coordinates(X”) = (w, y, xA), A = 1, 2, aredefinedas follows.(i) x° w is definedso that thesurfaces{w = constant}are thepast light conesC(q)of eventsq on

C (see fig. 3). We normalise w by requiring that it measureproper time t along C, and let w = w0correspondto theevent“hereandnow”. Oncetheconstantw0 is chosenthis uniquelydefinesw, whichis a C’ functionalmosteverywherewithin asimply convexnormalneighborhoodof thepoint w = w0 onC, but is not C

1 on C itself. Thus, the 1-form dw is notwell-definedon C, but thevectorfield 8/8w is. Infact,

8/öwjc= 8/8t~c. (29)

On theC(q)themselveswe havethepast-directednull geodesicvectorfield k, whoseintegral curves

arethe ruling geodesicsof C(q). If v is an affine parameterdown thesegeodesics,then

k = 8/öv, or k~= df/dv, (30a)

and

kaW.a, kak0=0. (30b)

The fact thatC itself is oneof a timelike congruenceof geodesicswith tangentvector

= df/dt, (31)

gives the importantrelationship

= _uaw.a= —u°= —dw/dt. (32a)

Thus,

lirn(u.k)=—1, (32b)

which meansthat

(8/8w). k)Ic = 1. (32c)


Imposingthecondition that thecentralworld line C shouldbe designatedby v = 0, eqs.(30a~and (32c)uniquely determinev.

(ii) x1 = y: The coordinatey is someparameter(not necessarilyaffine) down the null geodesicsruling C(q) such that y = 0 on C. Choosingy to be either r or z meansy is a directly observablecoordinate,which is useful for cosmographicanalysis,asin partI. However,a distinct disadvantageofbothr andz ascoordinatechoicesis that, asPenrosehaspointedout (privatecommunication),they arebothessentiallynon-localquantities.The monotonicityproblemmentionedin section3 arisesbecauseof this. In the restof this paper,we shall choosey = v.

(iii) xA: We label the null geodesicsruling the past light conesby the “angular coordinates”X”,

A = 1, 2. Thus,

Lk(Xd~~)= 04~(XA).ak~~0, (33a)so that

k~= (dy/dy)o~ (33b)

This determinesX’~up to a relabelling of thenull generators:

= xA’(w, xA). (34)

The coordinates(w, y, xj do not cover all of space—time.There is, first of all, the standardsingularityassociatedwith thexA. Further,asdiscussedin section3, thecoordinatesdo not necessarilygive a one-to-onecoveringof the regionof space—timethat is observablefrom C.

The choicey = v implies g0, = g°

1= —1, asfollows from eq. (33b)with dy/dy 1. In fact,

gia = g~$1b = ga,,kb = . gabw b = ~ (35a)

while

gaO = g0.l~,O = gw.b = gkb = = ~ (35b)

Thus thecoordinatecomponentsof themetric tensoraregiven by

g~—1jg~ g0,/100 ~\

g~= I (35c)g0~0~

and

/ 0 —1~ 0 0( —1 g11~g12 g’

3

gab =1 . (35d)

I n 12~u g AB

\ 0 g”~


The componentsgab and gab arerelatedby

~AB~ = (35e)glA = g

08gAB, (35f)

i.e.,

g12= (g

02g33— go3g,.~)h~,g’3 = (g

03g22— go2g,.~)h1,

g’1 = —g~+ go~g = —g~+ goAgoag. (35g)

Here h = detgAB.

A lessdirect but nonethelesssignificantadvantageof choosingy = v is that this is also thecoordinateusedin moststudiesof asymptoticallyflat space—timesemployingtheNP formalism.Forourpurposesoneof themostimportantof theseis theNewman—Unti (NU) analysisof thecharacteristicinitial valueproblemin asymptoticallyflat space—time(NewmanandUnti [60];seealso Dixon [61]).It turnsout that thereis acloseanalogy betweenthis problemand the cosmologicalcharacteristicfinal value problem we arestudying. Usingy = v enablesusto exploit this analogydirectly.

Of course,v is not an “observational”coordinate,sincethereis cosmographicallyno observationalmethodfor relatingv to either r or z. It is preciselyfor this reasonthat the resultsof ourcosmographicalanalysisin partI areratherlimited. However,if a setof field equationsis assumed,thenull Raychaudhuriequationmay be usedto determinev asafunctionof observablequantities(seesection12). Thus,in acosmologicalanalysis,thefact that v is not observationalposesno greatproblems.

Thecoordinates(xA) arenot asyet necessarilyobservational.For, supposeXA1 andXA2 label particular

null generatorswith tangentvectorsk1 andk2, respectively,on the pastlight conesC(p,)andC(p2)of twodistinctpointsPi andP2on C. Supposefurtherthat k, andk2 arerelatedby Fermi—WalkertransportalongC, i.e., theypoint in the “same”null directionwith respectto a non-rotatingframeon C. Thenthe(xA) willbe “observationalcoordinates”if this impliesXAI = XA2, i.e.,1ff generatorsstartingoff from Cin the“samenull direction” areassignedthesamecoordinatelabelsXA. In part I, thecoordinatefreedom(34)wasusedto make(xA) observational,by demandingthat theybebasedon thedirectioncosinesofk with respectto aparallellypropagatedorthonormaltetradon C(sinceC is assumedgeodesic,paralleltransportis equivalentto Fermi—Walkertransport).

In this partof thepaperweshall proceeddifferently. Insteadof using freedom(34) to makethe xA

observational,we shallusethisfreedomto choosesomeof thearbitrary functionsof integrationthatariseduring thederivationof thecentralconditionsin section 10.1. As far aspossible,we shall choosethesefunctions to havethe samevalues(up to a sign) they are given in the NU integration.A remarkableconsequenceof this strategyis that, aswill be shownin section10.2,thecoordinates(xA) obtainedin thiswayare,in fact, observationalin thesensedefinedabove.

9.3. The null tetrad

The most general null tetrad satisfying the normalisationconditions (27b), with k tangentto the

generatorsof thepast light conesC(q), is given by (Dixon [61]):

k” = (36a)

354 G.F.R. Ellis eta!., Idealobservationalcosmology= ~ + uo1a+ XAÔAa (36b)

ma = ~i5ia + ~ôAa, A = 2,3, (36c)

whereU andXA arereal,whereaso and~ arecomplexarbitraryfunctionsof thecoordinates,which arerelatedto themetric componentsby thecompletenessrelation (27b):

gU = 2(U— ox*3) (37a)

g1A=xA_(~A(;j+~ZAw) (37b)

gAB = _(eA~B+ ~A~B) (37c)

This tetradis well-definedonly off C. OnC it is singular,becausetherek pointsin a 2-sphere’sworthofnull directions.By (30) k is thehypersurfaceorthogonal(k[a;bJ = 0) null geodesic(k°ka= 0, k”;bk” = 0)vectorfield orthogonalto thenull hypersurfaces(w = constant)(w,0k” = 0), with v an affine parameteralong thesegeodesics.With this specification the following relationshipsobtain amongthe NP spincoefficients(seeappendixB):

K—O, e+~=~, p—~ö=O, rã+~8. (38)

Usingthis simplification, thecommutatorsbetweenthetetradderivativesD, 4 ando maybewritten as[58]:

4D—D4 (y+ ~)D—(r+~)~—(~+ir)ö (39a)

(39b)

M—4ô—iD+Aô+(jz—y+j~)S (39c)

(39d)

By applyingthesecommutatorsto eachofthecoordinatesw, v, XA in turn, the“metricvariables”U, XA, ~,

~A and the spin coefficientsmaybe relatedby the “metricequations”(Dixon [61]):

(40a)

(40b)

DXA = (f + ir)~A + (r + ~)5i~A (40c)

DU=(~+ir)w+(r+*)~i—(y+~) (40d)

(41a)

G.F.R. Ellis ci a!., Idealobservationalcosmology 35~

öU—Aw+~—y)w+A~—i~ (41b)

(41c)

(41d)

By applying theRicci identitiesto thetetradvectorsit, k, m, in, the 18 NPequationsareobtained(Dixon

[61]):

Dpp2+oô+’~bcxj (42a)

Do(2p+4e)o+1!’0 (42b)

Dr=fr+ i~)p+(i~+ir)u+2er+ ~‘i+ ~oi (42c)

(42d)

D$—&=(a+ir)q+(p+e)$+(~—ã)e+!P1 (42c)

y—4e(r+ff)a+(~?+ir)$—(~+~)e+rir+!P2—A+~,i (42f)

DA—~ir(p—4e)A+e1u+(ir+a—~)ir+~ (42g)

D~—öirp4u+o-A+1~*—(ã—$)1r+~‘2+2A (42h)

Dv— Air = (ir + ~)jt + (~ + r)A +(y — ~ )ir — 2ev+ !P3+ ~21 (42i)

4A—~v(~—3y—~t—~i)A+(2a+ir)v—!!‘4 (43a)

(43b)

(43c)

(43d)

(43e)

(43f)

(43g)

356 G.F.R.Ellis ci aL, Idea! observationalcosmology

4p — = (y + ~ — ,i)p — crA — 2ar — !P’2 — 2A (43h)

Aa—~y=(p+e)v—(r+$)A+(~—y—~i)a—!P3. (43i)

The NPequationsimply theBianchi identitiesandcontractedBianchi identities(Dixon [61]):

D(~0,— i!’,) S~oo±~f’o = (4a— ir)!Po— (4p+ 2e)!t’~+ (ii~— 2r)’Poo+ 2(p+ e)~o,+ ~ (44a)

D(P11+A —

+ ir4~+ (i~— 2d)~1o+2P~it+ o4~ (44b)

~

(44c)

D~t’4—~(~21+ ~!‘3)+4~~= —3A!t’2+2(a+2ir)~t’3+(p—4e)~t’4+2v~,o—2A~i,

+ (2~—27--~i)~ — 2/~P21+ °~~22 (44d)

4!!’o8(~o~+~

+ 2(~— $)~ + 2o4,~+ (p+ 4e)~~ (45a)

~ 14~OO

+ 2(fl — — y)t’P0, — 2A~10+ 2(r+ 2~ff)~11

+ (3a— ~ +2i~)~o~+ 4eP,2+ 2o42~ (45b)

A (3~!’~— 2~,,)— 8(~21+ 3~P3)+2S~12+ Db~= 6i’!P1 — 9~i1P2— 6äV’3+ 3o~!1’4— 2i4~~— 2i4~~~

+ 2(2~i— ~ +2A~—£~o+2(ir+ ~F—2~)~12

+ 2($+ r+ ~)~21 — (45c)

A(~I’3—~21)ô~4ô’~b22 3vW’2—2(y+21s)!P3+(4fl— r)!1’4

— ~+2A~12 +2(y +~i)~21—~ (45d)

D(~,,+3A)—o~10—ä.~0,+4~=(2y+2~—~—~ti)~oo+fr—2a~

+ (~ — 2ã— 2r)~jo.+4P~ii+ o4~

02+ cr’P~o (46a)

D~12—o(~11—3A)—g.~÷4~01=i~®+(2y—~—2~ti)~o1—A~,o+2(*—~

f (ir + — 3a)~o2+ (3p+ 2e)~,z+ O~21 (46b)

Do~21ä~P,2+4(~,,+3A)~

+ (2n-— f + 2~)~,2+ (2n~— r+2~)~2,+2p~n. (46c)

G.F.R. Ellis et aL, Idea!observationalcosmology 357

Theremainingtetrad freedomthat preservesall the relationsintroducedso far is (NewmanandUnti[60]):(i) a null rotationaboutk:

ka’=ka, fla’na+BBka+Bma+Brna (47a)

ma’=ma+Bka, (B(x”)EC); (4Th)

(ii) aspatialrotation in the (m, ,n)-plane:

ka’=k~~, n”=n”, m~z’=e~Crn(~,(C(xa)ER). (47c)

Equations(39—46) may besimplified in anumberof waysby furtherspecialisationof thenull tetrad(36). Throughoutthis paper,exceptin discussingtheBondi~-Sachsalternativeapproach(section 13.2),we demandthat thenull tetradbe parallelly propagatedalongthe integralcurvesof k,so that

VkCa=0. (48)

Thespin coefficientsthen satisfy thestrongerconditions

KO, e=0, ir=0, p—j5O, rã+f3. (49)

At the sametime the remainingtetrad freedomis given by (47) with B and C now independentof v,i.e.,

DB=0, DC=0. (50)

With this formalismin place,we cannow derivethelimiting behaviourof the spin coefficientsandof

themetric tensor.

10. Thecentralconditions

In section10.1, we derivewhat we havecalledthecentralconditionsof theNP spincoefficientsandof themetric tensorcomponents— their limiting valuesaswe approachthecentralworld line C alongany oneof the null geodesicgenerators of theC~(p).Afterwards,in section10.2, we provethat theangularcoordinateswe havechosenareobservational.Finally, we calculatein section10.3 thecentralbehaviourof the fluid 4-velocitycomponents.

10.1. Deriving the centralconditions

The factthat C is aregularworld line implies that all tetradcomponentsof thecurvaturetensor(see

appendixB) must be boundedas functions of v in the limit v-+0, i.e.,

•,,=o(1), A =o(1), c’0o(1), (51)

358 Q.F.R. Ellis ci aL, Idealobservationalcosmology

I,J=0, 1,2; 0=0, 1,...,4.Heretheordersymbolo( )isdefinedby

f(w, v, xA) = o[g(v)J if If(w, v, xA)I <g(v) F(w, xA), (52)

for someF(w, xA) and for all small v.For supposethat (51) doesnot hold. Then, from the transformationlaws for the Weyl and Ricci

tensorcomponentsgiven in appendixB, it follows that (51) would also be violated in any parallellypropagatednull tetrad.Therefore,it would be possibleto constructan orthonormalframefrom {ea} inwhich at leastoneof thecurvaturetensorcomponentsdiverges,contraryto the assumptionthat C isregular.This argument,of course,only worksbecausek hasbeendetermineduniquely.Thus, “boost”transformationsof theform k’~’= ska, ~4’ = are excludedfrom the groupof permissibletetradtransformations.

The next assumptionwe requirederivesfrom the fact that, in Minkowski space,a null cone(asopposedto any otherkind of null hypersurface)is characterisedby p = — 1/v, o = 0. Now Posadas[62]hasshownthat theonly solutionsof equations(42a,42b),which now takethe form

Dp=p2+~+o(1) (53a)

Du = 2p~+ o(1), (53b)

subjectto conditions(51) and (49) andhavingthe limiting behaviourp -~— 1/y, o -~0, are given by

p=—1/v+o(v) (54a)

o=o(v). (54b)

Sincethesurfaces{w = constant}arelight conesof eventsp on C, andC is regular,conditions(54)musthold.

Finally, we shall assumethat we may takeas manyangularderivatives(derivativeswith respecttoXA, A = 2, 3) of the quantitiesin (51) and (54) as may be required in our subsequentwork withoutspoiling the limiting behaviourgiven there:

~iJ.A,A2...A,=0(l); P.B,B2...B,,0(V), A1,B1=2,3, (55)

with similar expressionsobtainingfor A, !P0 andu.We now give two lemmaswhich will aid us in determiningthe central conditionsof the other

quantities.

Lemma 10.1 (NewmanandPenrose[58]):Let thecomplex(n X n) matrix O and the complexcolumnn-vectorh be givenfunctionsof s, with

O = o(s2) and l~= o(s2). (56)

Let the (n X n) matrix J~be independentof s and have no eigenvalueswith positive real part.Supposefurtherthat any eigenvalueof H with vanishingrealpart is regular,i.e., its multiplicity is equal

G.F.R. Ellis eta!., Ideal observationalcosmology 359

to the numberof linearlyindependenteigenvectorscorrespondingto it. Then all solutionsof

dl4Ids = (J~~_1+ O)u + (57)

areboundedfunctionsof s as s-+ ~, u = o(1).

Lemma10.2: SupposeG = o(v~)and h = o(v”’), where G is a complex (n x n) matrix and h is acomplex column n-vector.Then all the solutions of

Dx = (—Itr1 + G)x + h (58)

are suchthatx = o(v’).

Proof: Let s = ~ Then Dx = —(n + 1)s(ht+2~hI+1)dx/ds, andG = o(s~1~)andh = o(s~1~”~1~).

Thus,eq. (58) becomes

dx/ds = {1/(n + 1)}{(1f1 — Gs 2~1~n41~)~— hs2~”~’~}. (59)

Now, let x = uv~’= ~ With this substitution,eq. (59) takesthe form

du/ds= —{1/(n+ 1)}Gs~~ — {1/(n + 1)}hs”~3~”~1~.

But this is just

du/ds=Ou+l~,

where

O = —~G[o(s n+l))]

5 2)/(n+1) = 0(5—2)

and

= —~h[o(s~~~ = o(s2).

Therefore,by lemma10.1

u = o(1) and x = o(v1). Q.E.D.

It is now easyto showthat (Posadas[62])

—.~e 2~I4\ 2 —21 A

gAB—~ABv-rO~V,, ~A~—~A9~W,XUsing eq. (54) andcondition(49), we canwrite eq. (40a) as

D~A= [—v’ + o(v)]~A+ [o(v)]~. (61a)

360 G.F.R. Elliseta!., Idealobservationalcosmology

Lemma10.2now applies,with

X~~A

Hence~ = o(v1). Substitutingthis resultbackinto (61a) gives

D~A+ v~’~= o(1)

or

D(v~)= o(v). (61b)

Sincewe canintegrateordersymbols(but notdifferentiatethem), it follows that

~cA~Av_1+O(v), tAA(WXA) (62)

Therefore,by (37c)

gAB = _(~A~B+ ~0~)v_2+ o(1) jABv2+0(1). (63)

If we thereforedefine~ by

= ÔA, i.e., + ~ ô~

4, (64)

then(60) is an immediateconsequenceof (35e)and (63). Now (64)defines~AB if detgAB 4 0. In ordertoensurethat thisconditionis satisfied(asit mustbe,sinceC is regular)thecoordinatefreedom(34)isusedtoset

~2j~3_p, (65a)

where

P=P(w,~)40,with~vax2+ix3. (65b)

By using the tetradfreedom(4Th), it is furthermorepossibleto make

P=f’. (65c)

The coordinates(xA) arenow, in the limit v -+0, conformallyfiat (or isothermal):

jAB = diag(—2P2,—2P2), (66a)

= diag(—~P2,—~P2). (66b)

Thus,the 2-dimensionalcross-sectionsof thepastlight cones{w = constant,v = constant}havethe line

G.FR.Ellis eta!., Idealobservationalcosmology 361

element

dq2= —g,.~dXA dx~= (~P2d~d()v2+ o(v4). (66c)

The remaining tetrad freedom is (47a), and the remainingcoordinatefreedom is (Posadas[62])

a holomorphictransformation

(67)

Now considereqs.(42d,e). Using (61,64), thesemaybe written as

D() = (_v1 ~~)(;) + (o~,)o(~v))() + 0(1). (68)

Lemma10.2nowimplies a = o(v’), /3 o(v’). Substitutingtheseresultsbackinto (68), we find (see(61,62)):

a = a°v1+ o(v), a°= a°(w,xA) (69a)

/3 = $°v’+ o(v), /30 = /3°(w,xA). (69b)

Therefore,by (49),

r= r°v’+o(v), r°=ã°+/3° (69c)

The remainingtetradfreedomis now usedto set r~= 0, asfollows. Undera null rotation (47a), thespin coefficientsp ando~areleft invariant(seeappendixC), while r transformsas

= r+ Ba + Ep = (r°— .~)v’+ o(v).

By choosing~ = r°,we thereforehave(r’)°= 0. Hence,droppingprimes,wehave

r=o(v) (70a)

and

d0+$0~O (70b)

Equations(40b)and (42g,h)now read

Ow = [—v1+ o(v)Jw + [o(v)] ui + o(v) (71a)

DA = [—v~+ o(v)] A + [o(v)] ~i + o(1) (71b)

= [—v1+ o(v)] ~u+ [o(v)] A + 0(1). (71c)

362 G.F.R. Ellis et al., Idea!observationalcosmology

Hence

= w°v1+ o(v), w°= w°(w,xA) (72a)

A=A°v1+o(v), AO=AO(w,XA) (72b)

= ~°v1 + o(v), ~ = ~°(w,xA). (72c)

From (40c) and (42f, i) it follows that DXA, Dy and Dvareall 0(1), andthus

= 5~A(~xA)+ 0(v) (72d)

y = y°(w,xA) + o(v) (72e)

v= vo(w,xA)+o(v). (72f)

Before integrating(40d), we first substitutethe resultsobtainedthus far into (43b), which, with theaid of (28), (36) and(51), may be written as

wDp+ eAp — ~Da — (A~ = o(1). (73)

But by (51, 53, 54, 55)

Dp = [—v’ +o(v)J2+ o(1)= —v~2+ 0(1),

Do-=o(1), p,A0(V), a,~=o(v). (74)

Substitutingthese,togetherwith (62, 72a), into (73), we thencan showthat w°= 0, so that, by (72a),

w = o(v). But now (71a)givesDw + v’w = o(v),

which integratesto

w = o(v2). (75)

Hence,by (40d)and(72e),

U U°—(’y°+y°)v+o(v2), (76a)

whereU°is fixed, using (37a),by the requirementthat w measurespropertime alongC:

U0 = —i. (76b)

From (41a) we thenfind

G.F.R. Ellis ci a!., Idealobservationalcosmology 363

(~0+1)~A + £0~A= 0 (77a)

k22— k23 = + jk~,2 = 0 (77b)

2~°+ kAQogP),A — (log P),0=0. (77c)

Togetherwith (65), eq. (77a) implies

~ A°=0, (78a)

while (7Th) shows thatk2 + is holomorphic,and thus the coordinatefreedom(67) canbe usedto

set

kA_o (78b)

SinceP is real, (77c)now gives

y°=~°=~(logP),o. (78c)

The remainingcoordinatefreedomis then

(79)

In theNU integration,~° is set to zero by a coordinatetransformationinvolving w. In our casenosuch freedom exists, becausew is uniquely determined.Instead,we shall deducey°= 0 from thecondition thatC is geodesic.The sameargumentalso establishesthat v°= 0.

Let {Ea} = {E0, E,}, where E0 is timelike and E1 (i = 1,2,3) is spacelike,be an orthonormaltetradthat is well-definedon C. Clearly sucha tetradexists,sinceC is regular.Let E~cbe tangentto C,

(E0)=(a/aw), (80)

where we have introducedthenotation (valid for the remainderof this section)that roundbracketsdenotethe limit of aquantityon approachingC alonga particularnull generator{xA}. Becauseboth thenull tetrad (36) andthe null coordinatesystemaresingularon C, a quantity suchas(y) = y°will, ingeneral,be afunctionof theangularcoordinates(xA). In what follows, this functional dependencewillnot be indicatedexplicitly by writing (y~x”),it beingunderstoodthat all suchexpressionsare to beevaluatedby takingthelimit v-+0alongaparticularnull generator.Similarly,(k)denotesthetangentvector,on C, to somespecific null geodesic.

Since C is geodesic,V(~){(EO)}= 0. In addition, we demandthat (E) also be parallelly propagatedalongC. Thus

V(~){(Ea)}0. (81)

The null tetrad{ea} is relatedto {Ea} by

= eGbEb,for someea”. (82)

364 G.F.R. Ellis et aL, Idea! observationalcosmology

Now (32c) and(80) imply (ej°)= —1, i.e.,

(k) = —(E0) + (k’E,), (83a)

where,sincek is null, the3-dimensionalscalarproductk’k, satisfies

(k’k1)= (e1’ e17) = —1, (83b)

indicesbeingraisedandloweredby theorthonormaltetradmetric,gab = diag (1, —1, —1, —1).The singularity of the null tetrad {ea} is here reflected by the fact that k does not tend to a

well-definedlimit asC is approachedalongdifferent generators.From (36) andthecentrallimits derivedthus far, it is possibleto showthat

(E0) = (3/i9w) = —(n + ~k) (84a)

andhence

(n) = —~(E0)— ~(k’E1). (84b)

Thenk . m = 0 = n • m implies that (e20)= 0 = (e

30), and thus

(85a)

where

(m’m~)= 0 = (m’k1), (m

t,ñ~)= —1. (85b)

Using thedefinitionsof the spin coefficientsgiven in appendixB, a straightforwardcalculationshowsthat V(~){(Eo)}= 0 1ff

(y + ~)(k’)+ (v—~iXm’)—(i~—~r)(,ñ1)= 0.

By (83b, 85b), it follows that (y .+ ~) and (v — ~) vanish. But by (70a, 72e, f), (r) = 0, (y + ~) y°+ ~°

and (v) = i.’°. Thusby (78c),(86a)

P,0=0. (86b)

Now (76) gives

U=—~+o(v2). (87)

The explicit form of P may now bederivedfrom (41c), (43c) and (86b).From (41c) it is foundthat

(88)


so that (43c) implies

p32p/.~9~ (aP/a~)(8P/o() = ~. (89)In derivingtheseequations,many of thecentral limits deducedthus far havebeenused.

The general1-parameterfamily of solutionsof (89) is

P=a(1+b~~), a2b=1, (90a)

where a and b are real constants[recallthat, by (65b) and (86b)~P.0 = 0=P,1]. Using someof the

coordinatefreedom(79), wemay set b = 1, in which casea = 1/2\/2. Thus

P = (1/2V~)(1+ Ce), (90b)

andnow (66b)implies

dq02 lirn[—v2 g~dxA dx8] = 4(1+Co_i dCd~. (90c)

This is simply the line elementof the unit 2-sphereexpressedin termsof thestereographiccoordinate

C~e”cot(O/2), (91)

where(0, 4’) aretheusual 2-spherepolar coordinates.

The remainingcoordinatefreedomis (79) subjectto dC’/dC = ±1,i.e.

C’=±C+c (cEC). (92)

Theseare theso-called“rigid rotations”(Goldbergetal. [63]),which form a 3-parametersubgroupofthe full 6-parametergroupof conformalmappingsof theunit 2-sphereontoitself.

Collecting our results,the limiting behaviourof the spin coefficients(otherthan K, e and ir, whichsatisfy (49)) is therefore

a, r, v, A, y = o(v),

p=—v1+o(v), ~=—~v~+o(v),

a = a°v’+o(v), /3 = $°v1+o(v), (93a)

where

= _a0 = aP/a~(1/2V~)C. (93b)

Themetric variablessatisfy

U=—~+o(v2), w=o(v2), XA=o(v), ~A~~Av_i+o(v), (94a)

366 GF.R.Ellis ci aL, Idealobservationalcosmology

where

= i~= —P = —(1/2V2)(1 + CC). (94b)

The coordinatecomponentsof themetric tensorfollow from (35,37) and(94):

goa=_15~i, g

11=—1+o(v2), giAo(v),

gAB = _2PZÔABV_2+ o(1) (95a)

gia = ~8°a, g~= 1+o(v2), ~oA = o(v3),— 1o—2a 2~ a’ 4

gAB——2r OABV mOkV

10.2. Observational angular coordinates

We mustnow showthat the coordinates(x’~)are observationalcoordinatesin thesensedefinedinsection9.2. The orthonormaltetrad{E

0} introducedin section10.1 is completelyspecifiedon C onceaspaceliketriad{E~}hasbeenchosenatonepointon C. Thereremains,however,the freedomto choosehow this tetrad is propagatedoff C, and hencewe may set

V(~){(Ea)}= 0. (96)

The projection~k of the null vector k into the instantaneousrestspaceof an observerwith world

line C is, by (83a)= (k)+ (E0)= (k~E1). (97)

Thus the (k’) are simply the direction cosines of 1k with respect to the parallelly propagatedorthonormaltriad {E,}. But, from (84a) and from thecentrallimits (93), it is easyto showthat

= 0, (98a)

so that, by (81),

= (k

t).

0 = 0. (98b)

Thus the samedirection cosinesare assignedto the samenull directions (in the non-rotatingframe{E4}). It doesnot yet follow that thesamenull directionsareassignedthesamecoordinatelabelsXA. Inorderto showthat this is indeedthecase,we shall prove that wecan choose

(kt) = [sin0 sin 4’, sin 0 cos4’, cos0], (99)

where(0~ 4’) aredefinedby (91). It shouldbe noticed,however,that this choiceis not unique.In deriving (99), it is convenient to use the differential operator~ and the associatedclassof

functions,theso-calledspin-ssphericalharmonics(GoldbergCt al. [63];NewmanandPenrose[64]).Fora 2-spherewith line element


dq02= ~P2dCdO, (100)

wedefine

= 2V~P~ (P3~) (lOla)

= 2V~P1k’ ~(P~~) (lOib)

for any quantity ,~of spin weights, i.e.,if undera transformationof the form

~ja’~a~iC, (CER), (102a)

~ transformsas

(102b)

whereRe(,ñ)andIm(iñ) are two realvectorfields tangentto the2-sphere,andii = 0, in ñï = —1.Now Re(m)andIm(m) are tangentto the2-surfaces{w = constant,v = constant}.Equation(90) showsthat, in thelimit v-.0, these2-surfacesareconformalto theunit 2-sphere,with conformalfactor—v~2.Thus, for anyquantity ,~which is independentof v, we may replacein in thedefinition(102)of s by rnIt thenfollows from (94) that

= 4.J~(~A~ +2sã°~)(103)

= ~\/~ (~A~— 2sa°~).

The spin-ssphericalharmonics3Y~,,,(Goldberget al. [63])aredefinedby

= a~[(e—s)! (t+ s)!]_la (1+ C~’~ (é S) (~+fs+_Sm)CP(_()~m, (1~a)

with

= (—1)’m[(e+ m)! (1’— m)! (2e+ l)/4irJ”~,

where

JsJ�e, mJ_<(,

andsatisfy

= ~ (nosummationimplied), (104b)

= fte—s)(6+s+ 1)11/2~ Yen, (104c)

368 G.FR.Ellis ci aL, Idealobservationalcosmology

= —[(é+ s)(P—S+ 1)]h/2~_~Yem, (104d)

~,Yem = ~(f~ s)(�+ s+ 1) sYem. (104e)

They form a completeorthonormalset of functions for eachvalue of s; i.e., any function which isregularon the2-sphereandwhich hasspinweights canbe expandedin a seriesin ,Yem. [Forreference,thespin-weightsof theNPvariablesaregiven in appendixC.]

From thedefinition of the NP spin coefficients(appendixB)

Vmrn~ên+Ak+(fl—d)m (105a)

V_k=rk—pm—crñi. (105b)

It follows from theserelationshipsand from (36), (48), (83—85), (93 and94) and (101—105)that

(106a)

and

~(k’)= —‘s[2(mJ). (106b)

It mustbe pointedout herethat, accordingto eqs.(101),~ is definedasan operatoron scalars.Whenitoperateson thecomponentsof a vector,asit doesin eqs.(106), it thereforeis not a covariantoperator.Thus, thevectorbasisbeingusedmustbe specifiedat thesametime to avoid ambiguity.In this case,weareusing thebasis{Ea} introducedin section10.1.

Equations(106) canalsobe derivedusing theso-calledGHP formalism(Geroch,Held andPenrose[65];Ne! [57]).

Furthermore,similarly to (98), we have(m~),~= 0. Thus,by (104c, 106a),

(m1)=a~”1Y1~ (p101)

(with the summation convention holding), where a1” are complex constantsdeterminedby the

conditions(85b).Thus,we canmakethechoice(m’) = (1/V2)(1+ CC)~~[i(1 + (2), 1 — (2, 20], (107a)

and we can find a solution (but not the generalsolution) to (106b),with theconstantsof integration

beingdeterminedby (83b,85b).We obtain

(k’) = (1 + CC)_1 [i((—C), C+ ~ CC—i]. (10Th)

Finally, by substitutingfor C from (91) into (10Th), we obtain (99). Thus, thesamenull directionsare

assignedthesamelabelsXA.

10.3. Central conditionson thefluid 4-velocitycomponents

The centralbehaviourof thefluid 4-velocitycomponentscan now be calculated.Using the central

G.F.R.Ellis ci aL, Ideal observationalcosmology 369

limits (94), it is readily shown that (36) implies

8/Ow = —n + [—~ + o(v2)] k+ o(v2)rn + o(v2)Al (108a)

= k (lO8b)

8/Ox2 = o(v3)k — (l/2P) v [1+ o(v)] [rn+ in] (10&)

8/Ox3 = 0(v3)k— (i/2P)v [1+ o(v)] [m— in], (108d)

from which it follows that

u = —u°n+ {[—~+ o(v2)J u°+u’ + o(v3)u2+ o(v3)u3}k

+ {o(v2)u0_~v[1+o(v)][u2+iu3J}m + {o(v2)u0_~v[1+o(v)][u2_iu3]J1n. (109)

But, by (29) and(84a)

UIc = 8/8W~c= —(ii + ~k)~= EoJc. (110)

Comparisonof (109) and (110) gives

limu°=1 (lila)v-.0

limu1=O (ilib)

lim U” = i~”(wx8), (ilic)v-.O

wherei2A(w, x8) arearbitraryfunctions.Finally, thetotal energydensity,denotedby M from nowon, in order to distinguishit from the spin

coefficientj~,aswell asthepressurep, mustboth tendto well-defined limits (sincethey arescalars,andC is regular).Thus,

lim M = M°(w) (112a)v—DO

and

limp=p°(w). (ll2b)v—.O


11. Coiunoiogically ~gnificantobiervations

In order to study the cosmologicalcharacteristicfinal value problem,we must first review whatcosmologicallysignificant information canbe derivedfrom observations.Here we furtherdevelopandapplythe resultsof sections4 and5, constructingthemaximal dataseton our pastlight conein termsofthepresentnotation.As will be shownin section12, this datasetuniquely determinesthe solutiontothe Einstein field equationson that light cone, back to some limiting redshift z~.Undersuitableconditions,this solutioncanbepropagatedbackinsidethelight cone.We shall discussthataspectof theproblemsubsequentlyin section13.

11.1 Observationsofdistantgalaxies

(i) Reds/sift.When light emittedat awavelengthAe by somedistantgalaxy is observedon C to havea wavelengthA0, the redshiftof thesourceis definedto be

z = A0/A~—1.

As wesawin section4, it follows that

1+z=dw/dt=u°. (113)

Thus, the4-velocitycomponentu°is a directly observablequantity.

(ii) Propermotions.ThecomponentsUA arealsoobservable(in principle),andmaybedeterminedbymeasurementsof propermotions.Thecelestialsphereof an observerat p on C maybe definedas thesetof all unit spacelikevectorsin the instantaneousrestspaceof theobserver,i.e.,

S~,= {X E T1,,M:1 u =0,X . X = —1}.

But, from thediscussionin section10.2, it follows that S~,is simply thesetof projectednull directionsat

p, i.e.,

Sp={±kV):XAER),

where.,k is definedby (97).Now considera sourceon C(p)which is assignedangularcoordinates(xA~),i.e., asourcejoined to p

by a null geodesicwhosetangentvectorhasspatial projection1k(x”~).In general,thesecoordinateswillvary with time on C. At somelater time, correspondingto the eventp’ on C, thesamesourcewill beassigneddifferent angularcoordinates(x”~.).Supposethat p andp’ correspondto propertimes wo andw, respectively,on C. Thepropermotion of thesourceat w0 is definedto be

M” lim {x’~,— x”~}/(w— w0) (114a)w-.wo

or

G.F.R.Ellis eta!.,Idealobservationalcosmology 371

M” ~dxA/dwlw..wo. (114b)

ThusM” describestherateof changein positionof a galaxyon thecelestialsphereasobservedfrom C,relativeto a local non-rotatingreferenceframe.Of course,this definition only makessensebecausethe

are observationalcoordinates,as was shown in section10.2. An alternativedefinition of propermotions,usedin Kristian andSachs[1],is obtainedby consideringdk’/dw, wherethek’ aredefinedby(83). Thesetwo definitionsareclearlyequivalentand maybe relatedby (lO7b). For ourpurposes(114)is moreconvenient.

In orderto relatethe propermotionsM’~to the4-velocitycomponentsUA, we use(31) and (113)tofind

= (1 + z)_1MA. (115)

Thus the UA aredirectlyobservablequantities.The remaining 4-velocity component,u1, is not observablebut has to be found from the nor-

malisationconditionUaUa = 1:

(l+z~g00_2(1±z)ul+2(1+z)g0~uA+uAuBg~=1. (116)

From thecentrallimits (lllc) it follows that thepropermotionsdo not necessarilyvanishon C. Thisreflectsthefact that the fluid shearand thevorticity maybe non-zeroon C.

(iii) Observerareadistanceanddistortion.Both theshapeandsizeof theimageof anobservedobjectdependon thepaththenull geodesics,which conveyinformationfrom thesourceto theobserver,takethrough space—time.The Weyl curvature of space—timemay introduce distortion, and the Riccicurvaturemay causefocussing.The 2-dimensionalline elementat thesourceis givenby

d12 = gAB dXA dxB, (117)

where dP representsdistancesat the object perpendicularto the line of sight, and the dxA are thecorrespondingangulardisplacements.If theorientationand intrinsic size andshapeof distantobjectsare known from astrophysicalconsiderations,dP can be estimated;andthe dx’t aredirectly measur-able.By carryingout measurementsin all directionsdown to somelimiting redshiftz~= z*(xA), onecan thereforedetermine,in principle, themetric componentsg,.~asfunctionsof (z,xA).

It is well known (seee.g. Pirani [59] and Penrose[381)that, with thechoice of null tetrad (36), thefocussingeffect may be describedin termsof thespin coefficient p, while the distortion of the nullgeodesicsis represented,up to a phasefactor,by the null shearu. It follows from (40a) that

= ~ — ~3)_1 D(~2~— ~3) = ~D[log(detgAB)] (118a)

and

,. (~2~35E2~3)_1~ ~

3D~2]. (118b)

Now, sinceg~is measurable,(35e) implies that gAB is measurable.Thus, by (37c), the following

372 G.F.R.Ellis ci aL, Idealobservationalcosmology

quantitiesare measurable(i.e., determinedasfunctionsof (z,xA)): ~2~E2, ~3(3 (2~3 + ~2~3 Hence,ifwe set

= TA exp(ix4,(no sum), (119)

then rA and cosCy2— X3) aremeasurable.

But, by (118),

p = ~D[log(r2r3)]+ ~ cot(~y2— X3) D(y2— X3) (120a)

= —~ exp{i(~y2+ X3)} [D log(r3/r2)+ i D(~3— X2)] cosec(,y2— X3). (120b)

Herep dv/dzandu (dv/dz)exp{—i(~’2+ X3)} aremeasurable.The factordv/dzarisesbecausethereis noobservationalway of determining the relationship between v and z, whereas the phase factorexp[—i(~’2+ X3)] representsthearbitrarinessin thechoiceof the tetradvectorsm and iii, whoserealandimaginarypartsspan the 2-dimensionalscreenspaceorthogonalto thenull generators.If aparallellypropagatedtetradis used,so that, in additionto (38), conditions (49)also hold, this phasefactormay bedeterminedasfollows. From (40a) it follows that

D(~2~3)=2p~2~3+.&2

5E3+ ~2~3) (121)

andhence

D(y2+ X3) = D log(r2/r3), (122)

which integratesto

X2 + X3 = Iog(r2/r3)+ H(w, xA), for someH. (123)

But, by (65),

~2.... Pe1~v1+o(v) (124a)

= P e~’2v~+ o(v), (124b)

andhence

X2+ X3 = log(r2/r3)+ (2n + 1) ir/3, (n E N). (125)

SincerA is measurable,it follows that exp[—i(~’2+ X3)] is also measurable.Thus, measurementof g,.~implies that p dvldz and o dv/dzaremeasurable.

In practice,thesequantitieswill be determinedfirst, so that g,.~will then be deducedby reversingtheargumentgiven above.But evenp dvldzando dv/dzarenot theprimaryobservablequantities.Whatoneactuallymeasuresaretheobserverareadistanceandtheintegrateddistortion(seebelow),from whichp dv/dzando~dv/dzcanbe derivedby differentiationandgAB deduceddirectly.


The observerareadistancer may be definedby

d(logr2)/dv = ka;a = —2p, (126a)

rlc=0. (126b)

By (ll8a), theuniquesolutionof (l26a)subjectto (l26b) is given by

r4 = F(w, x’~)detgAB, (127a)

wherethe“constant”of integrationF(w, xA) mustbe foundby evaluating(127a)in the limit v -+0.

The limiting behaviourof r is obtainedby integrating(126a)using (Ma). We findr = v+ 0(v3). (12Th)

If the2-metric is expressedin termsof stereographiccoordinates(C~(), this givesF = 4P4,and thus

= 2P2 (det ~ (128a)

A morefamiliar expressionis obtainedif standardangularcoordinates(x’~’)= (0, ~),relatedto the XAby (91), areused.In this case

r2 sin 0= (detgA’B~Y~. (128b)

The observerareadistanceis usually definedby consideringa bundleof null geodesicsdivergingfrom the observer(Ellis [13]).If this bundle subtendsa solid angledfl at the observerand spansacross-sectionalareadSat somepoint alongthe rays,theobserverareadistanceof that point is definedby the relation

dS=r2d.O. (129)

The equivalenceof this definition and(126) is a consequenceof thecentralconditions(95b), (128a)and thearealaw of ray optics(see,e.g.,Penrose[38]),which may be written as

d(dS)/dv = (k’~a)(dS). (130)

Now, not only is r measurablein principle (as follows from (126) and the fact that pdv/dz ismeasurable),but also therearein factpracticalwaysof measuringit. Thesearediscussedin somedetailin Ellis andPerry[5]andEllis, PerryandSievers[6].We shallnot enterhereinto a descriptionof thesequite complex observationaltechniques,except to mention that they dependcrucially on a detailedknowledgeof sourcecharacteristicssuchassurfacebrightnessandevolution. It is customaryto expressthe resultsof suchmeasurementsby the so-called(r, z) or “observerareadistance”relation.We shallhenceforthrefer to r simply as the “areadistance”. Once r = r(z, xA) hasbeen determined,p dv/dzfollows from (126).

Similarly, o~dv/dzmayalso bedetermined(althoughpracticalobservationaltechniquesareonly now

374 G.F.R.Ellis ci aL, Idea!observationalcosmology

beingdeveloped,seeValdeset al. [66]),by measuringthedistortion presentin theimagesof objectsorgroups of objectswhoseintrinsic shapeand size are known. An intrinsically spherical object, forexample,will, in general,appearelliptical, with the ratio of themajor to theminor axisdeterminedbyu~and theorientationof the ellipse (with respectto the pre-assigned2-planeelementbasisvectorsRe(m) andIm(m)) determinedby argo~(seePenrose[38]).Whatone is really measuringin this wayisthe integratedshear

fadv= J(adv/dz)dz,

ratherthan theshearitself. But, by carrying out measurementsfor all valuesof z down a fixed nulldirection,we may deduceadv/dzby differentiation, therebyobtaining what we shall refer to as the(adv/dz,z), or “distortion-redshift”, relation.

(iv) Numbercounts.Supposethe observeron C countsgalaxiesseenin a solid angled~in thenulldirectionXA down to anaffine distancev. An affine incrementdv will includedN newgalaxiesin thecount,wheredN is thenumberof new sourcesdetected in thepropervolume (seeEllis [13])

dV= r2dfl(1+z)dv.

If the numberdensityof sourcesat a distancev is n, then n dV new sourceswill be includedin this

volume,andthus

dN = fn r2 di’2 (1 + z)(dv/dz)dz, (131)

wheref is theselectionfunction(seesection4.4)representingthefractionofsourcesin thevolumeelementd V thatareactuallydetected.Thus,the total numberof galaxiescountedwithin asolid angledown to aredshiftz is given by

N(z)=J(dN/dz’)dz’.

Providedthat theselectionfunctionf is known (in practice,onewould haveto estimatef from aknowledgeof galactic brightnessandspectra,andfrom the detectionlimits of the apparatusused),numbercountsof distantgalaxiesthereforedeterminer2 n~dt~dzasafunctionof redshiftanddirection.Thus,sincer maybe determinedby othermeans,n dv/dz is measurable.

11.2. The maximaldata set

Given the resultsof ideal observations,i.e., observationscarried out to indefiniteaccuracyin alldirections,thediscussionin part I andtheprevioussectionshowsthat the following quantitiesare, inprinciple,measurable:u°,w’~,p dv/dz,adv/dz,n dv/dz.

G.F.R. Ellis ci aL, Idealobservationalcoamoltigy 375

Becauseof limitations in the detectionapparatus(and becausethe universeprobably becomesopaqueat some era in the past), there will be limiting redshiftsbeyondwhich furtherobservationscannotbemade.Thesewill not necessarilybethesamefor all measurablequantities,nor eventhesamein all directionsfor any particularquantity.However,in orderto simplify our treatmentsomewhat,weshall assumethat theselimiting redshiftshavethesamevaluez*(xA) for all observables.Alternatively,z” is just a suitably definedminimum limiting redshift. In mostcalculations,it is convenientto workwith the (r, z), ratherthan with the(pdv/dz,z), relation. Weshall thereforein this partof thepapertakeasourfinal dataset

D(wo,z*)_={(uO,uA,r,adv/dz,ndv/dz):w=wo,0_<z�z*,xAER},

wherew0is thepropertime on C at which theobservationsareassumedto havebeenmade.Insteadof randadv/dz,wecould takeasimmediateobservationaldatag~,or/AD = gAB/r

2 andr (cf. section5.1).As mentionedpreviously, it is conceivablethat more information could be obtainedfrom direct

observation:In principle,,thereis no reasonwhy oneshouldnot be ableto measuretime variations inthe redshiftsof certainsources.However,any attempt to includesuchderivativesin the final datasetfacesthe severeproblemof having to decidewhensuchtime variationsare of a cosmological,ratherthanof an astrophysical,nature.

The data set D(wo, z*) will, therefore, be taken to representthe most detailedcosmologicalinformationwecould hopeto obtain by direct observationof distantgalaxies,andwill be referredto asthe maximaldata set. In theunlikely eventthat information not included in D(wo, z*) doescometolight, we shallusesuchinformationasacheckof both theprimarydataandthe field equationswe haveassumed.

11.3. The background radiation

For ourpurposes,themostimportantinformationto begainedby measurementsof the “backgroundradiation”,i.e. radiationreceivedfrom all unresolvedsourcesdown the line of sight, are its spectrumand its energydensityM,l~ at the point p of observation.Observationsappearto indicatethat theenergeticallydominantbackgroundradiation,the microwaveradiation,hasa well-defined blackbodytemperatureandis reasonablyisotropic aboutp. Then the radiationdensityon C(p) is given by [13]

M, = (M,l~) (1+z)4. (132)

In the remainderof this paperwe shall makethe assumptionthat, for dynamicalpurposes,theradiationcontributionto the total stress-energytensoris negligible— thegalacticfluid will be regardedasa pressure-freeperfect fluid. This assumptionis almostcertainly violated at earlytimes. A slightlymorerealisticmatter-energydescriptioncouldbe achievedby assumingthat thestress-energytensorT~consistsof separatelyconservedmatterandradiation components,representedby the sameaverage4-velocity utm, with equationsof state

Pmett,,r 0, Pra~iat~cx,= ~Mrwjjatjon. (133)

If this descriptionis used,theradiationdensityon CT(p) is an essentialpartof the requiredfinal data

376 G.F.R. Ellis er aL, Idealobservationalcosmology

for thefield equations.For now the informationprovidedby thebackgroundradiation will be relegatedto the role of indirect constraintdata.

11.4. Observationalspace—times

We shall notexamineherethepracticaldifficulties that arisewhenoneactually attemptsto measureany of thequantitiesdefinedin theprevioussections(thesearediscussedat lengthin Ellis [5] andEllis,PerryandSievers[6]).Forthepurposesof observationalcosmology,asit is conceivedin this paper,themajor problemis that neither the distortion nor the proper motions are likely to be measurableinpractice in the foreseeablefuture. Also, there is not muchdata currently availableconcerningtheangularvariationof the remainingelementsof D(wo, z*) (but seede VaucouleursandBollinger [67]and the referencescited therein).

To implementour observationalapproach,wecould proceedin a numberof ways. Two possibilitiesarethe following:

(i) All, or some,of those elementsof D(wo, z*) that cannotbe determinedby observationcouldsimplybe “carried” asunknownfunctionsin thesubsequentanalysis.Theextentto which thegeometryand the matter distribution are then undeterminedwould constitutea measureof “cosmologicaluncertainty”attributableto thedearthof currentlyavailableobservations.

(ii) All, or some, of the unknown quantitiescould be assumedto have a particular functionaldependence.This gives rise to various classesof “observational space—times”.The observationalspace—timeassociatedwith aparticulardatasetD(w9, z*) is theclassof perfectfluid generalrelativisticcosmologicalmodelsin which anull hypersurface£ maybe embeddedsuchthat £U{p} is thepastlightconeC(p) of somepoint p on a regulargeodesicworld line C, and the observationalrelationshipspredictedby this model are preciselythosedescribedby D(wo, z

Of course, the concept of an observationalspace—timeis applicable irrespectiveof whethertheelementsof D(w0,z*) are assumedor observedto havea particular functional form, and the mostinterestingcasesare likely to arisewhen one of the following setsof observational assumptions isadopted:

Areadistance, Propermotions,Numbercounts: Distortion:

IdealObservations(10) Measured MeasuredRestrictedIdealObservations(RIO) Measured UnknownSimplified IdealObservations(SIO) Measured VanishIsotropic IdealObservations(110) Isotropic Vanish

As an example,~considerthecaseof isotropic ideal observations.Thesearecharacterisedby (part I):

UA=O, r=r(wo,z), u=O, dN=dN(wo,z).

In addition,thegroupG of identificationsof observedobjectsis invariantunderrotations.Maartenshasshown(Maartens[50]andpaperII) that the only pressure-freeperfectfluid solutionsof Einstein’sfieldequationsconsistentwith theseassumptionsaresphericallysymmetricabouttheworld line C. Thus, theobservationalspace—timeassociatedwith 110 is just the Bondi—Tolmansolution(seeBondi [68]).Thisresultwill be usedin section15.

G.F.R.Ellis ci al., Idealobservationalcosmology 377

Supposeone is given a pressure-freeperfectfluid model,with C andpE C asabove.Supposefurtherthat observationalcoordinateshavebeenintroduced asin section9.2. The coordinatesystemis thenuniquely determinedup to the freedom(92), correspondingto a rigid rotation aboutonepoint on C ofthe orthonormal triad used to define the angular coordinates.For any given v~, the metric andstress-energytensorof themodel determineuniquely thequantitiesUa, M andg~on C(p, v*) whereC(p, v*) denotesthesubsetof C(p)consistingof pointslying within an affinedistancev’~ of p — that is,they determineuniquely theset

S(wo,v*)={(u0,M,g~)~w= w0,O�v:S V~,XA ER}.

On theotherhand,a necessarycondition for a null hypersurface.~ to beembeddablein themodel sothat .~U {p} is thepastlight coneof p, is that therebe a one—onemappingof C(p,v*) into .~U {p} andthat theelementsof S(wo,v*) take thesamevaluesat correspondingpointsof .X U {p} andC(p, v*).Any variation in the datain S(w0, v*) (modulothe remainingcoordinatefreedom)will correspondto areal differencein the space—timegeometryand the matterdistribution. We shall thereforeconsiderS(wo,v*) ascharacterisingthe local structureof themodel on the light cone(cf. section5.2).

Oneway of constructingthe observationalspace—timesassociatedwith a given setof observationsD(wo,z*) would thereforebe to try to deducethe local structureS(wo,Va) from the dataD(wo, z*),wherenow v~is the affinedistancecorrespondingto the limiting redshiftz~’, and thento propagatethesolutionoff C(p).

It was shown in part I that the deduction of S(w0,v*) from D(wo,z*) cannotbe madecosmo-graphically.In section12 we shallprovethat, given D(wo, z*), the local structureof thepastlight coneis uniquely determinedby Einstein’s field equations.In addition,weshall indicatebriefly in section13how thesolutionmay be propagatedinto the interior of C(p) without assumingany further final data.Section 14 considersthesituationnearthepoint of observationp, andsection15 thecaseof IsotropicIdeal Observations.

12. Solving thefield equationson the light cone

The main aim of this partof the paperis to show that the maximal dataset D(wo,z*) uniquelydeterminesthe local structureof the cosmologicalmodel on part of the pastlight conew = w0, whenEinstein’sfield equationsfor a pressure-freeperfectfluid,

Rat, = MuaUb+~Mga,,,

areassumed,with M> 0. That is, D(wo, z*) determinesthe set S(wo,Va) (seesection11.2),wherev~istheaffine distancedown thepastlight conecorrespondingto the limiting redshift f. Treatmentanddiscussionof propagatingthesolutionoff C(p)will follow in section13. In carryingthis out, weshall insection12.1 first provetheexistenceand uniquenessof solutionto the field equationsgiven D(wo, z*)by usingcontractionmapsandtheir extensions.Then, in section12.2, we shall showthat D(wo, z*) istheminimal datasetrequired.

12.1. The modelon the light cone

Einstein’sfield equations,with vanishingcosmologicalconstant,for a pressure-freeperfectfluid with


total energydensityM and4-velocity Ua maybe written as

Rth = - ~ M >0. (134)

From thedefinition (B3) [seeappendixB) it follows that

= ~M (1+z)2 (135a)

= —~M(1+z) YA~BgAB (135b)

~O2 = ~M (yA~Bg~~)2 (135c)

4ii = hM (1— 2Y”Y8g~) (135d)

~12 = ~M (1 + z)~(yA~Bg,,~)(yCyDg~,— 1) (135e)

= ~M (1 + z)2(yAyBg~~— 1)2 (135f)

where

(136)

The centrallimits (95a)and(112a)imply

~~=o(1), *~Po1=o(v), *1o~o(v

2),(137)

~ii = o(1), P12=0(v), ~‘~= o(1),

which areconsistentwith (51). The Ricci scalaris given by (see(B5))

A=21~R=~M (138a)

andhasthe limiting behaviour

A = o(1). (138b)

SupposeD(w0,z*) is specified,so that u°= (1+ z), uA, r, o dvldz and n dv/dz areall given as

functionsof (z,xA) on C~(p,z*). From thecondition that the fluid be pressure-free,it follows (Ellis[13])that.thetotal energydensityM isproportionalto the rest-massdensityof sources,which in turn isproportional to the numberdensityof sourcesn if we observea standardsetof sources.We shallassumethat theconstantof proportionalityof M to n is knownfrom local astronomicalobservations,sothat M dv/dzmaybe regardedasgiven on C(p, z*).

We shall first assumethat thecondition

dv/dz40 (139a)


is satisfied,andconsidervaluesof v suchthat

dr/dz40. (139b)

By substituting.for4~from (135a)into (42a)andusing(126),weobtain thenull Raychaudhuriequation

—(1/,)d2r/dv2= o~+ ~M(1+ z)2. (140)

(All referencesto eqs.(40—46) assumethat theyhavebeensimplified using (49).) BecauseM> 0, thisshowsthat almosteverywhere,

p4O4sdr/dv4O. (139c)

Using thedefinition of observerareadistance(126a)and(139a), wemay rewrite(140) in the form

fi(z, xA) ~ (~)+ f~(z,xA)(~)+ f3(Z, xA) = 0, (141a)

where

ft r1 (d*Iz) . (141b)

/2 ~ r’ d~r/dz2+ cvi (dv/dz)2 (141c)

f~ ~(Mdv/dz)(1+z)2. . . (141d)

Now (141a) constitutesa linear first order differential equationfor the unknownquantity dz/dv,providedft #0, acondition satisfiedby virtue of. (139b).The coefficients/~of this differential equationare all observablequantities,i.e., they are determinedby the data set D(wo, z*). From the central.conditions(lila) and(127b),it canbe shownthat

lim (dz/dv)=lim (dz/dr) Ho(xA), (142)v-~0 v-’O

whereH0 is theHubble“constant”. In general,H0 will vary with direction (see,e.g., de Vaucouleurs

andBollinger [67]).The uniquesolutionof (141a)subjectto (142) is given by

= F(z,xA) (Ho— Jf3(Z’~XA)F(zP,XA)dz~)_1 (143a)

where

F(z, xA) (~)exp(J [uc~(dr/dz)2](z’,xir(z’,xA) dz?) (143b)


A furtherquadraturenow determinesv = v(z,xA) uniquely, theconstantof integrationbeingfixed bythecentralcondition lim~..00z = 0. Sincedv/dz= (dv/dr)(dr/dz)~ 0 almost everywhere(a consequenceof (139)), this relation maybe invertedto give z = z(v, xA).

If dr/dz = 0 at somevalueof v, then (141a)determines(dz/dv) directly at that point (12 cannotalsobe zerothere,by (141a,d)). Thus we canevaluatev(z, xA) through suchpoints(which will be expectedto occur, seeHawking.and Ellis [15]; for example,in an Einstein—dc Sitter universethis refocussingoccurswhen z= ~). If (139a) holds almost everywherebut dz/dv= 0 at isolatedpoints, we can stillobtain the v(z)andz(v)relationspiecewise(althoughtheywill no longerbe 1—1). If howeverdz/dv= 0on anopeninterval (which occurs,forexample,in theEinsteinstaticuniverseand theGödeluniverse),z is not a good distanceindicatorandwe cannotuse(141).Howeverr is still a gooddistanceindicator,by (139c);andr(v) andhencev(r) canbe foundfrom theequation

d2r/dv2+ gi(dr/dv)2+g

2(dr/dv)= 0,

g1 rovi(dv/dr)2, g2 ~(Mdv/dr)r(1+ z)2,

which follows direct from (140); againthefunctionsg, arein principle directly observable.Oneshouldnotethat aftereither r or z havehada maximumvalue, the relevantdistanceindicator

for moredistantgalaxieswill havethesamevalueasfor closergalaxies.This causesno problemaslongastheothermeasureindicatesthe realdistanceto thesource.Howeversevereproblemscould ariseifboth measurabledistanceindicators(r andz) turn around;it might thenbe very difficult to distinguishvery distantobjectsfrom nearbyones.A warningexampleis thecaseof theBianchiI universes,whereat very early times bothindicatorswill havereversedwhenonemeasuresalongthe axis of the generic“cigar” singularity. Fortunatelyin this casethe behaviouris hiddenfrom us by theprimeval plasma.Similar behaviouris possiblein the static,sphericallysymmetricuniversesof Ellis, MaartensandNd[42](seecurves4 in fig. 3).

Thus,both in thegeneralcaseandin thespecialcaseswhere dv/dz is zero, thenull Raychaudhunequationdeterminestherelationshipbetweenthe affine.distancev andobservablemeasuresof distancesuch as z and r. It is precisely the inability to determinethis relationshipwithout the useof fieldequationsthat preventsthedeductionof S(wo,V*) from D(wo, z*) in thecosmographicanalysiscarriedout in partI. Theintegrationmaybe continueduntil oneof p, o~diverge(which mayoccuratacaustic,orat aspace—timesingularity)or M diverges(at amattersingularity).

Oncez hasbeendeterminedasa functionof (v, xA), the4-velocitycomponentsu°anduA (eqs.(113)and (115)), thespin coefficientsp andu and themetric variables~A (eqs.(118—120))aredeterminedon

v *). The Weyl tensorcomponent!P0 is nowgiven uniquely by

1I~0=D~y_2pg. (144)

At this stageof the integration,theonly Ricci curvaturecomponentsthat areknownareA and P00.The otherswill be knownoncey~hasbeenfound,that is, by (136), onceXA hasbeendetermined.Infact, by (135a,b), we have

= h1 — ~h2, (145a)

where

G.F.R.Ellis eta!., Ideal observationalcosmology 381

h1 = —~M (1+z) UA ~BgAB, (145b)

I~_vA~B I1A~n2~~k ç g~.

The identities

— Iç ç gAB—u, c c gAB——i,

which follow from (35e) and (37c), imply that, provideddet(gAB)� 0, the relation(145c) invertsto give

XA = _(hJA + /~A) (145d)

ThusXA, andhencey~,are determinedoncewe know h2. A differential equationfor h2 may be

derivedusing (35e),(37c), (40a,c) and (145). Wefind that—— CDi-’gAB— gAcgBDg

andhence

Dh2—ph2—ul—(a+13). (148)

Sincea and~ are also asyet unknown,an examinationof (40), (42) and (44) showsthat we needto

solve the following systemof coupleddifferential equations:Dcv = pa + ,Bö — ‘J)ooi~+ h1, (149a)

D$=pfl+acr+!t’1, (149b)

Dw=pw+cvti-ã-$, (149c)

Dh2 —ph2—0h2—a—fl, (149d)

D~P1= 4Ø’~+ 3ãP®- 4a~0+ 3f3~~~-

+ óiD!t’0 + h2(3p~Poo- D~) + 31i2c’~oo

+ (Dh1 — 2ph1— 2ri~~)+1A~I~0A— ~ (149e)

wherewe haveusedthe fact, which follows from (28) and (36), that

8 wD+;.4~.

In orderto show that (149), together with the appropriatecentral limits, uniquely determinetheunknownquantitiesa, fi, w, h2 and ~ we employ the techniqueof contractionmappings(see,forinstance,Bachmanand Narici [69]).The proof given below was suggestedby the slightly differentmethodof establishingtheexistenceanduniquenessof solutionsusedin Friedrich(70].

382 G.F.R. Ellis et al., Idealobservationalcosmology

The centrallimits (93a) imply

p = —v~+ ~5, a = a°v1+ a, /3 —ä°v1+ IL (150a)

where

lim~=0, limd=0, lim$=0, (150b)v-.0 v-.0 v—.0

and a°is definedby (93b).Since !P~= o(1), w = o(v2) and h

2 = o(v2)— from (94, 95) and(145c)—

lim h2= 0, urnv!!’1 = 0, lim w = 0. (150c)

v-’O v—~0 v-~O

Using(150a),we cannow write eqs.(149) in the form:

D(v&) = ,5(v&) + o(v~)— v~oo(h2)+ J1, (151a)

D(v~)= ñ(v~)+ o-(vd)+ v!!’1 + .12, (151b)

D(vw)= ~5(vw)+ ~(vá)— (vä)— (vg), (151c)

D(h2) = (v_i — ,5,Xh2)—crQi2)— v_1(vä)_ v~1(v~), (151d)

D(v!P1)=(~3v

1+4j5)(v!tc)+3(vä)~—4(vä)~

+ h2(—3~+ 3v13.Pva— vD4~o)+ 3o’14)oo(h2)+ J3. (iSle)

Here

Jl=ja0_uã0+~v, (152a)

J2 —~5ã°+cra°, (152b)

.13= v(Dhi — 2phi — 2crc1)+ v(~!t’o,A— e~00.A) + 4a°~P~ (152c)

areknown functionsat this stageof the integration.Theinitial conditionsfor eqs.(151)are

urn (vã, v~,vw, h2, v!Pi) = 0, (153a)v—.

0

and(152) satisfy

J1=o(v), J2=o(v), .13=0(1). (153b)

Now wearereadyto proceedwith theconstructionof acontractionmappingon themetric spacein

G.F.R. Ellis etaL, Idealobservailonalcosmology 383

which the unknown functions in the system (151) live. Let CEO, vs’] denote the set of continuousfunctionson [0,v*] that vanish at 0, anddenotethe 5-fold productby (xC[0, v*]r. Let ~, s/i E CEO, v*]and t’ (~‘, ~2,• . , ~), !P (s/’1, s/c2,. . ., s/ri)E (xC[O, v*])s. It is well-known that, if we definepositive definite metricse andE on CEO, v*] and (xC[0, v*])5, respectively,by

e(s/, s/i) = sup l~(v)— i/i(v)j (154a)vE(0,vJ

and

E(’~P,!P’) sup {J4’(v) — s/’t(v)I + J4~2(v)— s/’2(v)j + . . . + ~4~(v)— s/’5(v)}, (154b)vEfO. v1

then {C[O, v”], e} and {xC[O, v~])5, E} areBanachspaces.*A contractionmapF on a metric space(X, d) is amapwhich satisfies

d(Fx, Fy)�k d(x, y), for all x, y EX, (155a)

where k is a realnumberand

0<k<l. (155b)

Theprinciple of contractionmappingsstatesthat a contractionmappingon acompletemetric spacehasa uniquefixed point, i.e., apoint suchthat

Fx0 = x0.

Now, we wantto find a systemof integralequations

(156)

which are equivalentto the systemof differential equations(151), subjectto the centralconditions(153a).H will thenbe amapping

H: (XC[O, v*]r_s. (xC[O, va])$. (157)

We furtherdesirethat H bea contractionmapon thecompletemetricspace{(xC[0, v*1)s, E}. If wecanfind sucha mapping H, we therebyprove the existenceand uniquenessof a solution for (156),andthereforefor (151).

We definethemap H1: (xC[0, v*])S_~.(xC[0, v*])5 by

(H1~

1Xv)= v~3/2J (v’)3’2 {5~’+ ö~24 ~oo(v’)~ + (v’)3’2 J~}dv’, (158a)

‘Strictly, {cfO, v’), JI ~}is aBanachspace,with ~ = e(#,#)asanormon thelinearspace~O, v’J.


(Hi~2Xv)= v ~ J (v’)312 {~2 + ~ + (v’)1 ~ + (v’)312 J2} dv’, (158b)

(Hi~3)(v)= v3’2 J (v’)3’2 {~3 + ~3 — ~1 — ~2} dv’, (158c)

(Hi#4Xv) = v~1’2J{—~(v’)1’2~ — ~(v’)1’2~ — ~(v’)1’2(~1+ ~2)} dv’, (158d)

(H1~

5)(v)= v~712f (v’)7’2 {4~5+ 3~oo(v’)~ — 4~o(v’)~‘ + ~v’) ~ - (v’) D~00~

3

+ (v’) D!!’04

3 + 4(3,5(v’)3 ~ — 3(v’) 4)® — (v’)3 D’~P00}q~

+ l2cr(v’)2 ~oo4~+ (v’)~’2J

3} dv’. (158e)

Herethe .1 aregiven by (152). If we let

= (v1’2 a, v1’2 ~ë, v_l’2 w, ~v~3’2h

2, v”2 !t’j), (159)

we seethat (158) is e9uivalentto the original system(151), subject to the centralconditions (153a).Recall that & = o(v), /3 = o(v), as = o(v2), h

2 = o(v2) and !t’~= o(1). Thus, to establishan existenceand

uniquenessresult for (151), we need merely to provethe existenceanduniquenessof a solution for(158). In otherwords,we needto showthat H

1 is a contractionmapon {(xC[O, V *])5, E}.

Lemma12.1: The mapH1 definedby (158) maps(xC[0, v*1)3 into itself, i.e., H1~E (xC[O, v5])5 for all

(J) E (xC[0, V *])S.

Proof: Sinceall thefunctionsinvolved in thedefinition (158)arecontinuouson [0, v*], theonly point atwhich H

1~can possibly be discontinuousis at v = 0. But, from the limiting behaviourof thesefunctions,it follows that thereexistssomepositive realnumberB suchthat ii~I, IÔVI, ui, kPoovl, I ~1’oVI,vD~~I,vDV~’oI,JII, v’JiI and ivV2t areall boundedby B on [0,v”]. Therefore,we have

IH1~l� ~By [supj~1~+ supI~2i+4sup ~] + 21Bv”2, (160a)

H1c6

21� ~Bv[supI~2I+ supi~’i]+ ~sup~ + ~Bv1’2, (160b)

JHiçb3~� ~Bvsupi~i+ ~v (sup ~I + sup i~2i), (160c)

IH14i

4i � ~Bvsupl#~i+ ~(supi~~I+ sup1~2i), (160d)


IH1~5I� ~Bv[4supI4~I+7 sup ~ +3supI~2i+2supI~~i+ 12 supI4~I]

+?rv2B(28supIcb4t)+~vv2B. (160e)

Hereall supremumsaretakenover [0,v5].Because/i~(0)=0 andthe ~‘ arecontinuous,it follows that lim~....

0{supI~1(v)i}= 0. Therefore,(160)

shows that lim~~(H1~)(v)= (H1~)(O) = 0, as required. Q.E.D.

Theorem12.1: The mapH1 definedby (158) is a contractionmap on (xC[O, viJ)3 for somev1, satisfying0< v1~ v”.

Proof: Let B be defined as in Lemma 12.1. As in the proof of the lemma, we obtain boundsonIH14~ — H1s/i’i. This gives

iHicf~(v)— Hjs/i1(v)i ~ (~Bv+ ~v+~)supJ~— tfr~+ (~Bv+ ~v+ ~) supI~2 s/il

+ ~Bv supI~— ~I+ [~Bv + ~Bv2JsupJ~— + (~Bv+ ~s)supW — s/’51. (161)

From the definition of a contractionmap in (155),we can seeby examining (161) that H1 definesa

contractionmap for all v in the interval [0,vi], where v1 is suchthat all of the following hold for0< v1 � v

5:

1ffBv1+ ~v1<~, ~jBvj+ ~vi<~, ~Bv, < 1, (162)

~Bv1+~B(v1)3<1, ~

In otherwords,for k <1,

5 5

~ H1 cb~(v)— H1 s/”(v)I � k ~ sup ~‘(v) — s/i(v)I (163)5,—i 5=1 vE[O. vi]

for all v in [0,vi], wherev1 is determinedby (162). It follows immediatelynow from (154b)and from(163)that E(H1b, H1!!’) < kE(~,!!‘) on [0,v1], asrequired. Q.E.D.

So far, then,we haveproved theexistenceanduniquenessof a solution for (151) on ourpast lightconedown to v = v1. We canextendthis proof to v1 < v � v

5 by successivelyrepeatingthe followingprocedureuntil the interval,for which we havecontractionmappings,is [0, v5].

To do this, simply definea newmappingH2: (xC[O, v~])

5—s(xC[O, v5])3, which is like that given by(158),exceptthat the lower limits of the integralsarenow v

1, insteadof 0:

(H2~Xv)= v3’2 J (v’)3’2 {~1+ ~2 4 ~oo(v’) ~ + (v’)3’2 J

1} dv’, (164a)

35~ G.F.R.Ellis ci a!., Idealobservationalcosmology

(H2#2Xv)= v3’2 J (v’)3’2 {~2+ ~ + (v’)~#5+ (v’)3’2 J

2} dv’, (1Mb)

(H2#3)(v)= v~’2J (v’)3’2 {~#~+ - ~1 - ~2} dv’, (1~c)

(Hz#4)(v)= v~”2f { — ~ (v’)”2 ~ — u (v’)3’2 ~ — ~(v’)~’2~ + ~2)} dv’, (1~d)

(H2~)(v)= v~’2J (v’)

712{q~5+ 3 ~o(v’)~ -4 ~o(v’)#1 + 3 (v’) ~oo#2 - (v’) (D~oo)#~

+ (v’) (D’P0) ~ +4[3~(v’)3 tZ~— 3(v’) ~o — (v’)3 (Doo)] #~

+ 12cr(v’)3~~~oo+(v’)”2J

3} dv’. (164e)

H2 is definedfor the interval [v1,v5] and again is easily shown to be equivalentto the systemof

equations(151) for any v~ v1.

Applying thesameargumentsasusedin provingTheorem12.1 to theextensionmapH2, wefind thatH2 will be acontractionmapon the inverval [v1,v2], v1 < v2� v

5, wherev2 is suchthat:

‘~Bv2+ ~v2<~+ ~(v11v2)”2+ ~(v

1Iv2)3~’2Bv,+ ~(v

1Iv2)7’2Bv

1+ ~(v1/v2)312v,, (165a)

~Bv2+ ~v2< ~+ ~(v1/v2)

1’2÷~(v1/v2f’2Bv1+ ~(v,/v2)

3’2v,+ ~(v1/v2)

712Bv,, (165b)

j~Bv2< 1 + ~(v1/v2)

3’2Bv1+ ~(v1/v2)

712Bv1, (165c)

!Bv2 + ~(v2)3< 1+~(v1/u2)3’2Bv1+ ~(v1/v2)”2Bv1+ ~(v,/v2)712Bv

1+ ~(v1/v2)712Bv

12. (165d)

~Bv2<~+~(v1/v2)

3’2+~(vi/v2)

712Bv1. (165e)

Thus, the existenceand uniquenessof solution for (151) is proved on the interval [v1,v2J, which,togetherwith the previousresult, implies existenceand uniquenesson the interval [0,v2] A finitenumberof suchextensionmapprngsmay be usedto extendtheproof to [0, v5] From (165) it seemsclear that v,,> v~..1,andthat successiveextensionswill not approacha limit.

It shouldbepointedout that themap(158), togetherwith theequivalences(159), is not theonly onewhich will be a contractionmappingon somesub-intervalof [0,v

5]. There is a largefamily of suchmappings,of which (158) is onemember.

At this stageof theintegration,the following quantitiesare determinedon Ci,p, v5): p, u, a, /3, r

G.F.R. Ellis ci aL, Idea! observationalcosmology 387

a + ~), ~, as,XA, ~ (I, J = 0, 1,2), !!‘~, !P1. Thefollowing equationshavebeensatisfied: (40a—c),

(42a—e)and (44a).Note that (42c) is implied by (42d,e).Next weconsiderthegroupof equations(42g,h) and(44b):

(166a)

D~p,u+oA+!t’2+2A, (166b)

~ (166c)

Let

(167a)

andset

(#1, #2, #~)= (VA, V/i, 2v!P2). (16Th)

Thensubstitutionof (167) into (166) yields asystemof differential equationswhich is equivalentto the

systemof integralequationsk=F1P, (168a)

where ~ (#‘, #2,#~)and

F,: (xC[0, v*])3_* (xC[0, v*])S

is definedby

(F1#’Xv)= J ~ + ~ + G1} dv’, (168b)

(F1#2Xv)= J {~2~~ + ~ G

2}dv’, (168c)

(Fi#3)(v) = v2J (v’)2 {3~3 — 2~~#’+ 2~~#2+ G

3} dv’, (168d)

for v E (0, v5J and

F1P(0)=0. (168e)

388 G.F.R. Ellis ci aL, Idealobservationalcosmology

Here

G1 ~O + v0~= o(v), (168f)

G2/ij.~°+2vA=o(v), (168g)

G3 = 2vD(P11 + A) — 2v8c1310+ 2vö!t’1 — 4v!P1a+ 2~i°~00+ 4ãv~10— 4pv’P11— 2uv.~P~= o(1).(168h)

It is readily verified that F1 is a contractionmapon (xC[0, v])3 for v � v’~’ sufficiently small, andthus

A, ~ and !P’2 are uniquely determinedon C(,p, v1~),where0<v1~� v~.Extensionmaps similar to

thoseusedabovewill pushtheuniquesolution down to V5.

It shouldbenoticedthat thesecondsetof equations(166)includestermswhich arederivativesof thequantitieswhich were theunknownvariablesin thesystem(151),viz., D~

11,ScP~ciandô!P1. Fortunately,thesearefirst derivatives,so that theequations(151)and themappingH1 assureusthat they exist andare bounded.If they hadbeensecondor higher order derivatives,we would not haveknown if theyexistedorwerebounded.Of itself, acontractionmappingor fixed point theorem,thoughassuringus oftheexistenceanduniquenessof solution,tells usnothingaboutthederivativesof that solution.

Our integrationon the light coneis notyet complete.But the remainingquantitiesmaybe foundbyintegratingsequentiallythe following systemof uncoupleddifferential equations:

(169a)

DU~w+r~i—(y+~), (169b)

D~P3= D~21 — ö~2O+t5(~1’2+2A) 2A~P1+ 2Ø’3+ ~ 2(ã — /3)~o2P~21, (169c)

Dp=~+rA+~’3+~21, (169d)

D!P4= Ô(~’21+~!‘3)—3A!t’2+2a!P3+p!P4+2i41o—2A~1i

— ~ (169e)

Equations(169a, b, d) determiney, U and ii immediatelyby simple quadratures,the constantsofintegrationbeingdeterminedby thecentralconditions(93a) and (94a), whereas(169c) is a linear firstorder differential equationfor !P3.

D(v2!P

3) = 2~5(v2!l’

3)+ 2~iv2~

10+ 2v2(ã— /3 )4~ — 2pv2~

21— 2Av2!t’

1

+ v2D4

21—v2ô~

0+v2~(!t’

2+2A). (170)

Since limv...o v211’

3 = 0, (170) determines(v2~P

3)and hence !P3 uniquely. (If the central condition!1’3 = o(1) did nothold, !P3 would be determinedonly up to an unknowntermin v

2.)Similarly, (169e) determines!P4 uniquely eventhough the right-handside containsthe term 4’~P,.

0,which involvesderivativesin directionsoff the light cone.This is because,by (135),

— In. \2j’,f. \—1‘~O2— ~,W’Oi) k”OO)

G.F.R.Ellis eta!., Idealobservationalcosmology - 389

and 4’P00,4~o~areknownat this stageof the integration— they aregiven explicitly by thecontracted

Bianchi identities(46a,b).Collectingresults,we havethereforeshownthat, undertheassumptions(134) and (139), themaximal

dataset D(wo, z5)determinesuniquely all the spin coefficients, the metric variables and the tetradcomponentsof theRiemanntensoron C(p, v5). By substitutingthemetric variables(~,as, XA, U)into (37), themetrictensoron C(p, v5) maybefound.Furthermore,by (135a),M is known(oncezhasbeendeterminedasa functionof (v, xA)); theU’~aremeasurable;u°= (1+z) and u1 is determinedbythenormalisationcondition UaUQ = 1 [seeeq. (116)]. Thus S(wo,v5) is uniquely determined.

Our integrationdown thepastlight coneis valid andgives a uniquesolutionfor all v � min(v*, v~)wherev~is thevalueof v wherenull causticsfirst occur.Whencuspsappearon C(j,), i.e. pointswhereC(p) intersectsitself but whereall quantitiesnonethelessremain bounded,our solutionswill be validthrough thecusp.Howeversomeconsistencyconditionsmustnow hold; thesespecify that thesolutionat the cuspobtainedby integratingdown any generatorpassing through it must be identical to thatobtainedby integratingdown othergeneratorspassingthroughit from other directions;and that afterthecusp,thesolutiondeterminedon any generatormustagreewith that determinedby integratingbackfrom the final dataon the null cone, when v5 is sufficiently large that known datadeterminestheseconditions(afterthecusp,the null geodesicgeneratorof the light conelies in thechronologicalpastofthevertex).

12.2 D(wo, z5) as the minimal data set

In thecourseof deducingS(w0, v

5), the following equationshavebeensatisfied: (40), (42) and (44).While the argumentaboveestablishesthat D(wo, z5) is sufficient to determineS(wo, v5), it doesnotdemonstratethat it is also the minimal dataset requiredto determineS(wo, v5); for it is conceivablethat the elementsof D(wo, z5) are not independent.This would be the caseif any of the remainingnon-propagationequations(i.e., theequationswhich containtermsinvolving only derivativeswithin thepastlight cone)imply interrelationsbetweentheelementsof D(wo, z5). In orderto provethat this is notthecase,we shall showthat, oncethecentralconditions (93), (94) and(lila) and the radial equations(40), (42) and (44) havebeensatisfied,all theremainingnon-propagationequations— (41c,d) and (43b,c, d) — areautomaticallysatisfied.Theproof is similar to that usedin thevacuumcase(Friedrich [70])andwill thereforeonly be sketched.Let

a1~Sp—&r—pr+o’(3a—j~)+~_~o1 (171a)

a2~8a—ö$—,ap+Ai~—a&—$$+2afl+~P2—A— (171b)

a3öAÔ/.4/2T+A(3fla)+ !P3—~2l (171c)

~ — + (a — + (/3 — a)~A (l7id)

b2~—’~+(a—~ä)w+($—ã)~i+/i—5a, (171e)

sothat (41c, d) and (43b,c, d) aregiven, respectively,by b1~’= 0, b2 = 0, a1 = 0, a2 0 and a3 = 0. Label

theBianchi identities(44) similarly:B1 D(~01— !P~)— ô~® + ö!f’0 — 4a!f’0 + 4p!P1+ 2i4~~~— — 2o’~b10 (172a)

390 G.FR. Elks,ci aL, Ideal observationalcosmology

B2~D(’b11 + A — !t’2)+ 8!~— ô~—A!!’0— 2a!1’1 + 3~~J’2±~®+ 2ã~1o—~ — a~o (172b)

B3~D(’P21 ~ (172c)

Usingthecommutators(39), an extremelytediouscalculationgives

Da1= 3pa1— ~ra1+ B1 (173a)

Da2= 2pa2+ aa1— /3a1+B2 (173b)

Da3= 2pa3+ Aa1 — ~a1+ B3 (173c)

Db1A= 2pbiA+ aJA — (l73d)

Db2=2pb2+~ia1—wa1+ã2—a2. (173e)

Now, by (93) and(94), it follows that iim0...oa1v3 = 0. But (150) and (173a)imply

D(a1v

3)= 35(aiv3)— ~(ã1v

3)+ B1v

3. (174)

ProvidedB1 = 0 (which follows from theassumptionthat (44a) is satisfied),theuniquesolutionof (174)

subjectto limit0.~a1v3 = 0 is

a1v

3 = 0 on [0,v5]

andhence

a1 = 0 on (0, v

5].

Thus a1 is satisfiedon [0,v*] providedau0= 0. But this is preciselythecondition that liin~..,0vw = 0,

which is satisfiedby virtue of thecentralconditionas = o(v2).

In an entirely analogousfashion, it may be shown that a2, a3, b1A and b2 all vanish identically on

[0,v5J, i.e. (41c, d) and(43b,c, d)areautomaticallysatisfiedon [0,v5].

13. PropagatIngthesolutionoff Cip)

Now we needto propagatethesolution on C(p, v*) into the interior of the past light cone.Thisinvolvesa numberof difficult problems,which weshall-addressherein only apreliminaryway. Weshallshow, first of, all, that, when the data is analytic,we can deducethe w-derivativesof the dataset,without additional specificationof data. This hasgenerallybeen takento imply that one can thendeterminethe initial or final data “on a neighbouringhypersurfaceseparatedby an infinitesimalw.di~4acement”from theoriginal one (TamburinoandWinicour [71]),or “at thenext instantof w”(Bondi et al. [721). In this way,themodelmaybedeterminedon this “next” light coneby againcarryingout the hypersurfaceintegrationpresentedin section 12. Repeatedapplication of this processthenformally generatesasolution in someregion in the interior of C(p).


Themethodwe shall employ is similar to that of Bondi et al. [72],Sachs[27]andNewmanandUnti[60].Even thoughthis guaranteestheexistenceof a uniquesolutionoff C(p,v5)only in thecasewhenwehaveanalyticdata,it may neverthelessbe regardedasanecessaryfirst stepin obtainingarigorousexistenceanduniquenessproof for the non-analyticcase.Almost all such proofs start by assuminganalyticity andthenextendthe resultsto functionspacesin which the analyticfunctionsaredensewithrespectto the topologyinducedby asuitably chosennorm.

After outlining themethod,we shalldiscusssomeof the technicaldifficulties involved in thegeneralcaseandindicatehow thesemay be tackled.

13.1. Integration off C(p) in the analytic case

Supposethat the integrationproceduredescribedin section12 hasbeencarriedout, so that eqs.(40),(41c,d), (42), (43b, c, d) and (44) are all satisfied.Then the spin coefficients,metric variablesand thetetradcomponentsof themetric andcurvaturetensorsareknownon Ci,p, v5).

Now we considereqs.(41a) and (46). The first of thesegives

(175)

asa knownquantityon C(p, v5). SinceD~Aand5~Aarealsoknown,(28) and (36) showthat ~ is

determined.The contractedBianchi identities(46) maynow, using (135a,b, d) and (138a),bewritten asa systemof linear algebraicequationsfor the unknownquantities4M, ~1zand4 Y”:

Fx = G, (l76a)

where

/ —(l+z)2 —2M(l+z) 0 0

F~~-!I (1+z)YA~g~ yA~g~ M(1+z)~Jg~ M(1+z)~g3~

2~ (i+z)Y~4~fg~ yA

5tfg~ M(1+z)~j~g,.8 M(1+z)~g38

\ ~(y~Y8gAB — 1) 0 MY8g~ MY8g~

(176b)

(heresubscriptsR andI denoterealandimaginaryparts,respectively),

I 4z4y2 (l76c)

\4y3

and G is a(4x 4) realmatrix which is knownonce4~Ahasbeendeterminedusing (175).Thissystemof

equationshasauniquesolutionprovided that det F 4 0. A short calculationgives

detF=~iW(l+z~(detg~~40. (177)

392 G.F.R. Ellis et a!., Idealobservationalcosmology

[Moregenerally,if onedoesnotassumethe fluid to be pressure-free,thecontractedBianchi identitiescanbeshownto determine4M,4zand4 y~,providedtheequationof thestaterelatingthepressurepand the energydensityM is suchthat p 4 M]

By takingthe4-derivativeof (40c)and using (39a),(40c), (41a)and(43f, g), it can beshown that

D(4XA)= 2Re{~i~y+ S~+ pv + ~— 3(ã+ /3)A — ~i(a+ 2/3)

+ 2(y— fla + (y + y)(a + /3)— ,aa]+ 2TÔXA}. (178)

Since the right-handside of (178) is known, one may integratethis equation (the centralconditionsimply IiX A10 = 0) to determine4XA on C(p, v*). Once4XA and iiz areknown, it follows from (136)that AuA is determinedon C(p,v

5). Furthermore,(43g) gives 4o~directly on C(p, v*). Thus iiM, 4z,4UA and4cr areknown, and (36) showsthat M,

0, z,o, w4,

0 and o~areknownon C(p, v5).

Now it is clear from thediscussionin section12 that the set

I(wo, v*) = {(u°,U’~,o-, M): w = w0, 0 � v � v

5}

is a perfectly good (but not observable)final dataset for theNP equations.The integrationproceedsexactlyas before,except that, in the first step,the null Raychaudhuriequation(42a) is written, using(126a), as

r~1d2r/dv2 = o~+ ~M(1+ z)2. (179)

Sinceboth termson the right-handsideof (179)areknown (asfunctionsof v andxA) from I(wo, v*),this equationconstitutesa differential equationfor theunknownquantity r and thereforedeterminesthe relationshipbetweenareadistanceandaffine distance,cf. section12.1. [WhenthesetD(wo, z*) isusedasfinal data,(42a) determinesthe relationshipbetweenz and v.]

Thusthe final datasetD( W0, z 5) is usedto find thesolutionon C(j, v *). In particular,it determines

the final dataset I(wo, v 5) as well as the w-derivativesof all theelementsof I(wo, v5). Now, followingBondi et al. [72], onearguesthat thedataset I(wo, v*) can thereforebe determinedon someboundedsubsetC(p’, v’

5) of a “neighbouring”pastlight coneC(p’), wherep’ is a point on C to the pastof p,correspondingto thepropertime w = w

0 — dw on C. Once I(w, v?*) is knownon C(p, v~*),thesolutionmay be found on C(p’, v’

5) by performingthe hypersurfaceintegrationdescribedin section12. Thew-derivatives of the elementsof I(w, v’5) are again determined.Thus, an iterative procedureisestablishedwhich, it is claimed,determinesthesolution off thepastlight coneC(p) to the pastof p.

Thereare severalproblemsassociatedwith this approach.Firstly, assuminganalyticity, aswe havedone here, is unsatisfactory,since the hyperbolic nature of the field equationsseems to be afundamentalfeatureof generalrelativity. But, evenin the analytic case,the argumentwe havegivenaboveis not yet complete.It is still necessaryto show that the remaining equationswhich contain4-derivativesare now identically satisfied.The spin coefficient a, for example,may be determinedonC(p’) by integratingthesystemof eqs.(151)using the final dataset I(w, v’5). On theotherhand,it isalso possible,in theanalyticcase,to determineit on this light coneby expandingin aTaylorseriesin waboutpointsonC(p)andusing (43i), which determinesa

0 Unless(43i) is identically satisfiedby virtueof previouslydenvedrelations,it is possiblethat thesetwo methodsof finding a on C(p’) will leadtodifferent results. The samemethod that was used in section 12 to establish that the remainingnon-propagationequationsare automaticallysatisfiedcan be usedto verify that no such consistency

G.F,R.Ellis eta!., Idea!observationalcosmology 393

problemsdo, in fact, arise.However,this consistencyproof is considerablysimplifiedif oneworks onlywith the 10 field equationsandthecontractedBianchi identities,asin theBondi—Sachsapproach(Bondiet al. [72]andSachs[27]),which we shall describebriefly in section13.2. Nel [57] hascarriedout thisconsistencyproofin detail.

Thus, it is clear that the observationallydetermineddatais sufficient to determinethe solutiontowardsthe interiorof the light conein theanalyticcase.Thereis no reasonto think that thesituationwould be differentin thenon-analyticcase,particularlyin view of the results(a) by Muller zum HagenandSeifert[73]on the fluid characteristicinitial valueproblem(theydo notconsiderthecaseof a cone,but it shouldbe possibleto roundoff thevertexin thecasesthey do consider);and(b) by Friedrich [70]in thecaseof a cone, for the vacuumfield equations(obtainedby using harmoniccoordinates).Thus,while it would be useful to haverigorous proofs available,general argumentsfrom the hyperbolicnatureof the field equations,togetherwith the sufficiency of the data in the analytic case,suggestthat the solutionwill be uniquely determinedprecisely in the pastCauchydevelopmentof the setonwhich the datais known. From the structureof the field equationswe may suspectthat once thecontractedBianchi identities aresatisfied,no furtherconstraintswill ariseaswe integratetowardstheinterior. Not only will thedatabesufficient, but therewill be no superfluousdata.

13.2. TheBondi—Sachsmethod

In section13.1, wereferredto Nel’s [57] consistencyproof,which assuresus that, in theanalyticcase,all the remainingequationscontaining4-derivativesareidentically satisfied.Within theNPframeworktheproof involves a seriesof long calculations,becauseof the large numberof equations.Therefore,insteadof working within theNP formalism, Nel developedhis proof using the Bondi—Sachsmethod(Bondi et al. [721andSachs[27];see also ChelloneandWilliams [74]).Essentially,this allows one towork only with the ten independenttetradcomponentsof the Einstein field equations.Thesemay bedivided into threegroups:six main equations,one trivial equationandthreesupplementaryconditions.The six main equationsdivide further into four hypersurfaceequationsand two standardequations.This terminologyoriginatedwith Bondi et al. [72].

Now, thereis a lemma,which wasprovedby Bondi etal. [72],that in ourcasereads:

Lemma13.1: Let U bea simplyconvexnormalneighbourhoodof thepoint w = w0 on C. SupposethatthemainequationsandthecontractedBianchiidentitiesaresatisfiedin U, andthatp 4 0. Thenthetrivialequationis identicallysatisfiedin U, while thesupplementaryconditionsaresatisfiedon C(p)fl U iff, foreachnull generator~ of C(p)thereexistsa pointq E .~fl U suchthat thesupplementaryconditionsaresatisfiedatq.

This lemmashows that, oncethemain equationsand the contractedBianchi identities havebeensatisfiedin U, oneneedmerelycheckthat thesupplementaryconditionsare satisfiedon thehypersur-face(v = constant)in orderto insurethat they are satisfiedeverywhereon U.

A formal iterativeproceduresimilar to that presentedin section12 canthenbereadily setup and theintegration,or consistencyproof, as the casemaybe, carriedout in this equivalentframework.

13.3. Integration to the future of C(p)

It is clear from thehyperbolicnatureof the field equationsthat, in general,specificationof final or

394 G.F.R.Ellis eta!., Idea!observationalcosmology

initial dataon onenull hypersurfacecannotdeterminethesolutionthroughoutthe space—time(exceptin theanalyticcase),unlessoneassumes,apriori, that thespace—timesatisfiescertainglobal restrictionsthat renderit “deterministic”.

The problemof finding suchglobal restrictionsthat will allow oneto predict to the futureof C(p),given dataonlyon C(p),hasbeendiscussedby BudicandSachs[75].Theyconsiderspace—times(M, g)which satisfy theconditionthat, for eachpointp in M, everypast-endlesscausalcurvewhich intersectsthechtonologicaifutureof p also intersectsitschronologicalpast.Unfortunatelythis definitionseverelyrestricts the possiblespace—timestructure.For example,every deterministic space—timecontainscompactCauchysurfaces,andthe setof “small universes”(Ellis [52]) satisfy this condition; but, theuniversemay not belike this; indeed,as HawkingandEllis 1151 haveremarked,“there doesnot seemto be any physically compelling reasonfor believing that the universeadmits a Cauchy surface”.Furthermore,as Budic andSachs[75]haveshown, almost noneof thestandardcosmologicalmodelssatisfy their condition, andoneof theessentialfeaturesof theobservationalapproachoutlinedin thisseriesof papersis that onedoesnot wish to exclude, apriori, a whole classof possiblecosmologicalmodels.

Onealternativeis to impose,a priori, dataon somehypersurfaceS intersectingC(,p). This wouldenableone to predict to the future of C(p),but not to the whole space—timeunlessS is a Cauchysurface.In anycase,this would be contraryto the spirit of the observationalapproach,sincesuchdataoutsideCij,) would,of course,be completelyunobservable.

When the universedoesnot satisfy the “deterministic” conditions, in principle we cannotuseobservabledatato predict at all to the future of C(p) becauseof thehyperbolic natureof the fieldequations(physicallyspeaking,a shockwavecould be approachingthat will invalidateany predictionswe maymakeon thebasisof dataon C(p)) Howeverin practicewe assumewe maydo soand indeedcontinually havedoneso in thepast(eg predictingthemotion of theMoon, of themoonsof Jupiter,andthe tideson Earth) This is becausein practicea “no-interference”conditionholds (or rather,hasheld up to now)which in effect statesthat initial conditionsin theUniversearesuchthat they do notseriouslyinterferewith ourexpectationsin recenttimes (Ellis [52])

Theseexpectationsare basedon extrapolationof thedatathat we can observe The only caseinwhich thesecangivean unambiguouspredictioniswhenthesolutionis analytic(andeventhenwe mayeasily run into problemswhen we try to extendthe solution beyonda normalneighbourhoodof theorigin). Howeverit is noteasyto formulateeventhenecessaryconditionson theobservationaldatathatthe solutionbe analytic— supposingwe haveanalyticcharacteristicdataon C(,p), this doesnot implyexistenceof derivativeshigherthanthefirst of the “unknowns”in the two systemsto which weappliedthe contraction mappings Thus there is no guaranteethat thesevariablescan be expressedas aconvergentTaylor series(which involves thesehigher order derivatives),much less that the actualspace—timewill resemblethat predictedby suchan expansion.

The most useful way to apply a “no-interference”condition seemsto be to intersectC(p) by aspacehkesurfaceS (seefig 7), andthen attemptto usedataon J Ji,p)U S (which is determinedbyobservationaldataif S is sufficiently close to p, by the resultsof the previoussection)to determineconditions on the rest of S (i.e. on K = S— J). Thus if dataon J is analytic,we can at leastlocallyuniquely extendit toK andthenusestandardexistenceanduniquenessresults(HawkingandEllis [15])to predictconditionsin D”(S); this procedurewill predictconditionsto thefuture of Ci,p). Oneneedsto checkthat thepredictionis thenindependentof thechoiceof surfaceS; while it seemsclearthis willbe true,wehaveno formalproofof this feature.The“no-interference”condition is thensatisfiedif theactualdataon S is thesameas theanalytically extendeddata; in this case,the predictedand actualconditionsto thefuture of C(p)will agree.

G.ER.E!lis eta!., Idea!observaiionalcosmotogy 395

Fig. 7. Space—time diagram of prediction to the future of our past null cone Co from data on C0. We wish to predictwhat will happen at the event 0. The causal past J1 of 0 intersects a surface S in D = D1 fl D2 data on D is sufficient to determine what happens at0. Here D~is the part of D lying in the causal past Jo of P0 therefore data on D1 determines data on D3, the segment of C0 lying in Jj; in turn thedata on D1 is determined by data on C0. Hence given data on C0, we only lack data on 1)2 before we can determine what happens at 0. However D2is “much smaller” than Di: in addition, 1)2 is much further from 0 than most of D3, so the effect of data there is more attenuated by area distanceand redshift effects. Hence if (1)0 is “not too far to the future” of Co; (2) S is sufficiently far to the past of Po; and (3) data on D2 is not too differentfrom what we infer by simple extrapolation of conditions on D,, we can predict conditions at 0 to a good degree of accuracy.

We shall notattemptto formalisethis procedurein this paper.Ratherwemerelypoint out that it is apossibleway of determiningconditionsto the futureof p from dataon C(p),whensuitableanalyticityconditionsare satisfied.Its physicalusefulnessarisesbecauseone can then attempt to work out themagnitude of the errors in our predictions of conditions to the future of C(p), arising becauseconditionson S vary from ourexpectations(cf. Ellis [52]). In this application,onewould probablytakeS asthesurfaceof decouplingin theuniverse(the effectivesourceof thecosmicmicrowavebackgroundradiation).Onecould also extendit to caseswhere conditions on J are not analytic,but reasonableboundson expectationsof conditions on K can still be obtainedby suitableextrapolationprocedures.

14. Powerseriesapproximationsandtheangulardependenceof final data

The integrationprocedurepresentedin section12 assumesthat the maximal dataset D(wo,z*) isknown to indefiniteaccuracyon C’(p, v*). Herewe shall briefly discussthe slightly morerealistic casewhere the final data is specified in polynomial form. It turns out that, in this situation, the fieldequationscanbeintegratedsequentiallyin astraightforwardmanner,asKrlstian andSachs[1]were thefirst to do. Within this context,it is possibleto showthat whenthedatahasa polynomialform, onecanalso derive the general angular behaviourof all the expansioncoefficients in termsof the spin-ssphericalharmonicsintroducedin section10 (Nel [57]).

In practice,the final datawill ‘never be available in the explicit functional form required.Rather,from a discreteset of data points, it will always be necessaryto constructsmooth,C°°functionsbyinterpolatingprocedures,suchasfitting leastsquarespolynomialsto thedata.Whenthedatais given inthis form, i.e.,

a= a0-i-a1z+a2z2+~.+~,,z”+o(zfl~’), (180)

wherethepolynomialcoefficientsa, aregiven functionsof the XA — for example,

o~(dv/dz)(wo,z,xA)= (0 dv/dz)1(wo,xA)z + o(z

2), (181)

396 G.F.R.Ellis eta!., Idea! observationa!cosmology

the integrationproceduredetailedin section12 maybe simplifiedslightly andcarriedout explicitly. The

solutionwill thenbe expressedin the form

a = a°+a1v+a2v2+ ~+amvm+O(Vm+l), (182)

where the coefficients a’ of the solution are determinedby the a of the data through the fieldequations.Of course,the order in z to which a given memberof the dataset is knownwill be veryimportantfor the solution and,in general,will be different for different membersof the dataset. Forexample,we maybe ableto assumer up to orderz5, while the U” will be difficult to obtainevenup toorderz. Thereare practicalmethodsfor observationallydeterminingr(z), but uA(z)is unlikely to bedeterminedto any degreeof accuracyin the nearfuture. Our lack of observationsgiving u” willseverelylimit the order to which the solution can be calculated,no matter how accurately othermembersof thedatasetareknown.

Nel, in his thesis[57], hascarriedout in detail this integrationof the field equationsin NP form forpolynomialdata,obtainingthespin coefficients,metric variablesandRiemanntensorcomponentsup tovariousordersin the affine parameterv, asindicatedabove.He hasalso shownthat,for suchadataset,thenon-radialequationson Ci,p) are either identically satisfiedup to the orderv considered,or thatthey determine the w-derivatives of a consistent “final data set”.

Finally, Nd also derived [57] the explicit angular dependenceof the lower order expansioncoefficientsin termsof the spin-s sphericalharmonicsandhasshown how this canbe donefor higherorder coefficients.This work underscoresonceagainthe importanceof measurementsof the angular

variations of cosmologicallysignificantquantities.Evenif angularmeasurementsproducenothingmorethanan upperlimit on theanisotropyof thevariousobservables,this couldstill be usedto placeboundson the magnitude and anistropy of theseparameters,which, in turn, would imply boundson thepossibleanisotropyof space—timeitself.

As mentionedabove,suchcalculationswere first doneby Kristian andSachs[1].Nel’s resultsare inagreementwith theirsbut go to higherorder.

15. Sphericallysymmetricspace—time:an example

We notedin the Introductionto this paperthat theobservationalapproachto cosmologycanbeusedin conjunctionwith moreconventionalmethods.In this final partof our investigationweshall assume,apriori, that thespace—timeunderconsiderationis sphericallysymmetricabouta regulargeodesicworldline, andthen undertakean observationalanalysis.Thesespace—timesare of interest for a numberofreasons.Firstly, Maartens[50]hasshown that the classof pressure-free,isotropic ideal observationalspace—times(see section 11.2) is simply the Bondi—Tolman solution (cf. Bondi [681).Secondly,sphericallysymmetricmodelshaverecentlybeeninvestigatedaspossiblealternativesto the Friedmann—Robertson—Walker(FRW)models,but theseinvestigationshaveall proceededalongconventionallines(seeEllis, Maartensand Nel [42]).Finally, it would be useful to havesomeobservationaltest of thespatial homogeneityassumptionunderlyingthe FRW models.In section 15.3 we indicatesuch a testwithin thecontextof sphericallysymmetricspace—times:It is showntherethat asphericallysymmetric,pressure-freemodel is spatially homogeneousif the r(z) and the (M dvldzXz)relationstakepreciselythe form predictedby theFRW models(seealso in this connectionEllis [4]).

In section 15.1, the simplification of the metric and Riemanntensorcomponentsand the spin

G.F.R.Ellis ci a!., Idealobservationalcosmology 397

coefficientsinducedby the assumptionof sphericalsymmetrywill be derived,and the non-trivial NPequationsandBianchi identitiesisolated.The integrationschemeof section12.1 will then be examinedin section15.2. Finally, in section 15.3, we shall derive the form of the r(z) and the (M dv/dz)(z)relations under the assumptionthat the space—timeis, in addition, spatially homogeneous.Then theobservationalintegrationwill be carriedoutwith exactlytheseobservationalrelatipnsasfinal data,andit will be shownthat the resultingspace—timeis spatiallyhomogeneous.

15.1. Simplificationofthefield equations

Supposethe space—timeunderconsiderationis sphericallysymmetricaboutaregulargeodesicworldline C (seeHawking and Ellis [15]). The metric is then of type D and the null directionsn and k asspecified in (36) (Wainwright [76]),where now XA = 0, arerepeatedprincipal null directions,so that

~O~1~3!l~4O, (183a)

while

= !P2. (183b)

Clearly the proper motions must vanish, for otherwisethey would definepreferreddirectionson the

celestialsphere,contraryto theassumptionof sphericalsymmetry.Hence,by (135), (136) and (138a),

cPoi = ‘Po~= P12=0 (184a)

~Poo~M(1+z)2 (184b)

(184c)

t~P22=~M(1+z)

2. (184d)

Also, it follows from the symmetryassumedthat all physicalquantitiesmustbe functionsonly of w

and v, so that

r = r(w, v), M= M(w,v), z= z(w,v). (185)

As before,the spin coefficientssatisfy(49). Equation(42b)nowreads:

Thr=2po!,

or, since,by (126a),p = —Dr/r,

D(r2o)= 0. (186a)

But, by (93a) and (127a),

u=o(v), r=v+o(v3),

398 G.F.R. Ellisci a!., Idealobservationalcosmology

andhence(186a)implies

o~=0. (186b)

Thus,by (40a),

D(r~)=0,

so that

= r’F”(w, x0), for somefunctionsFA. (187)

Evaluating(187) in the limit v—~0, using (62) and (12Th),we obtain

(188)

where,asbefore,~ is definedby (65).

SinceXA = 0, (40c) implies

r0, (189a)

andhence,by (38),

a —~, (189b)

while (40b)gives

D(rw) = 0. (190)

Sinceat= o(v2), this meansthat

w=O. (191)

Similarly, (42g, i) imply

(192)

while (42d)maybe integratedto yield

a = a°r1, (193)

wherea°is definedby (93b).

Next, we notefrom (41d)and (191) that

(194)


and from (42f) and(183) that

D(y — ~) 0

and thus

v=.y (195)

(sincey = o(v)).

From(41a) it now follows that

whence,by (188),

/2=4r/r. (196)

[It is interestingthat, in this case,the expansionof the k-congruenceis p = —Vk(log r), while theexpansionof the is -congruenceis ~ = V (log r). This is related to the fact that r containsall theinformationaboutboth the intrinsic andextrinsiccurvatureof thepastlight conesof pointson C.]

Beforeconsideringthe non-radialequations,let us examinethemeaningof theequation

Note that, by (36), (188) and (191),

oi~=0 iffr_1~,~=0.

Thus, if i~is real, then

&~=0 iff~~=0 iff’~=i7(w,v), (197)

whereas,if i~is complex,then

817 = 0 ill [Re(17)],2+[Im(,7)]3= 0 (198a)

and[Re(~)],3— [Im(~)],2= 0, (198b)

which meansthat i~is analytic.

Applying (197)to (41b)gives

U= U(w, v), (199)

and,similarly, (44c)shows that

!J’2!P2(w,v). (200)

400 G.F.R.Ellis ci aL, Idealobservationalcosmology

Thenon-radialequations(43a,b, d, f, g, i), (45a, b, d),and the radial equations(44a, c, d)arenowall

trivially satisfied,andit is not difficult to showthat (41c) is also satisfiedif

a = —r~1~P/ö~,

which is anidentity by virtue of (193)and (93b).Thus, thenon-radialequations(41) areall satisfied.The remainingBianchi identities(44b)and (45c) may now be integratedas follows. From (42a,b),

(43e,h), (44b)and (45c) it follows that

D(pp~— !P2+ A + eP~)=2p(p~— !1’2+A + ~‘~) (201a)

4(p~—!P2+A+Pii)=—21z(pj.~—!P2+A+tP1j). (201b)

But, sincep = —D(log r) and~ = 4(logr), this gives

— !P2 + A + ~ = G(xA)r_2, (201c)

where,asusual,G(x”) is determinedby evaluating(201c)at thecentre.This yields

(202)

The Penrosecomplex curvatureinvariant K of a 2-surfaceS (seePenrose[77] andGeroch,Held and

Penrose[65]) is definedby

K—pp.—!1’2+A+cI.11. (203)

If we chooseS to be the2-surface(w = constant,v = constant),we havethe simple relationship

K = ~(rIs)’2. (204)

The realpart of K is relatedto the Gaussiancurvature,~ of S by’

(2)~(~+g)~_2, (205)

while the imaginarypart of K describesan extrinsiccurvatureinvariantof S in themanifold in which itis embedded(in this case,theonly non-trivial extrinsiccurvatureinvariant). Thus, theareadistancerdeterminescompletelyboth the intrinsic and theextrinsicgeometryof the2-dimensionalcross-sectionsSof thepastlight conesof pointson C.

From (202) it is readily shownthat (43c) is satisfiedif

~ (~P/a~-)(8P/o~)= 1/8,

an identity by virtue of (94b).Collecting results, the spin coefficients, metric variables,metric and Riemanntensorcomponents

havethereforebeenshownto satisfy the following conditions:

G.F.R. Ellis etaL, Idea!observationalcosmology 401

(206a)

a=_(öP/8~)r1, /3=—a, p—DQogr);

p.=4(logr); (206b)

at =0= XA, U = U(w, v), ~A = ~Ar_1; (206c)

where

~2...p ~:3=jp, P=(1/2V2X1+C(); (206d)

(206e)

‘P2=!P2(w,v), !1~2=~P2 (206f)

~O1=~O2’~12=0; (206g)

~oo=~M(1+ z)2, ~ = ~M, ~22 = bM(1+ z)2 (206h)

(206i)

g1’ = 2U, glA = 0, g~= diag(—r2,—r~2cosec20), (206j)

the last equationbeingaconsequenceof (37c), (91) and (206c,d).

Of the 37 equations(40—46),only the following still needto be satisfied:DU—2y (207)

Dp=p2+cP~ (208a)

D,up,a+!t’2+2A (208b)

(208c)

(209a)

4p=(2y—~)p—t~k2—2A (209b)

4~+ D(~11 + 3A) = (4y —

2/L)~ + 4p~ (210a)

4 (~ + 3A) + ~ = —4j~~+ 2p~n. (21Gb)

15.2 Theobservationalintegrationscheme

The observationalintegrationschemepresentedin section15.1 simplifies considerablyin thecaseof

402 GF.R.Ellis eta!., Idealobservationalcosmology

sphericalsymmetry.The only non-trivial elementsof the maximaldataset D(w0, z *) arenowthe area

distanceandnumbercounts—redshiftrelations,so that wemay define

D~(wo,z*)= {r(z), (M dvjdz) (z): w = w0, 0< z s z*} (211)

(whereS denotessphericalsymmetry)as the final dataset in this case.As before,(208a) is usedto determinethe relationbetweenthe redshiftz and the affine distancev.

Again, onerewrites(208a) as

fi~(~)+f2(~)+f3=0, (212)

wherenow

= r’ dr/dz (213a)

f2 = r~’d2r/dz2 (213b)

f~= ~(M dv/dz)(1+ z)2. (213c)

Equation(212) is noweasilyintegratedto give

~=(~){1—~Jr(M~)(1+z)2dz}_1 (214)

andthena simple quadraturedeterminesv = v(z)andhencez = z(v).Once(214)hasbeensolved,both r andM areknownasfunctionsof v on the final light cone,and

thusp, ~, a, /3 andall thenon-trivial componentsof theRicci tensormaybe determined.This leavesp~and y asthe only spin coefficientsstill to befound. But, by (202), (208b),

2p~+ ~ + 3A —

so that

~ =~J(~Mr2_~)dv (215a)

or

= J (~c~ r~— ~ dz. (215b)

G.F.R.Ellis ci aL, Ideal observationalcosmology 403

Oncej~hasbeendetermined,!P2 follows immediatelyfrom (202), while y is obtainedby integrating

(208c):

y=f(p~_~M_~r-2)dv, (216)

andnow (207) yields

U=—~_2Jydv. (217)

This completesthe radial integration:All the spin coefficients,metric variables,metric andRiemanntensorcomponentsare now determinedon the final pastlight coneC(p).But Bondi [68]hasshownthat apressure-free,sphericallysymmetricspace—timeis completelydeterminedby specificationof twofunctionsof a single variable.Thus, the requiredexistenceand uniquenessof the solution off Cip)follow from the fact that the dataset Ds(wo,z*) containspreciselytwo functionsof onevariable(see(211)).In fact,theentireradial integrationabovemay becarriedout in thecomovingcoordinatesystemusedby Bondi, and it can then be shown explicitly how Ds(wo,z*) determinesthe two arbitraryfunctionsof the Bondisolution.

Even thoughtheBondi solution is reasonablysimple whenexpressedin comovingcoordinates,wehavenotbeenableto reproduceit by integratingtheNPequationsorby transformingfrom comovingtonull observationalcoordinates.The basicproblem is that, even in comoving coordinates,it is notpossible to find explicitly the relation betweenz (or r) and the affine distancev. The differentialequationgoverningthis relationshiphasthus far proved intractable.And it is precisely this relationwhich is necessaryin orderto write down explicitly the requiredcoordinatetransformation.

It is not surprising that the propagationequations(209), (210) are difficult to integrate.The nullcoordinateswe havechosenaretied into thenull geometry,andhenceto theobservations,ratherthanto the fluid. Similarly, in comovingcoordinates,the observationalintegration is far morecomplicatedthan theschemepresentedhere.

15.3. Observationalintegration with FRWfinal data

In this section we show, by explicit integration,that a pressure-free,sphericallysymmetricspace—time is, in addition,spatiallyhomogeneousif the r(z) and (M dv/dz)(z) relationstakeexactly the formpredictedby thestandardFRW models(seeeqs.(229) and (231) below).

Suppose,first, that the universe is both spherically symmetric and spatially homogeneous.It iswell-known (seee.g.Penrose[78])that thespace-timeis then conformallyflat, so that

‘P2=0. - (218)

From (44b), (45c), (46c) and (206g,h, i) it thenfollows that

404 G.F.R.Ellis etaL, Idealobservationalcosmology

= ~ + 2pP1~ (219a)

2D~22= ~ + p~22. (219b)

But (206h)implies

= 4(P1~ (219c)

andhence

(220)

which integratesto give

= -~~ (221a)

where

a = 8(M0)213. (221b)

And now (219c) showsthat

‘P00 = 2a(~1i)

5~’3. (221c)

Thus, by (206h,)and(221),

(1 + z)4= *~~o(~~)1= ~a2(’Pi1)

413= (M/M

0)

413,

which meansthat

~ = M = M

0(1 + z)3. (222)

In orderto find (dz/dv)and(M dv/dz), we differentiate(219a) andsubstitutefor Dp andD~from

(208a,b). This gives ~

~ D21’11 — 5~D11)

2— 6a(~11)~’

3= 0, (223)

which, integratedonce,gives

(D~11)

2= (~

11)

10~(b+ 12a(~1j)

1’3). (224)

Hereb = b(w) is a “constant”of integrationwhosevaluewill bedeterminedlaterby evaluating(224) atthecentre.

By substitutingfor ~ = ~M from (222)andfor D4~~from


= (d~i1/dz)(dz/dv) = ~8M0(1+ z?(dz/dv) (225)

into (224), andevaluatingthe resultatthecentre,oneobtains

dz/dv= (1 + z)3(H

02+ ~Moz)~, (226)

where -

H0 = dz/dvJo (227a)

is just theusualHubbleparameter,orHubbleconstant.It is customaryto definetheso-calleddecelerationparameterq0 by

q0 = M0/6H0

2, (227b)

so that (224) assumesthemorefamiliar form

dz/dv= H0(1+ zr(1+ 2qoz)’~. (228)

Thus,by (222), the (M dv/dz)(z) relationbecomesin this case

Mdv/dz= 6qoHo(1 + 2qoz)_i~~2. (229)

The r(z) relationmay nowbe derivedby integrating(208a). Since

p = —(1/r)(Dr) = —(1/r) (dr/dz)(dz/dv),

while ~ is given by (221c) and (222),eq. (208a)now reads

d2r/dz2 + {3(1 + z)~+ q0(1+ 2qoz)

1}(dr/dz)+ 3q0(1+ z)’(l + 2qoz)~

1r= 0 (230a)

with centralconditions

r(0) = 0, (dr/dz)(0) = (dz/dv)(0)= H

0. (230b)

Now (230a)is a linear, homogeneousdifferential equationwhich is easilysolvedto give

r(z) = H~

1q ~2(1+ z)2{qoz+ (1— qo) + (q0— 1X1 + 2qoz)~}. (231)

Thus, (229) and (231) are the observationalrelations predictedif the universe is both spatiallyhomogeneousand isotropic, i.e., if space—timeis FRW.

Conversely,supposenow that the space—timeis isotropic about C, and that themaximal data setDs(wo,z*) is given by (229) and (231)~Thenfrom (214),

= (~){i — ~Jr(M~) (1 + z)2dz}1 = H~(1+ z)3(i + 2qoz)~. (232)

406 G.F.R.Ellis eta!., Idealobse,vationa!cosmology

By (215b),

2~r2=J(~M~?_~)dz.

Substitutingfor Mdv/dzfrom (229), for r from (231),and for dv/dz from (232), we find that

2~r2= 3q ~3H ~f(l — 2q0)

214+2q0(1— 2q0)13 + q~I2— ~q ~

+ (qo— 1)2J(1+ z)4(1 + 2qoz)112 dz — qo(qo— 1)(1+ z)2— ~(q

0— 1)(1 - 2q0)(1+ z)3}, (233a)

where

I~ J(1+ z)~(1+2qozyU2dz. (233b)

Using

= (1 + 2qoz)i~2 n-i + (2n— 3)qo (n � 2; q0 4~) (234a)

(n — 1)(2q0— 1)(1 + z) (n — l)(2q0 — 1)

and

J(1+ z)4(1+2qoz)U2dz= —~(1+ 2qoz)”2(1+ z)3 + ~q

0I3, (234b)

the integrationis readily performedandyields

2/Lr2 = —q~3H~(1 + z)3{[q

0z + (qo— iXqo— 2)1(1+ 2qoz)L’2+3q

0(q0— l)z+ (q0—i)(2—qo)},(235)

sincethe terms in I~,which involve logarithmsor hyperbolictangentsof z ~dependingon the sign of(1— 2qo)~ cancel. Even though (234a) is definedonly for 2q04 1, so that the case2q0 = 1 needsto beconsideredseparately,(235)is still valid for both cases.

Therefore,we have, finally

= —~q0H0(i+ z){(qo— 1) + (1+2qoz)2~}{qoz+ (1— qo)+ (qo— 1)(1+ 2qoz)1~}1. (236)

O.F.R. Ellis ci aL, Ideal observationalcosmology 407

Note that, forsmall valuesof z,

asrequiredby thecentral limit (93a) and the fact that

z= H0v+ o(v2).

Now, by (202),

so that

!P2r

2 = —~ar(di1dz) (dz/dv) + jP11r

2 — ~. (237)

But, by (206h),(229), (231),(232)and(236),

= —~q~(l+ z)~{(3qo—2— q~zX1+ 2qoz)+(qo—1)2(q0—2— 3q0z)

+ (qo — i)(4qo—4— 4qoz)(1+ 2qoz)’~}

_4.~ _2 1

—5’vjir —~.

Therefore,by (237),

~2l2~0. (238)

This provesthat ¶!‘2 vanisheseverywhere,exceptpossiblyatthecentre.Thatcanbeeasilyestablishedby looking at the results of the polynomial integrationschemedescribedin section 14 (Nel [57J)

combinedwith the factthat ~= 0 (seeeq. (183a)).Oncethis hasbeenestablished,it becomesa simplematter,using Bondi’s coinovingcoordinates,to showthat the two functionsthat determinethesolutiontakepreciselytheir FRW values,andthus that thespace-timeis spatiallyhomogeneous.This result isvalid in the region of space—timeinto which the final datais draggedby the fluid flow lines, and thusalsoto thefutureof C(p).‘It thereforefollows that, in this case,a “no-interference”conditionof thekind discussedin section13.3 is automaticallysatisflód.This is becausethecondition that thefluid beboth pressure-freeandsphericallysymmetricappears~tobreak thecharacteristicpropertyof thefieldequations.

16. Conclusionto pert 11

The principal resultof this part (presentedin section12) is that the maximal dataset D(wo,z*),consistingof redshifts,propermotions,observerareadistances,distortion measurementsand numbercountsof distant galaxies and quasarsdown to some limiting redshift z*, uniquely determinesthe

408 G.F.R.Ellis cia!., Idea! obsewaliona!cosmology

solution to the field equations,and thereforethe structureof space—time,on the observer’spast lightconeC(j,) down eachnull geodesicgeneratorto v = min(v*, v~), where v * is thevalue of the radialaffine distanceparameteron C(p) correspondingto z” and v~is thevalue correspondingto the firstcausticencounteredon that generator.This also constitutesthe minimal data set necessaryfor thatuniquedetermination.

It is worthwhile emphasizinghere the remarkablenatureof theseresults,which (consistentlywiththepioneeringwork by Kristian and Sachs)essentiallystatethat the ideal cosmologicalobservationaldatais preciselynecessaryandsufficient to determinethe space—timestructureon ourpastlight cone.Thereis no apriori reasonwhy this should be true. Space—timecould havebeenunder-determinedbysuchobservations,resultingin the theoreticalimpossibilityof determiningthegeometryof our pastlightconeby astronomicalobservations,no matterhow complete;this indeedis thesituationbeforethe fieldequationsare introduced. Alternatively, space—timecould havebeen over-determined,so that con-sistencyconditions would have to be fulfilled betweenthese ideal observationsif Einstein’s fieldequationsare to be satisfied.Neither is the case(except beyond cusps or caustics,where suchconsistencyconditions do indeedarise):given ideal observationaldata,it is preciselywhatis neededtodeterminethe space—timestructure.

In a muchless rigorousway we haveindicatedin section13 how thesolutionmaybe propagatedoffC(p) and into the interior, given analyticdata— arguingtheexistenceanduniquenessof asolution inthepastCauchydevelopmentof that part of C(p)on which final datais given.However,thereremaina numberof problemsand unansweredquestionsin this regard.Section 13 is a first step in thatdirection.Undoubtedly,theuseof Sobolevinequalities(seeMuller zum HagenandSeifert[73J)will benecessaryto solve this problemproperly. This is an areafor futurework. The way anextensionmay bemadeoff the past light cone Ci,p) to its future when the solution is analytic hasalso beenbrieflydiscussed;this also needsfurtherwork. The methodproposedherewill probablyallow an extensionforan estimationof errorswhen thesolution approximatesan analyticsolution.

When null causticsor otherunboundedbehaviouroccurson the past light coneat somevaluev~before v * is reached,such unboundedbehaviourwould preventuseof the analysispresentedabovebeyond vt At a null caustic, for instance,thenull expansionparameterp would be infinite. Furtherwork is needed,both in observationallycharacterisingcaustics(andcusps)— specifyingtheobservationalindicationswhich would signal theirpresence— andin carryingout the integrationof the field equationson thepastlight conebeyondthem, i.e., in finding awayof integrating“around” them.With regardtocuspson the light cone— setsof pointsor surfaceswhere thereis no unboundedbehaviouralong thenull geodeaicgeneratorsbut where the past light cone intersectsitself — it seemsthat our analysisshould be valid. Such cusps,however,would imposeadditional consistencyconditionson thesolutionobtained.Thesecould be usedas an internal checkon thevalidity andcorrectnessof our integrationprocedures.Thestructureof space—time— thesolutionof the field equations— determinedata cusponC(p) by dataon.one null generatorwith a given direction must be identical or equivalentto thatdeterminedby dataon generatorspassingthroughthecuspfrom otherdirections;andbeyondthecusp,mustagreewith any predictionsobtainedby integratingoff C(p) into its interior. Ultimately one wouldhopeto closethegapbetweenexistenceandsingularity theorems, proving that the solution does indeedexist and is regular until observationalconditions inevitably imply the existenceof a space—timesingularity.

Our concernto prove the existenceand uniquenessof the solution of the field equations,givencharacteristicfinal valuedataon C1,p),naturallyloadsto thequestionof thestability of thesesolutions.In thegèneralcase,smallerrorsin thedatawill belike perturbations.Will a small “perturbation” in the

G.F.R. Elliseta!., Idea!observationalcosmology 409

dataleadto a “stable” solutionasonepropagatesit backin time? Orwill it, instead,grow and leadto asolutionwhich divergesmore andmore from the “unperturbed”solution asoneruns the integrationback into the past?We are presentlybeginningwork on this question,first looking at perturbationsfrom FRW space—timesfitted by agiven maximal dataset.

Following upon our treatmentof the integrationon, and then off, C(p), we briefly indicatedinsection 14 how polynomial datawill uniquely determinethe solution to the field equationswithin acertainregionalongand within C(p).Finally, wespecialisedour generaltreatmentto the sphericallysymmetric,and sphericallysymmetric and spatiallyhomogeneous,cases,therebyindicating how ourobservationallyorientedtheoreticalapproachmay becombinedwith moreconventionalapproaches.Inparticular,we showedthat, if a pressure-free,sphericallysymmetricspace—timeis, in addition,spatiallyhomogeneous,the r(z) and (M dv/dz) (z) relationstake exactly the form predictedby the standardFRW models, and conversely.This provides a precise criterion for the spatial homogeneityofsphericallysymmetric,pressure-freecosmologicalmodels.

The observationalistwill of coursecry out that the ideal data mentionedin this paperdoes notresemblethat available in reality, e.g. the mass-to-lightratio and evolutionaryeffects (i.e. systematicchangesof source propertieswith redshift) are not accuratelyknown, and proper motionsof verydistantgalaxiesarehardly measurable.Weare awareof theseproblems;thenatureof theobservationalcosmologyprogrammein the light of theseconstraintsis the subjectof the next paperin this series(Maartens,Ellis and Nel [2]) and has been briefly discussedelsewhere(Ellis [4,79]). While theconclusionsthereare quite different from thosehere,we believe that their full implications are onlyclearwhen comparedwith the ideal observationalsituationconsideredin this paper.

Acknowledgements

We thankD. MatraversandA.E. Hwangfor useful discussions,andR. McLenaghanandR. Penrosefor useful comments.PartII is basedon thePhD thesisof S.D. Nd (1980), representingwork carriedout underthe direction of G.F.R. Ellis, partially in collaborationwith R. Maartens.That work wascarefully checked, corrected,and extendedby W. Stoegerand A.P. Whitman (a numerical errorresultedin an incorrect version of the contractionmapping in the thesis; correctedequationsarepresentedhere).

WRSwishesto thankthe Departmentof Applied Mathematicsof the Universityof CapeTown fortheir hospitality during the final stagesof this work, and APW acknowledgesthat of the VaticanObservatory.Financialsupport from the CSIR (RSA) and the University of CapeTown is gratefullyacknowledged.Oneof us (GFRE) would like to thank the Presidentof the University of Alberta forsupport from his NRC fund, and the Relativity Centre,University of Texas,for assistancewith theillustrations.Two of us (RM and SN) gratefully acknowledgesupportof AECI ResearchFellowshipsandRhodesScholarshipsatvariousstagesof this program.

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Appendix A: Centrallimits for themetric and4-velocity components

In this appendixwe adopt a technique used by Mannasseand Misner [35] to obtain centralexpansionsabouttheworld line C, in termsof thenull geodesicafilne parameterv, for themetrictensorcomponentsgu and matter4-velocity componentsu~.We do this in threesteps:first, we obtain theseexpansionsin termsof “null-geodesicbasednormal coordinates”,which are regularat C; then wetransformto “optical coordinates”,which arejust theobservationalcoordinatesof section3.1 with thechoicey = v; and finally, we transformto thegeneralobservationalcoordinatesof section3.1. Both thelattersetsof coordinatesare singularat C.

A.1. Null-geodesicbasednormalcoordinates

As in section3, thepastlight conesof the timelike geodesicC arelabelledby the function w, whichis normalisedby theconditionthat w measurespropertime alongC. Theaffineparameterfrom C downthepastnull geodesicis v, which is normalisedby thecondition:

lim(u4ka)= 1v-.O

where k°~ dx”/dv is the null-geodesictangentvector. Direction cosinesof the past null geodesics,relative to an orthonormaltriad of vectors e, on C which are parallel-propagatedalong C and areorthogonalto the tangentvectora/owI~of C, aredenotedby

�~(O,4,)an(sin0 sin 4,, sin 9cos4’, cos9)~~ (�k)2 = 1. (A.1)

412 G.F.R. Ellis ci aL, Idealobservationalcosmology

Then “null-geodesicbasednormalcoordinates”z(q)of a point q lying an afilne-parameterdistancevfrom C on thenull geodesicwith directioncosine

1k in thepastnull cone{w = constant},aredefinedtobe z’(q) (w — v, l~’v).In termsof thesecoordinates,the null geodesicy(w0, �~“)throughthepointw = w0 of C in direction jf0k is given by

z°(v)=w—v, f~(v)=4vv, (A.2)

and the tangentvectorfield k to y(wo, 4k) is given by

k~Edx’/dv = ~o~+ e0ks~~~k= —3/8z°+�~3/&”. (A.3)

It is thenapparentthat thebasisvectorsö/öz1 arelinearly independentorthonormalvectorson C, and

so thesecoordinatesare regularand well-definedin someneighbourhoodU of any interval I of theworld line C.

In fact on C, o/az~lie in the samedirectionsas the parallely propagatedtetrad ea (o/ow}~,er),which showsthat

g~J~c= n~, F’01 Ic 0. (A.4)

Further,on eachgeodesicy(w0, 4k), thevectorfield k satisfiesthegeodesicequation

d2z’/dv2+F Ik(dZ’/dV) (dz”/dv)= 0.

From (A.2), this shows that

rc~(w— v, �0°v)—2F~(w— v, 4’~’v)4K + F’~(w— v, �0~v)

6’~�O~= 0.

Takingthe limit as v —~0, onefinds (on using (A.3)),

r’ —

4

Thuswe have

F’Jk~C=O,~q,J,kICO. (A.5)

Equations(A.3) and (A.4) together imply the metric form on C up to the first order in v. Anexaminationof thegeodesicdeviationequationand thecurvaturetensorcomponentsshows ([50]; themethodis an adaptationof that in [35J)that when the second-ordertermsare included, the metrictensorexpansionis

— v, ékv) = —1— {ROVOM(w, 0)e” ~°‘~}v2+ 0(v3),

go,~w— v, �kv)= —~{RO~M(w,0)t~�“~}v2+ 0(v3), (A.6)

g~(w—v, /2kv) ~

G.F.R.Ellis eta!., Idealobservationalcosmology 413

In the null-geodesicbasednormal coordinates,the4-velocitycomponents.u’ takethe form

= ö~+ (u’,~(w,0) f”) v + 0(v2) (A.7)

becauseu’,. = 6’o, ii is smoothnearC, and the coordinatesz areregularnearC.

A.2. Optical coordinates

Optical coordinatesy” aredefinedby y” an(w, v, 0, 4,). Theyare relatedto thenull-geodesicbased

normalcoordinatesz’ by

w—v, x’=vsinOsinçb, x2=vsinocos4,, x3=vcos0. (A.8)

In these coordinates,one has exactly k = ö~, k, = 8,~, ~ = ô,° (as in section 3). Applying thetransformation(A.8) to the metric components(A.6), one finds that off the world line C onehasthemetric form given in eq. (6) with

a(w, v, X”) = —1 + a2(w, X”) v

2+ 0(v3),

,6(w,v,f)=1,

vj(w, V, X”) = v13(w,v, f) v

3 + 0(v4),

h71(w, v, f) = diag(1,sin

2 0) v2+ h114(w,x”) v

4 + 0(v5), (A.9)

where a2anROKO$(W, 0)1” 1’s, vz3an—~Ro~~,.~(w,0) 1K(3�073xI)1~,and h1j4an—~R~,~(w,0)x

(31r/3x1) IK (~t’°~/~x~)~‘ ((and the derivativesO1°/Dx’ follow from (A.1)). Theserelations imply the

centrallimits given in section3.2.Applying the transformation(A.8) to the fluid 4-velocity components(givenby (A.7)), we find

u°= 1 + u°1(w,0, 4,) v + 0(v

2),

u1 = u’1(w, 9, 4,)v + 0(v

2), (A.1O)

= u’o(w, 0, 475) + u’~(w,9, cc) v + 0(v2),

where the u’1(w, 9, 4,) and u~(w,0, qS) are arbitrary bounded,smooth functionsof their arguments.

Note that although one might have suspectedu’ -÷0as v -*0, this is not necessarilyso: basicallybecausealthoughu is smooth,thecoordinatebasis3/Dy is not regularon C (the vectorfields a/Dy’ goto zerothere).

A.3. Observationalcoordinates

Finally, we transformto observationalcoordinatesx” definedfrom the optical coordinatesy’ byan (w, y, 0, 4,) wherey y(w, v, 9, 4,) is suchthat Dy/öv(w,v, 0, 4,)� 0, y(w,0, 0, 4,)= 0. Applying this

414 G.F.R. Ellis el a!., Ideal observationalcosmology

transformationto (A.9), oneobtains

a(w,y,x’)=—1+2(D$o/Ow)y+cr2y2+0(y3),

13(w,y,x’)= $o—2($o3y2)Y+$2Y + O(Y3),

v1(w,y,x’)=(3f30/Dx’)y— (D[/3o

3y2]/Dx’)y

2+ v,3y

3+0(y4),

g11(w, y, x’) = /32 diag(1, sin

2 9) y2— E2$o4y2] diag(1,sin2 0)y3 + g114y

4 + 0(y5) (A.11)

wherein generalall coefficientsaresmoothfunctionsof w, x’; and $0, Y~relatey and v accordingto

y(w, v, x~)= [j9o(w,x’))1 v + y2(w,x’) v

2 + 0(v3). (A.12)

This implieseq. (8’) of section3.2.

Finally, thevelocity components(A.1O) in termsof thesecoordinatesare

ua(w,y, x’)— ,5a,ubo+ua1y + 0(y

2),

ua(w,y,xI)=1+ u°1y+0(y

2),

implying the limits (12) of section4.1.

AppendixB:NP~ncamce~*.andcurvaturecomponents

In the Newman—Penrose(NP) formalismdescribedin section9, the, 12 spincoefficientsare:

K = kabm’~k” p = fla;b1’fl°?Z

0’ = ka;bm~m” A = _fla;b’ña’ñtP

p = ka ;bm a,~b — ‘~a;bm am b

= ka;bm”n~’ IT fla;bm”kt’

$ = ~(ka;bfl~~mb — ma~brnm)

a 1(ka;bnmthb—

£ = ~(ka;bfl”k” — ma;brnk)

7 = ~(k4b?Z’fl”— ma.blñ”n”). (B.1)

The irreduciblepart of thecurvaturetensorR ,,~, arerepresentedasfollows: theWeyl tensorC~,is describedby

G.F.R. Ellis era!., Idealobservationalcosmology 415

= _C~kambk~~mt ~2 = _C~dnbk~~md

!P1= Ca,,cdk6n~~k~m�i !P’3= _C~dkaflbflC,ñd (B.2)

— f~ ab c—d

y4—~4nmflm

the tracefreepartof the Ricci tensorby

= _~R,.,,kakb ~ = —~R,,,,(k”n”+ m~,ñ”)

= ~~R~kam~ ~t2 _~R~nbmb

~=_~R~,,,m~1mb~n=~Rar,fl~~nb, (B.3)

with

I~J=&,, (I,J=0,1,2) (B.4)

while theRicci scalaris given by

A=~R. (B.5)

AppendixC: Transformationpropertiesof theNP spincoefficientsandcurvaturetensorcomponents

C.1. Transformation of the spin coefficientsand curvature tensor componentsunder null and spatialrotations

Considera null tetradn, k, m, ñi satisfying(27a) and (49). A null rotation aboutthetetradvectork isgiven by

ka’=ka, na’=na+BBka+Bma+Brna, m(~’=m~~+Bk~z. (C.1)

Weconsideronly thecasewhereB is independentof v (sincethis preservestheconditions(49)). Then

the spin coefficientstransformas (Carmeli[80])

p’p, a’a+Bp, A’A+2Ba+ÔB+pB2

K’K, 8’B, IT’IT, U’U, ,8’/3+Btr,

~‘~+2B$+B2q+ô~, r’r+Bo+Bp,

y’=74-Bfr-l-,8)+Ba+B

2o-+BBp,

p’= ~ (C.2)

Thetetradcomç~onentsof theWeyl tensortransformas

416 G.F.R.Ellis eta!., ideal observationalcosmology

!P~=!t’0, ~‘~‘o+!Pi, ~

!P~=B3~t’

0+3B2!t’

1+3t’2+V~’3,

~ !P4. (C.3)

Thetetradcomponentsof thetrace-freeRicci tensortransformas

cP~o=~oo, iB~I’oo+~oi, =B2~oo+2Bt~oi+cPo

2,

= B2B~IJ~+2B~

01+ BP~+ 2B~11+ B2P

10 + ~12,

= B2B2~POO+ 2BB2~

01+ ã24) + 2B2.~410+ 4BB4~’11+ 2B~12+ B2~+ 2B~

21+ ~22~ (C.4)

Undera spatialrotation in the (m, ni)-plane:

k’~’=k’~, ~ m~z’=e~’ma, (C.5)

the spincoefficientstransformas(Carmeli [80];again C = C(w, xA))

p’ = p, a’ = a +i~C, A’ = e~2~’~’A,iC? = e21~,

Ir’IT, e’=e, u~=e2~Crr, $‘=/3+i~C,

~LL’/L, r~=eiCr, ‘y’=y+i4C, v’=e~v. (C.6)

The tetradcomponentsof theWeyl tensortransformas

IP,=e2iC!Po, !tr=e”~1J.~1, !t’2=!1’2, V~=e~!t’3, ~=e

2’~P4. (C.7)

The tetradcomponentsof the trace-freeRicci tensortransformas

f~l — — iCd~ — 2iCñ,— ~ ~ — e ~ ~ — e “-~,

= ~, ~I2 = e’1~

12, ~‘ = ~I~zz. (C.8)

C.2. Spin-weightsof theNewman—Penrosevariables

The spin-weightsof theNPvariables(seesection10.2)areasfollows:

Spin coefficients

,c:2, ~:2, p:O, r:1, ~:—1, A:—2, ~u:0, ir:—l. (C.9)

Thespin coefficientsa, /3, A and~ do not, in general,havewell-definedspin-weights.However,if onerestrictsattentionto spatialrotationsin which thegroupparameterC (see (C.5))is independentof v,then s canbe assigneda spin-weightof 0.

G.F.R. Ellis et aL, Idealobservationalcosmology 417

Tetrad componentsof the Weyltensor

!t’~2, V’1:1, !P2:0, ~P3:~1, !t’4:~2. (C.10)

Tetrad componentsof the trace-freeRicci tensor

~ ~o~:1, ~O2:2, ~ ~12:1, ~ (C.11)

Note that if ,~hasweight s, then i~hasweight —s. Thus, since~, = i.,,, it follows that, e.g., ~ hasweight —1.

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