Mon.Not.R.Astron.Soc.000,000–000(0000) Printed3February2008 (MNLATEXstylefilev2.2) From planes to spheres: About gravitational lens magnifications O. Wucknitz1,2⋆ 1JointInstituteforVLBIinEurope,Postbus2,7990AADwingeloo,TheNetherlands 2Argelander-Institutfu¨rAstronomie,Universita¨tBonn,AufdemHu¨gel71,53121Bonn,Germany Accepted2008January24.Received2007December21 8 0 0 2 ABSTRACT We discuss the classic theorem according to which a gravitational lens always produces at n leastoneimagewithamagnificationgreaterthanunity.Thistheoremseemstocontradictthe a conservationoftotalfluxfromalensedsource.Thestandardsolutiontothisparadoxisbased J ontheexactdefinitionofthereference‘unlensed’situation,inwhichthelensmasscaneither 4 beremovedorsmoothlyredistributed. 2 Wecalculatemagnificationsandamplifications(inphotonnumberandenergyfluxden- ] sity)forgenerallensingscenariosnotlimitedtoregionsclosetotheopticalaxis.Inthisway h theformalismisnaturallyextendedfromtangentialplanesforthesourceandlensedimagesto p completespheres.WederivethelensingpotentialtheoryonthesphereandfindthatthePois- - sonequationismodifiedbyanadditionalsourcetermthatisrelatedtothemeandensityand o r to theNewtonianpotentialatthe positionsofobserverandsource.Thisnewtermgenerally t reducesthemagnification,tobelowunityfarfromtheopticalaxis,andensuresconservation s a ofthetotalphotonnumberreceivedonaspherearoundthesource. [ Thisdiscussiondoesnotaffectthevalidityofthefocusingtheorem,inwhichtheunlensed situation is defined to have an unchangedaffine distance between source and observer.The 1 v focusingtheoremdoesnotcontradictfluxconservation,becausethemeantotalmagnification 8 (oramplification)directlycorrespondstodifferentareasofthesource(orobserver)spherein 5 thelensedandunlensedsituation.Wearguethataconstantaffinedistancedoesnotdefinean 7 astronomicallymeaningfulreference. 3 Byexchangingsourceandobserver,weconfirmthatmagnificationandamplificationdif- . feraccordingtoEtherington’sreciprocitylaw,sothatsurfacebrightnessisnolongerstrictly 1 0 conserved.Atthislevelwealsohavetodistinguishbetweendifferentsurfacebrightnessdefi- 8 nitionsthatarebasedonphotonnumber,photonflux,andenergyflux. 0 Keywords: Gravitationallensing–Cosmology:miscellaneous : v i X r a 1 INTRODUCTION leadstoanincreaseofthetotalnumberofphotonsfromthesource. Thiswouldbeincontradictionwiththefactthatlensingcanneither Already Einstein (1936) calculated the magnification of a back- createnordestroyphotons.ItwasalreadypointedoutbySchneider ground star lensed by a foreground compact mass and found the (1984)thatsuchaninterpretationisnotcorrect.Inordertocompare classicresultforthetotalmagnificationofbothimages,whichde- thelensedwiththeunlensed situation,itisessentialtodefinethe creasesforlargeimpactangles(farawayfromtheopticalaxis)but latterinameaningfulway,whichisfarfromtrivial.Ingeneralrel- alwaysstaysaboveunity. ativity,thelenswillslightlydistortthegeometryofthewholeUni- Atthattime,gravitationallensingwasregardedasatheoret- verse,sothatitisplainlyimpossibletoaddalenswithoutchanging ical curiosity with very littlechance of leading to observable ef- theconfigurationofsourceandobserver.The‘referencesituation’ fects.Muchlater,afterthefirstactuallenssystemshadbeenfound, in the standard explanation is the one in which the mass of the Schneider (1984) presented and proved a theorem stating that a lensisredistributedevenlyovertheUniverse.Itcanbeshownthat gravitationallenswitharbitrarymassdistributionwillalwayspro- thecomparisonfindsnonetincreaseofphotonnumber then(e.g. duceatleastoneimagewithamagnificationgreaterthan1.Since Weinberg1976).Thereasonforthisisthattwodifferentlylensed thetheoremisthoughttobevalidforallconfigurationsofsource, scenariosarecompared. lens and observer, one might naively think that this necessarily Some level of uneasiness remained in the minds of several authors,becauseonemaythinkofsituationsinwhichthegeneral ⋆ E-mail:[email protected] validity of the theorem would still lead to a violation of photon (cid:13)c 0000RAS 2 O. Wucknitz number conservation. Avni&Shulami (1988) calculated the am- equationismodifiedbyanadditional sourceterm,whichensures plificationcausedbyapoint-masslensbydroppingthesmall-angle thevalidityofGauss’theoremontheclosedsphere. approximation of light rays with respect to the optical axis (de- The exact form of magnification matrices on the sphere for finedbyobserverandlens),whichisgenerallyusedinthestandard finite(andpossibly large)deflectionanglesisinvestigatedinAp- formalism. Onemight think thatfar fromthe optical axisthede- pendixB.Forourcurrent discussionweonlyneedtheresultthat flectionbecomessoweakthatthefullcalculationwouldaddonly themagnificationmatrixintermsofsecond-orderderivativesofthe higher-ordertermswhicharenotrelevantintheend.Still,slightly potentialdiffersfromtheusualformonlytosecondorderinthede- surprising, theseauthors findthat theamplificationactuallysinks flectionangle,sothatthisdifferencecanbeneglected. below1,whichcorrectsfortheamplificationexcessclosetotheop- In Sec. 5 we extend our calculations to finite values of the ticalaxis.Avni&Shulami(1988)workedwithapoint-massmetric sourcedistanceandexplicitlycalculatedeflectionanglesandmag- inSchwarzschildcoordinates,andfixedtheradialcoordinaterela- nificationsforpoint-masslenses.Byadaptingthepotentialtheory tive to the lens for the comparison of lensed and unlensed situa- tothissituation,wefindthatthecorrectiontermsinPoissonequa- tion.Schneider,Ehlers&Falco(1992)questionthevalueof such tionandmagnificationaredirectlyrelatedtotheNewtonianpoten- a comparison. They argue that there is generally no unique way tialatthesourceandobserver. tocomparemeanfluxesindifferentspace-times.Nevertheless,ap- ThesubjectofreciprocityiscoveredinSec.6,whereweex- proximatingthecalculationsofAvni&Shulami(1988)forregions changetheroleofsourceandobservertorelatemagnificationwith closetotheopticalaxiswoulddirectlyleadtothetangentialplane amplification.Thesetwoarenotthesameanymore,sothatthesur- approachofSchneider(1984).Itisnotentirelyclearwhy(ifatall) facebrightnessisnolongerconservedifwedefineitinastandard thetheoremlosesitsvalidityfarfromtheopticalaxis. physicalsense.Thisisadirectconsequenceofthereciprocitytheo- Jaroszyn´ski&Paczyn´ski (1996) make similar calculations, remofEtherington(1933)andofthespace-timemodificationatthe butkeepthemetricdistancebetweensourceandobserverconstant positionofsourceandobserver,causedbythegravitationalfieldof for the comparison. They find that the effect of the lens changes thelens.Inanticipationofthisresultwetrytomakeacleardistinc- thetotalareaofthespherearoundthesourceandcanthus(without tionbetweenamplificationandmagnificationfromthebeginning. contradictingphotonconservation)leadtochangesintheaveraged AfterabriefdiscussionoflighttraveltimesinSec.7wesum- amplification.Theyalsomakeadistinctionbetweenmagnification mariseanddiscussourresultsinSecs.8and9. and amplification, which becomes relevant at the accuracy levels requiredfarfromtheopticalaxis. Noneofthesepublicationsactuallypresentsthemagnification tosecond order (intermsofthedeflectionpotential) forarbitrary 2 PLANARLENSINGTHEORY anglestotheopticalaxis1.Theyeitherapproximateforsmallan- 2.1 Basics gles or include only first-order terms. One aim of our work is to extendthecalculationstoincludesecond-ordertermsalsoforlarge Hereweonly recall themost relevant aspects. Thecomplete the- impactangles.Evenmoreimportantisthediscussionofappropriate ory can be found elsewhere (e.g. Schneideretal. 1992). Close to unlensedsituationstoactasareferenceinordertodefinemagnifi- anopticalaxis(whichcanbedefinedarbitrarily,conventionallyby cationsandamplifications. observerandlenscentre),thegeometryofagravitationallenssys- We will also develop the formalism to treat off-axis lenses temcanbeprojectedonthetangentialplaneofthecelestialsphere. similarly to the standard small-angle approximation and learn Thisleadstothenotionofthesourceplaneforthetruesourcestruc- which part of the proof of the magnification theorem cannot be tureandthelensorimageplanefortheobservedlensedimage(s). generalisedforthissituation.Thiswillbedoneforarbitrarymass Intheseplaneswedefinetwo-dimensionalunit-lessvectorswhich distributionsandnotonlyforpoint-masslenses. are based on angular coordinates on the sphere. We now restrict Theoutlineofthispaperisasfollows.WestartinSec.2with the discussion to static, ‘thin’ lenses, where the light is deflected abrief review of thestandard planar lensing theory, which isde- onlyveryclosetothelensplane.Wecantheninterprettheaction scribed in a tangential plane at the position of the lens. This in- ofthelensasapotentialtime-delay,whichaffectsthelightwhenit cludesthemagnificationtheoremandasimpleversionofitsproof. crossesthelensplane.Thisdirectpotentialdelaycausesadeflec- InSec.3weexplainourapproachofdiscussingtheapparentmag- tionandthuschangesthegeometricallightpath,whichcreatesan nification in terms of solid angles. In this way we can avoid the additionalgeometricaldelay.Thetotaldelay∆tisusuallywritten explicit definition of an unlensed reference situation. Solid angle intermsoftheFermatpotential,whichcomprisesthegeometrical mustalwaysintegrateto4π,sothatthemagnificationtheorem(in andpotentialpartsinanormalizedversion:2 theseterms)cannotholdeverywhere.Thisdiscussionisequivalent (θ θ )2 tcoonasrteafnetr.encesituationinwhichtheareaofthesourcespherestays φθs(θ)= −2 s −ψ(θ) (1) D D Section4.1comprisescalculationsforapoint-masslensatin- ∆t= d sφθ (2) cD s finity. We explicitly find that the magnification sinks below 1 far ds fromtheopticalaxis.Generalmassdistributions(stillwithsource Here the vectors of the observed image and true source position atinfinity)arediscussedinSec.4.2.Therewedevelopthepoten- areθ =(θ ,θ )andθ ,respectively.Thedistanceparametersare x y s tialtheoryoflensesonthesphereasanextensionofthestandard angular-sizedistancesbetweenobserverandlens/source(D ,D) d s potential theory in the tangential plane. We find that the Poisson and between lens and observer (D ). For simplicity, we assume ds 2 Thepotentialisdefinedmoduloanarbitraryadditiveconstant.Wewill 1 Closetotheopticalaxis,thesecond-ordertermsbecomedominantfor generallyneglectthisconstant,evenifitdiverges.Withinequations,how- compactmasses. ever,consistentconstantsareused. (cid:13)c 0000RAS,MNRAS000,000–000 Fromplanestospheres 3 thatthelensisembeddedinaglobalMinkowskimetric.Inacos- directions3,nomatterwherethelensispositioned,raisestheques- mologicalsituation,Eq.(2)wouldhavetobescaledbyaredshift tionhowthiscanbeconsistentwithconservationoftotalflux.This factor1+z ,butremainsvalidapartfromthat. issueisdiscussedbySchneider(1984),Schneideretal.(1992)and d We define the apparent deflection angle α to fulfil the lens referencestherein.Thebasicargumentisthatonehastothoroughly equation definetheunlensedsituation,relativetowhichthemagnifications are calculated. Adding a lens modifies the geometry of the Uni- θ =θ α(θ) . (3) s − verse,sothatthecomparisonisnottrivial.Thiscanbeillustrated byathoughtexperimentinwhichweshrinkthesizeoftheUniverse Deflectionandtime-delayarerelatedbyFermat’stheorem:Images bysomefactor.Thiswouldmoveallsourcesclosertoallobservers, formatstationarypointsofthetime-delayfunction.Thesecanbe sothattheywouldallseeanincreasedfluxdensity.Still,thereisof minima,maximaorsaddle-points.Criticalimageswithdegenerate coursenocontradictionwithenergyorfluxconservationatall. Hessianmatrix(seebelow,e.g.Einsteinringsofpointsources)are Animportantexamplefortotalnetmagnificationsisthecase unstableandshallnotbediscussedhere.Wethustakethederivative of a point-mass lens, where the calculations were presented long oftheFermatpotentialinEq.(1)andsubstitutethelensequation beforethegeneraltheorem(Einstein1936;Refsdal1964).Theto- (3)tofindthatthedeflectionangleisthegradientofthepotential, talmagnification of thetwo images4 asafunction of theangular α(θ)=∇ψ(θ) . (4) separationθ betweenlensandsourceisgivenbythestandardex- s pression If we calculate the potential from the mass distribution, we can provethatitobeysaPoissonequation θ2+2m µ = s , (10) tot ∇2ψ(θ)=2κ(θ) . (5) θs θs2+4m whichisgreaterthanunityeverpywhere.Herewehavedefined TheconvergenceκistheprojectedsurfacemassdensityΣinunits ofthecriticaldensity m= 4GM Dds 4GM (forD D ,D D), c2 DdDs → c2Dd s ≫ d ds ≈ s Σ = c2 Ds . (6) (11) c 4πGD D d ds which is the square of the Einstein radius. We can calculate the Inmoredetail,wecanexpresstheHessianmatrix,consistingofthe excessmagnificationbyintegratingoverthetangentialplane: second-orderderivativesofψ,intermsoftheconvergenceandthe two-dimensionalexternalshearγ =(γx,γy): ∆µ= d2θs(µtot 1)=2πm (12) − ZZ ψ ψ κ+γ γ H= xx xy = x y (7) Theorderofmagnitudeofthisexcessisplausible,becausethelens ψ ψ γ κ γ „ yx yy« „ y − x« producessignificantadditionalmagnifications(oftheorder1)over aregioncorrespondingtotheEinsteinringwithradius√m. Wewilllaterfindthatthesolutionofthemagnificationpara- 2.2 Themagnificationtheorem dox hastodowithtermsof µ−1 that arelinear inthedeflection. ThistheoremwaspresentedbySchneider(1984)andisdescribed This can be anticipated by the fact that the excess in Eq. (12) is inabroadercontextbySchneideretal.(1992).Hereweonlywant alsolinear inm. Counting only thetermslinear inthe deflection to make a plausibility proof, without claiming to be fully mathe- (linearinψ,κ,γ)ofEq.(9),wefind maticallyrigorous.ThelocalmagnificationmatrixMistheinverse µ−1 1 H H =1 ∇2ψ=1 2κ . (13) Jacobianofthelensmapping,Eq.(3): ≈ − xx− yy − − ∂θ Thefirst-ordermagnificationcanonlybereducedbelow1ifeither M−1 = s =1 H (8) κisnegativeorthePoissonequationismodifiedinanappropriate ∂θ − way.Wedoexpectthelattertohappenwhenweleavethetangential Thescalarmagnificationµisthedeterminantofthismatrix, planeandconsiderthesphere.Thedivergence(sourcedensity)of µ−1 =det(1 H)=(1 κ)2 γ2 , (9) thedeflectionfield∇·α = ∇2ψcannotbepositiveeverywhere, − − − becauseofGauss’law.Theresimplyisnochanceforthefluxofα withγ = γ .Thesignofµdefinestheparityoftheimage. toescapefromtheclosedsphere,sothattheintegrateddivergence | | For large θ, the Fermat potential will be dominated by the mustvanish. firstterminEq.(1)forallrealisticmassdistributions,i.e.itgrows quadraticallywithθ.Localmassconcentrationswillproducemin- ima in ψ that can diverge for point masses, but can not lead to 3 SOLIDANGLES positivediscontinuouspeaksinψ.Itisthusplausiblethatatleast onelocalminimumofφexists.Thispointisasolutionofthelens Toavoidmostofthecomplicationsofarbitrarydefinitionsforref- equation and corresponds toareal image. Being aminimum, the erence situations, we want to interpret the apparent paradox in a HessianoftheFermatpotential,1 H,mustbepositivedefinite. − Theeigenvaluescanbeeasilyderivedtobe1 κ γ,whichmust − ± 3 Forthemomentweassumetheequivalenceofmagnificationandampli- thusbothbepositive.Ifweaddtheconditionthattheconvergence fication,whichcanbeproveninthisframework. ordensityκisnon-negative,wefind0<µ−1 <1.Inotherwords, 4 ArealSchwarzschildlenswouldadditionallyproduceaninfinitenumber this image has positive parity and a magnification µ > 1. As a ofimagesveryclosetothelens(e.g.Virbhadra&Ellis2000).Something result, the total magnification of all images corresponding to one similar,althoughnotquantitatively thesame,happenswiththedeflection sourcepositionisalwaysgreaterthanunity. angleaccordingtoEq.(24).Theseimagesareextremelydemagnifiedand Thefactthatlensingincreasestheobservedfluxdensityinall canbesafelyneglectedhere. (cid:13)c 0000RAS,MNRAS000,000–000 4 O. Wucknitz differentway.Insteadofdiscussingfluxes, wewanttoinvestigate how solid angles are mapped from the observed image sphere to thetruesourcesphere,whichweputataninfinitedistance,where the metricperturbation by the lens can be neglected. Using solid anglesinsteadofareasonthesourcesphere,wecanavoidapossi- blescalingofthetotalarea.Thetotalsolidanglealwaysremains 4π. Wedonotrestrictthediscussiontosmallregionsaroundthe opticalaxis(correspondingtothetangentialplane)butinsteadal- lowforarbitraryimpactangles. 3.1 Conservationofsolidangle Westartbyintegratingthetotalapparentsolidangleontheimage sphere andtransformthisintoanintegral over thesource sphere. For this we assume that all lines of sight end somewhere on the Figure1.Deflectedlinesofsightwiththeobserverinthecentreandthe sourcesphereandthateverypartofthesourcespherecanbeseen sourcesphereascircle.Theverticallinedefinestheopticalaxis,thehatched circleshowsthelens.Theregularlyspaceddotsonthesourcespherelabel in at least one direction, i.e. we neglect possible horizons which thecorrespondingimagepositions(straightprojectionsofthelinesofsight). wouldaddhigherordercorrections. 4π= dΩ(θ)= dΩ(θ )µ (θ ) (14) s s tot s magnification.ThemagnificationaccordingtoEq.(15)definesthe I I ratioofsolidanglesofaconeoflightraysmeasuredbytheobserver dΩ µtot(θs)= µ(θ) = dΩ (θ) (15) andatinfinity.Inastronomyweareinterestedintheratioofsolid θX(θs)˛ ˛ θX(θs)˛˛ s ˛˛ anglesofoneandthesamesourceseenwithandwithouttheaction ˛ ˛ ˛ ˛ ofthelens.Naively,onewoulddefinetheunlensedreferencesitu- Thesumistakenover allimages correspon˛ding tot˛herespective ationinsuchawaythatthedistancebetweenobserverandsource source position. We learn that the total magnification, defined in iskeptconstant,butdistancesarenotuniquelydefinedinacurved termsofsolidangleelements,averagedoverthesourcespheremust space-timesothatthisapproachisnotunique.Thepictureofsolid beequaltounity.Evenifitisvalidclosethetheopticalaxis,the anglesandaverylargesourcespherecanalsobeinterpretedasa magnificationtheoremcannotholdinotherregionswheretheex- referencesituationinwhichthetotalareaofthesourcesphereisthe cesshastobecompensated. sameinthelensedandunlensedsituation.Fluxconservationthen Weconcludethatifthemagnificationtheoremwouldholdin demandsameanmagnificationofunity,sothatgeneralvalidityof thissituation, wewould haveareal paradox that had to betaken themagnificationtheoremcannotbeexpected. seriously. Wewilllearnlaterthatthemagnificationtheoremdoes We want to postpone the discussion of an appropriate refer- indeedbecomeinvalidforlargeimpactangles.Intheseregionsthe encesituationbyredefiningtheproblem.Itiswell-knownthatthe primaryimage isonly veryweakly deflected. Any additional im- gravitational light deflection can be described as the action of a ageswillbestronglydeflected(invalidatingtheweak-deflectionap- refractivemediumthatfillsanEuclideanspace,whosecoordinates proximation),buthighlydemagnified.Inthefollowingdiscussion areidentifiedwiththoseusedinthestandardformoftheweak-field wewillmostlyneglecttheseimages,whichallowsustoconsider metricequation (16). Inthismodel, theunlensed situationwould the magnification as a function of image positions instead of the obviouslybetheonewithalldistancesunchanged buttherefrac- totalmagnification asafunction of sourceposition. Fig.3 shows tivemediumremoved.Solidanglescanthendirectlybeidentified thatthissimplificationisjustified. withareasonthesourcesphere,sothatthereasoningabout solid Figure 1 illustrates why the magnification theorem can be anglesinSec.3.1inevitablyleadstoafailureofthemagnification validaround the optical axisbut not everywhere. Weseethat the theoreminthissituation.SincethiscontradictstheresultofSec.2, lens increases the density of lines of sight on the source sphere even though the formalism isexactly the same, we have to iden- aroundthe(forward)opticalaxis,correspondingtoamagnification tifyandabandontheinadmissibleassumptionsorapproximations. greaterthanunity.Onthetangentialplane,thiswouldbepossible Oncetheformalismhasbeengeneralised,wecancomebacktothe everywhere,becauselinesofsight(orsolidangleonthesky)can definitionofreferencesituations. be‘borrowedfrominfinity’.Incontrasttothis,thetotalsolidangle AsanalternativetorefractionwecanrefertotheNewtonian mustbeconservedonthecompactsphere.Thisnecessarilyleadsto pictureofthedeflectionofmassiveparticlesmovingwiththespeed magnificationslessthanunitysomewhere,seeninthelowerparts of light. Modulo a factor of 2, the deflection follows exactly the of the diagram. Note that, for simplicity, a lens without multiple same law as in general relativity, but the geometry of space and imaging isshown here. Thelensmassdistributionwaschosen in timearenotaltered. suchawaythatthemagnificationscanbeseenalreadyinthistwo- dimensionalcross-sectionalview.Generallythedensityoflinesof sightonlyshowsinthefullthree-dimensionalgeometry. 4 SOURCESATINFINITEDISTANCES 3.2 Referencesituationsandtherefractionanalogue 4.1 Point-masslens Wenowknowthatforsolidangles,measuredforsourcesatinfin- Weknow thatclosetotheoptical axisthemagnificationtheorem ity, the magnification theorem cannot hold. However, we have to holds.FromthediscussionofFig.1,weexpectthatitbreaksdown beveryspecificaboutthesituationsthatarecomparedtodefinea atlargerimpactangles.Thereforewehavetogeneralizetheusual (cid:13)c 0000RAS,MNRAS000,000–000 Fromplanestospheres 5 formalismtoallowsourcepositionsfarfromthelenscentre.How- α ever,wewillkeeptheweak-fieldapproximationsothatdeflection anglesstillhavetobesmall. θ We describe the lens(es) as small perturbation of an asymp- s totic Minkowski metric instead of a more realistic Robertson- z x Walkermetric.Thissimplifiesthetreatmentwithoutobscuringthe M θ apparent magnification paradox or its solution. We use the static D weak-fieldmetricinisotropiccoordinates,sothatlocallyobserved d angles (and solid angles) can bederived directlyfrom thespatial Figure2.Geometry ofthe light deflection withasource atinfinity. The coordinatesr = (x,y,z).IntermsoftheNewtonianpotentialΨ, anglesθ,θsandαarepositiveinoursignconvention.Hereandinallfol- thelineelementcanbewrittenas lowinggeometricalconsiderations,wepretendthatspaceisEuclidean(and space-timeMinkowski)andtreatthelightdeflectionasanon-geometrical 2Ψ 2Ψ ds2 = 1+ c2dt2 1 dr2 . (16) effect.Thefiguresrepresentthegeometryofthecoordinatesusedtodefine c2 − − c2 themetricinEq.(16).Isotropyofthemetricensuresthatallanglescorre- „ « „ « spondtoreallocallyobservableangles.Straightlinesinthefiguresarenot Thisleadstothegeodesicequationforlightinfirstorderapproxi- necessarilygeodesicsinrealspaceorspace-time. mation r˙2 r¨= 2 ∇⊥Ψ , (17) 4.1.2 Potential − c2 where∇⊥ denotesthetransversalpart(relativetor˙)ofthegradi- ThedeflectionangleinEq.(24)canbewrittenasthederivativeof ent,anddotsstandforderivativeswithrespecttotheaffineparam- apotentialψ,whichisdefinedonthesphereas eter.Withinfirstorder(inΨ),thegeometricallengthorlocaltime θ m canalsobeusedequivalently. ψpm(θ)=mlnsin2 = 2 ln(1−cosθ)+const . (25) 4.1.1 Deflectionangle 4.1.3 Magnification WewanttocalculatethedeflectionangleofapointmassM with Consider as source a thin concentric annulus around the lens, as a source at infinity. We restrict ourselves to the plane defined by seenbytheobserver.Withradiusθsandwidthdθs,thesolidangle lens, source and observer, which contains the complete light ray. is dΩs = 2πsinθsdθs. This annulus will be observed at radius The coordinates have their origin at the observer, z is measured θ with width dθ and solid angle dΩ = 2πsinθdθ. We separate alongtheunperturbedlineofsight,xintheperpendiculardirection themagnificationintothetangentialandradialparts,MTandMR, (Fig.2).Thedeflectionangleisdefinedaccordingtothestandard respectively: formofthelensequation(3).Thelens,locatedat dΩ µ−1 = s =M−1M−1 (26) dΩ T R x = D sinθ , z =D cosθ , (18) 0 d 0 d − sinθ sin(θ α) M−1 = s = − =cosα cotθsinα (27) producesaNewtonianpotentialof T sinθ sinθ − dθ dα GM M−1 = s =1 (28) Ψ(x,z)= . (19) R dθ − dθ − (x x0)2+(z z0)2 m θ − − =1+ sin−2 (29) ThedeflectionangleiscalpculatedbyintegratingEq.(17)alongthe 4 2 unperturbedlineofsight.Itisdefinedtobepositivefordeflection The magnification we get from Eqs. (26–28) with the deflection inthenegativez-direction.Weusezasaffineparameter: anglefromEq.(24)isplottedinFig.3.Thedeflectionangleandits derivativearecorrectonlytofirstorderinm,sothatthefirstorder ∞ α= ∆x˙ = dzx¨(x=0,z) (20) isalsosufficientfortheradialandtangentialmagnifications5.Up − − Z0 tothisorder,wefind 2 ∞ = c2 Z0 dz∇⊥Ψ(x=0,z) (21) MT−1 =1−αcotθ , (30) 2GM ∞ −3/2 m m2 cosθ = x dz x2+(z z )2 (22) µ−1 =1+ . (31) − c2 0 0 − 0 2 − 4 (1 cosθ)2 Z0 h i − = 2GM 1+z0 x20+z02 −1/2 (23) For large θ, the m/2 term in Eq. (31) is dominant, whilst in the − c2 ` x0 ´ normallensingregime,them2termbecomesmoreimportant.Ex- m1+cosθ m θ = = cot (24) 2 D sinθ 2 2 d 5 Thisis not true forthe scalar magnification µ(the determinant ofthe Inthelimitofsmallθ,i.e.closetotheopticalaxis,werecoverthe magnificationmatrix),wherethesecondorderterms,resultingfromcombi- standardresultα=m/θ,withthedefinitionofmfromEq.(11). nations offirstordertermsinMT andMR,areresponsibleforthemag- Notethatforlargeθthedeflectionisnotconfinedtoregions nifications in the classical lensing regime close to the optical axis. It is smallcompared toDd,but takesplaceover arange x0. This ironic that our generalisation of the standard formalism introduces addi- ∼ | | meansthatnoteventhepoint-masslenscanbeconsideredasthin tional termsoffirstordertoµ,whilstthedominating classical partisof anymore. secondorder. (cid:13)c 0000RAS,MNRAS000,000–000 6 O. Wucknitz 3 µ µ −1 10 tot tot µ+ µ+−1 2 −µ− −µ− 1 1 0 −11[10] -1 -2 -3 0.1 1 10 0 500 1000 1500 2000 2500 θs [arcsec] θs [arcsec] Figure3.Magnification ofbothimages(positive andnegative parity) andthetotalmagnification asafunction ofthesourcepositionforalensofm = (1′′)2=2.35·10−11.Forsmallangles(left),themagnificationisindistinguishablefromtheclassicresult.Farfromtheopticalaxis(right),themagnification approaches1−m/2insteadof1.Themagnificationtheoremdoesnotholdinthisregion.Thetotalmagnificationandthemagnificationofonlythepositive parityimageareindistinguishableforverylargeimpactangles.Thisjustifiesourapproachofnotdiscussingthetotalmagnificationofallimagesbutonlythat oftheprimaryimage(µ+). pandedinpowersofθ,wefind tofind µ−1 =1+ m2 − mθ42 1+O θ2 . (32) ψ(θ)= 21π dΩ′(θ′)σ(θ′) ln 1−θ·θ′ +const . (36) I The corrections in the small-angle rhegime (`last´tierm) are not rel- Inthetangentialplane,thiscorrespon`dstothein´tegral evant in our context. The isotropic correction m/2, on the other 1 hand, leads to violations of the magnification theorem. Far away ψ(θ)= d2θ′σ(θ′) ln θ θ′ +const . (37) π | − | from the optical axis, the magnification sinks below unity. The ZZ transitionregion withamagnification of 1isaround θ = √42m, Eventhoughthisformalismisvalidforarbitrarymassdistributions, approximately the square root of the Einstein radius. For typical itiscorrectonlytofirstorderinρ.Thismeansitcannotbeusedfor galaxy lensing cases, this is a few arc-minutes from the lens, far multi-planestronglensing,wherethechangeofimpactparameter awayfromthestrong-lensingregimebutstillatasmallangle.For inonelensplaneasaresultofthedeflectioninanotherplanebe- microlensing,thetransitionregionisevencloser. comesrelevant. Thesmallcorrectionoftheorderm(10−10 intypicalgalaxy scalelenses)iscompletelyirrelevantforallpracticalcalculations. 4.2.2 Poissonequation Nevertheless,thistermisresponsiblefortheconservationoftotal flux or solid angle. Integrated over the complete celestial sphere, ThePoissonequationforthesphericalpotentialψcanbederived itleadstoadeficitof2πm+ (m2),whichexactlycompensates withverylittleformalcalculations.WedeterminethefluxF ofthe O fortheexcessfoundinEq.(12).Becausethemodificationisoffirst deflectionfieldαthroughacircleofradiusθaroundthepointmass orderinm,thefollowingdiscussionwillconcentrateonfirst-order m,correspondingtoσ(θ) = πmδ2(θ).Thisistheproductofα effects. withthecircumferenceofthecircle, F(θ)=α(θ)2πsinθ=πm(1+cosθ) . (38) 4.2 Generalmassdistributionsonthesphere In the limit of θ 0, this startswith F(0+) = 2πm and then 4.2.1 Potential decreaseslinearly→withtheenclosedareaAofthecircle: Knowingthepotentialforapoint-masslensfromEq.(25),wecan F(θ)=2πm mA(θ) A(θ)=2π(1 cosθ) (39) easily generalize for arbitrary mass distributions, defined by the − 2 − three-dimensionalmassdensityρ(r): WeknowfromGauss’theoremthatthisfluxequalstheintegrated 4G ρ(r′) ∡(θ,r′) divergence of the field(= Laplacian of thepotential), so that we ψ(θ)= c2 d3r′ r′ lnsin 2 (33) canwritethePoissonequationas ZZZ = 2cG2 d3r′ ρ(rr′′)ln 1− θr·′r′ +const (34) ∇2ψpm(θ)=2πmδ2(θ)− m2 . (40) ZZZ „ « Theδtermisthesameasonthetangentialplane.Themassactsin The direction is denoted by the unit vector θ with r = θr. We theusualwayassourceforthedeflectionfield.However,incontrast separatetheradialintegrationfromthetangentialpartonthesphere byintroducingthenormalizedsurfacemassdensity6 to the tangential plane, the sphere is closed, so that sources and sinksofthefieldmustcompensateeachother;thefieldlinescannot 4πG ∞ σ(θ)= drrρ(r) (35) extend to infinity. Without the second term in Eq. (40), the field c2 Z0 lineswould continue and meet at θ = π where theywould form anadditionalsingularityofmass m.Instead,thefield‘decays’to 6 Notethatinthisradialprojectiontheinfluenceofmasselementsatdis- ensureavanishingtotalintegrated−∇2ψ. tancesrdoesnotscalewithrbutwith1/r,seeEqs.(33–34). For an arbitrary mass distribution σ, the Poisson equation Fora thin shell around the observer at acertain distance Dd, wefind reads σ(θ) = (4πG/c2)DdR drρ(r) = Σ/Σc asinthestandardformalism. ∇2ψ(θ)=2[σ(θ) σ]=:2κ(θ) . (41) ComparewithEq.(6). − (cid:13)c 0000RAS,MNRAS000,000–000 Fromplanestospheres 7 Welearnthatnotσitselfisthesourceofthefieldbutthedifference ofσandthemeansurfacemassdensity z σ= 1 dΩ(θ)σ(θ) . (42) Ds α x 4π S O I InAppendixA1weshowhowthesphericallensingpotential Figure4.Theapparentdeflectionangleαisdefinedasthedifferencebe- canbewrittendirectlyasintegralovertheNewtonianpotential.It tweentheapparentandtruesourceposition.Changesofthedirectionofthe isthethree-dimensionalPoissonequationthatleadstothefirstterm lightpathwillthereforeaffectαscaledwithDds(z)/Ds. inEq.(41).ThesecondtermisrelatedtotheNewtonianpotential atthepositionoftheobserver.Thistellsusthatthedeviationsfrom thestandardformalismcanberelatedtolocaldistortionsofspace- sothatwecannotinfermagnificationsgreaterthanunityfromthe time. signsoftheeigenvalues.Eventhoughthedensity(evenincompar- isonwiththereferencesituation) isstillstrictlynon-negative, the 4.2.3 Themagnificationmatrix convergence is not, which invalidates one of the assumptions for theproofofthetheorem. The(inverse)magnificationmatrixisgivenbytheJacobianofthe Recall that,inorder todefine afirmfoundation forthisrea- lensmapping.Onthetangentialplane,theJacobianofEq.(3)leads soning, we refered to non-relativistic situations with unperturbed directlytotheHessianofthepotential,seeEq.(8).Thesituationis geometryinwhichthedeflectioniscausedeitherbyrefractionor morecomplicatedonthecurvedsphere,wherefinitedisplacements by Newtonian deflection. If wenow return tothe relativisticsce- cannotbetreatedasvectors.InAppendixBwecalculatethemag- nariobutdefinetheinfinitelylargesourcesphereinsuchawaythat nificationmatrixforarbitrarydeflectionfunctionsα(θ),including theareaisthesameforthelensedandunlensedsituation,wecome largedeflectionangles. Intermsofthissection,wherethedeflec- toexactlythesameconclusionforthatcase. tionangleαisthegradientofapotentialψ,theexactmagnification matrixfromEq.(B9)canbewrittenincoordinates( , )parallel k ⊥ andperpendiculartothenegativedeflectionangle α,as − 1 ψ ψ − k;k − k;⊥ 5 SOURCESATFINITEDISTANCES M−1 = . (43) 0−ψ⊥;ksinαα cosα−ψ⊥;⊥sinαα1 Inthefollowingweallowforsourcesatfinitedistances.Inthissit- @ A uation,themetriconthesourcesphereismodifiedbythelens,so Thelowerindicesofψdenotethesecond-order(covariant)deriva- thatthepreviouslyusedapproachisnotpossibleanymore,butas- tiveswithrespecttothecoordinateaxes.Thecorrectionsincosα sumptionsabouttheunlensedreferencesituationhavetobemade. andsinα/αareof secondorder inthedeflectionα.Theseterms Wedothisbyfixingthecoordinatedistances,whichwouldalsobe can be set to unity if the magnification matrix is needed only to theappropriateapproachforthemodelofrefractionorNewtonian firstorder.Theresultisthenequivalenttotheplanarformalismin deflectioninanEuclideanmetric.Modificationsoftheresultsfor Eqs.(7–8),butwiththemodifiedsphericalpotential. differentconventions(likefixedmetricdistances)willbediscussed Forapointmassatadistanceθ,theHessianofψisdiagonal later. with7 Mainaimofthissectionistheanalysisoflensdeflectionand ψ =ψ , ψ =ψ cotθ , (44) thepotentialtheoryonthesphereasopposedtotheplanartheory. k;k θ,θ ⊥;⊥ θ Forcompletenessthishastobegeneralisedtofinitedistances.Most whichleadsto ofthediscussionofthemagnificationtheorem,ontheotherhand,is m θ basedonthecaseofsourcesatinfinity,becausesomearbitrariness 1+ sin−2 0 M−1 = 4 2 . (45) indefiningareferencesituationcanbeavoidedthen. 0 1 0 cosα cotθsinα − @ A This matrix corresponds exactly to Eqs. (27–29). Neglecting the curvaturetermsinM−1correspondstothelinearapproximationin 5.1 Deflectionangle Eq.(30–31),whereonlythetermlinearinαwasincludedforthe tangentialmagnification. IfthesourceisatafinitedistanceDs fromtheobserver,thesim- plegeometryinFig.2hastobemodified.Ifwewanttokeepthe form of the lens equation (3), the apparent deflection angle α is 4.2.4 Magnificationtheorem nolongerdefinedastheanglebetweentheincomingandoutgoing light ray, but as the difference between apparent and true source Wehavenow,tofirstorder,derivedthemagnificationmatrixasthe position(Fig.4).Inthethin-lensapproximation, thisleadstothe Hessianmatrixofapotential,analogouslytothestandardformal- well-known correction factor of D /D to transform the true to ism.Wecanthusformallyfollow theproof for themagnification ds s the apparent deflection angle. As mentioned earlier, far from the theorem as described in Sec. 2.2 step by step. However, the as- opticalaxisnoteventhepoint-masslenscanbeconsideredasbe- sumption of positive (effective) κ isno longer true, seeEq. (41), ing thin, which means that D is not defined globally. If we in- ds stead consider how a local change of x˙ in Eqs. (20) and follow- 7 This can be derived from the covariant derivatives in (θ,φ) coordi- ing affects the difference between true and apparent source posi- nates, Tθ;θ = Tθ,θ,Tφ;φ = Tφ,φ +sinθcosθTθ,Tθ;φ = Tφ;θ = tion, we find that the integrands have to be scaled with the local Tθ,φ − cotθTφ, together with the scaling transformation ∂k = ∂θ, Dds(z)/Ds =(Ds−z)/Ds.Inaddition,theintegrationhastoend ∂⊥=∂φ/sinθ. atz =Ds.Withthesameapproachasbefore(λ=z),wefindfor (cid:13)c 0000RAS,MNRAS000,000–000 8 O. Wucknitz L towritethetangentialmagnificationM followingEq.(30)forthe T φ deflectionangleinEq.(49)as D D d ds GM 1 1 D +D φ M−1 =1+ d ds tan2 . (55) T c2 „Dd − Dds − DdDds 2« θ θ’ Theradialmagnificationcanbedeterminedfromthederivativeof O D S thedeflectionangle,seeEq.(28), s GM 1 1 D +D φ Fθ,igθu′,rean5d.DφeffionriatioconmopfathcetldeinsstaantcLe.pOarbasmerevteerrsaDndds,oDusrc,eDadrseadnedntohteedanbgyleOs MR−1 =1+ c2 „Dd − Dds + DddDdsds tan2 2« . (56) andS.Recallthat,inconcordancewithourgeneralphilosophy,thelengths TofirstorderinM,thescalarmagnificationisgivenby andanglesareinourconvention definedinanEuclidean metricwiththe samecoordinatesasinthegeneralmetricofEq.(16).(SeecaptionofFig.2.) 2GM 1 1 µ−1 1+ . (57) ≈ c2 „Dd − Dds« Notethat for these calculations weassumed that thecoordi- thedeflectionangleofapoint-masslens nates of source and observer are the same in the lensed and un- Ds D z lensedsituation. ForfiniteDs, thelenswillmodifythegeometry α(θ)= dz s− x¨(x=0,z) (46) of the source sphere, which makes the definition of the unlensed −Z0 Ds referencegeometrysomewhatarbitrary.Whilecoordinatescanbe =−2Gc2M x0Z0Dsdz DsD−s z hx20+(z−z0)2i−3/2 (47) nfisiexzneetdscofoonfrsetpxaonteitnnttdo-elaidkllesoowsuorufcorecrse.asF,moterhaitsnhioinssgenf,uowltecaopsmhporpouaplrdriisakoteene9pf.otrhethirepchoymsipcoa-l 2GM 1 (D z )2+x2 z (D z ) x2 FromthemetricinEq.(16)weinferthattheratioofareato = s− 0 0 + 0 s− 0 − 0 − c2 x0Ds »√x20+(Ds−z0)2 √x20+z02(48) – spooltiedntainalglΨes(aotnththeesosuorucreceposspihtieorne:) changes isotropically with the = 2GM Dds(θ)+Dscosθ−Dd , (49) dA = 1 2Ψs D2dΩ (58) c2 DdDssinθ s − c2 s s „ « withthecoordinatedistancebetweensourceandmass Usingthiswecandefinethetrueareamagnificationµ asafunc- (A) tionofthesolidanglemagnificationµ ,whichwecalledµbe- D (θ)= D2+D2 2D D cosθ . (50) (Ω) ds s d − d s fore: Inthelimitofsmallθthpisreducesto8 dΩ 1 dA 2Ψ µ−1 = s µ−1 = s = 1 s µ−1 (59) (Ω) dΩ (A) D2 dΩ − c2 (Ω) α= 4GM Ds−Dd Θ(Ds−Dd) + (θ) . (51) s „ « c2 » DdDs θ O – TheradialandtangentialmagnificationsMRandMT bothhaveto becorrectedwiththesamefactor,whichisthesquarerootofthat Forasourcebehindthelens,werecovertheclassicallimitingcase forthescalarmagnification,sothat withtheappropriatescalingoftheapparentdeflectionangle.Ifthe alleellnnyssibinsegfc)uo,rmtthheeesrsmianwugacuyhlatwrhitaeynakatethrθe,sj=uosu0trcavesa(enwxisphheieccshteawdn.edImnthatehyidsceacfllaelscbetaitochkneggrmeonauxenrid-- MMRR−(−(AΩ11)) = MMTT−(−(AΩ11)) =1− Ψc2s =1+ GcM2 D1ds . (60) mumeffectwillbereachedatsomefiniteθ. WenowapplythiscorrectiontothemagnificationsfromEqs.(55– Notethat incontrast tothestandardexpressions, Eq. (49) is 56): validforallcombinationsofdistances.EvenD D ispossible, includingthelimitofself-lensingwithDs =Dsd.≈ d MT−(A1) =1+ GcM2 „D1d − DDd+dDDdsds tan2 φ2« (61) GM 1 D +D φ M −1 =1+ + d ds tan2 (62) R(A) c2 D D D 2 5.2 Magnification „ d d ds « Thefirst-orderscalarmagnificationis Laterwewillcomparemagnificationwithamplification,whichcor- responds toanexchange ofobserver andsource, i.e.θ θ′ and 2GM 1 ↔ µ−1 1+ . (63) Dd ↔Dds(seeFig.5).Forthispurpose,weformulatethemagni- (A) ≈ c2 Dd ficationinaformthatexplicitlyshowsthesymmetries.Weusethe Allthesemagnificationsrefertoanunlensedsituationwiththe trigonometricrelations samecoordinatedistancetothesource.Asalternativeswediscuss D2 =D2+D2 2D D cosφ , (52) fixedaffineandmetricdistancesinAppendixC. s d ds− d ds D sinθ =D sinφ , (53) s ds D cosθ=D D cosφ , (54) 9 Note that Avni&Shulami (1988), who followed light rays from the s d ds − sourcetotheobserver,tookthechangedgeometryintoaccountontheside ofthesourcebutnotonthesideoftheobserver.Eventhoughtheyworkin Schwarzschild coordinates,wherethesurfaceofaspherewithaconstant 8 TheHeavisidefunctionisdefinedasΘ(x)=1forx>0andΘ(x)=0 radialcoordinaterisalways4πr2,thisshouldnotbeneglected,sincethe otherwise. comparisonsituationisaspherearoundthesourceandnotaroundthelens. (cid:13)c 0000RAS,MNRAS000,000–000 Fromplanestospheres 9 5.3 PotentialandPoissonequation AppendixA2–A3)convergetotheonesforinfiniteD derivedin s the previous sections and Appendix A1. The distinction between Thepotentialcanbederivedbyintegratingthepoint-massdeflec- µ andµ isnotnecessaryinthislimit. tionangleinEq.(49),oralternativelybyintegratingover theap- (A) (Ω) propriatelyscaledthree-dimensionalpotentialΨ,asshowninAp- pendixA3withtheresultinEq.(A17). 6 RECIPROCITYANDSURFACEBRIGHTNESS InAppendixA2wederivethePoissonequation byintegrat- ing over the three-dimensional potential derivatives, and find the ThemagnificationsinEqs.(61–63)arenotinvariantundertheex- relation changeofsourceandobserver.Writingµ (o,s) = µ forthe (A) (A) ∇2ψ(θ)=2σ(θ)+2Ψo−Ψs(θ) , (64) magnificationofthesourceasseenbytheobserverandµ(A)(s,o) c2 forthemagnificationoftheobserverasseenbythesource,wefind (byexchangingD D )thefollowingrelation,whichiscorrect whereΨoistheNewtonianpotentialattheobserver.Theprojected uptofirst-ordertedrm↔sintdhsetangentialandradialmagnifications. surfacemassdensityisnowdefinedas 4πG Ds D r µ(A)(s,o) =1+ 2GM 1 1 (69) σ(θ)= c2 drr sD− ρ(r) . (65) µ(A)(o,s) c2 „Dd − Dds« Z0 s Whatdoesthismeanphysically?Themagnificationµ (o,s)de- Theweightfunction inEq.(65)hastheshape ofaparabola with (A) finesthescalingoftheapparentsizeofamagnifiedsourceasseen maximum at r = D/2 and zeros at r = 0 and r = D. How- s s bytheobserver.Thereciprocalµ (s,o),ontheotherhand,isin- ever,sincetheapparentsizeofamassclumpscaleswith1/r2,the (A) verselyproportionaltotheareaintheobserverplanethatisspanned influenceofamasselementatrisproportionalto1/r 1/D. − s byacertainlightbundle;thatmeansitdefinestheamplificationin Inclassicallensingtheory,thedivergenceofthedeflectionan- terms of the number of photons received from the source by the gleleadstothelocalsurfacemassdensity.Farawayfromtheop- observerperdetectorareaelement. ticalaxisbutforsourcesatinfinity,thisischangedinthewaythat The ratio of amplification to magnification in Eq. (69) pro- weobtaininEq.(41)thedensitycontrastrelativetothemeanden- videsthegravitationalchangeof‘surfacebrightness’,measuredas sity,whichresultsfromthegravitationalpotentialatthepositionof photonnumberdensitypersolidangle.Inthissense,gravitational theobserver(seeAppendixA1).NowwefindinEq.(64)forfinite lensingdoesnotconservesurfacebrightness. D thatthedivergenceisalsoaffectedbythepotentialatthesource s Thismayseemsurprisingbutisinperfectagreementwiththe position(seeAppendixA2).Thesetwocontributionsdescribethe reciprocitytheoremderivedbyEtherington(1933)10.Ifonedefines influence of all masses, while the surface mass density only in- inanarbitraryspace-timefortwoeventsxandy,whicharecon- cludes masses inside of the source sphere. The projected surface nected by a null-geodesic, i.e. one can be seen by the other, the massdensityσ vanishescompletelyforD 6 D ,i.e.inthecase s d angularsizedistancesD forthedistanceofy asseenbyxand wherethesourceisclosertotheobserverthanthelens. xy viceversa,itcanbeshownthatthetwoarerelatedby WerecallfromSec.2.2thatthedivergenceofthedeflectionis directlyrelatedtothefirst-orderscalarmagnification.Thesameis Dyx = 1+zy , (70) truehere,butthecorrespondingmagnificationnowreferstosolid D 1+z xy x anglesinsteadofarea.WhenwritingEq.(64)forthepoint-mass, wherezaretheredshiftsasmeasuredbyanarbitraryobserver. wefindthatthepotentialtermsleaddirectlytothecorrectionterms Inour situation,thelensmagnificationsactascorrectionsto inEq.(57)ifwefollowthesamerecipeasinEq.(13). obtaintheeffectiveangularsizedistancesfromD: s As consistency check we should test if the integral over Eq. (64) vanishes, as required by the compact geometry of the 1 = µ(A)(o,s) 1 = µ(A)(s,o) (71) sphere.WeknowfromGauss’theoremthatthepotentialofaho- Do2s Ds2 Ds2o Ds2 mogeneoussphericalshellisequivalenttothatofapoint-massfor Thegravitationalredshiftsproducedbythefieldofthelenscanbe regions outside of the shell, and constant inside. Wecan turnthe derivedfromthemetric,Eq.(16),as argumentaround,tolearnthatthepotentialaveragedoverthesur- faceofanysphereisthesameasthatinthecentreofthesphere, cdt Ψ GM 1 1+z = =1 =1+ (72) ifmassesarelocatedonlyoutsideofthesphere.Thesemassesdo ds ˛dr=0 − c2 c2 D notinfluencetheintegralofthedifferenceΨo ΨsinEq.(64).For forapointmassatadist˛˛anceD. massesinsideofthesphere,ontheotherhand−,theaveragedpoten- ˛ WecanconfirmEq.(69)bytakingtheratioofEqs.(71)and tialisindependent of theirlocation, sothat thetotalmasscan be insertingEq.(70)andtheredshiftsfromEq.(72): thoughtofasbeinginthecentre: GM 1 2 <Ψs>=−GI dΩZ0Dsdrr2ρD(rs) +C (66) µµ((AA))((so,,os)) = DDos22os =„11++zzos«2 =011++GcM2 D1d 1 (73) Ψo =−G dΩ Dsdrr2ρ(rr) +C (67) 2GM 1 1B@ c2 DdsCA I Z0 =1+ (74) Thecontributionsfrommassesoutsideofthespherearedenotedas c2 „Dd − Dds« C.BycomparisonwithEq.(65),wefind So far, we have discussed the surface brightness defined as photonnumberpersolidangle.Ifwewanttodetermineitinterms <Ψ Ψ >= c2 <σ> . (68) o s − − Asrequired,theaveragesourcedensityinEq.(64)vanishes. 10 SeealsoEllis(2007)andthemoreeasilyavailablerepublicationofthe NotethatforDs Dd allexpressioninthissection(andin originalarticleasEtherington(2007). ≫ (cid:13)c 0000RAS,MNRAS000,000–000 10 O. Wucknitz ofenergyfluxdensityF,wehavetotakeintoaccountthatboththe anymore.The‘thickness’ofalensisnotdefinedbytheextentofthe energyoftheindividualphotonsandthearrivalrateattheobserver massdistributionbut bytheextentof thedeflectingpotentialand areaffected by theredshift. Thisprovides another factor of (1+ itsderivatives. The thickness of a point-mass lens isthus closely z )2/(1+z)2,sothatthesurfacebrightnessintermsofintensity related to the impact parameter and can be of the same order of o s scaleswith magnitudeasthedistancesinvolved. F 1+z 4 4GM 1 1 Toavoidambiguities,westartbyconsideringasourceatinfin- obs = o =1+ . (75) ity,sothatthegeometryonthesourcesphereisnotchangedbythe F0 „1+zs« c2 „Dd − Dds« actionofthelens,andtheareaofthesourcesphereisnotmodified. Thescalingofthesurfacebrightnesswith(1+z)−4is,ofcourse, Thismakesacomparisonwiththeunlensedsituationmeaningful. a well-known fact in cosmology. Here we have exactly the same Wecalculatemagnificationsbymappingsolidangleelementsfrom effect,butcausedbythemetricperturbationsofagravitationallens the lens sphere to the source sphere. Because we do not include atfinitedistance.Inanycase,weshouldkeepinmindthattheeffect horizonsaroundblackholes,whicharecorrectionsathigherorder, is extremely small, typically of the order m or the square of the thetotalsolidangleofthewholeskymustbe4πwithandwithout Einsteinradius.Fortypicalgalaxy-scalelenseswithEinsteinradii lensing. Lensing definesaone-to-many mapping fromthesource oftheorder 1′′thiscorrespondsto 10−11–10−10. spheretothelenssphere.Weseethebackgroundskyinalldirec- ∼ ∼ tions,nopartsofthebackground arehidden, butsomepartsmay bemultiplyimaged.Withthisargument,weconcludethatthetotal magnification cannot be larger than unity for all positions on the 7 LIGHTTRAVELTIME sourcesphere,incontradictiontothemagnificationtheorem.This Weseparatethelighttraveltimeintothreeparts:Theundisturbed thoughtissupportedbyasimplegeometricargument.Thedensity travel time Ds/c, and the geometrical and potential parts of the ofdeflectedlinesofsightonthesourcesphere(orequivalentlylight time-delay, ∆tgeom and ∆tpot, respectively. The potential part is raysonanobserver’s sphere around thesource) canbeincreased easytocalculate.WestartwiththemetricfromEq.(16)andwrite closetotheopticalaxis,becausealenswithpositivemasshasafo- the(coordinate)timeintervalforanull-curvealongthez-axisas cusingtendency.Intheapproximationofthetangentialplane,this can be true everywhere, since lines of sight (and solid angle ele- 2Ψ dz dt= 1 . (76) ments)canbeborrowedfrominfinity.Onceweconsiderthecom- − c2 c „ « pletesphere, thisisno longer possible. Solidangle elementsthat Forapoint-masslens,theintegralisbasicallythesameasforthe aremoved towards the optical axis must be re-moved from other metricdistanceshowninAppendixC2: parts of the sphere. The magnification cannot be larger than one everywhere.Forthisargumentitisessentialtoworkwithsolidan- D 2 Ds ∆t = dt s = dzΨ(x=0,z) (77) glesonasourcesphereatinfinity.Withasphereatfinitedistance, pot − c −c3 Z Z0 theareadensityofprojectedlinesofsightcanwellincreaseevery- = 2GM lnDds(θ)+Dd+Ds +const (78) where, simply by making the area of the sphere smaller. Masses c3 Dds(θ)+Dd Ds modify the geometry, so that mean magnifications above one are − We observe that this is proportional to the deflection poten- nolongerparadoxical.Thefurtherthesourcespheremovesaway, tial in Eq. (25) only in the limit D , where we have the weaker these distortions get. For a finite mass, they decrease s ∆t = D ψ/c.Inthegeneralcaseth→erei∞snoproportionality withoutlimitsandcanfinallybeneglected. pot d toEq.(A1−7). Thenon-relativisticpicturesofrefractionandNewtoniande- Infact,wedonotexpectaone-to-onerelationbetween∆t flectioninunperturbedgeometrysupportthisview.Inthesescenar- pot andthepotentialψasinthecaseofthinlenses.Theargumentused iosaparadoxcannotbeavoidedifnocorrectionstotheformalism in Sec. 2.1 to explain why the deflection angle is proportional to aremade. thegradient ofthepotentialtime-delaybreaksdown, becausethe Withthismotivation, wecalculate the deflection angle for a geometrical part of the time-delay does not keep its simple form point-masslensforarbitraryanglestotheopticalaxiswithsource of Eq. (1). It cannot be expressed in terms of image and source at infinity and derive magnifications from that. We find an addi- positionaloneanymore,becausethedeflectionisnotrestrictedtoa tional first-order term that lowers the magnification, violates the lensplane(orsphere).Inordertodeterminethegeometricaldelay, magnification theorem at some point, and assures total conserva- wehavetoknowthefullthree-dimensionalmassdistributionand tionofsolidangle.Inordertounderstand whichpartof theclas- notjustitsprojectionσortheprojecteddeflectionpotentialψ. sicalproof of thetheorembecomes invalidoncewegofromtan- Becauseitcannotbeeasilygeneralizedforarbitrarymassdis- gentialplanestofullspheres, wedevelop thepotential theoryfor tributions,wedonotpresentthecalculationof∆t forapoint- arbitrarymassdistributionsonthesphere.Asintheplane,wecan geom masslens. define a two-dimensional lensing potential that is a projection of thethree-dimensionalNewtonianpotential.Inthesamewaywede- fineaprojectedsurfacemassdensityonthesphere.Wefindthatthe Poissonequationismodifiedinaninterestingway.Intheplane,the 8 SUMMARY divergenceofthedeflectionisproportionaltothesurfacemassden- We discuss the classical magnification paradox in gravitational sity.Thiscannotbetrueonthesphere,becausefluxofthedeflec- lensing in order to better understand if and how a magnification tionfieldcannotescapefromthesphere.Theintegrateddivergence greater than unity everywhere can be consistent withglobal pho- must vanish inorder toobey Gauss’ law. Wedo indeed find that tonnumberconservation.Itisoffundamentalimportancetoallow thedivergenceis2(σ σ),thedensitycontrastrelativetotheaver- − forlargeanglesrelativetotheoptical axis.Thereforewedevelop ageddensityσ.Thefieldlinescorrespondingtothedeflectionfield a formalism of lens and source spheres instead of planes. Inthis ‘decay’withincreasingdistancetothemasses.Thisfactprovides situation,evenapoint-masslenscannotberegardedasathinlens averycleanandfirmbackgroundforacorrectionofthestandard (cid:13)c 0000RAS,MNRAS000,000–000