Table Of ContentMon.Not.R.Astron.Soc.000,000–000(0000) Printed2February2008 (MNLATEXstylefilev2.2)
On gravitational lensing by deflectors in motion
M. Sereno1,2,3⋆
1DipartimentodiScienzeFisiche,Universita`degliStudidiNapoli‘FedericoII’,ViaCinthia,MonteS.Angelo,80126Napoli,Italia
2IstitutoNazionalediAstrofisica-OsservatorioAstronomicodiCapodimonte,SalitaMoiariello,16,80131Napoli,Italia
3IstitutoNazionalediFisicaNucleare,Sez.Napoli,ViaCinthia,MonteS.Angelo,80126Napoli,Italia
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January10,2005
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a ABSTRACT
J
Gravitationallensingbyaspinningdeflectorintranslationalmotionrelativetotheobserveris
7 discussedintheweakfield,slowmotionapproximation.Theeffectofrotation,whichgener-
2 atesanintrinsicgravito-magneticfield,separatesfromthatduetoradialmotion.Corrections
tothelensequation,deflectionangleandtimedelayarederived.
1
v Keywords: gravitation–gravitationallensing–relativity
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0 1 INTRODUCTION tion. Whereas theeffect due toa translational motion isaconse-
5 quence of the existence of the standard gravito-electric field plus
0 Gravitationallensingisawellrecognizedpracticaltoolinmodern localinvariance,theproblemoflightdeflectionbyadeflectorwith
/ astrophysics.Itisisoneofthemostdeeplyinvestigatedphenom-
h angular momentum is related to the existence of the dragging of
ena of gravitation and progressive development of technological
p inertial frames and of the related gravito-magnetic field. In order
capabilitiesdemandstoleadafullanalysisonthebasisofhigher-
- toshowthatintrinsicgravito-magnetismandthelocalLorentzin-
o ordereffects(Sereno2003b).Correctionsduetothemotionofthe
variance on a static background are two fundamentally different
r deflectordeserve particularattentiononboththeoretical andphe-
t phenomena, space-time curvature by mass-energy currents rela-
s nomenological sides. The motion of the lens might play a role
tivetoother masscanbepreciselycharacterizedbyaframe-and
a in various astrophysical systems (Frittelli 2003b; Sereno 2003a;
: coordinate-independentmethod,basedonspace-timecurvaturein-
v Sereno&Cardone2002;Sereno2004).Themeasurementofanef-
variants(seeCiufolini&Wheeler1995,Section6.11).
i fectontimedelayofluminoussignals,duetoatranslationalmo-
X In this letter, we address the problem of light deflection by
tionofthedeflector,wasrecentlyclaimedinFomalont&Kopeikin
anextended lenswithangular momentum intranslationalmotion
r (2003) but a debate about correct theory and analysis behind the
a on a static background. To this aim, we adopt the usual frame-
experimentisstillunderway(Samuel2004;Kopeikin2005).Be-
work of gravitational lensing theory, i.e. i) weak field and slow
sidestranslationalmotiononastaticbackground,mass-energycur-
motion approximation for the lens and ii) thin lens hypothesis
rentsrelativetoother massesgeneratespace-timecurvature. This
(Schneideretal. 1992; Pettersetal. 2001). The paper is as fol-
phenomenon, knownasintrinsicgravito-magnetism,isanewfea-
lows. In Sec. 2, gravitational lensing in a stationary space-time
tureof general theory of relativityand other conceivable theories
isreviewed. In Sec. 3, we discuss the combined effect of a roto-
of gravity and is still waiting for an experimental confirmation
translationalmotionofthedeflectorontimedelay.Section4isde-
(Ciufolini&Wheeler1995).
votedtothediscussionofthebendingangleandtothewritingof
Various authors have investigated the problem of light de-
thelensequation.Section5containssomefinalconsiderations.
flection by lenses in motion. The effect of a translational mo-
tion of the deflector on a static background has been widely
discussed (Pyne&Birkinshaw 1993; Kopeikin&Scha¨fer 1999;
Frittelli 2003b,a; Heyrovsky 2004; Wucknitz&Sperhake 2004) 2 STATIONARYSPINNINGLENSES
and an early disagreement on the correction factor of first or-
Weareinterestedinthegravitational fieldbyaweakdeflector in
der in velocity has been solved (see Wucknitz&Sperhake 2004,
slowmotion. Mattervelocitiesaremuch lessthanc,thespeedof
and references therein). Gravitational lensing by spinning de-
lightinthevacuum,andmatterstressesarealsosmall.Themetricis
flectors has been also addressed with very different approaches
asymptoticallyflatanddeviatesonlyslightlyfromtheMinkowski
(Epstein&Shapiro 1980; Iba´n˜ez&Martin 1982; Iba´n˜ez 1983;
one. Up to leading order in c−3, such a metric can be written in
Dymnikova 1986; Glicenstein1999; Ciufolinietal. 2003; Sereno
Cartesiancoordinatesxα≡{t,x}1as
2004,2002,2003b,a;Sereno&Cardone2002).Thetwophenom-
enaarereallydifferentsincetheintrinsicangularmomentumofa φ V·dx φ
ds2≃ 1+2 c2dt2−8cdt − 1−2 dx·dx, (1)
bodycannot begenerated oreliminatedbyaLorentztransforma- c2 c3 c2
(cid:16) (cid:17) (cid:16) (cid:17)
⋆ E-mail:[email protected] 1 Greekindecesrunfrom0to3whereasLatinonesfrom1to3.
2 M. Sereno
where adot represents the Euclideanscalar product between two whereξ0 isalengthscaleinthelensplaneandΣistheprojected
three-dimensionalvectors.Letusconsiderastationaryspace-time. surfacemassdensityofthedeflector,
Here,φreducestotheNewtonianpotential,
φs(x)≃−GZℜ3 |xρ(−x′x)′|d3x′, (2) Σth(eξg)r≡aviZtolo-msρa(gξn,elt)icdlc;orrection to the potential time delay, u(p11to)
whereρisthemassdensityandGisthegravitationalconstant,and theorderv/c,canbeexpressedas(Sereno2002)
V is a vector potential taking into account the gravito-magnetic ′
fieldproducedbymasscurrents, ∆TsGRM≃ 8cG4 Zℜ2d2ξ′Σ(ξ′)hv·einilos(ξ′)ln|ξ−ξ0ξ |, (12)
Vs(x)≃−GZℜ3 (|ρxv−)(xx′′|)d3x′ (3) wehe,roefhthve·ecionmilposonisenttheofwtheeigvhetleodciatvyevraagleo,nagloeng,the line of sight
in in
wherevisthevelocityfieldofthemasselementsofthedeflector.
Intermsofanangularvelocityω,v=ω×x. hv·einilos(ξ)≡ R(v(ξ,l)·Σe(inξ))ρ(ξ,l)dl. (13)
Inaconformallystationaryspace-time,actuallightrayscanbe
determinedthroughtheFermat’sprinciple(Schneideretal.1992), Inthethinlensapproximation,theonlycomponentsoftheveloci-
whichstatesthatlighttraversesapathwhoseopticallengthissta- tiesparalleltothelineofsightentertheequationsofgravitational
tionarycomparedwithneighbouringpaths(Rossi1957).Acurved lensing.Weremindthatthetimedelayfunctionisnotanobserv-
space-timeembeddedwithastationarymetriccanbeinterpretedas able,butthetimedelaybetweentwoactualrayscanbemeasured.
aflatonewithaneffectiveindexofrefraction,ns(Schneideretal. In the above expressions for the time delays, we have neglected
1992;Sereno2003b).Alightsignalemittedatthesourcewillarrive somenotrelevantadditiveconstants.
after
1
∆T = nsdl, (4)
cZ 3 TIMEDELAYBYSHIFTINGANDSPINNINGLENSES
γ
whereγ isthe spatial projection of thelight curve. TheFermat’s Wewantnowtoconsideranadditionaltranslationalmotionofthe
principlecanbeexpressedas deflector.Thecorrespondingmetriccanbeobtainedbyapplyinga
coordinatetransformationtothemetricofthesamedeflectorwith
δ nsdl=0. (5) thecenter ofmassat rest,whichisexpressedbyEq.(1)withthe
Z potentialsinEqs.(2,3).Welimittoamotionwithslowvelocity,
For a stationary, slowly rotating, weak lens, the effective u ∼ O(v),andnegligibleacceleration.Hereafter,inthissection,
refraction index, ns, is given by (Schneideretal. 1992; Sereno theprimedcoordinateswillrefertotherestframe.Arigidmotion
2003b) along a path γ can be accounted for by a change of coordinates
x′ → x = x′ +γ(t′) (Frittelli 2003a). Limiting to negligible
2 4
ns ≃1− c2φs+ c3Vs·e, (6) athcacteldetra=tiondto′f+theγ˙·pdaxth′,/cth2e(cFhraitnteglelii2n0t0h3eat)i.mTehceomoredtirnicatteakisessutchhe
where e ≡ dx/dl is the unit tangent vector of a ray and dl ≡
sameforminEq.(1)with
δijdxidxj is the Euclidean arc length. The total travel time is
p(Sereno2002,2003b) φ(x,t) = φs(x−γ(t)) (14)
u
∆Ts ≃∆Tsgeo+∆TsSh+∆TsGRM (7) V(x,t) = Vs(x−γ(t))+ cφs(x−γ(t)), (15)
whereTgeoisthegeometricaltimedelay,∆TShistheShapirotime whereu≡γ˙.
s s
delay,i.e.thegravitationaldelayatthepost-Newtonianorder, ThemetricinEq.(1)withthepotentialsgiveninEq.(14,15)
isnomorestationaryandFermat’sprinciplecannot beappliedas
2
∆TsSh≡−c3 Zγφsdl (8) dpountaetioinnSoefct.h(e2g).raHvoitwateivoenra,lulenndseirngsuqituaabnletitaiesssucmanptpioroncs,eethdenceoamrly-
inthesameway.Infact,inusualastrophysicalsystems,thetransit
and∆TGRMisthegravito-magnetictimedelay
s timethroughthedeflectorcanbeconsideredsmallwithrespectto
4 thetotaltraveltime(Schneideretal.1992).Wecanassumethatin-
∆TsGRM≡ c4 Z Vs·edl. (9) teractionhappensinstantaneouslyatthemomentwhenthephoton
γ
passesthelens,t .Thepositionofthecenterofmassofthelens
d
Asusual,thelensisassumedtobethinandweak.Theactualpath att = 0locatestheoriginofthecoordinate system.Duringthe
d
of the photon deviates negligibly from the undeflected path. It is transittime,thevelocitiesareapproximatelyconstant.So,wecan
usefultoemploythespatialorthogonalcoordinates(ξ1,ξ2,l),cen- considerthecomponents ofthemetrictensorfixedatt = td and
tredonthelensandsuchthatthel-axisisalongtheincominglight proceedasinthestationarycase.Thecorrespondingadhocrefrac-
raydirectionein andthevectorξ spansthelensplane.So,inthe tionindexiswrittensolvingfordttheequationds2=0,andprop-
integrandfunctions,wecanusex = lein+ξin.Theunperturbed erlyconsideringthedifferencebetweentheproperarc-lengthina
photonimpactsthelensplaneinξin.Themaincontributiontothe curved space-time and the Euclidean arc length (Sereno 2003b).
potentialtimedelayis(Schneideretal.1992) Weobtain
∆TSh≃−4G d2ξ′Σ(ξ′)ln|ξ−ξ′|, (10) n ≃ 1− 1−2ulos(td) 2φs(x−γ(td))
s c3 Zℜ2 ξ0 (cid:18) c (cid:19) c2
Onthemotionofa gravitationallens 3
+ 4V (x−γ(t )), (16) 4 LENSEQUATION
c3 s,los d
Just as for a stationary space-time, where the Fermat’s principle
where again the subscript los denotes the component of a vector
exactlyholds,evenifthelensisinslowmotion,thebendingangle
along the line of sight. The index of refraction in Eq. (16) can
α andthegravitationaltimedelay,∆Tpot = ∆T −∆Tgeo,can
be also obtained from the index in the stationary case, through a
berelatedbyagradient(Frittelli2003a),
Lorentztransformation.AsfirstdemonstratedbyFizeauintheclas-
sicexperimentoflighttraversingamovingfluid,itis
α=−∇⊥(c∆Tpot), (24)
1 1 ucosϑ 1
≃ − 1− , (17)
n ns c (cid:16) n2s(cid:17) where∇⊥ ≡∇−ein(ein·∇).Thebendingangleturnsouttobe
where ϑ isthe angle between the direction of propagation of the
′
lightTanhdetchoemvpeulotactiitoynoofftthheemtiemdeiu-dme,laiy.ec.auncboespϑer=fourmloesd. bydefin- α(ξ) ≃ 4cG2 Zℜ2d2ξ′(cid:26)|ξξ−−ξξ′|2Σ(ξ′−u⊥td) (25)
inganewintegrationvariable,q(Frittelli2003b), × 1− ulos + 2hv·einilos(ξ′−u⊥td) .
q≡l−ein·γ(td); (18) (cid:16) c c (cid:17)o
intermsofthisvariable Onceagain,threetermscontributetoα.Themaintermisdueto
the static, gravito-electric field of the corresponding lens at rest.
x−γ(td)=qein+ξin−u⊥td (19) Thesecondtermderivesfromthespinandtakethesameformas
whereu⊥istheprojectedvelocityofthedeflectorinthelensplane, forastationarylenswithangular momentum. Thethirdeffect on
i.e.transversetothelineofsight,and thebendingangle,duetoatranslationalmotionofthedeflector,can
beinterpretedintermsofstandardaberrationoflightinanoptically
dq=(1−u /c)dl. (20)
los activemediumwithaneffectiveindexofrefractioninducedbythe
Thetimedelaybyamoving,spinningdeflectorcanbeexpressedin gravitationalfieldofthelens(Frittelli2003a).Thebendingangle
termsofthesamestationarydeflector.Bychangingtheintegration byapoint-likemovinglenscarriesapre-factor(1−ulos/c)with
variable,wefind respecttothebendingangleinthestaticcase(Frittelli2003a).For
a stationary spinning deflector, the path of the light-ray does not
∆T ≃∆Tgeo− 1− ulos 2φsdq+ 4Vsdq. (21) stayonaplane(Sereno2003a),asitisforasphericallysymmetric
(cid:16) c (cid:17)Zγ c3 Zγ c4 lens.However,sinceinourapproximations,deviationsinducedby
Solvingtheintegralsunderthehypothesisofathinlens,weget theangularmomentumofthelensaresmall,thesamearguments
basedonaberrationcanbeappliedasinFrittelli(2003a),providing
u
∆T ≃∆Tsgeo + 1− clos ∆TsSh(ξin−u⊥td) anindependentconfirmationtoEq.(25)asfarasaradialmotionis
(cid:16) (cid:17) concerned.
+ ∆TsGRM(ξin−u⊥td) (22) Letusfinallywritethelensequation.Weintroducetheangu-
The translational motion of the lens enters in two distinct ways. lar position of the source in the plane of the sky, β, and the an-
TheradialmotionaffectstheShapirotimedelaythroughanoverall gular position of the image in the lens plane θ = ξ/Dd, where
multiplicative factor, so that the correction to the time delay due Dd isthe angular diameter distance from the observer to the de-
tothetransversalmotionis∆Ttra =−(u /c)∆TSh.Thetrans- flector. For a system of point-like lenses, velocity effects do not
los s
verse motion affectsonly the relativeangular separation between alter the form of the lens equation with respect to the static case
thelensandthesource,i.etheimpactparameter,whichisnowtime (Kopeikin&Scha¨fer1999;Heyrovsky2004).Forthegeneralcase
dependent. of a spinning and shifting deflector, the lens equation takes the
Theweakfieldhypothesisandslowmotionapproximationare form,
enough to face almost all astrophysical systems (Schneideretal.
D
1992).Infact,peculiarvelocitiesofhighredshiftgalaxiesareonly β=θ− dsα(θ), (26)
atinyperturbationsontheHubbleflow,oforderof∼ 10−3c.Let Ds
usconsiderabackgroundquasarwhichislensedbyaforeground
whereandDsandDdsaretheangulardiameterdistancesfromthe
galaxy. We can model the lens as a singular isothermal sphere
observertothesourceandfromthedeflectortothesource,respec-
(Sereno2004).AnapproximaterelationholdsbetweentheShapiro
tively.Withrespecttothestaticcase,thepositionofthelensistime
timedelayandthegravito-magnetictimedelay(Sereno2004),
dependentandthebendingangleiscorrectedforfactorsduetothe
∆TGRM≃−9sinγ σv λ∆TSh, (23) radialmotionandtheangularmomentumofthedeflector.
(cid:16) c (cid:17) s Inthelimitofslowvelocities,translationalandrotationalmo-
whereγ istheanglebetweentheprojectionoftherotationaxisin tionsareseparatedandaddtogethertocorrectlensingquantities.In
the plane of the sky and the line trough the two nearly collinear general,thisisnottrue.Tofirstorderindeflection,thedeflection
images,λisthespinparameter,givinganestimateofthetotalan- anglefor alenswitharelativistictranslational motion canbedi-
gular momentumof thelens,andσv isthevelocitydispersionof rectlyobtainedfromthestationarydeflectionanglecausedbyalens
thegalaxy.Correctionsduetorotationandtranslationalmotionare atrestbyapplyingaLorentztransformation(Wucknitz&Sperhake
nearly of the same order for usual systems. For a typical lensing 2004).Thestationarydeflectionanglecanbewrittenasasumofthe
galaxy at zd = 0.5 withσv ∼ ulos ∼ 10−3, and a background gravito-electricandthegravito-magneticterms,αGRE +αGRM.
source at zs = 2.0, the time delay is ∼ 200 days for a source Forarelativistictranslationalvelocity,
nearlyinthemiddleoftheEinsteinradius,whereasthecorrections
are ∆Ttra ∼ 0.2 days and ∆TGRM ∼ 0.1-0.2×sinγ days for 1−u /c
α(u)= los αGRE+αGRM (27)
λ∼0.05-0.1. r1+u /c
los
(cid:0) (cid:1)
4 M. Sereno
Thescalingfactor inEq.(27),whichreduces to1−u /cfora Schneider P.,EhlersJ., FalcoE. E.,1992, Gravitational Lenses.
los
slowmotion,appliesinthesamewaytothemaintermandtothe Springer-Verlag,Berlin
gravito-magneticcorrection. SerenoM.,2002,Phys.Lett.A,305,7
SerenoM.,2003a,MNRAS,344,942
SerenoM.,2003b,Phys.Rev.D,67,064007
5 CONCLUDINGREMARKS SerenoM.,2004,MNRAS,inpress;astro-ph/0412108
SerenoM.,CardoneV.F.,2002,A&A,396,393
Wehaveconsideredtheeffectsongravitationallensingofboththe WucknitzO.,SperhakeU.,2004,Phys.RevD,69,063001
translationalmotiononastaticbackgroundandofanintrinsican-
gular momentum of the deflector. In the weak-field, slow motion
approximationtheeffectsonbendingandtimedelayoflightrays
areseparatedandaddstogether.Onlytheradialmotionofthecen-
tre of mass enters the corrections, whereas the transverse motion
hastobeconsideredonlywhendeterminingthelenspositionand
therelativeseparationbetweenlensandsource.Theradialandthe
angularvelocityenterthecorrectionswithadifferentscaling.The
internalrotationalvelocitymustbeweightedalongthelineofsight
andcarriesanextrafactor2withrespecttotheradialvelocity,that
mustbeevaluatedatthetimethephotonpassesthelens.Thisshows
thatresultsinSereno(2002);Sereno&Cardone(2002),whichre-
fertospinninglenses,arecorrect,differentlyfromwhatclaimedin
Wucknitz&Sperhake (2004). However, results in Sereno (2002)
cannotbeappliedtoaradialmotion.
Thanks to usual approximations, our analysis does not limit
to point-like lenses and extended deflector have been consid-
ered. The internal velocity fieldenters the equations. Thisallows
to solve the question whether the gravito-magnetic correction to
the deflection of light can probe the inner structure of a lens
(Wucknitz&Sperhake2004).Evenifthemaincontributionfora
light ray propagating outside the lens is due to the total angular
momentum,lightrayscrossingtheinnerstructureofthedeflector
probeonlya“partial”angularmomentum.
ACKNOWLEDGMENTS
Iwishtothankananonymousrefereeforthedetailedsuggestions.
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