Table Of ContentGravitational Lensing and Gravitomagnetic Time Delay
Ignazio Ciufolini
Dip. Ingegneria dell’Innovazione Universit`a di Lecce
Via Monteroni, 73100 Lecce, Italy
Franco Ricci
Dip. Fisica Universit`a di Roma “La Sapienza”
Piaz.le Aldo Moro 5, 00185 Roma, Italy
(Dated: February 7, 2008)
We derive the delay in travel time of photons due to the spin of a body both inside a rotating
shellandoutsidearotatingbody. Wethenshowthatthistimedelaybythespinofanastrophysical
objectmightbedetectedindifferentimagesofthesamesourcebygravitationallensing;itmightbe
relevantinthedeterminationoftheHubbleconstantusingaccuratemeasurementsofthetimedelay
between the images of some gravitational lens systems. The measurement of the spin-time-delay
3 mightalsoprovideafurtherobservabletoestimatethedarkmattercontentingalaxies, clusters,or
0
super-clusters of galaxies.
0
2
PACSnumbers: 04.,95.30.Sf,04.80.Cc,
n
a
J OneofthebasicconfirmationsofEinsteingeneralrela- its starting point, it finds itself advanced relative to a
8 tivityisthemeasurementofthedeflectionofpathofelec- clock kept there at ”rest”(in respectto ”distantstars”),
tromagnetic waves propagating near a mass due to the and a counter-rotating clock finds itself retarded rela-
1 spacetime curvature generated by a central body [1, 2]. tive to the clock at rest [2, 11]. Indeed, synchronization
v
To-day this effect is reported to agree with its general of clocks all around a closed path near a spinning body
0
relativisticvaluewithaccuracyofabout1.4partsin104, is not possible, and light co-rotating around a spinning
3
0 by using VLBI, Very Long Baseline radio Interferome- bodywouldtakelesstimetoreturntoafixedpointthan
1 try to observethe gravitationaldeflection of radio waves lightrotatingintheoppositedirection. Similarly,theor-
0 from quasars and radio galaxies produced by the Sun bital period of a particle co-rotating around a spinning
3
[3]. The VLBI measurement constrains the parameter body would be longer than the orbital period of a par-
0
ω of scalar-tensor theories to be larger than about 3500 ticle counter-rotating on the same orbit. Furthermore,
/
c
[4]. Depending from the relative position, distance and anorbiting particle arounda spinning body will have its
q
alignmentofsource,observeranddeflectingmass,several orbitalplane”dragged”aroundthe spinning body inthe
-
r images of the same source may be observed. This phe- same sense as the rotation of the body, and small gyro-
g
nomenonisthewellknowngravitationallensing. Several scopes that determine the axes of a local, freely falling,
:
v
examplesofgravitationallensinghavebeendiscovered;a inertial frame, where ”locally” the gravitational field is
i
X wellknowncaseistheGravitationalLensG2237+0305, ”unobservable,” will rotate in respect to ”distant stars”
r orEinsteinCross,wherethe lightbending produces four because of the rotation of the body. This phenomenon–
a imagesof the samequasaras observedfromEarth[5, 6]. called”draggingofinertialframes”or,”framedragging,”
asEinstein namedit–is alsoknownas Lense-Thirringef-
The other well known general relativistic effect due
fect. The Lense-Thirring effect has been observed in the
to the spacetime curvature generated by a central body
orbits of the LAGEOS satellites [12]. In general relativ-
is the delay in the travel time of electromagnetic waves
ity, all these phenomena are the result of the rotation of
propagatingnearthecentralmass,orShapirotimedelay
the central mass.
[7,8,9];thiseffectisto-daytestedwithaccuracyofabout
10 3 by observing the delay of radio waves propagating
− However, Einstein’s gravitational theory predicts pe-
neartheSunwithactivereflectionusingtransponderson
culiar phenomena also inside a rotating shell. In a well
the Viking spacecraft orbiting Mars or on its surface.
known paper of 1918, Thirring published a solution of
All these phenomena are due to a static mass and can Einstein’s field equation representing the metric inside a
be derived using the static Schwarzschild metric. If an rotatingshelltofirstorderin M,massoverradiusofthe
R
axially symmetric body has a steady rotation around shell,andtofirstorderinω,angularvelocityofthe shell
its symmetry axis an external solution is the stationary [13]. In 1966BrillandCohenderivedthe metric inside a
Kerr metric. In the weak-field and slow-motion limit, shell with arbitrary mass, this solution is a lowest order
the Kerr metric is, in Boyer-Lindquist coordinates, the series expansion in the angular velocity of the spherical
Lense-Thirring metric [2, 10]. Einstein’s general theory shell on the Scharzschild background of any mass, valid
of relativity [2] predicts that when a clock co-rotates ar- both inside and outside the shell [14]. An extension of
bitrarily slow around a spinning body and returns to the Brill-Cohen results to higher orders on ω was then
2
(φ,β,γ). Ifthedeflectingbodyissymmetricwithrespect
to Z, the shape of the body is invariant for rotations of
φ. We can thus choose φ = 0. Since we analyze the be-
haviourof photons, moving with speed c 1, we neglect
the U2 terms in the post Newtonian met≡ric, where U is
the Newtonian potential. In the new coordinate system
(x,y,z) we then have:
ds2 = ( 1+2U)dt2+(1+2U)δ dxidxj
ij
−
4J
+ (ycosβ zcosγsinβ)dtdx
r3 −
4J
(xcosβ zsinγsinβ)dtdy
− r3 −
4J
+ (xcosγsinβ ysinγsinβ)dtdz (3)
r3 −
in U, for simplicity, we have only included mass and
quadrupole moment of the deflecting body.
FIG. 1: Geometry of light rays propagating inside a rotating
We have chosen the quasi-Cartesian coordinates so
shell of radius R′ and around a central body.
thattheemittingandthedeflectingbodieshavethesame
x and y coordinates, (see fig.1), but a different z coor-
dinate: source, lens and observer are aligned. We have
published in 1985 by Pfister and Braun [15]. However,
chosen this simple configuration since, in this paper, we
the exact solution representing the spacetime geometry
areonlyinterestedtoanalyzethe timedelayduetospin.
insideashellwitharbitrarymassandrotatingwitharbi-
However, there is an additional time delay, called geo-
trary angular velocity is still unknown. In the following
metricaltime delay [20], due to the different geometrical
we consider only the weak-field and slow-motion metric
pathfollowedbydifferentrays. Dependingonthegeome-
derived by Thirring.
tryofthesystem,this additionaltermmaybe verylarge
The gravitomagnetic potential, i.e. the non-diagonal
andmaybethemainsourceoftimedelay. However,ifwe
partofthemetricinstandardisotropicPPNcoordinates
compare the time delay of photons that follow the same
[1],h0i,atthepost-Newtonianorder,outsideastationary
rotating body with angular momentum J, is [2]: geometricalpathwecanneglectthegeometricaltimede-
lay,as inthe caseoftwolightrayswith the sameimpact
2 J X parameter but on different sides of the deflecting object.
hext(X) = − × , (1)
∼ X3 For of a small deflection angle, the contribution to the
| |
traveltime delayfromthe differentpathlengthtraveled,
where X is the position vector. If J=(0,0,J), in spher- due to the small deflection, is of the order of U2 [2] and,
ical coordinates: he0xφt ∼= −2rJsin2θ. Inside a thin shell depending on the geometrical configuration considered,
of total mass M and radius R, with stationary rotation this delay may need to be included in the total time de-
around a Z-axis with small angular velocity ω, we have lay. In a following paper we shall analyze higher order
[2]: timedelayandcompareitwiththegravitomagnetictime
delay. Here, for simplicity, we neglect any geometrical
4M 4M 4M
hint(X)= ω X= ωY, ωX,0 (2) time delay. Thus, from the condition of null arc length,
−3R × (cid:18)3R −3R (cid:19) ds2 =0,solvingwithrespecttodt,wehaveatthelowest
Let us derive time delay and deflection of electro- order in U and g0z: dt ≃ g0zdz∓(1+2U)dz. Thus, by
integratingdtfrom z1toz2corresponding,respectively,
magnetic waves due to spin and quadrupole moment of −
tothepositionofsourceandobserver,ifz1 z2 z b
the central body; we keep only post-Newtonian terms ≃ ≡ ≫
(see fig. 1) we get:
in the following derivation. Deflection by spin was
analyzed in [16], and time delay by spin is consid- z¯ 4Jcos(α+γ) sinβ
ered in [17, 18]. Strong gravitational lensing due to a ∆T = 2z¯+4Mln 2 +
b b
(cid:16) (cid:17)
Schwarzschild black hole is treated in [19]. In the fol- 2MR2J2cos2(α+γ) sinβ2
lowing, the quasi-Cartesian coordinate system (x,y,z) (4)
b2
is a system of isotropic coordinates with the z-axis di-
rected towards the observer on Earth, while (X,Y,Z) In this expression the first term is the time that a radio
is the standard PPN system of isotropic coordinates at- pulse takes to travel from the source to Earth in the ab-
tached to the deflecting body. To relate the coordinates sence of a central mass: M = 0; the second term is the
(x,y,z)and(X,Y,Z)weusethe standardEuler’sangles Shapiro time delay; the third one is the gravitomagnetic
3
time delay and the last term is the additional time de- differs by about π, such as in the Einstein Cross,we can
lay due to the quadrupole moment, J2, of the deflecting directly eliminate the time delay due to the quadrupole
body. moment and thus determine the spin-time-delay.
Let us now calculate the time delay due to the spin
Together with the positive spin-time-delay of a
of some astrophysical sources. For the sun (M =1.477
counter-rotating photon relative to a co-rotating pho-
ton, there is a negative deflection of the path of the ksimde,rRin⊙g=tw6o.9l6ig·h1t05rakyms wanitdhJi2m⊙p∼=act1.p7a·r1a0m−7et[e2r]⊙s),bby=cRon-
counter-rotating photon, and a positive deflection of the ∼
and b, and, for simplicity, with γ = 0 and β = π⊙,
co-rotatingphoton,duetothespinofthelens. Thisneg- − 2
the gravitomagnetic and quadrupole-moment time de-
ativedeflectiongivesrisetoanegativetimedelay,dueto
lays,according to (4), are: ∆TJ = 8J =1.54 10 11sec,
the decreaseinthe pathtravelled,andto apositive time 12 b · −
delay due to the increase in the standard Shapiro delay and ∆T1J22 = 4J2Mb2R2 = 3.35·10−12sec. The time delay
by the mass of the lens due to the decrease of the dis- duetotheSunspincouldthen,inprinciple,bemeasured
tance fromthe centrallens. These twoadditionalcontri- usinganinterferometeratadistanceofabout8 1010km,
·
butions, negative geometricaldelay and positive Shapiro bydetecting bygravitationallensingphotonsemittedby
delay, are equal and opposite and cancel out so that the a laser on the side of the Sun opposite the detector, and
only remaining effect is the positive spin-time-delay of travelling on opposite sides of the Sun. To derive the
the counter-rotating photon and the negative spin-time- timedelayduetothelensinggalaxyoftheEinsteincross
delay of the co-rotating photon. [5, 6] we assume a simple model for rotation and shape
of the central object. Details about this model can be
To write the deflection of electromagnetic waves due
found in [21]. The angular separation between the four
to spin and quadrupole moment of the deflecting body
imagesisabout0.9 ,correspondingtoaradiusofclosest
we use the geodesic equation in the weak field approx- ′′
1
iaδimsr=etahta−ieolin4sgMtb.naen−dWd,a4hraJdefbstn2iednreβsflos−euocrmtc4iJeeo2,nMcdabRebly3cfl2uesalicnats2tipinβohg;newsrbihcowaederlyeottbhahjneeendcfitgoroesbfttsemt[re1var1mes]sr: asa∆hpsTsepul1Jrl2mowa=eic:thh8JbJot2fh≃≃aisb4ro0mau.1dtiinuRa,nsa≃dRnJd6is5∆≃0∼hT1−711J520.242p=3·c1k,40ma1J02n2Mhdhb2−7−7tR551h22,eM≃wm⊙e8a[sth6sh]r.ei.nnLTseihdhtaueuvsaes,
at least in principle, one could measure the time delay
M, the second term is the deflection by the angular mo-
due to the spin of the lensing galaxy by removing the
mentum, J, and the third one the additional deflection
larger quadrupole-moment time delay by the previously
due to the quadrupole moment, J2, of the central body.
described method; of course, as in the case of the Sun,
Letusnowstudythepossibilityofmeasuringthespin- one should be able to accurately enough model and re-
time-delay anddetermining the angularmomentum J of move all the other delays, due to other physical effects,
thecentraldeflectingbodybymeasuringtotaltimedelay. from the observedtime delays between the images. As a
We consider a simple case in which the source is behind thirdexampleweconsidertherelativetime delayofpho-
the lens and there are three lightrays with the same im- tons due to the spin of a typical cluster of galaxies; the
pactparameterb,propagatingalongoneaxis. Weassume precise calculations are shown in a following paper. We
to be able to measure, or determine, the following quan- consideraclusterofgalaxiesofmassM =1014M ,ra-
C ∼
tities: totaltime delaybetweenthethree rays,∆T12 and dius R = 5 Mpc and angular velocity ω = 10 ⊙18s 1;
C ∼ C ∼ − −
∆T13; deflection angles δ1, δ2 and δ3; equatorial radius dependingonthegeometryofthesystemandonthepath
Rofthedeflectingbodyanddistancesofsourceandlens followedbythephotons,wethenfindrelativetimedelays
from the observer. In this way we are able to determine ranging from a few minutes to several days [11].
the angle α for each light beam and the impact param- Let us now analyze the time delay in the travel time
eter b (see fig. 1), and we can write a system in which of photons propagating inside a rotating shell with ω =
theonlyunknownquantitiesare: angularmomentum,J, (0,0,ω). Inside the shellit is notpossible to synchronize
quadrupole moment, J2, mass, M, and Euler’s angle β clocks all around a closed path. Indeed, if we consider a
and γ. Solving this system we can, in principle, deter- clock co-rotating very slowly along a circular path with
mine the time delay due to the angular moment J and radius r, when back to its starting point it is advanced
the other unknown quantities. Detailed calculations are with respect to a clock kept there at rest (in respect to
givenin[11]. Ofcourse,forotherconfigurationsinwhich distant stars). The difference between the time read by
the source is not exactly aligned with lens and observer, the co-rotating clock and the clock at rest is equal to:
thedifferenceinpathtraveledandthecorrespondingdif-
ference in Shapiro time delay can be the main source of δT = g0i dxi = 8Mπωr2 (5)
relative time delay; one would then need to model and −I √−g00 3R
remove these delays between the different images on the Forashellwithfinitethicknesswejustintegrate(5)from
basis of the observed geometry of the system. Neverthe- the smaller radius to the larger one.
less, in special cases, for example if we observe four im- Let us now consider a co-rotating photon traveling
agesofthesourceandiftheangleαofeachdeflectedray with an impact parameter r on the equatorialplane of a
4
galaxy(seefig. 1). Thetime delaydueto therotationof ringcanprovidestrongconstraintsonthe massdistribu-
the external mass for every infinitesimal shell with mass tion in the lens; this, in turn, can be used in order to
dm=4πρR2dR and radius R r , is [11]: separate the time delays due to a mass distribution non-
′ ′ ′
≥| |
symmetrical with respect to the images. The accurate
∆tdm =Z √√RR′′22−rr22(h0x)dx= 8d3mωr√R′R2′−r2 (6) mgreaavsiutarteimoneanltleonfstshyestdeemlasyisbuestewdeeansathmeeitmhoadgetsoopfrosvomidee
− −
By integrating the second term in this expression from estimates of the Hubble constant, the time delay due to
r to the external shell radius R, we have: spin might then be relevant in the correspondingmodel-
| |
ingofthe delayinthe images. Themeasurementoftime
32π R
∆T = ωr ρR R2 r2dR (7) delay is possible, in the case of B0218+357,because the
3 Zr ′p ′ − ′ source is a strongly variable radio object, thus one can
| |
This is the time delay due to the spin of the whole ro- determine the delay in the variations of the images. In
tating mass of the external shell. From this formula we thissystemitispossibletoobserveclearvariationsinto-
can easily calculate the relative time delay between two talflux density,percentagepolarizationandpolarization
photons traveling on the equatorial plane of a rotating position angle at two frequencies. For B0218+357 the
shell, with impact parameters r1 and r2. measured delay is 10.5 ± 0.4 day [23]. Therefore. since
Finally, let us calculate the time delay corresponding the presentmeasurementuncertainty in the lensing time
to some astrophysical configurations. In the case of the delay is of the order of 0.5 day [24], the gravitomagnetic
”Einstein cross” [5], we assume, to get an order of mag- time delay might already be observable.
nitude of the effect, that the lensing galaxy has an ex- In conclusion, we have derived and studied the ”spin-
ternal radius R 5kpc; after some calculations based time-delay” in the travel time of photons propagating
≃
on the model given in ref. [21], the relative time delay neararotatingbody,orinsidearotatingshellduetothe
of two photons traveling at a distance of r1 650pc angular momentum We found that there may be an ap-
≃
and r2 650pc from the center, using (7) in the case preciabletimedelayduetothespinofthebody,orshell,
≃ −
r1 r2, is: ∆T 30min. If the lensing galaxy is in- thus spin-time-delay must be taken into account in the
≃ − ≃
side a rotating cluster, or super-cluster, to get an order modelingofrelativetimedelaysoftheimagesofasource
of magnitude of the time delay, due to the spin of the observedat a far point by gravitationallensing. This ef-
mass rotating around the deflecting galaxy, we use typi- fect is due to the propagationof the photons in opposite
cal super-cluster parameters: total mass M = 1015M , directionswith respectto the directionof the spin ofthe
radiusR=70Mpcangularvelocityω =2 10 18s 1[22⊙]. body, or shell. If other time delays can be accurately
− −
·
If the galaxyis in the center of the cluster and lightrays enoughmodeled andremovedfromthe observations,the
haveimpactparametersr1 15kpcandr2 15kpc(of larger relative delay due to the quadrupole moment of
≃ ≃−
the order of the Milky Way radius), the time delay, ap- thelensingbodycanberemoved,forsomeconfigurations
plying formula(7)in the caser1 r2 andρ=costant, of the images, by using special combinations of the ob-
≃−
is: ∆t 1day. servables;thus,onecoulddirectlymeasurethespin-time-
≃
If the lensing galaxy is not in the center of the clus- delayduetothegravitomagneticfieldofthelensingbody.
ter but at a distance r = aR from the center, with In order to estimate the relevance of the spin-time-delay
0 a 1 and R radius of the cluster, by integrating in some real astrophysical configurations, we have con-
≤ ≤
(7) between r = aR and R, when r1 r2 we have: sidered some possible astrophysical cases. We analyzed
∆t= 329πω(r1−r2)ρ(1−a2)1/2(1−4a2)R≃3. Thus, ifthe therelativetimedelayinthegravitationallensingimages
lensing galaxy is at a distance of 10 Mpc from the cen- causedbyatypicalrotatinggalaxy,orclusterofgalaxies.
ter of the cluster, the relative time delay due to the spin We then analyzed the relative spin-time-delay when the
of the external rotating mass between two photons with path of photons is inside a galaxy, a cluster, or super-
(r1 r2) 30kpc, is: ∆T 0.9day. cluster of galaxies rotating around the deflecting body;
− ≃ ≃
Promising candidates to observe time delay due to this effect should be large enough to be detected from
spin are systems of the type of the gravitational lens Earth. The measurement of the spin-time-delay, due to
B0218+357[23], where the separationanglebetweenthe the angular momentum of the external massive rotating
images is so small that also the delay between the im- shell, might be a further observable for the determina-
ages is very small. In B0218+357 the separation angle tionofthe totalmass-energyofthe externalbody,i.e. of
between the two images is 335 milliarcsec and the time thedarkmatterofgalaxies,clustersandsuper-clustersof
delayisabout10.5days. Insuchsystems,thetimedelay galaxies. Indeed, by measuring the spin-time-delay one
due to the spin of the external mass, or of the central candetermine the totalangularmomentum ofthe rotat-
object,mightbecomparableinsizetothetotaldelay. In ing body andthus, by estimating the contributionofthe
addition, in the system B0218+357 is observed an Ein- visible part, one can determine its dark-matter content.
stein ring the diameter of which is the same as the sepa- Theestimatespresentedinthispaperarepreliminarybe-
rationoftheimages. Insuchconfigurations,theEinstein cause we need to analyze the spin-time-delay in the case
5
of some particular, known, gravitational-lensing images; Theiss 1984 Gen. Relativ. Gravit. 16, 711.
furthermore, we need to estimate the size and the pos- [11] I. Ciufolini and F. Ricci (2001), to be published.
sibility of modeling other sources of time delay in these [12] I. Ciufolini and E. C. Pavlis et al., Science 279, 2100
(1998).
known systems. Nevertheless, depending on the geome-
[13] H. Thirring, Z. Phys. 19, 33 (1918).
try of the astrophysical system considered, the relative
[14] D.R.BrillandJ.M.Cohen,Phys.Rev.143,1011(1966),
spin-time-delay can be a quite large effect.
see also ref. 15.
[15] H. Pfister and K. Braun, Class. Quantum Grav. 3, 335
(1986).
[16] Ib´an˜ez et al, A&A 124, 175 (1983).
[17] S A Klioner, Sov. Astron. 35, 523 (1991).
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