PHYSICAL REVIEW D 69, 063004 ~2004!

Strong field limit analysis of gravitational retrolensing

Ernesto F. Eiroa*Instituto de Astronomı´a y Fısica del Espacio, C.C. 67, Suc. 28, 1428, Buenos Aires, Argentina

Diego F. Torres†

Lawrence Livermore National Laboratory, 7000 East Ave., L-413, Livermore, California 94550, USA~Received 4 November 2003; published 15 March 2004!

We present a complete treatment in the strong field limit of gravitational retrolensing by a static sphericallysymmetric compact object having a photon sphere. The results are compared with those corresponding toordinary lensing in similar strong field situations. As examples of the application of the formalism, a super-massive black hole at the Galactic center and a stellar mass black hole in the Galactic halo are studied asretrolenses, in both cases using Schwarzschild and Reissner-Nordstro¨m geometries.

DOI: 10.1103/PhysRevD.69.063004 PACS number~s!: 95.30.Sf, 04.70.Bw, 98.62.Sb

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I. INTRODUCTION

Gravitational lensing is an important tool in astrophysiObservable phenomena of lensing by stars and galaxiesbe explained in the weak field approximation, that is to skeeping only the first order term in the expansion of tdeflection angle@1#. But when the lens is a black hole, a fugeneral relativistic treatment is in order.

Recently, several articles studying strong field lenshave appeared in the literature. Virbhadra and Ellis@2# nu-merically analyzed the situation where the lens isSchwarzschild black hole placed in the center of the GalaThey obtained the lens equation with an asymptoticallymetric and found that, in addition to the primary and secoary images, two infinite sets of faint relativistic images aformed by photons that make complete turns~in both direc-tions of rotation! around the black hole before reaching tobserver. Fritelli, Kling, and Newman@3# found an exactlens equation without any reference to a background meand compared their results to those of Virbhadra and EBozzaet al. @4# developed an approximate analytical meth~generally called, as well, the strong field limit! by whichthey obtained the positions and magnifications of the relaistic images for the Schwarzschild black hole. SubsequenEiroa, Romero, and Torres@5# applied the strong field limitto the Reissner-Nordstro¨m black hole and discussed the posibility of detection of relativistic images in the next feyears. Bozza@6# extended the strong field limit to any statspherically symmetric lens and analyzed the case ocharged Brans-Dicke black hole. Bhadra@7# studied theGibbons-Maeda-Grafinkle-Horowitz-Strominger chargblack hole of string theory in the strong field limit. Virbhadand Ellis numerically investigated the lensing by naked sgularities @8#. Bozza and Mancini@9# used the strong fieldlimit to study the time delay between different relativistimages, showing that different types of black holes are chacterized by different time delays. Spinning black holesmore difficult to tackle. Bozza@10# analyzed the case oquasiequatorial lensing by rotating black holes and Va´zquez

*Email address: eiroa@iafe.uba.ar†Email address: dtorres@igpp.ucllnl.org

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and Esteban@11# studied the Kerr black hole where the oserver and source have arbitrary inclinations. Many of threcent advances~if not all! deal with theoretical implicationsof general relativity in a regime that is well beyond our curent technological capabilities for detection. Henceforth,emphasis of these studies is not as much in predictingphenomena that could be tested in a short period of timeis in gaining insight on how gravitational lensing in particlar, and gravity in general, behave in the strong field regim

In ordinary lensing situations, a lens is placed betweensource and the observer. But if the lens is a compact obwith a photon sphere and the observer is placed betweensource and the lens, it leads to the formation of images wdeflection angles closer to odd multiples ofp. The observersees the images in front and the source behind. This situais called retrolensing and was studied for the first timeHoltz and Wheeler@12#, who analyzed only the two strongeimages for a black hole placed in the Galactic bulge withsun as source. They also proposed retrolensing as amechanism for searching black holes. A similar idea wsuggested in the context of defocusing gravitational lensby Capozzielloet al. @13#. De Paoliset al. @14# recently ana-lyzed the retrolensing scenario for a bright star close tomassive black hole at the Galactic center.

In this paper we give a complete treatment of retrolensby a static, spherically symmetric lens, using the strong filimit. In Sec. II we obtain the lens equation for retrolensiand the deflection angle in the strong field limit. In Sec. Ithe position and magnification of the images are calculaIn Sec. IV, the results are compared with those for ordinlensing and astrophysical examples are given in Sec. V. Fconcluding remarks are given in Sec. VI. Throughout tpaper we use units such thatG5c51.

II. LENS EQUATION AND DEFLECTION ANGLE

Consider an observer~o! placed between a point sourcelight ~s! and a strong field object with a photon sphere~e.g.,a black hole!, which we will call the lens (l ). The line join-ing the observer and the lens define the optical axis. Tbackground space-time is considered asymptotically flwith the observer and the source immersed in the flat regWe callb the angular position of the source andu the angu-

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E. F. EIROA AND D. F. TORRES PHYSICAL REVIEW D69, 063004 ~2004!

lar position of the images~i! as seen by the observer. Thlens situation is shown in Fig. 1. We can takeb.0 withoutloosing generality. The lens equation for retrolensingslightly different from ordinary lensing~i.e., when the lenslies between the source and the observer!, due to the differ-ent relative positions of the lens, source, and observer:

tanb5tanu2dos

dls@ tan~a2u!1tanu#, ~1!

wheredos5Dos/2M and dls5Dls/2M are, respectively, theobserver-source and the lens-source distances in units oSchwarzschild radius, anda is the deflection angle. In lensing situations where the objects are highly aligned,anglesb andu are small anda is closer to an odd multipleof p. For the images at the opposite side of the source~seeFig. 1!, it can be written asa5(2n21)p1Dan , with nPN and 0,Dan!1. Then, the lens equation takes the fo

b5u2dos

dlsDan . ~2!

As in the case of ordinary strong field lensing, two infinsets of relativistic images are formed. To obtain the otherof images~at the same side of the source! we should take

FIG. 1. Lens diagram. The observer (o), the lens (l ), the source(s), and the projection on the lens plane of the first relativisimage~i! are shown. The angular position of the source isb, of theimage isu, anddol , dos , dls are, respectively, the observer-lenthe observer-source, and the lens-source distances. Two infiniteof relativistic images~with both directions of rotation around thlens! are obtained with deflection angles close to odd multipof p.

06300

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a52(2n21)p2Dan , so Dan must be replaced by2Dan in Eq. ~2!. In the case of perfect alignment, an infiniseries of concentric Einstein rings are obtained.

To calculate the deflection angle, consider a static sphcally symmetric metric

ds252 f ~x!dt21g~x!dx21h~x!dV2, ~3!

where x5r /2M is the radial coordinate in units of thSchwarzschild radius. We require this metric to have a pton sphere, the radiusxps of which is given by the greatepositive solution of the equation

h8~x!

h~x!5

f 8~x!

f ~x!, ~4!

where the prime means derivative with respect tox.The deflection angle corresponding to the images situa

at the opposite side of the source as a function of the clodistance of approachx0 is given by@15#

a~x0!5I ~x0!2p, ~5!

with

I ~x0!5Ex0

`

2S g~x!

h~x! D1/2S h~x! f ~x0!

h~x0! f ~x!21D 21/2

dx. ~6!

The impact parameterb ~in units of the Schwarzschildradius! is @15#

b~x0!5S h~x0!

f ~x0! D1/2

. ~7!

From the lens geometry we also have

b~x0!5dolsinu'dolu, ~8!

with dol5Dol/2M the observer-lens distance in units of thSchwarzschild radius.

Bozza@6# has shown that for a spherically symmetric lewith a photon sphere, light rays with impact parametebclose to the photon sphere have a deflection angle that caapproximated by

a52a1ln~b2bps!1a21O~b2bps!, ~9!

where bps5b(xps), and a1,2 are constants which depenonly on the type of lens. Provided the metric of the compobject that acts as a lens, the coefficientsa1,2 can be obtainedin a laborious but straightforward way using the formalisdeveloped in Ref.@6#. The method consists essentiallyseparating the integral of Eq.~6! in a divergent part thatgives the logarithmic term in Eq.~9!, and a regular part thagives the constanta2. This approximation, named the stronfield limit, gives very accurate results. For more details sRef. @6#.

Another way of calculating the constantsa1,2 is followinga similar approach to that used by Eiroa, Romero, and To@5#:

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STRONG FIELD LIMIT ANALYSIS OF . . . PHYSICAL REVIEW D 69, 063004 ~2004!

a15 limx0→xps

2@b~x0!2b~xps!#I 8~x0!

b8~x0!~10!

and

a25 ln~ limx0→xps

$eI (x0)2p@b~x0!2b~xps!#a1%!, ~11!

with the prime now meaning the derivative with respectx0. These limits can be evaluated numerically with standsoftware. The values of the coefficientsa1 and a2 forSchwarzschild and Reissner-Nordstro¨m lens geometries argiven in the Appendix.

III. POSITION AND MAGNIFICATION OF THE IMAGES

Inverting Eq.~9! to obtain the impact parameter

b5bps1expS a22a

a1D , ~12!

and using ofb5dolu, we find that the position of any imagis

u5bps

dol1

1

dolexpS a22a

a1D . ~13!

Making a Taylor expansion to first order arounda5(2n21)p, we can approximate the angular position of thenthimage by

un5un02

jn

dolDan , ~14!

with

un05

bps

dol1

1

dolexpFa22~2n21!p

a1G ~15!

and

jn51

a1expFa22~2n21!p

a1G . ~16!

Then from Eq.~14! we obtain

Dan52un2un

0

jndol , ~17!

and replacing it in the lens equation,

b5un1dosdol

dlsjn~un2un

0!. ~18!

The last equation can be written in the form

b2un05S 11

dosdol

dlsjnD ~un2un

0!, ~19!

and as the second term in large parentheses is much grthan one, we have

06300

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ter

b2un0'

dosdol

dlsjn~un2un

0!. ~20!

We finally obtain that the angular positions of the images

un5un01

dlsjn

dosdol~b2un

0!. ~21!

The second term in Eq.~21! is a small correction onun0 , so

all images lie very close toun0 . With a similar treatment, the

other set of relativistic images have positions given by

un52un01

dlsjn

dosdol~b1un

0!. ~22!

Whenb50 an infinite sequence of Einstein rings is formewith angular radius

unE5un

02dlsjn

dosdolun

0 . ~23!

Gravitational lensing conserves surface brightness, so thetio of the solid angles subtended by the image and the sogives the amplification of thenth image:

mn5U sinb

sinun

db

dunU21

'U b

un

db

dunU21

. ~24!

Using Eq.~21! we have

mn51

b Fun01

dlsjn

dosdol~b2un

0!G dlsjn

dosdol, ~25!

which can be approximated to first order indls /dosdol!1 by

mn51

b

dls

dosdolun

0jn . ~26!

The same result is obtained for the other set of images.first relativistic image is the brightest, and the magnificatiodecrease exponentially withn.

The total magnification, considering both sets of imagis m52(n51

` mn , which can be written as

m52

b

dlsa3

dosdol2 a1

, ~27!

where

a35bps

expS a22p

a1D

12expS 22p

a1D 1

expS 2a222p

a1D

12expS 24p

a1D . ~28!

Extended source. If the source is extended, we haveintegrate over its luminosity profile to obtain the magnifiction

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E. F. EIROA AND D. F. TORRES PHYSICAL REVIEW D69, 063004 ~2004!

m5

E ESImpdS

E ESIdS

, ~29!

whereI is the surface intensity distribution of the source amp is the magnification corresponding to each point ofsource. If the source is uniform, and we use polar coordin(R,w) in the source plane, withR50 in the optical axis, andwe consider the source as a diskD(Rc ,Rs) of radius Rscentered inRc , the last equation takes the form

m5

E ED(Rc ,Rs)

mpRdRdw

pRs2

, ~30!

and, usingb5R/Dos as the angular position of each pointthe source, we have

m5

E ED(bc ,bs)

mpbdbdw

pbs2

, ~31!

whereD(bc ,bs) is the disk with angular radiusbs centeredin bc . Then

mn5I

pbs2

dls

dosdolun

0jn , ~32!

and

m52I

pbs2

dlsa3

dosdol2 a1

, ~33!

with I 5**D(bc ,bs)dbdw. This integral can be calculated i

terms of elliptic functions@4#:

I 52 sgn@bs2bc#F ~bs2bc!ES p

2,

24bsbc

~bs2bc!2D

1~bs1bc!FS p

2,

24bsbc

~bs2bc!2D G , ~34!

where

F~f0 ,l!5E0

f0~12l sin2f!2 1/2df, ~35!

E~f0 ,l!5E0

f0~12l sin2f!1/2df, ~36!

are respectively, elliptic integrals of the first and second kwith the argumentsf0 andl.

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IV. COMPARISON WITH ORDINARY LENSING

In ordinary lensing the angular positions of the two setsrelativistic images, for a point source, are given by

un5un01

dosjn

dlsdol~b2un

0! ~37!

and

un52un01

dosjn

dlsdol~b1un

0!, ~38!

with

un05

bps

dol1

1

dolexpS a222np

a1D ~39!

and

jn51

a1expS a222np

a1D . ~40!

We see that for both cases~ordinary and retrolensing!, theimages are formed close toun

0 , which has values of the ordeof the angular radius of the photon sphere of the lens. FrEqs.~15! and ~39! we observe that the images for retrolening lies a bit farther of the photon sphere than in the ordincase.

The magnifications in ordinary lensing, for a point sourcof the relativistic images are~both sets!

mn51

b

dos

dlsdolun

0jn , ~41!

and the total magnification, summing the magnificationsboth sets of relativistic images, is

m52

b

dosa3

dlsdol2 a1

, ~42!

where

a35bps

expS a222p

a1D

12expS 22p

a1D 1

expS 2a224p

a1D

12expS 24p

a1D . ~43!

Comparing Eqs.~28! and~43! we see that the first term ofa3in retrolensing is greater by a factorep/a1 and the second bya factore2p/a1 with respect to the ordinary case. Then, takiby example the lensing by a Schwarzschild black hole, aconsidering a situation where the factor involving the dtances and the source angular position is the same, the mnification of the relativistic images in retrolensing is greaby a factor about 24 compared with ordinary lensing.

In addition, with ordinary lensing, besides the two setsrelativistic images, two weak field images with small defletion angles are produced by the lens. These weak fieldages do not appear in retrolensing, and it is the main dif

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STRONG FIELD LIMIT ANALYSIS OF . . . PHYSICAL REVIEW D 69, 063004 ~2004!

TABLE I. Black hole in the Galactic center as a gravitational retrolens for a star situated at 50 pc froEarth. For several values of the chargeq ~in units of the Schwarzschild radius!, the angular radii of the photonsphere (ups), the first Einstein ring (u1

E), and the limiting value of the Einstein rings (u`E), are given. The

magnification, for perfect alignment and taking the source as extended with radiusR( , of the first image is2m1, whereas the total magnification ism.

uqu 0 0.1 0.2 0.3 0.4 0.5

ups (marcsec) 9.755 9.667 9.395 8.899 8.080 6.503u1

E (marcsec) 17.39 17.28 16.95 16.37 15.53 15.04u`

E (marcsec) 16.90 16.78 16.43 15.80 14.78 13.012m1 (10210) 3.032 3.031 3.049 3.164 3.673 7.705m (10210) 3.037 3.037 3.055 3.172 3.687 7.784

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ence in both cases. The weak field images are ordermagnitude stronger than the relativistic ones@4,5# and lievery close to them. This makes detecting the relativisticages in ordinary lensing a very difficult task. In the caseretrolensing, instead, the faint relativistic images are stronand relatively easier to identify due to the absence ofbrighter weak field images.

V. EXAMPLES

In this section we apply the formalism to the Schwarchild and Reissner-Nordstro¨m lens geometries. Two caseare analyzed: a supermassive black hole situated at thelactic center and a stellar mass black hole in the Galahalo. In both cases a nearby star is taken as source.

There is strong evidence of the existence of a supermsive black hole at the center of our Galaxy@16#. We take aM;2.83106M ( black hole as an fiducial lens at a distanDol58.5 kpc. A star with radiusR5R( situated atDos550 pc is assumed as the source. In Table I the first~outer!Einstein ring angular position (u1

E) and the limiting angularvalue of the Einstein rings (u`

E) are given for the Schwarzschild and Reissner-Nordstro¨m black hole lenses. The imagelie very close to the Einstein angular radii. For extendsources, when there is complete alignment, instead of aof images for eachn ~one on each side!, an annular shapedimage is formed, with magnification 2mn . The magnificationof the first relativistic image and the total magnification, fperfect alignment, are also shown in Table I.

Low mass black holes are common in our Galaxy@17#.Let us now consider a stellar mass black hole with massM57M ( in the Galactic halo withDol54 kpc as retrolens anda star situated atDos550 pc with radiusR5R( as source.

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The results are given in Table II. In both examples, thependence of the angular positions and magnifications ofimages with the charge is weak, obtaining important chanonly for extreme values of charge.

As the images lie close to the photon sphere, their angpositions take very small values, which increase with larlens mass and smaller observer-lens distance. For the smassive black hole at the Galactic center, they are oforder of 10marcsec, and for the stellar mass black holethe Galactic halo they are about 1025 marcsec. Somethingsimilar happens with the magnifications, which also growith greater lens mass and smaller observer-lens distaFor the supermassive black hole at the Galactic centermagnifications are of order 10210 and for the low mass onein the Galactic halo of order 10221.

As long asdos!dol , the angular positions and amplifications of the images are not very sensitive to the value ofdos .So if we take the Sun or a star at 100 pc from the Earthsource, the order of magnitude of the results shown in TabI and II does not change. As the images appear highlymagnified, to improve the possibility of observing themstrong nearby sources with nearly perfect alignmentneeded. In all cases, the first image is the strongestcarrying about 99% of the total luminosity.

VI. CONCLUDING REMARKS

We have presented the strong field limit formalism fgravitational retrolensing. We have found the positions amagnifications of the relativistic images within the strofield limit, and studied some cases that might present aticular interest and be a benchmark point for compariswith the normal lensing situation. As in the case of ordina

star

TABLE II. Same as Table I for a black hole in the Galactic halo as a gravitational retrolens with asituated at 50 pc from the Earth as source.

uqu 0 0.1 0.2 0.3 0.4 0.5

ups (1025marcsec) 5.182 5.136 4.991 4.728 4.292 3.455u1

E (1025marcsec) 9.236 9.178 9.002 8.696 8.249 7.990u`

E (1025marcsec) 8.976 8.916 8.729 8.393 7.853 6.9102m1 (10221) 4.053 4.052 4.076 4.230 4.911 10.30m (10221) 4.060 4.060 4.084 4.240 4.928 10.41

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E. F. EIROA AND D. F. TORRES PHYSICAL REVIEW D69, 063004 ~2004!

lensing, the relativistic images are faint. However, if wcompare the magnification of retrolensed relativistic imawith those of the ordinary strong field lensing, we find ththe former are significantly stronger. Additionally, asbrighter weak field images appear in retrolensing, it wobe easier to detect them. The angular separation of thetivistic images is of course beyond the angular resolutioncurrent or foreseeable technologies, and not only angresolution but also sensitivity improvements are neededdetect them~see the discussion by Eiroa, Romero, and Tor@5#!. The image separations and magnifications presentues that make the observational identification of these rtivistic images a challenge for the future: we are not awareany current research, in any wavelength, that would althe resolution of the retrolensing images in the next yearsthe case of charged black holes, it is only for extreme valof charge that important changes are obtained.

ACKNOWLEDGMENTS

This work has been partially supported by UBA~UBA-CYT X-143, E.F.E.!. The work of D.F.T. was performed under the auspices of the U.S. DOE~NNSA!, by UC LLNLunder Contract No. W-7405-Eng-48.

APPENDIX: VALUES OF THE COEFFICIENTS a1 AND a2

Schwarzschild black hole

The metric functions are

f ~x!5g21~x!5121

x, h~x!5x2. ~A1!

Thenxps5

3

2, ~A2!

b~x0!5x0S 121

x0D 21/2

, bps53A3

2. ~A3!

The coefficients can be calculated exactly as@4,6#

a151, ~A4!

e

06300

st

dla-f

artosl-

a-f

Ins

a25 ln@324~7A3212!#2p. ~A5!

Reissner-Nordstrom black hole

In this case

f ~x!5g21~x!5121

x1

q2

x2, h~x!5x2, ~A6!

where q5Q/2M is the electric charge in units of thSchwarzschild radius. Then

xps53

4 F11S 1232

9q2D 1/2G , ~A7!

b~x0!5x0S 121

x01

q2

x2D 21/2

, ~A8!

bps5~31A9232q2!2

4A2A328q21A9232q2. ~A9!

The coefficienta1 can be calculated exactly as@6#

a15xpsAxps22q2

A~32xps!xps2 29q2xps18q4

, ~A10!

and the coefficienta2 can be approximated@6# by

a25c11c21a1ln bps2p ~A11!

with

c152 ln@6~22A3!#18

9$A3241 ln@6~22A3!#%q2

1O~q4!, ~A12!

c25a1lnF2~xps2q2!2~32xps!xps

2 29q2xps18q4

~xps22q2!3~xps2 2xps1q2!

G .

~A13!

.

a,

-

.

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