PHYSICAL REVIEW D, VOLUME 58, 044015

Perturbations in the Kerr-Newman dilatonic black hole background: Maxwell waves,the dilaton background, and gravitational lensing

R. CasadioDipartimento di Fisica, Universita` di Bologna, and I.N.F.N., Sezione di Bologna, via Irnerio 46, 40126 Bologna, Italy

B. HarmsDepartment of Physics and Astronomy, The University of Alabama, Tuscaloosa, Alabama 35487-0324

~Received 22 April 1998; published 22 July 1998!

In this paper we continue the analysis of one of our previous papers and study the affect of the existence ofa non-trivial dilaton background on the propagation of electromagnetic waves in the Kerr-Newman dilatonicblack hole space-time. For this purpose we again employ the double expansion in both the background electriccharge and the wave parameters of the relevant quantities in the Newman-Penrose formalism and then identifythe first order at which the dilaton background enters Maxwell equations. We then assume that gravitationaland dilatonic waves are negligible~at that order in the charge parameter! with respect to electromagnetic wavesand argue that this condition is consistent with the solutions already found in the previous paper. Explicitexpressions are given for the asymptotic behavior of scattered waves, and a simple physical model is proposedin order to test the effects. An expression for the relative intensity is obtained for Reissner-Nordstro¨m dilatonblack holes using geometrical optics. A comparison with the approximation of geometrical optics for Kerr-Newman dilaton black holes shows that at the order to which the calculations are carried out gravitationallensing of optical images cannot probe the dilaton background.@S0556-2821~98!07316-0#

PACS number~s!: 04.70.Bw, 04.50.1h, 11.25.Pm, 97.60.Lf

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I. INTRODUCTION

In Ref. @1# we started from the low energy effective actiodescribing the Einstein-Maxwell theory interacting withdilaton of arbitrary coupling constanta in four dimensions~we always setc5G51 unless differently stated!,

S51

16p E d4xA2gFR21

2~¹f!22e2afF2G , ~1.1!

and, on expanding the fields in terms of the charge-to-mratio Q/M of the source, we obtained the static solutionthe field equations corresponding to a Kerr-Newman dtonic ~KND! black hole rotating with arbitrary angular momentum. Our solution reduces to the Kerr-Newman~KN!metric ~see e.g.@2#! for a50 and to the Kaluza-Klein solution @3# for a5), but differs from the exact ReissneNordstrom dilatonic ~RND! black hole@4# in the zero angu-lar momentum limit@5#. In Ref. @6#, to which we refer for allthe definitions and notation, we used these solutions to wthe wave equations for the various field modes. Our metis to double expand each field in powers of the eleccharge and the wave parameter. Substituting these exsions into Maxwell’s equations, the dilaton equation and Estein’s equations then give~inhomogeneous! wave equationsfor the coefficients of the expansion to any desired accura

We have already given explicit expressions for Maxweequations for the coefficients linear in the wave paramefor the three lowest orders of the charge parameter and hfound the asymptotic form of the solutions linear in the waand charge parameters@order ~1,1! in the notation of@6##.The latter correspond to Maxwell waves produced by dilawaves scattered by the static electromagnetic backgroun

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In this paper we analyze the Maxwell waves producedthe scattering of electromagnetic waves by the static dilabackground which emerge at order~1,2!. Since Maxwell’sequations at order~1,2! contain waves of all kinds@7#, find-ing analytic solutions seems an intractable problem. We tpursue a suggestion already introduced in@6# and assumethat the dilaton waves and the gravitational waves are negible with respect to Maxwell waves, which allows usgreatly simplify the equations for electromagnetic waves.Sec. II we check the consistency of such a working ansand argue that it is valid at order~1,2! when one considersthe solutions at order~1,1! which we have found in@6#. InSec. III we compute the asymptotic behavior at a large dtance from the hole of both outgoing and ingoing modesthe reduced Maxwell equations and propose a methoddetecting the dilaton background in a binary system fromscattered pattern of radiation emitted by the star companIn Sec. IV we show an alternative way of computing tvariation of the flux of a null wave scattered by RND anKND black holes in the approximation of geometrical opticThis approach reflects the qualitatively different naturesthe two kinds of metrics and proves that light paths areaffected be the dilaton background in KND black holes atcomputed order in the charge-to-mass expansion. Wecompare this result with the wave approach of the previsections.

II. WAVE EQUATIONS AT ORDER „1,2…

We employ the double expansion of perturbations@6# ofthe static solution given in Ref.@1# in order to write the waveequations for the various field modes:

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R. CASADIO AND B. HARMS PHYSICAL REVIEW D58 044015

f~ t,r ,u,w!5(p,n

gpQnf~p,n!,

f i~ t,r ,u,w!5(p,n

gpQnf i~p,n! , i 50,1,2,

G~ t,r ,u,w!5(p,n

gpQnG~p,n!, ~2.1!

where g is the same wave parameter for the dilaton~f!,Maxwell (f i) and gravitational~collectively denoted byG)field quantities. Of course, we will only study the line(p51) case and setg51 from now on. We assume that thlinear perturbations have the following time and azimutdependence:

f~1,n!~ t,r ,u,w!5kdei vt1 imwf~1,n!~r ,u!,

f i~1,n!~ t,r ,u,w!5kEMei vt1 imwf i

~1,n!~r ,u!, i 50,1,2,

G~1,n!~ t,r ,u,w!5kGei vt1 imwG~1,n!~r ,u!, ~2.2!

wherekd , kEM and kG are parameters. We recall here theach function ofr and u on the right-hand sides~RHS’s!above implicitly carries an extra integer index,m, and acontinuous dependence on the frequencyv.

As shown in@6#, it is at order~1,2! that the effect inducedby the dilaton background appears in all of the equatioSince at this order the different kinds of waves do not distangle even fora50 @7#, we need a working ansatz to obtaa manageable set of equations. As already suggested in@6#,we shall tentatively assume

kEM@kd ,kG , ~2.3!

and neglect both dilaton and gravitational waves with respto Maxwell waves. This hypothesis is equivalent to assumthe existence of~space-time! boundary conditions such thathe gravitational and dilaton contents of the wave fieldnegligibly small when compared to the electromagnesources, a condition that is not automatically consistentwill be checked in the following for the whole set of fielequations.

For completeness, we observe that, in the casekEM;kd ,kG , the leading contributions to the electromagnewaves come from order~1,1! and have already been found@6#. Therefore the present paper together with@6# shouldcover most of the leading order electromagnetic physicsone can extract from the study of classical waves onKND background.

A. Gravitational equations

In @6# we considered the gravitational field to be detmined by the following three non-vacuum equations inNewman-Penrose~NP! formalism:

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~ d* 24a1p!C02~D24r22e!C123kC2

5~ d1p* 22a* 22b!R11

2~D22e22r* !R1212sR2122kR222k* R13

~D24g1m!C02~ d24t22b!C123sC2

5~ d12p* 22b!R12

2~D2e12e* 2 r* !R132l* R1122sR2222kR23

3~D2 r2 r* 23e1e* !s

2~ d2t1p* 2a* 23b!k2C050. ~2.4!

The Ricci tensor terms are given by the Einstein field eqtions,

Rab51

2f uaf ub12Tab

EM , ~2.5!

and the electromagnetic energy-momentum tensor is

Ti jEM5e2afFFikF j

k21

4gi j F

2G[e2afFFgS 121

4ggD ,

~2.6!

where, on the far RHS, we omit indices for brevity, so thatFrepresents any of the components of the Maxwell fistrength andg ~not to be confused with the wave paramethat we have set to 1! any component of the metric tensor

For the purpose of further simplifying the expressions,also observe that Eq.~2.4! above can be formally rewritten a

GG5GR, ~2.7!

whereG represents any differential operator with coefficienwhich depend on gravitational quantities only,G representsany gravitational quantity andR here stands for any component of the Ricci tensor. On omitting vanishing terms othen finds that at order~1,2! Eq. ~2.4! can be written

kG~G ~0,0!G~1,2!1G ~0,2!G~1,0!1G ~1,0!G~0,2!1G ~1,2!G~0,0!!

5G ~0,0!R~1,2!1kGG ~1,0!R~0,2!, ~2.8!

where

R~0,2!52F ~0,1!F ~0,1!g~0,0!S 121

4g~0,0!g~0,0!D

R~1,2!5kdf ua~1,0!f ub

~0,2!14kEMF ~0,1!F ~1,1!g~0,0!

3S 121

4g~0,0!g~0,0!D12F ~0,1!F ~0,1!

3FkGg~1,0!S 123

4g~0,0!g~0,0!D

2akdf~1,0!g~0,0!G . ~2.9!

If we now apply our ansatz, Eq.~2.3!, and neglect any termsproportional tokG ,kd with respect to terms proportional t

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PERTURBATIONS IN THE KERR-NEWMAN DILATONIC . . . PHYSICAL REVIEW D 58 044015

kEM , we find that the third equation in Eq.~2.4! is not af-fected, since it does not explicitly depend on electromagnwaves, while the first and second equations can apparebe reduced to

0.G ~0,0!R~1,2!.4kEMG ~0,0!F ~0,1!F ~1,1!g~0,0!

3S 121

4g~0,0!g~0,0!D . ~2.10!

These would be undesired constraints for the solutionsMaxwell’s equations at order~1,1! which we have alreadyfound in @6#. However, we recall here that those~particular!solutions of the inhomogeneous equations which we denby f i

(1,1) in Sec. IV of @6# are proportional tokd /kEM .ThereforeF (1,1);kd /kEM and, on substituting into the RHSof Eq. ~2.10!, one obtains expressions which are indeed pportional to kd . Thus it is not consistent to neglect termproportional tokG ,kd in Eq. ~2.8!, since there is no termproportional tokEM with respect to which they are small annone of the above gravitational equations is affected byapproximation in Eq.~2.3!. They are and remain three independent equations for the gravitational quantitiesG(1,2).

Of course, Eq.~2.8! is so involved that obtaining an analytic solution appears to be unlikely.

B. Dilaton equation

The equation for the dilaton field in the NP tetrad compnents is

@DD1DD2 d d* 2 d* d1~e1e* 2 r2 r* !D

1~m1m* 2g2g* !D1~t2p* 1a* 2b!d*

1~t* 2p1a2b* !d#f52ae2afF2, ~2.11!

and can be rewritten in analogy with Eq.~2.7! as

Gf52ae2afFFgg. ~2.12!

At order ~1,2! and neglecting terms which vanish identicallone then has

kdG ~0,0!f~1,2!1kdG ~0,2!f~1,0!1kGG ~1,0!f~0,2!

522akEMF ~1,1!F ~0,1!g~0,0!g~0,0!

22akGF ~0,1!F ~0,1!g~1,0!g~0,0!

1a2kdf~1,0!F ~0,1!F ~0,1!g~0,0!. ~2.13!

Again, we recall thatF (1,1);kd /kEM , so that no term pro-portional tokEM appears, and therefore the dilaton equatat order~1,2! does not lead to an undesired constraintlower order quantities.

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C. Maxwell equations

The equations for the Maxwell fields in the NP formalisare given by

~D22r !f12~ d* 1p22a !f01kf25J1

~ d22t!f12~D1m22g!f01sf25J3

~D2 r12e!f22~ d* 12p!f11lf05J4

~ d2t12b!f22~D12m!f11nf05J2 , ~2.14!

where the source terms are

J15a

2@~f11f1* !D2f0d* 2f0* d #f

J25a

2@~f11f1* !D2f2d2f2* d* #f

J35a

2@~f12f1* !d2f0D1f2* D#f

J45a

2@~f12f1* !d* 1f0* D2f2D#f.

~2.15!

We can expand to order~1,2! and make the same approxmation given in Eq.~2.3!. However, it is clear that nowthere are terms truly proportional tokEM , namely f i

(1,0)

and f i(1,2) , which will remain even after considerin

kEMf (1,1);kd as found in@6#. We are then allowed to neglect all terms proportional tokG , kd in Eqs. ~2.14! and~2.15! above, and the four Maxwell equations at order~1,2!then read

~D22r !~0,0!f1~1,2!2~ d* 1p22a !~0,0!f0

~1,2!5J1~1,2!

~ d22t!~0,0!f1~1,2!2~D1m22g!~0,0!f0

~1,2!5J3~1,2!

~D2 r !~0,0!f2~1,2!2~ d* 12p!~0,0!f1

~1,2!5J4~1,2!

~ d2t12b!~0,0!f2~1,2!2~D12m!~0,0!f1

~1,2!5J2~1,2! .

~2.16!

The currents on the right-hand sides~RHSs! are given by

J1~1,2!52 i S K

D D ~0,2!

f1~1,0!1

a

2 F ~f11f1* !~1,0!] r

21

&

S f0

r*1

f0*

rD ~1,0!

]uGf~0,2!

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R. CASADIO AND B. HARMS PHYSICAL REVIEW D58 044015

J2~1,2!52m~0,2!f1

~1,0!2a

2 F D0

2r2 ~f11f1* !~1,0!] r

11

&

S f2*

r*1

f2

rD ~1,0!

]uGf~0,2!

J3~1,2!52m~0,2!f0

~1,0!1a

2 F 1

& r~f12f1* !~1,0!]u

1S D0

2r2 f01f2* D ~1,0!

] r Gf~0,2!

J4~1,2!52 i S K

D D ~0,2!

f2~1,0!1

a

2 F 1

& r*~f12f1* !~1,0!]u

2S D0

2r2 f0* 1f2D ~1,0!

] r Gf~0,2!, ~2.17!

from which one concludes that the perturbationsf i(1,2)

couple to the~gradient of the! dilaton background throughthe free Maxwell wavesf i

(1,0) .

III. ASYMPTOTIC SOLUTIONS: SCATTERING FROMTHE BACKGROUND

The set of equations~2.16! together with the corresponding currents is still quite involved. However, we observe ththe electromagnetic wavesf i

(1,0) which enter the currents inEq. ~2.17! are actually input data: that is, we are free to chowhatever kind of electromagnetic waves we want to setoward the black hole and then compute the scattered paf i

(1,2) .Of course, one wants to consider a physically meaning

model. For instance, one can think of a double system mof a black hole and a companion star which is periodicaoccultated while revolving around the black hole. The ligof the star would thus periodically pass through the ergogion of the black hole where the dilaton background gradiis the strongest. This would allow for a comparison betwethe spectrum of the star when it is in front of the hole~andthe dilaton background effects are negligible! and the spec-trum of the star when it is just going behind the black ho~see Fig. 1!. We also note here that a classical wave scatteby the system is represented by a superposition of ingomodes coming from far away until it reaches the centerthe system~the black hole!, and then it switches to a supeposition of outgoing modes moving away from the blahole.

Asymptotically (r→`) the major contribution to theelectromagnetic wave field is given by

f2~1,0!.C2

6e6 i vr

r[F2

6 , ~3.1!

with a plus sign for ingoing modes and a minus sifor outgoing modes andC2

65C26(u) contains all the angu

lar dependence inu. Therefore we can assume th

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dgf

f0(1,0)5f1

(1,0)[0 ~the so-calledphantom gauge: see Ref.@2#!

in Eq. ~2.17!. This reduces the currents to

J1~1,2!50

J2~1,2!52

a

2&S F2*

r*1

F2

rD ]uf

J3~1,2!5

a

2F2* ] rf

J4~1,2!5 i

K

D02 F22

a

2F2 ] rf,

~3.2!where

f[f~0,2!52a

M

r

r2 . ~3.3!

We can obtain an equation forf2(1,2) alone from the third

and fourth equations in Eqs.~2.16!. Upon defining as usuaW2[2(r* )2f2

(1,2) andW1[& r* f1(1,2) we obtain

SD021

r* DW22SL01ia sin u

r* DW1

52~ r* !2F2S iK

D02 2

a

2] rf D

3SL1†2

ia sin u

r* DW22D0SD0†1

1

r* DW1

52ar* ~ r* F21 rF2* !]uf. ~3.4!

FIG. 1. A black hole binary system with the stellar companishown at two different positions in its orbit about the black hoWhen the star is occulting as seen from Earth, electromagneticdiation in the Earth’s direction will be scattered by the tensor ascalar components of the gravitational field.

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PERTURBATIONS IN THE KERR-NEWMAN DILATONIC . . . PHYSICAL REVIEW D 58 044015

Finally, upon using the commutation relation

FD0SD0†1

1

r* D ,SL01ia sin u

r* D G50, ~3.5!

one can eliminateW1 and get

@D0D0†D01L0L1

†12i vr#W25T2 , ~3.6!

where the source term is

T2[2D0SD0†1

1

r* D F ~ r* !2F2S iK

D02 2

a

2] rf D G

2aSL01ia sin u

r* D @ r* ~ r* F21 rF2* !]uf#.

~3.7!

To leading order in 1/r , the current is thus

T2652vC2

6~2M v1 ia2!3H e1 i vr ,r

Me2 i vr .

~3.8!

In the expression above the contribution from the dilabackground is the one proportional toa2, the remainingterms being purely KN. We can then write

T25TKN1Ta , ~3.9!

and separate the contributions for the two different sourcThe LHS of Eq.~3.6! can be expanded in powers of 1/r aswell upon assuming

W2.A6Z6e6 i vr

r n6, ~3.10!

whereZ65Z6(u). One then finds

A6Z6e6 i vr

r n621 2v@2M v1 i ~17n6!#5T26 , ~3.11!

from which it follows thatZ65C26 ,

n151

AKN1 51

Aa15 i

a2

2M v~3.12!

andn250

AKN2 52v

2M v21

114M2v2

Aa25

a2

M

112iM v

114M2v2. ~3.13!

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We can conclude that the amplitudes of the electromagnwaves scattered by the dilaton background become comrable to or greater than the intensity of the waves scatteby the KN background whenAa

2>AKN2 , that is when~we

restore the fundamental constants!

v<vc;a2c3

GM. ~3.14!

In string theorya51 andvc;1035 Hz/kg. For example, fora solar mass black hole, the critical frequency wouldabout 100 kHz.

We observe thatf2(1,2);1/r 2 is subleading with respect to

f2(1,0);1/r . However, we can apply the same argument t

we have formulated in@6#, Sec. V, and conclude that thscattered wavesf2

(1,2) are negligible with respect to frewavesf2

(1,0) only when the scattering process occurs at lar , which is quite sensible, since strong effects from the dton background are not expected far away from the horizTherefore, adjusting the above mentioned argument topresent context, we claim that testable electromagnwaves could be produced by the scattering of free Maxwwaves in a region~denoted byR in Fig. 1 and Ref.@6#! justoutside the horizon where the gradient of the dilaton baground is the strongest. Outside ofR the scattered wavethen propagate as free waves: thus one obtains

uf2~1,2!~r .r d!u.uf2

~1,2!~r d!ur d

r, ~3.15!

wherer d is the typical outer radial coordinate of the regioR.

The next step would now be to superpose a suitablelection of incoming modesf2

(1,0) to more realistically modelthe electromagnetic radiation coming from the stellar copanion of the black hole. Then, the corresponding supersition of excited modesf2

(1,2) would give us the scatterepattern and its time dependence due to the relative positof the source and the black hole with respect to the obserHowever, we feel that sensible results are out of reach ofanalytical methods which are all we want to consider inpresent work.

IV. GEOMETRICAL OPTICS EFFECTS

In the previous sections and in@6# we have studied thewave equations on the KND background. However, omight expect to obtain measurable effects on the propagaof light in such a background even at the level of geometrioptics @8#.

To start with, we consider a simpler case, the~exact!RND solution@4#,

ds252e2F dt21e2L dr21R2~du21sin2 udw2!,~4.1!

with

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R. CASADIO AND B. HARMS PHYSICAL REVIEW D58 044015

e2F5e22L5S 12r 1

r D S 12r 2

r D ~12a2!/~11a2!

R25r 2~12 r 2 /r !2a2/~11a2!

r 15M F11A12~12a2!Q2

M2Gr 25~11a2!

Q2

M F11A12~12a2!Q2

M2G21

,

~4.2!

and compute the deflection angleDw1 of a null ray comingfrom infinity with an impact parameterD15L1 /E1[D (L1is its angular momentum,E1 its energy! and approaching theouter horizonr 1 . From the equations of motion for nugeodesics one obtains an expression fordw/dr ~see@2# forthe details!. On integrating the latter fromr 51` to r 1 , theminimum radial position from the center of the hole reachby the ray, one finds

Dw152@„w~r 1!2w~`!…#2p. ~4.3!

For the metric in Eq.~4.1! the above expression becomes

Dw1.2M

r 1F11

4a223

11a2

Q2

4M2G , ~4.4!

where we have assumedr 1@r 2 and Q/M!1. Equation~4.4! already contains a dilatonic contribution and allowscomparison of the pure Reissner-Nordstro¨m case (a50)with the Schwarzschild case (Q50).

The radiusr 1 is one of the zeros ofdr/dw which in turnare the solutions of@2#

r 132D1

2~r 12r 1!50. ~4.5!

Of course one requiresr 1.r 1 for an unbound null ray thamust escape to infinity.

Let us now consider a second fiducial null ray startingaway from the hole which lies in the same plane and poin the same direction as the first one but has a slightlyferent impact parameterD25J2 /E25D1d, where d!D.The two rays, rotated around the axis passing throughcenter of the hole and asymptotically parallel to boththem, define an annulus of which we just consider a smportion whose area is

A~2`!;Q~D222D1

2!.2DdQ, ~4.6!

where Q is the relative angle between the two rays wrespect to the axis passing through the center of the hole~weassumeQ to be small!.

Each ray will then be deflected by the gravitational fieof the hole according to Eq.~4.4!. Since one hasr 1Þr 2 ,after having been scattered, the two fiducial rays will moat a relative angle

04401

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efll

e

D12[Dw22Dw152M F114a223

11a2

Q2

4M2G S 1

r 22

1

r 1D .

~4.7!

Therefore the area that they define will change according

A~r !;r 2 Q uD12u. ~4.8!

Although Eq. ~4.5! is exactly soluble, the solutions arcomplicated: thus we will just consider the following aproximations. Since we are interested in rays which traclose to the horizon, we assumer i5r 11xi , with xi!r 1

and, from Eq.~4.5! we get

Di2.

r 13

xi@r 1

2 . ~4.9!

On solving the above equation forxi and substituting intoEq. ~4.7! one finally obtains

D12.16M2d

D3 F112a414a225

11a2

Q2

8M2G , ~4.10!

where the approximationDi@r 1 ,d has been used, bud;r 1 is allowed.

The rate of decrease of the intensityI can then be com-puted for any massless linear wave moving ‘‘betweentwo rays,’’

I ~r !

I ~2`!5

A~2`!

A~r !;

D4

M2 F122a414a225

11a2

Q2

8M2G 1

r 2 .

~4.11!

The above result depends on the form of the metric onand one might think of repeating the same computationthe KND case. However, as we pointed out in@6#, up to theorder at which we have computed it, the KND metric is juthe KN metric. Therefore, no effect on the null rays canexpected from the dilaton background at any order beQ4/M4 when the black hole is rotating@9#.

A final remark is due to clarify the difference between tapproach shown in this section and the previous one.solutions to the wave equation that we have found in Secdisplay the full wave nature of the electromagnetic radiatand they also depend on the direct coupling betweendilaton field and the Maxwell field as formulated in the ation ~1.1!. However, the results in the present section relythe approximation of geometrical optics, thus neglectingspecific nature of the null rays which includes both the wacharacter and the coupling to the dilaton. Therefore it issurprising that, in the geometrical optics picture, no effectthe Maxwell rays is found for the KND geometry, while ithe wave picture and at the same order of approximationthe charge-to-mass expansion the dilaton does affect elemagnetic waves.

By comparing the two approaches, one can concludethe presence of a non-trivial background dilaton in KNblack holes~at the order included in our computation! affectsonly the intensity of the electromagnetic radiation and noteikonal paths. This means that the calculation of grav

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tional lensing effects for comparison with measurementstests for the existence of non-trivial dilaton fields mustcarried to higher order in the expansion, and this can prably be done only numerically.

V. CONCLUSIONS

We have shown that to the order at which we are workin perturbation theory the neglect of the gravitational adilatonic waves with respect to the electromagnetic waveinternally consistent. This assumption has allowed us totain asymptotic expressions for ingoing and outgoing moin the phantom gauge and the dilatonic contributions to thmodes. Our analysis shows that electromagnetic wavestered by the dilaton background have a critical frequencyapproximately 0.1 MHz. This is in a frequency range whiis not currently being studied by astrophysicists, becamost stars emit relatively small amounts of energy in trange. On the other hand, if a detector were to be construto observe black hole binaries in this frequency range,signal should be relatively clean, and any enhancementhis frequency in the intensity of the radiation from an oculting star would be a clear indication of the presence oscalar component of gravity.

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At any given frequency a decrease in the intensity ofradiation should be expected for an occulting star due toscattering of the light by the gravitational field. However,exact expression for the intensity will have to wait until ware able to realistically model the electromagnetic radiatfrom the stellar companion of a black hole. This involves tsuperposition of the incoming Maxwell scalar modes, athis will probably entail a numerical investigation of thvariation of the intensity of the star as a function of its orbiposition.

In the geometrical approximation for a charged, norotating ~Reissner-Nordstro¨m! dilaton black hole the inte-grated intensity decreases during occultation if 2a414a2

25.0. In string theorya51, and a decrease in the intensiof light from the occulting star as compared to its intenswhen it eclipses the black hole would be evidence for a slar component of gravity as predicted by string theory.

ACKNOWLEDGMENTS

We wish to thank Y. Leblanc for his contributions to thearly stages of this work. This work was supported in partthe U.S. Department of Energy under Grant No. DE-FG096ER40967.

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@1# R. Casadio, B. Harms, Y. Leblanc and P. H. Cox, Phys. Rev55, 814 ~1997!.

@2# S. Chandrasekhar,The Mathematical Theory of Black Hole~Oxford University Press, Oxford, 1983!.

@3# V. Frolov, A. Zelnikov and U. Bleyer, Ann. Phys.~Leipzig!44, 371 ~1987!.

@4# G. W. Gibbons and K. Maeda, Nucl. Phys.B298, 741 ~1988!;G. T. Horowitz and A. Strominger,ibid. B360, 197 ~1991!.

@5# The fact that the addition of even a small amount of angumomentum drastically changes the qualitative features ofmetric was already pointed out by J. H. Horne and G.Horowitz, Phys. Rev. D46, 1340~1992!.

@6# R. Casadio, B. Harms, Y. Leblanc and P. H. Cox, Phys. Rev

D

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56, 4948~1997!.@7# Gravitational and electromagnetic waves do not decouple e

in the KN backgroundwithout the dilaton@2#. Of course herethe situation is even more involved because there is one mfield.

@8# For the details of this approach see C. W. Misner, K. S. Thoand J. A. Wheeler,Gravitation ~Freeman, New York, 1973!,Chap. 25.6.

@9# This conclusion seems to hold also for the metric in thecalled string framewhich is obtained from the one we havbeen using so far by a conformal transformation. R. Casaand B. Harms, ‘‘Charged Dilatonic Black Holes: String Framvs Einstein Frame,’’ gr-qc/9806032.

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