The physical deflection angle of a light ray propagating in a space-time
supplied with an asymptotically flat metric has to be expressed in terms of the
impact parameter.Comment: 11 pages, 1 figur

Gerard, J. -M.

Pireaux, S.

English

arXiv

11 pages, 1 figure, gr-qc/9907034The physical deflection angle of a light ray propagating in a space-time supplied with an asymptotically flat metric has to be expressed in terms of the impact parameter

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The observable light deflection angle

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The physical deflection angle of a light ray propagating in a space-time
supplied with an asymptotically flat metric has to be expressed in terms of the
impact parameter.
 Comment: 11 pages, 1 figur

Gérard, Jean-Marc

DIAL UCLouvain

arXiv:gr-qc/9907034v1  8 Jul 1999UCL-IPT-99-08The observable light deflection angleJ.-M. Ge´rard and S. PireauxInstitut de Physique The´oriqueUniversite´ catholique de LouvainB-1348 Louvain-la-Neuve, BelgiumTo the memory of Jacques DemaretAbstractThe physical deflection angle of a light ray propagating in a space-time supplied withan asymptotically flat metric has to be expressed in terms of the impact parameter.1 IntroductionLight deflection by a gravitational source played a crucial role in the early establishment ofGeneral Relativity. Nowadays, gravitational optics is systematically used to search for elusivedark matter candidates (such as white dwarfs) through microlensing effects [1].¿From a theoretical point of view, light deflection also provides us with a rather uniquetool to test relativistic theories of gravity at various distance scales. In most of these the-ories, the exact value of the predicted deflection angle requires some numerical integration.Indeed, elliptic functions of the first kind are already necessary in the simple case of GeneralRelativity. Therefore, approximate analytical expressions are often used in the literature,both in the weak and strong field limits. In this short note, we would like to point out thatonly expressions in terms of the physical impact parameter are meaningful. Although thisstatement is rather academic in the case of General Relativity, we illustrate how it mighthave quantitative consequences for extended theories of gravity.2 The deflection angle ...Let us consider a static and spherically symmetric space-time defined by the line elementds2 = A2(r)c2dt2 −B2(r)dr2 −D2(r)r2(dθ2 + sin2 θdϕ2). (1)The null geodesics equations in the θ = pi2plane lead to the following equation for the photontrajectories : (dudϕ)2+D2B2u2 =1b2D4A2B2(2)withu ≡ 1r.2If space-time is asymptotically flat, b is simply the impact parameter which can be expressedin terms of the radial coordinate of the point (r0, ϕ0) of closest approachb =D(r0)A(r0)r0. (3)Moreover, in such a space-time, the deflection angle αˆ is usually defined byαˆ(r0) ≡ 2∫ ϕ00dϕ− π= 2∫∞r0B(r)D(r){( rr0)2 D2(r)D2(r0)A2(r0)A2(r)− 1}−12 drr− π (4)and requires, in general, a numerical integration.In the weak field approximation, the standard Eddington-Robertson parametrizationis defined byA2(ρ) ≃ 1 + 2α(Vc2)+ 2β(Vc2)2+32ξ(Vc2)3B2(ρ) = D2(ρ)≃ 1− 2γ(Vc2)+32δ(Vc2)2(5)withV (ρ) ≡ −GMρ,the gravitational potential expressed in terms of the isotropic radial coordinate ρ. In thisapproximation, the deflection angle as a function of the closest approach distance readsαˆω(ρ0) =4GMc2ρ0{(α+ γ2)+[−(α+ γ)22+(8α2 + 8αγ − 4β + 3δ)16π]GMc2ρ0}+O(V 3c6). (6)In particular, one easily proves the absence of light deflection in a space-time supplied witha conformally flat metricgµν =(1 +Vc2)2ηµν . (7)33 ... in General Relativity ...The Schwarzschild coordinates are more appropriate to study light deflection in the vicinityof a strong gravitational field. In the case of General Relativity, they correspond to the choiceA2(r) = B−2(r) = 1− 2mrD2(r) = 1 (8)withm ≡ GMc2.Assumingr0 ≥ 3m (9)to ensure deflection, we obtain from Eqs. (4) and (8) the angle as an exact function of theclosest distance of approach [2]αˆ(r0) = 4(r0q) 12[F(π2, k)− F (σ0, k)]− π. (10)Here, the elliptic integral of the first kindF (σ, k) ≡∫ σ0dx{1− k2 sin2 x} 12(11)is characterized by the amplitude σ withσ0 = arcsin{q − r0 + 2mq − r0 + 6m} 12(12)and the modulusk ={q − r0 + 6m2q} 12(13)whereq ≡{(1− 2mr0)(1 +6mr0)} 12r0.4In the weak field approximationǫ ≡ mr0≪ 1 (14)and the modulus is smallk2 = 4ǫ(1− 3ǫ) +O(ǫ3). (15)Expanding the elliptic integralsF(π2, k)≃ π2(1 + ǫ− 34ǫ2)F (σ0, k) ≃ π4(1 + ǫ− 34ǫ2)− ǫ (16)we obtainαˆω(r0) ≃ 4GMc2r0{1 +[−1 + 1516π]GMc2r0}. (17)in the Schwarzschild coordinates. On the other hand, Eq. (6) impliesαˆω(ρ0) ≃ 4GMc2ρ0{1 +[−2 + 1516π]GMc2ρ0}(18)in the isotropic coordinates, since the Eddington-Robertson parameters defined in Eq. (5)are conventionally normalized to unity for General Relativity.The apparent second order discrepancy between Eqs. (17) and (18) can be understoodin the following way. The deflection angle being by definition observable, it has to be fullyexpressed in terms of measurable, i.e. coordinate-independent, quantities. Here, the closestdistance of approach (r0 in Schwarzschild coordinates, ρ0 in isotropic coordinates) is obviouslynot such a measurable quantity. Its corresponding substitution by the impact parameter baccording to Eq. (3)b ≃ r0 +m≃ ρ0 + 2m (19)5reconciliates our results of calculations performed with two physically equivalent forms of themetric. In General Relativity, the observable deflection angle is correctly given byαˆω(b) =4GMc2b{1 +[0 +15π16]GMc2b}+O(m3b3). (20)This is the analog of what has been noted [3] for the radar-echo experiment where theobservable transit time has to be expressed in terms of the measurable orbital parameters(periods and eccentricities).In the strong field approximationǫ ≡ 1− 3mr0≪ 1 (21)and the modulus is close to onek2 = 1− 43ǫ+O(ǫ2). (22)Expanding again the elliptic integralsF(π2, k)≃ ln 4√1− k2F(arcsin1√3, 1)≃ 0.65848, (23)we obtain a simple analytical expression for the deflection angle of a light ray grazing a blackhole[2] :αˆs(b) ≃ ln(3.482mb− 3√3m). (24)The Eqs. (20) and (24) provide us with useful (weak and strong field) approximations tostudy lensing effects in terms of the observable deflection angle αˆ(b). They readily replace theunpractical exact expression obtained from Eq. (10) after substitution of r0 by b accordingto Eqs. (3) and (8). However, the second order contribution to the deflection angle in Eq.(20) only improves the weak field approximation in a narrow domain of the impact parameter(see Fig. 1). In that sense, our comment on the ambiguous coordinate dependence of αˆ(r0)defined in Eq. (4) is rather academic as far as General Relativity is concerned.64 ... and beyondLet us now consider the coupled Einstein-improved scalar actionS = − 12κ∫d4x√−gR+ 12∫d4x√−g[gµνΦ,µΦ,ν +16RΦ2](25)withκ ≡ 8πGc4.The non-minimal coupling of the massless scalar Φ to the curvature invariant ensures thevanishing of the stress tensor’s trace, as required by conformal symmetry [4].It is however well-known [5] that an appropriate rescaling of the metricg˜µν =(1− κ6Φ2)gµν (26)together with a redefinition of the scalar fieldΦ =(6κ) 12tanh[(κ6) 12Φ˜](27)recast the theory into the minimal formS˜ = − 12κ∫d4x√−g˜R˜+ 12∫d4x√−g˜g˜µνΦ˜,µΦ˜,ν. (28)The most general static, spherically symmetric and asymptotically flat exact solution to thecorresponding equations of motionR˜µν = κ Φ˜,µΦ˜,ν⊓⊔ Φ˜ = 0 (29)is given by [6]A2(r) = B−2(r) =(1− 2mλr)λD2(r) =(1− 2mλr)1−λ(30)7for the metric, andΦ˜ =(1− λ22κ) 12ln(1− 2mλr)(31)for the scalar field. If we assume a positive Newton constant G (i.e. κ > 0), then0 < λ ≤ 1 (32)and we recover the standard Schwarzschild solution of General Relativity when λ goes to one.Light deflection requires nowr0 >(2 +1λ)m. (33)In the weak field approximation, Eqs. (4) and (30) implyαˆω(r0) ≃ 4GMc2r0{1 +[−2 + 1λ+(1− 116λ2)π]GMc2r0}(34)On the other hand, the Eddington-Robertson parameters of the minimal Einstein-masslessscalar theory readα = 1β = 1γ = 1δ =43(1− 14λ2)(35)such thatαˆω(ρ0) ≃ 4GMc2ρ0{1 +[−2 +(1− 116λ2)π]GMc2ρ0}. (36)Since a deviation from General Relativity only arises at the second order, the use ofthephysical impact parameter b according to Eq. (3)b ≃ r0 +(2− 1λ)m≃ ρ0 + 2m (37)8is now crucial to obtain the observable deflection angleαˆω(b) =4GMc2b{1 +(1− 116λ2)πGMc2b}+O(m3b3). (38)The same result is obtained by working directly in the improved action basis characterizedbyα = 1β =56+16λ2γ = 1δ =109− 19λ2. (39)The authors of ref. [7] only gave the coordinate-dependent Eq. (4) to analyse the quantitativemodifications of lensing characteristics in the presence of the massless scalar field Φ˜. We havejust argued that Eqs. (4) and (3) should always be carefully handled to get rid of the closestdistance of approach dependence. The deflection angle αˆ is then correctly expressed in termsof the physical impact parameter and does not depend on the arbitrary choice of coordinatesystem.5 ConclusionAny observable has to be expressed in terms of measurable quantities which are coordinate-independent. The use of the impact parameter is therefore mandatory for the deflection angleof a light ray propagating in a space-time supplied with an asymptotically flat metric. Toour knowledge, this point has been overlooked in the literature.AcknowledgementsWe would like to thank Elisa Di Pietro and Luc Haine for discussions.9References[1] See for example, S. Refsdal and J. Surdej, Rep. Prog. Theor. Phys. 56 (1994) 117.[2] C.W. Misner, K.S. Thorne and J.A. Wheeler, Gravitation (Freeman; San Francisco,1973).[3] D.K. Ross and L.I. Schiff, Phys. Rev. 141 (1966) 1215.[4] C.G. Callan, S. Coleman and R. Jackiw, Ann. Phys. 59 (1970) 42; S. Deser, Ann. Phys.59 (1970) 248.[5] S. Deser, Phys. Lett. B134 (1984) 419.[6] A.I. Janis, E.T. Newman and J. Winicour, Phys. Rev. Lett. 20 (1968) 878; M. Wyman,Phys. Rev. D24 (1981) 839; K.S. Virbhadra, Int. J. Mod. Phys. A12 (1997) 4831.[7] K.S. Virbhadra, D. Narasimha and S.M. Chitre, A & A 337 (1998) 1.10Figure 1: The light deflection angle αˆ predicted by General Relativity, as a function of the impact parameterb. The smooth curve corresponds to the exact expression based on Eq. (10) after substitution of the closestdistance of approach r0 by b according to Eqs. (3) and (8). Crosses stand for the strong field approximationgiven in Eq. (24). Diamonds and circles represent respectively the first and second order approximations inthe weak field limit defined by Eq. (20).11

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