General Relativistic Aberration Equation and Measurable Angle of Light Ray in Kerr--de Sitter Spacetime
aa r X i v : . [ g r- q c ] O c t General Relativistic Aberration Equation and Measurable Angleof Light Ray in Kerr–de Sitter Spacetime
Hideyoshi ARAKIDA ∗ College of Engineering, Nihon University,Koriyama, Fukushima 963-8642 JAPAN (Dated: October 23, 2018)
Abstract
As an extension of our previous paper, instead of the total deflection angle α , we will mainlyfocus on discussing the measurable angle of the light ray ψ at the position of the observer inKerr–de Sitter spacetime which includes the cosmological constant Λ. We will investigate thecontributions of the radial and transverse motions of the observer which are connected with theradial velocity v r and transverse velocity bv φ ( b is the impact parameter) as well as the spinparameter a of the central object which induces the gravitomagnetic field or frame dragging andthe cosmological constant Λ. The general relativistic aberration equation is employed to take intoaccount the influence of the motion of the observer on the measurable angle ψ . The measurableangle ψ derived in this paper can be applied to the observer placed within the curved and finite-distance region in the spacetime. The equation of the light trajectory will be obtained in sucha way that the background is de Sitter spacetime instead of Minkowski spacetime. If we assumethat the lens object is the typical galaxy, the static terms O (Λ bm, Λ ba ) are basically comparablewith the second order deflection term O ( m ), and they are almost one order smaller that of theKerr deflection − ma/b . The velocity-dependent terms O (Λ bmv r , Λ bav r ) for radial motion and O (Λ b mv φ , Λ b av φ ) for transverse motion are at most two orders of magnitude smaller than thesecond order deflection O ( m ). We also find that even when the radial and transverse velocitieshave the same sign, their asymptotic behavior as φ approaches 0 is differs, and each diverges tothe opposite infinity. PACS numbers: 95.30.Sf, 98.62.Sb, 98.80.Es, 04.20.-q, 04.20.Cv ∗ E-mail:[email protected] . INTRODUCTION The cosmological constant problem is an old but unsolved issue in astrophysics andcosmology that is closely related to the general theory of relativity; see reviews by, e.g.,[1, 2]. After the establishment of the general theory of relativity in 1915–1916, Einsteinincorporated the cosmological term Λ g µν into the field equation in order to represent thestatic Universe. Although the discovery of cosmic expansion by Hubble caused Einstein towithdraw the cosmological term from the field equation, nowadays it is widely consideredthat the cosmological constant Λ, or the dark energy in a more general sense, is the mostpromising candidate for explaining the observed accelerating expansion of the Universe [3–5]despite the fact that its details are not at all clear. One straightforward way to tackle thisproblem from another viewpoint is to investigate the effect of the cosmological constant Λon the bending of a light ray. In fact, the bending of a light ray is the basis of gravitationallensing which is a powerful tool used in astrophysics and cosmology; see, e.g., [6, 7] and thereferences therein.The influence of the cosmological constant Λ on light deflection, especially on the totaldeflection angle α , had been the subject of a long debate and was investigated mainlyunder the static and spherically symmetric vacuum solution, namely the Schwarzschild–deSitter/Kottler solution. Islam [8] first showed that the trajectory of a light ray is not relatedto the cosmological constant Λ because the second-order differential equation of the light raydoes not depend on Λ. On the basis of the result obtained by Islam, it was thought for a longtime that the cosmological constant Λ does not affect the bending of a light ray. However, in2007, a significant indication was provided by Rindler and Ishak [9] who pointed out that thecosmological constant Λ does contribute to the bending angle of a light ray in terms of theinvariant cosine formula. The important point of [9] is that though the trajectory equationof the light ray admittedly does not depend on the cosmological constant Λ, the angleshould be determined via the metric tensor g µν : because Schwarzschild–de Sitter spacetimeis not asymptotically flat, the metric g µν plays an important role in determining the angleas well as the length [42]. Inspired by this paper, many authors intensively discussed itsappearance in diverse ways; see [10] for a review article, and also see, e.g., [11–24] and thereferences therein. Moreover, several authors discussed light deflection in Kerr–de Sitterspacetime which is the stationary and axially symmetric vacuum solution and includes the2pin parameter a of the central object as well as the cosmological constant Λ; see, e.g.,[25–28] and the references therein. The same consideration is further extended to the moregeneral Kerr-type solutions, see, e.g., [29–36].However, in spite of the intensive discussions and various approaches, a definitive conclu-sion has not yet emerged. One of the main reasons for this is that because the Schwarzschild–de Sitter and Kerr–de Sitter solutions are not asymptotically flat unlike the Schwarzschildand Kerr solutions, it is ambiguous and unclear how the total deflection angle α should bedefined in curved spacetime. Although to overcome this difficulty a method for calculatingthe total deflection angle is independently investigated and proposed on the basis of theGauss–Bonnet theorem by [23, 24], it seems that further consideration is needed to settlethe argument. Because of the difficulty of defining the total deflection angle α in curvedspacetime, our argument is described in Appendix A of [37].As described briefly above, the concept and definition of the total deflection angle ofthe light ray α is a counterintuitive and difficult problem; however it is always possible todetermine the measurable angle ψ at the position of the observer P which can be describedas the intersection angle between the tangent vector k µ of the light ray Γ k that we investigateand the tangent vector w µ of the radial null geodesic Γ w connecting the center O and theposition of observer P . See FIG. 1 in section III. We investigated in [37] the measurableangle of the light ray ψ at the position of the observer in the Kerr spacetime on the basis ofthe general relativistic aberration equation [20] (and see also [38, 39]) which enables us tocompute the effect of the motion of the observer more easily and straightforwardly becausethe equations of the null geodesic of Γ k and Γ w do not depend on the motion of the observer;the velocity effect is incorporated in the formula as the form of the 4-velocity of the observer u µ .In present paper, we will extend our discussion of the measurable angle ψ at the position ofobserver P to Kerr–de Sitter spacetime containing the cosmological constant Λ as well as thespin parameter of the central object a . Our purpose is to examine not only the contributionof the cosmological constant Λ and the spin parameter a of the central object but also theeffect of the motion of the observer on the measurable angle ψ . As in our previous paper,the 4-velocity of the observer u µ is converted to the coordinate radial velocity v r = dr/dt and coordinate transverse velocity bv φ = bdφ/dt ( b is the impact parameter and v φ = dφ/dt denotes the coordinate angular velocity), respectively.3his paper is organized as follows: in section II, the trajectory of a light ray in Kerr–deSitter spacetime is derived from the first-order differential equation of the null geodesic.In section III, the general relativistic aberration equation is introduced and in section IVthe measurable angle ψ in Kerr–de Sitter spacetime is calculated for the cases of the staticobserver, the observer in radial motion and the observer in transverse motion. Finally,section V is devoted to presenting the conclusions. II. LIGHT TRAJECTORY IN KERR–DE SITTER SPACETIME
The Kerr–de Sitter solution — see Eqs. of (5.65) and (5.66) in [40], and also, e.g., [25–28]— in Boyer–Lindquist type coordinates ( t, r, θ, φ ) can be rearranged as ds = g µν dx µ dx ν = − ∆ r − ∆ θ a sin θρ Ξ dt + ρ ∆ r dr + ρ ∆ θ dθ − a sin θρ Ξ (cid:2) ∆ θ ( r + a ) − ∆ r (cid:3) dtdφ + sin θρ Ξ (cid:2) ∆ θ ( r + a ) − ∆ r a sin θ (cid:3) dφ , (1)where ∆ r = r + a − mr − Λ3 r ( r + a ) , (2)∆ θ = 1 + Λ3 a cos θ, (3) ρ = r + a cos θ, (4)Ξ = 1 + Λ3 a . (5) g µν is the metric tensor; Greek indices, e.g., µ, ν , run from 0 to 3; Λ is the cosmologicalconstant; m is the mass of the central object; a ≡ J/m is a spin parameter of the centralobject ( J is the angular momentum of the central object) and we use the geometrical unit c = G = 1 throughout this paper.For the sake of brevity, we restrict the trajectory of the light ray to the equatorial plane θ = π/ , dθ = 0, and rewrite Eq (1) in symbolic form as ds = − A ( r ) dt + B ( r ) dr + 2 C ( r ) dtdφ + D ( r ) dφ , (6)4here A ( r ) , B ( r ) , C ( r ), and D ( r ) are A ( r ) = (cid:18) a (cid:19) − (cid:20) − mr − Λ3 ( r + a ) (cid:21) , (7) B ( r ) = (cid:20)(cid:18) − Λ3 r (cid:19) (cid:18) a r (cid:19) − mr (cid:21) − , (8) C ( r ) = − (cid:18) a (cid:19) − a (cid:20) mr + Λ3 ( r + a ) (cid:21) , (9) D ( r ) = (cid:18) a (cid:19) − (cid:20) ( r + a ) (cid:18) a (cid:19) + 2 ma r (cid:21) . (10)Two constants of motion of the light ray, the energy E and the angular momentum L , aregiven by E = A ( r ) dtdλ − C ( r ) dφdλ , (11) L = C ( r ) dtdλ + D ( r ) dφdλ , (12)where λ is the affine parameter. Solving for dt/dλ and dφ/dλ , we obtain two relations: dtdλ = ED ( r ) + LC ( r ) A ( r ) D ( r ) + C ( r ) , (13) dφdλ = LA ( r ) − EC ( r ) A ( r ) D ( r ) + C ( r ) . (14)From the null condition ds = 0 and Eqs. (6), (13), and (14), the geodesic equation of thelight ray can be expressed as (cid:18) drdφ (cid:19) = A ( r ) D ( r ) + C ( r ) B ( r )[ bA ( r ) − C ( r )] [ − b A ( r ) + 2 bC ( r ) + D ( r )] , (15)where b is the impact parameter defined as b ≡ LE . (16)Using Eqs. (7), (8), (9), (10), and (15), the first-order differential equation of the light raybecomes (cid:18) drdφ (cid:19) = (cid:20)(cid:18) − Λ3 r (cid:19) ( r + a ) − mr (cid:21) r (cid:18) a (cid:19) (cid:20) b − mr ( b − a ) − Λ3 ( b − a )( r + a ) (cid:21) × (cid:20) r + a − b + 2 mr ( b − a ) + Λ3 ( b − a ) ( r + a ) (cid:21) . (17)5utting u = 1 /r , Eq. (17) is rewritten as [43] (cid:18) dudφ (cid:19) = (cid:20)(cid:18) − Λ3 u (cid:19) (1 + a u ) − mu (cid:21) (cid:18) a (cid:19) (cid:20) b − mu ( b − a ) − Λ3 u ( b − a )(1 + a u ) (cid:21) × (cid:20) a − b ) u + 2 mu ( b − a ) + Λ3 ( b − a ) (1 + a u ) (cid:21) . (18)Expanding Eq. (18) up to the order O ( m , a , am, m Λ , a Λ , Λ ), we have (cid:18) dudφ (cid:19) = 1 b + Λ3 − u + 2 mu − a u + 3 a u b − amub + 2Λ a b u + O ( ε ) . (19)Note that for the sake of simplicity, we introduced the notation for the small expansionparameters m , a and Λ as ε ∼ m ∼ a ∼ Λ , ε ≪ , (20)then O ( ε ) in Eq. (19) denotes combinations of these three parameters. Henceforth we usethe same notation to represent the order of the approximation and residual terms.It is instructive to discuss how to choose a zeroth-order solution u of the light trajectory u . If m = 0 and a = 0, then Eq. (19) reduces to the null geodesic equation in de Sitterspacetime (cid:18) dudφ (cid:19) = 1 b + Λ3 − u , (21)which can be also derived immediately from Eqs. (17) and (18). Because we assume anonzero cosmological constant Λ a priori , we cannot take Λ to be zero; in fact the action S = Z (cid:20) c πG ( R − L M (cid:21) √− gd x, (22)and the field equation R µν − g µν R + Λ g µν = 8 πGc T µν , (23)include the cosmological constant Λ explicitly where g = det( g µν ), L M denotes the La-grangian for the matter field; R µν and R are the Ricci tensor and Ricci scalar, respectively;and T µν is the energy-momentum tensor. Because m and a are the integration or arbitraryconstants in the Kerr–de Sitter solution, it is possible to put m = 0 and a = 0. Therefore6q. (21) cannot be reduced to the null geodesic equation in Minkowski spacetime and thezeroth-order solution of the u of the light ray should be taken as the form u = sin φB , B ≡ b + Λ3 (24)instead of u = sin φ/b . Note that Eq. (24) should be evaluated in de Sitter spacetime ds = − (cid:18) − Λ3 r (cid:19) dt + (cid:18) − Λ3 r (cid:19) − dr + r dφ , (25)instead of in the Minkowski spacetime. The choice of the zeroth-order solution is key toobtaining the measurable angle ψ ; see section IV A below.Let us obtain the equation of the light trajectory in accordance with the standard per-turbation scheme. We take the solution u = u ( φ ) of the light trajectory as u = sin φB + δu + δu , (26)where δu and δu are the first order O ( ε ) and second order O ( ε ) corrections to the zeroth-order solution u = sin φ/B , respectively. Inserting Eq. (26) into Eq. (19), then expanding,integrating and collecting the same order terms, the equation of the light trajectory inKerr–de Sitter spacetime is given up to second-order O ( ε ) by1 r = sin φB + m B (3 + cos 2 φ )+ 116 B b (cid:8) b (cid:0) m + 2 a (cid:1) (3 sin φ − sin 3 φ ) + 4 b (cid:2) b m + 6 a ( B − b ) (cid:3) sin φ + (cid:0) b m − a b + 12 B a b (cid:1) ( π − φ ) cos φ − B am (cid:9) − Λ B a (cid:0) φ − (cid:1) b sin φ + O ( ε ) , (27)where the integration constants of δu and δu are chosen so as to maximize u (or to minimize r ) at φ = π/ dδu dφ (cid:12)(cid:12)(cid:12)(cid:12) φ = π/ = 0 , dδu dφ (cid:12)(cid:12)(cid:12)(cid:12) φ = π/ = 0 . (28)Eq. (27) contains two constants B and b , and leads to a complicated expression for themeasurable angle ψ . To avoid the complex expression for ψ , we expand B in Λ and express B by b , obtaining1 r = (cid:18) b + b Λ6 − b Λ (cid:19) sin φ + m φ ) (cid:18) b + Λ3 (cid:19) + 116 b (cid:8) m [37 sin φ + 30( π − φ ) cos φ − φ ] + 8 a sin φ − am (cid:9) − Λ a (cid:0) φ − (cid:1) φ + O ( ε ) . (29)7ote that if we use the approximate solution of Eq. (19) as [44] u = sin φb + δu + δu , (30)the following terms in Eq. (29) disappear: (cid:18) b Λ6 − b Λ (cid:19) sin φ, m Λ6 (3 + cos 2 φ ) . (31)The existence of the above terms in Eq. (29) reflects the fact that the background spacetimeis de Sitter spacetime instead of Minkowski spacetime.Before concluding this section, it is noteworthy that unlike Schwarzschild–de Sitter space-time, the trajectory equation of the light ray in Kerr–de Sitter spacetime depends on thecosmological constant Λ; from the condition dudφ (cid:12)(cid:12)(cid:12)(cid:12) u = u = 0 , u = 1 r , (32)and Eq. (19), we have following relation:1 B = 1 b + Λ3 = 1 r − mr + 2 a r − a b r + 4 amb r − ar b + O ( ε ) , (33)in which r is the radial coordinate value of the light ray at the point of closest approach(in our case φ = π/ r can be obtained by the observation in principle as thecircumference radius ℓ = 2 πr . It is also possible to obtain a similar relation from Eqs.(18) and (19); but the expression becomes more complicated. Eq. (33) means that unlikethe Schwarzschild–de Sitter case, B cannot be expressed only by r , m , and a ; Λ and b tooare required. As a result, the trajectory equation of the light ray in Kerr–de Sitter spacetimedepends on the cosmological constant Λ and b ; whereas the equation of the light trajectoryin Schwarzschild–de Sitter spacetime is independent of Λ and b ; in fact setting a = 0 in Eqs.(17) and (19) yields (cid:18) dudφ (cid:19) = 1 b + Λ3 − u + 2 mu , (34)thus using u = 1 /r , we obtain 1 B = 1 b + Λ3 = 1 r − mr . (35)Eq. (35) shows that the constant B SdS in Schwarzschild–de Sitter spacetime can be deter-mined without knowing Λ and b . 8 II. GENERAL RELATIVISTIC ABERRATION EQUATION
The general relativistic aberration equation is given by [20]; also see [38, 39]:cos ψ = g µν k µ w ν ( g µν u µ k ν )( g µν u µ w ν ) + 1 , (36)where k µ is the 4-momentum of the light ray Γ k which we investigate, w µ is the 4-momentumof the radial null geodesic Γ w connecting the center O and the position of observer P , u µ = dx µ /dτ is the 4-velocity of the observer ( τ is the proper time of the observer), and ψ is the angle between the two vectors k µ and w µ at the position of observer P . The detailsof the derivation of Eq. (36) are described in section V B of [20]. Because Eq. (36) includesthe 4-velocity of the observer u µ , it enables us to calculate the influence of the motion ofthe observer on the measurable angle ψ . See FIG. 1 for the schematic diagram of the light FIG. 1: Schematic diagram of light trajectory. Γ k (bold line) is the trajectory of the light ray weinvestigate, Γ w (dotted line) is the radial null geodesic connecting the center O and the positionof observer P and the measurable angle ψ is the intersection angle between Γ k and Γ w at P . Twobold vectors u r and u φ at P indicate the directions of the r (radial) and φ (transverse) componentsof the 4-velocity u µ . The direction of time component u t is perpendicular to this schematic plane. trajectory.Eq. (36) can be written as the tangent formula (see Eq. (22) in [37]) which gives thesame approximate solution of the measurable angle ψ as with Eq. (36) but requires tediousand lengthy calculations. Hence, we will present the results obtained from Eq. (36).9 V. MEASURABLE ANGLE IN KERR–DE SITTER SPACETIME
Henceforth, in accordance with the procedure described in our previous paper [37], wecalculate the measurable angle ψ at the position of observer P . From now on, only importantequations are summarized; see section IV in [37] for the details of the derivation.Because we are working in the equatorial plane θ = π/ , dθ = 0 as the orbital plane ofthe light ray, the components of the tangent vectors k µ and w µ are k µ = ( k t , k r , , k φ ) , (37) w µ = ( w t , w r , , . (38)As k µ and w µ are null vectors, from the null condition g µν k µ k ν = 0 and g µν w µ w ν = 0, k t and w t are expressed in terms of k r , k φ and w r , respectively as k t = C ( r ) k φ + p [ C ( r ) k φ ] + A ( r )[ B ( r )( k r ) + D ( r )( k φ ) ] A ( r ) , (39) w t = s B ( r ) A ( r ) w r , (40)where we chose the sign of k t and w t to be positive. The inner product g µν k µ w ν is given interms of A ( r ), B ( r ), C ( r ), and D ( r ) by g µν k µ w ν = ( − s B ( r ) A ( r ) A ( r ) D ( r ) + C ( r ) bA ( r ) − C ( r )+ p B ( r )[ A ( r ) D ( r ) + C ( r )][ − b A ( r ) + 2 bC ( r ) + D ( r )] bA ( r ) − C ( r ) ) k φ w r . (41) A. Measurable Angle by Static Observer
In the case of the static observer, the component of the 4-velocity of the observer u µ becomes u µ = ( u t , , , , (42)and the condition for the time-like observer g µν u µ u ν = − u t as u t = 1 p A ( r ) , (43)10here we take u t to be positive. The inner products g µν u µ k ν and g µν u µ w ν become g µν u µ k ν = − p A ( r ) A ( r ) D ( r ) + C ( r ) bA ( r ) − C ( r ) k φ , (44) g µν u µ w ν = − p B ( r ) w r , (45)Inserting Eqs. (41), (44), and (45) into Eq. (36), we havecos ψ static = s A ( r )[ − b A ( r ) + 2 bC ( r ) + D ( r )] A ( r ) D ( r ) + C ( r ) . (46)Further, substituting Eqs. (7), (8), (9), (10) and (29) into Eq. (46), and expanding up tothe order O ( ε ), ψ static for the range 0 ≤ ψ ≤ π/ ψ static = φ + 2 mb cos φ + 18 b (cid:8) m [15( π − φ ) − sin 2 φ ] − ma cos φ (cid:9) − Λ b φ + Λ b φ (cid:2) m (1 + csc φ ) + 2 a csc φ (cid:3) − Λ b
288 csc φ sin 4 φ + O ( ε ) . (47)and for the range π/ ≤ φ ≤ πψ static = π − φ − mb cos φ − b (cid:8) m [15( π − φ ) − sin 2 φ ] − ma cos φ (cid:9) + Λ b φ − Λ b φ (cid:2) m (1 + csc φ ) + 2 a csc φ (cid:3) + Λ b
288 csc φ sin 4 φ + O ( ε ) . (48)As in [37], we divided the expression for ψ static into two cases, Eqs. (48) and (47). Thepurpose of this was to utilize trigonometric identities such as p − sin φ = cos φ for 0 ≤ φ ≤ π/ p − sin φ = − cos φ for π/ ≤ φ ≤ π . Henceforth we adopt a similarprocedure when calculating angle measured by the observer in radial motion, ψ radial andin transverse motion ψ transverse below. Although this procedure may not be necessary forcomputing ψ static and ψ radial , it is required when computing ψ transverse ; see Eq. (76) andobserve the case for φ → π .The first and second lines in Eqs. (47) and (48) are in agreement with the measurableangle of the static observer in Kerr spacetime derived in [37], and the third lines in Eqs.(47) and (48) are due to the influence of the cosmological constant Λ.11ere, let us estimate how the cosmological constant Λ contributes to the measurableangle of the light ray. We assume that the observer is located within the range 0 ≤ φ ≤ π/ M gal , radius R gal , and angularmomentum J gal ; the impact parameter b is comparable with R gal (see TABLE I). Because TABLE I: Numerical values. We use the following numerical values in this paper. As the value ofthe total angular momentum J gal , we adopt that of our Galaxy J gal ≃ . × kg m /s from [41].Name Symbol ValueMass of the Galaxy M gal = 10 M ⊙ . × kg m = GM gal /c . × mImpact Parameter b = R gal = 26 kly 2 . × mAngular Momentum of the Galaxy [41] J = J gal . × kg m /sSpin Parameter a = J/ ( M gal c ) 1 . × mCosmological Constant Λ 10 − m − Hubble Constant H = c p Λ / . × − s − Distance from Lens Object D = 1 . . × mRecession Velocity v H = H D . × m / sRadial Velocity v r = v H /c . bv φ = v H /c . the Kerr contributions appearing in Eqs. (47) and (48) are examined in our previous paper[37], we extract the terms concerning the cosmological constant Λ from Eq. (47) and put ψ static ( φ ; Λ , b ) = − Λ b φ − Λ b
288 csc φ sin 4 φ, (49) ψ static ( φ : m, a, Λ , b ) = Λ b φ (cid:2) m (1 + csc φ ) + 2 a csc φ (cid:3) , (50)To compare the contribution of the cosmological constant Λ with the result of the Kerrcase, we compute the following terms of the total deflection angle α in Kerr spacetime (seee.g., [37]) using the values summarized in TABLE I:4 mb ≈ . × − rad , (51)15 πm b ≈ . × − rad , (52) − mab ≈ ∓ . × − rad for ± a. (53)12IGs. 2 and 3 show the φ dependences of Eqs. (49) and (50), respectively. From -2-1.5-1-0.5 0 ψ s t a t i c ( ϕ ; Λ , b ) [ r ad ] ϕ Static Observer in Kerr-de Sitter Spacetime
FIG. 2: φ dependence of Eq. (49). FIG. 2, Eq. (49) is a monotonic function of φ and increases rapidly, diverging to negativeinfinity as φ approaches 0. This property is due to the existence of the de Sitter horizonat r dS ≈ p / Λ. The order O (Λ b, Λ b ) terms in Eq. (49) take a negative value whichdiminishes the measurable angle ψ .Eq. (50) contains the order O (Λ bm, Λ ba ) terms and its magnitude is O (10 − ) which isalmost comparable to the second order deflection angle, Eq. (52). In accordance with thesign of the spin parameter a , Eq. (50) takes a different sign; for a >
0, Eq. (50) is positiveand vice versa. However, regardless of the sign of the spin parameter a , Eq. (50) divergesto positive infinity as φ approaches 0.Before concluding this section, we note that if Eq. (30) (see also Eq. (31)) is adoptedinstead of Eqs. (26) and (29), the measurable angle ˜ ψ static for the range 0 ≤ φ ≤ π/ -10 -10 -10 -10 -10 -9 ψ s t a t i c ( ϕ ; m , a , Λ , b ) [ r ad ] ϕ Static Observer in Kerr-de Sitter Spacetime a > 0 a < 0 FIG. 3: φ dependence of Eq. (50). becomes ˜ ψ static = φ + 2 mb cos φ + 18 b (cid:8) m [15( π − φ ) − sin 2 φ ] − ma cos φ (cid:9) − Λ b φ sec φ + Λ b m (cot φ csc φ − sec φ ) + 2 a cot φ ] − Λ b φ csc φ + O ( ε ) , (54)where the first and second lines are also in agreement with the measurable angle of the staticobserver in the Kerr spacetime as derived in [37].Comparing Eqs. (47) and (54), we find the following: first, in spite of the differentcorrection terms due to the cosmological constant Λ, the measurable angles ψ static and ˜ ψ static take a large value rapidly and diverge to negative infinity when φ approaches 0. This propertyis related to the existence of the de Sitter horizon. Second, when φ → π/
2, Eq. (47) leads tothe result ψ static → π/
2, which is consistent with the initial condition in our case, Eq. (28).Note that at φ = π/
2, the two null geodesics k µ and w µ are orthogonal. However, Eq. (54)diverges, and ˜ ψ static → ∞ , which contradicts the initial condition Eq. (28). Therefore, Eq.(24) should be employed as the zeroth-order solution of u ; as a consequence Eq. (27) or at14east Eq. (29) should be used as the trajectory equation of the light ray when investigatinglight bending in Kerr–de Sitter spacetime. The same holds for Schwarzschild–de Sitterspacetime. Until now, how a zeroth-order solution u is to be chosen was not consideredcarefully, the above indication is one of the important suggestions in this paper. B. Measurable Angle by Observer in Radial Motion
The component of the 4-velocity u µ of the radially moving observer is u µ = ( u t , u r , , , (55)and from the condition g µν u µ u ν = − u t can be expressed in terms of u r as u t = s B ( r )( u r ) + 1 A ( r ) , (56)where u t is taken to be positive. The inner products g µν u µ k ν and g µν u µ w ν are given by g µν u µ k ν = ( − s B ( r )( u r ) + 1 A ( r ) A ( r ) D ( r ) + C ( r ) bA ( r ) − C ( r )+ u r p B ( r )[ A ( r ) D ( r ) + C ( r )][ − b A ( r ) + 2 bC ( r ) + D ( r )] bA ( r ) − C ( r ) ) k φ , (57) g µν u µ w ν = n − p B ( r ) [ B ( r )( u r ) + 1] + B ( r ) u r o w r . (58)Here, instead of u r , we introduce the coordinate radial velocity v r as v r = drdt = dr/dτdt/dτ = u r u t , (59)and substituting Eq. (56) into Eq. (59), we find, u r = v r p A ( r ) − B ( r )( v r ) . (60)Using Eq. (60), we rewrite Eqs. (57) and (58) in terms of v r g µν u µ k ν = ( − A ( r ) D ( r ) + C ( r )[ bA ( r ) − C ( r )] p A ( r ) − B ( r )( v r ) + v r p B ( r )[ A ( r ) D ( r ) + C ( r )][ − b A ( r ) + 2 bC ( r ) + D ( r )][ bA ( r ) − C ( r )] p A ( r ) − B ( r )( v r ) ) k φ , (61) g µν u µ w ν = − p A ( r ) B ( r ) − B ( r ) v r p A ( r ) − B ( r )( v r ) w r . (62)15ere we impose the slow motion approximation for the radial velocity of the observer, v r ≪
1. Next, following the same procedure used to obtain Eqs. (47) and (48), we insertEqs. (7), (8), (9), (10), (41), (61) and (62) into Eq. (36), obtaining ψ radial for the range0 ≤ φ ≤ π/ O ( ε , ε v r ) ψ radial = φ + v r sin φ + 2 mb (cos φ + v r )+ 18 b (cid:0) m [15( π − φ ) − sin 2 φ ] − am cos φ (cid:1) + v r b (cid:8) m [30( π − φ ) cos φ + 95 sin φ − sin 3 φ ] − am (1 + cos 2 φ ) − a (3 sin φ − sin 3 φ ) (cid:9) − Λ b φ − Λ b v r
12 (cos 2 φ −
3) + Λ b φ (cid:2) m (1 + csc φ ) + 2 a csc φ (cid:3) + Λ bv r φ [ a + m csc φ + ( a − m csc φ ) cos 2 φ ] − Λ b
288 csc φ sin 4 φ − Λ b v r
144 (cos 2 φ −
7) cot φ csc φ + O ( ε , ( v r ) ) , (63)and for the range π/ ≤ φ ≤ πψ radial = π − φ + v r sin φ + 2 mb ( − cos φ + v r ) − b (cid:0) m [15( π − φ ) − sin 2 φ ] − am cos φ (cid:1) + v r b (cid:8) m [30( π − φ ) cos φ + 95 sin φ − sin 3 φ ] − am (1 + cos 2 φ ) − a (3 sin φ − sin 3 φ ) (cid:9) + Λ b φ − Λ b v r
12 (cos 2 φ − − Λ b φ (cid:2) m (1 + csc φ ) + 2 a csc φ (cid:3) + Λ bv r φ [ a + m csc φ + ( a − m csc φ ) cos 2 φ ]+ Λ b
288 csc φ sin 4 φ − Λ b v r
144 (cos 2 φ −
7) cot φ csc φ + O ( ε , ( v r ) ) . (64)The first four lines in Eqs. (63) and (64) coincide with the measurable angle in Kerr space-time which has already been investigated in [37], and the remaining terms, lines 5 to 7, arethe correction due to the cosmological constant Λ.Now we concentrate on investigating the influence of the cosmological constant Λ and theradial velocity v r . Then we extract the order O (Λ b v r , Λ b v r ) and O (Λ bmv r , Λ bav r ) terms16rom Eq. (63) except the contributions of Eqs. (49) and (50) and put ψ radial ( φ ; Λ , b, v r ) = − Λ b v r
12 (cos 2 φ − − Λ b v r
144 (cos 2 φ −
7) cot φ csc φ, (65) ψ radial ( φ ; m, a, Λ , b, v r ) = Λ bv r φ [ a + m csc φ + ( a − m csc φ ) cos 2 φ ] . (66)Because the background spacetime of Kerr–de Sitter is de Sitter spacetime, we assume thatradial velocity obeys Hubble’s law: v r ≈ v H = H D, H = c r Λ3 , (67)where D is the distance between the lens (central) object O and observer P , and we take D ≈ ≃ . × m which is the typical distance scale of the galaxy lensing. -0.01-0.005 0 0.005 0.01 ψ r ad i a l ( ϕ ; Λ , b , v r ) [ r ad ] ϕ Radial Motion in Kerr-de Sitter Spacetime v r > 0 v r < 0 FIG. 4: φ dependence of Eq. (65). FIGs. 4 and 5 illustrate the φ dependence of Eqs. (65) and (66), respectively. Eq. (65)diverges to positive or negative infinity when φ approaches 0 depending on the sign of thevelocity v r . On the other hand, when φ → π/
2, Eq. (65) approaches Λ b v r / v r >
0, Eq. (66) becomes negative infinity when φ approaches0, while for the negative velocity v r <
0, Eq. (66) diverges to positive infinity. Theseproperties are independent of the sign of the spin parameter a . Next, when φ → π/ -11 -5x10 -12 -12 -11 ψ r ad i a l ( ϕ ; m , a , Λ , b , v r ) [ r ad ] ϕ Radial Motion in Kerr-de Sitter Spacetime a > 0, v r > 0 a < 0, v r > 0 a > 0, v r < 0 a < 0, v r < 0-1x10 -12 -12 π/4 π/2 FIG. 5: φ dependence of Eq. (66). Eq. (66) converges to Λ bmv r which depends on the radial velocity v r but is independentof the spin parameter a . The magnitude of the velocity-dependent part, Eq. (66), is atmost O (10 − ) which is two orders of magnitude smaller then the second order contribution O ( m ) in Eq. (52). C. Measurable Angle by Observer in Transverse Motion
As the third case, let us investigate the observer in transverse motion which is the motionin a direction perpendicular to the radial direction in the orbital plane. The component ofthe 4-velocity of the observer u µ is u µ = ( u t , , , u φ ) , (68)and the condition g µν u µ u ν = − u t = C ( r ) u φ + p [ C ( r ) u φ ] + A ( r )[ D ( r )( u φ ) + 1] A ( r ) , (69)18n which we chose the sign of u t to be positive. g µν u µ k ν and g µν u µ w ν are computed as g µν u µ k ν = A ( r ) D ( r ) + C ( r ) A ( r ) ( u φ − p [ C ( r ) u φ ] + A ( r )[ D ( r )(( u φ ) + 1)] bA ( r ) − C ( r ) ) k φ , (70) g µν u µ w ν = − s B ( r ) A ( r ) { [ C ( r ) u φ ] + A ( r )[ D ( r )( u φ ) + 1] } w r . (71)In the same way as was done for Eqs. (61) and (62), we rewrite Eqs. (70) and (71) in termsof the coordinate angular velocity v φ which is determined by v φ = dφdt = dφ/dτdt/dτ = u φ u t , (72)and using Eq. (69), u φ is obtained by means of v φ as, u φ = v φ p A ( r ) − C ( r ) v φ − D ( r )( v φ ) . (73)Inserting Eq. (73) into Eqs. (70) and (71), g µν u µ k ν and g µν u µ w ν are rewritten as g µν u µ k ν = [ A ( r ) D ( r ) + C ( r )]( − bv φ )[ bA ( r ) − C ( r )] p A ( r ) − C ( r ) v φ − D ( r )( v φ ) k φ , (74) g µν u µ w ν = − s B ( r ) A ( r ) A ( r ) − C ( r ) v φ p A ( r ) − C ( r ) v φ − D ( r )( v φ ) w r . (75)Because v φ = dφ/dt is the coordinate angular velocity, we regard bv φ as the coordinatetransverse velocity which allows us to employ the slow motion approximation bv φ ≪
1. Aswas done when deriving Eqs. (63) and (63), we substitute Eqs. (7), (8), (9), (10), (41), (74)and (75) into Eq. (36), and expand up to the order O ( ε , ε bv φ ), obtaining ψ transverse for19 ≤ φ ≤ π/ ψ transverse = φ + bv φ tan φ mb cos φ (cid:18) bv φ φ (cid:19) + 18 b (cid:8) m [15( π − φ ) − sin 2 φ ] − am cos φ (cid:9) + bv φ b (1 + cos φ ) ( m (cid:20) π − φ ) −
16 sin φ + 7 sin 2 φ + 16 tan φ (cid:21) +8 ma (1 − φ − cos 2 φ ) ) − Λ b φ − Λ b v φ cot φ φ ) + Λ b φ (cid:2) m (1 + csc φ ) + 2 a csc φ (cid:3) + Λ b v φ [ m (1 − φ + cos 2 φ ) − a (sin φ + sin 2 φ )]6 b (cos φ − φ ) − Λ b
288 csc φ sin 4 φ − Λ b v φ φ (1 − φ + 3 cos 2 φ ) csc φ φ O ( ε , ( bv φ ) ) , (76)and for π/ ≤ φ ≤ π : ψ transverse = π − φ + bv φ cot φ − mb cos φ (cid:18) bv φ − cos φ (cid:19) − b { m [15( π − φ ) − sin 2 φ ] − am cos φ } + bv φ b (1 − cos φ ) ( m (cid:20) − π − φ ) −
16 sin φ − φ + 16 cot φ (cid:21) +8 am (1 + 2 cos φ − cos 2 φ ) ) + Λ b φ + Λ b v φ cot φ − cos φ ) − Λ b φ (cid:2) m (1 + csc φ ) + 2 a csc φ (cid:3) − Λ b v φ [ m (1 + 4 cos φ + cos 2 φ ) − a (sin φ − sin 2 φ )]6 b (cos φ − (1 + cos φ )+ Λ b
288 csc φ sin 4 φ + Λ b v φ φ + 3 cos 2 φ ) csc φ φ O ( ε , ( bv φ ) ) . (77)As is the case of Eqs. (63) and (64), the first four lines in Eqs. (76) and (77) are equivalentto the measurable angle in Kerr spacetime obtained in [37], and the remaining terms, lines5 to 7, are the additional terms due to the cosmological constant Λ.Even here, we pay attention to the contribution of the cosmological constant Λ and take20he parts of the cosmological constant Λ and the transverse velocity bv φ : ψ transverse ( φ ; Λ , b, bv φ ) = − Λ b v φ cot φ φ ) − Λ b v φ φ (1 − φ + 3 cos 2 φ ) csc φ φ , (78) ψ transverse ( φ ; m, a, Λ , b, bv φ ) = Λ b v φ [ m (1 − φ + cos 2 φ ) − a (sin φ + sin 2 φ )]6 b (cos φ − φ ) , (79)and we assume that the transverse velocity is comparable with the recessional velocity v H bv φ ≈ v H , (80)see also TABLE I.FIGs. 6 and 7 show the φ dependences of Eqs. (78) and (79), respectively. Eq. (78)consists of the order O (Λ b v φ , Λ b v φ ) terms, however unlike Eq. (65), for the positivetransverse velocity bv φ >
0, Eq. (78) diverges to negative infinity when φ approaches 0 andvice versa. When φ → π/
2, Eq. (78) converges to 0 regardless of the sign of the transversevelocity bv φ .From FIG. 7, we find that regardless of the sign of the spin parameter a , Eq. (79)becomes positive infinity for the positive transverse velocity bv φ > bv φ < φ approaches 0. When φ → π/
2, Eq.(79) converges to Λ b av φ / a and the transversevelocity bv φ unlike Eq. (66). As in Eq. (66), the magnitude of Eq. (79) is at most O (10 − )and it is two orders of magnitude of smaller than the second order contribution O ( m ) inEq. (52). D. Comparison of Static, Radial and Transverse Cases
Here, we summarize the asymptotic behavior, φ → φ → π/
2, of Eqs. (49), (50)(65), (66), (78), and (79) and their sign within the range 0 < φ < π/ φ →
0, Eq. (65) diverges to positive infinity whereas Eq.(78) diverges to negative infinity for the positive velocities v r > bv φ >
0. The samesituation can be observed in the case of Eqs. (66) and (79).When φ → π/
2, Eqs. (49), (50), and (79) converge to 0 whereas the results of Eq. (65)depend on the radial velocity v r . Despite the fact that Eq. (66) includes the spin parameter21 ψ t r an sv e r s e ( ϕ ; Λ , b , b v ϕ ) [ r ad ] ϕ Transverse Motion in Kerr-de Sitter Spacetime v ϕ > 0 v ϕ < 0 FIG. 6: φ dependence of Eq. (78). -1x10 -11 -5x10 -12 -12 -11 ψ t r an sv e r s e ( ϕ ; m , a , Λ , b , b v ϕ ) [ r ad ] ϕ Transverse Motion in Kerr-de Sitter Spacetime a > 0, v ϕ > 0 a < 0, v ϕ > 0 a > 0, v ϕ < 0 a < 0, v ϕ < 0 FIG. 7: φ dependence of Eq. (79). ABLE II: Asymptotic behavior of Eqs. (49), (50), (65), (66), (78), and (79).Motion of Observer Eq. Number φ → < φ < π/ φ → π/ −∞ Negative 0Eq. (50) ∞ Positive for a > a < ∞ for v r > v r > b v r / −∞ for v r < v r < −∞ for v r > a > bmv r Mostly Negative for a < ∞ for v r < a > a < −∞ for v φ > v φ < ∞ for v φ < v φ < ∞ for v φ > a > b av φ / a < −∞ for v φ < a > a < a and radial velocity v r , it depends only on v r and is independent of a . The results of Eq.(79) depend on both the transverse velocity bv φ and the spin parameter a .Within the range 0 < φ < π/
2, the sign of Eqs. (65), (66), (78), and (79) depends on thesign of the velocity v r , bv φ and the spin parameter a ; e.g., for the positive velocity v r > bv φ >
0, these equations have a (mostly) positive value for a > a < V. CONCLUSIONS
In this paper, instead of the total deflection angle α we mainly focused on discussingthe measurable angle of the light ray ψ at the position of the observer in Kerr–de Sitterspacetime which includes the cosmological constant Λ. We investigated the contributionsof the radial and transverse motions of the observer which are related to the radial velocity23 r and the transverse velocity bv φ as well as the influence of the gravitomagnetic field orframe dragging described by the spin parameter a of the central object and the cosmologicalconstant Λ.The general relativistic aberration equation was employed to incorporate the effect of themotion of the observer on the measurable angle ψ . The expressions for the measurable angle ψ derived in this paper apply to the observer placed within the curved and finite-distanceregion in the spacetime.To obtain the measurable angle ψ , the equation of the light trajectory was obtained insuch a way that the background is de Sitter spacetime instead of Minkowski spacetime.At the end of section IV A, we showed that the choice of the zeroth-order solution u isimportant and a zeroth-order solution u in Kerr–de Sitter and Schwarzschild–de Sitterspacetimes should be chosen in such a way that the background is de Sitter spacetime, Eq.(24). Further, Eq. (27) or at least Eq. (29) should be used as the trajectory equation of thelight ray.We find that even when the radial and transverse velocities have the same sign, theirasymptotic behavior when φ approaches 0 is differs, and each diverges to the oppositeinfinity.If we assume that the lens object is the typical galaxy, the static terms O (Λ bm, Λ ba )in Eq. (50) are basically comparable with the second order deflection term O ( m ) but al-most one order smaller than the Kerr deflection − ma/b . The velocity-dependent terms O (Λ bmv r , Λ bav r ) in Eq. (66) for radial motion and O (Λ b mv φ , Λ b av φ ) in Eq. (79) fortransverse motion are at most two orders of magnitude smaller than the second order de-flection O ( m ). Therefore, if the second order deflection term O ( m ) becomes detectableby gravitational lensing, it may be possible to detect the cosmological constant Λ from thestatic terms in Eq. (50). [1] S. Weinberg, Rev. Mod. Phys., , 1-23 (1989).[2] S. Carroll, Living Rev. Relativity , , 1 (2001).[3] A. G. Riess, A. V. Filippenko, P. Challis, A. Clocchiatti, A. Diercks, P. M. Garnavich, R. L.Gilliland, C. J. Hogan, S. Jha, R. P. Kirshner, B. Leibundgut, M. M. Phillips, D. Reiss, B. P. chmidt, R. A. Schommer, R. C. Smith, J. Spyromilio, C. Stubbs, N. B. Suntzeff, J. Tonry,Astron. J., , 1009-1038 (1998).[4] B. P. Schmidt, N. B. Suntzeff, M. M. Phillips, R. A. Schommer, A. Clocchiatti, R. P. Kirshner,P. Garnavich, P. Challis, B. Leibundgut, J. Spyromilio, A. G. Riess, A. V. Filippenko, M.Hamuy, R. C. Smith, C. Hogan, C. Stubbs, A. Diercks, D. Reiss, R. Gilliland, J. Tonry, J.Maza, A. Dressler, J. Walsh, R. Ciardullo, Astrophys. J., , 46-63 (1998).[5] S. Perlmutter, G. Aldering, G. Goldhaber, R. A. Knop, P. Nugent, P. G. Castro, S. Deustua, S.Fabbro, A. Goobar, D. E. Groom, I. M. Hook, A. G. Kim, M. Y. Kim, J. C. Lee, N. J. Nunes,R. Pain, C. R. Pennypacker, R. Quimby, C. Lidman, R. S. Ellis, M. Irwin, R. G. McMahon,P. Ruiz-Lapuente, N. Walton, B. Schaefer, B. J. Boyle, A. V. Filippenko, T. Matheson, A. S.Fruchter, N. Panagia, H. J. M. Newberg, W. J. Couch, Astrophys. J., , 565-586 (1999).[6] P. Schneider, J. Ehlers, E. E. Falco, Gravitational Lenses , (Springer Verlag, Berlin, Heidelberg,New York, 1999).[7] P. Schneider, C. Kochanek, J. Wambsganss,
Gravitational Lensing: Strong, Weak and Micro ,(Springer, Berlin, Heidelberg, New York, 2006).[8] J. N. Islam, Phys. Lett. A, , 239-241 (1983).[9] W. Rindler, M. Ishak, Phys. Rev. D, , id. 043006 (2007).[10] M. Ishak, W. Rindler, Gen. Rel. Grav., , 2247 (2010).[11] K. Lake, Phys. Rev. D, , id. 087301 (2002).[12] M. Park, Phys. Rev. D, , id. 023014 (2008).[13] I. B. Khriplovich, A. A. Pomeransky, Int. J. Mod. Phys. D, , 2255 (2008).[14] F. Simpson, J. A. Peacock, A. F. Heavens, On lensing by a cosmological constant, MNRAS, , 2009 (2010).[15] A. Bhadra, S. Biswas, K. Sarkar, Phys. Rev. D, , id. 063003 (2010).[16] H. Miraghaei, M. Nouri-Zonoz, Gen. Rel. Grav. , 2947 (2010).[17] T. Biressa, J. A. de Freitas Pacheco, Gen. Rel. Grav., , 2649 (2011).[18] H. Arakida, M. Kasai, Phys. Rev. D, , id. 023006 (2012).[19] F. Hammad, F. Mod. Phys. Lett. A, , 1350181 (2013).[20] D. Lebedev, K. Lake, arXiv:1308.4931 (2013).[21] D. Batic, S. Nelson, M. Nowakowski, Phys. Rev. D, , id 104015 (2015).[22] H. Arakida, Universe, , 5 (2016).
23] A. Ishihara, Y. Suzuki, T. Ono, T. Kitamura, H. Asada, Phys. Rev. D, , id.084015 (2016).[24] H. Arakida, Gen. Rel. Grav., , id. 48 (2018).[25] G. V. Kraniotis, Class. Quant. Grav., , 4391-4424 (2005).[26] G. V. Kraniotis, Class. Quant. Grav., , id. 085021 (2011).[27] J. Sultana, Phys. Rev. D, , id. 042003 (2013).[28] D. Charbul´ak, Z. Stuchl´ık, Eur. Phys. J. C, , id.897 (2017).[29] L. J. Goicoechea, E. Mediavilla, J. Buitrago, F. Atrio, Mon. Not. R. Astron. Soc. , 281-292(1992).[30] S. V. Iyer, E. C. Hansen, Phys. Rev. D, , id 124023 (2009).[31] G. V. Kraniotis, Gen. Rel. Gravit., , id.1818 (44 pages) (2014).[32] G. He, W. Lin, Class. and Quant. Grav., , id. 095007 (2016).[33] G. He, W. Lin, Class. and Quant. Grav., , id. 029401 (2017).[34] G. He, W. Lin, Class. and Quant. Grav., , id. 105006 (2017).[35] C. Jiang, W. Lin, Phys. Rev. D, , id.024045 (2018).[36] R. Uniyal, H. Nandan, P Jetzer, Phys. Lett. B, , 185-192 (2018).[37] H. Arakida, submitted to PRD, arXiv:1808.03418 (2018).[38] K. R. Pechenick, C. Ftaclas, J. M. Cohen, ApJ, , 846-857 (1983).[39] D. Lebedev, K. Lake, arXiv:1609.05183 (2016).[40] B. Carter, in Black holes (Les astres occlus) , 57-214 (1973).[41] I. Karachentsev,
DOUBLE GALAXIES , (Izdatel’stvo Nauka, Moscow, 1987),https://ned.ipac.caltech.edu/level5/Sept02/Keel/frames.html Translated by William Keeland Nigel Sharp.[42] If one obtains the deflection angle using the equation of the light trajectory only , it meansthat the angle is evaluated in flat spacetime because only in flat spacetime, do r and φ havethe meaning of the length and angle, respectively. On the other hand, in curved spacetime, r and φ are just “the coordinate values” and then the angle and length should be determinedby the metric.[43] We mention that the term (cid:0) Λ3 a (cid:1) in the denominator is missing in Eq. (4) of [27] however,their result is not affected by this missing term because they expanded Eq. (4) up to the secondorder in m , a , and Λ, whereas Λ a corresponds to the third order.[44] This solution is used by [27].corresponds to the third order.[44] This solution is used by [27].