Braneworld Black Hole Gravitational Lensing

Transcription

1 Commun. Theor. Phys Vol. 67, No. 4, April, 27 Braneworld Black Hole Gravitational Lensing Jun Liang 梁钧 College of Arts and Sciences, Shannxi University of Science and Technology, Xi an 72, China Received November 2, 26; revised manuscript received January 23, 27 Abstract A class of braneworld black holes, which I called as Bronnikov Melnikov Dehen BMD black holes, are studied as gravitational lenses. I obtain the deflection angle in the strong deflection limit, and further calculate the angular positions and magnifications of relativistic images as well as the time delay between different relativistic images. I also compare the results with those obtained for Schwarzschild and two braneworld black holes, i.e., the tidal Reissner-Nordström R-N and the Casadio Fabbri Mazzacurati CFM black holes. PACS numbers: 4.7.-s, Sb,.25.-w DOI:.88/253-62/67/4/47 Key words: braneworld black hole, gravitational lensing, strong deflection limit Introduction The study of the black holes as gravitational lenses has received great attention in the last few years, mainly due to the evidence of the presence of a super-massive black hole at the center of our galaxy. [] Gravitational lensing by an ordinary star or galaxy can be analyzed in the weak field approximation. [2] This approximation is only valid for photons with large impact parameter see Fig.. While for gravitational lensing by a black hole, a full strong field treatment is needed, which is because large deflection angles are possible for photons passing close to the photon sphere see below for detail about the photon sphere. These photons could even perform one or more complete loops, in both directions of rotation, around the black hole before ultimately reaching an observer. As a consequence, besides the primary and secondary weak field images formed by photons performing no loops, a theoretically infinite number of images called relativistic images are produced on either side of lens, corresponding to successive winding numbers around the black hole see Fig. 2. In 22, Bozza [3] developed an approximate analytical method called the strong deflection limit for obtaining the deflection angle for a general class of static, spherically symmetric black hole. Before Bozza s paper was published, some works studying strong field lensing scenarios had been done. See, for example, Refs. [4 ]. He applied this method to Schwarzschild, R-N and Janis Newman Winicour black holes. Subsequently, many works of gravitational lensing by using Bozza s method have been done for different types of static spherically symmetric black holes and rotating black holes. [2 44] In addition, Bozza s method can also be used to calculate time delays between relativistic images. [45 47] On the other hand, in recent years, brane world received a considerable attention from the physical community by reason of the hierarchy problem. [48] In brane world scenario, our world is a 4-dimensional submanifold called brane embedded in a higher-dimensional spacetime called bulk, and the Standard Model particles are assumed to live on the brane, as the gravitational field is free to propagate in the bulk. [49] Gravitational collapse of matter on a brane will produce a black hole on the brane. [48] Static and spherically symmetric brane world black hole solutions can be obtained by solving the gravitational field equations on the brane. [5] It is an interesting topic to explore black holes in the braneworld framework of extra dimensions. In the past few years, several works of brane world black holes as gravitational lenses have been done. [6 8] In this paper, I am going to study the relativistic images produced by a class of brane world black holes, which I call as the Bronnikov Melnikov Dehen BMD black holes, [5] by using the strong deflection limit method. The paper is organized as follows: In Sec. 2, we obtain the deflection angle formula in the strong deflection limit in BMD metric. In Sec. 3, we will discuss some observables. Section 4 is a brief summary. 2 BMD Black Hole Gravitational Lensing in the Strong Deflection Limit BMD black hole is described by the metric [5] ds 2 = Ardt 2 Brdr 2 CrdΩ 2 2, Ar = Br = 2M r 2M r /γ, 2 2+/γ, 3 Supported by Natural Science Foundation of Education Department of Shannxi Provincial Government under Grant No. 5JK77, and Doctorial Scientific Research Starting Fund of Shannxi University of Science and Technology under Grant No. BJ2-2 c 27 Chinese Physical Society and IOP Publishing Ltd

2 48 Communications in Theoretical Physics Vol. 67 Cr = r 2. 4 Here γ N. and dω 2 2 denotes the metric of a unit twodimensional sphere. For γ =, Eq. reduces to the Schwarzschild line element. For convenience, we take the horizon radius r h = 2M as the measure of distances, then, the line element has the form ds 2 = Axdt 2 Bxdr 2 CxdΩ 2 2, 5 Ax = Bx = x /γ, 6 x 2+/γ, 7 Cx = x 2. 8 The horizon is located at x h =. Consider a photon approaching the BMD black hole from infinity see Fig.. In this paper, I shall focus on the situation in which the orbit of the photon is confined to the equatorial plane, and the lens as well as the observer is in the same orbit plane. From the null geodesics equation, the deflection angle of the photon can be given by [52] αx = Ix π, 9 x is the minimum distance of the photon to the black hole and B dx Ix = C CA /C A, x 2 A and C are the values of Ax and Cx at x = x, respectively. Fig. A photon approaches near the lens L at a minimum distance x from the source S with an impact parameter u, and is deflected by an angle α. The observer O sees an image I of the source. D OS is the distance between the observe and the source, D LS is the distance between the lens and the source, and D OL is the distance between the observe and the lens. Defining the line joining the observer and the lens as the optical axis, δ as the angular position of the source and θ as the angular position of the image. Fig. 2 The source S, The lens L, the observer O, the primary image I P, the secondary image I S, and the relative images I R are shown I only show the first with clockwise winding and the first with counterclockwise winding. When we decrease the impact parameter u and consequently x, see Fig., the deflection angle increases. When u decreases to a certain value, the deflection angle will exceed 2π, resulting in a complete loop around the black hole see Fig. 2. Decreasing u further, the photon will wind several times before eventually reaching the observer. Finally, when x = x m corresponding to an impact parameter u = u m, the deflection angle will diverge see below, and the photon will be captured. Here x m is the radius of the photon sphere which is defined as the largest root of the following photon sphere equation [3] C x Cx = A x Ax, the prime denotes the differentiation with respect to x. Solving Eq., we get x m = + 2γ. 2 Following Ref. [3], we can get the deflection angle close to the divergence as u αθ = log + u b + Ou u m. 3 m From Fig., θ sin θ = u/d OL, therefore, Eq. 3 can be finally transformed to θdol αθ = log + u b + Ou u m. 4 m Here the impact parameter u is given by [3] C u =, 5 A and its value at x m is Cm x u m = = m A m /xm, 6 γ

3 No. 4 Communications in Theoretical Physics 49 the subscript m of a quantity denotes its value at x = x m. Substituting Eq. 2 into Eq. 6, u m can be rewritten as u m = 2γ The coefficients and b are + 2γ +/2γ. 7 = + γ, 8 { b = + 2γ γ /2γ 2+γ/γ log 2 + 2γ [ /γ ] 2 } + π + b R γ The details of the calculation of the coefficients and b can be found in Appendix A. 3 Observables in the Strong Deflection Limit The lensing setup is shown in Fig.. The lens lies between the observer and the source. This case is called standard lensing. The other two are called retrolensing in which the source lies the observer and the lens or the observer lies between the source and the lens, see, for example Ref. [9]. The spacetime described by 5 is assumed asymptotically flat, and both the observer and the source are in the flat region. From Fig., the Virbhadra Ellis lens equation [5] tan δ = tan θ D LS D OS [tanα θ + tan θ] 2 can be obtained. When the source, lens and observer are highly aligned, the relativistic images are most prominent. [4] In this case, both δ and θ are small angles and α is close to a multiple of 2π. Writing α = 2nπ + α n, n is the number of loops that photon winds the lens, and then substituting it into Eq. 2, the lens equation becomes δ = θ D LS D OS α n. 2 As doing in Ref. [3], using Eqs. 4 and 2, we can obtain three observables as b S = θ θ = θ exp 2π, 22 θ = u m, D OL 23 R = µ 2π = exp, µ n 24 n=2 S is the angular separation between the outermost image and the others, we consider the simplest situation, in which only the outermost relativistic image θ is resolved as a single image, while all others are packaged at θ. and R is the ratio of flux between the first image and all the others. The details of the derivation of these formulae can be found in appendix B. Equations 22 and 24 can be easily inverted to give = 2π log R, 25 RS b = log. 26 θ Thus, if observables S, R and θ can be obtained, by comparing the experimental coefficients with those calculated according to different black hole models, we will be able to identify the nature of the black hole lens. Time delay results are also important for relativistic images. Time delays between an n -loop and an n 2 -loop relativistic image can be obtained [53] by using a similar technique of deriving the deflection angle formula in the strong deflection limit given in Appendix A as T n,n 2 = 2πn n 2 u m + 2 Bm um A m c 2 exp b 2 [ exp n 2π exp n ] π, 27 c 2 is given by Eq. A22. In Table, I show the estimates for the lens obervables and deflection angle coefficients for the central black hole of our galaxy with mass M = M and the distance from the Earth D OL = 8.33 kpc, [54] in the case of BDM braneworld black hole metric. I also show these obtained in the case of Schwarzschild black hole metric [3] and in the cases of two different braneworld black hole metrics, i.e., the tidal R-N [6 7,55] and the CFM [7,56 59] black hole metrics, for comparison. The metric describing the tidal R-N braneworld black hole is [6,55] ds 2 = x + q x 2 dt 2 x + q dx 2 x 2 x 2 dω Equation 28 resembles the R-N black hole metric, but with a positive or negative tidal charge q. The observational solar system data on the perihelion shift of mercury gives the tightest restriction q [6 62] The formulae of the impact parameter and the coefficients of the strong deflection angular for the tidal R-N braneworld black hole can be obtained from those corresponding to the R-N black hole by making the substitution q q as [3] q 2 u m = q q, 29 = x m xm 2 q 3 xm x 2 m 9 qx m + 8 q 2, 3 { 2xm q b 2 [3 x m x 2 = log m 9 qx m + 8 q 2 ] } x m 2 q 3 x 2 m x m + q π + 8[ log2 6 3] q [ log2 6 3] q 2 + O q 3, 3 x m = q 4. 32

4 4 Communications in Theoretical Physics Vol. 67 The metric describing the CFM black hole is [56 57] ds 2 = dt 2 x 3/4/x /x[ 3/4/x + 4/9η] dx2 x 2 dω When the parameter η =, Eq. 33 reduces to the Schwarzschild metric, and short distance tests of Newtonian gravity yield η 4 from solar system tests. [57 59] The formulae of the impact parameter and the coefficients of the strong deflection angular for the CFM braneworld black hole can be obtained with the method given in Appendix A as u m = 3 3 2, 34 3 =, 9 4η 35 b = log 6 π + 2 log { 5 log [ ]} log η [ tanh log2 6 ] 3 η 2 + Oη From Table, we see that, the parameter γ has significant effects on the deflection angle coefficients and the observables. With increase of γ, the angular position θ, the ratio of flux between the first image and all the others R m, the impact parameter u m and the deflection angle coefficient b decrease, but the angular separation S and the deflection angle coefficient increase for the BMD black hole. The relativistic images for the BMD black hole have smaller θ, R m and T 2,, but larger S than those for the Schwarzschild and the other two braneworld black holes with the same mass and distance. Table Estimates for the lens obervables and deflection angle coefficients for the central black hole of our galaxy with mass M = M and the distance from the Earth D OL = 8.33 kpc, in the case of Schwarzschild black hole metric [3] and the cases of three different braneworld black hole metrics, i.e., the BMD, [5] the tidal R-N [6 7,55] and the CFM [7,56 59] black hole metrics. The unit of θ S is micro-arcsecond µas, the unit of T 2, is minute min,, b, R m and u m/r S are dimensionless. Here R m = 2.5 log R is R converted to magnitudes and R S = 2 GM/c 2. BMD With parameter γ θ /µas S/µas R m /mag T 2, /min u m /R S b γ = γ = γ = γ = Schwarzschild Tidal R-N With charge q q = q = q = q = CFM With parameter η η = η = We can find that, for three braneworld black hole gravitational lenses, there are some common properties for the dependence of the deflection angle coefficients and the observables on the parameters γ, q and η. Similar to the case of the BMD black hole, with increase of the parameter q η, S and increase, and R m decreases for the tidal R-N CFM black hole. On the other hand, the different effects of the parameter on the deflection angle coefficients and observables for three braneworld black holes can also be found. Similar to the case of the BMD black hole, with increase of the parameter q, θ and u m decrease for the tidal R-N black hole, while for the CFM black hole, θ and u m are independent on the parameter η. In addition, similar to the case of the BMD black hole, with increase of the parameter η, b decreases for the CFM black hole, while for the tidal R-N black hole, with increase of the parameter q, b increases. Since the resolution of observations of VLBI is about

5 No. 4 Communications in Theoretical Physics 4 3 µas at present, [63] which is unable to tell difference among all the black holes in the Table. I hope that the measurement of θ for the BMD black hole can be made as for the Schwarzschild and the other two braneworld black holes in a not-so-far future since µas resolution is in principle attainable by VLBI project. [3] The measurement of the angular separation S for the BMD black hole can also be hopeful to be made with upcoming instruments since Millimetron mission will have an angular resolution of.3 µas and MAXIM project is expected to have about. µas angular resolution. [5] While the time delay between the second and the first images is of the order of minutes for the central black hole of our galaxy, so we have a little chance to observe such short time delay for a reasonable reference value of time exposure of hours. [53] In order to yield measurable time delays, we need consider more massive black hole than the black hole in the center of our galaxy. In Table 2, I give estimates for the time delay for supermassive black holes located at the center of several nearby galaxies [64] in the case of BMD braneworld black hole metric. Clearly, all of the black holes in Table 2 would yield measurable time delays. Table 2 Estimates for the time delay for supermassive black holes located at the center of several nearby galaxies in the case of BMD braneworld black hole metric. The unit of T 2, is hour h, and T 2, γ γ = 2, 3, 4, 5 denotes the time delay between the second and the first images in the case of BMD metric γ = 2, 3, 4, 5. Galaxy Mass/M Distance/Mpc T 2, 2/h T 2, 3/h T 2, 4/h T 2, 5/h NGC4486 M NGC NGC4374 M NGC NGC4486B M NGC Summary In summary, a class of BMD braneworld black holes are studied as gravitational lenses. I obtain the deflection angle in the strong deflection limit, and also calculate the angular positions and magnifications of relativistic images as well as the time delay between different relativistic images. I compare the results with those obtained for the Schwarzschild black hole, the tidal R-N and the CFM braneworld black holes. It is found that the parameter γ has a significant effect on the observables. With increase of γ, the angular position of the relativistic images θ and the relative magnification R m decrease, but the angular separation between the outermost image and the others S increases. The relativistic images for the BMD black hole have smaller angular position θ, relative magnification R m and the time delay T 2,, but larger angular separation S than those for the Schwarzschild and the other two braneworld black holes with the same mass and distance. The measurement of the angular position θ and the angular separation S for the black hole at the center of our galaxy can be made in principle with upcoming instruments in the case of BMD braneworld metric. However, in order to yield measurable time delays, we need to consider more massive black hole than the black hole in the center of our galaxy. Appendix A: Calculation of the Deflection Angle in the Strong Deflection Limit We define two variables [5] y = Ax, A z = y y y, A2 y = A. The integral Ix becomes Ix = Rz, x fz, x, Rz, x = 2 Bxy CxA x y C, fz, x = y [ y z + y ]C /Cx. A3 A4 A5 Here, it should be stressed that the functions without the subscript, Bx, Cx, and A x are evaluated at x = A [ y z + y ]. Rz, x is regular for all values of z and x, while fz, x is divergent for z. In order to find the order of the divergence of A3, we expand the argument of the square root in fz, x to the second order in z: fz, x fz, x =, A6 ζz + χz 2 ζ = y C A C y C A, A7 χ = y 2 2C 2 [2C C A 2 A 3 + C C 2C 2 y A C C y A ]. A8 Equation A7 clearly shows that ζ vanishes at x = x m, so fz, x /z and Ix and consequently the deflection angle αx diverges logarithmically. Then, follow-

6 42 Communications in Theoretical Physics Vol. 67 ing Ref. [5], we split A3 into two pieces Ix = I D x + I R x, A9 I D x is divergent, while I R x is regular, they are I D x = I R x = R, x m f z, x dz, gz, x dz, gz, x = Rz, x fz, x R, x m f z, x. I D x can be evaluated exactly as I D x = R, x m 2 χ log A A A2 χ + ζ + χ ζ. A3 We further expand ζ in powers of x x m, getting ζ = 2χ ma m y m x x m + Ox x m 2, A4 χ m = C m y m 2 C my m C m A m 2ymC 2 m 2. A5 Substituting Eqs. A4 and A5 into Eq. A3, we get x I D x = a log + b D + Ox x m, A6 x m a = R, x m, A7 χm b D = R, x m χm log 2 y m A mx m. A8 We can also expand gz, x in powers of x x m to get gz, x = n! x x m n n g. A9 x =x m n= x n Substituting Eq. A9 into Eq. A, and just retain the n = term, we get I R x = Expanding Eq. 5, we get the coefficient c 2 is c 2 = C my m C m A m 4 y 3 mc m = χ m gz, x m dz + Ox x m. u u m = c 2 x x 2, ym C 3 m A2 A2 C 2 m 2 y m 2. A22 Using Eqs. A2, A22, and θ u/d OL, we can finally express the deflection angle as a function of θ as θdol αθ = log + u b + Ou u m, A23 m = R, x m 2. A24 χ m b R see Eq. A2. b = π + br + log 2χ m, A25 y m is defined as b R = I R x m = gz, x m dz, A26 Appendix B: Derivation of the Formulae of Some Observables Setting αθ n = 2nπ, and solving Eq. 4 with αθ = 2nπ, one can obtain θ n = u m D OL + e n, e n = e b 2nπ/. A27 A28 Then expanding αθ around θ = θ n, and using Eq. 4, we obtain α n = D OL u m e n θ n, θ n = θ θ n. A29 A3 Substituting these equations into Eq. 2, the lens equation becomes δ = θn DOL D LS + θ n + θ n. A3 u m e n D OS The second term in the righ hand side of Eq. A3 is negligible when compared to the last term due to u m D OL. Thus, we get θ n = δ θ nu m e n D OS D LS D OL, A32 and the position of an n-loop relativistic image is given θ n = θ n + u me n δ θ nd OS D LS D OL. A33 Obviously, the second term in Eq. A33 is a small correction on θn. With similar treatment, the position of the n-th relativistic image on the opposite side of the source can be also obtained. The magnification of the n-loop relativistic image is given by µ n = sin δ/sin θ δ/ θ δ/θ δ/ θ. A34 θ n θn From Eq. A3 we have δ = + D OL D LS. θ u m e n D OS θ n A35 Substituting Eq. A35 the first term in Eq. A35 is small compared to the second and can be neglected and Eq. A27 into Eq. A34, the magnification can be finally written as u 2 µ n = e m + e n D OS n δdol 2 D. A36 LS From Eq. A36 and noticing Eq. A28, clearly, µ n decreases very quickly in n.

7 No. 4 Communications in Theoretical Physics 43 From Eq. A27 in the limit n, the minimum impact parameter u m can be expressed as u m = D OL θ, A37 θ represents the asymptotic angular position approached by a set of images. As has been pointed out in Sec. 3, we consider the simplest situation, in which only the outermost relativistic image θ is resolved as a single image, while all others are packaged together at θ. Two observalbes can be defined as S = θ θ, A38 µ R = n=2 µ. n A39 Here S is the angular separation between the outermost image and the others, and R is the ratio of flux between the first image and all the others. Using Eqs. A27, A28, and A37, S can be expressed as b S = θ exp 2π. A4 From Eq. A36, and noticing e 2π/ and e b/ is of order one, we can derive out R = µ 2π = exp. A4 µ n n=2 References [] R. Schödel, et al., Nature London [2] P. Schneider, J. Ehlers, and E. E. Falco, Gravitational Lenses, Springer-Verlag, Berlin 992. [3] V. Bozza, Phys. Rev. D [4] K. S. Virbhadra and G. F. R. Ellis, Phys. Rev. D [5] K. S. Virbhadra and G. F. R. Ellis, Phys. Rev. D [6] S. Frittelli, T. P. Kling, and E. T. Newman, Phys. Rev. D [7] C. Darwin, Proc. Roy. Soc. London A [8] J. P. Luminet, Astron. Astrophys [9] S. Chandrasekhar, The Mathematical Theory of Black Holes, Oxford University Press, Oxford 983. [] H. C. Ohanian, Am. J. Phys [] V. Bozza, et al., Gen. Rel. Grav [2] E. F. Eiroa and C. M. Sendra, Class. Quantum Grav [3] E. F. Eiroa and C. M. Sendra, Phys. Rev. D [4] E. F. Eiroa and C. M. Sendra, Phys. Rev. D [5] E. F. Eiroa, Eur. Phys. J. C [6] R. Whisker, Phys. Rev. D [7] A. S. Majumdar and N. Mukherjee, Int. J. Mod. Phys. D [8] E. F. Eiroa and C. M. Sendra, Phys. Rev. D [9] E. F. Eiroa, Phys. Rev. D [2] S. S. Zhao and Y. Xie, J. Cosmol. Astropart. Phys [2] S. Chen and J. Jing, Phys. Rev. D [22] H. Cheng and J. Man, Class. Quantum Grav [23] C. Ding, S. Kang, and C. Y. Chen, Phys. Rev. D [24] C. Ding and J. Jing, J. High Energy Phys [25] C. Ding, et al., Phys. Rev. D [26] N. Tsukamoto, et al., Phys. Rev. D [27] S. W. Wei, K. Yang, and Y. X. Liu, Eur. Phys. J. C [28] H. Sotani and U. Miyamoto, Phys. Rev. D [29] S. W. Wei, Y. X. Liu, and C. E. Fu, Adv. High Energy Phys [3] A. Bhadra, Phys. Rev. D [3] K. Sarkar and A. Bhadra, Class. Quantum Grav [32] N. Mukherjee and A. S. Majumdar, Gen. Rel. Grav [33] T. Ghosh and S. SenGupta, Phys. Rev. D [34] G. N. Gyulchev and I. Zh. Stefanov, Phys. Rev. D [35] S. Chen and J. Jing, J. Cosmol. Astropart. Phys [36] Y. Huang, S. Chen, and J. Jing, Eur. Phys. J. C [37] V. Bozza, Phys. Rev. D [38] S. E. Vazquez and E. P. Esteban, Nuovo. Cimento. B Ser [39] V. Bozza, et al., Phys. Rev. D [4] V. Bozza, Phys. Rev. D [4] S. Chen and J. Jing, Class. Quantum Grav [42] S. Chen, Y. Liu, and J. Jing, Phys. Rev. D [43] S. Chen and J. Jing, Phys. Rev. D [44] S. Wang, S. Chen, and J. Jing, J. Cosmol. Astropart. Phys [45] K. S. Virbhadra and C. R. Keeton, Phys. Rev. D [46] J. Man and H. Cheng, J. Cosmol. Astropart. Phys [47] X. Lu, F. W. Yang, and Y. Xie, Eur. Phys. J. C [48] A. Chamblin, S. W. Hawking, and H. S. Real, Phys. Rev. D

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