Gravitational lensing for a boosted Kerr black hole in the presence of plasma

We obtain the deflection angle for a boosted Kerr black hole in the weak field approximation. We also study the behavior of light in the presence of plasma by considering different distributions: singular isothermal sphere, non-singular isothermal gas sphere, and plasma in a galaxy cluster. We find that the dragging of the inertial system along with the boosted parameter $\Lambda$ affect the value of the deflection angle. As an application, we studied the magnification for both uniform and SIS distributions.


Introduction
The revolutionary detection of gravitational waves from the coalescence of two black holes showed the formation of rapidly rotating black hole boosted with linear velocity [1,2,3]. The possible observation of the electromagnetic counterpart from black hole merger could provide more information about angular and linear momentum of the black hole in such systems [4,5]. This fact indicates the importance of the inclusion of the boost parameter to Kerr spacetimes in order to study the effects of the boost velocity to the geometry (gravitational field) around a black hole. The solution of Einstein's vacuum field equations describing a boosted Kerr black hole relative to an asymptotic Lorentz frame at future null infinity was obtained in [6]. The electromagnetic structure around a boosted black hole has been studied in [4]. The author of Ref. [5] has considered the solution of Maxwell equations in the background geometry of a boosted black hole. In the present paper, we study a e-mail: bambi@fudan.edu.cn weak gravitational lensing around a boosted black hole described by the solution in [6].
In the present paper, we study weak lensing around a boosted black hole in the presence of plasma. The paper is organized as follow. In Section 2, we briefly review the optics in curved spacetime and describe the procedure to obtain the deflection angle in the weak field approximation following [23,26]. In Section 3, we present the boosted Kerr metric in both diagonal and non-diagonal cases (non-rotating and slowly rotating cases, respectively). In Subsections 4.1 and 4.2, we find the expression for the deflection angle. Finally, we study the deflection angle in the presence of plasma, both for uniform and non-uniform distributions. For the inhomogeneous case, we consider three distribution models: singular isothermal sphere (SIS), non-singular isothermal sphere (NSIS), and the case of a plasma in a galaxy cluster (PGC).
Throughout the paper we use the convention in which Greek indices run from 0 to 3, while Latin indices run from 1 to 3. Moreover, with the exception of Section 2, we use geometrized units, where c = G = 1.

Optics in a curved space-time
In this section, we review the optics in a curved spacetime developed by Synge in 1960 [7]. Let us consider a static spacetime metric describing a weak gravitational field in an asymptotically flat spacetime. The metric coefficients can be written as [23,26,68] where η αβ is the metric of the Minkowski spacetime, h αβ 1, h αβ → 0 for x α → ∞, and h αβ = h αβ .
Using this approach for the static case, the phase velocity 1 u and the 4−vector of the photon momentum p α are related by the following equation [7] In order to obtain the photon trajectories in the presence of a gravitational field, one can modify the Fermat's least action principle for the light propagation by considering a dispersive medium [7]. Then, using the Hamiltonian formalism, it is easy to show that the variational principle with the condition leads to the following system of differential equations that describes the trajectories of photons 1 The phase velocity is defined as the minimum value of [7] where u is the velocity of a fictitious particle riding on the wave front relative to a time-like world-line C (intersecting the wave) of an observer with 4-velocity V µ (see [7] for details).
with the affine parameter λ changing along the light trajectory. Note that the scalar function W (x α , p α ) has been defined by means of Eq. (4).
In the Refs. [23,26], it has been considered a static inhomogeneous plasma with a refraction index n which depends on the space location x i where ω(x i ) is the frequency of the photon that, due to gravitational redshift, depends on the space coordinates x 1 , x 2 , x 3 , e is the electron charge, m is the electron mass, ω e is the electron plasma frequency, and N (x i ) is the electron concentration in an inhomogeneous plasma [23].
According to Synge [7], for the case of a static medium in a static gravitational field, one can express the photon energy as Using Eq. (7) one can express the scalar function W (x α , p α ) in the following form where is the Planck's constant. The scalar function expressed in Eq. (10) has been used in Refs. [23,26] to find the equations of light propagation for diagonal and non-diagonal spacetimes.
In contrast with the case of a flat spacetime in vacuum, where the solution for photon's trajectory is a straight line, the presence of an arbitrary medium in curved spacetimes makes photons move along bent trajectories. However, taking into account only small deviations, it is possible to use the components of the 4-momentum of the photon moving in a straight line along the z−axis as an approximation. This components are given by (see, e.g. [23,26]) Eqs. (11) and (12) are known as the null approximation.

Equations of light propagation in a diagonal spacetime
First, we consider the spacetime with a diagonal metric tensor. In this spacetime, the components of the metric tensor g αβ vanish for α = β. Hence, after using Eq. (10), the system in (6) can be expressed as [23] dx Then, with the aid of the null approximation, the first equation in (14) reduces to In the null approximation, the 3−vector in the direction of the photon momentum is written as e i = e i = (0, 0, 1). Therefore p i can be expressed as Hence, the second equation in (14) can be expressed by Then, after using Eq. (15) and differentiating, the last expression takes the form In Ref. [23], only those components of the 3−vector that are perpendicular to the initial direction of propagation were taken into account. In this sense, the contribution to the deflection of photons is due only to the change in e 1 and e 2 . Hence, after using the null approximation e i = 0 along with the assumption of week gravitational field, Eq. (18) reduces to for i = 1, 2.
The deflection angle is determined by the change of the 3−vector e i . This means that Then, using Eq. (19), the deflection angle becomeŝ for i = 1, 2. In the last expression ω e and n are evaluated at infinity, and ω(∞) = ω [23]. In terms of the impact parameter b, Eq. (21) takes the form [23] where r = √ b 2 + z 2 .

Equations of light propagation in a non-diagonal spacetime
Now we consider a spacetime with a non-diagonal metric tensor; that is, the components of metric tensor g αβ do not vanish for α = β. Therefore, the scalar function W (x α , p α ) in Eq. (10) can be expressed as [26] W ( x α , p α ) = 1 2 g 00 p 2 0 + 2g 0l p 0 p l + g lm p l p m + Hence, the system of differential equations in (6) takes the form Then, using Eq. (15) and assuming that the gravitational field is weak, we obtain Therefore, following the procedure in Subsection 2.1, the deflection angle for a non-diagonal spacetime in the weak limit has the form

Boosted Kerr metric
The boosted Kerr metric, which describes a boosted black hole relative to an asymptotic Lorentz frame, is a solution of Einstein's vacuum field equations obtained in [6]. This solution has three parameters: mass, rotation and boost. In Kerr-Schild coordinates, the line element reads where a = J/M is the specific angular momentum of the compact object with total mass M , α = cosh γ, β = sinh γ, and γ is a constant related to the boost velocity v by the formula v = tanh γ = β/α. Note that the metric in (28) exactly reduces to the Kerr one when Λ = 1 (v = 0).
To study the deflection angle for the boosted Kerr metric in the presence of a medium, we consider both the non-rotating and the slowly rotating cases. In this sense, following the ideas in [23] and [26], we devote this section to find the form of the line element (28) in each case.

Boosted Kerr metric: non-rotating case
The non-rotating case is obtained by setting a = 0. Hence, the metric (28) reduces to In Ref. [23], Cartesian coordinates have been used to find the terms h ik . Nevertheless, before changing the coordinates, we want to write the form of the metric in Eq. (32) for small values of the velocity (v 1). In order to do so, we express 1/Λ and 1/Λ 2 in terms of v and consider a Taylor expansion up to first order. Therefore, the metric (32) takes the form Now, to transform the line element (33) into Boyer-Lindquist coordinates, we use the relation (see [69], page 15) from which one can easily obtain In the weak field limit, the approximation is done by considering 2M /r 1. In this sense, according to [23], the main idea is to express the line element in Eq. (35) as where is the flat space-time, and ds 2 is the part of the metric containing the perturbation terms h ik . Therefore, after considering the weak approximation, the line element (35) has the form Eq. (35) is the non-rotating boosted Kerr metric in the weak field approximation expressed in Boyer-Lindquist coordinates. In order to identify the components h ik , we need to express the line element in Eq. (38) in Cartesian coordinates. After following the procedure described in Appendix I, we found that h 00 and h 33 are

Boosted Kerr metric: rotating case
The spacetime describing a slowly rotating massive object was obtained in [70]. However, in this work, we use the form of the metric reported in [26]. Using geometrized units, this metric takes the form where ω LT = 2M a/r 3 = 2J/r 3 is the Lense-Thirring angular velocity of the dragging of inertial frames.
For the case of the boosted Kerr metric, the line element has the same form. Introducing the notation ω LT = 2J/r 3 , where J = J/Λ, one may obtain the "modified" metric of slowly rotating boosted velocity. Finally, the spacetime around boosted slowly rotating objects can be expressed by the following metric When v = 0, the expression in (42) reduces to that in [26].

Deflection of light for the non-rotating case
In Subsection 2.1, we discussed the procedure in [23] to obtain Eq. (22). Now, we apply this result to find the deflection angle for the boosted Kerr metric in the presence of a uniform plasma. We first consider the nonrotating case. From Eqs. (39) and (40) we have that b r Then, recalling that cos θ = z/r, r = √ b 2 + z 2 , and using Eq. (22), the deflection angle iŝ Thus, after integration, we obtain In the last expression, we took into account that the deflection angle is defined as the difference between the initial and the final ray directions; that is,α = e in − e out . Therefore, it has the opposite sign (see [8]).
From Eq. (46) we note that, at first order,α b does not depend on the velocity. This is due to the fact that the second and third integrals in Eq. (45), which contain the dependence on v, vanish. If we consider a uniform plasma (ω e constant), and the approximation 1 − n ωe ω , Eq. (46) reduces to [23] In Fig. 1 we plotα b as a function of b/M for different values of ω 2 e /ω 2 . It has the usual behavior: the deflection angle decreases when the impact parameter increases, and it becomes small when b/2M → ∞. Moreover, for bigger values of ω 2 e /ω 2 the contribution toα b is grater.
On the other hand, Fig. 2 showsα b as a function of ω 2 e /ω 2 for different values of b/2M . The plot shows thatα b has the value 4M/b when there is not plasma (ω e = 0).

Deflection angle for the slowly rotating case
Due to the presence of non-diagonal terms in the line element (42), we use the form of the deflection angle in Eq. (27). According to [26], the effect of dragging   of the inertial frame contributes toα only by means of the projection J r of the angular momentum. Hence, after the introduction of polar coordinates (b, χ) on the intersection point between the light ray and the xyplane, where χ is the angle between J r and b , we find that [26] (see Fig. 3) Since Eq. (48) depends on χ and b, the deflection angle contains two contributions: the partial derivatives Thus, Eq. (27), for both contributions, takes the form whereα bS is the deflection angle for Schwarzschild (see Eq. (46)). Therefore, considering an homogeneous plasma (constant value of ω e ), these contributions reduce tô where n was replaced by 1 − ω 2 e ω 2 . It is important to point out that Eq. (53) is only valid for ω > ω e , because waves with ω < ω e do not propagate in the plasma [23,71].  In Fig. 4, we plotα bS andα b for the slowly rotating case as a function of the impact parameter b/2M . From this figure, we can see that there is a difference between both angles. This means thatα b for a boosted Kerr black hole is greater thanα bS . This is due to the   5 showsα b as a function of ω 2 e /ω 2 . The behavior is very similar to that of Schwarzschild. It tends to 2M/b + 2J r sin χ/(b 2 Λ) when there is not plasma (ω 2 e = 0).
In Fig. 6 we plotted Eq. (53) as a function of Λ for different values of J r . We took into account the condition in which 0 < Λ ≤ 1 in order to give the values. In this figure, for different values of Λ, we see thatα b is bigger when Λ → 0. Moreover, for Λ = 1, the deflection angle reduces to the valueα bS + 2J r sin χ/nb 2 .

Models for the boosted Kerr metric with non-uniform plasma distribution
The deflection angle for a boosted Kerr metric in a non-uniform plasma was calculated in Subsection 4.2.
Eq. (55) is quite similar to that obtained in [23]. In this equation, we also find the vacuum gravitational deflectionα S1 , the correction to the gravitational deflection due to the presence of the plasmaα S2 , the refraction deflection due to the inhomogeneity of the plasmaα S3 , and its small correctionα S4 . Nevertheless, when the boosted Kerr metric is considered, three more terms appear:α B1 ,α B2 , andα B3 . These are contributions due to the dragging of the inertial frame. The former is a constant that appears in all models considered, while the others two depend on the plasma distribution.
From now on, let us suppose that the vectors J r and b are perpendicular to each other (cos χ = 0). Therefore, the contributionα χ vanishes (see Eq. (54)) and sin χ = 1. Furthermore, sinceα S4 is small, we neglect its contribution (see [23]).

Singular isothermal sphere
In this subsection, we consider the model for a singular isothermal sphere proposed in [72,73,74]. In this model, often used in lens modelling of galaxies and clusters, the density distribution has the form where σ 2 v is a one-dimensional velocity dispersion. The concentration of the plasma has the form where m p is the proton mass and κ is a non-dimensional coefficient which is related to the dark matter contribution [23]. Using Eqs. (7) and (56) the plasma frequency is Then, from Eqs. (55) and (58), and the well known property of the Γ -function [75] (see Appendix II), the contributions to the deflection angle can be found in the form Where . Hence deflection angle takes the form In Fig. 7, we plotα SIS as a function of b for different values of Λ. The figure does not show any difference for values of b/2M greater than 10. However, for values of b/2M near to 10, we see a small difference. This means thatα SIS is greater when Λ is small. For Λ = 1 (v = 0), we have the case of a slowly rotating massive object. Therefore, the parameter Λ has a small effect on the deflection angle. This tendency can be seen clearly in Fig. 8, where we plotted the behavior of the deflection angle as a function of Λ for different values of J r . Note that the boosted parameter is constrained to be in the interval 0 < Λ ≤ 1. In Fig. 9, on the other hand, we plotα SIS as a function of J r for different values of Λ. From this figure we conclude that, not only the dragging of the inertial system, but also the boosted parameter Λ contribute to the deflection angle: the greater the values of J r (plus small values of Λ) the greater the value of the deflection angleα SIS .

Non-singular isothermal gas sphere
Now we consider a gravitational lens model for an isothermal sphere. For this model, the singularity at the origin is replaced by a finite core and the density distribution is given in [76] where r c is the core radius. Therefore, after substitution of Eq. (61) in Eqs. (57) and (58), the plasma frequency is expressed as Then, from Eqs. (55) and (58), the contributions to the deflection angle are (see Appendix II) where ω 2 c =  In Fig. 10 we plotα N SIS as a function of b for different values of Λ. In the plot, we have b r c because we are in the weak field limit. According to the figure, the behavior is quite similar to that of the deflection angle in the case of a singular plasma distribution: there are small differences inα N SIS when small values of Λ are considered, and no there is no difference in the deflection angle when the impact parameter b takes values greater than 10. Fig. 11 helps to see this behavior clearly. Note that the order of magnitude for NSIS is smaller than SIS when we compare with Fig. 8.
In Fig. 12 we plot the deflection angle as a function of J r for different values of Λ. Once again, the dragging of the inertial system along with small values of the boosted parameter Λ play an important role when compared with the slowly rotating case [26].

Plasma in a galaxy cluster
In a galaxy cluster, due to the large temperature of the electrons, the distribution of electrons may be homogeneous. Therefore, it is proper to suppose a singular isothermal sphere as a model for the distribution of the gravitating matter. Using this approximation, and without considering the mass of the plasma, Bisnovatyi-Kogan and O. Yu. Tsupko solved the equation of hydrostatic equilibrium of a plasma in a gravitational field finding that the plasma density distribution has the  form [23].
and the plasma frequency is equal to Hence, using Eqs. (55) and (58) once again, the contributions to the deflection angle are (see Appendix II) where In Figs. 13, 14, and 15 we plotα P GC as a function of b, Λ, and J r , respectively. In order to obtain these plots we considered the case s << 1 [23]. According to Figs. 13 and 14, differences in the deflection angle can be seen clearly for the PGC distribution when compared with the previous distributions. Furthermore, Fig. 15 shows that the deflection angle increases due to the dragging and small values of Λ.
On the other hand, in Fig. 16, we plotted the behavior of the deflection angle for all distributions as a function of the impact parameter b. Note that the values ofα for the PGC distribution are grater than the  Finally, in Fig. 17 we plottedα as a function of ω 2 c /ω 2 (for SIS and NSIS) and ω 2 f /ω 2 (for PGC). This figure clearly show that the deflection angle is more affected by the plasma for the PCG distribution than the other two for values of ω 2 f /ω 2 greater than 0.4.

Conclusion
In this work we have studied the deflection angle for the boosted Kerr metric in the presence of both homogeneous and non-homogeneous plasma, and in the latter case three different distributions have been considered.
In Subsection 4.1 we investigated the behavior of the deflection angle for the non-rotating case in the pres-ence of uniform plasma (ω e = costant) by considering small values of v. According to Eq. (46) we found that α b does not dependent, at least at first order, on the velocity v. It was also found that, after the approximation 1 − n ωe ω , the deflection angle in Eq. (45) reduces to that obtained in [23] (see Eq. (46)). As a consequence, the optics for the non-rotating boosted Kerr metric is the same as Schwarzschild. In this sense, the bending of light, due to the presence of a uniform plasma, is greater than the Schwarzschild case in vacuum for values of ω 2 e /ω 2 smaller than unity.
In Subsection 4.2, we studied the rotating case by considering a uniform distribution. Following the ideas of [26], we found that the expression for the deflection angleα b in Eq. (53) contains two terms: the Schwarzschild angleα bS , and the contribution due to the dragging of the inertial frameα bD . The result is quite similar to that of V.S Morozova et al.. However, in contrast with their result, Eq. (53) also depends on the parameter Λ. This dependence is shown in Fig. 6. Form this figure we found that the smaller the values of Λ (constrained to the interval 0 < Λ ≤ 1) the greater is the deflection angle. In this sense, not only the dragging and the presence of a plasma, but also the motion of the black hole will contribute to the lensing. Therefore, since no effect was found in the previous case, we may concluded that α b depends on v only when the dragging of the inertial frame takes place.
In Section 5, we consider the deflection angle in terms of b, Λ, and J r for different distributions. As shown in our figures,α is affected by the presence of plasma and is greater when compared with vacuum and uniform distributions. Furthermore, we found again thatα increases not only due to the dragging, but also when small values of the boosted parameter Λ are considered.
In this work, we also found some important constraints for two of the models. In the case of NSIS, for example, the radius of the core r c must have values greater than 6M . If the core radius is smaller than this limit the deflection angle becomes negative at some point and will not agree with the usual behavior when b → ∞. On the other hand, regarding the PGC, we found that s must be different from −1 or −3 as can be seen from Eq. (69). Nevertheless, this condition is fulfilled since we consider positive values of s << 1.
Finally, no important difference between the models was found. In the case of SIS and NSIS, for example, the behavior was very similar. Therefore, under the weak field approximation, it is not possible to distinguish these two distributions. In this sense, we believe that no important result will be obtained from the study of magnification if we consider them. Nevertheless, the deflection angle is affected considerably when we consider a plasma in a galaxy cluster. The values of the deflection angle are greater than those obtained with the other two models. This behavior is clearly shown in Fig. 16. Furthermore, according to Fig. 17, we found that the deflection angle affected by the plasma when the PGC distribution is considered.

Appendix I: Transformation to cartesian coordinates
The transformation relations for the non rotating case (a = 0) are (see [69]) t = t x = r sin θ cos φ y = r sin θ sin φ z = r cos θ.