Effect of the hair on deflection angle by asymptotically flat black holes in Einstein-Maxwell-dilaton theory

TL;DR

This paper investigates how the hair on asymptotically flat charged black holes in Einstein-Maxwell-dilaton theory affects the deflection angle of light in weak gravitational lensing, considering both vacuum and plasma medium effects.

Contribution

It provides an analytical study of the influence of black hole hair on photon deflection angles using Gauss-Bonnet theorem in Einstein-Maxwell-dilaton spacetime.

Findings

Hair significantly affects the deflection angle in weak field limits.

Plasma medium modifies the gravitational lensing effect.

Analytical expressions for deflection angles are derived.

Abstract

In this paper, we are interested in a model of exact asymptotically flat charged hairy black holes in the background of dilaton potential. We study the weak gravitational lensing in the spacetime of hairy black hole in Einstein-Maxwell theory with a non-minimally coupled dilaton and its non-trivial potential. In doing so, we use the optical geometry of the flat charged hairy black hole for some range of parameter $\gamma$. For this purpose, by using Gauss-Bonnet theorem, we obtain the deflection angle of photon in a spherically symmetric and asymptotically flat spacetime. Moreover, we also investigate the impact of plasma medium on weak gravitational lensing by asymptotically flat charged hairy black hole with a dilaton potential. Our analytically analyses show the effect of the hair on the deflection angle in weak field limits.

Equations

$${\iint }_{D}K\mathrm{d}S+{\oint }_{\partial D}\kappa \mathrm{d}t+\sum _i{\theta }_i=2\pi \chi \left(D\right),$$

$$\hat{\alpha }=-\int {\int }_{D_{\infty }}K\mathrm{d}S.$$

$$I[g_{\mu \nu },A_{\mu },\varphi ]=\frac{1}{2k}{\int }_{\mathcal{M}}{d}^{4}x\sqrt{-g}[R-e^{\gamma \phi }{F}^2-\frac{1}{2}(\partial \phi {)}^2-V(\phi )],$$

$$F^2=F_{\mu \nu }{F}^{\mu \nu },(\partial \phi {)}^2={\partial }_{\mu }\phi {\partial }^{\mu }\phi ,$$

$$R_{\mu \nu }-\frac{1}{2}{g}_{\mu \nu }R$$

$${\partial }_{\mu }(\sqrt{-g}{e}^{\gamma \phi }{F}^{\mu \nu })$$

$$\frac{1}{\sqrt{-g}}{\partial }_{\mu }(\sqrt{-g}{g}^{\mu \nu }{\partial }_{\nu }\phi )$$

$$T_{\mu \nu }^{\phi }\equiv \frac{1}{2}{\partial }_{\mu }\varphi {\partial }_{\nu }\phi -\frac{1}{2}{g}_{\mu \nu }[\frac{1}{2}(\partial \varphi {)}^2+V(\varphi )],$$

$$T_{\mu \nu }^{EM}\equiv 2e^{\gamma \phi }(F_{\mu \alpha }{F}_{\nu }^{\alpha }-\frac{1}{4}{g}_{\mu \nu }{F}^2).$$

$$V(\phi )=2\alpha (2\phi +\phi \cosh \phi -3\sinh \phi ),$$

$$d{s}^2$$

$$F$$

$$\Omega (x)=\frac{x}{{\eta }^2(x-1{)}^2},$$

$$f(x)=\alpha \left[\frac{x^2-1}{2x}-\ln (x)\right]+\frac{{\eta }^2(x-1{)}^2}{x}\left[1-\frac{2q^2(x-1)}{x}\right].$$

$$d{t}^2=\frac{{\eta }^2}{x^{2}f(x{)}^2}d{x}^2+\frac{1}{f(x)}d{\phi }^2.$$

$$\mathcal{K}=\frac{R_{icciScalar}}{2},$$

$$\mathcal{K}\approx -{\eta }^{2}x-\frac{{\eta }^2}{x}+1/4\frac{{\eta }^2}{x^2}+3/2{\eta }^2+1/4x^2{\eta }^2-1/2x\alpha \ln \left(x\right)-1/2\frac{\alpha \ln \left(x\right)}{x}+1/4\alpha x^2-1/4\frac{\alpha }{x^2}-x^2{\eta }^2{q}^2$$

$$+5{\eta }^2{q}^{2}x-11\frac{{\eta }^2{q}^2}{x^2}+16\frac{{\eta }^2{q}^2}{x}+3\frac{{\eta }^2{q}^2}{x^3}-12{\eta }^2{q}^2+5\frac{\alpha q^2}{x}-3/2\frac{\alpha q^2}{x^3}$$

$$-4\frac{\alpha q^2\ln \left(x\right)}{x^2}+3\frac{\alpha q^2\ln \left(x\right)}{x}+x\alpha \ln \left(x\right)q^2-3\alpha q^2-1/2x^2\alpha q^2+1/2x\alpha q^2-1/2\frac{\alpha q^2}{x^2}.$$

$$\int {\int }_{S_R}\mathcal{K}dS+{\oint }_{\partial S_R}kdt+\sum _j{\alpha }_j=2\pi \chi (S_R),$$

$$\int {\int }_{S_R}\mathcal{K}dS+{\oint }_{C_r}kdt{=}^{\rho \to \infty }\int {\int }_{S_{\infty }}kdS+{\int }_0^{\pi +\Theta }d\phi =\pi .$$

$$k(C_R)=|{\nabla }_{{\dot{C}}_r{\dot{C}}_r}|,$$

$$\underset{R\to \infty }{\lim }{\int }_0^{\pi +\Theta }\left[k_g\frac{d\sigma }{d\phi }\right]{|}_{C_R}d\phi =\pi -\underset{R\to \infty }{\lim }\int {\int }_{S_R}\mathcal{K}dS.$$

$$k(C_R)dt=d\phi .$$

$$\underset{R\to \infty }{\lim }{k}_g\frac{d\sigma }{d\phi }{|}_{C_R}=1.$$

$$\Theta =-\underset{R\to \infty }{\lim }{\int }_0^{\pi }{\int }_{b/\sin \phi }^R\mathcal{K}dS,$$

$$KdS\approx 1/4\frac{{\eta }^2}{x^2}+1/4\frac{\alpha }{x^2}-\frac{{\eta }^2{q}^2}{x^2}-1/2\frac{\alpha q^2}{x^2}.$$

$$\tilde{\alpha }\simeq -1/2\frac{{\eta }^2}{b}-1/2\frac{\alpha }{b}+2\frac{{\eta }^2{q}^2}{b}+\frac{\alpha q^2}{b}.$$

$$n(x)=\sqrt{1-\frac{{\omega }_e^2}{{\omega }_{\infty }^2}\left(\frac{xf(x)}{{\eta }^2(x-1{)}^2}\right)},$$

$$d{\tilde{\sigma }}^2=g_{jk}^{opt}d{x}^{j}d{x}^k=\frac{n^2(x)}{f(x)}\left(\frac{{\eta }^2}{x^{2}f(x)}d{x}^2+d{\phi }^2\right).$$

$$det{g}_{x\phi }^{opt}=\frac{xf(x){\omega }_e^4-2{\eta }^2{\omega }_{\infty }^2{\omega }_e^2(x-1{)}^2}{{\eta }^2{\omega }_{\infty }^{4}xf(x{)}^2(x-1{)}^4},$$

$$f(x)=\alpha \left[\frac{x^2-1}{2x}-\ln (x)\right]+\frac{{\eta }^2(x-1{)}^2}{x}\left[1-\frac{2q^2(x-1)}{x}\right].$$

$$\frac{d\tilde{\sigma }}{d\phi }=n(x){\left(\frac{{\alpha }^2{x}^2}{f(x)}\right)}^{1/2},$$

Full text

Relativity and gravitation; Classical black hole; Deflection angle; Gauss-Bonnet; Non-linear electrodynamics.

I Introduction

At the darkest points in the universe, their boundaries perilous and invisible, space warps. The singularity constitutes the centre of a black hole and is hidden by the object’s “surface,” the event horizon. A black hole is a location in space that possesses so much gravity, nothing can escape its pull, even light. It is said that fact is sometimes stranger than fiction, and nowhere is this more true than in the case of black holes. Since the first image of a black hole by Event Horizon Telescope Akiyama:2019cqa , physicist now try to take even sharper images so that Einstein’s theory of general relativity can be tested and also see the properties of the black holes R29 ; Konoplya:2019sns ; Bambi:2019tjh ; Shaikh:2019fpu ; Abdikamalov:2019ztb ; Abdujabbarov:2017pfw ; Abdujabbarov:2016efm ; Abdujabbarov:2016hnw ; Atamurotov:2013sca because there are many theoretically obtained black hole solutions with different properties.

According to Einstein, a black hole (BH) can be described only in terms of its mass and spin, which is known as ”no-hair” theorem. No-hair means that information about the physical state of matter must be lost as the matter is sucked into a BH, otherwise, this information would distinguish one BH from another. In 1974, Hawking made the landmark conjecture that BHs do not simply suck in everything, but rather behave as black bodies that emit radiation as well as absorbing it Hawking:1974sw . He calculated the black-body temperature of a BH known as Hawking temperature. Having a distinct temperature implies that a BH has entropy, which Hawking also calculated. Entropy is a measure of the number of different ways the microscopic constituents of a BH can arrange themselves. This goes against the no-hair theorem, which says that a BH can only be arranged in one way – as defined by its mass and spin. Recently, Hawking with his colleagues suggested that information-preserving massless particles known as “soft hair” could surround BHs. They have calculated the entropy of a BH that has a certain kind of soft hair, which leads to the Hawking’s original calculation of BH entropy H1 .

Einstein-Maxwell-dilaton (EMD) theory is an arousing and highly motivated theory to examine the affect of new essential degrees of freedom in order to prove the no-hair theorem because of the existence of hairy BH solutions in contact with the vector, scalar and tensor spectrums. It is to be observed that, in the background of asymptotically anti de Sitter (AdS) geometries, there must exist exact scalar-hairy BH solutions with respect to specific scalar potentials Sudarsky:2002mk ; Nucamendi:1995ex .

At the back of no-hair conjecture, the simplest physical visualization is: after the gravitational collapse, the matter fields left over in the exterior region would finally be immersed by the BH itself or be radiated off to infinity. Although, the hair of a BH is located at the outer side of event horizon, the question then arises that how there exist a possibility that the matter can hover in an immense gravitational field without collapsing entirely? Intuitively, the respond to the question could be that this is possible if the internal pressure is large up to a sufficient degree. On the other hand, such type of intuition does work only close to the event horizon and it must be noted that the non-linear interaction of matter fields provides the basis of the existence of hairy BHs Nunez:1996xv .

Geometric spontaneous scalarisation phenomenon has recently been examined Doneva:2017bvd ; Silva:2017uqg ; Cunha:2019dwb ; Blazquez-Salcedo:2018jnn ; Herdeiro:2018wub . It is noted that in gravitational field, matter models where a real/scalar field couples minimally to the curvature squared Gauss-Bonnet (GB) combination under particular options of the coupling function, there exists two possibilities of solutions. The standard vacuum BH solutions (bald) of general relativity (GR) and very recent ”hairy” BH solutions with a scalar field characterization, both types of solutions deceiving no-scalar hair conjecture. Moreover, it is suggested that the hairy BHs could form by means of spontaneous scalarisation, since the standard vacuum BH solutions were shown to be perturbatively unstable. Following the previous analysis, Herdeiro and Radu Herdeiro:2015waa ; Herdeiro:2019yjy have examined the basics of the spontaneous scalarisation phenomenon for BHs. They have elaborated this phenomenon within the quantum framework by giving two examples of spherically symmetric static asymptotically flat BHs in effective field theory. One of the example is that the trace anomaly occurs in the matter sector: provides a generalized form of the Reissner-Nordström BH solution incorporating $F^{4}$ correction. While, in second case it arises from the geometry sector: provides geometrically a non-commutative generalized Schwarzschild BH. In comparison, they have also discussed the scalarisation phenomenon of Einstein-Maxwell-dilaton BHs. Because of a non-minimal coupling, they have investigated the connection of the scalarisation mechanism with the quantum instabilities Mendes:2013ija .

An important fundamental predication of GR is the phenomenon of gravitational lensing, which is the deflection of light in the presence of gravitational fields of compact objects. Solender $(1801)$ proposed the gravitational lensing due to the Sun in the background of Newton’s theory, while the absolute value of deflection angle $\delta=4M_{\odot}/R_{\odot}\simeq 1.75$ arc-sec was obtained in the background of GR by Chowlson $(1929)$ and Einstein $(1936)$. In depth, the study of gravitational lensing of compact objects initiated with the learning of facts of the double quasar $Q0957+561$ having redshift $\mathscr{Z}=0.39$ (Young et al. $1980$) young ; VallsGabaud:2012xz ; Bartelmann:1999yn . They focused on the effects of the weak gravitational lensing, i.e., the lensing for which the deflection angle is very small, equal to a few arc-seconds. Generally, there exist a differentiation between lensing on non-cosmological as well as on cosmological distances. Moreover, the assumption of luminosity distances which are model-dependent leads us to the opportunity to examine the mass distribution of dark matter haloes, in particular the inner density slopes Cunha:2015yba ; Bozza:2009yw ; Bartelmann:2010fz . Schunck et al. Schunck:2006rk have investigated the deflection angle of a spherically symmetric halo built from a precisely solvable scalar field model incorporating Emden type self-interaction. They have examined the gravitational lensing impact by considering bosonic configuration and obtained the normalized projected mass as well as the corrections with respect to the pressure and analyze the weak-field dimension-less lens equation.

In this paper, we try to understand the effect of the hair on deflection angle by asymptotically flat black holes in Einstein-Maxwell-dilaton (EMD) theory, where is derived from a string theory at low energy limits C1 . The EMD black holes have a scalar hair, in addition to mass, rotation and charge so that they offers an fascinating theoretical black hole models to check experimentally in black hole experiments, and investigate possible differences between Einstein gravity and modified gravity theories C1 ; B17 ; A4 .

Further, let us now briefly review the GBT which connects the topologically surfaces. First, using Euler characteristic $\chi$ and a Riemannian metric $g$, one can choose the subset oriented surface domain as $(D,\chi,g)$ to find the Gaussian curvature $K$. Then the Gauss-Bonnet theorem is defined as follow A6

[TABLE]

where $\kappa$ is the geodesic curvature for $\partial D:\{t\}\rightarrow D$ and $\theta_{i}$ is the exterior angle with $i^{th}$ vertex. Following this approach, global symmetric lenses are considered to be Riemannian metric manifolds, which are geodesic spatial light rays. In optical geometry, we calculate the Gaussian optical curvature $K$ to find the asymptotic bending angle which can be calculated as follows A6 :

[TABLE]

Note that this equation is an exact result for the deflection angle. In this equation, we integrate over an infinite region of the surface $D_{\infty}$ which is bounded by the light ray. By assumption, one can use the above relation only for asymptotically Euclidean optical metrics. Therefore it will be interesting to see the form of the deflection angle in the case of non-asymptotically Euclidean metrics. This method has been applied in various papers for different types of spacetimes A7 ; Crisnejo:2019xtp ; R5 ; Crisnejo:2018ppm ; R6 ; Arakida:2017hrm ; Ono:2018ybw ; Jusufi:2017vta ; Ovgun:2018xys ; Jusufi:2017mav ; Ono:2017pie ; Jusufi:2017vew ; Ono:2018jrv ; Jusufi:2017lsl ; Ovgun:2019wej ; Jusufi:2017hed ; Sakalli:2017ewb ; Ovgun:2018prw ; Jusufi:2017uhh ; Kumaran:2019qqp ; Jusufi:2018jof ; Ovgun:2018fnk ; Ovgun:2018ran ; Ovgun:2018oxk ; Ovgun:2018fte ; Ovgun:2018tua ; Javed:2019qyg ; Javed:2019a ; Javed:2019b .

This paper is organized as follows: In section 2, we review some basic concepts about asymptotically flat hairy BH. In section 3, we compute the Gaussian optical curvature for deflection angle and calculate the deflection angle by using GBT for $\gamma=1$ also find the deflection angle in a plasma medium. In section 4, we calculate the deflection angle for $\gamma=\sqrt{3}$ also find the deflection angle in a plasma medium. The last section comprises of concluding remarks and results obtained from graphical analysis.

II Asymptotically flat black holes in Einstein-Maxwell-dilaton theory

In this section, we briefly review the asymptotically flat hairy BH in EMD theory in the background of dilaton potential. We consider the following action of Einstein-Maxwell-dilaton theory C1 ; B17 :

[TABLE]

where

[TABLE]

$V(\varphi)$ is the dilaton potential, and $c=G_{N}=4\pi\epsilon_{0}=1$ by using the convention $k=8\pi$. The equations of motion for gauge field, dilaton and metric are following:

[TABLE]

where the stress tensors of matter fields are defined as

[TABLE]

[TABLE]

For exact regular hairy BH solutions, for a general scalar potential there is a new method in a number of papers A4 , by using a specific ansatz. For flat spacetime, we should apply the similar special ansatz for the metric and dilaton. For the sake of simplicity, we are going to focus on two particular cases for which the exponent coefficient of the dilaton coupling with the gauge field takes the values $\gamma=1$ and $\gamma=\sqrt{3}$, but for more general solution there exist a literature B17 .

III Weak Deflection Angle of Calculation of photon lensing for $\gamma=1$ by Gauss-Bonnet Theorem

For this solution we consider the following scalar field potential:

[TABLE]

here $\alpha$ is an arbitrary parameter. For $\gamma=1$ the static hairy BH metric and gauge field, yields that:

[TABLE]

where $\eta$ and $q$ are defined as independent parameters of the solution and correspond to the mass and charge of this BH. $d\Sigma^{2}=d\theta^{2}+\sin^{2}\theta d\phi^{2}$ is spherical line element, and the coordinate $x$ is restricted to be positive, $x\in[0,\infty)$. We can suppose that $\eta>0$. One can use the conformal factor as follows:

[TABLE]

and then check that the equations of motion are satisfied for the following spacetime metric function:

[TABLE]

It is appropriate to first find the black hole optical metric by imposing the null condition $ds^{2}=0$, and solving the space-time metric for $dt$ and also set the metric into equatorial plane with $\theta=\frac{\pi}{2}$, which yield as:

[TABLE]

The optical geometry is in two dimensions, and is obtained for thermodynamically stable asymptotically flat hairy BH with a dilaton potential as follows A6 . By using Gauss-Bonnet theorem, initially we find the Gaussian curvature $\mathscr{K}$ of the optical spacetime, as

[TABLE]

[TABLE]

For multiple images, we use the global theory (Gauss-Bonnet theorem) to relate with the local feature of the space-time such that Gaussian optical curvature.

The above equation will be apply to calculate the deflection angle by taking a non singular domain $S_{R}$ outside of the light ray (along with boundaries $\partial S_{R}=\gamma_{g}\cup C_{R}$) with Euler characteristic $\chi(S_{R})$, Gaussian curvature $\mathscr{K}$, geodesic curvature $k$ and exterior jump angles $\alpha_{i}=(\alpha_{O},\alpha_{S})$ at vertices.

[TABLE]

at weak limit approximation $(\rho\rightarrow\infty)$, $\alpha_{O}+\alpha_{S}\rightarrow\pi$. Then GBT becomes

[TABLE]

Now, by geodesic property, the geodesic curvature vanishes $k(\gamma_{g})=0$, and we get

[TABLE]

with $C_{r}:=\rho(\varphi)=r=$constant. Then GBT reduces

[TABLE]

Now, for radial distance

[TABLE]

Therefore,

[TABLE]

In the weak field regions, the light ray follows a straight line approximation, so that we can use the condition of $r=b/\sin\varphi$ at zero order.

[TABLE]

where

[TABLE]

After simplification, we find the deflection angle of photon for asymptotically flat hairy BH in leading order terms as

[TABLE]

Therefore, we can say that GBT provides a globally as well as topologically effect, this method is very useful for quantitative tool and can be apply in any asymptotically flat metrics.

III.1 Photon lensing in a plasma medium

In this section, we analyze the effect of plasma medium on the photon lensing by asymptotically hairy black hole. The refractive index for hairy black hole is as follows A7 ,

[TABLE]

then, the corresponding optical metric yields that

[TABLE]

The determinant of above optical metric is:

[TABLE]

where the metric function is given as

[TABLE]

Now, we have

[TABLE]

hence we get differently which goes to $\alpha$:

[TABLE]

We use straight line approximation $r=b/\sin\varphi$, for the limit $x\rightarrow\infty$, then GBT stated as

[TABLE]

where

[TABLE]

Note that we only consider the first order terms. After simplification, we obtain the deflection angle in weak field limits as follows:

[TABLE]

The above results shows that the photon rays are moving in a medium of homogeneous plasma.

IV New Asymptotically flat black holes in Einstein-Maxwell-dilaton theory

In this section, we analyzed the exact asymptotically flat charged hairy BH’s in the background of dilaton potential. There exist a literature related to this BH recently discussed by Astefanesei et al. B17 . We are interested in a action of of Einstein-Maxwell-dilaton theory ($\kappa=8\pi G_{N}$):

[TABLE]

where the gauge coupling and potential are functions of the dilaton. The corresponding equations of motion are

[TABLE]

[TABLE]

[TABLE]

where $T_{\mu\nu}^{\phi}$ and $T_{\mu\nu}^{EM}$ are the stress tensors of the matter fields and represented as follows

[TABLE]

Now, we consider only asymptotically flat solutions. To implement this condition, we require

[TABLE]

on this account, we fix $f_{0}$ to obtain B17

[TABLE]

IV.1 Solutions with a non-trivial dilaton potential ($\gamma=1$)

For this solution we consider the following scalar field potential B17 :

[TABLE]

[TABLE]

Here we study only the $\gamma=1$ case, which is smoothly connected with a solution of $\mathscr{N}=4$ supergravity which gives the solutions of metric function as follows:

[TABLE]

[TABLE]

[TABLE]

Note that the dilaton field is vanishing at the boundary, $x=1$ and also the dilaton potential is vanishing when $\alpha=0$. At the boundary $x=1$, one can find a asymptotically flat spacetime. After make a change of coordinates using:

[TABLE]

which is given for, the $x<1$ black holes, by

[TABLE]

The spacetime metric becomes asymptotically flat B17

[TABLE]

[TABLE]

It is noted that the scalar field potential is regular everywhere, except at the spacetime singularities.

V Deflection angle of photons by asymptotically flat hairy black holes in Einstein-Maxwell-dilaton theory

Initially we find the Gaussian curvature $\mathscr{K}$ of the optical spacetime, as

[TABLE]

and

[TABLE]

In weak field limits,

[TABLE]

For multiple images, we use the global theory (Gauss-Bonnet theorem) to relate with the local feature of the space-time such that Gaussian optical curvature.

In the weak field regions, the light ray follows a straight line approximation, so that we can use the condition of $r=b/\sin\phi$ at zero order.

[TABLE]

Now by using Eq. $(18)$, the deflection angle of photon by exact asymptotically flat charged hairy black hole with a dilaton potential in weak field limit is found as:

[TABLE]

V.1 Deflection angle of photons in plasma medium by asymptotically flat hairy black holes in Einstein-Maxwell-dilaton theory

In this section, we analyze the effect of plasma medium on the photon lensing by asymptotically hairy black hole. The refractive index for hairy black hole is as follows A7 ,

[TABLE]

then, the corresponding optical metric yields that

[TABLE]

The determinant of above optical metric is:

[TABLE]

Now, by using Eq. $17$ and $29$, in weak field limit the Gaussian optical curvature stated as follows:

[TABLE]

where

[TABLE]

Now, we have

[TABLE]

hence we get differently which goes to $\alpha$:

[TABLE]

We use straight line approximation $r=b/\sin\varphi$, for the limit $x\rightarrow\infty$, then GBT stated as

[TABLE]

After simplification, we obtain

[TABLE]

The proposed deflection angle shows that the photon rays are moving in a medium of homogeneous plasma.

VI Conclusion

In this paper, we obtain the deflection angle of photon to the spherically symmetric and asymptotically flat spacetime of hairy BH with Einstein-Maxwell-dilaton system in weak field limit. To this end, we set the photon rays on the equatorial plane in the black hole spacetime. For this purpose, we have used the GBT and obtain the deflection angle of photon by integrating a domain outside the impact parameter. Moreover, we also found the deflection angle of photon by asymptotically flat hairy BH in plasma medium. We examined that the proposed deflection angle shows that gravitational lensing can be affected from the hair of the black hole and it is a global effect as well as a valuable tool to study the nature of singularities of black holes.

In our analysis, we obtain the deflection angle of photon to the spherically symmetric and asymptotically flat spacetime of hairy BH with Einstein-Maxwell-dilaton (EMD) system in the weak-field limit by using Gauss-Bonnet theorem. The computed deflection angle defined by (55) is stated as follows

[TABLE]

It is noted that, for the selection of mass term $\eta=m$, $p=2$, in the absence of charge $Q=0$, and $\alpha=0$ the above equation reduces up to the first order term of deflection angle for Schwarzschild black hole A6

[TABLE]

where $m$ is black hole mass. The significance of obtained result in weak-field approximation is that the deflection of a light ray is evaluated by taking domain outside of the lensing region, which implies that the impact of the gravitational lensing is a global effect in such a way that there are multiple light rays converging among the source and observer.

Furthermore, by considering the homogeneous plasma medium we evaluate the deflection angle of photon given by (64) for asymptotically flat hairy BH, which is given below

[TABLE]

The above equation can be expressed as

[TABLE]

For $Q=0$ and $p=2$, one can obtain the following form

[TABLE]

Due to the presence of plasma medium, the gravitational deflection angle increases, depends upon the frequency of photons. In homogeneous plasma medium, the photons having smaller frequency or greater wavelengths, and are deflected by a larger angle near the gravitating center. For $\omega\rightarrow\omega_{e}$, the influential difference in the gravitational deflection angles is substantial for longer wavelengths which is possible only for the radio waves. In this regard, the gravitational lens in plasma acts as a radio spectrometer BisnovatyiKogan:2008yg . Crisnejo and Gallo A7 have studied the dynamics of light rays in a cold non-magnetized plasma medium. For this purpose, they have obtained the deflection angle for Schwarzschild spacetime in homogenous plasma medium by using Gauss-Bonnet theorem, i.e.,

[TABLE]

In comparison with the deflection angle obtained in A7 , with the choice of parameters $\eta=m$, $p=2$, $\alpha=0$ and $Q=0$, our proposed deflection angle (67) approximates the deflection angle of Schwarzschild black hole (70). In the future, the astrophysical observations might shed light on the effect of hair on deflection angle. Any discovery of the hair would be an important signal beyond the general relativity.

Acknowledgements.

The authors are grateful to anonymous referees for their valuable comments and suggestions to improve the paper. This work is supported by Comisión Nacional de Ciencias y Tecnología of Chile (CONICYT) through FONDECYT Grant N$\mathrm{o}$ 3170035 (A. Ö.).