# Strong gravitational lensing by a charged Kiselev black hole

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## Abstract

We study the gravitational lensing scenario where the lens is a spherically symmetric charged black hole (BH) surrounded by quintessence matter. The null geodesic equations in the curved background of the black hole are derived. The resulting trajectory equation is solved analytically via perturbation and series methods for a special choice of parameters, and the distance of the closest approach to black hole is calculated. We also derive the lens equation giving the bending angle of light in the curved background. In the strong field approximation, the solution of the lens equation is also obtained for all values of the quintessence parameter \(w_q\). For all \(w_q\), we show that there are no stable closed null orbits and that corrections to the deflection angle for the Reissner–Nordström black hole when the observer and the source are at large, but finite, distances from the lens do not depend on the charge up to the inverse of the distances squared. A part of the present work, analyzed, however, with a different approach, is the extension of Younas et al. (Phys Rev D 92:084042, 2015) where the uncharged case has been treated.

## 1 Introduction

It is predicted by general relativity (GR) that in the presence of a mass distribution, light is deflected. However, it was not entirely a new prediction by Einstein, in fact, Newton had obtained a similar result by a different set of assumptions. In 1936, Einstein [1] noted that if a star (lens), the background star (source) and the observer are highly aligned then the image obtained by the deflection of light of a background star due to another star can be highly magnified. He also noted that optical telescopes at that time were not sufficiently capable to resolve the angular separation between images.

In 1963, the discovery of quasars at high redshift gave the actual observation to the gravitational lensing effects. Quasars are central compact light emitting regions which are extremely luminous. When a galaxy appears between the quasar and the observer, the resulting magnification of images would be large and hence well separated images are obtained. This effect was named macro-lensing. The first example of gravitational lensing was discovered (the quasar QSO 0957 + 561) in 1979 [2].

The weak field theory of gravitational lensing is based on the first order expansion of the smallest deflection angle. It has been developed by several authors such as Klimov [3], Liebes [4], Refsdal [5], Bourassa [6, 7, 8], and Kantowski [9]. They were succeeded in explaining astronomical observations up to now (for more details see [10]).

Due to a highly curved space-time by a black hole (BH), the weak field approximation is no longer valid. Ellis and Virbhadra obtained the lens equation by studying the strong gravitational fields [11]. They analyzed the lensing of the Schwarzchild BH with an asymptotically flat metric. They found two infinite sets of faint relativistic images with the primary and secondary images. Fritelli et al. [12] obtained exact lens equation and they compared them with the results of Virbhadra and Ellis. By using the strong field approximation, Bozza et al. [13] gave analytical expressions for the magnification and positions of the relativistic images.

From recent observational measurements, we can see that our Universe is dominated by a mysterious form of energy called “Dark Energy”. This kind of energy is responsible for the accelerated expansion of our Universe [14, 15]. Dark energy acts as a repulsive gravitational force so that usually it is modeled as an exotic fluid. One can consider a fluid with an equation of state in which the state parameter *w*(*t*) depends on the ratio of the pressure *p*(*t*) and its energy density \(\rho (t)\), such as \(w(t) = \frac{p(t)}{\rho (t)}\). So far, a wide variety of dark energy models with dynamical scalar fields have been proposed as alternative models to the cosmological constant. Such scalar field models include quintessence [16, 17, 18, 19], k-essence [20], quintom [21, 22], phantom dark energy [23] and others.

Quintessence is a candidate of dark energy which is represented by an exotic kind of scalar field that is varying with respect to the cosmic time. The solution for a spherically symmetric space-time geometry surrounded by a quintessence matter was derived by Kiselev [16]. There is little work focused on studying the Kiselev black hole (KBH). Thermodynamics and phase transition of the Reissner–Nordström BH surrounded by quintessence are given in [17, 18, 24]. The thermodynamics of the Reissner–Nordström–de Sitter black hole surrounded by quintessence has been investigated by one of us [18] and has led to the notion of two thermodynamic volumes. The properties of a charged BH surrounded by the quintessence were studied in [18, 25]. New solutions that generalize the Nariai horizon to asymptotically de Sitter-like solutions surrounded by quintessence have been determined in [18]. The detailed study of the photon trajectories around the charged BH surrounded by the quintessence is given in [26]. Recently, Younas et al. worked on the strong gravitational lensing by Schwarzschild-like BH surrounded by quintessence [27].

We will extend that work by adding a charge *Q* (charged KBH). We will consider the lensing phenomenon only for the case of non-degenerate horizons. By computing the null geodesics, we examine the behavior of light around a charged KBH. We analyze the circular orbits (photon region) for photons. Furthermore, we observe how both the quintessence and the charge parameters affects the light trajectories of massless particles (photons), when they are strongly deflected due to the charged KBH. We will not restrict the investigation to the analytically tractable cases \(w_{q}=-1/3\) and \(w_{q}=-2/3\), as some work did [25, 26, 28]; rather, we will consider the full range of the quintessence parameter \(w_q\) and we will rely partly on the work done by one of us [18].

The paper is structured as follows: in Sect. 2, we study the charged KBH geometry and we derive the basic equations for null geodesics. Additionally, in that section we write down the basic equations for null geodesics in charged Kiselev space-time along with the effective potential and the horizons. In Sect. 3, the analytical solution of the trajectory equation is obtained via the perturbation technique. Section 4 is devoted to the study of the lens equation to derive the bending angle. The strong field approximation of the lens equation is discussed as well. Finally, we provide a conclusion in Sect. 6.

Throughout this paper, we adopt the natural system of units where \(c = G = 1\) and the metric convention is \((+,-,-,-)\).

## 2 Basic equations for null geodesics in charged Kiselev space-time

*M*is the mass of the BH, \(w_{q}\) is the quintessence state parameter (having range between \(-1\le w_{q} <-1/3\)), \(\sigma \) is a positive normalization factor and

*Q*is the charge of the BH. The equation of state for the quintessence matter with isotropic negative pressure \(p_{q}\) is linear of the form

### 2.1 Horizons in charged Kiselev black hole

### 2.2 Equations of motion for a photon

*E*and

*L*are constants known as the energy and angular momentum per unit mass. Using the condition for null geodesics \(g_{\mu \nu }u^{\mu }u^{\nu }=0\), we obtain the equation of motion for photons

*b*is the impact parameter which is a perpendicular line to the ray of light converging at the observer from the center of the charged KBH. Further, photons experience a gravitational force in the presence of the gravitational field. This force can be expressed via the effective energy potential which is given by (\(\dot{r}^2+V_\text {eff}=E^2\))

*u*, \(V_\text {eff}\) reads

*n*local minima with \(n\in \mathbb {N}\). These extreme values are determined by the constraint \(dV_\text {eff}/du=0\), which reads (\(-2\le 3w_q+1<0\))

Therefore there is no stable closed orbit for the photons. If \(E^2=V_\text {eff max}\), the photons describe unstable circular orbits. If \(E^2<V_\text {eff max}\), the motion will be confined between the event horizon and the smaller root of \(V_\text {eff}=E^2\) or between the cosmological horizon and the larger root of \(V_\text {eff}=E^2\). If \(E^2>V_\text {eff max}\), the motion will be confined between the event and cosmological horizons.

In Figs. 2 and 3, the effective potential \(V_\text {eff}\), i.e. Eq. (16), is plotted to study the behavior of photons near a charged KBH for the non-extreme case where \(0<\sigma <0.17\) and \(0<Q<1\). We observe that in each curve, there are no minima. In these graphs each curve corresponds to the maximum value \(V_\text {max}\), which means that, for photons, only an unstable circular orbit exists. In these two figures, the effective potentials of Kiselev (19), Reissner–Nordström (20) and Schwartzschild (21) black holes are displayed as references. In Fig. 2 (Fig. 3), the quintessence parameter \(\sigma \) is varying (fixed) and the charge *Q* is fixed (varying). Both graphs are reciprocal to each other. We observe that by increasing the value of \(\sigma \) (*Q*), the photon has more (less) possibility to fall into the black hole.

### 2.3 The *u*–\(\phi \) trajectory equation

## 3 Solution to the trajectory equation

*c*from the reduced expression of (23) upon taking \(M=0\) and \(Q=0\):

In bending-angle problems the parameter *b* is assumed to be large to allow for series expansions in powers of 1 / *b*. Since quintessence is not supported observationally, we make the statement that \(\sigma \ll 1\), which we will make clearer in the next section (Eq. (47)).

*b*and \(r_{n}\); rather, they use loosely a common notation

*R*for

*b*and \(r_{n}\). This remains more or less justified as far as quintessence is not taken into consideration where one may write \(b \gtrsim r_{n}\). As we mentioned earlier, in bending-angle problems the parameter

*b*is assumed to be large to allow for series expansions in powers of 1 /

*b*, so in the presence of quintessence, one has to further assume \(b\sigma =\tfrac{L\sigma }{E}\ll 1\) (Eq. (47)) in order to have \(b \gtrsim r_{n}\). In the presence of quintessence, corrections in the expression of \(r_n\) are needed: if \(\sigma \ll 1\) and \(b\sigma \ll 1\) we obtain to the first order in 1 /

*b*(see Eq. (39) for further orders of approximation)

*c*is given by (31) and the coefficients (\(C_1,C_2,C_3\)) are related to the coefficients (\(B_1,B_2,B_3\)), which were first evaluated in Ref. [26], by \(C_1=B_1-\sin \phi \), \(C_2=B_2\), and \(C_3=B_3\). We have

*M*contributes to the second order while \(Q^2\) contributes to the third order of the series expansion in powers of 1 /

*b*.

## 4 Lens equation: bending angle

*r*, as depicted in Fig. 4, is given by [31]

*b*, we will employ \(u_n\) as an independent parameter around which we expand the deflection angle \(\alpha \).

*F*(

*u*) denote the function on the r.h.s. of (23),

*independent*parameters (\(u_o\ll 1,u_s\ll 1,u_n\ll 1\)) are small compared to unity but are not 0.

*independent*parameters (\(\sigma \ll 1,u_o\ll 1,u_s\ll 1,u_n\ll 1\)).

We will not assume the location of the observer to correspond to \(\phi _o=\pi /2\), as some authors did [26, 33], for this introduces a wrong term [34] in the series expansion^{1} of \(\alpha \).

## 5 Strong deflection limit

### 5.1 Case \(w_{q}=-2/3\)

The power series in the r.h.s. of (48) has been determined as follows. The series expansions of the arcsin terms in (46) in powers of (\(\sigma ,u_o,u_s,u_n\)) is straightforward; the series expansion of the first line in (46) has been done in Appendix A of Ref. [35]. In this work we show how to derive the series expansion of the first term in the second line of (46); the series expansion of the second term is obtained by mere substitution \(u_o\leftrightarrow u_s\). In all calculations the series expansions are obtained in the order given in (47); that is, we first expand with respect to \(\sigma \) to order 1, then expand with respect to (\(u_o,u_s\)) to order 2, and finally we expand with respect to \(u_n\) to order 2 too. In the final expansion (48) we have kept all the terms with order not exceeding 2.

*x*is straightforward.

### 5.2 Case for all \(-1\le w_{q}< -1/3\)

*u*in (23) are positive or zero. Since \(-1\le w_{q}< -1/3\), the new constant lies between \(0\le \gamma < 2\). Now, let us compute each term of Eq. (46) separately. For the sake of simplicity we will denote each term of (46) as follows:

#### 5.2.1 Computing \(I_{1}\)

*x*will depend on \(\gamma \) so that it is not possible to write down an explicit result for \(I_{1}\) for a general \(\gamma \). Therefore, for \(0\le \gamma \le 1\), we can write \(I_{1}\) as follows:Note that \(\displaystyle \lim _{\gamma \rightarrow 0}\Gamma ((\gamma +1)/2)/\Gamma (\gamma /2)=0\) and \(\displaystyle \lim _{\gamma \rightarrow 0}\Gamma ((\gamma +1)/2)/(\gamma \Gamma (\gamma /2))=\sqrt{\pi }/2\) are finite, so that the above expression is well defined for \(\gamma =0\). Now, for the range \(1<\gamma < 2\), the integral becomes

#### 5.2.2 Computing \(I_{2}\)

*x*and then compute the second integral by changing \(u_{o}\) by \(u_{s}\) we arrive at

#### 5.2.3 Computing \(I_{3}\)

#### 5.2.4 Computing \(\alpha \)

*x*in Eqs. (63) to (65) do converge and could be given in closed forms, however, for some values of \(\gamma \) only. For instance, for \(\gamma =3/2\) the integral in (65) is given in terms of the complete elliptic integral

*E*(

*m*) and the complete elliptic integral of the first kind

*K*(

*m*)

## 6 Conclusion

The motion of photons around black holes is one of the most studied problems in black hole physics. The behavior of light near black holes is important to study the structure of space-time near black holes. In particular, if the light returns after circling around the black hole to the observer, it cause a gravitational lens phenomenon. Light passing by the black hole will be deflected by angle which can be large or small depending on its distance from the black hole.

In present paper, we have extended our previous work for the Kiselev black hole by including the effects of the electric charge. This extra parameter enriches the structure of space-time with an additional horizon. By solving the geodesic equations, we have obtained the null geodesic structure for this black hole. Moreover, the lens equation provides the information as regards the bending angle. For a general \(w_q\), we managed to find an analytical expression for the bending angle in the strong deflection limit considering quintessence as a perturbation to the Reissner–Nordström. Since this geometry is non-asymptotically flat, one needs to be very careful to compute the bending angle since the standard approach, i.e. using the bending formula (see [29]), cannot be applied any more.

Instead of this approach, by using perturbation techniques and series expansions (assuming some physical conditions on the parameters), we directly integrate Eq. (23) for all \(w_q\) to find the bending angle. The final expression of the bending angle in the strong limit Eqs. (63) to (65) and (48) contain some corrections to the deflection angle obtained by a Reissner–Nordström black hole, which are proportional to the normalization parameter \(\sigma \), as well as corrections due to the finiteness of the distances of the source and observer to the lens.

It is instructive to compare the results of deflection in the presence of quintessence with those in the presence of phantom fields. In Ref. [35] light paths of normal and phantom Einstein–Maxwell-dilaton black holes have been investigated. It was emphasized that, in the presence of phantom fields, light rays are more deflected than in the normal case. Adopting the Bozza formalism [36], the authors of Ref. [37] have shown that the lensing properties of the phantom field black hole are quite similar to that of the electrically charged Reissner–Norström black hole, i.e., the deflection angle and angular separation increase with the phantom constant. A similar approach was adopted in [38] to study lensing by a regular phantom black hole. These authors have demonstrated that the deflection angle does not depend on the phantom field parameter in the weak field limit, whereas the strong deflection limit coefficients are slightly different form that of Schwarzschild black hole (see also [39]). In our case, \(C_{\sigma }\) (66) is positive for \(1<\gamma <2\). This means that the deflection angle is a bit larger if quintessence is present.

As future work, one can also study the lensing for other interesting configurations such as Nariai BHs, ultra cold BHs, and also for rotating black holes surrounded by quintessence matter. This type of work might be important to study highly redshifted galaxies, quasars, supermassive black holes, exoplanets and dark matter candidates, etc.

## Footnotes

- 1.
This is similar to finding a series expansion to the third order in

*x*of, say, \(\ln (1+\sin x)\). Expanding \(\ln (1+\sin x)\) as \(\sin x-\sin ^2 x/2\) produces the wrong answer: \(\ln (1+\sin x)\simeq x -x^2/2-x^3/6\). The correct step is to expand \(\ln (1+\sin x)\) by \(\ln (1+\sin x)\simeq \sin x-\frac{\sin ^2 x}{2}+\frac{\sin ^3 x}{3}\), which yields \( \ln (1+\sin x)\simeq x-\frac{x^2}{2}+\frac{x^3}{6}\).

## Notes

### Acknowledgements

SB is supported by the Comisión Nacional de Investigación Científica y Tecnológica (Becas Chile Grant No. 72150066). The authors would like to thank Azka Younas for useful discussions and her initial efforts in this work.

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