1. Introduction
In general relativity (GR), black holes (BHs) were initially predicted to be formed by the gravitational collapse of compact massive bodies, usually referred to as solutions to the field equations of GR. However, the modern observations, such as recent gravitational wave detectors [1, 2] and the BlackHoleCam and EHT collaborations [3, 4], have confirmed their existence in the Universe, opening attractive avenues for future research to examine important insights remarkable properties of BHs in GR. Observations have also predicted that BHs can be considered rotating black holes [5–10]. Hence, BHs can at least possess mass M and spin a. However, BHs can also have an electric charge that was proposed by various mechanisms, such as the balance of gravitational and Coulomb forces [11, 12] and the irradiating photons [13], which may provide a positive net electric charge at the near surface of BHs. Additionally, magnetic field lines can lead to a induced charge under the Wald mechanism due to the frame dragging effect [14]. Then the exact solution of the rotating Schwinger dyon was obtained in the form of a charged BH with mass M and electric and magnetic charges, together with spin a [15, 16]. Furthermore, solutions of charged BHs without physical singularity were proposed as regular BH solutions with nonlinear electrodynamics (NED) and extended to various contexts [17–27]. It is worth noting that GR was successfully tested in addressing the low energy limit of string theory with the dilaton scalar field presented by the Einstein–Hilbert action as a combination of the axion, gauge and dilaton fields [28, 29]. Later, the heterotic string theory with the scalar dilaton field associated with the electromagnetic field tensor was also proposed in [30]. This also led to an increase in activity where various authors investigated the causal structures and thermodynamic properties of BH solutions with dilaton field in various scenarios [29–34]. Additionally, there are investigations that suggested a means of obtaining BH solutions within the string theory and extended theories [35–39]. Later on, the quantum aspects have also been considered in BH solutions, and their effects on the background geometry have been examined in the context of extended theories of gravity [40–44].
The EHT collaboration [3, 4] first reported a BH's image/shadow recently. With the recent discovery of BH images/shadows, many researchers have been focusing on theoretical modeling of BH shadows. It is envisaged by the fact that the light can be highly bent and even travels in circles near the BH, as no light escapes from the BH, resulting in the BH appearing as a dark disk, usually referred to as the BH's shadow. The point to be noted is that the first study of the light bending around the Schwarzschild BH was conducted by Synge [45], and later its image was simulated by Luminet [46]. In this context, the study of the light bending and the shadow analysis allow one to test the spacetime geometry with its accretion disk. To this end, Amarilla et al first conducted the shadow analysis for a Kaluza–Klein rotating dilaton BH [47]. Following this analysis, it was then extended to the Einstein–Maxwell–Dilaton–Axion BH [48]. The BH shadow was also examined around the rotating charged BH including a scalar dilaton field [49]. An investigation also exists showing that the BH shadow can be significantly influenced by the impact of the deformation parameters [50]. Also, the BH shadow analysis can be applied for scalar boson and Proca stars, showing the geometry of such alternative compact objects that may exhibit departures from BH geometries [51–53]. Other theoretical modeling of BH shadows have also been proposed in recent years, and thus it has sparked increased research activity, with various authors investigating BH shadows in different scenarios [54–69].
It must also be emphasized that, together with BH shadow analysis, the gravitational lensing plays a decisive role in probing unknown aspects of astrophysical compact objects. It is envisaged that the light can be highly deflected, thus changing its path around a BH. This deflection reflects the gravitational lensing in GR. This is because GR was widely examined first through the gravitational lensing effects [70]. Therefore, gravitational lensing can allow one to test the background geometry at the BH's close vicinity and provide information regarding distant sources and compact objects. Additionally, it may provide the possibility to exhibit departures in the geometry and structure of astrophysical compact objects. This also led to an increase in activity, where various authors have been extending the gravitational lensing effects to examine theories of gravity in various situations [see, e.g. [71–80]]. Additionally, an extensive analysis has been implemented on these lines to study gravitational lensing effects in the weak-field regime in the presence of a plasma medium [81–87] and in the context of modified theories of gravity [88–90].
It is believed that the plasma medium can exist in the environment surrounding a BH in the astrophysical scenario. As stated, BHs are very intriguing gravitational and geometric objects that showcase regions where spatial curvature dominates the description of gravity. Therefore, it would be valuable to explore BHs in different contexts, including in EMS theory. In this regard, investigating a solution describing a charged BH in EMS theory as an extension of the Reissner–Nordström (RN) solution involving a dilaton field could provide an interesting alternative for testing optical phenomena in strong gravity field regimes. This solution in EMS theory offers unique features compared to the Schwarzschild BH in Einstein's theory and other dilaton BH solutions. Therefore, studying BHs in EMS theory with a plasma medium and examining their effects on optical phenomena can enhance our understanding of its optical properties and provide insights into deviations from other BH solutions in astrophysical observations. In this paper, our research not only contributes to theoretical knowledge but also sets the stage for future observational and experimental endeavor to investigate extreme environments around BHs. With this motivation, in this paper, we consider an interesting solution describing a charged black hole in the EMS theory of gravity, together with its properties. We aim to study the shadow and gravitational lensing effects, including the magnification of lensed images in the presence of the plasma medium. We then investigate the plasma effects along with the BH parameters on these astrophysical events, enhancing our understanding of their implications in explaining astrophysical observations.
The paper is organized as follows. We briefly review the solution of the charged BH in EMS theory, together with the photon motion and its dynamics, which is followed by the study of the BH shadow in the presence of plasma medium in section 2 . Further, we consider the weak gravitational lensing in section 3 and the magnification of the gravitationally lensed images in section 4 in the presence of the plasma medium. We summarize our results in section 5 . Throughout this paper, we use (−, +, +, +) signature for the spacetime metric and system of units in which we set G = c = 1.
2. Plasma impact on black hole shadow
2.1. Charged black hole in Einstein–Maxwell-scalar theory
Here, we consider the metric that can describe a static charged BH in EMS theory in Schwarzschild coordinates (t, r, θ, φ) can be written as (see details in [29, 91])
$\begin{eqnarray*}S=\int {{\rm{d}}}^{4}x\sqrt{-g}\left[R-2{{\rm{\nabla }}}_{\alpha }\phi {{\rm{\nabla }}}^{\alpha }\phi -K(\phi ){F}_{\alpha \beta }{F}^{\alpha \beta }-V(\phi )\right],\end{eqnarray*}$
where φ is the massless scalar field and K(φ) the scalar field function in the action and K(φ) the coupling function acting as the relation for the electromagnetic Fαβ and dilaton fields. Note that the term V(φ) is the potential associated with the cosmological constant Λ, $V(\phi )=\tfrac{{\rm{\Lambda }}}{3}\left({{\rm{e}}}^{2\phi }+4+{{\rm{e}}}^{-2\phi }\right)$, usually referred to as the de-Sitter BH solution including the dilaton field in the EMS theory [92]. We can then further consider the case, V(φ) = 0, which allows one to describe a spherically symmetric charged BH as follows [91] $\begin{eqnarray}{{\rm{d}}{s}}^{2}=-U(r){{\rm{d}}{t}}^{2}+\displaystyle \frac{{{\rm{d}}{r}}^{2}}{U(r)}+f(r)\left({\rm{d}}{\theta }^{2}+{\sin }^{2}\theta {\rm{d}}{\varphi }^{2}\right),\end{eqnarray}$
where U(r) and f(r) are defined by $\begin{eqnarray}\begin{array}{rcl}f(r) & = & {r}^{2}\left(1+\displaystyle \frac{\gamma {Q}^{2}}{{Mr}}\right),\\ U(r) & = & 1-\displaystyle \frac{2M}{r}+\displaystyle \frac{\beta {Q}^{2}}{f(r)},\end{array}\end{eqnarray}$
where M is the mass and Q is the BH's electric charge. The radial functions U(r) and f(r) include dimensionless constants such as β and γ in the EMS theory. It can be straightforward that one can recover the Schwarzschild and the RN BH solutions from f(r) and U(r) depending on parameters β and γ [29, 30], e.g. the solution with β → 0 and γ → 0 reduces to the Schwarzschild, γ = 0 and β = 1 to the Reissner–Nordström and β = 0 and γ = −1 to the dilation BH solutions [93, 94]. We further wish to examine the null geodesics around the charged EMS black hole in the presence of plasma medium.To this end, we consider the photon motion and its dynamics with the usage of the Hamilton–Jacobi equation. Hence, we need to write the Hamiltonian for the null geodesics around the BH with the plasma distribution background, i.e., it is given by [95]3 ), n is the refractive index, i.e., n = ω/k with k the wave number, which can be written as [96]9 ) and (10 ) to obtain the orbit equation as follows:
$\begin{eqnarray}{ \mathcal H }({x}^{\alpha },{p}_{\alpha })=\displaystyle \frac{1}{2}\left[{g}^{\alpha \beta }{p}_{\alpha }{p}_{\beta }-({n}^{2}-1){\left({p}_{\beta }{u}^{\beta }\right)}^{2}\right],\end{eqnarray}$
where xα is the spacetime coordinates, uβ and pα the four-velocity and momentum of the photon. Note that, in equation ( $\begin{eqnarray}{n}^{2}=1-\displaystyle \frac{{\omega }_{{\rm{p}}}^{2}}{{\omega }^{2}},\end{eqnarray}$
with plasma frequency ${\omega }_{{\rm{p}}}^{2}({x}^{\alpha })=4\pi {{\rm{e}}}^{2}N({x}^{\alpha })/{m}_{{\rm{e}}}$, where me and e are the electron mass and charge and N the number density of the electrons. Following ${\omega }^{2}={\left({p}_{\beta }{u}^{\beta }\right)}^{2}$, the photon frequency, ω, can then be defined by $\begin{eqnarray}\omega (r)=\displaystyle \frac{{\omega }_{0}}{\sqrt{U(r)}},\qquad {\omega }_{0}=\mathrm{const}.\end{eqnarray}$
For the metric function, U(r) → 1 is satisfied as r → ∞ and ω( ∞ ) = ω0 = − pt that manifests the photon energy at spatial infinity [97]. Using the photon geodesics ${ \mathcal H }=0$, ω0 can be restricted as follows: $\begin{eqnarray}\displaystyle \frac{{\omega }_{0}^{2}}{U(r)}\gt {\omega }_{{\rm{p}}}^{2}(r).\end{eqnarray}$
The physical meaning of this restriction is that the photon frequency at a given point, ω(r), must be greater than the plasma frequency at the same point. This rule consistently applies to light propagation in a plasma. Therefore, the BH shadow can have various forms compared to the vacuum case, i.e., ωp = 0. For the light geodesics in the presence of a plasma medium the Hamiltonian then yields as [95, 98] $\begin{eqnarray}{ \mathcal H }=\displaystyle \frac{1}{2}\left[{g}^{\alpha \beta }{p}_{\alpha }{p}_{\beta }+{\omega }_{{\rm{p}}}^{2}\right].\end{eqnarray}$
The light ray equations for the photon then take the following forms in the equatorial plane θ = π/2 $\begin{eqnarray}\dot{t}\equiv \displaystyle \frac{{\rm{d}}{t}}{{\rm{d}}\lambda }=\displaystyle \frac{-{p}_{t}}{U(r)},\end{eqnarray}$
$\begin{eqnarray}\dot{r}\equiv \displaystyle \frac{{\rm{d}}{r}}{{\rm{d}}\lambda }={p}_{r}U(r),\end{eqnarray}$
$\begin{eqnarray}\dot{\phi }\equiv \displaystyle \frac{{\rm{d}}\phi }{{\rm{d}}\lambda }=\displaystyle \frac{{p}_{\phi }}{f(r)},\end{eqnarray}$
with the relation ${\dot{x}}^{\alpha }=\partial { \mathcal H }/\partial {p}_{\alpha }$. We use equations ( $\begin{eqnarray}\displaystyle \frac{{\rm{d}}{r}}{{\rm{d}}\phi }=\displaystyle \frac{{g}^{{rr}}{p}_{r}}{{g}^{\phi \phi }{p}_{\phi }}.\end{eqnarray}$
For the photon geodesics ${ \mathcal H }=0$, the above equation can yield $\begin{eqnarray}\displaystyle \frac{{\rm{d}}{r}}{{\rm{d}}\phi }=\sqrt{\displaystyle \frac{{g}^{{rr}}}{{g}^{\phi \phi }}}\sqrt{{\gamma }^{2}(r)\displaystyle \frac{{\omega }_{0}^{2}}{{p}_{\phi }^{2}}-1},\end{eqnarray}$
where the following relation holds well $\begin{eqnarray}{\gamma }^{2}(r)\equiv -\displaystyle \frac{{g}^{{tt}}}{{g}^{\phi \phi }}-\displaystyle \frac{{\omega }_{p}^{2}}{{g}^{\phi \phi }{\omega }_{0}^{2}}.\end{eqnarray}$
For a light ray that comes from infinity, reaches a minimum at a radius rps, and goes out to infinity again. From a mathematical point of view, it is a turning point of the γ2(r) function. Therefore, the radius of the photon sphere can be determined from the following equation $\begin{eqnarray}{\left.\displaystyle \frac{{\rm{d}}({\gamma }^{2}(r))}{{\rm{d}}{r}}\right|}_{r={r}_{\mathrm{ps}}}=0.\end{eqnarray}$
The above equation solves to give the photon radii that we further explore numerically. In figure 1, we depict the photon radii for various possible cases. It can be observed from figure 1 that the radius of the photon sphere always increases in size with the increase in plasma frequency, while it decreases with the increase in the BH charge Q. It should also be noted that the curves of the photon radius sphere shift down to its small values as the parameter β increases. They do however shift up to its larger values under the effect of the parameter γ, depending on its sign, as seen in the top-right panel of figure 1.
Figure 1. Top-left panel: the dependence of the radius of the photon sphere on the plasma frequency for different values of the β parameter. BH charge and γ parameter are Q/M = 0.5 and γ = 0, respectively. Top-right panel: the radius of the photon sphere as a function of the plasma frequency for different values of the γ parameter. BH charge and β parameter are Q/M = 0.5 and β = 1, respectively. Bottom panel: the dependence of the radius of the photon sphere on the BH charge for different values of the β parameter. The other parameters are ${\omega }_{{\rm{p}}}^{2}/{\omega }^{2}=0.5$ and γ = 0. |
2.2. Black hole shadow in plasma
Now we consider the radius of the shadow of the charged BH in the presence of plasma. To be more informative, we show the trajectory of the photon in figure 2. It can be seen from figure 2 that the angle α approaches the angular radius of the shadow, αsh, as R → rps. On the usage of this figure, we can further explore the BH shadow. The angular radius αsh of the BH can be defined as [55, 97]13 ) we can easily find the γ2(rps) and γ2(ro). If the observer is located at a sufficiently large distance from the BH then it can be approximated the radius of BH shadow using equation (15 ) as [97]13 ), at spatial infinity for both models of plasma along with a constant magnetic field. The top-left panel of figure 3 shows the dependence of the BH shadow on the plasma frequency for different values of the β parameter. One can see from this figure that the radius of the BH shadow decreased with an increase of the plasma frequency. Also, under the influence of the β parameter the radius of the BH shadow decreased. It can be seen from the top-right panel of this figure that the radius of the BH shadow depends on the sign of the γ parameter. Similarly, the radius of the BH shadow decreased with an increase of the BH charge as demonstrated in the bottom panel of figure 3. Now we consider the assumption that the compact objects Sgr A* and M87* are static, spherically symmetric objects, even though the observation obtained by the EHT collaboration does not support the assumption made here. However, we try to theoretically investigate the lower limits of the BH charge Q of the BH in EMS theory, using the data provided by the EHT collaboration project. For constraint, we chose the BH charge Q and the plasma frequency. One can use the observational data provided by the EHT collaboration regarding the shadows of the supermassive black holes Sgr A* and M87* in order to constrain these two quantities Q and ${\omega }_{p}^{2}/{\omega }^{2}$. The angular diameter θM87* of the BH shadow, the distance from Earth and the mass of the BH at the center of the M87* are θM87* = 42 ± 3μas, D = 16.8 ± 0.8 Mpc and MM87* = 6.5 ± 0.7 × 109M⊙ [3], respectively. For Sgr A*, the data provided by the EHT collaboration are θSgrA* = 48.7 ± 7μ D = 8277 ± 9 ± 33pc and MSgrA* = 4.297 ± 0.013 × 106M⊙ (VLTI) [99]. From this information, we can calculate the diameter of the shadow caused by the compact object per unit mass as follows [100]
$\begin{eqnarray}\begin{array}{rcl}{\sin }^{2}{\alpha }_{\mathrm{sh}} & = & \displaystyle \frac{{\gamma }^{2}({r}_{\mathrm{ps}})}{{\gamma }^{2}({r}_{{\rm{o}}})},\\ & = & \displaystyle \frac{f({r}_{\mathrm{ps}})\left[\tfrac{1}{U({r}_{\mathrm{ps}})}-\tfrac{{\omega }_{p}^{2}({r}_{\mathrm{ps}})}{{\omega }_{0}^{2}}\right]}{f({r}_{{\rm{o}}})\left[\tfrac{1}{U({r}_{{\rm{o}}})}-\tfrac{{\omega }_{p}^{2}({r}_{{\rm{o}}})}{{\omega }_{0}^{2}}\right]},\end{array}\end{eqnarray}$
where rps and ro represent the locations of the photon sphere and the observer, respectively. From equation ( $\begin{eqnarray}\begin{array}{rcl}{R}_{\mathrm{sh}} & \simeq & {r}_{{\rm{o}}}\,\sin \,{\alpha }_{\mathrm{sh}},\\ & = & \sqrt{f({r}_{\mathrm{ps}})\left[\displaystyle \frac{1}{U({r}_{\mathrm{ps}})}-\displaystyle \frac{{\omega }_{{\rm{p}}}^{2}({r}_{\mathrm{ps}})}{{\omega }_{0}^{2}}\right]},\end{array}\end{eqnarray}$
where we have used the fact that γ(r) → r, which follows from equation ( $\begin{eqnarray}{d}_{{\rm{sh}}}=\displaystyle \frac{D\theta }{M}.\end{eqnarray}$
We know that from the expression dsh = 2Rsh, we can easily obtain the expression for the diameter of the BH shadow. It is worth noting that the distance D is considered a dimension of M [3, 4]. Thus, the diameter of the BH shadow ${d}_{{\rm{sh}}}^{{{\rm{M}}87}^{* }}=(11\pm 1.5){\rm{M}}$ for M87* and ${d}_{{\rm{sh}}}^{{{\rm{Sgr}}}^{\ast }}=(9.5\pm 1.4){\rm{M}}$ for Sgr A*. From observational EHT data, we can find the lower limits on the quantities Q and ${\omega }_{{\rm{p}}}^{2}/{\omega }^{2}$ for the supermassive BHs at the centers of the galaxies Sgr A* and M87*. It is demonstrated numerically in figure 4.
Figure 2. The angle α approaches the angular radius of the BH shadow αsh as R approaches rps. |
Figure 3. Top-left panel: the dependence of the radius of the BH shadow on the plasma frequency for different values of the β parameter. BH charge and γ parameter are Q/M = 0.5 and γ = 0, respectively. Top-right panel: the radius of the BH shadow as a function of the plasma frequency for different values of the γ parameter. BH charge and β parameter are Q/M = 0.5 and β = 1, respectively. Bottom panel: the dependence of the radius of the BH shadow on the BH charge for different values of the β parameter. The other parameters are ${\omega }_{{\rm{p}}}^{2}/{\omega }^{2}=0.5$ and γ = 0. |
Figure 4. The constrained values of BH charge Q and ${\omega }_{{\rm{p}}}^{2}/{\omega }_{0}^{2}$ for supermassive BHs sitting at the center M87 and Sgr A⋆ galaxies. Here, we note that we set β = 1 and γ = 0 for both panels. |
3. Weak gravitational lensing for black hole
Here, we consider the weak gravitational lensing for uniform and non-uniform plasma cases. For that we first represent a weak-field approximation that is defined by [75, 101]
$\begin{eqnarray}{g}_{\alpha \beta }={\eta }_{\alpha \beta }+{h}_{\alpha \beta },\end{eqnarray}$
where ηαβ and hαβ are respectively introduced to define the expressions for Minkowski spacetime and the perturbation gravity field that delineates the EMS theory and is defined by $\begin{eqnarray}\begin{array}{rcl}{\eta }_{\alpha \beta } & = & {\rm{diag}}(-1,1,1,1),\\ {h}_{\alpha \beta } & \ll & 1,{h}_{\alpha \beta }\to 0{\rm{under}}{x}^{\alpha }\to \infty ,\\ {g}^{\alpha \beta } & = & {\eta }^{\alpha \beta }-{h}^{\alpha \beta },{h}^{\alpha \beta }={h}_{\alpha \beta }.\end{array}\end{eqnarray}$
Following to the fundamental equation we are able to represent the deflection angle around the EMS BH as follows [101] $\begin{eqnarray}\begin{array}{rcl}{\hat{\alpha }}_{{\rm{b}}} & = & \displaystyle \frac{1}{2}{\displaystyle \int }_{-\infty }^{\infty }\displaystyle \frac{b}{r}\left(\displaystyle \frac{{{\rm{d}}{h}}_{33}}{{\rm{d}}{r}}+\displaystyle \frac{1}{1-{\omega }_{{\rm{p}}}^{2}/{\omega }^{2}}\displaystyle \frac{{{\rm{d}}{h}}_{00}}{{\rm{d}}{r}}\right.\\ & - & \left.\displaystyle \frac{{K}_{{\rm{e}}}}{{\omega }^{2}-{\omega }_{{\rm{p}}}^{2}}\displaystyle \frac{{\rm{d}}{N}}{{\rm{d}}{r}}\right){\rm{d}}{z},\end{array}\end{eqnarray}$
where ω is the photon frequency and ωp the plasma frequency. We further expand the metric functions into a Taylor series for calculations. The line element is written as $\begin{eqnarray}\begin{array}{rcl}{{\rm{d}}{s}}^{2} & \approx & {{\rm{d}}{s}}_{0}^{2}+\left(\displaystyle \frac{2M}{r}-\displaystyle \frac{\beta {{MQ}}^{2}}{r\left({Mr}+\gamma {Q}^{2}\right)}\right){{\rm{d}}{t}}^{2}\\ & & \,+\left(\displaystyle \frac{2M}{r}-\displaystyle \frac{\beta {{MQ}}^{2}}{r\left({Mr}+\gamma {Q}^{2}\right)}\right){{\rm{d}}{r}}^{2},\end{array}\end{eqnarray}$
with ${{\rm{d}}{s}}_{0}^{2}=-{{\rm{d}}{t}}^{2}+{{\rm{d}}{r}}^{2}+{r}^{2}({\rm{d}}{\theta }^{2}+{\sin }^{2}\theta {\rm{d}}{\phi }^{2})$. In the following, we define the components of hαβ as the perturbations which are written as follows: $\begin{eqnarray}{h}_{00}=\displaystyle \frac{2M}{r}-\displaystyle \frac{\beta {{MQ}}^{2}}{r\left({Mr}+\gamma {Q}^{2}\right)},\end{eqnarray}$
$\begin{eqnarray}{h}_{{ik}}=\left(\displaystyle \frac{2M}{r}-\displaystyle \frac{\beta {{MQ}}^{2}}{r\left({Mr}+\gamma {Q}^{2}\right)}\right){n}_{i}{n}_{k},\end{eqnarray}$
$\begin{eqnarray}{h}_{33}=\left(\displaystyle \frac{2M}{r}-\displaystyle \frac{\beta {{MQ}}^{2}}{r\left({Mr}+\gamma {Q}^{2}\right)}\right){\cos }^{2}\chi ,\end{eqnarray}$
with ${\cos }^{2}\chi ={z}^{2}/({b}^{2}+{z}^{2})$ and r2 = b2 + z2. The first derivatives of h00 and h33 with respect to the radial coordinate then give to write as follows: $\begin{eqnarray}\begin{array}{l}\displaystyle \frac{{{\rm{d}}{h}}_{00}}{{\rm{d}}{r}}\\ \ \ =\,\displaystyle \frac{M\left(-2{M}^{2}{r}^{2}+2{{MQ}}^{2}r(\beta -2\gamma )+\gamma {Q}^{4}(\beta -2\gamma )\right)}{{r}^{2}{\left({Mr}+\gamma {Q}^{2}\right)}^{2}},\end{array}\end{eqnarray}$
$\begin{eqnarray}\displaystyle \frac{{{\rm{d}}{h}}_{33}}{{\rm{d}}{r}}=\displaystyle \frac{{{Mz}}^{2}\left(-6{M}^{2}{r}^{2}+4{{MQ}}^{2}r(\beta -3\gamma )+3\gamma {Q}^{4}(\beta -2\gamma )\right)}{{r}^{4}{\left({Mr}+\gamma {Q}^{2}\right)}^{2}}.\end{eqnarray}$
Taking the above expressions together, the deflection angle can be written as $\begin{eqnarray}\hat{{\alpha }_{b}}=\hat{{\alpha }_{1}}+\hat{{\alpha }_{2}}+\hat{{\alpha }_{3}},\end{eqnarray}$
with $\begin{eqnarray}\begin{array}{rcl}\hat{{\alpha }_{1}} & = & \displaystyle \frac{1}{2}{\displaystyle \int }_{-\infty }^{\infty }\displaystyle \frac{b}{r}\displaystyle \frac{{{\rm{d}}{h}}_{33}}{{\rm{d}}{r}}{\rm{d}}{z},\\ \hat{{\alpha }_{2}} & = & \displaystyle \frac{1}{2}{\displaystyle \int }_{-\infty }^{\infty }\displaystyle \frac{b}{r}\displaystyle \frac{1}{1-{\omega }_{{\rm{p}}}^{2}/{\omega }^{2}}\displaystyle \frac{{{\rm{d}}{h}}_{00}}{{\rm{d}}{r}}{\rm{d}}{z},\\ \hat{{\alpha }_{3}} & = & \displaystyle \frac{1}{2}{\displaystyle \int }_{-\infty }^{\infty }\displaystyle \frac{b}{r}\left(-\displaystyle \frac{{K}_{e}}{{\omega }^{2}-{\omega }_{{\rm{p}}}^{2}}\displaystyle \frac{{dN}}{{\rm{d}}{r}}\right){\rm{d}}{z}.\end{array}\end{eqnarray}$
We further explore the impact of plasma density distributions on the deflection angle. This is what we wish to examine in the following subsections.3.1. Uniform plasma
Here, we consider a uniform plasma surrounding the BH in EMS theory and examine its impact on the gravitational deflection angle. To this end, we recall equation (27 ) and rewrite the form of the deflection angle as follows:24 ), (27 ) and (28 ) helps to define the deflection angle influenced by a uniform plasma medium. We then further analyze the deflection angle around the BH in EMS theory. The profile of the deflection angle as a function of the impact parameter b is shown in figure 5 for various combinations of BH charge and parameters β and γ and ${\omega }_{{\rm{p}}}^{2}/{\omega }^{2}$. The left panel reflects the role of parameter β on the deflection angle, whereas the right panel reflects the role of the BH charge while keeping fixed parameters β and γ. As can observed from figure 5, the deflection angle ${\hat{\alpha }}_{{\rm{uni}}}$, sensitive to the effect of the impact parameter, gets decreased as a consequence of an increase in the values of parameters β and Q. In figure 6, we show the role of the plasma parameters on the deflection angle. It should be noted that the increasing rate of the deflection angle is much more sensitive and increases rapidly as a consequence of an increase in the value of the plasma frequency, as seen in figure 6. As inferred from the increasing rate of the deflection angle, the plasma plays a pivotal role in changing the light geodesics.
$\begin{eqnarray}{\hat{\alpha }}_{{\rm{uni}}}={\hat{\alpha }}_{{\rm{uni}}1}+{\hat{\alpha }}_{{\rm{uni}}2}+{\hat{\alpha }}_{{\rm{uni}}3}.\end{eqnarray}$
The use of equations (
Figure 5. The plots shows the deflection angle ${\hat{\alpha }}_{\mathrm{uni}}$ against the impact parameter b for various combinations of parameter β (left panel) and BH charge Q (right panel). |
Figure 6. The plot shows the deflection angle ${\hat{\alpha }}_{\mathrm{uni}}$ profile against the impact parameter b for various values of the plasma parameters. Here we have set Q/M = 0.5, β = 1 and γ = 0. |
3.2. Non-uniform plasma
Here, we consider the non-singular isothermal sphere (SIS) that is the foremost favorable model acting as a useful tool for testing the photon geodesics around BHs, usually referred to as a spherical gas cloud with a singularity sitting at the center of infinite density. SIS density distribution is usually defined by [101]24 ), (28 ), and (33 ) allows one to obtain the form of the deflection angle, as well as to bring out an additional plasma constant ${\omega }_{c}^{2}$ with its explicit form [101]
$\begin{eqnarray}\rho (r)=\displaystyle \frac{{\sigma }_{\nu }^{2}}{2\pi {r}^{2}},\end{eqnarray}$
where ${\sigma }_{\nu }^{2}$ represents a one-dimensional velocity dispersion. Keeping this in mind, the analytic expression for the plasma concentration reads as $\begin{eqnarray}N(r)=\displaystyle \frac{\rho (r)}{{{km}}_{{\rm{p}}}},\end{eqnarray}$
where mp is proton mass and k a dimensionless constant. It is then straightforward to define the plasma frequency as follows: $\begin{eqnarray}{\omega }_{{\rm{e}}}^{2}={K}_{{\rm{e}}}N(r)=\displaystyle \frac{{K}_{{\rm{e}}}{\sigma }_{\nu }^{2}}{2\pi {{km}}_{{\rm{p}}}{r}^{2}}.\end{eqnarray}$
For further analysis of non-uniform plasma (SIS) effect, one needs to write out explicit form for the deflection angle around the BH. It is then written as $\begin{eqnarray}{\hat{\alpha }}_{{\rm{SIS}}}={\hat{\alpha }}_{{\rm{SIS}}1}+{\hat{\alpha }}_{{\rm{SIS}}2}+{\hat{\alpha }}_{{\rm{SIS}}3}.\end{eqnarray}$
The usage of equations ( $\begin{eqnarray}{\omega }_{c}^{2}=\displaystyle \frac{{K}_{e}{\sigma }_{\nu }^{2}}{2\pi {{km}}_{{\rm{p}}}{R}_{S}^{2}},\end{eqnarray}$
where RS = 2M. In figure 7, we have plotted the behavior of the deflection angle around BH in EMS theory against the impact parameter b for various values of β and Q for fixed γ and ${\omega }_{{\rm{p}}}^{2}/{\omega }^{2}$. One can observe from figure 7 that the deflection angle αsis decreases with the increase in the value of the impact parameter b/M similar to that of the uniform plasma case, while its shape slightly shifts down to its smaller values with increasing the parameter β and the BH charge. It must be noted that the changing rate of the deflection angle may have the opposite behavior as a consequence of the non-uniform plasma effect. For that we turn to analyze the impact of non-uniform plasma on the light's deflection angle, see figure 8. From figure 8 the deflection angle αsis decreases in magnitude with the increase in non-uniform plasma parameter. To gain a deeper understanding in relation to the explicit distinction between the role of the uniform and non-uniform plasma on the light's deflection angle, as seen in figure 9. The light ray's deflection angle acts in a different way, i.e. the uniform and non-uniform plasma have the opposite effects on the deflection angle.
Figure 7. The plot shows the deflection angle ${\hat{\alpha }}_{\mathrm{sis}}$ against the impact parameter b for various values of parameter β (left panel) and BH charge Q (right panel) in the existence of non-uniform plasma. |
Figure 8. The plot shows the deflection angle ${\hat{\alpha }}_{\mathrm{sis}}$ against the impact parameter b for various values of the plasma parameters. Note that we set Q/M = 0.5, β = 1 and γ = 0. |
Figure 9. The plot shows the deflection angle ${\hat{\alpha }}_{{\rm{b}}}$ against the impact parameter b (left panel) as well as the plasma parameters ${\omega }_{{\rm{p}}}^{2}/{\omega }^{2}$ (right panel). |
4. Magnification of the gravitationally lensed image
Here, we focus on the study of the magnification of the gravitationally lensed image around the BH in EMS theory, placed in the plasma, with the usage of the deflection angle of the light. To this end, we introduce the following equation as a combination of the light angles, $\hat{{\alpha }_{b}}$, θ and β around the black hole [71, 75, 102]35 ), the source's angular position, β, reads as
$\begin{eqnarray}\theta {D}_{{\rm{s}}}=\beta {D}_{{\rm{s}}}+\hat{{\alpha }_{b}}{D}_{\mathrm{ds}},\end{eqnarray}$
where Ds is the distance between the source and the observer, Dd is the lens and the observer, Dds is the source and the lens. Also, θ is the image's angular position and β the source's angular position, respectively. From equation ( $\begin{eqnarray}\beta =\theta -\displaystyle \frac{{D}_{\mathrm{ds}}}{{D}_{{\rm{s}}}}\displaystyle \frac{\xi (\theta )}{{D}_{{\rm{d}}}}\displaystyle \frac{1}{\theta },\end{eqnarray}$
with $\xi (\theta )=| {\hat{\alpha }}_{b}| \,b$ with b = Ddθ. It must be noted that the image's shape can be determined as Einstein's ring with the radius Rs = Dd θE provided that its appearance is a ring. Here, the corresponding angular part θE is given by $\begin{eqnarray}{\theta }_{E}=\sqrt{2{R}_{{\rm{s}}}\displaystyle \frac{{D}_{{ds}}}{{D}_{{\rm{d}}}{D}_{{\rm{s}}}}}.\end{eqnarray}$
The magnification of brightness then yields $\begin{eqnarray}{\mu }_{{\rm{\Sigma }}}=\displaystyle \frac{{I}_{\mathrm{tot}}}{{I}_{* }}=\displaystyle \sum _{k}\left|\left(\displaystyle \frac{{\theta }_{k}}{\beta }\right)\left(\displaystyle \frac{{\rm{d}}{\theta }_{k}}{{\rm{d}}\beta }\right)\right|,\quad k=1,2,\cdot \cdot \cdot ,j,\end{eqnarray}$
where Itot is the total brightness and I* the unlensed brightness of the source, Then, the source's magnification is given by [103–105] $\begin{eqnarray}{\mu }_{+}^{\mathrm{pl}}=\displaystyle \frac{1}{4}\left(\displaystyle \frac{x}{\sqrt{{x}^{2}+4}}+\displaystyle \frac{\sqrt{{x}^{2}+4}}{x}+2\right),\end{eqnarray}$
$\begin{eqnarray}{\mu }_{-}^{\mathrm{pl}}=\displaystyle \frac{1}{4}\left(\displaystyle \frac{x}{\sqrt{{x}^{2}+4}}+\displaystyle \frac{\sqrt{{x}^{2}+4}}{x}-2\right),\end{eqnarray}$
where x = β/θE is a dimensionless quantity and ${\mu }_{+}^{\mathrm{pl}}$ and ${\mu }_{-}^{\mathrm{pl}}$ the images. As a consequence, the total magnification is given as a liner combination of the images, i.e., $\begin{eqnarray}{\mu }_{\mathrm{tot}}^{\mathrm{pl}}={\mu }_{+}^{\mathrm{pl}}+{\mu }_{-}^{\mathrm{pl}}=\displaystyle \frac{{x}^{2}+2}{x\sqrt{{x}^{2}+4}}.\end{eqnarray}$
Furthermore, we turn to analyze the source's magnification with the usage of two different plasma cases, such as uniform and non-uniform plasma distributions which surround BH in EMS theory. This is what we intend to examine in the next subsections.4.1. Uniform plasma
In this subsection, we begin to consider the uniform plasma effect on the magnification of the lensed image. Hence, we recall equation (41 ) with uniform plasma in the EMS BH environment we examine the total magnification of the image, ${\mu }_{{\rm{tot}}}^{{\rm{pl}}}$, that reads as
$\begin{eqnarray}{\mu }_{{\rm{tot}}}^{{\rm{pl}}}={\mu }_{+}^{{\rm{pl}}}+{\mu }_{-}^{{\rm{pl}}}=\displaystyle \frac{{x}_{{\rm{uni}}}^{2}+2}{{x}_{{\rm{uni}}}\sqrt{{x}_{{\rm{uni}}}^{2}+4}},\end{eqnarray}$
We further explore ${\left({\theta }_{E}^{{\rm{pl}}}\right)}_{{\rm{uni}}}$ numerically. Here, the images ${\left({\mu }_{+}^{{\rm{pl}}}\right)}_{{\rm{uni}}}$ and ${\left({\mu }_{-}^{{\rm{pl}}}\right)}_{{\rm{uni}}}$ are defined by $\begin{eqnarray}{\left({\mu }_{+}^{{\rm{pl}}}\right)}_{{\rm{uni}}}=\displaystyle \frac{1}{4}\left(\displaystyle \frac{{x}_{{\rm{uni}}}}{\sqrt{{x}_{{\rm{uni}}}^{2}+4}}+\displaystyle \frac{\sqrt{{x}_{{\rm{uni}}}^{2}+4}}{{x}_{{\rm{uni}}}}+2\right),\end{eqnarray}$
and $\begin{eqnarray}{\left({\mu }_{-}^{{\rm{pl}}}\right)}_{{\rm{uni}}}=\displaystyle \frac{1}{4}\left(\displaystyle \frac{{x}_{{\rm{uni}}}}{\sqrt{{x}_{{\rm{uni}}}^{2}+4}}+\displaystyle \frac{\sqrt{{x}_{{\rm{uni}}}^{2}+4}}{{x}_{{\rm{uni}}}}-2\right),\end{eqnarray}$
with $\begin{eqnarray}{x}_{{\rm{uni}}}=\displaystyle \frac{\beta }{{\left({\theta }_{{\rm{E}}}^{{\rm{pl}}}\right)}_{{\rm{uni}}}}.\end{eqnarray}$
In figure 10, we plotted the total magnification of the image ${\mu }_{\mathrm{tot}}^{\mathrm{pl}}$ in the presence of uniform plasma that exists in the BH environment in the EMS theory for various values of β in the left panel while Q in the right panel. It is obvious from figure 10 that the curves of the total magnification shift down to its smaller values as the parameter β and the BH charge Q increase. We also plotted the total magnification against x0 for the uniform plasma having suitable parameters, as shown in the top-left panel of figure 12. We explicitly show in figure 12 that the total magnification shifts towards up to its larger magnitude with the increase in the uniform plasma parameters while keeping fixed b/M.
Figure 10. The plot shows the total magnification μtot against the plasma parameter ${\omega }_{p}^{2}/{\omega }^{2}$ for different values of β (left panel) and Q (right panel) while keeping fixed b = 6M. |
4.2. Non-uniform plasma
Here, we consider non-uniform plasma (as SIS medium) and examine its effect on magnification. For non-uniform plasma it is given by46 ), one is able to determine the total magnification as a function of the plasma parameter. It can be observed from figure 11 that the magnification decreases with the increase of the plasma parameter, while its shape also shifts down to its small values as a consequence of β and Q. Furthermore, we also plotted the profile of total magnification as a function of x0 under the effect of non-uniform plasma while keeping fixed the impact parameter b and the BH parameters, see in the top-right panel of figure 12. Interestingly, we observe that the total magnification decreases under the impact of non-uniform plasma in comparison with one for the uniform plasma. To understand more deeply, we also provide a comparison of uniform and non-uniform plasma cases, which is shown in the bottom panel of figure 12. It can clearly be seen from the comparison that the magnification of uniform plasma distribution is more sensitive than the non-uniform plasma one, i.e. the uniform plasma distribution is larger than the non-uniform plasma one.
$\begin{eqnarray}{\left({\mu }_{{\rm{tot}}}^{{\rm{pl}}}\right)}_{{\rm{SIS}}}={\left({\mu }_{+}^{{\rm{pl}}}\right)}_{{\rm{SIS}}}+{\left({\mu }_{-}^{{\rm{pl}}}\right)}_{{\rm{SIS}}}=\displaystyle \frac{{x}_{{\rm{SIS}}}^{2}+2}{{x}_{{\rm{SIS}}}\sqrt{{x}_{{\rm{SIS}}}^{2}+4}},\end{eqnarray}$
with $\begin{eqnarray}{\left({\mu }_{+}^{{\rm{pl}}}\right)}_{{\rm{SIS}}}=\displaystyle \frac{1}{4}\left(\displaystyle \frac{{x}_{{\rm{SIS}}}}{\sqrt{{x}_{{\rm{SIS}}}^{2}+4}}+\displaystyle \frac{\sqrt{{x}_{{\rm{SIS}}}^{2}+4}}{{x}_{{\rm{SIS}}}}+2\right),\end{eqnarray}$
$\begin{eqnarray}{\left({\mu }_{-}^{{\rm{pl}}}\right)}_{{\rm{SIS}}}=\displaystyle \frac{1}{4}\left(\displaystyle \frac{{x}_{{\rm{SIS}}}}{\sqrt{{x}_{{\rm{SIS}}}^{2}+4}}+\displaystyle \frac{\sqrt{{x}_{{\rm{SIS}}}^{2}+4}}{{x}_{{\rm{SIS}}}}-2\right),\end{eqnarray}$
where xSIS is $\begin{eqnarray*}{x}_{{\rm{SIS}}}=\displaystyle \frac{\beta }{{\left({\theta }_{{\rm{E}}}^{{\rm{pl}}}\right)}_{{\rm{SIS}}}}.\end{eqnarray*}$
By equation (
Figure 11. The plot shows the total magnification μtots against the plasma parameters ${\omega }_{c}^{2}/{\omega }^{2}$ for different values of β (left panel) and Q (right panel) while keeping fixed b = 6M. |
Figure 12. The profile of the image of magnification against x0. Top-left/right panel: the magnification is plotted under the effect of uniform/SIS. Bottom panel: the comparison of magnification is plotted for uniform and non-uniform plasma cases for the fixed ${\omega }_{{\rm{p}}}^{2}/{\omega }^{2}={\omega }_{{\rm{p}}}^{2}/{\omega }^{2}=0.9$. We note that we have set b/M = 3, β = 1 and γ = 0. |
5. Conclusions
In this paper, we have investigated the optical properties of the BH in the EMS theory in the presence of plasma medium for various situations. From the performed research we can summarize our main results as follows:
| • | We have investigated the photon motion around the BH surrounded by a plasma. We have obtained numerical results on the dependence of the radius of the photon sphere on the plasma frequency (see figure 1). It has been shown that the radius of the photon sphere increases with the increase of the plasma frequency. Also, the value of the photon sphere radius decreases under the influence of the β parameter and the BH charge. The radius of the photon sphere depends on the sign of the γ parameter. |
| • | We have also studied the shadow of the BH in plasma. The radius of the BH shadow has been calculated by numerical method. The dependencies of the radius of the BH shadow on the plasma frequency and BH parameter have been demonstrated in figure 3. One can see from this figure that the radius of the BH shadow decreased with the increase of the plasma parameter. The effects of the parameters β and γ and the BH charge for the BH shadow are the same as for the photon sphere. |
| • | In figure 4, we have demonstrated that the size of the shadow depends on the BH and the gravity theory parameters in the future the obtained results can be applied to the images of Sgr A* and M87* SMBHs to get constraints on EMS gravity parameters. Additionally, the observational data of the gravitational lensing in [106–108] may be further used to get constraints on the spacetime parameters of EMS and get estimation the plasma characteristics. |
| • | Furthermore, weak gravitational lensing for the BH in the EMS theory has been investigated. For this, we considered that the BH is surrounded by uniform and non-uniform plasma. We have found the deflection angle for every case independently. Figures 5 and 6 correspond to the uniform plasma case. We can see from these figures that the value of the deflection angle increases with the increase of the plasma frequency. Also, there is a slight decrease with the increase of the β parameter and the BH charge. The obtained results for the non-uniform plasma case are illustrated in figures 7 and 8. It is clear from these figures that the β parameter and the BH charge have the same effect in the case of non-uniform plasma as in the case of uniform plasma. However, the value of the deflection angle decreases with the increase of the non-uniform plasma frequency. |
| • | In addition, we have compared the deflection angle of light for uniform and non-uniform plasma in figure 9. We can easily see from this figure that the value of the deflection angle of light for uniform plasma is greater than for non-uniform plasma. |
| • | Finally, we have studied the total magnification of the images as a function of uniform and non-uniform plasma, and the dependencies have been plotted in figures 10 and 11. It can be seen from these figures that the value of the total magnification of the image for uniform plasma increased with the increase of plasma parameter and for non-uniform plasma vice versa. Also, we have investigated the image magnification for both cases. The results were demonstrated in figure 12. |
From an astrophysical viewpoint, it is important to explore optical phenomena around the BHs in different contexts, including in the EMS theory. That is why investigating a solution describing a charged BH in EMS theory as an extension of the RN solution involving a dilaton field could provide an interesting alternative for testing optical phenomena in strong gravity field regimes. Our findings can enhance our understanding of the optical properties of BHs and provide insights into deviations from other BH solutions in astrophysical observations by studying BHs in the EMS theory with a plasma medium and examining their effects on optical phenomena. These theoretical findings not only contribute to theoretical knowledge but also set the stage for future observational and experimental endeavors to investigate extreme environments around BHs.