1. Introduction
Albert Einstein (1916) described his general theory of relativity (GR), which explains gravity. In his GR theory, Einstein predicted gravitational waves' existence and gravitational lensing [1]. Black holes (BHs) are fascinating objects in the Universe. The gravitational wave of a laser interferometer observatory experiment in 2015 provided experimental confirmation that several forms of BHs exist [2] in the cosmos. The Event Horizon Telescope cooperation demonstrated the BH’s existence using its shadow in [3]. Gravitational waves are mainly detected via the physics of BHs [2], and BH physics is additionally used to understand the information paradox and entropy [4]. Furthermore, it plays a very important role in the fascinating gravitational lensing feature. According to Einstein, a large object like a BH causes light to bend, acting as a lens for anything behind it. While gravity does not form a distant galaxy image with minor lensing, it has a perceivable and supportive effect. Strong lensing produces curves and Einstein’s rings, such as Einstein’s ring, as well as weak lensing, which offers an accurate method to detect dark energy and the universe’s accelerating expansion. The Universe, dark matter, dark energy and galaxies can be explained using gravitational lensing [5]. A significant amount of work on gravitational lensing has been examined for BHs, cosmic strings, wormholes and different types of spacetime [6–29]. This work turned into the main gravitational lensing perspective of Eddington. The geodesic methodology [30–35] has been considered since Einstein concentrated on gravitational lensing.
The Gauss–Bonnet (GB) theorem was explained in 2008 by the Gibbons–Werner [36] approach, which demonstrated a new approach for computing the weak deflection angle (DA). In spherically and static symmetric spacetime, such as a Schwarzschild BH, they discovered light’s weak gravitational deflection by using the GB theorem [36]. Werner illustrated that by using the GB theorem, it is also possible to determine the stationary BHs of weak DAs, such as the Kerr BH. The optical metrics of Gaussian curvature can be integrated to provide the DA of light in the GB theorem. We can use a space in the GB theorem in which the photon beam is located, together with a boundary curve of the circular that is located at the photon beams and focal point that intersects the observer and source. The observer and origin should both be at the focal point coordinate distance in space DR. The GB theorem is formatted as follows1 ) associates the integral of curvature quantities with the Euler feature of region DR. When exploring the topological and geometric properties of surfaces, the GB theorem is an effective tool [37]. The photon DA may be significantly influenced by the magnetic charge’s physical characteristics, and the greybody bound of the BH in Einstein GB gravity in [38]. It can also have an impact on the photons in the plasma medium. The contribution of the magnetic charge to the BH's greybody component will then be of greater interest, and it has also been concluded that the magnetic charge parameter in the Einstein GB gravity BH affects the field limitations of the weak DA. The weak DA is then determined using the expression shown in [36, 39]
$\begin{eqnarray}\int {\int }_{{{\rm{D}}}_{R}}{\bf{K}}{\rm{d}}S+{\oint }_{\partial {{\rm{D}}}_{R}}k{\rm{d}}t+\displaystyle \sum _{{\rm{i}}}{\epsilon }_{{\rm{i}}}=2\pi \varepsilon ({{\rm{D}}}_{R}),\end{eqnarray}$
where K stands for the optical metric Gaussian curvature, k is geodesic curvature and dS represents an areal factor. Taking into account the characteristics of Euler ϵ(DR) = 1, the jump angle is obtained by applying the approximation of straight light, and the GB term is ∑iεi = π. Equation ( $\begin{eqnarray}\psi \approx -{\int }_{b/\sin \phi }^{\infty }{\int }_{0}^{\pi }{\bf{K}}\sqrt{| g| }{\rm{d}}\phi {\rm{d}}r,\end{eqnarray}$
where ψ represents the angle of light deflection, and g gives the determinant of the optical metric. Numerous researchers have studied the angle of weak deflection by using the GB theorem for different spacetimes. The angle of light deflection for BHs and wormholes was examined in [40–48]. The BH's thermodynamic properties [49–53] and the DA [54, 55], such as the Hawking temperature, are significant when discussing BH geometry. The majority of researchers claim that supermassive BHs are present at the galactic center of the Milky Way and they are looking for their shadow. Furthermore, the new findings may give physicists a previously unattainable view of BH dynamics, enabling researchers to test GR [56–60].A gauge potential is present in an accelerating charged AdS BH, and the permanent configurations that follow the rules of physics are charged BHs. Due to their substantial charges, they possess certain fascinating characteristics. Because they are thought to be accurate representations of stellar mass and supermassive BHs, charged BHs have garnered a lot of interest in astrophysics. The extraction of electromagnetic energy from rotating charged BHs has been approached in many ways. Studies have been performed on Hawking radiation and the motion of charged particles around charged BHs. It was demonstrated that the super-radiant instability of BHs and the intensity of evaporation are enhanced in the presence of a charged field. Recent research has suggested that specific particle collisions in the presence of an accelerating weakly charged BH can generate particles with high center-of-mass energies. Charged BHs and gauge potentials may occasionally be employed as particle magnetizers. The Einstein–Maxwell equation’s exact solutions [61] may explain BH physics. In four-dimensional spacetime, magnetized BH solutions were created using the Harrison transformation. A set of Einstein–Maxwell and Einstein–Maxwell-dilaton equations that describe black particles in external magnetic fields in five dimensions have recently been added to them. Since only the most basic magnetic solution can reveal a BH in a gravitational field, we think it is essential to obtain more charged or magnetic solutions.
Many authors study the BH shadow, which is a BH with a stunning ring of encompassing radiation produced by rapidly rotating and extremely hot gas. Through gravitational light deflection, BHs create their shadow. A photon’s impact parameter determines the trajectory of the particle in a vacuum. To form the shadow, the photon sphere is very important. In contrast to the rotating BH shadow, which is simply a circle, the stationary BH shadow is bent and oblate. The literature contains numerous examples of scientific investigations into BH shadows.
The main purpose of this article is to examine the DA and shadow in the background of an accelerating charged Ads BH in a non-magnetic plasma absence/presence medium. To do so, we apply the Gibbons–Werner technique to calculate the DA in the context of plasma/non-plasma. Moreover, we also calculate the DA by using the Keeton and Petter approach and also graphically represent the DA. We examine the shadow of the BH under the non-magnetized plasma influence using graphic interpretation. We study the graphic behavior of shadows under the effects of different parameters. The paper is divided into the following sections: in section 2 , we discuss the metric for the BH. In section 3 , we use the Gibbons–Werner and Keeton–Petters approaches to determine the DA in the absence/presence of plasma medium. In section 4 , we check the BH shadow in the presence of non-magnetized plasma. Additionally, in section 5 , we examine the graphic behavior to examine the weak shadow when the plasma medium is present. We finally conclude our findings in section 6 .
2. Introductory review of accelerating charged AdS black hole
We see the geometry of the gauge-potential-like AdS BH with accelerating and charged parameters. We proceed by examining the mechanics of the gauge-potential-like AdS BH with accelerating and charged parameters, showing the relationship between physical factors and their solutions to numerical parameters. The spacetime as a gauge-potential-like AdS BH with accelerating and charged parameters can be described in the form [62, 63]
$\begin{eqnarray}\begin{array}{l}{\rm{d}}{s}^{2}=\displaystyle \frac{1}{{{\rm{\Omega }}}^{2}}\left[-f\left(r\right){\rm{d}}{t}^{2}+\displaystyle \frac{{\rm{d}}{r}^{2}}{f\left(r\right)}\right.\\ \left.+{r}^{2}\left(\displaystyle \frac{{\rm{d}}{\theta }^{2}}{g\left(\theta \right)}+g\left(\theta \right){\sin }^{2}\theta \displaystyle \frac{{\rm{d}}{\phi }^{2}}{{k}^{2}}\right)\right],\end{array}\end{eqnarray}$
with $\begin{eqnarray}\begin{array}{l}f\left(r\right)=\left(1+\displaystyle \frac{{e}^{2}}{{r}^{2}}-\displaystyle \frac{2m}{r}\right)\left(1-{A}^{2}{r}^{2}\right)+\displaystyle \frac{{r}^{2}}{{l}^{2}},\\ g\left(\theta \right)=1+{e}^{2}{A}^{2}{\cos }^{2}\theta +2{mA}\cos \theta ,\end{array}\end{eqnarray}$
and the factor of conformal ${\rm{\Omega }}={Ar}\cos \theta +1$ defines the conformal infinity or AdS BH boundary. The e and m define the electric charge and the mass, respectively. In addition, A > 0 and $l=\sqrt{\tfrac{-{\rm{\Lambda }}}{3}}$ describe the measure of the AdS radius and the acceleration, respectively. This particular approach to describing the spacetime implies an AdS BH with observable continuity and shows how the acceleration changes the spherical surfaces, such as the horizon explained by the polar coordinates (see [64] for an account of possible coordinates of the C metric).In equation (4 ), we can check that the acceleration parameter contends with the cosmological $\tfrac{{r}^{2}}{{l}^{2}}$ term in the potential of the Newtonian; alternatively, the acceleration effect is canceled out by the AdS space’s negative curvature. The form of F shows that $A\lt \tfrac{1}{l}$ depicts a single BH hanging in the AdS BH, with the only BH horizon [65]. For $A\gt \tfrac{1}{l}$, there are two (oppositely charged) BHs classified by the horizon with an acceleration parameter [66, 67]; the condition of $A=\tfrac{1}{l}$ is unique and was discovered in [68]. We further constrain ${mA}\lt \tfrac{1}{2}$ to ensure that our angular coordinates match the standard two-sphere coordinates. For a description of general C spacetimes in the AdS field and its holographic implications see [69]. Observing the spacetime of the angular part and the g(θ) trajectory at the poles, θ+ = 0 and θ− = π, indicates cosmic string existence. The spacetime pole regularity implies
$\begin{eqnarray}g\left({\theta }_{\pm }\right)={k}_{\pm }=1+{e}^{2}{A}^{2}\pm 2{mA}.\end{eqnarray}$
It is obvious that for mA ≠ 0 it is impossible to fix K in such a way that we have pole regularity, and the lack of regularity is precisely an indicant of a conical singularity. The K is usually chosen to regularise one pole, while leaving either a conical deficiency or a conical excess along the other. We adopt the fact that our BH is regular on the north pole at θ = 0, setting k+ = k = 1 + A2e2 + 2mA, as a conical excess would be sourced via an object of negative energy, and so there is a conical deficiency in the direction θ− = π of a south pole as
$\begin{eqnarray}\delta =\left(1-\displaystyle \frac{{g}_{-}}{{k}_{+}}\right)2\pi =\displaystyle \frac{8\pi {mA}}{1+{e}^{2}{A}^{2}+2{mA}},\end{eqnarray}$
that is equivalent to a cosmic string under tension $\mu =\tfrac{\delta }{8\pi }$. The five physical parameters that make up the C-metric solution are, in short, the acceleration A, the mass m, the cosmological constant l, the charge e and the tension of the cosmic strings on each axis, which is conveyed by the periodicity of the angular coordinate.3. Light-deflected angle by Gibbons and Werner approach
The Gibbons–Werner technique is a powerful tool for exploring light deflection in curved spacetimes. Applying the GB theorem to integral regions in a two-dimensional manifold allows geometry-based expression and calculation of light deflection. The Gibbons–Werner technique for accelerating charged AdS BHs has two issues: (a) the integral region is generally infinite, which is ill-defined for some non-flat spacetimes with a singular metric, and (b) the intricate double and single integrals cause complicated calculations, especially for highly accurate results and complex spacetimes. To address these concerns, a generalized Gibbons–Werner technique is proposed, in which the infinite region is substituted by a flexible region that eliminates the singularity and provides a more detailed explanation of light deflection in complex spacetime. Light deflection was examined in [70] to gain an in-depth understanding of this mechanism and its implications for astrophysics. The impact parameter affects the BH light angle and the distance from the compact object. The impact parameter is the nearest choice among the photons surrounding the BH. This contributes to the understanding of how weak light deflection, particularly at a schematic view of the DA using the GB theorem [29, 70], affects the amplification of an image source when plasma is present. The spacetime of the charged accelerating AdS BH given in equation (3 ) is re-expressed as follows7 ), to derive the optical metric in the following form9 ) can be rewritten as follows10 ) and their derivatives into equation (11 ), we compute the Ricci scalar in the given form13 ), the BH of Gaussian curvature is computed as follows9 ), it can be shown in the form dt = Tdφ. So, as a formation, we get15 ) and (21 ), the DA can be obtained from equation (2 ) as well as using the leading order terms of the Gaussian curvature, and the DA is computed as
$\begin{eqnarray}\begin{array}{l}{\rm{d}}{s}^{2}=-F(r,\theta ){\rm{d}}{t}^{2}+G(r,\theta ){\rm{d}}{r}^{2}\\ +H(r,\theta ){\rm{d}}{\theta }^{2}+R(r,\theta )d{\phi }^{2},\end{array}\end{eqnarray}$
where $\begin{eqnarray}\begin{array}{l}F(r,\theta )=\displaystyle \frac{f(r)}{{{\rm{\Omega }}}^{2}},\,\,\,\,G(r,\theta )=\displaystyle \frac{1}{f(r){{\rm{\Omega }}}^{2}},\\ H(r,\theta )=\displaystyle \frac{{r}^{2}}{g(\theta ){{\rm{\Omega }}}^{2}},\,\,\,\,R(r,\theta )=\displaystyle \frac{g(\theta ){r}^{2}{\sin }^{2}\theta }{{k}^{2}{{\rm{\Omega }}}^{2}}.\end{array}\end{eqnarray}$
Now, we utilize the GB theorem to examine the BH bending angle in a non-plasma medium. We consider the equatorial plane $\theta =\tfrac{\pi }{2}$, then $g(\tfrac{\pi }{2})=1$, Ω = 1 and dθ = 0, if the observer, source and null photon are all turned up in the tropical region. We set ds2 = 0 in equation ( $\begin{eqnarray}{\rm{d}}{t}^{2}=\displaystyle \frac{1}{{f}^{2}(r)}{\rm{d}}{r}^{2}+\displaystyle \frac{{r}^{2}}{f(r){k}^{2}}{\rm{d}}{\phi }^{2}.\end{eqnarray}$
The optical metric equation ( $\begin{eqnarray}{\rm{d}}{t}^{2}=\tilde{F}{\rm{d}}{r}^{2}+\tilde{G}{\rm{d}}{\phi }^{2},\end{eqnarray}$
where $\tilde{F}=\tfrac{1}{{f}^{2}(r)}$ and $\tilde{G}=\tfrac{{r}^{2}}{f(r){k}^{2}}$. The Ricci scalar (${ \mathcal R }$) of optical metrics is determined using the following formula $\begin{eqnarray}{ \mathcal R }=\displaystyle \frac{2\tilde{F}^{\prime\prime} (r)\tilde{G}(r)\tilde{F}(r)-\tilde{F}^{\prime} (r)\tilde{G}^{\prime} (r)\tilde{F}(r)-\tilde{F}{{\prime} }^{2}(r)\tilde{G}(r)}{2{\tilde{G}}^{2}(r){\tilde{F}}^{2}(r)}.\end{eqnarray}$
After putting the values of the metric functions from equation ( $\begin{eqnarray}\begin{array}{l}{ \mathcal R }\approx -\displaystyle \frac{4m}{{r}^{3}}-\displaystyle \frac{20{{mA}}^{2}{e}^{2}}{{r}^{3}}\\ +\displaystyle \frac{6{e}^{2}}{{r}^{4}}-\displaystyle \frac{12{{me}}^{2}}{{r}^{5}}+O({m}^{2},{A}^{4},{e}^{4}).\end{array}\end{eqnarray}$
According to the definition of the Gaussian curvature, we have $\begin{eqnarray}{\bf{K}}=\displaystyle \frac{{ \mathcal R }}{2}.\end{eqnarray}$
Using equation ( $\begin{eqnarray}\begin{array}{l}{\bf{K}}\approx -\displaystyle \frac{2m}{{r}^{3}}-\displaystyle \frac{10{{mA}}^{2}{e}^{2}}{{r}^{3}}+\displaystyle \frac{3{e}^{2}}{{r}^{4}}\\ -\displaystyle \frac{6{{me}}^{2}}{{r}^{5}}+O({m}^{2},{A}^{4},{e}^{4}).\end{array}\end{eqnarray}$
In the non-singular domain YT region, the DA for the accelerating AdS BH solution, by utilizing the GB theorem [71, 72], can be found by applying the following expression $\begin{eqnarray}\int {\int }_{{Y}_{T}}{\bf{K}}{\rm{d}}S+{\oint }_{\partial {Y}_{T}}k{\rm{d}}t+\displaystyle \sum _{{\rm{i}}}{\epsilon }_{{\rm{i}}}=2\pi \varepsilon ({Y}_{T}).\end{eqnarray}$
In the above formulation, k shows geodesic curvature, expressed as $\bar{g}({{\rm{\nabla }}}_{\dot{\eta }}\dot{\eta },\ddot{\eta })=k$, and $\bar{g}(\dot{\eta },\dot{\eta }),\ddot{\eta }=1$ denote the unit acceleration vector and εii states the exterior angle at the ith vertex. As T → ∞ , the jump angles become $\tfrac{\pi }{2}$ and we find θO + θs → π. As ϵ(YT) = 1, this is a Euler characteristic. So, $\begin{eqnarray}\int {\int }_{{Y}_{T}}{\bf{K}}{\rm{d}}S+{\oint }_{\partial {Y}_{T}}k{\rm{d}}t+{\epsilon }_{{\rm{i}}}=2\pi \varepsilon ({Y}_{T}),\end{eqnarray}$
where εi = π shows the angle. If T → ∞ , a result of the geodesic curvature is $\begin{eqnarray}k({D}_{T})=| {\rm{\nabla }}\dot{{}_{{D}_{T}}}\dot{{D}_{T}}| .\end{eqnarray}$
In this case, the radial geodesic curvature component equals $\begin{eqnarray}{\left({{\rm{\nabla }}}_{\dot{{D}_{T}}}\dot{{D}_{T}}\right)}^{r}=\dot{{D}_{T}^{\phi }}{\partial }_{\phi }\dot{{D}_{T}^{r}}+{{\rm{\Gamma }}}_{\phi \phi }^{r}{\left({\dot{D}}_{T}^{\phi }\right)}^{2}.\end{eqnarray}$
For the T value, ${D}_{T}:= r(\phi )=T=\mathrm{constant}$, then the formation is $\begin{eqnarray}{\left({{\rm{\nabla }}}_{\dot{{D}_{T}}}\dot{{D}_{T}}\right)}^{r}\to \displaystyle \frac{1}{T}.\end{eqnarray}$
It has no topological flaw in the geodesic curvature; then, k(DT) → T−1. In the optical metric equation ( $\begin{eqnarray}k({D}_{T}){\rm{d}}t={\rm{d}}\phi .\end{eqnarray}$
Now, by utilizing the previous formation one can find the following expression $\begin{eqnarray}\int {\int }_{{Y}_{T}}{\bf{K}}{\rm{d}}S+{\oint }_{\partial {Y}_{T}}k{\rm{d}}t=\int {\int }_{T\infty }k{\rm{d}}S+{\int }_{0}^{\pi +\psi }{\rm{d}}\phi ,\end{eqnarray}$
the zero-order ray in the limits of the weak field is computed as $r(t)=\tfrac{b}{\sin \phi }$. By using equations ( $\begin{eqnarray}\begin{array}{l}\psi \approx \displaystyle \frac{4m}{b}-\displaystyle \frac{20{{mA}}^{2}{e}^{2}}{b}-\displaystyle \frac{3{e}^{2}\pi }{4{b}^{2}}\\ +\displaystyle \frac{8{{me}}^{2}}{3{b}^{3}}-\displaystyle \frac{9{{mA}}^{2}\pi }{16{b}^{4}}+O({m}^{2},{A}^{4},{e}^{4}).\end{array}\end{eqnarray}$
The ψ depends on the mass m, gauge potential A, charge e and impact parameter b. It can be concluded that the ψ is inverse to the impact parameter. Moreover, we analyze the graphical behavior of ψ with impact parameters under the effects of the gauge potential A and charge e. As an assumption, it is possible to get around this problem using a more exact relation for the trajectory of a light ray in the integration domain. Additionally, our predicted DAs for the first order are reduced to the Einstein DA [73] or Schwarzschild (${\psi }_{{\rm{Schw}}}=\frac{4m}{b}$).
Figure 1 shows the graphical conduct of the DA ψ via the impact parameter b. The left plot represents the behavior of ψ for varying values of the gauge potential A and for fixed m = 1, e = 0.6. One can observe that the DA decreases with the increasing impact parameter values. The DA also decreases for the rising values of the gauge potential A. Therefore, we can deduce that the DA is in an inverse relation with the gauge potential A and impact parameter b. Moreover, we observe a strong deflection at the low values of impact parameter b at b = 1.004 for different variations of plots. We also observe that the deviating curve represents that the DA is less than the Schwarzschild case in the presence of the gauge potential.
Figure 1. The DA ψ versus impact parameter b for (left) different values of A and fixed m = 1 and e = 0.6, and (right) for different values of e and fixed m = 1 and A = 0.5. |
The plot on the right depicts the behavior of ψ for varying values of charge e and for fixed m = 1, A = 0.5. One can observe that the DA is decreasing with the rising values of b. For the increasing values of charge e, the DA decreases. Therefore, it can be observed that the DA is in an inverse relation with the impact parameter b and charge e. A strong deflect can be observed at the low values of impact parameter b at b = 1.03 for different variations of plots. We also notice that when there is an electric charge, the DA is less than that in the Schwarzschild case.
3.1. Light-deflected angle effect under the influence of the non-magnetic plasma
The calculation of the DA in a non-magnetic plasma medium serves as the foundation for this section. Let v be the speed of light through hot ionized gas to encompass the effects of non-magnetic plasma. The refractive index n(r) = c/v is represented [74] by24 ) can be rewritten as follows25 ) and their derivatives into equation (26 ), we compute the Ricci scalar in the presence of a plasma medium as follows13 ) and the value of the Ricci scalar in the presence of a plasma medium, we obtain the optical Gaussian curvature as follows2 ), the DA of the accelerating AdS BH in a non-magnetic plasma for the terms in leading order can be computed as29 ) can be converted into equation (22 ) by ignoring the plasma effects, i.e. $\tfrac{{\omega }_{0}^{2}}{{\omega }_{\infty }^{2}}\to 0$. Moreover, we check the graphical behavior of the impact parameter on the DA under the effects of non-magnetic plasma frequencies.
$\begin{eqnarray}n=\sqrt{1-\displaystyle \frac{{\omega }_{0}^{2}}{{\omega }_{\infty }^{2}}F}.\end{eqnarray}$
When the observer’s predicted photon frequency at infinity is ${\omega }_{\infty }^{2}$ and the electron plasma frequency is denoted by ${\omega }_{0}^{2}$, the associated optical metric can be expressed as $\begin{eqnarray}{\rm{d}}{t}^{2}={n}^{2}\left(\displaystyle \frac{G}{F}{\rm{d}}{r}^{2}+\displaystyle \frac{R}{F}{\rm{d}}{\phi }^{2}\right).\end{eqnarray}$
The optical metric equation ( $\begin{eqnarray}{\rm{d}}{t}^{2}=U{\rm{d}}{r}^{2}+V{\rm{d}}{\phi }^{2},\end{eqnarray}$
where $U=\tfrac{{n}^{2}G}{F}$ and $V=\tfrac{{n}^{2}R}{F}$. The Ricci scalar (${ \mathcal R }$) of optical metrics is calculated using the following formula $\begin{eqnarray}{ \mathcal R }=\displaystyle \frac{2U^{\prime\prime} (r)V(r)U(r)-U^{\prime} (r)V^{\prime} (r)U(r)-U{{\prime} }^{2}(r)V(r)}{2{V}^{2}(r){U}^{2}(r)}.\end{eqnarray}$
After putting the values of the metric functions from equation ( $\begin{eqnarray}\begin{array}{l}{ \mathcal R }\approx -\displaystyle \frac{4m}{{r}^{3}}-\displaystyle \frac{20{{mA}}^{2}{e}^{2}}{{r}^{3}}+\displaystyle \frac{6{e}^{2}}{{r}^{4}}\\ -\displaystyle \frac{12{{me}}^{2}}{{r}^{5}}-\displaystyle \frac{6m{\omega }_{0}^{2}}{{r}^{3}{\omega }_{\infty }^{2}}-\displaystyle \frac{132{{mA}}^{2}{e}^{2}{\omega }_{0}^{2}}{{r}^{3}{\omega }_{\infty }^{2}}\\ -\displaystyle \frac{10{e}^{2}{\omega }_{0}^{2}}{2{r}^{4}{\omega }_{\infty }^{2}}+\displaystyle \frac{52{{me}}^{2}{\omega }_{0}^{2}}{{r}^{5}}+O({m}^{2},{A}^{4},{e}^{4}).\end{array}\end{eqnarray}$
Using the formula for Gaussian curvature from equation ( $\begin{eqnarray}\begin{array}{l}{\bf{K}}\simeq -\displaystyle \frac{2m}{{r}^{3}}-\displaystyle \frac{10{{mA}}^{2}{e}^{2}}{{r}^{3}}+\displaystyle \frac{3{e}^{2}}{{r}^{4}}\\ -\displaystyle \frac{6{{me}}^{2}}{{r}^{5}}-\displaystyle \frac{3m{\omega }_{0}^{2}}{{r}^{3}{\omega }_{\infty }^{2}}-\displaystyle \frac{66{{mA}}^{2}{e}^{2}{\omega }_{0}^{2}}{{r}^{3}{\omega }_{\infty }^{2}}\\ -\displaystyle \frac{5{e}^{2}{\omega }_{0}^{2}}{2{r}^{4}{\omega }_{\infty }^{2}}+\displaystyle \frac{26{{me}}^{2}{\omega }_{0}^{2}}{{r}^{5}}+O({m}^{2},{A}^{4},{e}^{4}).\end{array}\end{eqnarray}$
We derive the angle by utilizing the GB thereom; for this role, we employ a straight-line approach $r(t)=\tfrac{b}{\sin \phi }$ at zero-order. By substituting the values of K and the determinant into the above equation ( $\begin{eqnarray}\begin{array}{l}\psi \approx \displaystyle \frac{4m}{b}-\displaystyle \frac{20{{mA}}^{2}{e}^{2}}{b}-\displaystyle \frac{3{e}^{2}\pi }{4{b}^{2}}\\ +\displaystyle \frac{8{{me}}^{2}}{3{b}^{3}}+\displaystyle \frac{6m{\omega }_{0}^{2}}{b{\omega }_{\infty }^{2}}+\displaystyle \frac{76{{mA}}^{2}{e}^{2}{\omega }_{0}^{2}}{b{\omega }_{\infty }^{2}}-\displaystyle \frac{5{e}^{2}\pi {\omega }_{0}^{2}}{2{b}^{2}{\omega }_{\infty }^{2}}\\ +\displaystyle \frac{104{{me}}^{2}{\omega }_{0}^{2}}{9{b}^{3}}+O({m}^{2},{A}^{4},{e}^{4}).\end{array}\end{eqnarray}$
The ψ depends on the mass m, gauge potential A, charge e and impact parameter b, as well as the non-magnetic plasma frequencies ${\omega }_{0}^{2}$ and ${\omega }_{\infty }^{2}$. It can be observed that the DA is in an inverse relation with the impact parameter. We check that equation (Figure 2 shows the graphical conduct of the DA ψ versus impact parameter b. The left panel represents the behavior of ψ for varying values of gauge potential A and for fixed m = 1, e = 0.5 and ${\omega }_{0}^{2}/{\omega }_{\infty }^{2}=0.1$. One can observe that the DA is decreasing with the increasing impact parameter values. For the rising values of the gauge potential A, the DA decreases. Therefore, we can say that the DA is in an inverse relation with impact parameter b and gauge potential A. One can observe a strong deflection at the low values of impact parameter b in the region 1.12 ≤ b ≤ 1.20 for different variations of graphs. It is also notable that for variations of the gauge potential in the presence of non-magnetic plasma, the DA is lower than compared to the Schwarzschild case.
Figure 2. The DA ψ versus impact parameter b for (left) different values of gauge potential A and fixed m = 1, e = 0.5 and frequency ratios ${\omega }_{0}^{2}/{\omega }_{\infty }^{2}=0.1$, and (right) for different frequency ratios ${\omega }_{0}^{2}/{\omega }_{\infty }^{2}$ and fixed m = 1, A = 0.5 and e = 0.2. |
The right panel shows the behavior of ψ for varying values of frequency ratio ${\omega }_{0}^{2}/{\omega }_{\infty }^{2}$ and for fixed m = 1, A = 0.5 and e = 0.2. One can observe that the DA is decreasing with the rising values of b. With the increasing frequency ratio values, the DA is also increasing. Therefore, it can be deduced that the DA is in an inverse relation with impact parameter b and in a direct relation with the frequency ratio ${\omega }_{0}^{2}/{\omega }_{\infty }^{2}$. It can be seen that a strong deflection occurs at the low values of impact parameter b in the domain 1 ≤ b ≤ 1.02 for distinct graphs. In this case, the DA is greater than the Schwarzschild case due to the presence of a plasma medium.
3.2. Light-deflected angle analysis with the Keeton and Petters approach
The Keeton and Petters method is used to compute the DA for an accelerating charged AdS BH solution. Keeton and Petters developed that technique to calculate the expression [75–77] and to make it more feasible with their approach. The limit of weak deflection is defined as an expansion of series with a single mass m variable in the post-post Newtonian (PPN) framework, which is an easy way to deal with gravity theories in weak field limits. This idea was compared to the third order [69, 74]. The spacetime for the BH can be given as33 ), the coefficients represent the invariant expression for the bending angle as7 ) for the accelerating AdS BH only in terms of r in the form34 ), we find39 ) is reduced to the DA of Schwarzschild (ψSchw = 4m/b).
$\begin{eqnarray}{\rm{d}}{s}^{2}={A}^{{kz}}{\rm{d}}{t}^{2}+{B}^{{kz}}{\rm{d}}{r}^{2}+{r}^{2}{\rm{d}}{{\rm{\Omega }}}^{2}.\end{eqnarray}$
The metric functions in the form of a PPN series can be written as $\begin{eqnarray}\begin{array}{rcl}{A}^{{kz}} & = & 1+2{u}_{1}\left(\displaystyle \frac{\alpha }{{c}^{2}}\right)+2{u}_{2}{\left(\displaystyle \frac{\alpha }{{c}^{2}}\right)}^{2}+2{u}_{3}{\left(\displaystyle \frac{\alpha }{{c}^{2}}\right)}^{3}+...,\\ {B}^{{kz}} & = & 1-2{v}_{1}\left(\displaystyle \frac{\alpha }{{c}^{2}}\right)+4{v}_{2}{\left(\displaystyle \frac{\alpha }{{c}^{2}}\right)}^{2}-8{v}_{3}{\left(\displaystyle \frac{\alpha }{{c}^{2}}\right)}^{3}+...,\end{array}\end{eqnarray}$
where α is a three-dimensional Newtonian potential, and it can be defined [72] as $\begin{eqnarray}\displaystyle \frac{\alpha }{{c}^{2}}=-\displaystyle \frac{m}{r},\end{eqnarray}$
with c, m and r denoting the speed of light in vacuum, mass of BH and radius of BH, respectively. In an expansion of a series, the light deflection ψ is specified [76] as $\begin{eqnarray}\begin{array}{l}\psi \approx {S}_{1}\left(\displaystyle \frac{m}{b}\right)+{S}_{2}{\left(\displaystyle \frac{m}{b}\right)}^{2}\\ +{S}_{3}{\left(\displaystyle \frac{m}{b}\right)}^{3}+O{\left(\displaystyle \frac{m}{b}\right)}^{4}.\end{array}\end{eqnarray}$
In expression ( $\begin{eqnarray}\begin{array}{rcl}{S}_{1} & = & 2({u}_{1}+{v}_{1}),\\ {S}_{2} & = & \left(2{u}_{1}^{2}-{u}_{2}+{u}_{1}{v}_{1}+\displaystyle \frac{{v}_{1}^{2}}{4}+{v}_{2}\right)\pi ,\\ {S}_{3} & = & \displaystyle \frac{2}{3}\left(35{u}_{1}^{3}+15{u}_{1}^{2}{v}_{1}\right.\\ & & -3{u}_{1}(10{u}_{2}+{v}_{1}^{2}-4{v}_{2})+6{u}_{3}\\ & & \left.+{v}_{1}^{3}-6{u}_{2}{v}_{1}-4{v}_{1}{v}_{2}+8{v}_{3}\right).\end{array}\end{eqnarray}$
Now, we recall the metric equation ( $\begin{eqnarray}{{\rm{d}}{s}}^{2}=-F(r){{\rm{dt}}}^{2}+G(r){{\rm{d}}{r}}^{2}+{r}^{2}{\rm{d}}{{\rm{\Omega }}}^{2},\end{eqnarray}$
where $\begin{eqnarray}\begin{array}{l}F(r)=\displaystyle \frac{1}{G(r)}=1-\displaystyle \frac{2m}{r}+\displaystyle \frac{{e}^{2}}{{r}^{2}}\\ -{A}^{2}{r}^{2}-2{A}^{2}{mr}-{A}^{2}{e}^{2}+\displaystyle \frac{{r}^{2}}{{l}^{2}}.\end{array}\end{eqnarray}$
Therefore, we equate the F(r) with Akz and G(r) with Bkz and we can limit the coefficients of the PPN as follows $\begin{eqnarray}\begin{array}{l}{u}_{1}=1,\,\,\,{u}_{2}=\displaystyle \frac{{e}^{2}}{2{m}^{2}},\,\,\,{u}_{3}=0,\\ {v}_{1}=1-{A}^{2}{e}^{2},\,\,\,{v}_{2}=\displaystyle \frac{{e}^{2}}{4{m}^{2}},\,\,\,{v}_{3}=-\displaystyle \frac{{e}^{2}}{4{m}^{2}}.\end{array}\end{eqnarray}$
After inserting of all the coefficients into equation ( $\begin{eqnarray}\begin{array}{l}{S}_{1}=4(1-{A}^{2}{e}^{2}),\\ {S}_{2}=\displaystyle \frac{13\pi }{4}-\displaystyle \frac{3{e}^{2}\pi }{4{m}^{2}}-3{A}^{2}{e}^{2}\pi ,\\ {S}_{3}=22-16{A}^{2}{e}^{2}+\displaystyle \frac{22{e}^{2}}{3{m}^{2}}.\end{array}\end{eqnarray}$
As an expression, the DA of the charged accelerating AdS BH solution utilizing the Keeton and Petters method can be estimated as follows $\begin{eqnarray}\begin{array}{l}\psi \approx 4(1-{A}^{2}{e}^{2})\left(\displaystyle \frac{m}{b}\right)\\ +\left(\displaystyle \frac{13\pi }{4}-\displaystyle \frac{3{e}^{2}\pi }{4{m}^{2}}-3{A}^{2}{e}^{2}\pi \right){\left(\displaystyle \frac{m}{b}\right)}^{2}\\ +\left(22-16{A}^{2}{e}^{2}+\displaystyle \frac{22{e}^{2}}{3{m}^{2}}\right){\left(\displaystyle \frac{m}{b}\right)}^{3}+O{\left(\displaystyle \frac{m}{b}\right)}^{4}.\end{array}\end{eqnarray}$
After simplification and getting results only for leading order terms, we get our results in this form $\begin{eqnarray}\begin{array}{l}\psi \approx \displaystyle \frac{4m}{b}-\displaystyle \frac{4{{mA}}^{2}{e}^{2}\pi }{b}-\displaystyle \frac{3{e}^{2}\pi }{4{b}^{2}}\\ +\displaystyle \frac{22{{me}}^{2}}{3{b}^{3}}+O({m}^{2},{A}^{4},{e}^{4}).\end{array}\end{eqnarray}$
The ψ depends on the mass m, charge e, gauge potential A and impact parameter b. We can observe that we get almost the same result for the DA as the non-plasma medium. The difference is only in the coefficients due to the approximate result. Moreover, by setting e = 0 = A, the obtained angle equation (Figure 3 represents the graphical conduct of ψ via impact parameter b. One can observe that the DA is compared with the Schwarzschild case when the gauge potential and charge are neglected (i.e. A = 0 = e) and the graph in the Schwarzschild case intersects and goes asymptotically flat until b → ∞ with the case when the charge e and potential show variant values. Furthermore, we can see that in the Schwarzschild case, the DA is stronger at the low value of b while, in the presence of the gauge potential A and charge e, the DA is at its highest point for the low value of b and it becomes weaker with the increment in b.
Figure 3. The DA ψ via impact parameter b for (left) different values of A and fixed m = 1 and e = 0.6, and (right) for different values of e and fixed m = 1 and A = 0.5. |
4. Evaluation of shadows in non-magnetized plasma
In this section, we check the shadow from an accelerating charged AdS BH under the influence of a non-magnetized plasma medium. The interior of the so-called apparent critical curve or boundary is depicted by a BH shadow. The critical curve is defined when a distant observer follows it back to the BH: the light rays that are a part of its technique are a photon asymptotically bound orbit. Huang et al employed [78] a new ray-tracing idea to study the photon’s velocity around a BH in the presence of plasma (whose density is a function of the radius coordinate), and the impact of the plasma on the BH’s shadow. The discussion, technique and physical analysis have all been altered, and significant information has been added to the BH solution, which now compares to their previous published results. To calculate the shadow for the accelerating AdS BH, the metric equation (7 ) can be rewritten in the expression42 ) derivation from Maxwell’s equations with a two-fluid source. The light rays are the Hamilton equation solution in the form44 ), it follows that jt and jφ of motion are constants. We write ω:= − jt. The frequency ω of a static observer measured by a function of r is the expression
$\begin{eqnarray}\begin{array}{l}{\rm{d}}{s}^{2}=-F(r,\theta ){\rm{d}}{t}^{2}+G(r,\theta ){\rm{d}}{r}^{2}+H(r,\theta )\\ \times \left({\rm{d}}{\theta }^{2}+\displaystyle \frac{R(r,\theta )}{H(r,\theta )}{\rm{d}}{\phi }^{2}\right).\end{array}\end{eqnarray}$
The equation for the Hamilton Jacobi approach [77] can be defined as $\begin{eqnarray}\begin{array}{l}H=\displaystyle \frac{1}{2}\left({g}^{{ik}}{j}_{i}{j}_{k}+{\omega }_{\infty }({r}^{2})\right)=\displaystyle \frac{1}{2}\\ \times \left(-\displaystyle \frac{{j}_{t}^{2}}{F(r,\theta )}+\displaystyle \frac{{j}_{r}^{2}}{G(r,\theta )}+\displaystyle \frac{{j}_{\phi }^{2}}{H(r,\theta )}+{\omega }_{\infty }^{2}(r)\right),\end{array}\end{eqnarray}$
where ω∞ denotes the photon frequency. We mention the introduction given in the literature for a Hamiltonian equation ( $\begin{eqnarray}\dot{{j}_{i}}=-\displaystyle \frac{\partial H}{\partial {x}^{i}},\,\,\,\dot{{x}^{i}}=-\displaystyle \frac{\partial H}{\partial {j}_{i}},\end{eqnarray}$
which implies that $\begin{eqnarray}\begin{array}{c}\dot{{j}_{t}}=-\displaystyle \frac{\partial H}{\partial t}=0,\,\,\,\,\,\dot{{j}_{\phi }}=-\displaystyle \frac{\partial H}{\partial \phi }=0,\\ \dot{{j}_{r}}=-\displaystyle \frac{\partial H}{\partial r}=0,\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}\dot{{j}_{r}}=-\displaystyle \frac{\partial H}{\partial r}=\displaystyle \frac{1}{2}\left(-\displaystyle \frac{{j}_{t}^{2}F^{\prime} (r,\theta )}{F{\left(r,\theta \right)}^{2}}\right.\\ \left.+\displaystyle \frac{{j}_{r}^{2}G^{\prime} (r,\theta )}{G{\left(r,\theta \right)}^{2}}+\displaystyle \frac{{j}_{\phi }^{2}H^{\prime} (r,\theta )}{H{\left(r,\theta \right)}^{2}}-\displaystyle \frac{d}{{dr}}{\omega }_{\infty }^{2}(r)\right),\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}\dot{t}=\displaystyle \frac{\partial H}{\partial {j}_{t}}=-\displaystyle \frac{{j}_{t}}{F(r,\theta )},\\ \dot{\phi }=\displaystyle \frac{\partial H}{\partial {j}_{\phi }}=-\displaystyle \frac{{j}_{\phi }}{H(r,\theta )},\\ \dot{r}=\displaystyle \frac{\partial H}{\partial {j}_{r}}=-\displaystyle \frac{{j}_{r}}{G(r,\theta )},\end{array}\end{eqnarray}$
by considering H = 0, we get $\begin{eqnarray}0=-\displaystyle \frac{{j}_{t}^{2}}{F(r,\theta )}+\displaystyle \frac{{j}_{r}^{2}}{G(r,\theta )}+\displaystyle \frac{{j}_{\phi }^{2}}{H(r,\theta )}+{\omega }_{\infty }^{2}(r),\end{eqnarray}$
where a dot and prime represent differentiation w.r.t. λ (affine parameter) and differentiation w.r.t. r, respectively. From equation ( $\begin{eqnarray}\omega (r)=\displaystyle \frac{{\omega }_{0}}{\sqrt{F(r,\theta )}}.\end{eqnarray}$
For the orbit expression, we use equation (46 ) to find the values47 ), this results in50 ), can be rewritten using equation (52 ) as follows55 ) into equation (56 ), we get57 ), we get the shadow angular radius as51 ) for many applications, and we can consider that the observer region has a very low plasma density. Then, equation (51 ) implies that58 ) gets the form60 ) obtains the final result in the form
$\begin{eqnarray}\displaystyle \frac{{\rm{d}}{r}}{{\rm{d}}\phi }=\displaystyle \frac{\dot{r}}{\dot{\phi }}=\displaystyle \frac{H(r,\theta ){j}_{r}}{G(r,\theta ){J}_{\phi }}.\end{eqnarray}$
By substituting for jr from equation ( $\begin{eqnarray}\displaystyle \frac{{\rm{d}}{r}}{{\rm{d}}\phi }=\pm \displaystyle \frac{\sqrt{H(r,\theta )}}{\sqrt{G(r,\theta )}}\sqrt{\displaystyle \frac{{\omega }_{0}^{2}z{\left(r\right)}^{2}}{{j}_{\phi }^{2}}-1},\end{eqnarray}$
where the ω0 is a static observer frequency, and we have defined the function $\begin{eqnarray}z{\left(r\right)}^{2}=\displaystyle \frac{H(r,\theta )}{F(r,\theta )}\left(1-F(r,\theta )\displaystyle \frac{{\omega }_{\infty }^{2}}{{\omega }_{0}^{2}}\right).\end{eqnarray}$
As X corresponds to the trajectory turning point, the condition $\tfrac{{\rm{d}}{r}}{{\rm{d}}\phi {| }_{X}}=0$ has to hold. This expression relates to X to the constant of motion $\tfrac{{j}_{\phi }}{{\omega }_{0}}$ as $\begin{eqnarray}z{\left(X\right)}^{2}=\displaystyle \frac{{j}_{\phi }^{2}}{{\omega }_{0}^{2}}.\end{eqnarray}$
Now, we consider a light beam projected from the observer’s point at r0 into the past at an angle θ relative to the radial direction in the given form $\begin{eqnarray}\begin{array}{l}\cot \theta ={\left.\pm \displaystyle \frac{\sqrt{{{\bf{g}}}_{{rr}}}}{\sqrt{{{\bf{g}}}_{\phi \phi }}}\displaystyle \frac{{\rm{d}}{r}}{{\rm{d}}\phi }\right|}_{r={r}_{0}}\\ ={\left.\pm \displaystyle \frac{\sqrt{H(r,\theta )}}{\sqrt{G(r,\theta )}}\displaystyle \frac{{\rm{d}}{r}}{{\rm{d}}\phi }\right|}_{r={r}_{0}}.\end{array}\end{eqnarray}$
If the ray reaches a minimum radius R, the orbit expression, equation ( $\begin{eqnarray}\displaystyle \frac{{\rm{d}}{r}}{{\rm{d}}\phi }=\pm \displaystyle \frac{\sqrt{H(r,\theta )}}{\sqrt{G(r,\theta )}}\sqrt{\displaystyle \frac{z{\left(r\right)}^{2}}{z{\left(X\right)}^{2}}-1}.\end{eqnarray}$
For the angle θ, we obtain $\begin{eqnarray}{\cot }^{2}\theta =\displaystyle \frac{z{\left(r\right)}^{2}}{z{\left(X\right)}^{2}}-1,\end{eqnarray}$
where $\begin{eqnarray}1+{\cot }^{2}\theta =\displaystyle \frac{1}{{\sin }^{2}\theta }.\end{eqnarray}$
By inserting equation ( $\begin{eqnarray}{\sin }^{2}\theta =\displaystyle \frac{z{\left(r\right)}^{2}}{z{\left(X\right)}^{2}}.\end{eqnarray}$
Rays spiral asymptotically towards a circular orbit with radius rph to find the boundary of the shadow θ. As a result, by sending X → rph in equation ( $\begin{eqnarray}{\sin }^{2}\theta =\displaystyle \frac{z{\left({r}_{{\rm{p}}{\rm{h}}}\right)}^{2}}{z{\left({r}_{0}\right)}^{2}},\end{eqnarray}$
where z(r) is devoted by the expression equation ( $\begin{eqnarray}z{\left(r\right)}^{2}=\displaystyle \frac{H({r}_{0},\theta )}{F({r}_{0},\theta )},\end{eqnarray}$
and equation ( $\begin{eqnarray}\begin{array}{l}{\sin }^{2}\theta =\displaystyle \frac{H({r}_{{\rm{p}}{\rm{h}}},\theta )F({r}_{0},\theta )}{F({r}_{{\rm{p}}{\rm{h}}},\theta )H({r}_{0},\theta )}\\ \times \left(1-\displaystyle \frac{F({r}_{{\rm{p}}{\rm{h}}},\theta ){\omega }_{\infty }^{2}({r}_{{\rm{p}}{\rm{h}}})}{{\omega }_{0}^{2}({r}_{{\rm{p}}{\rm{h}}})}\right).\end{array}\end{eqnarray}$
After inserting the values of the metric functions F and H at the equatorial plane (θ = π/2), equation ( $\begin{eqnarray}{\sin }^{2}\theta =\displaystyle \frac{{r}_{{\rm{p}}{\rm{h}}}^{2}\left(\tfrac{{r}_{0}}{{l}^{2}}+(1-{A}^{2}{r}_{0}^{2})(1+\tfrac{{e}^{2}-2{{mr}}_{0}}{{r}_{0}^{2}})\right)\left(1-\tfrac{{\omega }_{\infty }^{2}({r}_{{\rm{p}}{\rm{h}}})\left(\tfrac{{r}_{{\rm{p}}{\rm{h}}}}{{l}^{2}}+(1-{A}^{2}{r}_{{\rm{p}}{\rm{h}}}^{2})\left(1+\tfrac{{e}^{2}+2{{mr}}_{{\rm{p}}{\rm{h}}}}{{r}_{{\rm{p}}{\rm{h}}}^{2}}\right)\right)}{{\omega }_{0}^{2}({r}_{{\rm{p}}{\rm{h}}})}\right)}{{r}_{0}^{2}\left(\tfrac{{r}_{{\rm{p}}{\rm{h}}}}{{l}^{2}}+(1-{A}^{2}{r}_{{\rm{p}}{\rm{h}}}^{2})\left(1+\tfrac{{e}^{2}-2{{mr}}_{{\rm{p}}{\rm{h}}}}{{r}_{{\rm{p}}{\rm{h}}}^{2}}\right)\right)}.\end{eqnarray}$
The shadow of the BH depends upon the gauge potential A, charge e, mass m, AdS radiuslphoton radius rph, observer’s radius r0, and plasma frequencies ω0 and ω∞. It is also worth mentioning that when we set e = 0, A = 1 = l in the above equation, our results reduced into the shadow radius of the Schwarzschild BH in a plasma medium [79].5. Graphical analysis of shadow under the influence of non-magnetized plasma
This section is related to the geometrical analysis of shadow from an accelerating charged AdS BH. We analyze the effects of non-magnetic plasma frequencies on the shadow of the BH.
Figure 4 shows the behavior of the shadow θ via the observer radius r0. The left plot shows the conduct for varying frequency ratios ${\omega }_{0}^{2}/{\omega }_{\infty }^{2}$ and fixed m = 1, e = 0.2, rph = 0.5 and l = 0.5. The θ slowly increases in the domain 2.2 ≤ r0 ≤ 2.8. We can observe that the radius size for the accelerating charged AdS BH is greater compared to the Schwarzschild BH.
Figure 4. The shadow θ versus observer radius r0: (left) for different frequency ratios ${\omega }_{0}^{2}/{\omega }_{\infty }^{2}$ and fixed m = 1, e = 0.2, rph = 0.5 and l = 0.5; (right) for different values of charge e and fixed $m=1,\,A=0.1,\,l=0.2,\,{r}_{{\rm{p}}{\rm{h}}}=0.05,\,\mathrm{and}\,{\omega }_{0}^{2}/{\omega }_{\infty }^{2}=0.4$. |
The right plot depicts the conduct of θ for fixed $m=1,\,A=0.1,\,l=0.2,\,{r}_{{\rm{p}}{\rm{h}}}=0.05,\,\mathrm{and}\,{\omega }_{0}^{2}/{\omega }_{\infty }^{2}=0.4$, as well as for varying e. One can observe that the θ constantly decreases and takes an asymptotically flat form until r0 → ∞ in the region 0.3 ≤ r0 ≤ 1.2. Moreover, with the increasing values of e, the θ is decreasing. The shadow radius for the Schwarzschild case is greater than the accelerating charged AdS BH.
The contour plots for the BH shadow can be represented in the form of the celestial coordinates α and β.
Figure 5(i) represents the contour plots of shadow for distinct values of the photon radius rph. We can observe that the effective size of the shadow rises with the increase in rph, while figure 5(ii) depicts the contour plots of shadow for varying r0. One can observe that the shadow size decreases with the increase in r0.
Figure 5. The contour plots for the shadow of the BH for: (i) fixed $m=1,\,A=0.5,\,{r}_{0}=0.1,\,{\omega }_{0}^{2}/{\omega }_{\infty }^{2}=0.3,\,e=0.2$ and different rph = 0.01 (purple), 0.02 (yellow), 0.03 (green), 0.04 (orange), 0.05 (blue), 0.06 (red); and (ii) for different r0 = 0.05 (purple), 0.06 (yellow), 0.07 (green), 0.08 (orange), 0.09 (blue), 0.1 (red) and fixed $m=1,\,A=0.5,\,{r}_{{\rm{p}}{\rm{h}}}=0.01,\,{\omega }_{0}^{2}/{\omega }_{\infty }^{2}=0.3,\,e=0.2$. |
6. Conclusions
In this study, we have used Gaussian curvature to first determine the weak DA of the accelerating charged AdS BH in the absence/presence of non-magnetic plasma. To do this, we calculated the DA using the GB theorem, as first suggested by Gibbons and Werner. In a non-plasma medium, the DA depends on the mass m, gauge potential A, charge e and impact parameter b. It can be concluded that the DA is in an inverse relation with the impact parameter b. Additionally, when we neglected the potential and charge parameters, our result in equation (22 ) reduced into the DA of a Schwarzschild BH $(\tfrac{4m}{b}$). We have analyzed our results using graphic interpretation under the effects of different parameters. We can conclude that, for varying values of gauge potential and charge, the DA is in an inverse relation with the gauge potential A, charge e and parameter b. We also noticed that in the presence of the gauge potential and charge, there is less deflection compared to the Schwarzschild case. We also observed a strong deflection at the low values of impact parameter b. Moreover, the DA in the non-magnetic plasma medium depends on the mass m, gauge potential A, charge e, impact parameter b and non-magnetic plasma frequencies ${\omega }_{0}^{2}$ & ${\omega }_{\infty }^{2}$. It can be concluded that the DA is also in an inverse relation with the impact parameter b in the presence of a non-magnetic plasma medium. We observed that equation (29 ) can be converted into equation (22 ) by neglecting the effects of non-magnetic plasma, i.e. $\tfrac{{\omega }_{0}^{2}}{{\omega }_{\infty }^{2}}\to 0$. Furthermore, via graphical interpretation of the DA in the non-magnetic plasma medium, we have explained that, for varying values of the gauge potential A, the DA is in an inverse relation with the impact parameter b and also shows low values when we increase the values of the gauge potential A. We have concluded that in the presence of a non-magnetic plasma medium, there is less deflection compared to the Schwarzschild case with variation of the gauge potential. Moreover, for varying values of the plasma frequency ratio ω0/ω∞, the DA is in an inverse relation with the impact parameter b and a direct relation with the frequency ratio. Furthermore, in the presence of a non-magnetic plasma medium, the deflection is higher compared to the Schwarzschild case.
Moreover, by using the Keeton and Petter approach, we have discovered the DA in the non-plasma scenario to verify our results with the Schwarzschild case. We have concluded that the DA depends on the mass m, charge e, gauge potential A and impact parameter b. By setting e = 0 = A, the obtained angle in equation (39 ) is reduced to the DA of the Schwarzschild BH. We have also compared our results using graphs.
Finally, by investigating the ray-tracing approach of an accelerating charged AdS BH, we also discovered the shadow for the BH. The shadow’s image is then depicted in the distant observer’s sky once we have determined the shadow’s radius. The shadow of the BH depends upon the gauge potential A, charge e, mass m, AdS radius l, photon radius rph, observer’s radius r0 and plasma frequencies ω0 and ω∞.
Furthermore, we have studied the effects of the plasma medium at the shadow’s radius in the regime of the observer’s radius r0; at first, the θ slowly decreases in the domain 2.2 ≤ r0 ≤ 2.8 with variations of the plasma frequency ratio. One can also detect that the radius size of the accelerating charged BH is greater compared to the Schwarzschild case. Moreover, it has been observed that the θ constantly decreases and takes the an asymptotically flat form until r0 → ∞ in the region 0.3 ≤ r0 ≤ 1.2. Moreover, with the increasing charge values, the shadow radius decreases. The shadow radius of the Schwarzschild case is more than the accelerating charged AdS BH for variations of charge. We also plotted contour graphs of the accelerating charged BH for different variations of photon and observer radius, and concluded that the effective size of the shadow rises with the increase in rph while the shadow size decreases with the increase in r0.