1. Introduction
Observing the shadow of a black hole provides a significant opportunity to study its characteristics and parameters. The paths of light particles around a black hole are impacted by their angular momentum. Photons with higher angular momentum scatter away from the black hole and can reach a distant observer at infinity. In contrast, photons with lower angular momentum are drawn into the black hole, resulting in a dark area that defines the shadow region around the horizons of the black hole. The black hole shadow forms near the event horizon, providing insight into the basic geometrical structure of horizons. Recent astronomical observations from the Event Horizon Telescope (EHT) confirm the existence of the black hole Sagittarius A* in our galaxy Milky Way [1-6] and a supermassive black hole at the galactic center of Messier 87 [7-12]. The shadow of a black hole appears as a two-dimensional dark region surrounded by electromagnetic radiation and matter particles. The dark region occurs due to the infalling photons inside the event horizon while the photons that form orbits can be visible to an outside observer. The shadow figure depends on black hole parameters as well as the surrounding matter field around the black hole. Therefore, the study of the shadow figure is a valuable tool for detecting its spin and other deformation parameters, as its shape and size reflect the geometry surrounding the black hole. Astrophysical black hole candidates exhibit a significant amount of rotation. This drives our interest in studying rotating black holes and their shadows. Since regular black holes include the Kerr metric as a special case, we view them as prototypes for a broader category of non-Kerr families. In these cases, the metric retains the same form, but the mass parameter M is replaced with a function m(r). This method is very useful for studying the various parameters associated with black holes in the general theory of relativity, as well as in different theories of gravity. It also provides an opportunity to test these theories using astrophysical black holes. In general relativity, the shape and size of a black hole’s shadow are determined solely by its spin and mass [13, 14]. However, in alternative theories of gravity, the shadow of a black hole can be influenced by various other parameters. Therefore, it is worthwhile to investigate the properties of black hole shadows within different modified gravitational theories. Over the past few years, various researchers have explored this topic in modified theories of gravity such as [15-52]. The properties of black hole shadows are affected by nearby matter fields, which has led to investigations into these shadow characteristics in plasma conducted by [53-55]. The intriguing aspect of a black hole's shadow has been investigated in higher-dimensional spacetime [56-60]. Cunha et al [61, 62] developed a ray tracing method to better understand photon rays in the vicinity of Kerr spacetime, both with and without scalar hair.
In this article, we consider an observer positioned in an arbitrary location rather than at infinity, and we trace various shadow images cast by regular anti-de Sitter (AdS) black holes as seen by that observer. The articles [34, 63] examine the shadow images cast by Kerr-Newman-NUT-AdS black holes and rotating AdS black holes within the braneworld scenario. Additionally, [64, 65] investigated the shadows of superentropic and Bardeen black holes in AdS spacetime. The current cosmological observations verified that our Universe is accelerating and considering that accelerating, [66, 67] investigated the shadows of rotating black with the cosmological constant. In this article, we present a detailed analysis of the shadows of rotating regular AdS black holes. Our focus is primarily on how the spin and deformation parameters affect the characteristics of the black hole shadow. Compared to the shadows produced by Kerr [13] and nonsingular [21] black holes, the shadow of a rotating regular AdS black hole is smaller and appears more distorted.
This paper is organized as follows: section 2 describes the properties of the rotating regular AdS black hole. In section 3 , we calculate the equations of motion for the geodesics of photons. Following that, in section 4 , we present various shadow images. Additionally, we estimate the energy emission rate and conclude with our final remarks in section 6 .
2. A brief overview of the rotating regular AdS black hole
This section focuses on the spacetime structure of regular black holes that are minimally coupled to a nonlinear source term in the presence of a negative cosmological constant. We will start by discussing the static spacetime metric, which possesses spherical symmetry. Following this, we will review the rotating counterparts of these solutions in Anti-de Sitter (AdS) spacetime. We will describe a four-dimensional static black hole that maintains spherical symmetry as defined by 3 ) reads [69, 72],4 ) with the energy-momentum tensor (5 ) and the Lagrangian (9 ), we get the metric function with q2 = 2Mk, having the form [68, 69, 72]11 ). The equation of gauge potential for a nonlinear rotating charged black hole is given by [73, 74]12 ) in the rotational case, have Maxwell field invariant [73]
$\begin{eqnarray}{\rm{d}}{s}^{2}=-f(r){\rm{d}}{t}^{2}+\frac{{\rm{d}}{r}^{2}}{f(r)}+{r}^{2}{\rm{d}}{{\rm{\Omega }}}_{2}^{2},\end{eqnarray}$
where function f(r) is the radial dependent metric function and the line element on a two-dimensional sphere with unit radius is given by ${\rm{d}}{{\rm{\Omega }}}_{2}^{2}={\rm{d}}{\theta }^{2}+{\sin }^{2}\theta {\rm{d}}{\phi }^{2}$. The action that describes the regular AdS black hole solution of our interest is best formulated as follows, as defined in the works by [68, 69] $\begin{eqnarray}{ \mathcal I }=\int {{\rm{d}}}^{4}x\sqrt{-g}[R-{ \mathcal L }\left({ \mathcal F }\right)+6{l}^{-2}].\end{eqnarray}$
The action of the black hole, as mentioned in the equation above, has a Ricci scalar quantity R and a radius of curvature l that is related to the cosmological constant by Ʌ = - 3/l2 and ${ \mathcal L }({ \mathcal F })$ is the Lagrangian that includes a nonlinear electromagnetic source and can be described as [68, 69] $\begin{eqnarray}{ \mathcal L }\left({ \mathcal F }\right)={{\rm{e}}}^{\left[-s{\left(2{g}^{2}{ \mathcal F }\right)}^{\frac{1}{4}}\right]}{ \mathcal F },\end{eqnarray}$
with ${ \mathcal F }={F}_{\mu \nu }{F}^{\mu \nu }/4$, and Fμν = 2∇[μAν]. We choose for the sake of clarity, s = q/2M, where q and M are the electric charge and mass of the black hole respectively [70, 71]. The following equations can be obtained by the verification of black hole action ${ \mathcal I }$ concerning the metric tensor gμν and vector potential Aμ as $\begin{eqnarray}\begin{array}{rcl}{G}_{\mu \nu }-\frac{3}{{l}^{2}} & = & {T}_{\mu \nu },\\ {{\rm{\nabla }}}_{\mu }\left(\frac{d{ \mathcal L }\left({ \mathcal F }\right)}{d{ \mathcal F }}{F}^{\nu \mu }\right) & = & 0,\\ {{\rm{\nabla }}}_{\mu }(* {F}^{\mu \nu }) & = & 0,\end{array}\end{eqnarray}$
where the stress-energy tensor, Tμν for the nonlinear source concerned is given as $\begin{eqnarray}{T}_{\mu \nu }=2\left[\frac{d{ \mathcal L }\left({ \mathcal F }\right)}{d{ \mathcal F }}{F}_{\mu \alpha }{F}_{\nu }^{\alpha }-{g}_{\mu \nu }{ \mathcal L }\left({ \mathcal F }\right)\right].\end{eqnarray}$
The form of the Faraday tensor Fμν is written as the following ansatz [70] $\begin{eqnarray}{F}_{\mu \nu }=2{\delta }_{[\mu }^{\theta }{\delta }_{\nu ]}^{\phi }q\left(r\right)\sin \theta .\end{eqnarray}$
The dual electromagnetic field equation, ∇μ(*Fμν) = 0 alternatively follows dF = 0, and that follows $q^{\prime} (r){\rm{d}}r\wedge {\rm{d}}\theta \wedge {\rm{d}}\phi \,=0$, and leads to q(r) = constant = q [70]. The resulting Maxwell tensor can have only one component [70], $\begin{eqnarray}{F}_{\theta \phi }=-q\sin \theta ,\end{eqnarray}$
and $\begin{eqnarray}{ \mathcal F }=\frac{{q}^{2}}{2{r}^{4}}.\end{eqnarray}$
Therefore, the Lagrangian ( $\begin{eqnarray}{ \mathcal L }\left({ \mathcal F }\right)=\frac{{q}^{2}}{2{r}^{4}}{{\rm{e}}}^{-\frac{{q}^{2}}{2Mr}}.\end{eqnarray}$
When we solved the zeroth component of the Einstein field equations ( $\begin{eqnarray}f(r)=1-\frac{2M\,{{\rm{e}}}^{-\frac{k}{r}}}{r}+\frac{{r}^{2}}{{l}^{2}}.\end{eqnarray}$
Through the following calculations, we obtained the regular black hole metric in AdS spacetime, which is given by [68, 69] $\begin{eqnarray}\begin{array}{c}\begin{array}{rcl}{{\rm{ds}}}^{2} & = & -\left(1-\displaystyle \frac{2M\,{{\rm{e}}}^{-\displaystyle \frac{k}{r}}}{r}+\displaystyle \frac{{r}^{2}}{{l}^{2}}\right){\rm{d}}{t}^{2}\\ & & +\displaystyle \frac{1}{\left(1-\displaystyle \frac{2M\,{{\rm{e}}}^{-\displaystyle \frac{k}{r}}}{r}+\displaystyle \frac{{r}^{2}}{{l}^{2}}\right)}{\rm{d}}{r}^{2}+{r}^{2}{\rm{d}}{{\rm{\Omega }}}_{2}^{2}.\end{array}\end{array}\end{eqnarray}$
Next, we derive the rotating solution for the regular AdS spacetimes, as described in equation ( $\begin{eqnarray}{A}_{\mu }=-\displaystyle \frac{g\,a\,\cos \theta }{\Upsilon }{\delta }_{\mu }^{t}+\displaystyle \frac{g({r}^{2}+{a}^{2})\cos \theta }{\Upsilon }{\delta }_{\mu }^{\phi }.\end{eqnarray}$
In the absence of the spin parameter, the expression of the gauge potential reduces for the spherically symmetric spacetime as follows ${A}_{\mu }=g\cos \theta {\delta }_{\mu }^{\phi }$. The gauge potential ( $\begin{eqnarray}{ \mathcal F }=\frac{2{g}^{2}\left({\left(\Upsilon -2{r}^{2}\right)}^{2}-4{a}^{2}{r}^{2}{\cos }^{2}\theta \right)}{{\Upsilon }^{4}}.\end{eqnarray}$
We derive the expressions for the Lagrangian and its derivative with respect to the Maxwell field invariant [73] by solving any two independent components of the Einstein field equations given by $\begin{eqnarray}\begin{array}{rcl}{ \mathcal L }(r) & = & \displaystyle \frac{4{r}^{2}\left(2{r}^{2}\Upsilon \,{m}^{{\prime\prime} }+5\,m^{\prime} \left({\left(\Upsilon -2{r}^{2}\right)}^{2}-8{r}^{2}\Upsilon +4{r}^{2}\right)\right)}{{\Upsilon }^{4}},\\ \displaystyle \frac{{\rm{d}}{ \mathcal L }(r)}{{\rm{d}}{ \mathcal F }} & = & \displaystyle \frac{r\Upsilon \,{m}^{{\prime\prime} }+2m^{\prime} \left(\Upsilon -2{r}^{2}\right)}{2{g}^{2}},\end{array}\end{eqnarray}$
with the $^{\prime} $ and ″ denoting, are the first and second derivative with respect to the radial coordinate r. The regular rotating black holes are defined by their mass, spin, and nonlinear parameters. We consider the rotating regular black hole in Boyer-Lindquist coordinates. In these coordinates, the black hole takes the form of the Kerr metric with additional parameters.Subsequent discussion is devoted to the stationary axisymmetric regular AdS black hole solutions. The rotating regular black hole metric in AdS spacetime, expressed in Boyer-Lindquist coordinates, is best described by 15 ) is parameterized by four distinct parameters that are its mass M, spin a, the free parameter k, and the curvature radius l, and in the absence of cosmological constant the black metric reduces to the rotating nonsingular spacetime [75]. The rotating black hole metric (15 ) simplifies to the Kerr-AdS black hole when k = 0, and further simplifies to the Kerr black hole when Ʌ = 0. The black hole spacetime metric (15 ) shows singularity at ϒ = 0 and ∈r = 0, but ∈r = 0 is coordinate-dependent singularity and can be easily removable by taking a proper coordinate transformation. For k ≪ r the black hole metric (15 ) is reduced for the Kerr-Newman-AdS black hole metric such that [76]15 ) has the form 15 ) has no time t and azimuthal coordinates Φ, hence it corresponds two communicating Killing vectors, ξμ(t) = δμt, and ${\xi }_{(\phi )}^{\mu }={\delta }_{\phi }^{\mu }$ and that satisfies the Killing equation, ξμ;ν + ξν;μ = 0. The various metric components are evaluated as a scalar product of these Killing vectors such that
$\begin{eqnarray}\begin{array}{rcl}{{\rm{ds}}}^{2} & = & -\displaystyle \frac{{\in }_{r}}{\Upsilon }{\left({\rm{d}}t-\displaystyle \frac{a{\sin }^{2}\theta }{{\rm{\Xi }}}{\rm{d}}\phi \right)}^{2}+\displaystyle \frac{\Upsilon }{{\in }_{r}}{\rm{d}}{r}^{2}+\displaystyle \frac{\Upsilon }{{\in }_{\theta }}{\rm{d}}{\theta }^{2}\\ & & +\displaystyle \frac{{\in }_{\theta }{\sin }^{2}\theta }{\Upsilon }{\left(a{\rm{d}}t-\displaystyle \frac{{a}^{2}+{r}^{2}}{{\rm{\Xi }}}{\rm{d}}\phi \right)}^{2},\end{array}\end{eqnarray}$
with $\begin{eqnarray}\Upsilon ={r}^{2}+{a}^{2}{\cos }^{2}\theta ,\,\,{\in }_{r}=\left({r}^{2}+{a}^{2}\right)\left(1+\displaystyle \frac{{r}^{2}}{{l}^{2}}\right)-2Mr{{\rm{e}}}^{-k/r},\end{eqnarray}$
and $\begin{eqnarray}{\in }_{\theta }=1-\displaystyle \frac{{a}^{2}}{{l}^{2}}{\cos }^{2}\theta ,\,\,{\rm{\Xi }}=1-\displaystyle \frac{{a}^{2}}{{l}^{2}}.\end{eqnarray}$
The above black hole metric ( $\begin{eqnarray}{\in }_{r}=\left({r}^{2}+{a}^{2}\right)\left(1+\frac{{r}^{2}}{{l}^{2}}\right)-2Mr+{q}^{2}+{ \mathcal O }({k}^{2}/{r}^{2}).\end{eqnarray}$
The determinant of the metric ( $\begin{eqnarray}\sqrt{-g}=\frac{{\rm{\Upsilon }}\sin \theta }{{\rm{\Xi }}}.\end{eqnarray}$
Since the spacetime metric equation ( $\begin{eqnarray}{\xi }_{(t)}^{\mu }{\xi }_{{(t)}_{\mu }}={g}_{tt}=-1+\frac{2Mr{{\rm{e}}}^{-k/r}}{\Upsilon }-\frac{{r}^{2}+{a}^{2}{\sin }^{2}\theta }{{l}^{2}},\end{eqnarray}$
$\begin{eqnarray}{\xi }_{(t)}^{\mu }{\xi }_{{(\phi )}_{\mu }}={g}_{t\phi }=-\frac{a{\sin }^{2}\theta }{{\rm{\Xi }}}\left(\frac{2Mr{{\rm{e}}}^{-k/r}}{\Upsilon }-\frac{{r}^{2}+{a}^{2}}{{l}^{2}}\right),\end{eqnarray}$
$\begin{eqnarray}{\xi }_{(\phi )}^{\mu }{\xi }_{{(\phi )}_{\mu }}={g}_{\phi \phi }=\frac{{\sin }^{2}\theta }{{\rm{\Xi }}}\left(\left({r}^{2}+{a}^{2}\right){\rm{\Xi }}+\frac{2Mr{a}^{2}{\sin }^{2}\theta {{\rm{e}}}^{-k/r}}{\Upsilon }\right).\end{eqnarray}$
Now, let us consider the Einstein field equations 26 ) simplifies to equation (25 ). In the asymptotic limit, the equation of the angular velocity does not vanish and yields a finite value as follows26 ) and (27 ), we derive a more relevant expression for the angular velocity of the black hole in the frame that is static at asymptotic infinity.28 ) is one of the most important characteristics of rotating AdS black holes. It plays a crucial role in deriving a consistent first law of black hole thermodynamics, particularly in relation to the Smarr relation. It can also be demonstrated that the angular velocity, as described in equation (28 ), aligns with the angular velocity of the boundary Einstein universe. This provides a relevant basis within the context of the AdS/CFT duality concerning any rotating AdS black hole.
$\begin{eqnarray}{G}_{\mu \nu }={R}_{\mu \nu }-\frac{1}{2}R{g}_{\mu \nu }+{\rm{\Lambda }}{g}_{\mu \nu }=8\pi {T}_{\mu \nu },\end{eqnarray}$
and by examining the above field equations, we have the Einstein tensor as [77] $\begin{eqnarray}\begin{array}{rcl}{G}_{tt} & = & \displaystyle \frac{2[{r}^{4}{m}^{{\prime} }-2{r}^{3}{m}^{{\prime} }m+{a}^{2}{r}^{2}{m}^{{\prime} }-{a}^{4}{\sin }^{2}\theta {m}^{{\prime} }{\cos }^{2}\theta ]}{{\Upsilon }^{3}}\\ & & -\displaystyle \frac{r{a}^{2}{\sin }^{2}\theta {m}^{{\prime\prime} }}{{\Upsilon }^{2}}+{\rm{\Lambda }}\displaystyle \frac{{a}^{2}{\sin }^{2}\theta -{\in }_{r}}{\Upsilon },\\ {G}_{rr} & = & -\displaystyle \frac{2{r}^{2}{m}^{{\prime} }}{\Upsilon {\in }_{r}}+{\rm{\Lambda }}\displaystyle \frac{\Upsilon }{{\in }_{r}},\\ {G}_{\theta \theta } & = & -\displaystyle \frac{2{a}^{2}{\cos }^{2}\theta {m}^{{\prime} }}{\Upsilon }-r{m}^{{\prime\prime} }+{\rm{\Lambda }}\Upsilon ,\\ {G}_{t\phi } & = & \displaystyle \frac{2a{\sin }^{2}\theta [({r}^{2}+{a}^{2}){m}^{{\prime} }({a}^{2}{\cos }^{2}\theta -{r}^{2})]}{{\Upsilon }^{3}}\\ & & -\displaystyle \frac{r{a}^{2}{\sin }^{2}\theta ({r}^{2}+{a}^{2}){m}^{{\prime\prime} }}{{\Upsilon }^{2}}+{\rm{\Lambda }}\displaystyle \frac{a{\sin }^{2}\theta [{\in }_{r}-{r}^{2}-{a}^{2}]}{\Upsilon },\\ {G}_{\phi \phi } & = & -\displaystyle \frac{{\sin }^{2}\theta [({r}^{2}+{a}^{2}){a}^{2}({a}^{2}+(2{r}^{2}+{a}^{2})\cos 2\theta )+{r}^{3}2{\sin }^{2}\theta m)]}{{\Upsilon }^{3}}\\ & & -\displaystyle \frac{r{\sin }^{2}\theta {m}^{{\prime\prime} }{({r}^{2}+{a}^{2})}^{2}}{{\Upsilon }^{2}}\\ & & +{\rm{\Lambda }}\displaystyle \frac{{\sin }^{2}\theta [{({r}^{2}+{a}^{2})}^{2}-{a}^{2}{\in }_{r}]}{\Upsilon }.\end{array}\end{eqnarray}$
The black hole spacetime has Killing vector fields ${\xi }_{(t)}^{\mu }$ and ${\xi }_{(\phi )}^{\mu }$, which arise from the symmetries of the spacetime. These Killing vectors represent null generators of the event horizon and are also tangent to the null surfaces of the event horizon. The angular velocity Ω+ at the event horizon of a black hole is expressed in the following form $\begin{eqnarray}{{\rm{\Omega }}}_{+}=\frac{a{\rm{\Xi }}}{{a}^{2}+{R}_{+}^{2}}.\end{eqnarray}$
The angular velocity of the observer moving on the fixed orbits with constant radius can be derived using the four-velocity uμ such that ${u}^{\mu }{\xi }_{{(\phi )}_{\mu }}=0$ and that gives $\begin{eqnarray}{\rm{\Omega }}=-\frac{{g}_{t\phi }}{{g}_{\phi \phi }}=\frac{a{\rm{\Xi }}\left(\left({r}^{2}+{a}^{2}\right){\in }_{\theta }-{\in }_{r}\right)}{{\left({r}^{2}+{a}^{2}\right)}^{2}{\in }_{\theta }-{a}^{2}{\sin }^{2}\theta {\in }_{r}}.\end{eqnarray}$
As the horizon radius approaches the black hole's outer horizon, the equation for angular velocity ( $\begin{eqnarray}{{\rm{\Omega }}}_{\infty }=-\frac{a}{{l}^{2}}.\end{eqnarray}$
By using equations ( $\begin{eqnarray}{\omega }_{+}={{\rm{\Omega }}}_{+}-{{\rm{\Omega }}}_{\infty }=\frac{a}{{a}^{2}+{R}_{+}^{2}}\left(1+\frac{{R}_{+}^{2}}{{l}^{2}}\right).\end{eqnarray}$
The angular velocity discussed in equation (The metric (15 ) belongs to the prototype of a non-Kerr black hole encompassing the Kerr black holes as a special case (k = 1/l2 = 0). The regular rotating black holes modeled as the non-Kerr black holes put some bounds on the spin parameter and the deviation parameter k to behave like a Kerr black hole mimickers. Consequently, the spacetime geometry around the black hole candidates modeled as the Kerr black holes using the radio and x-ray data [78-81] shows us a possible deviation from the black hole in general relativity. Proceeding along this line we consider the regular rotating AdS black hole candidates which may discard or constrain possible deviations from the Kerr background.
3. Null geodesics
In this section, we derive the null geodesics around regular AdS spacetime. For this purpose, we consider the Hamilton-Jacobi equation given by [82]: 30 ) in the Hamilton-Jacobi equation (29 ) we obtain the complete geodesics equations for the electromagnetic radiations around the black hole as 33 ) and (34 ), the expressions of R and Θ are defined as 33 ), the equation takes the following form 33 ) 38 ) simultaneously and that follows 39 ) and (40 ). In the limit of Ʌ = 0, these impact parameters reduce to the expressions for nonsingular spacetime [21] as
$\begin{eqnarray}\frac{\partial { \mathcal S }}{\partial \sigma }=-\frac{1}{2}{g}^{\alpha \beta }\frac{\partial { \mathcal S }}{\partial {x}^{\alpha }}\frac{\partial { \mathcal S }}{\partial {x}^{\beta }}.\end{eqnarray}$
Here in the above Hamilton-Jacobi equation ϒ is the affine parameter. The black hole metric has two conserved quantities from the symmetries of spacetime, which have the dimensions of energy (E) and angular momentum (L), respectively. To derive all the equations of motion, we select a separable solution defined as [83] $\begin{eqnarray}{ \mathcal S }=\frac{1}{2}{{m}_{0}}^{2}\sigma -Et+L\phi +{{ \mathcal S }}_{r}(r)+{{ \mathcal S }}_{\theta }(\theta ).\end{eqnarray}$
Where in the above separable solution m0 is the mass of the test particle and null geodesics its value is zero. By using the separable solution ( $\begin{eqnarray}\begin{array}{rcl}\Upsilon \displaystyle \frac{{\rm{d}}t}{{\rm{d}}\sigma } & = & \displaystyle \frac{{r}^{2}+{a}^{2}}{{\in }_{r}}\left[E\left({r}^{2}+{a}^{2}\right)-a{\rm{\Xi }}L\right]\\ & & -\displaystyle \frac{a}{{\in }_{\theta }}\left(aE{\sin }^{2}\theta -{\rm{\Xi }}L\right),\end{array}\end{eqnarray}$
$\begin{eqnarray}\Upsilon \frac{{\rm{d}}\phi }{{\rm{d}}\sigma }=\frac{a{\rm{\Xi }}}{{\in }_{r}}\left[E\left({r}^{2}+{a}^{2}\right)-a{\rm{\Xi }}L\right]-\frac{{\rm{\Xi }}}{{\in }_{\theta }}\left(aE-{\rm{\Xi }}L{\csc }^{2}\theta \right),\end{eqnarray}$
$\begin{eqnarray}{\Upsilon }^{2}\frac{{{\rm{d}}}^{2}r}{{\rm{d}}{\sigma }^{2}}={R}_{r},\end{eqnarray}$
and $\begin{eqnarray}{\Upsilon }^{2}\frac{{{\rm{d}}}^{2}\theta }{{\rm{d}}{\sigma }^{2}}={{\rm{\Theta }}}_{\theta },\end{eqnarray}$
in the above geodesics equations ( $\begin{eqnarray}{R}_{r}={\left[({a}^{2}+{r}^{2})E-a{\rm{\Xi }}L\right]}^{2}-\left[{\left(L-aE\right)}^{2}+{ \mathcal K }\right]{\in }_{r,}\end{eqnarray}$
$\begin{eqnarray}{{\rm{\Theta }}}_{\theta }=-{\left(a{\sin }^{2}\theta -\xi {\rm{\Xi }}\right)}^{2}{\csc }^{2}\theta -\left[{\left(\xi -a\right)}^{2}+{ \mathcal K }\right]{\in }_{\theta .}\end{eqnarray}$
For the separation of radial and angular geodesics equations, we use a separating constant ${ \mathcal K }$ that is also called the Carter separable constant. The Carter separable constant has the dimensions of the angular momentum. The above geodesic equations describe the motion of all frequencies of electromagnetic radiation around the rotating AdS black hole spacetimes. Now we are interested in analyzing these radiations and for that, it's required to define two impact parameters: ξ = L/E and $\eta ={ \mathcal K }/{E}^{2}$. After substituting these impact parameters in equation ( $\begin{eqnarray}{R}_{r}={\left[({a}^{2}+{r}^{2})-a\xi {\rm{\Xi }}\right]}^{2}-\left[{\left(\xi -a\right)}^{2}+\eta \right]{\in }_{r}\,.\end{eqnarray}$
To obtain the innermost unstable orbits we maximize the radial equation of motion equation ( $\begin{eqnarray}{ \mathcal R }(r)=\frac{{\rm{d}}{R}_{r}}{{\rm{d}}r}=0,\end{eqnarray}$
we can separate the impact parameters by solving the above equation ( $\begin{eqnarray}\xi ({r}_{ph})=\frac{-4{r}_{ph}{\in }_{r}+\left({r}_{ph}^{2}+{a}^{2}\right){\in }_{r}^{\prime} }{a{\rm{\Xi }}{\in }_{r}^{\prime} },\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}\eta ({r}_{ph})\\ =\frac{-16{r}_{ph}^{2}{\in }_{r}^{2}-{\left({r}_{ph}^{2}+\frac{{a}^{4}}{{l}^{2}}\right)}^{2}{\in }_{r}{{\prime} }^{2}+8{r}_{ph}{\in }_{r}\left(2{a}^{2}{r}_{ph}{{\rm{\Xi }}}^{2}+\left({r}_{ph}^{2}+\frac{{a}^{4}}{{l}^{2}}\right){\in }_{r}^{\prime} \right)}{{a}^{2}{{\rm{\Xi }}}^{2}{\in }_{r}{{\prime} }^{2}},\end{array}\end{eqnarray}$
where the term ${\in }_{r}^{\prime} $ denotes the derivative of ∈r with respect to r at point r = rph and it can be written as $\begin{eqnarray}\begin{array}{rcl}{\in }_{r}^{\prime} & = & -\frac{2}{3}\left(3\left(M{{\rm{e}}}^{-k/{r}_{ph}}(k+{r}_{ph})/{r}_{ph}\right)\right.\\ & & \left.-3{r}_{ph}-\frac{1}{3{l}^{2}}{r}_{ph}({a}^{2}+2{r}_{ph}^{2})\right).\end{array}\end{eqnarray}$
The geometry of electromagnetic radiations of all frequencies around regular AdS black holes can be traced by the impact parameters given in equations ( $\begin{eqnarray}\xi =\frac{3M\left({a}^{2}(k+{r}_{ph})+{r}_{ph}^{2}(k-3{r}_{ph})\right)+3{r}_{ph}^{2}\left({a}^{2}+{r}_{ph}^{2}\right){{\rm{e}}}^{k/{r}_{ph}}}{3aM(k+{r}_{ph})-3a{r}_{ph}^{2}{{\rm{e}}}^{k/{r}_{ph}}},\end{eqnarray}$
$\begin{eqnarray}\eta =-\frac{{r}_{ph}^{4}\left(4{a}^{2}M{{\rm{e}}}^{k/{r}_{ph}}(k-{r}_{ph})+{\left({r}_{ph}\left({r}_{ph}{{\rm{e}}}^{k/{r}_{ph}}-3M\right)+kM\right)}^{2}\right)}{{a}^{2}{\left({r}_{ph}\left(M-{r}_{ph}{{\rm{e}}}^{k/{r}_{ph}}\right)+kM\right)}^{2}},\end{eqnarray}$
and which in addition, for k = 0 exactly reduces to the Kerr spacetime such that $\begin{eqnarray}\xi =\frac{{a}^{2}(M+{r}_{ph})+{r}_{ph}^{2}({r}_{ph}-3M)}{a(M-{r}_{ph})},\end{eqnarray}$
$\begin{eqnarray}\eta =-\frac{{r}_{ph}^{3}\left({r}_{ph}{({r}_{ph}-3M)}^{2}-4{a}^{2}M\right)}{{a}^{2}{(M-{r}_{ph})}^{2}}.\end{eqnarray}$
One can analytically study the photon orbits by using the equations (39 ) and (40 ) around regular black holes in AdS spacetimes. To obtain the figures of black hole shadow observed at some finite distance by an arbitrary observer, one has to use the coordinates, say (rO, νO), and these coordinates are related to the orthonormal tetrads.
4. Rotating regular AdS black hole shadow
In this section, we figure out the images of the black hole shadow and perform related analysis. We fix the observer position somewhere between the outer and cosmological horizons in Boyer-Lindquist coordinates (rO, νO). We define the orthonormal tetrads at (rO, νO) as [34, 63, 84, 85]48 ) and (49 ), the scale factor p can be calculated as 48 ) and (49 ), we have
$\begin{eqnarray}{e}_{0}={\left.\frac{\left({r}^{2}+{a}^{2}\right){\partial }_{t}+a{\rm{\Xi }}{\partial }_{\phi }}{\sqrt{{\in }_{r}\Upsilon }}\right|}_{({r}_{O},{\vartheta }_{O})},\,\,{e}_{1}={\left.\sqrt{\frac{{\in }_{\theta }}{\Upsilon }}{\partial }_{\theta }\right|}_{({r}_{O},{\vartheta }_{O})},\end{eqnarray}$
and $\begin{eqnarray}{e}_{2}=-{\left.\frac{{\rm{\Xi }}{\partial }_{\phi }+a{\sin }^{2}\theta {\partial }_{t}}{\sqrt{{\in }_{\theta }\Upsilon }\sin \theta }\right|}_{({r}_{O},{\vartheta }_{O})},\,\,{e}_{3}=-{\left.\sqrt{\frac{{\in }_{r}}{\Upsilon }}{\partial }_{r}\right|}_{({r}_{O},{\vartheta }_{O})},\end{eqnarray}$
where e0 and e3 denote the four velocities of the observer which gives the spatial direction towards the black hole and the combination e0 ± e3 is tangential to the outgoing and ingoing principal null congruences of the metric. The coordinates §(s) = (r(s), θ(s), Φ(s), t(s)) be assigned for each light ray which in turn gives its tangent vector as $\begin{eqnarray}\dot{\S }=\dot{r}{\partial }_{r}+\dot{\theta }{\partial }_{\theta }+\dot{\phi }{\partial }_{\phi }+\dot{t}{\partial }_{t},\end{eqnarray}$
and the tangent vector is $\begin{eqnarray}\dot{\S }=p\left(-{e}_{0}+\cos \beta \sin \alpha {e}_{1}+\sin \beta \sin \alpha {e}_{2}+\cos \alpha {e}_{3}\right).\end{eqnarray}$
From equations ( $\begin{eqnarray}\S =n(\dot{\S },{e}_{0})=n\left(\dot{\lambda },\frac{{r}^{2}+{a}^{2}}{\sqrt{{\in }_{r}\Upsilon }}{\partial }_{t}+\frac{a{\rm{\Xi }}}{\sqrt{{\in }_{r}\Upsilon }}{\partial }_{\phi }\right)={\left.\frac{-E({r}^{2}+{a}^{2})+a{\rm{\Xi }}L}{\sqrt{{\in }_{r}\Upsilon }}\right|}_{({r}_{O},{\vartheta }_{O})},\end{eqnarray}$
and by comparing the corresponding coefficients in equations ( $\begin{eqnarray}\alpha ={\rm{a}}{\rm{r}}{\rm{c}}\sin \left({\left.\frac{\sqrt{\left[{\left(\xi ({r}_{ph})-a\right)}^{2}+\eta ({r}_{ph})\right]{\in }_{r}}}{{r}^{2}+{a}^{2}-a{\rm{\Xi }}\xi ({r}_{ph})}\right|}_{({r}_{O},{\vartheta }_{O})}\right),\end{eqnarray}$
and $\begin{eqnarray}\beta =\arcsin \left({\left.\frac{\sqrt{{\in }_{r}}\sin \theta }{\sqrt{{\in }_{\theta }}\sin \alpha }\left[\frac{a-{\rm{\Xi }}\xi ({r}_{ph}){\csc }^{2}\theta }{a{\rm{\Xi }}\xi ({r}_{ph})-\left({r}^{2}+{a}^{2}\right)}\right]\right|}_{({r}_{O},{\vartheta }_{O})}\right).\end{eqnarray}$
In order to obtain the apparent shape of the shadow, we have considered the stereographic projection of the celestial sphere onto a plane, as shown in figure 4, where the Cartesian coordinates in this plane take the form
$\begin{eqnarray}\begin{array}{rcl}X & = & -2\tan \left(\frac{\alpha ({r}_{ph})}{2}\right)\sin \left(\beta ({r}_{ph})\right),\\ Y & = & -2\tan \left(\frac{\alpha ({r}_{ph})}{2}\right)\cos \left(\beta ({r}_{ph})\right),\end{array}\end{eqnarray}$
which satisfies the relation $\begin{eqnarray}{X}^{2}+{Y}^{2}=4{\tan }^{2}\left(\frac{\alpha ({r}_{ph})}{2}\right).\end{eqnarray}$
One can obtain the figure of the black hole shadow by simply taking the contour of equation (54 ). The figure of the black hole shadow depends primarily on its spin parameter a, curvature radius 1/l2, and the free parameter k. We show the figures of black hole shadow in figures 1, 2, and 3 for different values of a, 1/l2, and k. The figure of the black hole shadow emphasizes that the size and distortion of the shadow images increases with the 1/l2 (cf figure 1) while its size decreases with k (cf figure 2). We also plot the spherical (a = 0) regular AdS black hole shadow in figure 3 which appears as a perfect circle and increases with the increasing values 1/l2.
Figure 1. Plot showing the black hole shadow in rotating regular AdS spacetime at rO = 50 and for different values of k, a and 1/l2. |
Figure 2. Plot showing the black hole shadow in rotating regular AdS spacetime at rO = 50 and for different values of k, a, and 1/l2. |
Figure 3. Non-spinning regular AdS black hole shadow with observable Rs for different values of k with 1/l2 at rO = 50. |
Figure 4. The celestial coordinates α and β are introduced to trace the paths of photon rays in the observer's frame. Using stereographic projection, we define a map from the celestial sphere to the observer [86]. |
For the analysis of the shadow figures we define observable: one is the shadow radius Rs and another is the distortion parameter of the shadow figure δs given by [23]
$\begin{eqnarray}{R}_{s}=\frac{{({X}_{t}-{X}_{r})}^{2}+{{Y}_{t}}^{2}}{2| {X}_{t}-{X}_{r}| },\end{eqnarray}$
where (Xt, Yt) and (Xr, Yr) are the topmost and the rightmost positions of the celestial coordinates from where the contour passes from the axes. The second observable δs reads $\begin{eqnarray}{\delta }_{s}=\frac{{D}_{s}}{{R}_{s}}.\end{eqnarray}$
The variation of the shadow radius and the distortion parameter is shown in figure 5. The shadow radius and distortion parameter increase monotonically with the radius of curvature parameter 1/l2 for fixed a (cf. figure 5). In figure 6 we plot the shadow radius Rs and distortion parameter δs in (a, 1/l2) plane. The point of intersection of Rs and δs lines gives the spin parameter a and radius of curvature parameter 1/l2. If we compare our results with those of nonsingular black hole shadows [21], we find that in AdS spacetime, the size of the black hole shadow appears larger and becomes more distorted as the values of the parameter 1/l2 increase (cf figure 1). Our results indicate that the shadow size of a regular AdS spacetime is larger compared to that of the Kerr black holes [83]. Table 1 provides detailed numerical values for shadow radius and distortion parameters across various spin parameter and free parameter values. The angular diameter of the black hole can be estimated via $\begin{eqnarray}{\theta }_{{\rm{d}}}=\frac{2}{{r}_{o}}\sqrt{\frac{A}{\pi }},\end{eqnarray}$
where ro = 50M and A shadow area, and can be estimated via the shadow radius. Figure 7, illustrates how the area of the shadow image varies with the parameters 1/l2 and k. The area of the black hole shadow increases for the increasing values of 1/l2 while it decreases with the parameter k. The variation in the angular diameter of the black hole shadow is illustrated in figure 8 for a static observer within the domain of outer communication. The angular diameter of regular AdS black hole shadow increases with 1/l2 and varies in the range of 2 ± 0.52° for 0.05≤1/l2≤ 0.2 with 0.05≤k≤ 0.2.
Figure 5. Plots showing the variation of observables Rs and δs with 1/l2 for different values of a and k at rO = 50. |
Figure 6. The contours of Rs (solid red lines) and δs (dashed gray lines) in (a, 1/l2) plane. The point of intersection of Rs and δs lines gives the values of black hole parameters. |
Table 1. Variation of shadow radius and distortion parameter with parameter M2/l2. |
| a = 0.5M, k = 0.1M | ||
|---|---|---|
| Radius of curvature (M2/l2) | Shadow radius (Rs/M) | Distortion parameter (δs) |
| 0.05 | 0.90 | 0.06 |
| 0.10 | 1.11 | 0.08 |
| 0.15 | 1.22 | 0.10 |
| 0.20 | 1.30 | 0.12 |
| | ||
| a = 0.6M, k = 0.2M | ||
| | ||
| Radius of curvature M2/l2 | Shadow radius (Rs/M) | Distortion parameter (δs) |
| | ||
| 0.05 | 0.89 | 0.11 |
| 0.10 | 1.09 | 0.17 |
| 0.15 | 1.21 | 0.23 |
| 0.20 | 1.28 | 0.27 |
Figure 7. Variation of the Area of the black hole shadow with the black hole parameters. |
Figure 8. Variation of the angular diameter of the black hole shadow with the black hole parameters. |
5. Energy emission rate
As a good approximation, we can use the limiting constant value of the absorption cross-section for a nearly spherically symmetric black hole as
$\begin{eqnarray}{\sigma }_{{\rm{lim}}}\approx \pi {R}_{s}^{2}.\end{eqnarray}$
We can use such limiting value of the geometric cross-section in order to calculate the energy emission rate $\begin{eqnarray}\frac{{{\rm{d}}}^{2}E(\omega )}{{\rm{d}}\omega {\rm{d}}t}=\frac{2{\pi }^{2}{R}_{s}^{2}}{{{\rm{e}}}^{\omega /T}-1}{\omega }^{3},\end{eqnarray}$
where ω is the frequency of the emitted photon and the temperature T of the nonsingular-AdS black hole in terms of the event horizon radius is expressed as $\begin{eqnarray}T={T}_{{\rm{Kerr-AdS}}}-\frac{k\left(1-{R}_{+}^{2}/{l}^{2}\right)}{4\pi {R}_{+}^{2}({R}_{+}^{2}+{a}^{2})},\end{eqnarray}$
where TKerr-AdS is the corresponding temperature for the Kerr-AdS black holes $\begin{eqnarray}{T}_{{\rm{Kerr-AdS}}}=\frac{{R}_{+}^{2}\left(1+3{R}_{+}^{2}/{l}^{2}\right)-{a}^{2}\left(1-{R}_{+}^{2}/{l}^{2}\right)}{4\pi {R}_{+}\left({R}_{+}^{2}+{a}^{2}\right)}.\end{eqnarray}$
In figure 9 we have plotted the energy emission rate of the black hole with ω for different values of curvature radius term. It is shown in the figure that there exists a peak of energy emission rate, which decreases as we increase 1/l2 term and other black hole parameters.
Figure 9. Plot showing the energy emission rate of the photon region. |
6. Conclusion
In rotating Kerr spacetimes the shapes of black hole shadows get distorted mainly by the spin parameter a. In other theories of gravity viz. the MGs, some other parameters may enhance or decrease the shadow of the black hole. In this paper, we have investigated the shadow of the rotating regular black hole in the presence of a cosmological constant. The size of the black hole shadow and distortion parameter increases with the increasing values of curvature radius parameter 1/l2. The black hole shadow appears smaller and more distorted in comparison to the regular and Kerr spacetime (cf figure 2). For a better understanding of black hole shadow, we have also calculated observables of the shadow radius Rs and the distortion parameter δs. The observables Rs and δs increase with the increasing values of 1/l2 and the effective size of the shadow radius decreases with the free parameter k. We also calculated the angular diameter of the black hole θd = 2 ± 0.52° for 0.05≤1/l2≤ 0.2 with 0.05≤k≤ 0.2. From the study of energy emission rate, we conclude that the electromagnetic spectrum emitted by the shadow region of regular black holes in AdS spacetime decreases for the increasing values of the parameter 1/l2.