1. Introduction
Despite the vast amount of experimental and observational data that supports the standard theory of gravity, known as general relativity, it still faces some fundamental challenges. One such challenge is its incompatibility with the quantum field theory, as well as the presence of singularities at the source’s origin. To overcome these difficulties, scientists may employ alternative theories or modified theories of gravity. The problem of singularity also exists in classical electrodynamics, which Born and Infield attempted to resolve through the inclusion of nonlinear terms while preserving the theory’s relativistic and gauge invariant properties [1]. One may obtain regularization of the electrodynamics using the polarization of the vacuum (see, for example, Euler-Heisenberg nonlinear electrodynamics [2]). Other nonlinear effects have been developed which remove the singularity from the solution of field equations for the electromagnetic field. These nonlinear effects coupled with general relativity may exclude the singularity from the spacetime. The first regular solution of the Einstein equation without singularity was obtained by Bardeen [3]. Later, other methods of regularization were used in order to obtain regular black hole (BH) solutions [4, 5].
One of the attempts to obtain a regular BH solution is proposed in [6–10]. In particular, the authors of [11] found new regular BH solutions within the so-called ModMax electrodynamics background. The dyonic solution has been obtained taking into account SO(2) invariance for electric and magnetic fields. The authors of [12] introduced a Galilean cousin of the ModMax theory, written in a covariant formalism and showed that it is invariant under Galilean conformal symmetries. The nonlinear effects of ModMax on the Lorentz force, the Coulomb law, the Lienard–Wiechert fields, Dirac’s and Schwinger’s quantization of electric and magnetic charges, and the Compton effect have been explored in [13]. The generalized ModMax model of nonlinear electrodynamics coupled to general relativity has been studied in [14]. The question regarding triviality or being a stealth field of stationary nonlinear electromagnetic field within the ModMax model has been discussed in [15]. BH thermodynamics in the presence of nonlinear electromagnetic fields described by the ModMax model have been explored in [16]. The deformation of the ModMax theory, as a unique Lagrangian of nonlinear electrodynamics preserving both conformal and electromagnetic-duality invariance has been considered in [17]. Here, we plan to explore optical properties of the spacetime around a rotating BH in the presence of ModMax electrodynamics.
Recent revolutionary discoveries of gravitational waves [18] and observation of the shadows of M87 [19, 20] and Sgr A* [21] opened new doors to probe the modified or alternative theories of gravity. The accuracy of the observations and detection of gravitational radiation may be used to rule out the modifications and get proper constraints on metric parameters. The photon motion in curved spacetime may be used as a useful tool to test and probe the gravity. Gravitational lensing, as well as the shadow cast by the BH, is the consequence of the photon motion around a gravitating object [22]. The idea of the formation shadow of the BH was first proposed by Bardeen [23] and later developped by Luminet [24]. A review of the photon motion and shadow formation can be found in [22, 25]. The physical mechanism of accretion onto BHs is investigated in various modified gravities in [26, 27].
Due to the capture of some of the photons by the BH located between the source of the electromagnetic waves and the observer, the latter can detect a black spot on the celestial plane. This spot is referred to as the BH shadow and its shape and size are defined by the parameters of the spacetime [22]. The main observables of the BH shadow may provide information about the nature of gravity and parameters of the BH. Theoretical studies of BH shadows for different gravity models and BH solutions can be found in [28–90]. After the first ever observation of the BH shadows of M87 and Sgr A* in 2019 [19] and in 2022 [21], the observations have been developed by various authors [91–98].
In this paper we plan to study the shadow of a regular BH described within the ModMax model. The paper is organized as follows: in section 2 we have briefly reviewed the ModMax BHs and studied the horizon structure and the equation of motion of photons around a ModMax BH. Observable quantities of the BH shadow have been considered in section 3 . The emission energy from a ModMax BH has been investigated in section 4 . Finally, detailed discussion of the results obtained is presented in section 5 . We have used the geometric units system that fixes the speed of light and the gravitational constant via G = c = 1.
2. Rotating ModMax black hole
The motivation behind the development and exploration of ModMax electrodynamics theory arises from the inherent challenges posed by singularities in both Einstein’s general relativity theory and Maxwell’s electrodynamics. Singularities at the beginning of the Universe and in BH solutions have been persistent issues. To address these challenges, modifications such as Born–Infeld (BI) non-linear electrodynamics (NED) and Euler–Heisenberg (EH) NED were introduced, offering solutions with finite self-energy and SO(2) invariance.
More recently, a generalized form of Maxwell electrodynamics called ModMax electrodynamics was introduced, exhibiting a low-energy limit as a one-dimensional parameter generalization of BI. The condition where γ = 0 yields Maxwell’s equations. Flores-Alfonso et al discovered new BH solutions within ModMax electrodynamics [11, 99]. The SO(2) invariance concerning electric and magnetic fields leads to dyonic solutions. The impact of ModMax electrodynamics on BH spacetimes, mediated by the screening factor γ, acts to shield the actual charges.
2.1. Brief review of ModMax electrodynamics
The Lagrangian of ModMax electrodynamics is given by [11, 99, 100]
$\begin{eqnarray}{L}_{\mathrm{ModMax}}(x,y)=-x\cosh \gamma +{\left({x}^{2}+{y}^{2}\right)}^{1/2}\sinh \gamma ,\end{eqnarray}$
with the electromagnetic Lorentz invariants $x:= \tfrac{1}{4}{F}_{\mu \nu }{F}^{\mu \nu },y:= \tfrac{1}{4}{F}_{\mu \nu }{\tilde{F}}^{\mu \nu },$ and the electromagnetic tensor is Fμν with its dual ${\tilde{F}}_{\mu \nu }:= \tfrac{1}{2}{\epsilon }_{\mu \nu \sigma \rho }{F}^{\sigma \rho }$. Let us introduce a tensor quantity connecting the Lagrangian and the electromagnetic tensor, namely [101] $\begin{eqnarray}\begin{array}{rcl}{P}_{\mu \nu } & := & -{L}_{x}{F}_{\mu \nu }-{L}_{y}{\tilde{F}}_{\mu \nu }\\ & = & \left[\cosh \gamma -\displaystyle \frac{x}{{\left({x}^{2}+{y}^{2}\right)}^{1/2}}\sinh \gamma \right]{F}_{\mu \nu }\\ & & -\displaystyle \frac{y\sinh \gamma }{{\left({x}^{2}+{y}^{2}\right)}^{1/2}}{\tilde{F}}_{\mu \nu },\end{array}\end{eqnarray}$
with its dual $\begin{eqnarray}\begin{array}{rcl}{\tilde{P}}_{\mu \nu } & = & \left[\cosh \gamma -\displaystyle \frac{x}{{\left({x}^{2}+{y}^{2}\right)}^{1/2}}\sinh \gamma \right]{\tilde{F}}_{\mu \nu }\\ & & +\displaystyle \frac{y\sinh \gamma }{{\left({x}^{2}+{y}^{2}\right)}^{1/2}}{F}_{\mu \nu }.\end{array}\end{eqnarray}$
then one can calculate electromagnetic invariants as $\begin{eqnarray}\begin{array}{rcl}{\rm{\Pi }} & := & -\displaystyle \frac{1}{4}{P}_{\mu \nu }{P}^{\mu \nu }\\ & = & -x\cosh 2\gamma +{\left({x}^{2}+{y}^{2}\right)}^{1/2}\sinh 2\gamma ,\\ {\rm{\Lambda }} & := & -\displaystyle \frac{1}{4}{P}_{\mu \nu }{\tilde{P}}^{\mu \nu }=-y.\end{array}\end{eqnarray}$
ModMax electrodynamics can be considered as a low-energy approximation of the generalized BI structure that was proposed in the aforementioned paper [99]. Then it is found that $\begin{eqnarray}\begin{array}{rcl}x & = & -{\rm{\Pi }}\cosh 2\gamma +{\left({{\rm{\Pi }}}^{2}+{{\rm{\Lambda }}}^{2}\right)}^{1/2}\sinh 2\gamma ,\\ y & = & -{\rm{\Lambda }}.\end{array}\end{eqnarray}$
The dual of ModMax is given as $\begin{eqnarray}{\hat{L}}_{\mathrm{ModMax}}:= -\displaystyle \frac{1}{2}{P}^{\mu \nu }{F}_{\mu \nu }-{L}_{\mathrm{ModMax}}.\end{eqnarray}$
When expressed in terms of the variables x and y, the ModMax Lagrangian, denoted by ${\hat{L}}_{\mathrm{ModMax}}$, is equal to the Lagrangian ${L}_{\mathrm{ModMax}}$. However, in terms of the variables Π and Λ, the Lagrangian exhibits a different form as $\begin{eqnarray}\begin{array}{rcl}{\hat{L}}_{\mathrm{ModMax}}(s,t) & = & -x\cosh \gamma +{\left({x}^{2}+{y}^{2}\right)}^{1/2}\sinh \gamma \\ & = & -x\cosh \gamma +\displaystyle \frac{{\rm{\Pi }}+x\cosh 2\gamma }{\sinh 2\gamma }{\left({{\rm{\Pi }}}^{2}+{{\rm{\Lambda }}}^{2}\right)}^{1/2}\\ & = & {\rm{\Pi }}\cosh \gamma -{\left({{\rm{\Pi }}}^{2}+{{\rm{\Lambda }}}^{2}\right)}^{1/2}\sinh \gamma .\end{array}\end{eqnarray}$
It is important to note that both Maxwell and BI electrodynamics possess an electric–magnetic duality symmetry under the SO(2) group. In other words, if an electric solution of the theory is provided, it is possible to generate a magnetic solution to the equations of motion by performing a Hodge dualization operation.
The field equations of the ModMax field are2 ).
$\begin{eqnarray}{{\rm{\nabla }}}_{\mu }{P}^{\mu \nu }=0.\end{eqnarray}$
The Einstein equations take a specific form when combined with the expression for P given by equation ( $\begin{eqnarray}{R}^{\mu }{}_{\nu }-\displaystyle \frac{1}{2}{\delta }_{\nu }^{\mu }R+{\rm{\Lambda }}{\delta }_{\nu }^{\mu }=8\pi {T}^{\mu }{}_{\nu }.\end{eqnarray}$
The expression for the energy momentum tensor T is provided by [101] $\begin{eqnarray}\begin{array}{rcl}4\pi {T}^{\mu }{}_{\nu } & = & -{F}^{\mu \beta }{P}_{\beta \nu }+{\delta }_{\nu }^{\mu }L\\ & = & {F}^{\mu \beta }({L}_{x}{F}_{\beta \nu }+{L}_{y}{\tilde{F}}_{\beta \nu })+{\delta }_{\nu }^{\mu }L,\\ & = & {\hat{L}}_{{\rm{\Pi }}}{P}^{\mu \beta }{P}_{\nu \beta }+{\delta }_{\nu }^{\mu }(2s{\hat{L}}_{{\rm{\Pi }}}+t{\hat{L}}_{\lambda }-\hat{L}),\end{array}\end{eqnarray}$
with ${\hat{L}}_{{\rm{\Pi }}}:= \partial \hat{L}/\partial {\rm{\Pi }}$ and ${\hat{L}}_{{\rm{\Lambda }}}:= \partial \hat{L}/\partial {\rm{\Lambda }}$.2.2. Rotating ModMax black hole spacetime
We consider a spherically symmetric BH solution described by the line element written in Boyer–Linquist coordinates [11, 102]
$\begin{eqnarray}{\rm{d}}{s}^{2}=-f(r){\rm{d}}{t}^{2}+\displaystyle \frac{1}{f(r)}{\rm{d}}{r}^{2}+{r}^{2}({\rm{d}}{\theta }^{2}+{\sin }^{2}\theta {\rm{d}}{\varphi }^{2}),\end{eqnarray}$
with $\begin{eqnarray}f(r)=1-\displaystyle \frac{2M}{r}+\displaystyle \frac{{Q}^{2}{{\rm{e}}}^{-\gamma }}{{r}^{2}},\end{eqnarray}$
where M is the BH mass, Q is charge and γ is screening factor proposed in [11, 102]. The field equations’ solution yields spacetime metric of an electrically charged BH. In this case, the electromagnetic invariant is used as t = 0 and ${\hat{L}}_{\mathrm{ModMax}}=(\cosh \gamma -\sinh \gamma )s={{\rm{e}}}^{-\gamma }s={{\rm{e}}}^{-\gamma }{\hat{L}}_{\mathrm{Maxwell}}$. Given that ${\hat{L}}_{\mathrm{Maxwell}}={Q}^{2}/2{r}^{4}$, where Q represents the charge of the Reissner–Nordström BH, it follows that the ModMax BH has an effective electric charge of e−γ/2Q, resulting in a metric function that can be expressed simply.The event horizon of the ModMax BH is located at:
$\begin{eqnarray}{r}_{\pm }=M\pm \sqrt{{M}^{2}-{Q}^{2}{{\rm{e}}}^{-\gamma }}.\end{eqnarray}$
There is an extremal case of the ModMax BH at r+ = r−:
$\begin{eqnarray}{M}_{\mathrm{ext}}=| Q| {{\rm{e}}}^{-\gamma /2},\end{eqnarray}$
where γ > 0.
Using the Newman–Janis algorithm (see, for example, [103, 104]) one may get the rotating ModMax BH solution using the static solution, equation (11 ) with equation (12 ). The spacetime around the rotating ModMax BH with the spin parameter a can be represented using the line element
$\begin{eqnarray}\begin{array}{rcl}{\rm{d}}{s}^{2} & = & -\displaystyle \frac{{\rm{\Delta }}}{{\rm{\Sigma }}}{\left({\rm{d}}t-a{\sin }^{2}\theta {\rm{d}}\phi \right)}^{2}+\displaystyle \frac{{\rm{\Sigma }}}{{\rm{\Delta }}}{\rm{d}}{r}^{2}+{\rm{\Sigma }}{\rm{d}}{\theta }^{2}\\ & & +\displaystyle \frac{{\sin }^{2}\theta }{{\rm{\Sigma }}}{\left(a{\rm{d}}t-({r}^{2}+{a}^{2}){\rm{d}}\phi \right)}^{2},\end{array}\end{eqnarray}$
where the metric functions are defined as $\begin{eqnarray}{\rm{\Sigma }}={r}^{2}+{a}^{2}{\cos }^{2}\theta ,\end{eqnarray}$
$\begin{eqnarray}{\rm{\Delta }}={r}^{2}+{a}^{2}-2{Mr}+{Q}^{2}{{\rm{e}}}^{-\gamma }.\end{eqnarray}$
Figure 1. The separatrix lines indicate the border corresponding to extreme BHs which separates the BH from no BH for selected values of the γ parameter. |
Now we will discuss the properties of the horizon structure of the charged ModMax BH. We also plan to explore the effect of spacetime parameters on the horizon structure of the BH. Using equation (17 ) and solving Δ = 0, one may obtain the values of the BH horizon. The graphical representation of the numerical solutions for the BH horizon is shown in figure 2. Left and right panels show the dependence of inner and outer horizons of the BH on the spin parameter for different values of either γ or Q for a fixed value of the other parameter, Q = 0.5 or γ = 0.2. From figure 2, one may obtain information about the inner (Cauchy) and the outer radius of the horizon: how the radius is modified by varying charge Q and screening factor γ. According to figure 2, the horizon radius increases with increasing γ for fixed Q = 0.5, and for fixed γ = 0.2, the radius decreases with increading Q. Figure 3 indicates extreme values of the horizon for different values of γε, aε and Qε for fixed a and Q, γ and Q, and a and γ, respectively. For a charged rotating ModMax BH, the horizon exists at γ > γε, a < aε and Q < Qε, and there is naked singularity at γ < γε, a > aε and Q > Qε. γε and aε grow with increasing Q and γ, respectively, and Qε decreases with decreasing γ parameter. The inner (Cauchy) horizon radius decreases and the outer horizon radius increases with the increase of γ for fixed Q and a, and the inner horizon radius increases and outer horizon radius decreases with increasing a and Q, for fixed γ and Q, and a and γ, respectively.
Figure 2. Dependence of horizon radius rh on BH spin parameter a for fixed values of parameters Q and γ where M = 1. |
Figure 3. Dependence of delta function on the radial coordinate r for fixed values of the BH parameters a, Q and parameter γ with M = 1. |
2.3. Null geodesics
Now we will discuss the geodesic structure of a massless particle in the vicinity of a charged rotating ModMax BH. Using the Hamilton–Jacobi formulation, one may obtain the equations of motion of a photon around a BH in ModMax gravity [105] as19 ), Sr(r) and Sθ(θ) are the functions of radial r and angular θ, respectively. Substituting equation (19 ) into equation (18 ), one can obtain the geodesic equations of the massless particle as
$\begin{eqnarray}\displaystyle \frac{\partial S}{\partial \tau }=-\displaystyle \frac{1}{2}{g}^{\mu \nu }\displaystyle \frac{\partial S}{\partial {x}^{\mu }}\displaystyle \frac{\partial S}{\partial {x}^{\nu }},\end{eqnarray}$
here, τ is an affine parameter and gμν is the metric tensor of the spacetime. The action S for the photon can be written using the separation of variables method as $\begin{eqnarray}S=-{ \mathcal E }t+{ \mathcal L }\phi +{S}_{r}(r)+{S}_{\theta }(\theta ),\end{eqnarray}$
where the conserved quantities corresponding to energy of the massless particle ${ \mathcal E }$ and angular momentum of the massless particle ${ \mathcal L }$ can be defined as $\begin{eqnarray}{ \mathcal E }=-{g}_{t\mu }{\dot{x}}^{\mu },\end{eqnarray}$
$\begin{eqnarray}{ \mathcal L }={g}_{\phi \mu }{\dot{x}}^{\mu }.\end{eqnarray}$
From equation ( $\begin{eqnarray}\begin{array}{rcl}{\rm{\Sigma }}\displaystyle \frac{{\rm{d}}t}{{\rm{d}}\tau } & = & a({ \mathcal L }-a{ \mathcal E }{\sin }^{2}\theta )\\ & & +\displaystyle \frac{{r}^{2}+{a}^{2}}{{\rm{\Delta }}}\left({ \mathcal E }({r}^{2}+{a}^{2})-a{ \mathcal L }\right),\end{array}\end{eqnarray}$
$\begin{eqnarray}{\rm{\Sigma }}\displaystyle \frac{{\rm{d}}r}{{\rm{d}}\tau }=\pm \sqrt{{ \mathcal R }},\end{eqnarray}$
$\begin{eqnarray}{\rm{\Sigma }}\displaystyle \frac{{\rm{d}}\theta }{{\rm{d}}\tau }=\pm \sqrt{{\rm{\Theta }}},\end{eqnarray}$
$\begin{eqnarray}{\rm{\Sigma }}\displaystyle \frac{{\rm{d}}\phi }{{\rm{d}}\tau }=({ \mathcal L }{\csc }^{2}\theta -a{ \mathcal E })+\displaystyle \frac{a}{{\rm{\Delta }}}\left({ \mathcal E }({r}^{2}+{a}^{2})-a{ \mathcal L }\right),\end{eqnarray}$
with ${ \mathcal R }$ and Θ defined as $\begin{eqnarray}{ \mathcal R }={\left[\left({r}^{2}+{a}^{2}\right){ \mathcal E }-a{ \mathcal L }\right]}^{2}-{\rm{\Delta }}\left[{ \mathcal K }+{\left({ \mathcal L }-a{ \mathcal E }\right)}^{2}\right],\end{eqnarray}$
$\begin{eqnarray}{\rm{\Theta }}={ \mathcal K }+{\cos }^{2}\theta \left({a}^{2}{{ \mathcal E }}^{2}-{{ \mathcal L }}^{2}{\sin }^{-2}\theta \right),\end{eqnarray}$
where ${ \mathcal K }$ is the Carter constant [105].Now we can rewrite the radial equation of motion of the photon around a BH in ModMax theory using the expression dr/dτ as
$\begin{eqnarray}{\left({\rm{\Sigma }}\displaystyle \frac{{\rm{d}}r}{{\rm{d}}\tau }\right)}^{2}+{V}_{\mathrm{eff}}=0,\end{eqnarray}$
where the effective potential of radial motion in the equatorial plane (θ = π/2) reads as $\begin{eqnarray}{V}_{\mathrm{eff}}=\displaystyle \frac{{\rm{\Delta }}\left({ \mathcal K }+{\left({ \mathcal L }-a{ \mathcal E }\right)}^{2}\right)-{\left(({a}^{2}+{r}^{2}){ \mathcal E }-a{ \mathcal L }\right)}^{2}}{2{r}^{4}}.\end{eqnarray}$
Using equation (29 ), we have plotted the dependence of the effective potential on the radius coordinate for different values of parameters a, Q and γ in ModMax gravity. The general behaviour of the effective potential Veff with reference to photon orbits for different values of BH parameters is illustrated in figure 4. It can be seen that the peak value of the effective potential, which refers to photon orbit, increases and shifts to the left with increasing charge Q for fixed γ and a. As result fixed γ contracts region between two potential lines expands. As the values of the screening factor γ and the spin parameter a increase, the effective potential peak value decreases and there is also a shift to the right for fixed a and Q, and Q and γ, respectively.
Figure 4. Radial dependence of the effective potential Veff for different values of a, Q and γ, where M = 1. |
3. Black hole shadow
Introducing the following conserved parameters $\xi ={ \mathcal L }/{ \mathcal E }$ and $\eta ={ \mathcal K }/{{ \mathcal E }}^{2}$, one may investigate the apparent shape of the shadow of the BH. First, one needs to rewrite the expression for ${ \mathcal R }(r)$ in terms of ξ and η
$\begin{eqnarray}{ \mathcal R }={\left[({r}^{2}+{a}^{2})-a\xi \right]}^{2}-{\rm{\Delta }}\left[\eta +{\left(\xi -a\right)}^{2}\right].\end{eqnarray}$
Unstable circular orbits of photons satisfy the following conditions [44]:
$\begin{eqnarray}{ \mathcal R }=0,\quad \displaystyle \frac{\partial { \mathcal R }}{\partial r}=0,\quad \displaystyle \frac{{\partial }^{2}{ \mathcal R }}{\partial {r}^{2}}\lt 0.\end{eqnarray}$
The shape and borders of the BH shadow is determined via conserved parameters ξ and η. From equations (30 ) and (31 ), one may obtain the expressions for ξ and η as described by Hioki and Maeda [44, 69]
$\begin{eqnarray}\xi =\displaystyle \frac{\left({a}^{2}+{r}^{2}\right){\rm{\Delta }}^{\prime} -4{\rm{\Delta }}r}{a{\rm{\Delta }}^{\prime} },\end{eqnarray}$
$\begin{eqnarray}\eta =\displaystyle \frac{{r}^{2}\left(16{\rm{\Delta }}\left({a}^{2}-{\rm{\Delta }}\right)-{r}^{2}{\rm{\Delta }}{{\prime} }^{2}+8{\rm{\Delta }}r{\rm{\Delta }}^{\prime} \right)}{{a}^{2}{\rm{\Delta }}{{\prime} }^{2}},\end{eqnarray}$
together with the condition $\begin{eqnarray}r+2\displaystyle \frac{{\rm{\Delta }}}{{\rm{\Delta }}{{\prime} }^{2}}({\rm{\Delta }}^{\prime} -r{\rm{\Delta }}^{\prime\prime} )\gt 0.\end{eqnarray}$
Now we investigate the shadow of a charged rotating ModMax BH introducing the celestial coordinates α and β in the following way [44, 47]:35 ) and (36 ), we obtain the following expression for the celestial coordinates
$\begin{eqnarray}\alpha =\mathop{\mathrm{lim}}\limits_{r\to \infty }\left({\left.-{r}^{2}\sin \theta \displaystyle \frac{{\rm{d}}\phi }{{\rm{d}}r}\right|}_{\theta \to {\theta }_{0}}\right)=-\xi \csc {\theta }_{0},\end{eqnarray}$
$\begin{eqnarray}\begin{array}{rcl}\beta & = & \mathop{\mathrm{lim}}\limits_{r\to \infty }\left({\left.{r}^{2}\displaystyle \frac{{\rm{d}}\theta }{{\rm{d}}r}\right|}_{\theta \to {\theta }_{0}}\right)\\ & = & \pm \sqrt{\eta +{a}^{2}{\cos }^{2}{\theta }_{0}-{\xi }^{2}{\cot }^{2}{\theta }_{0}},\end{array}\end{eqnarray}$
where θ is inclination angle. We determine the apparent shape of the BH shadow in an equatorial plane (θ0 = π/2) and, from equations ( $\begin{eqnarray}\alpha =-\xi ,\end{eqnarray}$
$\begin{eqnarray}\beta =\pm \sqrt{\eta }.\end{eqnarray}$
The line defined by the parametric equations (37 )–(38 ) in terms of ξ and η in celestial coordinates gives the silhouette of the charged rotating BH shadow. Figure 5 displays the shadows of the charged BH for fixed values of the ModMax BH parameters: spin a, charge Q and screening factor γ. From the results obtained (see figure 5), one can see that the size of the BH shadow is enlarged with an increase in the value of parameter γ (upper panel) for fixed a and Q. On the other hand, with increasing Q, the size of the BH shadow is reduced, for fixed a and γ (middle panel). The lower panel illustrates the decrease in size of the BH shadow with increasing spin parameter a (for fixed Q and γ). Moreover, the shape of the BH shadow deforms towards the right under the influence of increasing values of BH spin parameter a.
Figure 5. BH shadow for different values of ModMax parameters, where M = 1. |
As we mentioned before, as a result of the effects of BH parameters, the apparent shape of the BH shadow is distorted. Factually, any symmetric BH has a shadow in a perfectly circular form. To fully analyze the deformation of a BH's shadow, two parameters, distortion δs and the radius Rs, have been introduced by Hioki and Maedia [44].
So, considering the shape of the rotating shadow is not a pure circle as shown in figure 6 (see, for example, [43]), the radius of the shadow, Rs, of the BH is defined as follows in terms of a reference circle [43]: the shape of the silhouette as a circle passes through the topmost point (A), the bottommost point (B), and the furthest point left (C) of the boundary of the circle as shown in figure 6. The coordinates of these points are (αt, βt), (αb, βb) and (αr, 0), respectively. The most left point C on the circle corresponds to the orbit seen by an observer in the equatorial plane. The average radius Rs of the shadow of the object is approximated in the following form [44]
$\begin{eqnarray}{R}_{s}=\displaystyle \frac{{\left({\alpha }_{t}-{\alpha }_{r}\right)}^{2}+{\beta }_{t}^{2}}{2({\alpha }_{t}-{\alpha }_{r})}.\end{eqnarray}$
The second observable parameter δs which defines rate of distortion has been introduced as [44] $\begin{eqnarray}{\delta }_{s}=\displaystyle \frac{{D}_{{cs}}}{{R}_{s}},\end{eqnarray}$
where Dcs is the distance of the deviation of the shape of the silhouette of the BH from a pure circle (note that the distortion parameter is δs = 0 for the non-rotating BH shadow).
Figure 6. Schematic view of the observables: radius Rs for the BH and the distortion parameter δs = Dcs/Rs as described in [43]. |
In figure 7, the observable parameter Rs is represented as a function of BH parameters γ and charge Q. One may see that the average radius Rs of the BH shadow increases with increasing screening factor γ (upper) and decreases with the increase of charge Q (lower). We also observe in figure 8 (upper) that there is less distortion with increasing γ (δs reduces with increase of γ) and the shape of the shadow is more distorted with increasing Q (δs increases with Q, lower panel).
Figure 7. Dependence of average radius Rs of the charged rotating ModMax BH shadow on its parameters, where M = 1. The upper panel is the dependence on the screening factor γ for fixed a and selected Q. The lower panel is the dependence on the Q parameter for fixed a and selected γ. |
Figure 8. Dependence of the distortion (deviation) δs parameter of the charged rotating ModMax BH shadow on its parameters, where M = 1. The upper panel is the dependence on the screening factor γ for fixed a and selected Q. The lower panel is the dependence on charge Q for fixed a and selected γ. |
By using the data for M87* and SgrA*, which was provided by the EHT Collaboration, we aim to find the upper limit for Q charge with different fixed values of γ, according to our metric. We can determine the diameter of the shadow using [106]:
$\begin{eqnarray}{d}_{s}=\frac{D\theta }{M},\end{eqnarray}$
here, θ, D and M are the angular diameter of the BH shadow, the distance of the BH from Earth and mass of the BH, respectively. For M87*, these quantities are θM87* = 42 ± 3μas, DM87* = 16.8Mpc and MM87* = 6.5 ± 0.90x109M⊙, and for Sgr A*, θSgrA* = 48, 7 ± 7μ, DSgrA* = 8277 ± 33pc and MSgrA* = 4.3 ± 0.013x106M⊙. The diameters of the BH shadows are calculated from data ${d}_{s}^{M87* }=(11\pm 1.5)M$ and ${d}_{s}^{{SgrA}* }=(9.5\pm 1.4)M$ and we know that the radius of the BH shadow can be found from the expression ds = 2Rs. From this, the calculated minimum radii for M87* and Sgr A* are ${R}_{s}^{M87* }=4.75M$ and ${R}_{s}^{{SgrA}* }=4.05M$, respectively. In table 1, we have calculated, for fixed screening factor γ and spin a = 0.5, estimated maximum values of charge parameter Q which conformed to the radii from the provided data. It can be seen that the value of Q increases with increasing γ, corresponding with figure 7.Table 1. Some values of Q for the minimum radii of M87* and SgrA* which come from observational data for different fixed γ. |
| γ , a = 0.5 | Q , a = 0.5 | |
|---|---|---|
| −0.5 | 0.46114 | |
| M87* | 0 | 0.59212 |
| 0.5 | 0.76293 | |
| −0.5 | 0.672056 | |
| SgrA* | 0 | 0.8629381 |
| 0.5 | 1.08034 |
4. Emission energy
There is a black-body radiation theory, well known as Hawking radiation theory, which describes how the mass and rotational energy of a BH slowly reduce until it completely annihilates because of relativistic quantum effects. Here, we evaluate the energy emission rate of the charged ModMax rotating BH by using the relation [69, 70]
$\begin{eqnarray}\displaystyle \frac{{{\rm{d}}}^{2}{ \mathcal E }(\omega )}{{\rm{d}}\omega {\rm{d}}t}=\displaystyle \frac{2{\pi }^{2}{\sigma }_{{\text{}}\mathrm{lim}}}{\exp \omega /T-1}{\omega }^{3},\end{eqnarray}$
where ${\sigma }_{{\text{}}\mathrm{lim}}$ is the limiting constant value; in our case, the area of the shadow of the BH is about equal to the high-energy absorption cross section and for a BH the absorption cross section oscillates around ${\sigma }_{\mathrm{lim}}$. We know that T = κ/2π is the Hawking temperature and κ is the surface gravity. Using the average radius Rs of the BH shadow, the limited constant value ${\sigma }_{{\text{}}\mathrm{lim}}$ calculated as [69, 70] $\begin{eqnarray*}{\sigma }_{{\text{}}{lim}}\approx \pi {R}_{s}^{2}.\end{eqnarray*}$
Hence, $\begin{eqnarray*}\displaystyle \frac{{{\rm{d}}}^{2}{ \mathcal E }(\omega )}{{\rm{d}}\omega {\rm{d}}t}=\displaystyle \frac{2{\pi }^{3}{R}_{s}^{2}}{{{\rm{e}}}^{\omega /T}-1}{\omega }^{3}.\end{eqnarray*}$
The variation of energy emission (using ${{ \mathcal E }}_{\omega t}\,={{\rm{d}}}^{2}{ \mathcal E }(\omega )/{\rm{d}}\omega {\rm{d}}t$) with screening factor γ and charge Q is shown in figure 9, where it can be seen that the peak of the energy emission rate drops with the increase in the value of the γ parameter (left, Q is fixed) and increases with the increase in the value of the charge Q of the BH (right, γ is fixed).
Figure 9. Energy emission from the BH varying with the frequency for different values of the charge Q (left panel) and parameter γ (right panel), where M = 1. |
5. Conclusion
This work presents a detailed analysis of the shadow of a charged rotating ModMax BH described by the line element, equation (11 ). The horizon structure and equations of motion of photons in the vicinity of the BH are studied, and their effects on the effective potential and the shape of the shadow of the BH are examined. Additionally, the influence of the ModMax BH parameters, such as spin a, charge Q and screening factor γ, on the energy emission process is investigated. The results are summarized in the following statements:
| • | From figure 1, separatrix lines which are the borders corresponding to extreme BHs separate the BH region from the no BH region. Additionally, we have demonstrated that increasing the γ parameter causes the BH region to expand while the non-BH region narrows. |
| • | The event horizon structure of the charged rotating ModMax BH is modified, according to figure 2, by varying the charge Q and screening factor γ parameter with the following results: the horizon radius increases with increase in the value of γ for fixed Q, and, for fixed γ, the radius decreases with increasing Q. From figure 3 the BH horizon exists at γ > γε, a < aε and Q < Qε, and there is naked singularity at γ < γε, a > aε and Q > Qε. γε and aε increase with increase of Q and γ, respectively, and Qε decreases with decreasing γ parameter. The inner (Cauchy) horizon radius decreases and the outer horizon radius increases with increasing γ, and inner horizon radius increases and outer horizon radius decreases with increase of a and Q. |
| • | We have also studied the general behaviour of effective potential with respect to photon orbits for different values of BH parameters a, Q and γ, by plotting in figure 4 the dependence of the effective potential and radius on the parameters. According to figure 4, a peak of the effective potential which correspond to photon orbit rises and shifts towards the left direction with the increasing charge Q and fixed γ sees decline region between two potential lines expands. In addition, with increasing screening factor γ and spin parameter a, the peak value of effective potential falls away and there is also a shift to the right. |
| • | Using the Hamilton–Jacobi formulation and the separation of variables method, we have analytically derived the equation of motion of the photon around a BH. Using the geodesic equations, we have investigated the apparent shape of the BH shadow in celestial coordinates, figure 5. The size of the BH shadow is enlarged with an increase in parameter γ and decreased with increasing Q and decreasing a. As a result of increasing values of BH spin parameter a, the shape of the BH shadow is distorted towards the right. |
| • | We have analyzed the rate of distortion δs and average radius Rs, which are BH shadow parameters, in figures 7 and 8. With the increase in parameter γ, radius Rs increases but distortion rate δs reduces. An increase in Q charge causes radius Rs to decrease but distortion δs increases. In table 1, we have calculated maximum values of Q which correspond to the radii of BH shadows from EHT data. |
| • | Finally, in figure 9, we evaluate the effects of γ and Q parameters on the energy emission rate of the charged rotating ModMax BH. According to the figure, the peak of the energy emission rate falls away with increasing γ and increases with increasing charge Q. |