Ergosphere and shadow of a rotating regular black hole
EErgosphere and shadow of a rotating regular black hole
Sushant G. Ghosh,
1, 2, 3, ∗ Muhammed Amir, † and Sunil D. Maharaj ‡ Centre for Theoretical Physics, Jamia Millia Islamia, New Delhi 110025, India Multidisciplinary Centre for Advanced Research and Studies (MCARS),Jamia Millia Islamia, New Delhi 110025, India Astrophysics and Cosmology Research Unit, School of Mathematics,Statistics and Computer Science, University of KwaZulu-Natal,Private Bag X54001, Durban 4000, South Africa (Dated: June 16, 2020)
Abstract
The spacetime singularities in classical general relativity predicted by the celebrated singularity theoremsare formed at the end of gravitational collapse. Quantum gravity is the expected theory to resolve the singu-larity problem, but we are now far from it. Therefore attention has shifted to models of regular black holesfree from the singularities. A spherically symmetric regular toy model was obtained by Dymnikova (1992)which we demonstrate as an exact solution of Einstein’s field equations coupled to nonlinear electrodynamicsfor a Lagrangian with parameter b related to magnetic charge. We construct rotating counterpart of thissolution which encompasses the Kerr black hole as a special case when charge is switched off ( b = 0). EventHorizon Telescope has released the first image of supermassive black hole M87 ∗ , revealing the structurenear black hole horizon. The rotating regular black hole’s shadow may be useful to determine strong fieldregime. We investigate ergosphere and black hole shadow of rotating regular black hole to infer that theirsizes are sensitive to charge b and have a richer chaotic structure. In particular, rotating regular black holepossess larger size, but less distorted shadows when compared with Kerr black holes. We find one to onecorrespondence between ergosphere and shadow of the black hole. ∗ Electronic address: [email protected], [email protected] † Electronic address: [email protected] ‡ Electronic address: [email protected] a r X i v : . [ g r- q c ] J un . INTRODUCTION The elegant theorems of Hawking and Penrose imply that spacetime singularities are pervasivefeatures of general relativity, so that the theory itself predicts its own failure to describe thephysics of these extreme situations. The spacetime singularities that arise in gravitational collapseare always hidden inside black holes, which is the essence of the weak cosmic censorship conjecture,put forward 40 years ago by Penrose [1]. This signature is still one of the most important openquestions in general relativity. We are far away from any robust and reliable quantum theory ofgravity capable of resolving the singularities in the interior of black holes. Hence, there is significantattention towards models of black hole solutions without the central singularity. The earliest ideaof regular models dates back to the pioneering work of Sakharov [2] and Gliner [3] where theyproposed that the spacetime in the highly dense central region of a black hole should be de Sitter-like at r (cid:39)
0. Thus spacetime filled with vacuum could provide a proper discrimination at the finalstage of gravitational collapse, replacing future singularity [3]. The prototype of these regular blackholes is the Bardeen metric [4], which can be formally obtained by coupling Einstein’s gravity to anonlinear electrodynamic field [5], thought to be an alteration of the Reissner-Nordstr¨om solution.There has been enormous attentions to obtain regular black holes [6–14]; most of these regular blackholes were driven by the Bardeen idea [4]. However, these are only nonrotating black holes, whichcan be hardly tested by observations, as black hole spin is an important parameter in astrophysicalprocesses.Further, a study of rotating regular solutions is important as the astronomical observationspredict that the astrophysical black hole might be a Kerr black hole [15]. However, the real natureof these astrophysical black holes is still to be verified [15]. Analyzing the iron kα lines and thecontinuum fitting method are possible procedures, which can probe the geometry of spacetimearound astrophysical black holes candidates [16]. The rotating solutions of the Bardeen metricshave been obtained in Ref. [15]. There exists a number of rotating regular black holes [17–21]which were discovered with the help of the Newman-Janis algorithm [22], and also by using othertechniques [23–26]. The authors in Refs. [27–31] demonstrated that the regular black holes can beconsidered as the particle accelerator. The quasinormal modes of test fields around the regular blackholes has been discussed explicitly in [32]. An interesting study on electromagnetic perturbationof the regular black holes has been discussed thoroughly in [33–35].The black hole shadow is a dark zone in the sky and the shadow is a useful tool for measuringblack hole parameters because its shape and size carry impression of the geometry surrounding2he black hole. Observing the black hole shadow is a possible method to determine the spin ofthe black hole, and this subject is popular nowadays. This triggered theoretical investigation ofthe black holes for wide variety of spacetimes [36–64]. Apart from black holes, the theoreticalinvestigation of shadow of wormholes has also been discussed in [65–73]. Hioki and Maeda [40]discuss a simple relation between the shape of shadow, inclination angle and spin parameter forthe Kerr black hole. A coordinate-independent method was proposed to distinguish the shape ofblack hole shadow from other rotating black holes [56]. From the observational point of view, thestudy of shadow is significant because of the Event Horizon Telescope . It was setup with an aimto image the event horizon of supermassive black holes Sgr A ∗ and M87 ∗ . They have successfullyreleased the first image of M87 ∗ which is in accordance with the Kerr black hole predicted bygeneral relativity [74–76]. The photon ring and shadow have been observed which open a newwindow to test general relativity and modified theories of gravity in strong field regime.The purpose of this paper is to construct the rotating counterpart or Kerr-like regular black holefrom the spherically symmetric regular black hole proposed by Dymnikova [77]. The black hole issingularity free or regular black hole with an additional parameter b , and for definiteness, we cancall it the rotating regular black hole. We investigate several properties including the shadow ofthis black hole to explicitly bring out the effect of parameter b , and compare with the Kerr blackhole. It turns out that the parameter b leaves a significant imprint on the black hole shadow, andalso affects the other properties.The paper is organized as follows. In Sec. II, we show that the spherically symmetric regularblack hole obtained by Dymnikova [77] is an exact solution of general relativity coupled to nonlinearelectrodynamics. We introduce a line element of the rotating regular black hole spacetime anddiscuss associated sources with it in Sec. III. The weak energy conditions is the subject of Sec. IV.In Sec. V, we discuss some basic properties of the rotating regular black hole. Shadow of rotatingregular black hole is the subject of Sec. VI where we derive analytical formulae for the shadow andwe conclude in the Sec. VII. We consider the signature convention ( − , + , + , +) for the spacetimemetric and use the geometric unit G = c = 1 throughout the paper. https://eventhorizontelescope.org I. NONLINEAR ELECTRODYNAMICS AND EXACT REGULAR BLACK HOLE SO-LUTION
The spherically symmetric metric proposed by Dymnikova [77] represents a regular black holewith de Sitter core instead of a singularity. The stress-energy tensor responsible for the geometrydescribes a smooth transition from the standard vacuum state at infinity to isotropic vacuumstate through anisotropic vacuum state in the intermediate region. The spacetime metric of theDymnikova solution reads ds = − (cid:34) − M (1 − e − r /b ) r (cid:35) dt + (cid:34) − M (1 − e − r /b ) r (cid:35) − dr + r d Ω , (1)where d Ω = dθ + sin θdφ , M is the black hole mass, and the parameter b is given by b = r M .The corresponding energy-momentum tensor of metric (1) is [77] T = T = 3 r e − r /b ,T = T = 3 r (cid:18) − r b (cid:19) e − r /b . (2)The metric (1) admits two horizons, i.e., the event horizon and the Cauchy horizon, but there isno singularity. We know that the horizons are zeros of g rr = 0, which are given [77] by r + ≈ M (cid:20) − O (cid:18) exp( − M r ) (cid:19)(cid:21) ,r − ≈ r (cid:104) − O (cid:16) exp( − r M ) (cid:17)(cid:105) . The metric (1) is regular, everywhere including at r = 0, which is evident from the behavior of thecurvature invariants. Thus the metric (1) that for large r coincides with the Schwarzschild solution,for small r behaves like the de Sitter spacetime and describes a spherically symmetric regular blackhole. In what follows, we show the metric (1) as an exact solution of general relativity coupled toan appropriate nonlinear electrodynamics.We start with action of general relativity coupled to the nonlinear electrodynamics which isgiven by S = 116 π (cid:90) d x √− g [ R − L ( F )] , (3)where R is the Ricci scalar and L ( F ) is an arbitrary function of F = F µν F µν with F µν = 2 ∇ [ µ A ν ] ,and A µ denotes the electromagnetic potential. The Einstein field equations are derived from the4ction (3) which read simply G µν = 2 (cid:18) ∂ L ∂F F µα F αν − g µν L (cid:19) , (4) ∇ µ (cid:18) ∂ L ∂F F µν (cid:19) = 0 . (5)We further consider the spherically symmetric line element of the following form ds = − (cid:20) − M ( r ) r (cid:21) dt + (cid:20) − M ( r ) r (cid:21) − dr + r d Ω . (6)For the spherically symmetric spacetime the only nonzero components of F µν appropriate to mag-netic charge are F θφ and F φθ with F θφ = − F φθ such that F θφ = 2 b sin θ [78]. Now it is straightforward to compute F = 2 b r = 2 B , (7)where B is the absolute value of magnetic field. The regular solution [77] in which we are muchinterested comes from particular nonlinear electrodynamics source L ( F ) = 3 sb exp (cid:34) − (cid:18) b F (cid:19) / (cid:35) , (8)where the parameter s is given by s = | b | /M with b and M are arbitrary constant can be related tothe magnetic charge and the black hole mass, respectively. The ( t, t ) component of the Einstein’sfield equations (4) gives M (cid:48) ( r ) = r L ( F ) . (9)On substituting (8) into (9) and then integrating it results in (cid:90) ∞ r M (cid:48) ( r ) dr = M − M ( r ) = M e − r /b , (10)where the integration constant is purposely chosen as M , thus the above equation yields M ( r ) = M (1 − e − r /b ) . (11)When we substitute (11) in (6) eventually we get the spherically symmetric Dymnikova regularspacetime (1). Thus, we can say that the Dymnikova spacetime (1) can be obtained exactly withsource from a particular nonlinear electrodynamics where the source is given by the Lagrangian(8). 5 II. ROTATING REGULAR BLACK HOLES
In this section, we aim to study the rotating counterpart of the Dymnikova spacetime whichis a generalization of the Kerr black hole. We start by constructing a rotating counterpart of theregular spacetime (1) which in the Boyer-Lindquist coordinates reads ds = − (cid:18) − M ( r ) r Σ (cid:19) dt − aM ( r ) r sin θ Σ dtdφ + Σ∆ dr + Σ dθ + (cid:18) r + a + 2 a M ( r ) r sin θ Σ (cid:19) sin θdφ , (12)where Σ = r + a cos θ, ∆ = r + a − rM ( r ) . Here a represents the rotation parameter and M ( r ) is the mass function given in (11). Henceforth,we use the term rotating regular black hole for the spacetime metric (12). It is noticeable that inthe absence of charge ( b = 0), we obtain the Kerr spacetime. The spacetime (12) has independenceon t and φ coordinates that means t and φ coordinates are cyclic coordinates in the Lagrangianwhich corresponding to two Killing vectors, namely, the time translation Killing vector χ a , and theazimuthal Killing vector ζ a .Now we are going to check the validity of the rotating solution by computing the source nonlinearelectrodynamics equations. The action and corresponding field equations are explicitly expressedin the previous section. The vector potential in case of spherically symmetric spacetime is givenby A µ = − b cos θδ φµ . The vector potential for the rotating spacetime gets modify [21] and given asfollows A µ = (cid:20) ba cos θ Σ , , , − b ( r + a ) cos θ Σ (cid:21) . (13)It is easy to compute the field strength ( F ) for the rotating spacetime, which turns out to be infollowing form F = 2 b [( r − a cos θ ) − a r cos θ ]Σ . (14)We can immediately recover F = 2 b /r while substitute a = 0 in (14). In order to compute thesource for the rotating spacetime, we solve the Einstein field equations, G µν = T µν with respect to L and ∂ L /∂F . Consequently, the Lagrangian density reads simply L = 3 M r e − r /b [ r b + 2 a r ( b − r ) cos θ + 3 a (3 b − r ) cos θ ] b Σ , (15)6nd ∂ L /∂F has the form ∂ L ∂F = 6 M r e − r /b [3 r − a (4 b − r ) cos θ ]8 b . (16)Interestingly, when we substitute a = 0 into Eqs. (15) and (16) as a result we obtain L = 3 M e − r /b b , ∂ L ∂F = 9 M r e − r /b b . (17)These expressions are exactly similar to that of the spherically symmetric spacetime. IV. ENERGY CONDITIONS
Now we are much interested in computing the orthonormal basis also known as tetrads of thespacetime (12) so that we can analyze the matter associated with the rotating regular spacetime[20, 79]. The computation of the orthonormal basis gives the following form [20, 79]: e ( a ) µ = (cid:112) ∓ ( g tt − Ω g tφ ) 0 0 00 √± g rr √ g θθ g tφ / √ g φφ √ g φφ , (18)where Ω = g tφ /g φφ is the angular velocity of the rotating regular black hole. We further determinethe components of the energy-momentum tensor by using the following relation [20, 79] T ( a )( b ) = e ( a ) µ e ( b ) ν G µν . When we compute the components of the energy-momentum tensor T ( a )( b ) , then we realize that ithas only diagonal components [20, 79], i.e., T ( a )( b ) = diag( ρ, P , P , P ) , (19)where ρ and P are the matter density and the pressure, respectively. In case of the rotating regularblack hole, the quantities ρ and P that appearing in (19) have the forms as follows ρ = 6 M r e − r /b b Σ = − P ,P = − M r e − r /b (cid:2) − r Σ + 2 b ( r + 2 a cos θ ) (cid:3) b Σ = P . (20)When we compare them with the components of static counterpart (2), we find that the componentsof (20) are different because of the black hole rotation. This also happens when we consider rotating7 IG. 1: Radial and angular dependence of matter density ρ for the rotating regular black hole, for thegiven values of the rotation parameter a = 0 . , .
6, the parameter b = 0 . , .
5, and θ = π/ x = cos θ ). Vaidya solution [80], it has in addition null radiation some another source. Having the expressionsof matter density ρ and pressure P i , we can verify the weak energy condition for the spacetime(12) that requires the inequalities ρ ≥ ρ + P i ≥
0, ( i = 1 , ,
3) to be satisfied. Moreover, astraight forward calculation gives ρ + P = ρ + P = 3 M r e − r /b (cid:2) r Σ − a b cos θ (cid:3) b Σ , (21)which shows that the energy condition may be violated and it can be seen in the plots (cf. Fig. 1and 2). We find that the matter density is always positive for all values of a and b as can beseen from the Fig. 1. It turns out that the rotating regular black hole solution may violate energyconditions; however, this happens for all the rotating regular black holes in the literature (see, e.g.,[17, 20]). Despite small violation of such solutions are important from phenomenologically and also8 IG. 2: Radial and angular dependence of ρ + P for the rotating regular black hole, for the given valuesof the rotation parameter a = 0 . , .
6, the parameter b = 0 . , .
5, and θ = π/ x = cos θ ). they are important as astrophysical black holes are rotating. V. PROPERTIES OF ROTATING REGULAR SPACETIME
In this section, we will discuss some important properties of the rotating regular black holesolution, for instance, the curvature scalars, the horizons, and the ergospheres etc. It is importantto study these basic properties of the spacetime from physical point of view.
A. Curvature invariants
In order to check the curvature singularity within the spacetime, we compute the curvatureinvariants or scalars of the rotating regular black hole spacetime. We find that the curvature9
IG. 3: The behavior of Ricci square and Riemann square for the rotating regular black hole. invariants of the spacetime (12) have cumbersome forms, but in the limit θ → π/
2, they reduce tothe following expressionslim θ → π/ R = 18 M (9 r − r b + 8 b ) e − r /b b , lim θ → π/ K = 12 M e − r /b b r (cid:104) b (1 + e r /b ) − r b + 2 r b + 2 b ) e r /b +27 r + 24 r b + 8 r b (cid:105) , lim θ → π/ R = − M (3 r − b ) e − r /b b . (22)We further consider the limit r → r → lim θ → π/ R = 144 M b , lim r → lim θ → π/ K = 96 M b , lim r → lim θ → π/ R = 24 Mb . (23)The curvature invariants have finite values for M, b (cid:54) = 0, therefore, we can say that the metric(12) is regular everywhere as can be seen from the Fig. 3. The figure shows that the Kretschmannscalar and the square of Ricci tensor are well behaving for different values of the parameters b and a . It is noticeable that the presence of the exponential factor in the mass of the black hole (12),i.e., e − r /b , makes the black hole singularity free. B. Horizons
Now we wish to discuss the effect of charge on the structure of horizons and ergosphere. It turnsout that as like the Kerr black hole, the spacetime has two surfaces, viz., static limit surfaces andhorizons. The static limit surface is a surface on which the time translation Killing vectors ( χ a )10 .0 0.5 1.0 1.5 2.0 - - r g tt a = Θ = Π b = b = b = b = b = b = - - - - r g tt a = Θ = Π b = b = b = b = b = b = FIG. 4: Plots showing the variation of static limit surface with radius r for different values of the parameter b and the rotation parameter a .TABLE I: Radius of EHs, SLSs and δ a = r + SLS − r + EH for different values of parameter b a = 0 . a = 0 . a = 0 . a = 0 . b r + H r + sls δ . r + H r + sls δ . r + H r + sls δ . r + H r + sls δ . becomes null or χ a χ a = 0. It requires that the g tt component to be vanished, r + a cos θ − M (1 − e − r /b ) r = 0 . (24)The typical behavior of the static limit surface is depicted in Fig. 4 for various values of theparameters (see also Table I). It turns out that the behavior is similar to that of the Kerr blackholes. However, the radii of the static limit surface shrink with increasing values of charge b (Table I).Since the spacetime (12) has a coordinate singularity at ∆ = 0, which corresponds to the horizonof the rotating regular black holes. Actually the event horizon is located at the larger root ( r + ) of∆ = r + a − M (1 − e − r /b ) r, (25)where (24) coincides with (25) when either θ = 0 or π . Clearly, the radii of horizons depend on the11 .0 0.5 1.0 1.5 2.0 - r D a = b = b = b = b = b = b = - r D a = b = b = b = b = b = b = FIG. 5: Plots showing the variation variation of ∆ with radius r for different values of the parameter b andthe rotation parameter a .TABLE II: Radius of inner horizon and outer horizon for different values of parameter b and aa = 0 . a = 0 . a = 0 . a = 0 . b r − H r + H r − H r + H r − H r + H r − H r + H parameter b and they are different from the Kerr black hole. We solve (25) for horizons numericallyas well as plot them in Fig. 5 for different values of parameters a and b . It turns out that thehorizons of the rotating regular black hole have similar behavior like the Kerr black hole, but thereexist several extremal black holes corresponding to the various values of charge b .The Fig. 5 and Table II shows the existence of two roots, for a set of values of parameters a and b , which corresponds to the Cauchy horizon (smaller root) and event horizon (larger root).We find that for a given value of a , there exists a critical value of parameter b = b c , where the twohorizons coincide ( r = r ± ) corresponding to the extremal black holes (Fig. 5 and 6). When b < b c ,we have rotating black hole with two horizons, and for b > b c no black hole will form (Fig. 5 and6). 12 - - - r sin Θ cos Φ r c o s Θ a = b = - - - - r sin Θ cos Φ r c o s Θ a = b = - - - - r sin Θ cos Φ r c o s Θ a = b = - - - - r sin Θ cos Φ r c o s Θ a = b = - - - - r sin Θ cos Φ r c o s Θ a = b = - - - - r sin Θ cos Φ r c o s Θ a = b = - - - - r sin Θ cos Φ r c o s Θ a = b = - - - - r sin Θ cos Φ r c o s Θ a = b = - - - - r sin Θ cos Φ r c o s Θ a = b = - - - - r sin Θ cos Φ r c o s Θ a = b = - - - - r sin Θ cos Φ r c o s Θ a = b = - - - - r sin Θ cos Φ r c o s Θ a = b = - - - - r sin Θ cos Φ r c o s Θ a = b = - - - - r sin Θ cos Φ r c o s Θ a = b = - - - - r sin Θ cos Φ r c o s Θ a = b = - - - - r sin Θ cos Φ r c o s Θ a = b = FIG. 6: Plots showing the behavior of the ergoregion in the xz-plane for different values of the parameters b and a . C. Ergosphere
An ergosphere is bounded by the two above discussed surfaces, namely, the static limit surfaceand the event horizon. It lies outside the black hole. The ergoregion of the rotating black holes isa region of spacetime where the time translation Killing vector χ a becomes spacelike, and also thatevery timelike vector acquires rotational ( φ ) counterpart. Hence, a particle can enter into ergoregion13nd can leave again, but it cannot remain stationary without the ergosphere. The ergospheres forthe rotating regular black hole are shown in the Fig. 6 for different values of parameters b , whichare polar plots of (24) and (25).Interestingly, we note that ergosphere becomes more prolate with increasing values of charge b (see Fig. 6 horizontally) and with increasing spin a (see Fig. 6 from top to bottom). It meansthe faster rotating and charged black holes are more prolate thereby increasing the area of theergosphere. Furthermore, we observe from the Fig. 6 that we can find a critical parameter b = b c where two surface shrinks to one, and for b > b c , we have no ergoregion. It turns out that energycan be extracted from ergoregion via Penrose process [81] and the magnetic charge shall affect thePenrose process. VI. BLACK HOLE SHADOW
In this section, we are going to study one of the most interesting astrophysical phenomena forthe rotating regular black hole known as black hole shadow. This can be understood as follows.When a black hole is placed in between an observer and the bright background source, the photonswith a small angular momentum fall into the black hole without reaching the observer create adark spot in the sky, which is called the black hole shadow. Thus, to obtain the apparent shape orshadow of the black hole and discuss the properties of shadow, we must discuss the geodesic of thephoton in the background of the rotating regular black hole. These geodesics can be determinedby the Hamilton-Jacobi formulation [82] given by ∂S∂σ = − g µν ∂S∂x µ ∂S∂x ν , (26)which has a solution in the separable form as follows S = 12 m σ − Et + L z φ + S r ( r ) + S θ ( θ ) , (27)where m corresponds to the rest mass of the particle, and S r ( r ) and S θ ( θ ) are respectively functionsof r and θ only. Note that in case of photon the rest mass, m = 0. By following the standardprocedure of the Hamilton-Jacobi method, we obtain the following forms of the geodesic equations,Σ dtdσ = − a (cid:0) aE sin θ − L z (cid:1) + ( r + a ) (cid:2) ( r + a ) E − aL z (cid:3) r + a − M (1 − e − r /b ) r , (28)Σ dφdσ = − (cid:0) aE − L z csc θ (cid:1) + a (cid:2) ( r + a ) E − aL z (cid:3) r + a − M (1 − e − r /b ) r , (29)14 drdσ = ±√R , (30)Σ dθdσ = ±√ Θ , (31)where R and Θ can be expressed as follows R = (cid:2) ( r + a ) E − aL z (cid:3) − (cid:104) r + a − M (1 − e − r /b ) r (cid:105) (cid:2) K + ( L z − aE ) (cid:3) , Θ = K + cos θ (cid:0) a E − L z csc θ (cid:1) . (32)These geodesic equations represent first-order differential equations with respect to the affine pa-rameter σ and contain three constants of the motion: the energy E , the angular momentum L z ,and the Carter constant K [82]. If we substitute b = 0, in the geodesic equations, they reduce tothe Kerr spacetime case [40]. Now in order to get the shadow of the rotating regular black hole, weintroduce the two independent dimensionless quantities or impact parameters such that ξ = L z /E and η = K /E . Equation R = 0 explores the radial turning points of the photons and the sphericalphoton orbits around the black hole can be determined by the following relations [79] R = 0 and d R dr = 0 . (33)On solving (33) for impact parameters ξ and η , we immediately get their forms as follows ξ = M (cid:2) a ( b − r ) − r ( b + r ) (cid:3) − b e r /b (cid:2) a ( r + M ) + r ( r − M ) (cid:3) aM ( b − r ) + ab ( r − M ) e r /b ,η = 4 a b r M e r /b (cid:104) b ( e r /b − − r (cid:105) − r (cid:104) b e r /b ( r − M ) + 3 M ( b + r ) (cid:105) (cid:2) aM ( b − r ) + ab ( r − M ) e r /b (cid:3) . (34)It turns out that the impact parameters have a dependency on charge b . If charge b is set to bezero, then Eq. (34) reduces to the Kerr black hole case [40]. This calculation is helpful to determinethe celestial coordinates for a distant observer in order to get the shadow of the rotating regularblack hole. The celestial coordinates for a distant observer in ( α , β )-plane are given [83] by α = lim r →∞ (cid:18) − r sin θ dφdr (cid:19) ,β = lim r →∞ (cid:18) r dθdr (cid:19) , (35)where r is the distance from the black hole to the observer and θ represents the inclination anglebetween the direction to observer and the rotation axis of the black hole. By using (28), (29), (30),(31), and (32), the celestial coordinates (35) transform into α = − ξ csc θ ,β = ± (cid:112) η + a cos θ − ξ cot θ . (36)15 - - - - β a = = θ = ° θ = ° θ = ° θ = ° - - - - - β a = = θ = ° θ = ° θ = ° θ = ° - - - - - β a = = θ = ° θ = ° θ = ° θ = ° - - - - - β a = = θ = ° θ = ° θ = ° θ = ° FIG. 7: Plots showing the shapes of shadow with variation of inclination angle θ for fixed values of b and a . - - - - - α β b = θ o = o a = a = a = a = - - - - - α β b = θ o = o a = a = a = a = - - - - - α β b = θ o = o a = a = a = a = - - - - - α β b = θ o = o a = a = a = a = FIG. 8: Plots showing the shapes of shadow for different values of the parameter b and a . Equation (36) represents a direct relationship between the celestial coordinates ( α, β ) and theimpact parameters ( ξ, η ). Every single photon that approaches the observer determines a pointon the ( α , β )-plane of the image, which can be visible with the help of telescope. The size andshape of the shadow depends on the parameters of black hole as well as on the inclination angle θ .The pictorial representation of the rotating regular black hole’s shadow can be seen from Figs. 7,8 and 9. We obtain several images of the rotating regular black hole’s shadow for the particularchoice of the parameters a , b , and θ . A distortion in the shape of the black hole’s shadow arisesthat increases with higher values of a as well as θ . Fig. 9 depicts the effect of parameter b on theshadow of the rotating regular black hole, which includes the case of the Kerr black hole ( b = 0)for a comparison. Moreover, the actual size and distortion in shape of the rotating regular blackhole’s shadow can be determined by the observables, namely, R s and δ s [40]. Here R s and δ s correspond to the actual size or radius of the shadow and distortion in the shape of the shadow,respectively. In order to compute the radius of the shadow, we consider a circle passing throughthe three different points, namely, top, bottom, and rightmost corresponding to ( α t , β t ), ( α b , β b ),and ( α r , - - - - α β a = , θ o = o - - - - - - b = b = b = - - - - - α β a = , θ o = o - - - - - - - b = b = b = - - - - - α β a = , θ o = o - - - - - - - b = b = b = - - - - - α β a = , θ o = o - - - - - - - - b = b = b = FIG. 9: Plots showing the shapes of shadow for different values of the parameter b . and Maeda [40]: R s = ( α t − α r ) + β t | α t − α r | , (37) δ s = ( ˜ α p − α p ) R s , (38)where ( ˜ α p ,
0) and ( α p ,
0) are the points where the reference circle and the silhouette of the shadowcut the horizontal axis at the opposite side of ( α r ,
0) (cf., Fig. 10). This special characterizationhelps us to find the effect of charge b on radius and distortion in the shape of the rotating regularblack hole’s shadow. The typical behavior of these observables with charge b can be seen in Fig. 11.We examine that an increase in the magnitude of charge b increases the radius of the black hole’s17 ) O R s ( ,0)( ,0) ~ ( ) FIG. 10: Illustration of the observable of the black hole shadow [59]. b R s a = b δ s a = FIG. 11: Plots showing the behavior of observable R s and δ s with parameter b . shadow ( R s ). On the other hand, the distortion in the shape of the black hole’s shadow ( δ s )decreases with an increase in magnitude of charge b (cf. Fig. 11). When a comparison of ourresults with the Kerr-Newman and the braneworld Kerr spacetimes is taking into account, wediscover that the effect of nonlinear charge (our case) is opposite that of electric charge q (Kerr-Newman) and tidal charge (braneworld Kerr). In our case, the radius of the black hole’s shadowincreases with charge instead of being decreased at the same time the distortion decreases withcharge instead of being increase. 18 II. CONCLUDING REMARKS
We have shown that the regular black hole metric (1) is an exact solution of the Einstein’s fieldequations coupled to the nonlinear electrodynamics associated with the Lagrangian (8) correspondsto a magnetic charge b . We have also constructed a rotating counterpart of the regular black hole(1) containing an additional parameter b , which encompasses the Kerr black hole in the particularcase when b = 0, and therefore belongs to a family of non-Kerr black holes. The source of therotating regular black hole has also been computed in order to validate the solution. We havefurther discussed the energy conditions of the rotating regular black hole. The rotating regularspacetime describes the extremal black holes with degenerate horizons for a critical amount ofcharge b = b c , the non-extremal black holes with two distinct horizons for b < b c corresponding tothe Cauchy and the event horizon. We have made a comprehensive analysis of horizon structureand ergosphere of the rotating regular black holes and explicitly brings out the effect of charge b . The ergosphere is sensitive to the charge b , which enlarges the ergoregion and becomes moreprolate with an increasing magnitude of b .The immediate target of the Event Horizon Telescope is observations of the image by super-massive black hole Sgr A ∗ and M87 ∗ (shadow) which is a major attempt of understanding thenature of these black holes and explain the strong field gravity. Motivated from this, we havestudied the shapes of the shadow from rotating regular black hole to discuss the effect of charge b on the Kerr black hole shadow. We have done a detailed analysis of particle motion in the rotatingregular black hole spacetimes. In order to achieve the goal, we have derived necessary analyticalexpressions to obtain the black hole shadow. The shadow is completely characterized by the twoobservables, namely, radius R s and distortion δ s . It turns out that the radius R s increases with anincrease in the magnitude of charge b , it results in a larger shadow of the rotating regular black holethan the Kerr black hole shadow. The rotating regular black hole is less distorted when comparedwith the Kerr black hole shadow as distortion δ s decreases with increasing charge b . We have alsocompared our results with the Kerr-Newman and braneworld Kerr spacetimes; as a consequence,we found that in our case, the charge increases the radius of shadow and decreases the distortionin the shadow of the black hole. The results obtained in our study may be useful in the light ofthe observational outcome of the Event Horizon Telescope. The black hole shadow may help toconclude that whether the Kerr metric is an accurate description of astrophysical black holes orthere is room for non-Kerr black holes like one we have discussed.19 cknowledgments We thank the DST INDO-SA bilateral project for Grant No. DST/INT/South Africa/P-06/2016. M.A. acknowledges that this research work is supported by the National Research Foun-dation, South Africa. S.D.M. acknowledges that this work is based upon research supported bySouth African Research Chair Initiative of the Department of Science and Technology and theNational Research Foundation. We would like to thank IUCAA, Pune, for hospitality, where apart of this work was done. [1] R. Penrose, Riv. Nuovo Cim. , 252 (1969) [Gen. Rel. Grav. , 1141 (2002)].[2] A.D. Sakharov, Sov. Phys. JETP, , 241 (1966).[3] E.B. Gliner, Sov. Phys. JETP, , 378 (1966).[4] J. Bardeen, in Proceedings of GR5 (Tiflis, U.S.S.R., 1968)[5] E. Ay´on-Beato and A. Garc´ıa, Phys. Lett. B , 149 (2000)[6] I. Dymnikova, Class. Quantum Gravity , 4417 (2004).[7] K.A. Bronnikov, Phys. Rev. D , 044005 (2001).[8] S. Shankaranarayanan and N. Dadhich, Int. J. Mod. Phys. D , 1095 (2004).[9] S.A. Hayward, Phys. Rev. Lett. , 031103 (2006).[10] S. Ansoldi, arXiv:0802.0330 .[11] H. Culetu, Int. J. Theor. Phys. , 2855 (2015).[12] L. Balart and E.C. Vagenas, Phys. Lett. B , 14 (2014).[13] L. Balart and E.C. Vagenas, Phys. Rev. D , 124045 (2014).[14] L. Xiang, Y. Ling, and Y.G. Shen, Int. J. Mod. Phys. D , 1342016 (2013).[15] C. Bambi, Mod. Phys. Lett. A , 2453 (2011).[16] C. Bambi, Phys. Lett. B , 59 (2014).[17] C. Bambi and L. Modesto, Phys. Lett. B , 329 (2013).[18] B. Toshmatov, B. Ahmedov, A. Abdujabbarov, and Z. Stuchl´ık, Phys. Rev. D , 104017 (2014).[19] S.G. Ghosh and S.D. Maharaj, Eur. Phys. J. C , 7 (2015).[20] J.C.S. Neves and A. Saa, Phys. Lett. B , 44 (2014).[21] B. Toshmatov, Z. Stuchl´ık, and B. Ahmedov, Phys. Rev. D , 084037 (2017).[22] E.T. Newman and A.I. Janis, J. Math. Phys. , 915 (1965).[23] M. Azreg-A¨ınou, Eur. Phys. J. C , 2865 (2014).[24] M. Azreg-A¨ınou, Phys. Lett. B , 95 (2014).[25] M. Azreg-A¨ınou, Phys. Rev. D , 064041 (2014).[26] S.G. Ghosh, Eur. Phys. J. C , 532 (2015).
27] S.G. Ghosh, P. Sheoran, and M. Amir, Phys. Rev. D , 103006 (2014).[28] M. Amir and S.G. Ghosh, J. High Energy Phys. 07 (2015) 015.[29] S.G. Ghosh and M. Amir, Eur. Phys. J. C , 553 (2015).[30] M. Amir, F. Ahmed and S.G. Ghosh, Eur. Phys. J. C , 532 (2016).[31] F. Ahmed, M. Amir and S.G. Ghosh, Astrophys. Space Sci. , 10 (2019).[32] B. Toshmatov, A. Abdujabbarov, Z. Stuchl´ık, and B. Ahmedov, Phys. Rev. D , 083008 (2015).[33] B. Toshmatov, Z. Stuchl´ık, J. Schee, and B. Ahmedov, Phys. Rev. D , 084058 (2018).[34] B. Toshmatov, Z. Stuchl´ık, and B. Ahmedov, Phys. Rev. D , 085021 (2018) .[35] B. Toshmatov, Z. Stuchl´ık, B. Ahmedov, and D. Malafarina, Phys. Rev. D , 064043 (2019).[36] S. Chandrasekhar, The Mathematical Theory of Black Holes (Oxford University Press, New York,1992).[37] H. Falcke, F. Melia, and E. Agol, Astrophys. J. , L13 (2000).[38] R. Takahashi, J. Korean Phys. Soc. , S1808 (2004) [Astrophys. J. , 996 (2004)].[39] R. Takahashi, Publ. Astron. Soc. Jap. , 273 (2005).[40] K. Hioki and K.i. Maeda, Phys. Rev. D , 024042 (2009).[41] C. Bambi and K. Freese, Phys. Rev. D , 043002 (2009).[42] C. Bambi and N. Yoshida, Class. Quant. Grav. , 205006 (2010).[43] L. Amarilla, E.F. Eiroa, and G. Giribet, Phys. Rev. D , 124045 (2010).[44] L. Amarilla and E.F. Eiroa, Phys. Rev. D , 064019 (2012).[45] A. Yumoto, D. Nitta, T. Chiba, and N. Sugiyama, Phys. Rev. D , 103001 (2012).[46] A.F. Zakharov, F. De Paolis, G. Ingrosso, and A.A. Nucita, New Astronomy Reviews (2012) 64-73.[47] L. Amarilla and E.F. Eiroa, Phys. Rev. D , 044057 (2013).[48] F. Atamurotov, A. Abdujabbarov, and B. Ahmedov, Astrophys. Space Sci. , 179 (2013).[49] F. Atamurotov, A. Abdujabbarov, and B. Ahmedov, Phys. Rev. D , 064004 (2013).[50] S.W. Wei and Y.X. Liu, J. Cosmol. Astropart. Phys. 11 (2013) 063.[51] A. Abdujabbarov, F. Atamurotov, Y. Kucukakca, B. Ahmedov, and U. Camci, Astrophys. Space Sci. , 429 (2013).[52] T. Johannsen, Astrophys. J. , 170 (2013).[53] A. Grenzebach, V. Perlick, and C. L¨ammerzahl, Phys. Rev. D , 124004 (2014).[54] Z. Li and C. Bambi, J. Cosmol. Astropart. Phys. 01 (2014) 041.[55] U. Papnoi, F. Atamurotov, S. G. Ghosh, and B. Ahmedov, Phys. Rev. D , 024073 (2014).[56] A.A. Abdujabbarov, L. Rezzolla, and B.J. Ahmedov, Mon. Not. Roy. Astron. Soc. , 2423 (2015).[57] C. Goddi et al. , Int. J. Mod. Phys. D , 1730001 (2016).[58] A. Abdujabbarov, M. Amir, B. Ahmedov, and S.G. Ghosh, Phys. Rev. D , 104004 (2016).[59] M. Amir and S.G. Ghosh, Phys. Rev. D , 024054 (2016).[60] Z. Younsi, A. Zhidenko, L. Rezzolla, R. Konoplya, and Y. Mizuno, Phys. Rev. D , 084025 (2016).[61] A. Grenzebach, The Shadow of Black Holes: An Analytic Description , Springer Briefs in Physics, pringer, Heidelberg (2016).[62] M. Amir, B.P. Singh, and S.G. Ghosh, Eur. Phys. J. C , 399 (2018).[63] R. Kumar, B.P. Singh, M.S. Ali, and S.G. Ghosh, arXiv:1712.09793 .[64] B.P. Singh and S.G. Ghosh, Ann. Phys. , 127 (2018).[65] P.G. Nedkova, V.K. Tinchev, and S.S. Yazadjiev, Phys. Rev. D , 124019 (2013).[66] C. Bambi, Phys. Rev. D , 107501 (2013).[67] M. Azreg-A¨ınou, J. Cosmol. Aastro. Phys. 1507 (2015) 037.[68] T. Ohgami and N. Sakai, Phys. Rev. D , 124020 (2015).[69] A. Abdujabbarov, B. Juraev, B. Ahmedov, and Z. Stuchlk, Astrophys. Space Sci. , 226 (2016).[70] R. Shaikh, Phys. Rev. D , 024044 (2018).[71] T. Ohgami and N. Sakai, Phys. Rev. D , 064071 (2016).[72] M. Amir, A. Banerjee and S.D. Maharaj, Annals Phys. , 198 (2019).[73] M. Amir, K. Jusufi, A. Banerjee and S. Hansraj, Class. Quantum Gravity , 215007 (2019).[74] K. Akiyama et al. [Event Horizon Telescope Collaboration], Astrophys. J. , L1 (2019).[75] K. Akiyama et al. [Event Horizon Telescope Collaboration], Astrophys. J. , L5 (2019).[76] K. Akiyama et al. [Event Horizon Telescope Collaboration], Astrophys. J. , L6 (2019).[77] I. Dymnikova, Gen. Relativ. Grav. , 235 (1992).[78] S. Fernando, Int. J. Mod. Phys. D , 1750071 (2017).[79] J.M. Bardeen, W.H. Press, and S.A. Teukolsky, Astrophys. J. , 347 (1972).[80] M. Carmeli and M. Kaye, Annals Phys. , 97 (1977).[81] R. Penrose and R.M. Floyd, Nature (London) , 177 (1971).[82] B. Carter, Phys. Rev. , 1559 (1968).[83] J.M. Bardeen, in Black holes, in Proceedings of the Les Houches Summer School, Session 215239 , editedby C. De Witt and B.S. De Witt (Gordon and Breach, New York, 1973)., editedby C. De Witt and B.S. De Witt (Gordon and Breach, New York, 1973).