Optical Features of Rotating Black Hole with Nonlinear Electrodynamics M. Zubair1Muhammad Ali Raza1yand Ghulam Abbas2z 1Department of Mathematics COMSATS University Islamabad Lahore Campus Lahore Pakistan

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Optical Features of Rotating Black Hole with Nonlinear Electrodynamics

M. Zubair,1, ∗Muhammad Ali Raza,1, †and Ghulam Abbas2, ‡

1Department of Mathematics, COMSATS University Islamabad, Lahore Campus, Lahore, Pakistan

2Department of Mathematics, The Islamia University of Bahawalpur, Pakistan

In this article, we considered the strong ﬁeld approximation of nonlinear electrodynamics black

hole and constructed its rotating counterpart by applying the modiﬁed Newman-Janis algorithm.

The corresponding metric function in the strong ﬁeld limit of the static black hole is identiﬁed in

order to study the radius of photon sphere. However, the metric function for the rotating counterpart

in the strong ﬁeld limit is considered in order to study the horizon radius w.r.t spin parameter. We

considered the Hamilton-Jacobi method to derive the geodesic equations for photon and constructed

an orthonormal tetrad for deriving the equations for celestial coordinates in the observer’s sky.

Shadows, distortions and energy emission rates are investigated and the results are compared for

diﬀerent values of nonlinear electrodynamics parameter, charge and spin. It is found that the

presence of the nonlinear electrodynamics parameter aﬀects the shape and size of the shadows and

thus the distortion in the case of rotation. It is also found that the nonlinearity of electrodynamics

diminishes the ﬂatness in the shadow due to the eﬀect of spin and other parameters.

Keywords: General Relativity, Black Hole, shadow, nonlinear electrodynamics, Newman-Janis

algorithm

I. INTRODUCTION

General Relativity (GR) predicts regions of ultra strong gravity known as black holes (BHs) having immense

curvature. The photons will deviate from its straight path while moving in the gravitational ﬁeld of the BHs and

other massive objects. Such concept was veriﬁed by Arthur Eddington and his teammates in 1919 [1] that ultimately

veriﬁed GR. It is called gravitational lensing which is a general term assigned to the study of eﬀects that are caused

by the deﬂection of photons and an outstanding progress has been made in the ﬁeld over the years [2–9].

The photons moving close enough to the BH will get trapped, otherwise they will scatter away. This trapping

region is called photon sphere in which photons orbit the BH in circular paths. The scattered photons present in the

unstable circular orbits will reach our eyes giving a glowing image but falling in photons will be lost, giving us a dark

2D image called shadow. The falling in trajectory of the photons in the outermost photon region and the scattering

trajectory in the innermost region will deﬁne the boundary of the shadow [10]. Usually the shape of the static BH

shadows is diﬀerent from the rotating ones due to the fact that the photons may move in any direction around a

BH in the photon region of certain width. Without loosing generality, photons in the both extreme orbits can be

considered moving in opposite directions. Thus, one side of the shadow appears ﬂattened as compared to the other.

Shadow of Kerr BH is one such example [10, 11].

The shadow study remains one of the foremost topics related to the BHs because whenever a BH solution is

discovered the ﬁrst arising question is that what would likely be its physical appearance? Some earlier studies related

to visual appearance of BHs are [12–17]. In these studies, the visual appearance of the BH was termed with various

names such as optical appearance, escape cone, cross section, cone of gravitational capture of radiation etc. Over the

years, some mathematical techniques [18–20] were developed to study the shadows analytically. The authors of [18]

and [19] investigated the shadows by considering the observer at a position (r0, ϑ0). Using the orthonormal tetrad

at this position, the authors have constructed the mathematical framework for celestial coordinates in the observer’s

sky. This method is widely applicable for both distant and nearer observers. Bardeen [20] studied the geodesics for

Kerr BH by considering two impact parameters. He developed his own method for studying the optical properties in

the vicinity of a BH. However, this method is applicable only for a distant observer. Despite its limitation, Bardeen’s

method is the most commonly used procedure for the shadow study, see [21–30]. Atamurotov et al. [25] investigated

the shadows of Kerr-Newman BH immersed in perfect ﬂuid dark matter by considering the Bardeen’s method. They

found that the shadow is ﬂattened by increasing the value of spin and the size of the shadow decreases with increase

in the values of charge and perfect ﬂuid dark matter.

∗Electronic address: drmzubair@cuilahore.edu.pk; mzubairkk@gmail.com

†Electronic address: maliraza01234@gmail.com

‡Electronic address: ghulamabbas@iub.edu.pk

arXiv:2210.13750v1 [gr-qc] 21 Oct 2022

By the discovery of more general BH solutions, it became diﬃcult to deal with the mathematical structure manually.

As a result, various software tools were used in order to suggest the possible appearance of such complicated BH

solutions. Some notable studies are [31–43]. Hioki and Maeda [32] calculated the shadows and the related observables

by the use of software, especially the contour plots therein. The BH solutions formulated in various theories of gravity

depend upon diﬀerent BH parameters. These parameters have diﬀerent sensitivity level. The shadows are highly

inﬂuenced by more sensitive BH parameters. As said above, spin parameter ﬂattens the shadow on one side for an

equatorial observer and the presence of cosmological constant, perfect ﬂuid dark matter, quintessence etc. have a

signiﬁcant impact on the size and shape of the shadows, for more details see [44–49]. Haroon et al. [50] studied the

eﬀects of perfect ﬂuid dark matter (α) and cosmological constant (Λ) on the shadows of rotating BH. They found that

for negatively increasing α, the shadow size increases while the shadow shrinks for positively increasing α. Moreover,

the distortion due to high spin value diminishes for α&0.8. The shadow size increases for anti-de Sitter case and

decreases for de Sitter case.

In Maxwell’s electrodynamics, there exist a singularity at the position of the point charge and has an inﬁnite

self-energy. To overcome this problem, Born and Infeld [51] developed a nonlinear electromagnetic ﬁeld. Motivated

by this, the coupling of GR and Born-Infeld ﬁeld has been studied in order to deal with the singularity problem of

the BHs along with some other properties, see [52–55]. Demianski [52] derived an asymptotically ﬂat solution which

becomes regular spacetime when the internal mass is considered zero. Cai et al. [54] studied the BH solution and

its thermodynamic properties in Born-Infeld theory. The coupling of GR and some other nonlinear electrodynamics

models have also been studied providing us with some useful results in BH physics [56–60]. Javed et al. [60] considered

a magnetized nonlinear electrodynamics BH and studied the eﬀect of nonlinear electrodynamics parameter βon the

deﬂection angle in the vicinity of BH. Kruglov [61] constructed a BH solution with nonlinear electromagnetic ﬁeld.

This BH solution is asymptotically Reissner-Nordstr¨om and the electric ﬁeld has ﬁnite value at the origin which does

not possess singularity at r= 0. Recently, Uniyal et al. [62] considered the same solution developed by Kruglov and

studied the photon sphere and shadow in the weak and strong ﬁeld limits.

In this paper, we consider the nonlinear electrodynamics BH solution as in Refs. [61, 62] and apply the modiﬁed

Newman-Janis algorithm [63, 64] to the eﬀective metric in the strong ﬁeld approximation of the BH. We work for

the horizons, shadows and related physical observables to examine the eﬀect of nonlinearity of electromagnetic ﬁeld.

The paper is presented as: In section 2, the nonlinear electrodynamics BH solution is presented. The approximated

metric in the strong ﬁeld limit is considered for further analysis and the radius of photon sphere is studied for the

non-rotating case. In section 3, rotating metric is constructed and the corresponding horizon radii are studied. In

section 4, the governing equations for shadows are developed using the Hamilton-Jacobi formalism and the method of

orthonormal tetrads. The shadows, distortions and energy emission rates are presented for the observer at diﬀerent

locations. We summarize the results in the last section. Note that the units G=c= 1 have been used.

II. 4D NONLINEAR ELECTRODYNAMICS BLACK HOLE

We start by the review of the non-rotating nonlinear electrodynamics BH. The gravitational action containing the

Lagrangian of nonlinear electromagnetic ﬁeld [61, 62] is given as

SG=Zd4x√−gR

2κ2+Lem,(1)

where g=det(gµν ), Ris Ricci scalar, κ−1is the reduced Planck mass and Lem is the nonlinear electromagnetic

Lagrangian [65] given as

Lem =−F

2βF+ 1,(2)

such that βis the corresponding parameter of nonlinear electromagnetic ﬁeld with the dimension of (length)4and

is related to the upper bound of the electric ﬁeld, F=Fµν Fµν

4and Fµν is the Maxwell tensor. The Lagrangian (2)

corresponds to classical linear electrodynamics when β= 0. The nonlinearity of electrodynamics in various forms,

coupled with gravity can be useful in achieving some of the unanswered questions. Furthermore, for the possible

quantum gravity corrections to Maxwell’s electrodynamics, the parameter βis introduced as in [65]. The function

(2) is formulated in such a way that the correspondence principal is not broken and the model works eﬃciently with

usual dielectric permittivity ε= 1 and magnetic permeability µ= 1. Hence, the action (1) in [61, 62] is the possible

coupling of the function (2) with GR. In this model, the energy momentum tensor has non-zero trace and at the

origin, the electric ﬁeld has a ﬁnite value without singularities, for more details, see [61, 65]. For a 4D spacetime in

spherical symmetry, we have

ds2=−f(r)dt2+dr2

f(r)+r2(dϑ2+ sin2ϑdϕ2),(3)

the solution turns out to be [61, 62]

f(r)=1−2M

r+Q2

r2−C2κ2

2r2+C2κ2

30r2(5ζ3−22ζ2+ 32ζ),(4)

where M,Qand Care mass, charge and integration constant, respectively. Here,

ζ=12√3√βr2−βCλ3/4

12βCλ1/4,(5)

whereas

λ=63

√6r2(3

√2β2/33

√Cγ2/3−83

√3βC)

β4/3C5/33

√γ,(6)

γ=√3p256βC2+ 27r4+ 9r2.(7)

Keeping in view the complexity of ζ, the simple forms of metric function f(r) in strong and weak ﬁeld limits are

obtained by applying series expansion for r→0 and r→ ∞ respectively, such that

f(r)s= 1 −2M

r+Q2

r2−C2κ2

2r2+16C3/2κ2

15β1/4r,(8)

f(r)w= 1 −2M

r+Q2

r2−βC4κ2

10r6.(9)

Since, we aim to study the horizon, photon sphere and shadow, so it is impossible to proceed with the metric

function (4) due to its complication and the required mathematical structure. Also, the horizons, photon spheres and

shadows are strong ﬁeld phenomena that exist only in the vicinity of the BHs. Thus, to avoid the complexity of the

metric function (4), we will consider the metric function (8) explicitly in further analysis, treating the metric as an

eﬀective metric. This eﬀective metric, when stretched out again in the entire universe, becomes asymptotically ﬂat as

r→ ∞. It will raise some extra sources and will not satisfy the ﬁeld equations exactly the same way as the metric (4)

because both of these metrics are not exactly equal. Due to this reason, their properties will also be slightly diﬀerent.

However, without caring for the diﬀerence in properties, we will proceed with the eﬀective metric in order to explore

the eﬀect of nonlinear electrodynamics in the strong ﬁeld limit which is the intent of this study.

As we move closer to the horizon, the eﬀect of nonlinear electrodynamics is highly impactful and can not be

removed. Apparently, it seems that the eﬀect of nonlinear electrodynamics can be removed in the strong ﬁeld region

by considering the limit β→ ∞ in f(r)s. However, this is physically not a realistic limit, since under this limit, the

Lagrangian (2) will be vanished. Hence, under a ﬁnite value of the parameter β, if we consider C= 0, we will get the

Reissner-Nordstr¨om metric which is again not a useful limit. Hence, we deduce that nonlinear electrodynamics eﬀect

can not be removed in the vicinity of the BH.

Note that the index soccurring as the superscript or subscript throughout the discussion corresponds to the

functions and variables in the strong ﬁeld limit. As mentioned above, the symbol Cis just a dimensionless constant

and has no physical meaning discussed in the Ref. [61, 62]. So, we will ﬁx its value in our work because we mainly aim

at the parameter β. However, the constant Cplays an important role in distinguishing the metric in strong ﬁeld limit

from Reissner-Nordstr¨om metric because inserting C= 0 in Eq. (8), we will get the standard Reissner-Nordstr¨om

metric. So, for a rigorous analysis, the non-zero value of Cmust be considered. To avoid the imaginary numbers, the

negative values of Ccan not be considered. Hence, we will consider C= 1 in our calculations in the same way as we

consider M= 1 quite often for simplicity.

In the function f(r)s, we can see that there exist two extra terms of O(1

r) and O(1

r2) and apparently the metric (8)

seems identical to the Reissner-Nordstr¨om metric by shifting the constants and parameters in f(r)s. However, the

terms containing β,Cand κmake this metric distinct from the Reissner-Nordstr¨om metric because of their physical

nature lying therein. Since, the Lagrangian (2) consists a function of Maxwell’s tensor but with the presence of β,

deﬁning the nonlinear electromagnetic theory. Then the resulting function f(r)sdepends upon M,Q,β,Cand

κ. These parameters and constants are responsible for deﬁning the strong ﬁeld metric in nonlinear electromagnetic

theory. However, the Maxwell’s Lagrangian results in Reissner-Nordstr¨om BH whose metric function depends upon M

and Qonly and is linearly electrically charged. Whereas, the metric under consideration consists of charge with strong

nonlinear eﬀects. As we know that the trace of energy-momentum tensor is zero for the Maxwell’s Lagrangian that

gives Reissner-Nordstr¨om BH. However, for the nonlinear electrodynamics case, the trace of energy-momentum tensor

is non-zero for the Lagrangian (2). This non-zero value of the trace contains βand further contains Cin the function

h0(r) as given in Refs. [61, 62]. By considering β= 0, the trace becomes zero. Also, by taking the value C= 0 in the

trace and in the metrics (4) and (8), the trace becomes zero and the metrics reduce to Reissner-Nordstr¨om metric.

Hence, the non-zero ﬁnite values of βand Cdeﬁne our metric that is distinguishable from the Reissner-Nordstr¨om

metric i.e. by the comparison of the trace, the strong ﬁeld metric is distinguishable from the Reissner-Nordstr¨om

metric.

Suppose that after the shifting, we get the parameters m=M−8C3/2κ2

15β1/4as eﬀective mass and q=qQ2−C2κ2

as eﬀective charge in f(r)s. Then the resulting metric would appear as Reissner-Nordstr¨om metric with parameters

mand qas said above. This would be signiﬁcant only if we use the parametric values of mand qwith a physical

description. However, there is no physical description of mand qapart from calling them as eﬀective mass and

eﬀective charge, respectively. Moreover, we can not use the values of mand qindependently in the calculations

because if we put m= 0.5 and q= 0.6, these are actually the combination of parametric values of M,Q,β,Cand

κ. So, if we have to use the values of M,Q,β,Cand κat the end, then considering mand qis of no use. Else,

if we use the values of mand q, then the required explanation of the results would be related to M,Q,β,Cand κ

which is inconsistent with what is used in the calculations. Furthermore, using mand qwould suppress the actual

objective of this study because we aim to focus on the eﬀect of nonlinear electrodynamics parameter on the photon

sphere, horizon and shadow. As said earlier, the nonlinear electrodynamics eﬀects can not be removed or neglected

in the strong ﬁeld region and the nonlinear electrodynamics is not the characteristics of Reissner-Nordstr¨om metric.

Hence, the metric (8) is diﬀerent from the Reissner-Nordstr¨om metric for all of the above mentioned reasons. Later,

this fact will also be demonstrated through the results of photon sphere, horizon and shadow.

To discuss the radius of photon sphere in the strong ﬁeld limit, we assume an observer near the BH. Then the

general condition for the radius of photon sphere is [10]

dr r2

f(r)sr=rs

= 0,(10)

which gives

p=3

2M−8C3/2κ2

15β1/4+1

2s9M−8C3/2κ2

15β1/42

−8Q2+ 4C2κ2.(11)

Figure 1presents the plots for the radius of photon sphere vs βand Qand shows a comparison of results with

Reissner-Nordstr¨om BH. Note that, for all plots, we have used M= 1, C= 1 and κ= 1 for simplicity. We have kept

the same values and variations for βfor a rigorous comparison. The negative values of βhave not been considered

because there exist no photon sphere for β < 0 in the strong ﬁeld limit. So, βis kept positive for all calculations.

Also, βbeing less sensitive as compared to other parameters, is not varied with small step size in the intervals such

as (0,1). Instead, we vary it in the interval (0,100] keeping in view the sensitivity level. Moreover, it is still possible

to do calculations for β∈(0,1). For example, there will be ﬁnite values of radius of photon sphere for β∈(0,1).

However, the results will not be convergent to rs

p= 3 (Schwarzschild BH) for small variation in the values of βand for

Q= 0. For a comprehensive analysis, it is important that the values of radius of photon sphere should approach the

value of Schwarzschild BH. So, as β100 and Q= 0, the value of radius of photon sphere approaches 3. Whereas,

for a small step size of β, the radius of photon sphere will be approximately around 1.5, which is not a signiﬁcant

value to compare the results.

In the left plot, it can be seen that when β→0, the radius rs

pdrops asymptotically for diﬀerent values of charge

Qfor nonlinear electrodynamics case. However, for Q= 1, minimum value of rs

p∼1. Also, the radius increases

rapidly till β∼10 and for other values of β, radius increases slowly and approaches the corresponding values of

Reissner-Nordstr¨om BH shown with dashed curves. Moreover, the outermost curve corresponds to Q= 0 and by

increasing Q, the curves are shifted inwards. The dashed curves show a constant value of radii for each ﬁxed value

of Qbecause the Reissner-Nordstr¨om BH has no dependence upon β. In the right plot, it is found that the radius

decreases as charge increases for each curve. Also the curves are shifted outwards for increasing β. As we know

that certain parameters weaken the gravity and hence the photon regions become smaller by increasing the values of

those parameters. Here, such behavior is also shown by the charge parameter Qas if it is responsible for weakening

the gravity. The outermost orange curve corresponds to the Reissner-Nordstr¨om BH and all of the other curves

approach it under the limit β→ ∞. However, under this limit, these curves will not exactly match the curve for

Reissner-Nordstr¨om BH unless C= 0.

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OpticalFeaturesofRotatingBlackHolewithNonlinearElectrodynamicsM.Zubair,1,MuhammadAliRaza,1,yandGhulamAbbas2,z1DepartmentofMathematics,COMSATSUniversityIslamabad,LahoreCampus,Lahore,Pakistan2DepartmentofMathematics,TheIslamiaUniversityofBahawalpur,PakistanInthisarticle,weconsideredthestrongeldappro...

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Optical Features of Rotating Black Hole with Nonlinear Electrodynamics M. Zubair1Muhammad Ali Raza1yand Ghulam Abbas2z 1Department of Mathematics COMSATS University Islamabad Lahore Campus Lahore Pakistan

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