Deflection angle and shadow of the Reissner-Nordström black hole with higher-order magnetic correction in Einstein-nonlinear-Maxwell fields_2

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Citation: Kumaran, Y.; Övgün, A.

Deﬂection angle and shadow of the

Reissner-Nordström BH with

higher-order magnetic correction in ENM

ﬁelds. Preprints 2022,1, 0.

https://doi.org/

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Attri- bution (CC BY) license (https://

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4.0/).

Article

Deﬂection angle and shadow of the Reissner-Nordström black hole

with higher-order magnetic correction in Einstein-nonlinear-Maxwell

ﬁelds

Yashmitha Kumaran 1,†, and Ali Övgün 1,†

1Physics Department, Eastern Mediterranean University, North Cyprus via Mersin 10, Famagusta 99628, Turkey ;

yashmitha.kumaran@emu.edu.tr (Y.K.); ali.ovgun@emu.edu.tr (A.O.)

*Correspondence: ali.ovgun@emu.edu.tr

†These authors contributed equally to this work.

Abstract: Nonlinear electrodynamics is known as the generalizations of Maxwell electrodynamics at strong ﬁelds

and presents interesting features such as curing the classical divergences present in the linear theory when coupled

to general relativity. In this paper, we consider the asymptotically ﬂat Reissner-Nordström black hole solution with

higher-order magnetic correction in Einstein-nonlinear-Maxwell ﬁelds. We study the effect of the magnetic charge

parameters on the black hole, viz. weak deﬂection angle of photons and massive particles using Gauss-bonnet

theorem. Moreover, we apply the Keeton-Petters formalism to conﬁrm our results of the weak deﬂection angle.

Apart from vacuum, their inﬂuence in the presence of different media such as plasma and dark matter are probed

as well. Finally, we examine the black hole shadow cast using the null-geodesics method and investigate its

spherically in-falling thin accretion disk. Our inferences show how the magnetic charge parameter

affects the

other physical quantities; so, we impose some constraints on this parameter using the observations from the Event

Horizon Telescope.

Keywords:

Relativity; Gravitation Lensing; Black hole; Nonlinear electrodynamics; Gauss-Bonnet Theorem;

Deﬂection angle; Plasma medium; Shadow

PACS: 95.30.Sf, 98.62.Sb, 97.60.Lf

1. Introduction

Gravity is the weakest force in the universe that we live in today. While it can pull everything

reachable inwards, its effect decreases with (the square of) distance until where it is no longer signiﬁcant,

according to the Newtonian physics. However, when the mass of the gravitating object increases, gravity

starts to behave differently. With a mass high enough, collapsing on itself and concentrated at a single

point, it can not only overcome the other three natural forces, but also crush the familiar laws of physics

that are known to govern our universe. This object of extreme mass, inﬁnite density, no volume and

dominating gravity – so strong that nothing that goes in comes out – is a black hole [

]. Black holes have

been particularly of interest since their discovery by the Event Horizon Telescope [2,3].

A black hole is surrounded by an accretion disk in which matter, dust and photons are stuck in

unstable orbits around it [

]. The not-so-circular photon sphere gives rise to the phenomenon of

gravitational lensing. When a massive cluster falls in the light trajectory aimed at an observer, the

gravitational ﬁelds of the clusters act as a lens by deﬂecting the rays of light, causing distortions of the

light source in its background [

]. This fascinating phenomenon was particularly prominent in the ﬁrst

images from the James Webb Space Telescope, especially of the galaxy cluster called SMACS 0723,

which was reportedly due to an astronomical quantity of matter in view on the speck of sky almost as big

as a sand grain at arm-length [7].

Gravitational lensing can be classiﬁed into strong lensing and weak lensing; this paper is built

on the latter. It majorly depends on the mass distribution of the lensing cluster. Weak lensing is a

arXiv:2210.00468v1 [gr-qc] 2 Oct 2022

2 of 26

consequence of general relativity arising from minor distortions that are too small to be detected in terms

of magniﬁcation, yet sufﬁcient enough to distinguish between various mass distributions [

–

]. It is

known in astrophysics that distances have a dominant role in obtaining the properties of astrophysical

objects. However, Virbhadra proved that just observation of relativistic images can also say an incredibly

accurate value for the upper bound to the compactness of massive dark objects [

] and then Virbhadra

showed that there exists a distortion parameter such that the signed sum of all images of singular

gravitational lensing of a source identically vanishes by testing this with images of Schwarzschild case

(SC) lensing in weak and strong gravitational ﬁelds [19].

Weak lensing utilizes the ﬁne property of differential deﬂection exhibited by the bending of light to

explore the structures of the cosmic deeper. In order to achieve this, the angle of deﬂection is calculated

using the optical geometry derived from the Gauss-Bonnet theorem given by [20]

Z ZDKdS+Z∂Dκdt+∑

αi=2πχ(D),(1)

where,

is the Euler characteristic of the topology,

is a Riemannian metric of the manifold of the

symmetric lens,

(D,χ,g)

represent the domain of the surface,

is the Gaussian curvature,

is the

geodesic curvature, and

αi

is the exterior angle at the

ith

vertex. In the literature, there are various studies

of this method on black holes, wormholes and other spacetimes [21–58].

The scope of this paper extends to evaluate the weak deﬂection angle through two different ap-

proaches: the Gibbons and Werner (GW) method [

] and the Keeton-Petters formalism [

]. Moreover,

we study the shadow cast of the black hole with thin-accretion disk.

The accretion disk, along with the lensing effect, creates the appearance of a shadow of the black

hole. This is due to the emission region that is geometrically thick but optically thin and is accompanied

by a distant, homogeneous, isotropic emission ring [60–62].

The shadow is essentially illustrated as the critical curve interior which separates the capture orbits

that spiral into the black hole from the scattering orbits that swerve away from the black hole, i.e. entering

versus exiting photon orbits. Although the size of the shadow is primarily dependant on the intrinsic

parameters of the black hole and its contour is determined by the orbital instability of the light rays from

the photon sphere, it merely appears to be a dark, two-dimensional disk for a distant observer illuminated

by its bright, uniform surrounding [30,31,45–48,63–118].

A factor of interest that affects the radius of the shadow is the effect of magnetic charge especially

since it cannot be neutralized with regular matter unlike electric charge in a conductive medium [

119

The presence of magnetic charge tends to increase the curvature of the spacetime, resulting in more

photons being pulled into the black hole and hence, decreasing the radius of the shadow [

]. Along

with black hole spin, magnetic charge is found to create two distinct horizons, namely the inner and the

outer horizons, deﬁned by stable and unstable photon orbits [

]. For a negligible charge, they are one

and indistinguishable. But as the value of the charge increases towards a critical value, Sun et al. have

obtained that these horizons become more prominent as the inner horizon appears from the center, with

the outer horizon existing sensitive to the charge. Other studies have shown the inﬂuence of magnetic

charge on the shadow for different cases of black holes [62–64,77,120].

Here, we will proceed to discuss how magnetic charges affect the shadow of a black hole starting

with the black hole solution from a new model of nonlinear electrodynamics proposed by [

121

] coupled

in Einstein’s gravity. The paper is organized as follows: the black hole solution is introduced in section 2.

This is followed by calculating the weak deﬂection angle using the Gauss-Bonnet theorem in section 3

along with determining the deﬂection angle and the observables using Keeton-Petters formalism. Then,

the weak deﬂection angle for massive particles is computed in section 4with the help of the Jacobi

metric. Furthermore, the weak deﬂection angle is calculated in the presence of plasma and dark matter in

section 5. With inferences and comments about shadows in section 6, we ﬁnish with concluding remarks

about our results in section 7.

3 of 26

2. Brief Review of Reissner-Nordström black hole with higher-order magnetic correction in

Einstein-nonlinear-Maxwell ﬁelds

From the beginning of the universe to the black holes, singularity has been a matter of question,

hindering general relativity from being uniﬁed to the other models of the universe. Researchers have been

working on ﬁnding the solution for a black hole without singularities pioneered by Bardeen [122,123].

Building on this attempt to eliminate singularity, the black hole solution presented by [

121

] brings

nonlinear electrodynamics into play. They have provided an analytical black hole solution much like the

Born-Infeld-type corrections to the linear Maxwell’s theory in the weak ﬁeld limit [

124

]. The governing

equations for a pure magnetic ﬁeld are

Vector potential: Aφ=−pcos θ,

Spacetime tensor ﬁeld: Fµν =∂αAβ−∂βAα,

Electromagnetic invariant: F ≡ 1

4Fµν Fµν,

Lagrangian density of the new model: L=−1

βln hcos2p−βFi ,F>−π2

4β,

Action coupling Lminimally to Einstein’s gravity: I=Zd4xp−gR

16πG+L,

Energy-momentum tensor: Tv

µ=1

4πLδv

µ−LFFµλ Fvλ,

(2)

where,

is the magnetic charge construed as a magnetic monopole,

is a dimensional constant

with a dimension of

[Length]4

is the metric,

is the Ricci scalar,

is the Newton’s gravitational

constant in four-dimensional spacetime and LF=∂L

∂F.

The line element of a spherically symmetric spacetime is written as

ds2=−f(r)dt2+1

f(r)dr2+r2dθ2+sin2θdφ2.(3)

Given a radial magnetic ﬁeld, Br=p/r2, using the set of equations from (2)

F=p2

2r4,(4)

and new nonlinear electrodynamics Lagrangian is proposed in [

121

] by Mazharimousavi and Halilsoy -

satisfying the Maxwell-nonlinear equations F[αβ,γ]=0and ∂µ(√−gLFFµν)=0- to be

L=−1

βln"cosh2 rβ

r2!#.(5)

Then the corresponding energy-momentum tensor for the nonlinear electrodynamics is calculated

as follows

t=−1

4πβ ln cosh2 rβ

r2!!,(6)

on the other hand, using the spherically symmetric spacetime in Eq.3, the Einstein’s tensor

Gν

is obtained

as follows

Gν

µ=diagr f 0+f−1

r2,r f 0+f−1

r2,r f 00 +2f0

2r,r f 00 +2f0

2r.(7)

4 of 26

Then using the relation between the Einstein’s tensor and energy momentum tensor

Gν

µ=κ2Tν

µ,(8)

where κ2=8πG,tt components of the Einstein’s ﬁeld equations become

−1+r f 0(r) + f(r)

r2=−1

4πβ ln cosh2 rβ

r2!! (9)

which gives the metric function f(r)

f(r) = 1−2GM

r−2G

βrZr

x2ln"cosh2 rβ

x2!#dx,(10)

that reduces in the weak ﬁeld limit to the non-asymptotic behavior of magnetic Reissner-Nordström

black hole [121]

f(r) = 1−2GM

r+Gp2

r2−βGp4

60r6+β2Gp6

810r10 −Oβ3.(11)

This magnetic black hole solution will be the center of analyses henceforth. Fig. 1illustrates the effect of

the magnetic charge on incoming light rays. Note that the Reissner-Nordström metric belongs to the class

of Stackel spaces. Geodesic equations can be exactly reintegrated (or reliably solved approximately)

only in such spaces, because the Hamilton-Jacobi equation (for light rays - eikonal equation) admits

in these spaces a complete separation of the variables and Obukhov present the method of complete

separation of variables can be found in [132].

Figure 1. Illustration of the function f(r)in the Cartesian coordinates. The ﬁrst ﬁgure shows the top view and the

second ﬁgure shows an angular view. The white circles represent the Schwarzschild case (SC); the yellow, red and

purple circles correspond to different values of magnetic charge, p=1.4,p=1.45 and p=1.5 respectively.

The event horizon radius

of a black hole is the larger root of the above equation - where the

outer horizon is located - with

f(r) = 0

. As shown in Fig.

(2)

, the number of horizons are dependant on

the parameters of f.

The 4-velocity is determined by

u=ut∂t,(12)

accompanied by the satisfying normalization condition

1=uµuµ,(13)

5 of 26

β=2β=3 SC

01234567

-6

-4

-2

f(r)

Figure 2. The lapse function f(r)as a function of rfor M=2,p=1,G=1and for the different values of β.

where ut=1/√gtt. Correspondingly, the particle acceleration aµ

pis given by [125]

aµ=−gµv∂vln ut,(14)

as the metric components exist as functions of rand θ. The surface gravity (κsg)is deﬁned as

κsg =lim

r→rhpaµaµ

ut,(15)

which helps in ﬁnding the black hole temperature

κsg =1

2∂rf2r+

,T=κsg

2π,(16)

as shown in Fig.3for different values of β. The horizon area is plainly calculated as

ABH =Z2π

0dϕZπ

0p−gdθ=4πr+,(17)

giving the black hole entropy to be

SBH =ABH

4=πr+.(18)

As for the thermodynamic properties of a black hole, the mass of a black hole described by

g00|r=r+=0is

M(r+) = r+

2G+β2p6

1620r9

+−βp4

120r5

+p2

2r+.(19)

At the limit of the Gauss-bonnet coupling and the magnetic parameters vanishing

β=p=0

, the

black hole mass and temperature reduce to the Schwarzschild case,

MSC =r+/2G

and

TSC =1/8πM

respectively.

3. Weak Deﬂection Angle using Gauss-Bonnet Theorem

For the equatorial plane

θ=π/2

, applying null geodesics to the line element in eq. (3) yields the

optical metric

dt2=dr2

f(r)2+r2

f(r)dφ2,(20)

where the determinant of the optical metric

g=r2/f(r)3

. In order to calculate the deﬂection angle while

accounting for the optical geometry using a domain outside the light rays’ trajectory, the Gauss-Bonnet

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Citation:Kumaran,Y.;Övgün,A.DeectionangleandshadowoftheReissner-NordströmBHwithhigher-ordermagneticcorrectioninENMelds.Preprints2022,1,0.https://doi.org/Publisher'sNote:MDPIstaysneutralwithregardtojurisdictionalclaimsinpublishedmapsandinstitutionalafl-iations.Copyright:©2022bytheauthors.Submitted...

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Deflection angle and shadow of the Reissner-Nordström black hole with higher-order magnetic correction in Einstein-nonlinear-Maxwell fields_2

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