Optical properties of Kerr-Newman spacetime in the presence of plasma
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Noname manuscript No.
(will be inserted by the editor)
Optical properties of Kerr-Newman spacetime in the
presence of plasma
Gulmina Zaman Babara,1 , Abdullah Zaman Babarb,2 , Farruh
Atamurotovc,3,4 ,
1 Schoolof Natural Sciences, National University of Sciences and Technology, Sector H-12, Islamabad, Pakistan
2 Department of Electrical Engineering, Air University, Islamabad, Pakistan
3 Inha University in Tashkent, Ziyolilar 9, Tashkent 100170, Uzbekistan
4 Ulugh Beg Astronomical Institute, Astronomy St. 33, Tashkent 100052, Uzbekistan
arXiv:2008.05845v2 [gr-qc] 27 Jul 2021
the date of receipt and acceptance should be inserted later
Abstract We have studied the null geodesics in the hole and the radial coordinate where the observer is
background of the Kerr-Newman black hole veiled by located. Unlike a static black hole, the shadow of a ro-
a plasma medium using the Hamilton-Jacobi method. tating black hole is not a circular disk. The first, fore-
The influence of black hole’s charge and plasma param- most accurate calculations of the shadow were done by
eters on the effective potential and the generic photon Bardeen considering the Kerr space time [4]. So far, the
orbits has been investigated. Furthermore, our discus- latter feature of the black hole has been widely investi-
sion embodies the effects of black hole’s charge, plasma gated for various gravities adopting a similar approach
and the inclination angle on the shadow cast by the using classical method [5–24].
gravity with and without the spin parameter. We ex-
The influence of plasma medium on the events tak-
amined the energy released from the black hole as a
ing place in the black hole vicinity contributes an ad-
result of the thermal radiations, which exclusively de-
ditional insight into its physical properties. The rela-
pends on the size of the shadow. The angle of deflection
tivistic effects of plasma tracing light rays in the sur-
of the massless particles is also explored considering a
roundings of compact objects are thoroughly studied
weak-field approximation. We present our results in jux-
in [25]. A detailed discussion about gravitational lensing
taposition to the analogous black holes in General Rel-
in the presence of a non-uniform plasma is carried out
ativity, particularly the Schwarzschild and Kerr black
by Bisnovatyi-Kogan and Tsupko in [26]. Later on, they
hole.
extended their research for the Schwarzschild space-
time [27–29]. One may get specific details from [30–36]
in reference to the above mentioned analysis.
1 Introduction
Nowadays, shadow of the black hole in the pres-
The existence of super massive black holes has been in- ence of plasma has become the field of interest for re-
vestigated extensively for nearly two decades, through searchers. Recently, a profound examination has been
various esoteric astrophysical phenomena. Recently, The established to study the shadow of the Schwarzschild
Event Horizon Telescope(EHT) project has been ob- and Kerr space-time coupled with a plasma medium
served first direct image of M87* black hole [1, 2] us- in the following papers [37, 38] using the Synge for-
ing very long baseline interferometer(VLBI). The phys- mulism [39] and the performance of the plasma medium
ical structure of black holes is well apprehended by the work was studied using a different approach in [40]. We
shadow imaged by it, which is created when the black shall put forth the Synge formulism analysis in analogy
hole confronts a luminous source. Synge [3] studied the to the aforementioned papers to retrace the influence
shadow of the Schwarzschild black hole, which was then of plasma on the Kerr-Newman space-time. It is a sta-
termed as the “escape cones” of light. The radius of the tionary and an axisymmetric solution to the Einstein-
shadow was calculated in terms of mass of the black Maxwell equations depending on the mass, angular mo-
a e-mail: gulminazamanbabar@yahoo.com mentum and electrical charge of the black hole. The
b e-mail: abdullahzamanbabar@yahoo.com surface geometry of the Kerr-Newman metric and its
c e-mail: atamurotov@yahoo.com, fatamurotov@gmail.com physical properties are well described in [41]. After this2
work was published, several works were performed in a 2.0
charged black hole [42–44].
The rest of our paper is organized as follows. In
Sec. (2), we consider the equations of motion of photons 1.8
around an axially symmetric black hole in the presence
of a plasma. In Sec. (3) the effective potential along
with the generic photon orbits are studied. Formalism 1.6
for the shadow cast by the space-time under consider-
r /M
ation is set-up in Sec. (4). The subsections of Sec. (4)
incorporate the analysis of the shadow and energy emis- 1.4
sion by taking into account a non rotating charged black
hole. Sec. (5) includes an elaborate analysis of the de-
flection angle caused by the deviation of photons in a 1.2
weak-field approximation. Finally, in Sec. (6) we sum-
marize our main results.
1.0
0.0 0.2 0.4 0.6 0.8 1.0
2 Photon Motion Around the Charged Black a
hole in the Presence of a Plasma
Fig. 1 The spin parameter a dependence of the radial coordi-
nate r for the different values of electric charge Q. The values
In Boyer-Lindquist coordinates the charged rotating Kerr- assigned to Q (top to bottom) are 0,0.2,0.4,0.6 and 0.8.
Newman spacetime, an exact solution of the Einstein-
Maxwell field equations, is characterized by the line el-
ement [42–44], vacuum case when n = 1. The Hamilton-Jacobi equa-
tion for a black hole surrounded by a plasma is
∆ ρ2
ds2 = − 2 (dt − a sin2 θdφ)2 + dr2 + ρ2 dθ2
ρ ∆
2 µ 1 µν 2 ν 2
sin θ H(x , p µ ) = g p p
µ ν + (n − 1)(p ν u ) . (4)
+ 2 [adt − (r2 + a2 )dφ]2 , (1) 2
ρ
where ∆ = r2 − 2M r + a2 + Q2 and ρ2 = r2 + Now we use the Hamilton-Jacobi equation which defines
2 2
a cos θ. The parameters M , Q and a corresponds to the equation of motion of the photons for a given space-
the total mass, electric charge and specific angular mo- time geometry
mentum of the black hole, respectively. The Kerr metric
2
is recovered when the charge Q is annulled. Roots of the
p
1 µν
function ∆, H(xµ , pµ ) = g pµ pν − (n2 − 1) p0 −g 00 . (5)
2
p
r± = M ± M − a2 − Q2 , (2) The equations x˙µ = ∂H/∂pµ and p˙µ = ∂H/∂xµ de-
determine the radii of the inner and outer horizons fine the trajectories of the photon in the plasma medium,
of the black hole. The inner horizon r− and the outer given as below
horizon r+ are generally termed as the Cauchy horizon
and the event horizon, respectively. The event horizon (a2 + r2 ) 2
ρ2 ṫ = n E(a2 + r2 ) − aL
acts as a threshold from where no turning back is possi-
∆
ble. The charge Q has an evident influence on the hori-
L
zon of the black hole which moves to a farther position +a sin2 θ − an 2
E , (6)
sin2 θ
at Q = 0, shown in Fig. (1).
We consider a static inhomogeneous plasma in the a L
ρ2 φ̇ = (E(a2 + r2 ) − aL) + − aE , (7)
gravitational field with a refractive index n, the ex- ∆ sin2 θ
√
pression of which was formulated in general terms by ρ2 ṙ = R, (8)
√
Synge [39], ρ2 θ̇ = Θ. (9)
pµ pµ The overdot denotes differentiation with respect to
n2 = 1 + , (3) the particle’s proper time τ . The functions R(r) and
(pν uν )2
Θ(θ) admit the following expressions,
pµ and uν refers to the four-momentum and four- 2
velocity of the massless particle. One may obtain the R = −∆(K − 2aLE) + nE(a2 + r2 ) − aL3
+2aLE(n − 1)(a2 + r2 ), (10) charge Q of the black hole gives an additional strength
1 to it, as illustrated in Fig.(2). For the Kerr Black hole at
L2 + a2 n2 E sin4 θ .
Θ =K− (11)
sin2 θ Q=0, the effective potential attains its maximum value
Here, K is the constant of separation. E and L are with the highest potential barrier. It is observed that
the conserved quantities acting along the axis of sym- the minimum values of Q yields an extra shield to the
metry termed, respectively, as the energy and angular photons against the black hole gravity, hence, enhanc-
momentum of the photon. ing their stability. The plasma density increases with
an increase in the refractive index, consequently, the
photon’s strength depletes to carry on its motion in the
3 Effective Potential and Photon Sphere black hole vicinity. It is also incurred that the barrier
is sufficiently reduced when n = 1 (vacuum case) [38].
For a systematic rational reasoning, it is necessary to We follow the formulism of [37] to find the pho-
introduce a specific form of the refractive index n [26, ton orbits, by considering a pressureless inhomogeneous
27], defined as follows plasma in the equatorial plane, θ = π/2. Using (5) we
ωe 2 obtain the flight path of the photons in a unique way,
n2 = 1 − . (12)
ω2 2
−g rr (g tt E 2 + g φφ L2 − 2g tφ LE + we 2 )
Here, ωe is the plasma electron frequency and ω is the dr
= . (17)
photon frequency perceived by a distant observer. The dφ (g φφ L − g tφ E)2
photon frequency depends on the spatial coordinates For a circular photon orbit rph we have dr/dφ=0,
xµ due to the gravitational field. The light propaga- which further yields a distinguished special parameter
tion through the plasma medium is possible, provided L/E in the form of a function
that ω 2 > ωe 2 . The plasma frequency has the following
analytic expression q
we 2
L g tφ + (g tφ )2 − g φφ (g tt + E2 )
ξ(r) = = , (18)
4πe2 N (r) E g φφ
ωe = , (13)
m here, γ = wEe is the dimensionless plasma constant.
where e, m and N (r) are the charge, mass and num- Note that γ=0 corresponds to the vacuum case. The
ber density of the electron, respectively. By the impli- working out of dξ(r)/dr = 0 gives a general expression,
cation of radial law density [31] the roots of which provide the radius of the photon
orbits,
2
0 = (∆ − a2 ) r2 2Q2 + r(−3M + r) − γ 2 ∆ − a2
N0 −
N (r) = , (14)
rh
p
a(Q2 − M r 2r ∆(r2 − γ 2 (∆ − a2 )) +
where h ≥ 0, the plasma frequency becomes a((∆ − a2 )γ 2 − 2r2 ) .
(19)
We have solved numerically (19) to examine the influ-
2 k ence of plasma and charge parameter on the spherical
ωe = h. (15)
r photon orbits. The left panel of Fig.(3) exhibits the de-
For simplicity, we shall take h = 1 [31, 38]. The pendence of the radius on the plasma parameter for
radial potential can be directly evaluated from (8) to different values of Q. The rph increases in the presence
study the generic photon behaviour in the presence of of γ, hence, the orbit shifts at a far distance due to rise
plasma, in the plasma factor. The right panel of Fig.(3) shows
the effect of charge parameters on the radius for dif-
ferent values of γ. Thus, it is inferred that the orbits
a2 L2 − aE(2L − an2 E) + ∆(2aLE − K)
Veff =− come closer to the black hole due to the electric field
r4 intensity. It is noticed that the radius of the orbits is
2 2 2 2
2aE(L − an E) − n E r generally larger at Q=0. It is worth mentioning that
+ . (16)
r2 the silhouette of the black hole, elaborated in the sub-
The circular orbits exist at ṙ = 0 and the massless sequent section is observed only when rph > r+ [5], i.e,
particle attains its stability at a fixed stationary posi- the radius of the spherical photon orbit must be greater
tion r under the constraint ∂r Veff . The particle may fol- than that of the event horizon. An infinitesimal gravi-
low a marginally stable circular motion between the rel- tational perturbation would drive the massless particles
ative extrema satisfying the condition ∂r2 Veff [45]. The into the black hole or toward spatial infinity.4
a/M=k /M=0.5 a/M=Q/M=0.5
3 3
2 2
1 1
Veff
Veff
0 0
-1 -1
-2 -2
-3 -3
1.0 1.5 2.0 2.5 3.0 1.0 1.5 2.0 2.5 3.0
r /M r /M
Fig. 2 Radial potential for photons with fixed parameters E = 0.9 and L = 4. In the left panel the values ascribed to Q (top
to bottom) are 0,0.2.0.4,0.6 and 0.8 and in the right panel n (top to bottom) has the values 0.2,0.4,0.6,0.8 and 1.
a Q/M -./1 γ
*+,
3
'()
$%&
rph /M
rph /M
!"#
2
1.5
0.0 0 0.6 0.8 0.0 0.1 0.5 0.6
γ Q/M
Fig. 3 The photon orbits by varying γ and Q. In the left and right panel from top to bottom the values of Q and γ are
0,0.1,0.2,0.3,0.4,0.5 and 0.6.
4 Shadow of Kerr-Newman Black hole in the impact parameters and are further used to explore
Presence of Plasma the contour of the black hole shadow.
p
We first present an ansatz for the analysis of the shadow ξ = A+ A2 − B, (20)
cast by the Kerr-Newman gravity in the presence of a 2 2
2aM ξ − (a + r ) 2rn + nn (a + r )
2 ′
2 2
plasma. Let us consider the black hole between a bright η= . (21)
M −r
source of light and an observer situated at fixed a Boyer-
Lindquist coordinate (r0 , θ0 ), where θ0 is the inclination where,
M (a2 − r2 ) + rQ2
angle between the rotation axis of the black hole and
A= ,
the line of sight of the observer and withal r0 → ∞. a(M − r)
The light waves emitted by the source reach the ob- ′
(a2 + r2 ) M (a2 − r2 ) + rQ2 + r∆ n2 + ∆nn (a2 + r2 )
server after gravitational deflection whereas, the pho- B= .
a2 (M − r)
tons with comparably less impact factor gets absorbed
(22)
by the black hole. As a result, a dark patch in the space
is created which is called the shadow. The boundary of The contour of the shadow in terms of celestial co-
this shadow provides us details regarding the intrinsic ordinates, (α, β) is given as below,
configuration of the black hole. Utilizing the conserved
quantities along with the constant K, one can conve- dφ
L
α = lim − sin θ0 r02
, (23)
niently introduce the impact parameters ξ = E and r0 →∞ dr
K
η = E 2 . With reference to (10) the orbits must satisfy
dθ
the conditions, R(r) = ∂r R(r) = 0 which are fulfilled by β = lim r02 . (24)
r0 →∞ dr5
π π π π
Q/M=0.5A θ0= Q/M=0.6P θ0= Q/M=0.8_ θ0= Q/M=opqr θ0=
B Q ` s
6 6 6 6
@ O ^ n
? N ] m
β /M
β /M
β /M
β /M
0 0 0 0
=> LM [\ kl
;< JK YZ ij
: 6 I 6 X 6 h 6
45 67 0 8 9 6 8 CD EF 0 G H 6 8 RS TU 0 V W 6 8 bc de 0 f g 6 8
α /M α /M α /M α /M
π π π π
Q/M=0.5 θ0= Q/M=0.6 θ0= Q/M=0.8 θ0= Q/M=®¯°± θ0=
²
6 6 6 6
¬
β /M
β /M
β /M
β /M
0 0 0 0
}~ ª«
{| ¨©
z 6 6 6 § 6
tu vw 0 x y 6 8 0 6 8 0 6 8 ¡¢ £¤ 0 ¥ ¦ 6 8
α /M α /M α /M α /M
π π π π
Q/M=0.5À θ0= Q/M=0.6Ï θ0= Q/M=0.8Þ θ0= Q/M=íîïð θ0=
Á Ð ß ñ
6 6 6 6
¿ Î Ý ì
¾ Í Ü ë
β /M
β /M
β /M
β /M
0 0 0 0
¼½ ËÌ ÚÛ éê
º» ÉÊ ØÙ çè
¹ 6 È 6 × 6 æ 6
³´ µ¶ 0 · ¸ 6 8 Âà ÄÅ 0 Æ Ç 6 8 ÑÒ ÓÔ 0 Õ Ö 6 8 àá âã 0 ä å 6 8
α /M α /M α /M α /M
π π π π
Q/M=0.5ÿ θ0= Q/M=0.6, θ0= Q/M=0.8 θ0= Q/M=0%&' θ0=
6 6 6 6
6 6 6 6
þ $
ý #
β /M
β /M
β /M
β /M
0 0 0 0
ûü !"
ùú
ø 6 6 6 6
òó ôõ 0 ö ÷ 6 8 - 4 2 0 6 8 0 6 8 0 6 8
α /M α /M α /M α /M
Fig. 4 The shadow of the black hole surrounded by a plasma for the different values of the charge parameter Q, inclination
angle θ0 , and the refraction index n. The solid lines in the plots correspond to the vacuum case, while for dotdashed lines the
plasma frequency is k/M =0.5 and a=0.4.
p
Calculating dφ/dr and dθ/dr using equations (7-9), η − ξ 2 csc2 θ − a2 n2 sin2 θ
β=± . (26)
we have n
In Fig. (4) the shadow of the rotating black hole for
the different values of black hole charge parameter Q,
ξ csc θ inclination angle θ0 and the plasma factor is shown.
α=− , (25) We made a special choice of the plasma frequency in
n6
the form ωe /ω = k/r. It is clearly seen in Fig. (4), the the case when the black hole is static and spherically
size and shape of the rotating black hole surrounded symmetric. The limiting constant σlim defines the value
by the plasma gradually gets modified as the charge of the absorption cross section vibration for a spheri-
and plasma refractive index variate. From a physical cally symmetric black hole
perspective the change in refractive index occurs as a
result of the gravitational red shift phenomenon. σlim ≈ πRsh 2 , (29)
where, Rsh is computed from (27). Therefore, (28)
4.1 Shadow of a non rotating charged black hole in takes the form [8]
Presence of Plasma d2 E(ω) 2π 3 Rsh 2 ω 3
= ω . (30)
dωdt e T −1
Now we focus on the static charged black hole, aiming The energy radiation of a black hole in the pres-
to understand the charge and plasma effects thoroughly. ence of plasma is directly proportional to the size of its
Using (25,26) the radius of the static black hole walled shadow. The dependence of the energy emission rate
in by a plasma is obtained as, on the frequency for the different values of charge and
1 plasma parameters is illustrated in Fig. (7). It is ob-
2M r(Q2 − M r) − 2n2 r3 (M − r) −
Rsh =
n(M − r) served that the rate of emission is higher for the small
′
nn r4 (M − r) + 2M r{ Q2 (Q2 − 2M r) + M 2 r2 − charge values, thus, at Q = 0, comparatively a large
amount of energy is liberated. In case of the plasma
n2 r(M − r) 2Q2 + r(−3M + r) −
parameter the emission rate exhibits rise with the in-
′ 1/2
nn r2 (M − r) Q2 + r(−2M + r) }1/2
(27) creasing values of the plasma. In the absence of plasma
k/M = 0, a less amount of energy release is detected.
For Q = 0, at rph = 3M we retain the radius of √ the
shadow for Schwarzschild black hole, i.e, Rsh = 3 3M
[46, 47]. The influence of charge and plasma on the ra- 5 Lensing in Weak Field In the Presence of
dius of the shadow for a static charged black hole is Plasma
demonstrated in Fig. (5). The relative values of rph are
obtained numerically using (19). The results attained We first present a model of weak-field for the non rotat-
are analogous to [48], the increase in charge makes the ing Kerr Newman black hole to study the gravitational
shadow appear smaller to the distant observer while lensing more precisely. The weak-field approximation is
for the plasma parameter we notice contrary effects on given by the relation
the radius. When k/M = 0, the shadow emerges in a gµν = ηµν + hµν , (31)
much smaller size. In Fig. (6) the evolution of shadow
ηµν and hµν refer to the Minkowski metric and per-
is represented visually with respect to various charge
turbation metric, respectively. They satisfy the below
and plasma parameters. In the left panel, as the value
mentioned properties
of charge adds up the shadow radius is seen to shrink
down. It is clearly shown in the right panel that the ra- ηµν = diag(−1, 1, 1, 1) ,
dius becomes larger with the increasing plasma factor. hµν ≪ 1, hµν → 0 under xi → ∞ ,
g µν = η µν − hµν , hµν = hµν . (32)
4.2 Emission energy of a non rotating charged black Taking into account the weak-field approximation
hole and weak plasma strength, for photon propagation along
z direction, one can easily obtain the angle of deflection
Black holes are known to emit thermal radiations which complying the steps in [34], we have
lead to a slow decrease in mass of the black hole until
∞
h00 ω 2 − ωe 2
1
Z
it completely annihilates [49]. We are interested to ex- α̂k = h33 + dz .
amine the energy emission of the Kerr-Newman black 2 −∞ ω 2 − ωe2 ,k
hole in the presence of plasma using the relation for the (33)
Hawking radiation at the frequency ω
Note that negative and positive sign for α̂b indicate
respectively deflection towards and away from the cen-
d2 E(ω) 2π 2 σlim ω 3 tral object. At large r, the black hole metric could be
= ω , (28)
dωdt e T −1 approximated to [35]
where, T = κ/2π is the Hawking temperature and κ Q2 Q2
2 2 2M 2 2M
is the surface gravity. Here, for convenience, we consider ds = ds0 + − 2 dt + − 2 dr2 ,(34)
r r r r7
6.0 6.5
5.5 6.0
h /M
Rsh /M
s
5.0 5.5
R
5.0
135
0.0 ()* +./ 0.6 0.8 0.0 678 9:; 0.6 0.8
Q/M k/M
Fig. 5 The radius of the shadow as a function of charge and plasma parameter. In the left panel the solid, dashed and dotted
plots correspond to k/M = 0,0.5 and 0.7, respectively. In the right panel the solid, dashed and dotted plots correspond to
Q/M =0,0.5, and 0.7, respectively.
π π
k/M= ? θ0 = Q /M= @AB C θ0 =
2 2
4 4
2 2
β /M
β /M
0 0
-2 -2
-4 -4
-4 -2 0 2 4 -4 -2 0 2 4
α /M α /M
Fig. 6 A visualization of the silhouette for distinct charge and plasma parameters. In the left panel the solid, dashed and
dotted plots correspond to Q/M =0,0.5 and 0.7, respectively. In the right panel the solid, dashed and dotted plots correspond
to k/M =0,0.5 and 0.7, respectively.
where ds20 = −dt2 + dr2 + r2 (dθ2 + sin2 θdφ2 ). Using the above expressions in the formula (33), one
In the Cartesian coordinates the components hµν can compute the light deflection angle for a black hole
can be written as in plasma
Z ∞ "
Q2 z2
Rg Q2
∂ Rg
h00 = − 2 , α̂b = √ − 2 2
r r 0 ∂b b2 + z 2 b +z b + z2
2
Q2
#
Rg Q2
hik = − 2 ni nk , 1 Rg
r r + √ − 2 dz , (36)
1 − ωe2 /ω 2 b2 + z 2 b + z2
Q2
Rg
h33 = − 2 cos2 χ , (35) where b2 = x21 + x22 is the impact parameter, and x1
r r
and x2 are the coordinates on the plane orthogonal to
the z axis, and the photon frequency at large r is given
where Rg = 2M . by8
k/M= MNO ` /M= abc
UVW
DEF PST
t _
L ^
2.0 2.0
K ]
J \
( )/
( )/
1.5
I 1.5 [
H Z
1.0
2
2
d Y
1.0
0.5
0.5
0.0
0.0 0.2 0.4 0.6 0.8 0.0 0.2 0.4 0.6 0.8 1.0
G X
Fig. 7 Energy emission of the black hole for distinct charge and plasma parameters. In the left panel the solid, dashed and
dotted plots correspond to Q/M =0,0.5 and 0.7, respectively. In the right panel the solid, dashed and dotted plots correspond
to k/M =0,0.5 and 0.7, respectively.
emission and the weak field lensing in the background
2 of the Kerr-Newman gravity walled in by a plasma
ω∞
ω2 = Rg Q2
. (37) medium. The results are recovered for the Schwarzschild
(1 − r + r2 ) metric when the charge and spin parameter are ex-
Here, ω∞ is the asymptotic value of photon frequency. cluded. The impact of the charge and plasma parame-
In the approximation of the charge and large distance, ters on the aforesaid properties have been investigated
the expression (12), after expanding in series on the explicitly. It is construed that the shadow size viewed
powers of 1/r, can be approximated to by a distant observer is smaller as the charge parame-
−1 ter is increased and by supplementing the plasma factor
ωe2 4πe2 N0 r0 4πe2 N0 r0 Rg
2 the size of the shadow appears to be larger. Since, the
n = 1− 2 ≃1+ 2 r
− 2 r2
(38)
.
ω mω∞ mω∞ energy liberated from the black hole depends on the ra-
Using this approximation one can easily find the de- dius of the shadow, therefore, rate of energy emission
flection angle α̂b of the light around a black hole in from the black hole is higher when the black hole is
presence of plasma surrounded by a plasma. As far as angle of deflection is
concerned, the photons are observed to experience an
π 2 e2 N0 r0 4πe2 N0 r0 Rg
2Rg increase in the deviation as the plasma factor gradually
α̂b = 1+ 2 b
− 2 b2
b mω∞ mω∞ adds up. While on the other hand, angle of deflection
Q2 4πe2 N0 r0
3πRg sufficiently reduces when the amount of charge param-
− 2 3π + 8 − . (39)
4b mω∞ 2 b b eter rises. Nevertheless, it is investigated that the Kerr-
Newman black hole experiences a contradictory influ-
We get αˆb = 2Rg /b in the absence of charge and
ence of the charge and plasma parameters.
plasma for the Schwarzschild black hole [29]. The de-
pendence of the angle of deflection αˆb on the impact
parameter b for various charge and plasma parameters
is demonstrated in Fig. (8). In the left panel as the value Acknowledgements
of charge increases the angle of deflection decreases and
it is seen that αˆb is maximum when the charge is turned F.A. was supported by Grants No. VA-FA-F- 2-008,
off, i.e., Q = 0. We observe that the deflection angle αˆb No.MRB-AN-2019-29 of the Uzbekistan Ministry for
increases with the gradual supplement in the plasma pa- Innovative Development and INHA University in Tashkent.
rameter (right panel). Also, one can clearly notice that
the deviation of photons is smaller when the plasma
factor is removed from the black hole background. References
1. K. Akiyama and et al., ‘First M87 Event Horizon
Telescope Results. VI.The Shadow and Mass of the Central
6 Conclusion
Black Hole’, Ap. J. 875 (2019) 44.
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4 π eg r 0 Ν0
i
= jkl
m wh M Q=0.5
2.0 1.4
1.8
yz{
1.6
1.4 1.0
âb
âb
1.2
vwx
1.0
0.8 qru
0.6
2 e 4 f 6 m 4 n o 7 p
b/M b/M
Fig. 8 Deflection angle α̂b as a function of the impact parameter b for different charge and plasma values. In the left panel
πe2 N0 r0
the values of Q (top to bottom) are 0,0.5,0.7 and 0.9 and in the right panel the values 4mω 2 M (bottom to top) are 0,0.5,0.7
∞
and 0.9.
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