The Black hole shadow in Modified Gravity

Signatures of Regular Black Holes
from the shadow of M87* and Sgr A*
Published: JCAP 09 (2022) 066 AAPCOS-2023
Subhadip Sau
January 23, 2023
Subhadip Sau (JRC & IACS) Shadow of Regular Black Hole January 23, 2023 1 / 28

Overview
1 Black Hole Shadow: Brief Review
2 Regular black holes and their rotating counterpart
3 Shadow of rotating black hole
4 Ghosh-Culetu black hole and its shadow
5 Observations and Results
6 conclusion
Subhadip Sau (JRC & IACS) Shadow of Regular Black Hole January 23, 2023 2 / 28

Black Hole Shadow: Brief Review

Black Hole Shadow: Brief Review Black hole
Black Hole
The Schwarzschild spacetime is given by the metric:
ds2
=

1 Rs
r

dt2
+ dr2

1 Rs
r
+r2
d 2
(1)
Credit: cnx.org.
Figure 1: The space distortion becomes more noticeable
around increasingly larger masses.
Subhadip Sau (JRC IACS) Shadow of Regular Black Hole January 23, 2023 3 / 28

Black Hole Shadow: Brief Review Photon Sphere
Photon sphere
The light particles with sufficient angular momentum avoid being pulled into the black
hole by traveling in a nearly tangential direction known as an exit cone (orange path).
Figure 2: Photon sphere (Credit: RealClearScience)

Black Hole Shadow: Brief Review Photon Sphere
Photon sphere
The light particles with sufficient angular momentum avoid being pulled into the black
hole by traveling in a nearly tangential direction known as an exit cone (orange path).
Figure 2: Photon sphere (Credit: RealClearScience)
Figure 3: You can see back of your head !!!

Black Hole Shadow: Brief Review Shadow
Shadow
Credit: Nicolle Rager Fuller/NSF.
Figure 4: Photon Sphere of Schwarzschild BH
While matter accretes
and accumulates
around the central
supermassive black
hole in a galaxy, it
heats up and emits
light. That light then
gets bent by the black
hole’s gravity,
creating a “ring” of
radio light from any
external perspective.

Photon Sphere of a Black Hole
Credit: Volker Perlick et al.
Figure 5: Formation of black hole shadow

Photon sphere

Photon sphere
Figure 6: Euclidean vs. angular size

Regular black holes and their
rotating counterpart

Regular black holes and their rotating counterpart Non-linear electrodynamics
Gravity coupled to non-linear electrodynamics (NLED)
• Gravity coupled to NLED ...
S =
Z
d4
x
p
h

R
16
L( F)

• Equation of motion
r

@L
@f
F

= 0
r (?F) = 0
F = @A @A
f = F = FF

S =
Z
d4
x
p
h

R
16
L( F)

r

@L
@f
F

= 0
r (?F) = 0
F = @A @A
f = F = FF
Einstein’s Equation
G = 8

4 @L
@f
gFF gL(f)


S =
Z
d4
x
p
h

R
16
L( F)

r

@L
@f
F

= 0
r (?F) = 0
F = @A @A
f = F = FF
Einstein’s Equation
G = 8

4 @L
@f
gFF gL(f)

Static spherically symmetric solution:
ds2
=

1 2m(r)
r

dt2
+

1 2m(r)
r
1
dr2
+r2
d2
+r2
sin2
d2
(2)
Eloy Ayon Beato and Alberto Garcia [Phys. Rev. Lett. 80, 5056 (1999), Phys. Lett. B 493 (2000)]

Regular black holes and their rotating counterpart mass function
Mass function
• Existence of event horizon ! mass
function should be positive definite i.e
m(r) 0 for r 0.

Mass function
m(r) 0 for r 0.
• regularity of m(r) =) at least three
times differential and approaches zero
sufficiently fast in the limit r ! 0

Mass function
m(r) 0 for r 0.
• m000(r) is finite at the origin r = 0.

Mass function
m(r) 0 for r 0.
• Curvature polynomials...
R = 4m0
r2
+ 2m00
r
RR = 8m02
r4
+ 2m002
r2
RR = 48m2
r6
16m
r3

4m0
r2
m00
r

+4 8m02
r4
4m0m00
r3
+ m002
r2
!
Zhong-Ying Fan and Xiaobao Wang [Phys. Rev. D 94, 124027]

Mass function
m(r) 0 for r 0.
• Curvature polynomials...
R = 4m0
r2
+ 2m00
r
RR = 8m02
r4
+ 2m002
r2
RR = 48m2
r6
16m
r3

4m0
r2
m00
r

+4 8m02
r4
4m0m00
r3
+ m002
r2
!
Zhong-Ying Fan and Xiaobao Wang [Phys. Rev. D 94, 124027], Hideki Maeda [JHEP 11 (2022), 108]

Deternination of mass function
• For spherically symmetric spacetime
the non-zero components of F !
Ftr and F

Ftr and F
• For pure magnetic charge
A = ~
g cosd =) F = ~
g sin

Ftr and F
A = ~
g cosd =) F = ~
g sin
2m0(r)
r2
+2L(f) = 0
m00(r)
r
+2L(f) 2~
g2
r4
@L
@f
= 0

Ftr and F
A = ~
g cosd =) F = ~
g sin
2m0(r)
r2
+2L(f) = 0
m00(r)
r
+2L(f) 2~
g2
r4
@L
@f
= 0
Bardeen black hole
L(F) =
j~
gj
M
~
g2
2~
g2
F
+3
4
n
1+(2~
g2F)=4
o1+

m(r) = Mr
(r + ~
g)

Eloy Ayon Beato and Alberto Garcia [Phys. Rev. Lett.
80, 5056 (1999), Phys. Lett. B 493 (2000)]

Ftr and F
A = ~
g cosd =) F = ~
g sin
Culetu black hole
L(F) = Fe (2~
g2F)1=4
; = ~
g=(2M) (3)
m(r) = Me =r; = ~
g2
=2M (4)
Hristu Culetu [ Int.J.Theor.Phys. 54 (2015)]
2m0(r)
r2
+2L(f) = 0
m00(r)
r
+2L(f) 2~
g2
r4
@L
@f
= 0
Bardeen black hole
L(F) =
j~
gj
M
~
g2
2~
g2
F
+3
4
n
1+(2~
g2F)=4
o1+

m(r) = Mr
(r + ~
g)

Eloy Ayon Beato and Alberto Garcia [Phys. Rev. Lett.
80, 5056 (1999), Phys. Lett. B 493 (2000)]

Regular black holes and their rotating counterpart Gürses-Gürsey metric
Rotating black hole in NLED
Gürses-Gürsey metric
ds2
=

1 2m(r)r

dt2 4m(r)arsin2

dtd +
dr2
+ d2
+

(r2
+a2
)+ 2m(r)r
a2
sin2

sin2
d2
= r2
+a2
cos2

= r2
+a2
2rm(r)

Shadow of rotating black hole
Shadow of rotating black hole: HJ Equation
• For stationary, axisymmetric
spacetime, the lagrangian for
test particle is
g _
x _
x = gtt _
t2
+2gt _
t _

+g _
2
+grr _
r2
+g _
2
= 2L
(5)
• Hamilton Jacobi equation
H

x;
@S
@x

+ @S
@
= 0 (6)
S = Et +L +Sr(r)+S()

test particle is
g _
x _
x = gtt _
t2
+2gt _
t _

+g _
2
+grr _
r2
+g _
2
= 2L
(5)
H

x;
@S
@x

+ @S
@
= 0 (6)
Radial part

dSr
dr
2
= R(r)
2
(7)
R(r) =
h
C (L aE)2
i
+
n
r2
+a2

E aL
o2
(8)

test particle is
g _
x _
x = gtt _
t2
+2gt _
t _

+g _
2
+grr _
r2
+g _
2
= 2L
(5)
H

x;
@S
@x

+ @S
@
= 0 (6)
Radial part

dSr
dr
2
= R(r)
2
(7)
R(r) =
h
C (L aE)2
i
+
n
r2
+a2

E aL
o2
(8)
Angular part
dS
d
!2
= () (9)
() = C +cos2

E2
a2 L2
sin2

#
(10)

Analysis of angular part
S = Et +L +
Z p
R(r)
dr +
Z q
()d (11)
EoM
_
r =
p
R(r)

_
=
p


S = Et +L +
Z p
R(r)
dr +
Z q
()d (11)
• Assume u = cos, then
EoM
_
r =
p
R(r)

_
=
p


S = Et +L +
Z p
R(r)
dr +
Z q
()d (11)

E
2
_
u2
= u2
( +2
a2
) a2
u4
= F(u())
(12)
EoM
_
r =
p
R(r)

_
=
p

8

:
= C=E2
= L=E2

S = Et +L +
Z p
R(r)
dr +
Z q
()d (11)

E
2
_
u2
= u2
( +2
a2
) a2
u4
= F(u())
(12)
• F(u(max)) = 0
u2
0 = ( +2
a2
)
p
( +2 a2)2 +4a2
2a2
(13)
EoM
_
r =
p
R(r)

_
=
p

8

:
= C=E2
= L=E2

|u0|
χ0
χ=1, η=0.9, a=0.8
0.0 0.2 0.4 0.6 0.8 1.0
-0.5
0.0
0.5
1.0
u
F(u)
(a)
χ0
χ=-0.1, η=0.3, a=0.8
0.0 0.2 0.4 0.6 0.8 1.0
-0.10
-0.05
0.00
0.05
u
F(u)
(b)
Figure 7: The above figure depicts the variation of the angular potential F(u) with u for 0
and 0.

χ=0 ; η2
a2
χ=0, η=0.9, a=0.8
0.0 0.2 0.4 0.6 0.8 1.0
-0.8
-0.6
-0.4
-0.2
0.0
u
F(u)
(a)
1 -
η2
a2
χ=0 ; η2
a2
χ=0, η=0.5, a=0.8
0.0 0.2 0.4 0.6 0.8 1.0
-0.15
-0.10
-0.05
0.00
0.05
u
F(u)
(b)
Figure 8: The above figure depicts the variation of the angular potential F(u) with u for = 0.
We note that when 2 a2, F(u) is negative.

Analysis of radial part
The differential equation associated with the radial part is given by

E
2
_
r2
=

L
E
a
2
#
+

r2
+a2
a
2
V (r) (15)


E
2
_
r2
=

L
E
a
2
#
+

r2
+a2
a
2
V (r) (15)
For spherical photon orbits one needs to satisfy the condition: V (r) = 0 = V 0(r) .


E
2
_
r2
=

L
E
a
2
#
+

r2
+a2
a
2
V (r) (15)
= r4
a2
= a2
+r2
a


E
2
_
r2
=

L
E
a
2
#
+

r2
+a2
a
2
V (r) (15)
= r4
a2
= a2
+r2
a
=

r3

4a2
rm0(r) 4a2
m(r)+r3
m0(r)2
+2r3
m0(r)
6r2
m(r)m0(r) 6r2
m(r)+9rm(r)2
+r3

=

a2
rm0(r)+m(r) r
2

(17a)
=a2
rm0(r)+a2
m(r)+a2
r +r3
m0(r) 3r2
m(r)+r3
a(rm0(r)+m(r) r) (17b)

Celestial coordinates
= lim
r0!1

r2
0 sin0
d
dr

(18a)
= lim
r0!1

r2
0
d
dr

(18b)
i = csco (19a)
i =
q
+a2 cos2 0 2 cot2
0
(19b)

Ghosh-Culetu black hole and its
shadow

Ghosh-Culetu black hole and its shadow shadow structure
Shadow of Ghosh-Culetu black hole
θ=π/3; a/M=0.2
k/M=0.0
k/M=0.3
k/M=0.6
-6 -4 -2 0 2 4 6 8
-6
-4
-2
0
2
4
6
α/M
β/M
(a) Variation of BH shadow with metric
parameter k. Here the inclination angle is
taken to be = 60 and the spin is
assumed to be a = 0:2.
θ=π/4; a=0.5
k=0.1
k=0.2
k=0.3
k=0.4
-6 -4 -2 0 2 4 6 8
-6
-4
-2
0
2
4
6
α
β
(b) Variation of BH shadow with metric
parameter k. Here the inclination angle is
taken to be = 45 and the spin is
assumed to be a = 0:5.

Ghosh-Culetu black hole and its shadow shadow structure
Shadow of Ghosh-Culetu black hole
θ=π/4; k=0.1
a=0.1
a=0.2
a=0.3
a=0.4
-6 -4 -2 0 2 4 6 8
-6
-4
-2
0
2
4
6
α
β
(c) Variation of BH shadow with
spin-parameter a. Here the inclination
angle is taken to be = 45 and k = 0:1
a=0.5; k=0.1
π/2
π/4
π/6
π/10
-6 -4 -2 0 2 4 6 8
-6
-4
-2
0
2
4
6
α
β
(d) Variation of BH shadow with inclination
angle . Here the spin is taken to be
a = 0:5 and k = 0:1

Ghosh-Culetu black hole and its shadow Observables
Observables
Effects beyond GR ! Need Observables
βt
βb
αl αr
-4 -2 0 2 4 6 8
-6
-4
-2
0
2
4
6
α
β

Observables
Angular Diameter
= GM
c2D

βt
βb
αl αr
-4 -2 0 2 4 6 8
-6
-4
-2
0
2
4
6
α
β

Observables
Angular Diameter
= GM
c2D

Axis Ratio
A =

βt
βb
αl αr
-4 -2 0 2 4 6 8
-6
-4
-2
0
2
4
6
α
β

Contact with observations: Observables
Deviation from circularity βt
βb
αl αr
-4 -2 0 2 4 6 8
-6
-4
-2
0
2
4
6
α
β

Deviation from circularity
Average radius
βt
βb
αl αr
-4 -2 0 2 4 6 8
-6
-4
-2
0
2
4
6
α
β

Average radius
Distance from geometric centre
βt
βb
αl αr
-4 -2 0 2 4 6 8
-6
-4
-2
0
2
4
6
α
β

Average radius
`() =
q
f () cg2
+ 2()
βt
βb
αl αr
-4 -2 0 2 4 6 8
-6
-4
-2
0
2
4
6
α
β

Average radius
Ravg =
s
1
2
Z 2
0
d `2()
`() =
q
f () cg2
+ 2()
βt
βb
αl αr
-4 -2 0 2 4 6 8
-6
-4
-2
0
2
4
6
α
β

C = 1
Ravg
s
1
2
Z 2
0
df`() Ravgg2
Average radius
Ravg =
s
1
2
Z 2
0
d `2()
`() =
q
f () cg2
+ 2()
βt
βb
αl αr
-4 -2 0 2 4 6 8
-6
-4
-2
0
2
4
6
α
β

Observables
For M87*
1 Angular diameter: (423)as
2 Axis Ratio: . 4
3
3 Deviation from circularity:
C . 10%
For Sgr A*
1 Angular diameter: (48:77)as
2 Axis Ratio: Prediction...
3 Deviation from circularity:
Prediction...

Observations and Results M87*
Angular diameter [M87*]
28
30
32
34
35.1
36
-0.5 0.0 0.5
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
a
k
26.0
28.0
30.0
32.0
34.0
35.1
36.0
(e) Contours illustrating the dependence of
the angular diameter of the shadow of M87*
on k and a assuming M ' 6:2109M and
distance D ' 16:8 Mpc
15
16
17
18
19
20
21
-0.5 0.0 0.5
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
a
k
14
15
16
17
18
19
20
21
(f) Contours illustrating the dependence of
the angular diameter of the shadow of M87*
on k and a assuming M ' 3:5109M and
distance D ' 16:8Mpc

Observations and Results M87*
Axis ratio and C
0
0.05
0.10
0.15
(g) The above figure shows the variation of C
with k and a assuming an inclination angle of
17 corresponding to M87*.
1.000
1.005
1.010
1.015
(h) The above figure shows the variation of axis
ratio with k and a assuming an inclination angle
of 17 corresponding to M87*.

Observations and Results Sgr A*
Angular diameter [Sgr A*]
38
40
41.7
42
44
46
48
48.7
49.5
50
-0.5 0.0 0.5
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
a
k
34.0
38.0
40.0
41.7
42.0
44.0
46.0
48.0
48.7
49.5
50.0
(i) The above figure depicts the angular diameter
in the k a plane assuming M = 3:951106M
and D = 7:935 kpc.
38
40
41.7
42
44
46
48
48.7
49.5
50
-0.5 0.0 0.5
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
a
k
34.0
38.0
40.0
41.7
42.0
44.0
46.0
48.0
48.7
49.5
50.0
(j) The above figure depicts the angular diameter
and D = 7:959 kpc.

Angular diameter [Sgr A*]
38
40
41.7 42
44
46
48
48.7
49.5 50
51.8
-0.5 0.0 0.5
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
a
k
38.0
41.7
44.0
48.0
49.5
51.8
(k) The above figure depicts the angular
diameter in the k a plane assuming
M = 4:261106M and D = 8:2467 kpc.
38
40
41.7 42
44
46
48
48.7
49.5
50
51.8
-0.5 0.0 0.5
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
a
k
38.0
41.7
44.0
48.0
49.5
51.8
(l) The above figure depicts the angular diameter
and D = 8:277 kpc.

Axis ratio and C
0.10
0.15
0.20
0.25
(m) The above figure shows the variation of C
134 corresponding to Sgr A*.
1.0000
1.0025
1.0050
1.0075
1.0100
1.0125
(n) The above figure shows the variation of A
134 corresponding to Sgr A*.

conclusion
Conclusions
• The observation of regular black hole shadows can also help us test the theory of
general relativity in the strong gravity regime.
• Finding the rotating counterpart of the regular black hole with electric charge.
• Other type of regular black hole ?
• Can the study of regular black hole shadows help us understand the evolution and
formation of these objects?
• How do the properties of regular black hole shadows compare to those of
supermassive black holes and intermediate-mass black holes?

conclusion Thanks
Thank You!

The Black hole shadow in Modified Gravity

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