Rotating dirty black hole and its shadow

Reggie C. Pantig1, ∗ and Emmanuel T. Rodulfo†1Physics Department, De La Salle University-Manila, 2401 Taft Ave., 1004 Manila Philippines

In this paper, we examine the effect of dark matter to a Kerr black hole of mass m. The metric is derived usingthe Newman-Janis algorithm, where the seed metric originates from the metric of a Schwarzschild black holesurrounded by a spherical shell of dark matter with mass M and thickness ∆rs. We analyzed both the time-likeand null geodesics and found out that if the dark matter density is considerably low, time-like geodesics showsmore deviations from the Kerr case compared to null geodesics. Furthermore, energy extraction via the Penroseprocess remains unchanged. A high concentration of dark matter near the rotating black hole is needed to haveconsiderable deviations on the horizons and photonsphere radius. With the dark matter configuration used in thisstudy, we found that deriving an analytic estimate to determine the condition for dark matter to have a notablechange in the shadow radius is inconvenient.

I. INTRODUCTION

Perhaps one of the most interesting objects in the universeis a black hole, at least for theoretical physicists, as it providesa theoretical playground in finding hints about the possibleunion of quantum theory and Einstein’s general relativity. Inthis quest, a remarkable breakthrough happened in 2019, wherethe Event Horizon Telescope collaborative efforts successfullyimaged the silhouette of the supermassive black hole at theheart of galaxy M87 using a technique called Very Long Base-line Interferometry (VLBI). Future improvements in visualiz-ing black holes might reveal the true geometry of black holes[1–52].

Recently, there are also efforts in analyzing black holeshadow influenced by astrophysical environments such as darkmatter, dark energy, or gravitational waves [53–58]. Thiscoined the term ”dirty black hole” [59, 60]. In these envi-ronments, dark matter remains an elusive entity because it doesnot interact with the electromagnetic field. Its only manifes-tation is through gravitational interaction with normal matter,and many believe in its existence because of its gravitationallensing effects [61–65]. If dark matter can also distort the blackhole shadow by distorting the black hole geometry, then it ispossible to use black hole shadows to detect dark matter thatsurrounds it indirectly. This idea can serve as an alternative toEarth-based dark matter detection experiments [66, 67].

A study in Ref. [54] analytically estimated the specificcondition for dark matter effects to occur notably in the shadowof a Schwarzschild black hole. In this paper, we follow thesame model for the dark matter configuration, and motivationsto investigate a more realistic scenario - dark matter effectson a rotating black hole. The dark matter configuration isdescribed only through its mass, and span that can be adjustedto determine dark matter density, hence, making it less model-dependent.

For the rest of this paper, Sect. II introduces theSchwarzschild metric surrounded by dark matter as modeledin Ref. [54]. In Sect. III, we derive the rotating solution byusing the seed metric introduced in Sect. II. In Sections IV to

∗ reggie.pantig@dlsu.edu.ph† emmanuel.rodulfo@dlsu.edu.ph

VII, we investigate the effect of dark matter on the horizons,time-like and null circular orbits, black hole shadow, and itsobservables. Sect. VIII is devoted to summarizing the resultsand recommendations for future studies. Lastly, we considerthe +2 metric signature, and G = c = 1.

II. SCHWARZSCHILD BLACK HOLE SURROUNDED BYDARK MATTER

The metric for a static, spherically-symmetric, uncharge,and non-rotating black hole is expressed as

ds2 = −f(r)dt2 + f(r)−1dr2 + r2(dθ2 + sin2 θdφ2

)(1)

where the metric function f(r) is given by

f(r) = 1− 2m

r. (2)

In an attempt to make some estimates as to what extent doesdark matter can affect the black hole geometry, a mass functionincorporating dark matter as an effective mass is introduced inRef. [54]. The metric function reads

f(r) = 1− 2

r(m+MG(r)) (3)

where M is the dark matter mass and G(r) describes its config-uration. If the dark matter’s configuration is a spherical shellthat surrounds the black hole, then G(r) can be expressed as

G(r) =

(3− 2

r − rs∆rs

)(r − rs∆rs

)2

. (4)

Here, rs is the inner radius of the dark matter shell and ∆rs isits thickness. Indeed, (4) is chosen so that for a given value ofrs and ∆rs, a piecewise mass function can be realized:

m(r) =

m, r < rs;

m+MG(r), rs ≤ r ≤ rs + ∆rs;

m+M, r > rs + ∆rs

(5)

In this way, when observing certain black hole phenomena,theorists can have insights or alternative perspectives where toattribute the cause of deviations from a theory [68, 69].

arX

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[gr

-qc]

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Mar

202

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2

It is easy to see that (3) reduces to the Schwarzschild caseif M = 0 and if r < rs due to (5). In the presence of darkmatter, ∆rs can never be zero, but it can have a value thatis much larger than rs. In this way, the dark matter densitycan be adjusted by tweaking the values of M and ∆rs. Also,note that M < 0 and rs = 0 are allowed. In this paper, werestrict the analysis to M > 0 and rs = 2m. Results in Ref.[54] estimated the thickness of dark matter in order have anoticeable effect on the shadow of Schwarzschild black hole:

∆rs ∼√

3mM. (6)

III. KERR BLACK HOLE SURROUNDED BY DARKMATTER

Using the Newman-Janis algortihm [70–77], we generalize(1) with metric function given by (3) to a Kerr black hole thatis surrounded by spherical shell of dark matter. The standardformalism starts with the conversion of the coordinates in (1)to a horizon-penetrating coordinates (also known as Eddington-Finkelstein coordinates):

du = dt− dr∗ = dt− dr

f(r)(7)

and we obtain

ds2 = −f(r)du2 − 2dudr + r2dθ2 + r2 sin2 θdφ2. (8)

The components of the contravariant metric tensor gµν in lineelement (8) can be expressed in terms of the null tetrad vectorcomponents which is

gµν = −lµnν − lνnµ +mµmν +mνmµ (9)

where

l = lµ∂

∂xµ= δµ1

∂

∂xµ,

n = nµ∂

∂xµ=

(δµ0 −

f(r)

2δµ1

)∂

∂xµ,

m = mµ ∂

∂xµ=

1√2r

(δµ2 +

i

sin θδµ3

)∂

∂xµ,

m = mµ ∂

∂xµ=

1√2r

(δµ2 −

i

sin θδµ3

)∂

∂xµ. (10)

We then do a basic complex coordinate transformation byimplementing

x′µ

= xµ+ia(δµr−δµu) cos θ →

u′ = u− ia cos θ,r′ = r + ia cos θ,θ′ = θ,φ′ = φ

(11)

so that f(r)→ f(r, r). Also, along with this transformation isthe transformation of the null tetrad vector components via

eaµ → e′a

µ=∂x′

µ

∂xνeaν ≡

(l′µ, n′

µ,m′

µ, m′

µ). (12)

In particular, the transformation matrix in (12) is given by∂u′

∂u∂u′

∂r∂u′

∂θ∂u′

∂φ∂r′

∂u∂r′

∂r∂r′

∂θ∂r′

∂φ∂θ′

∂u∂θ′

∂r∂θ′

∂θ∂θ′

∂φ∂φ′

∂u∂φ′

∂r∂φ′

∂θ∂φ′

∂φ

=

1 0 ia sin θ 00 1 −ia sin θ 00 0 1 00 0 0 1

(13)

and hence, the null tetrad vector components are now the fol-lowing:

l′µ = δµ1 ,

n′µ =

(δµ0 −

f(r, r)

2δµ1

),

m′µ =1√2r

[(δµ0 − δ

µ1 ) ia sin θ + δµ2 +

i

sin θδµ3

],

m′µ =1√2r

[− (δµ0 − δ

µ1 ) ia sin θ + δµ2 −

i

sin θδµ3

]. (14)

The components of the new contravariant metric tensor cannow be constructed using

g′µν

= −l′µn′ν − l′νn′µ +m′µm′

ν+m′

νm′

µ (15)

which results to

g′µν

=

a2 sin2 θ

Σ −1− a2 sin2 θΣ 0 a

Σ

−1− a2 sin2 θΣ F + a2 sin2 θ

Σ 0 − aΣ

0 0 1Σ 0

aΣ − a

Σ 0 1Σ sin2 θ

(16)

where Σ = r2 + a2 cos2 θ and F is a function of r and θ:

F =∆(r)− a2 sin θ2

r2 + a2(1− sin2 θ). (17)

We get the inverse metric as

g′µν =

−F −1 0 a sin2 θ (F − 1)−1 0 0 a sin2 θ0 0 Σ 0

a sin2 θ (F − 1) a sin2 θ 0 A sin2 θ

(18)

where A = Σ + a2 (2− F ) sin2 θ. The final step in theNewman-Janis procedure is to revert back to Boyer-Lindquistcoordinates by using the coordinate transformation

dt = du′ − r′2

+ a2

∆(r)dr′, dφ = dφ′ − a

∆(r)dr′ (19)

where ∆(r) is defined in terms of the complexified metricfunction f(r, r). In fact, following Ref. [73] on how r iscomplexified,

∆(r) = r2 − 2m(r)r + a2 (20)

where m(r) = m+MG(r). Thus, the line element of a rotat-ing, uncharged, and axially-symmetric black hole surrounded

3

by a spherical shell of dark matter is given by

ds2 = −(

1− 2m(r)r

Σ

)dt2 − 4am(r)r sin2 θ

Σdtdφ

+Σ

∆(r)dr2 + Σdθ2

+ sin2 θ

[r2 + a2

(1 +

2m(r)r sin2 θ

Σ

)]dφ2. (21)

In what follows, we will always treat the inner radius ofthe spherical shell of dark matter, rs, to always coincide withthe event horizon. Hence, it is expected for the value of rs tochange because the location of the event horizon depends onthe spin parameter a.

0.0 0.5 1.0 1.5 2.0 2.5r/m

0.0

0.2

0.4

0.6

0.8

1.0

1.2

(r)

M = 50mM = 100mM = 150mM = 200m

1.12 1.13 1.14 1.15 1.160.02

0.00

0.02

(a) a = 0.99m,∆rs = 100m, rs = 1.14m.

0.0 0.5 1.0 1.5 2.0 2.5r/m

0.5

0.0

0.5

1.0

1.5

(r)

M = 50mM = 100mM = 150mM = 200m

0.10 0.12 0.140.02

0.00

0.02

(b) a = 0.50m,∆rs = 100m, rs = 1.87m.

FIG. 1. Behavior of ∆(r) = 0.

IV. HORIZONS

We now examine the horizons of the metric given in Eq.(21). Since the metric blows up when ∆(r) = 0, the locationsof the horizons can be found by solving

r2 − 2 [m+MG(r)] r + a2 = 0. (22)

Figure 1 shows two plots about the behavior of ∆(r) for spe-cific spin parameter and as dark matter mass M varies. thelocation of the horizons for different values of spin parametera. For both cases, the event horizon is unaffected regardlessof dark matter density. The radius of the Cauchy horizon de-creases as dark matter mass increases. The overall effect isto increase the separation distance between the inner at outerhorizons. Without dark matter, we know that the extreme con-dition occurs when a = m where the two horizons coincide.As Fig. 1(a) indicates, dark matter makes it possible for thetwo horizons to coincide even at a > m.

2 4 6 8 10r/m

1.00

0.75

0.50

0.25

0.00

0.25

0.50

0.75

g tt=

0

M = 0mM = 50mM = 100mM = 150m

2.0 2.10.02

0.00

0.02

(a) a = 0.99m,∆rs = 100m, rs = 1.14m.

2 4 6 8 10r/m

1.00

0.75

0.50

0.25

0.00

0.25

0.50

0.75

g tt=

0

M = 0mM = 50mM = 100mM = 150m

(b) a = 0.50m,∆rs = 100m, rs = 1.87m.

FIG. 2. Behavior of gtt = 0.

For the radius of the ergosphere, one can find its location by

4

solving r as gtt = 0. In particular,

1− 2(m+MG(r))r

Σ= 0. (23)

If θ = π/2, the Kerr black hole without dark matter surround-ing it will give only one value of the ergoregion, which is atr = 2m because gtt becomes independent of a. When thereis dark matter, the influence of the spin parameter remainsbecause of rs. Hence, we observe the dark matter effect on theergosphere in Fig. 2(a) as the black hole is near extremal. Ata = 0.50m, the dark matter effect is not very obvious, as Fig.2(b) shows. Dark matter can also produce (though unphysical)a second ergoregion at large values of r. The significance ofthis ergoregion becomes important if M >> ∆rs.

V. TIME-LIKE CIRCULAR ORBITS

The geodesics of both particles and photons can be studiedusing the Hamilton-Jacobi approach to general relativity. TheHamilton-Jacobi equation reads

∂S

∂λ= −H (24)

where S is the Jacobi action and defined in terms of an affineparameter λ and coordinates xµ. In general relativity, theHamiltonian is given by

H =1

2gµν

∂S

∂xµ∂S

∂xν(25)

and it follows that

∂S

∂λ= −1

2gµν

∂S

∂xµ∂S

∂xν. (26)

The metric in Eq. (21) is independent on t, φ, and λ, thuswe can use the separability anzats given by

S =1

2µ2λ− Et+ Lφ+ Sr(r) + Sθ(θ), (27)

where µ is proportional to the particle’s rest mass and Sr(r) +Sθ(θ) are both functions of r and θ. The equations of motions

are then derived by combining (26) and (27). The results are

Σdt

dλ=r2 + a2

∆(r)P(r)− a(aE sin2 θ − L),

Σdr

dλ=√R(r),

Σdθ

dλ=√

Θ(θ),

Σdφ

dλ=

a

∆(r)P(r)−

(aE − L

sin2 θ

), (28)

with P(r),R(r) and Θ(θ) given by

P(r) = E(r2 + a2)− aL,R(r) = P(r)2 −∆(r)[Q+ (aE − L)2 + µ2r2],

Θ(θ) = Q−[a2(µ2 − E2

)+

L2

sin2 θ

]cos2 θ, (29)

withQ being the Carter constant: Q ≡ K− (L− aE)2 and Kis another constant of motion.

In order to see the effect of dark matter to the innermoststable circular orbit (ISCO) of particles, we introduce the fol-lowing impact parameters [78–80]:

ξ =L√

E2 − µ2, η =

QE2 − µ2

. (30)

Using the substitution ε = E√E2−µ2

, the second line in Eq.

(29) can be expressed as

R(r) =[ε(r2 + a2

)− aξ

]2−∆(r)

[η + (εa− ξ)2

+µ2ε2

E2

](31)

For circular orbits, the conditions

R(r) =dR(r)

dr|r=ro= 0 (32)

must be satisfied. The simultaneous equations will allow usto derive the expressions for ξ and η. By setting η = 0, itis possible to obtain solutions for the energy per unit massE of the particle. The results are the following cumbersomeequations (∆′(r) denotes a derivative with respect to r):

ξ =ε[E∆′(r)

(a2 + r2

)+√

2√A− 2E∆(r)r

]Ea∆′(r)

, (33)

η =8rε2

∆′(r)2E2a2

{−1

8∆′(r)2r

(a2µ2 + E2r2

)− 1

2E√

2

(a2 +

∆′(r)r

2−∆(r)

)√A

+1

2∆′(r)∆(r)

[µ2

2

(∆(r)− a2

)+ E2r2

]+ E2∆(r)r

(a2 −∆(r)

)}, (34)

E =

[a2∆′(r)2r2 ± 2

√2√B − 2∆′(r)∆(r)r

(a2 −∆(r)

)− 8∆(r)

(a2 −∆(r)

)216∆(r)r2 (a2 −∆(r))−∆′(r)2r4 + 8∆′(r)∆(r)r3

]1/2

. (35)

5

where ∆′(r) denotes a derivative with respect to r,

A = ∆(r)2r[2E2r −∆′(r)µ2

]and

B = a2∆(r)2[2(a2 −∆(r)

)+ ∆′(r)r

]3.

The location of ISCO, which dictates the innermost part ofan accretion disk, can be found by differentiating E2 with re-spect to r and solving the result for r. Unfortunately, analyticsolution is inconvenient given that ∆(r) depends on G(r) in(4) which gives additional complexity to the equation. Noticein Figure 3 that even if the dark matter density is very low, atime-like particle is very sensitive to dark matter effects. Atvery large ∆rs, one can see that the ISCO radius is increasedrelative to its value when M = 0 and specific spin parameter.Abnormalities arises as dark matter density increases becausea second and large ISCO radius is produced. This larger ISCOradius is irrelevant because of how ISCO is defined. Never-theless, this can constrain the extent as to what should be thevalue of dark matter density which provides physical sense andclosely agrees with prevailing results.

Other types of circular orbits such as bound, stable, andunstable circular orbits can be studied qualitatively using theeffective potential method. Following Ref. [81], the effectivepotential in terms of angular momentum per unit mass is givenby

V± =2m(r)aL

r3 + a2 (2m(r) + r)

±

{∆(r)

[(r2 + a2)2 − a2∆(r) + r2L2

][r3 + a2 (2m(r) + r)]

2

}1/2

. (36)

Figure 4 tells us plenty of information about other types ofcircular orbits. Here, the vertical line represents the locationof the event horizon. The maxima of the effective potentialrepresents the unstable circular orbit in which any perturbationin the particle’s orbit will dictate whether it will fall into theblack hole or escape to infinity. The higher the spin of theblack hole, the higher the energy requirement in this unstableorbit. The inset plot reveals that increasing dark matter massdecreases slightly the energy required in such an orbit. Fur-thermore, the radius where the peak is located decreases. Thereverse happens in the stable circular orbit in which the radiuswhere the minima occurs increases. Also, like the unstableorbit, the energy requirement decreases more obviously. Forthe low spin parameter, the available energy for elliptic boundorbits to occur is minimal, hence its easier to plunge into theblack hole.

By convention, particles that revolve clockwise on a blackhole have negative angular momentum L. Figure 5 shows thatin a Schwarzschild case, the effective potential is always posi-tive, and negative energy is not even allowed. If the black holeis rotating, negative effective potentials (or negative energies)of particles are allowed for aL < 0 and energy extraction fromthe black hole is allowed via the Penrose process. Regardlessof dark matter density, the location where the Penrose processshould occur remains unchanged. For a black hole that has a

60 80 100 120 140 160 180 200rs/m

8

9

10

11

12

13

14

r/m

M = 0M = 2mM = 4mM = 6m

(a) a = 0.99m, rs = 1.14m.

40 60 80 100 120 140 160 180 200rs/m

7

8

9

10

11

12

13

14

r/m

M = 0M = 2mM = 4mM = 6m

(b) a = 0.50m, rs = 1.87m.

FIG. 3. Location of innermost stable circular orbit (ISCO).

very high spin (Fig. 5(a)), the particle has more negative energycompared to a black hole that spins slowly (Fig. 5(b)). Hence,any deviations in the Penrose process due to dark matter is verynegligible.

VI. NULL CIRCULAR ORBITS AND BLACK HOLESHADOW

Null geodesics are of importance in studying the contourof a black hole silhouette. In particular, we need to determineand locate the unstable circular orbit for photons and do thebackward ray tracing method to plot the contour of the result-ing shadow in the celestial coordinates of a remote observer.For this case, we use (30) and take µ = 0. Doing the sameprocedures on how we derived (33) and (34), we obtain thefollowing:

ξ =∆′(r)(r2 + a2)− 4∆(r)r

a∆′(r), (37)

6

0 2 4 6 8 10 12 14 16 18 20 22 24r/m

0.900.910.920.930.940.950.960.970.980.991.001.011.021.031.04

V + o

r EM = 0M = 4mM = 8m

1.59 1.69 1.791.03401.03451.03501.03551.03601.03651.03701.03751.0380

(a) a = 0.90m, rs = 1.44m.

0 2 4 6 8 10 12 14 16 18 20 22 24r/m

0.900.910.920.930.940.950.960.970.980.991.001.011.021.031.04

V + o

r E

M = 0M = 4mM = 8m

2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.70.9340.9350.9360.9370.9380.9390.9400.941

(b) a = 0.75m, rs = 1.66m.

FIG. 4. Effective potential for ∆rs = 100m and L = 2.75m.

η =−r4∆′(r)2 + 8r3∆(r)∆′(r) + 16r2∆(r)(a2 −∆(r))

a2∆′(r)2

(38)which appears to be general. IfM = 0, one can obtain the verywell known analytic formula for the prograde and retrogradeorbit radii. However, it can be tedious or inconvenient to obtainan analytic formula for a Kerr black hole with dark matterconfiguration given in (4) since the result of η = 0 involves a5th power polynomial:

16∆(r)r2(a2 −∆(r)

)−∆′(r)2r4 + 8∆′(r)∆(r)r3 = 0.

(39)This inconvenience is also true even with the approximationas ∆rs → ∞, which reduces the above equation to the 4thpower. By numerical calculations and satisfying (32) we canlocate the unstable photon orbits and obtain some insights as towhat happens when ∆rs →∞ (i.e., low dark matter density).Unlike the time-like particles, Figure 6 reveals that high darkmatter densitiy is needed in order to see deviations in the nullorbits. In the near extremal case, Fig. 6(a) shows that the

1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5r/m

1.0

0.8

0.6

0.4

0.2

0.0

0.2

0.4

0.6

0.8

V + o

r E

a = 0.90m, M = 4ma = 0.90m, M = 100ma = 0, M = 4

1.42 1.44 1.46

0.96

0.95

0.94

0.93

(a) L = −3.464m,∆rs = 100m, rs = 1.44m.

1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5r/m

1.0

0.8

0.6

0.4

0.2

0.0

0.2

0.4

0.6

0.8

V + o

r E

a = 0.75m, M = 4ma = 0.75m, M = 100ma = 0, M = 4

1.65 1.70 1.750.55

0.50

0.45

(b) L = −3.464m,∆rs = 100m, rs = 1.66m.

FIG. 5. Effective potential for −L.

prograde is nearly unaffected, while the dark matter effect onthe retrograde radius is to decrease its value relative to the Kerrcase where M = 0. In Fig. 6(b), the change in the progradeorbit is evident. These changes, that the photon radius mustdecrease due to the presence of dark matter, agrees with theresult in Ref. [54]. As explained, the decrease in radius isdue the dark matter’s full effect (both under and above thephotonsphere) causing a new orbital equilibrium.

Any perturbations can lead the photons in the unstable or-bit to escape the rotating black hole’s gravitational influence.These photons will travel in the intervening space between theblack hole and the remote observer. In our case, the photonswill pass through the dark matter configuration. Hence, weexpect a difference in the resulting shadow when vacuum, andspace with dark matter, are compared. The method on deriv-ing the celestial coordinates with respect to the Zero AngularMomentum Observers (ZAMO) is very well established. The

7

60 80 100 120 140 160 180 200rs/m

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5r/m

M = 50mM = 100mM = 150mM = 200m

(a) a = 0.99m, rs = 1.14m.

60 80 100 120 140 160 180 200rs/m

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

r/m

M = 50mM = 100mM = 150mM = 200m

(b) a = 0.50m, rs = 1.87m.

FIG. 6. Location of unstable photon orbit (Photonsphere).

celestial coordinates, in general, are given by [82]

α = −r0ξ

ζ√gφφ

(1 +

gtφgφφ

ξ) ,

β = r0±√

Θ(i)

ζ√gθθ

(1 +

gtφgφφ

ξ) (40)

and in the limit r →∞, (40) reduces to

α = −ξ csc θ0,

β = ±√η + a2 cos2 θ0 − ξ2 cot2 θ0 (41)

where θo is the polar orientation of the remote observer withrespect to the equatorial plane, while ξ and η are given by (37)and (38). Figure 7 shows how different dark matter densityaffects the black hole shadow. If there is no dark matter, wefind the almost D-shaped contour of the Kerr black hole when

4 2 0 2 4 6 8 108

6

4

2

0

2

4

6

8M=0M=25mM=50mM=75m

(a) a = 0.99m,∆rs = 100m, rs = 1.14m.

4 2 0 2 4 6 88

6

4

2

0

2

4

6

8M=0M=50mM=100mM=150m

(b) a = 0.50m,∆rs = 100m, rs = 1.87m.

FIG. 7. Black hole shadow (θo = π/2).

the spin parameter is near extremal. When dark matter ispresent, the contour that represents the retrograde photon orbitbulges more as dark matter density increases. These contoursperfectly agree with Fig. 6(a). In effect, this increases theradius of the shadow. The contour that represents the progradeorbit deviates less as dark matter density increases. With thegiven values of dark matter mass in the contour plot, it seemsthat the change in the size of the shadow is kind of exaggerated.It is only to demonstrate, however, how dark matter changesthe size of the shadow. The D-shaped contour is not changedat all, hence, the fundamental properties of the rotating blackhole is retained in the presence of dark matter. Fig. 7(b) showsthe shadow contour when the black hole spin is a bit lower.

When we consider different values of the polar angle θo,Figure 8 shows how the remote observer sees the rotating blackhole. As the observer gets near the poles, the shadow contouris becoming more of an ellipse-shaped.

8

7.5 5.0 2.5 0.0 2.5 5.0 7.5 10.08

6

4

2

0

2

4

6

8= /2= /4= /6= /8

(a) a = 0.99m,∆rs = 100m, rs = 1.14m.

7.5 5.0 2.5 0.0 2.5 5.0 7.5 10.08

6

4

2

0

2

4

6

8= /2= /4= /6= /8

(b) a = 0.50m,∆rs = 100m, rs = 1.87m.

FIG. 8. Black hole shadow for different polar angle (M = 100m).

VII. SHADOW RADIUS, RADIUS DISTORTION, ANDENERGY EMISSION

Here, we show briefly a derivation of the shadow radius, andintroduce the expression for the radius distortion parameter,which are known to be observables useful in extracting infor-mation about black hole shadows [83, 84]. In order to haveinsights about the shadow radius, a schematic diagram likein Figure 9 is very helpful (see also [85, 86]). Here, callingxc = Oαo, we can see thatRs = αr−xc. A right triangle willbe formed and by inspection, R2

s = β2t + (αt − xc)2. After

some basic algebra,

Rs =β2t + (αt − αr)2

2|αt − αr|. (42)

As mentioned, Rs corresponds to the line αrαo, and thisradius is used for the reference circle. The shadow distortionis then defined as ds = αl − αl. Thus, the radius distortion

FIG. 9. Schematic diagram of black hole shadow.

parameter can be expressed as

δs =dsRs

=αl − αlR

(43)

Figure 10 shows how dark matter affects the shadow radius.The black dotted horizontal line represents the Schwarzschildcase (M = 0). Indeed, dark matter increases the shadowradius and such increase is also amplified by the black hole’sspin parameter a. For both cases in the figure, the curve isasymptotic to the Schwarzschild case when ∆rs →∞. Dueto (39) and the complexity looming in (42), we emphasizeagain that it is inconvenient to derive a formula to estimatethe effective radius of the dark matter halo in order to haveconsiderable effect on the shadow radius. This is unlike theSchwarzschild scenario where the estimate ∆rs =

√3mM

was easily attained because rph can be derived analytically aswell as the expression for the shadow radius.

The radius distortion parameter is plotted in Figure 11. Here,we see the agreement in Figure 7 because as the spin parameterdecreases, the more the shadow becomes close to a perfectcircle. The radius distortion is indeed greater when the blackhole spin is near the extremal case, which is also amplified bydark matter effect.

We can also use the shadow radius to determine the angulardiameter of the rotating black hole. The angular radius is givenby

θs = 9.87098× 10−3Rsm

D(44)

where m must be measured in terms of solar mass and D inparsec. Let’s consider supermassive black hole in M87 galaxywith mass m = 6.9 × 109M� and its distance from Earth isD = 16.8Mpc. Figure 12 shows the plot with and withoutdark matter. In general, not only the dark matter influencesthe increase in angular diameter, but also the spin parameter.

9

50 75 100 125 150 175 200 225rs/m

5.00

5.25

5.50

5.75

6.00

6.25

6.50

6.75

7.00R s

/mM = 50mM = 100m

(a) a = 0.99m, rs = 1.14m.

50 75 100 125 150 175 200 225rs/m

5.00

5.25

5.50

5.75

6.00

6.25

6.50

6.75

R s/m

M = 50mM = 100m

(b) a = 0.50m, rs = 1.87m.

FIG. 10. Shadow radius.

Even for M = 50m and ∆rs = 100m, the angular diameterincreases drastically as a increases.

The energy emission rate of a black hole is defined as

d2E

dσdt= 2π2 Πilm

eσ/T − 1σ3. (45)

where T is the black hole temperature. Following [56], thetemperature is given by

T =rh

4π(r2h + a2)2

[2a2(f(rh)− 1) + rh(r2

h + a2)f ′(rh)]

(46)in which rh is the event horizon radius and f(rh) = −gttin the metric (21). For a remote observer, the area of theshadow is approximately equal to the high energy absorptioncross-section which oscillates around a constant, Πilm = πR2

s .Figure 13 shows how the energy emission rate changes ifthe rotating black hole is surrounded by dark matter. Thedeviation from the Kerr case is not obvious when M = 50m.

50 75 100 125 150 175rs/m

0.00

0.05

0.10

0.15

0.20

0.25

s

M = 50mM = 100m

(a) a = 0.99m, rs = 1.14m.

50 75 100 125 150 175rs/m

0.000

0.005

0.010

0.015

0.020

0.025

0.030

s

M = 50mM = 100m

(b) a = 0.50m, rs = 1.87m.

FIG. 11. Radius distortion parameter.

However, in the third plot (M = 100m), the peak is slightlyhigher. Hence, the effect of dark matter is to increase theenergy emission rate near the event horizon. Dark matter alsohas a negligible effect on the photon’s peak frequency becauseshifting to lower or higher frequency is not so evident, even inthe case of high dark matter density.

VIII. CONCLUSION

In this paper, we extended the study in Ref. [54] to a ro-tating black hole by utilizing the Newman-Janis algorithm.Considering only the case where the initial configuration of thedark matter starts at the event horizon, we analyzed changesin some basic black hole properties such as the horizons, andthe ergoregion. Along with the null geodesics, we have shownthrough numerical analysis that these require high dark mat-ter density in order to have considerable changes. With thenature of how dark matter is configured, it is inconvenient to

10

5.1960 5.1965 5.1970 5.1975 5.1980 5.1985 5.1990 5.1995Rs

42.135

42.140

42.145

42.150

42.155

sKerr case

(a) M = 0.

5.2 5.4 5.6 5.8 6.0 6.2 6.4 6.6 6.8Rs

44

46

48

50

52

54

s

M = 50m

(b) a = 0.99m, rs = 1.14m.

5.20 5.25 5.30 5.35 5.40 5.45 5.50 5.55 5.60Rs

42.5

43.0

43.5

44.0

44.5

45.0

45.5

s

M = 50m

(c) a = 0.50m, rs = 1.87m.

FIG. 12. Angular diameter.

0.0 0.1 0.2 0.3 0.4 0.5

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

d2 Ed

dt

a = 0.2ma = 0.6ma = 0.9m

(a) M = 0.

0.0 0.1 0.2 0.3 0.4 0.5

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

d2 Ed

dt

a = 0.20ma = 0.60ma = 0.90m

(b) M = 50.

0.0 0.1 0.2 0.3 0.4 0.5

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.16

d2 Ed

dt

a = 0.20ma = 0.60ma = 0.90m

(c) M = 100.

FIG. 13. Energy Emission rate.

11

provide an analytic estimate using the radius of the black holeshadow to determine notable changes due to the dark mattereffect. The size of the shadow is seen to enlarge while theshape is maintained. We also showed that time-like geodesicsare very sensitive to dark matter effects because the locationof the ISCO radius drastically change even at low-density con-figuration. While the Penrose process remained unaffected,other types of orbits are seen to be affected by dark matter.

Following this study, prospects might include the assumptionof a hypothetical situation where dark matter penetrates theevent horizon: coinciding with the Cauchy horizon, or eventhe case where rs = 0. This study showed that using theshadow, finding the analytic estimate similar to what is donein the Schwarzschild case, remains an inconvenient task. It isinteresting to find out whether finding such estimate is possibleby using a different black hole phenomena.

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