Shadows of Sgr Alowast Black Hole Surrounded by Superfluid Dark Matter Halo

Kimet Jusufi1 2 lowast Mubasher Jamil3 4 5 dagger and Tao Zhu3 5 Dagger

1Physics Department State University of Tetovo Ilinden Street nn 1200 Tetovo North Macedonia2Institute of Physics Faculty of Natural Sciences and Mathematics

Ss Cyril and Methodius University Arhimedova 3 1000 Skopje North Macedonia3Institute for Theoretical Physics and Cosmology Zhejiang University of Technology Hangzhou 310023 China

4Department of Mathematics School of Natural Sciences (SNS)National University of Sciences and Technology (NUST) H-12 Islamabad 44000 Pakistan

5United Center of Gravitational Wave Physics (UCGWP)Zhejiang University of Technology Hangzhou 310023 China

In this paper we construct a black hole solution surrounded by superfluid dark matter (BH-SFDM)and baryonic matter and study their effects on the shadow images of the Sgr Alowast black hole Toachieve this goal we have considered two density profiles for the baryonic matter described by thespherical exponential profile and the power law profile including a special case describing a totallydominated dark matter galaxy (TDDMG) Using the present values for the parameters of the super-fluid dark matter and baryonic density profiles for the Sgr Alowast black hole we find that the effects of thesuperfluid dark matter and baryonic matter on the size of shadows are almost negligible comparedto the Kerr vacuum black hole In addition we find that by increasing the baryonic mass the shadowsize increases considerably This result can be linked to the matter distribution in the galaxy namelythe baryonic matter is mostly located in the galactic center and therefore by increasing the baryonicmatter can affect the size of black hole shadow compared to the totally dominated dark matter galaxywhere we observe an increase of the angular diameter of the Sgr Alowast black hole of the magnitude10minus5microarcsec

I INTRODUCTION

It is widely believed that the region at the centre ofmany galaxies contain black holes (BHs) They are as-trophysical objects which perform manifestations of ex-tremely strong gravity such as formation of gigantic jetsof particles and disruption of neighboring stars Fromtheoretical perspective BHs serve as a perfect lab to testgravity in strong field regime Recently the Event Hori-zon Telescope (EHT) Collaboration announced their firstresults concerning the detection of an event horizon ofa supermassive black hole at the center of a neighbor-ing elliptical M87 galaxy [1] Due to the gravitationallensing effect the bright structure surrounding the blackhole also known as the accretion disk appears distortedMoreover the region of accretion disk behind the blackhole also gets visible due to the bending of light by blackhole The shadow image captured by EHT is in goodagreement with the predictions of the spacetime geom-etry of black hole described by the Kerr metric Thestudies of the shadow images of various black holes to-gether with the current and future observations are ex-pected to provide an important approach to understandthe geometric structure of black holes or small devia-tions from Kerr metric in the strong field regime

On the other hand it is believed that up to 90 ofmatter in the host galaxy of a central black hole con-

lowastElectronic address kimetjusufiuniteedumkdaggerElectronic address mjamilzjuteducn(corresponding author)DaggerElectronic address zhut05zjuteducn

sists of dark matter Thus it is natural to expect that thedark matter halo which surrounds the central black holecould lead to small deviations from Kerr metric nearthe black hole horizon With this motivation the blackhole solutions with dark matter halo and their effectson the black hole shadow has been studied in [2ndash7] Inthis paper we consider a scenario of a BH solution sur-rounded by a halo which contains both the dark matterand baryonic matter For dark matter we assume a su-perfluid dark matter model recently proposed in [8] andconstruct the corresponding black hole solution with thehalo of dark matter and baryonic matter As a specificexample we are going to elaborate the shadow imagesof Sgr Alowast to estimate its angular diameter with the ef-fect of the halo

One aim here is to study the shadows of rotating blackholes surrounded by SFDM and baryonic matter As wementioned shadows possess interesting observationalsignatures of the black hole spacetime in the stronggravity regime and in the future we expect that shadowobservations can impose constrains on different gravitytheories It is interesting to mention that the characteris-tic map of a shadow image depends on the details of thesurrounding environment around the black hole whilethe shadow contour is determined only by the space-time metric itself In light of this there have been ex-tensive efforts to investigate shadows cast by differentblack hole and compact object spacetimes The shadowof a Schwarzschild black hole was first studied in [9] andlater in [10] and the same for Kerr black hole was studiedby [11] Since then various authors have studied shad-ows of black holes in modified theories of gravity andwormholes geometries [12ndash51]

arX

iv2

005

0529

9v1

[gr

-qc]

8 M

ay 2

020

2

The plan of the paper is as follows In Sec II we usethe superfluid density profile to obtain the radial func-tion of a spherically symmetric spacetime In Sec IIIwe study the effect of baryonic and SFDM in the space-time metric using the exponential and power law pro-file for the baryonic matter In Sec IV we find a spheri-cally symmetric black hole metric surrounded by SFDMAnd then in Sec V we first construct the spinning blackhole surrounded by dark matter halo based the spheri-cally symmetric spacetime we obtained by applying theNewman-Janis method With this spinning black holewe then study null geodesics and circular orbits to ex-plore the shadows of Sgr Alowast black hole In Sec VI wepresent the main conclusions of this paper We shall usethe natural units G = c = h = 1 throughout the paper

II SUPERFLUID DARK MATTER AND SPHERICALLYSYMMETRIC DARK MATTER SPACETIME

A Superfluid dark matter

From the view of field theory one can describe the su-perfluid by the theory of a spontaneously broken globalU(1) symmetry in a state of finite U(1) charge densityIn particular at low energy regime the relevant degreeof freedom is the Goldstone boson for the broken sym-metry of the phonon field ψ On the other hand theU(1) symmetry acts non-linearly on ψ as a shift symme-try ψ rarr ψ + c In the non-relativistic regime at finitechemical potential micro the most general effective theoryis described by [8]

LT=0 = P(X) (1)

in which X = micro minus mΦ + ψ minus (~nablaψ)22m Note that mis the particle mass and Φ represents the Newtoniangravitational potential The main idea put forward in[8] is that the DM superfluid phonons are described bythe modified Newtonian dynamics (MOND) type La-grangian

LDM T=0 =2Λ(2m)32

3Xradic|X| (2)

At first sight the fractional power of X may seemstrange if (2) represents for example a scalar field how-ever from the theory of phonons one can argue that ex-ponent is crucial since it determines the superfluid equa-tion of state On the other hand to obtain a mediate aforce between baryons and the DM phonons one musthave the coupling

Lint = αΛψ ρB (3)

where α is a constant while ρB gives the baryonic den-sity Let us point out that at zero temperature this su-perfluid theory has three parameters namely the parti-cle mass m a self-interaction strength parameter Λ and

finally the coupling constant α between phonons andbaryons Starting from the action composed of the (2)and (3) one can obtain a MOND type force law In par-ticular the equation of motion for phonons is the follow-ing [8]

~nabla middot

(~nablaψ)2 minus 2mmicroradic(~nablaψ)2 minus 2mmicro

~nablaψ

=αρb

2 (4)

with micro equiv micro minus mΦ Using the limit (~nablaψ)2 2mmicro andignoring the curl term one has

|~nablaψ|~nablaψ α~aB (5)

with ~aB being the Newtonian acceleration due tobaryons only While the mediated acceleration resultingfrom (3) is

~aphonon = αΛ ~nablaψ (6)

In this way it can be shown that

aphonon =radic

α3Λ2 aB (7)

From this result it is easy to see well known MOND ac-celeration by identifying

a0 = α3Λ2 (8)

From the galactic rotation curves for a0 it is found

aMOND0 12times 10minus8 cms2 (9)

As we saw the MOND type law obtained in the su-perfluid model is not exact value and only applies in theregime (~nablaψ)2 2mmicro More importantly the total ac-celeration experienced by a test particle is a contributionof~aB~aphonon and~aDM

B Spacetime metric in the presence of superfluid darkmatter

The density profile can be given in terms of dimen-sionless variables Ξ and ξ defined by [8]

ρ(r) = ρ0Ξ12 r =radic

ρ0

32πΛ2m6 ξ (10)

where ρ0 equiv ρ(0) is the central density From the Lane-Emden equation

1ξ2

ddξ

(ξ2 dΞ

dξ

)= minusΞ12 (11)

and using the boundary conditions Ξ(0) = 1 andΞprime(0) = 0 the numerical solution yields

ξ1 275 (12)

3

The last equation determines the size of the condensate

R =

radicρ0

32πΛ2m6 ξ1 (13)

A simple analytical form that provides a good fit is

Ξ(ξ) = cos(

π

2ξ

ξ1

) (14)

The central density is related to the mass of the halo con-densate as follows

ρ0 =M

4πR3ξ1

|Ξprime(ξ1)| (15)

From the numerics we find Ξprime(ξ1) minus05 Substitut-ing (13) in Eq (15) we can solve for the central density

ρ0 (

MDM

1012M

)25 ( meV

)185(

ΛmeV

)6510minus24 gcm3

(16)Meanwhile the halo radius is

R (

MDM

1012M

)15 ( meV

)minus65(

ΛmeV

)minus2545 kpc

(17)Remarkably for m sim eV and Λ sim meV we obtain

DM halos of realistic size The mass profile of the darkmatter galactic halo is given by

MDM (r) = 4πint r

0ρDM

(rprime)

rprime2drprime (18)

To solve the last integral we use Eqs (10)-(14) and byrewriting the mass profile in terms of the new coordi-nate ξ In particular we obtain

MDM (r) = sin(πr

2R

) πr2

2Rρ2

08Λ2m6 r le R (19)

From the last equation one can find the tangential ve-locity v2

tg (r) = MDM(r)r of a test particle moving inthe dark halo in spherical symmetric space-time

v2tg (r) = sin(

πr2R

)πr2R

ρ20

8Λ2m6 (20)

In this section we derive the space-time geometry forpure dark matter To do so let us consider a static andspherically symmetric spacetime ansatz with pure darkmatter in Schwarzschild coordinates can be written asfollows

ds2 = minus f (r)dt2 +dr2

g(r)+ r2

(dθ2 + sin2 θdφ2

) (21)

in which f (r) = g(r) are known as the redshift andshape functions respectively

v2tg (r) =

d lnradic

f (r)d ln r

(22)

we find

f (r)DM = eminusρ2

0 cos( π r2 R )

4Λ2m6 (23)

with the constraint

limρ0rarr0

(eminus

ρ20 cos( π r

2 R )

4Λ2m6

)= 1 (24)

In the standard cold dark matter picture a halo ofmass MDM = 1012 M In addition for m = 06 eVand Λ = 02 meV we find ρ0 = 002 times 10minus24gcm3 =92 times 10minus8eV4 Note that here we have used the conver-sion 10minus19 gcm3 04 eV4

III EFFECT OF BARYONIC MATTER ANDSUPERFLUID DARK MATTER

In a realistic situation galaxies consist of a baryonic(normal) matter (consisting of stars of mass Mstar ion-ized gas of mass Mgas neutral hydrogen of mass MHIetc) the dark matter of mass MDM which we assumeto be in the form of a superfluid The total mass of thegalaxy is therefore MB = Mstar + Mgas + MHI as the to-tal baryonic mass in the galaxy But as we saw by intro-ducing the baryonic matter we must take into account aphonon-mediated force which describes the interactionbetween SFDM and baryonic matter

To calculate the radial acceleration on a test baryonicparticle within the superfluid core this is given by thesum of three contributions [52]

~a(r) =~aB(r) +~aDM(r) +~aphonon(r) (25)

where the first term is just the baryonic Newtonian ac-celeration aB = MB(r)r2 The second term is the grav-itational acceleration from the superfluid core aDM =MDM(r)r2 The third term is the phonon-mediated ac-celeration

~aphonon(r) = αΛ ~nablaψ (26)

The strength of this term is set by α and Λ or equiva-lently the critical acceleration

aphonon(r) =radic

a0aB (27)

which is nothing but the phonon force closely matchesthe deep-MOND acceleration a0 given by Eq(8) Thusin total the centrifugal acceleration of the particle of atest particle moving in the dark halo in spherical sym-metric space-time is given by

v2tg (r)

r=

MDM(r)r2 +

MBM(r)r2 +

radica0

MBM(r)r2 (28)

Outside the size of the superfluid rdquocorerdquo noted as Rthere should be a different phase for DM matter namely

4

it is surrounded by DM particles in the normal phasemost likely described by a Navarro-Frenk-White (NFW)profile or some other effective type dark matter profilemost probably described by the empirical Burkert pro-file (see for example [53ndash59])

However as was argued in [52] we do not expect asharp transition but instead there should be a transitionregion around at which the DM is nearly in equilibriumbut is incapable of maintaining long range coherence Inthe present work we are mostly interested to explore theeffect of the SFDM core and therefore we are going toneglect the outer DM region in the normal phase givenby the NFW profile

A Exponential profile

As a toy model for describing the baryonic distribu-tion we consider a spherical exponential profile of den-sity given by [52]

ρB(r) =MB

8πL3 eminusrL (29)

where the characteristic scale L plays the role of a radialscale length and MB is the baryonic mass In that casethe total mass can be written as a sum of masses of darkmatter and the disc The mass function of the stellar thindisk is written as

MB(r) =MB

[2L2 minus

(2L2 + 2rL + r2) eminus

rL

]2L2 (30)

We observe from these profiles that by going in theoutside region of core ie r gtgt L the mass MB(r) re-duces to a constant MB In that way the last term in(28) becomes independent of r and the empirical TullyndashFisher relation which describes the rotational curves ofgalaxies In our analyses we shall simplify the calcu-lations by considering a fixed baryonic mass MB in thelast term in (28) Let us introduce the quantity

v20 =

radicMBa0 (31)

and use the tangent velocity while assuming a spheri-cally symmetric solution resulting with

f (r)BM+DM asymp C r2 v20 eminus

ρ20 cos( π r

2 R )

4Λ2m6 (32)

times exp[minusMB

rL(2Lminus (2L + r)eminus

rL )

]

valid for r le R In the present work we shall focues onthe HSB type galaxyes such as the Milky Way galaxyWe can use MB = 5minus 7times 1010M and L = 2minus 3 kpc[60]

FIG 1 Schematic representation of the superfluid DM coreand baryonic matter surrounding a black hole at the center

Outside the superfluid core DM exists in a normal phaseprobably described by the NFW profile or Burkert profile

B Power law profile

As a second example we assume the baryonic matterto be concentrated into an inner core of radius rc andthat its mass profile MB(r) can be described by the sim-ple relation [61]

MB(r) = MB

(r

r + rc

)3β

(33)

in the present paper we shall be interested in the caseβ = 1 for high surface brightness galaxies (HSB) Mak-ing use of the tangent velocity and assuming a spheri-cally symmetric solution for the baryonic matter contri-bution in the case of HSB galaxies we find

f (r)BM+DM asymp C(πr

2R

)2 v20 eminusMB(2r+rc)

(r+rc)2 eminusρ2

0 cos( π r2 R )

4Λ2m6 (34)

Note that the above analyses holds only inside the su-perfluid core with r le R

C Totally dominated dark matter galaxies

There is one particular solution of interest which de-scribes a totally dominated dark matter galaxies (TD-DMG) or sometimes known as dark galaxies To do thiswe neglect the baryonic mass MBM rarr 0 and thereforea0 rarr 0 yielding

f (r)TDDMG = C eminusρ2

0 cos( π r2 R )

4Λ2m6 (35)

valid for r le R This result is obtained directly from Eqs(32) and (34) in the limit MB = 0 In other words this so-lution reduces to (23) as expected Note that the integra-tion constant C can be absorbed in the time coordinateusing dt2 rarr C dt2

5

IV BLACK HOLE METRIC SURROUNDED BYBARYONIC MATTER AND SUPERFLUID DARK

MATTER

Let us now consider a more interesting scenario byadding a black hole present in the SFDM halo Toachieve this aim we shall use the method introduced in[6] Namely we need to compute the spacetime metricby assuming the Einstein field equations given by [6]

Rνmicro minus

12

δνmicroR = κ2Tν

micro (36)

where the corresponding energy-momentum tensorsTν

micro = diag[minusρ pr p p] where ρ = ρDM + ρBM encodesthe total contribution coming from the surrounded darkmatter and baryonic matter respectively The space-time metric including black hole is thus given by [6]

ds2 = minus( f (r) + F1(r))dt2 +dr2

g(r) + F2(r)+ r2dΩ2 (37)

where dΩ2 = dθ2 + sin2 θdφ2 is the metric of a unitsphere Moreover we have introduced the followingquantities

F (r) = f (r) + F1(r) G(r) = g(r) + F2(r) (38)

In terms of these relations the space-time metric de-scribing a black hole in dark matter halo becomes [6]

ds2 = minus exp[int g(r)

g(r)minus 2Mr

(1r+

fprime(r)

f (r))drminus 1

rdr]dt2

+ (g(r)minus 2Mr

)minus1dr2 + r2dΩ2 (39)

In the absence of the dark matter halo ie f (r) =g(r) = 1 the indefinite integral reduces to a constant[6]

F1(r) + f (r) = exp[int g(r)

g(r) + F2(r)(

1r+

fprime(r)

f (r))minus 1

rdr]

= 1minus 2Mr

(40)

yielding F1(r) = minus2Mr In that way one can obtain theblack hole space-time metric with a surrounding matter

ds2 = minusF (r)dt2 +dr2

G(r) + r2dΩ2 (41)

Thus the general black hole solution surrounded bySFDM with F (r) = G(r) are given by

F (r)exp = C r2 v20 eminus

ρ20 cos( π r

2 R )

4Λ2m6 (42)

times exp[minusMB

rL(2Lminus (2L + r)eminus

rL )

]minus 2M

r

and

F (r)power = C(πr

2R

)2 v20 eminus

ρ20 cos( π r

2 R )

4Λ2m6 eminusMB(2r+rc)

(r+rc)2 minus 2Mr

(43)

As a special case we obtain a black hole in a totallydominated dark matter galaxy given by

F (r)TDDMG = C eminusρ2

0 cos( π r2 R )

4Λ2m6 minus 2Mr

(44)

valid for r le R Note that M is the black hole mass Inthe limit M = 0 our solution reduces to (23) as expected

V SHADOW OF SGR Alowast BLACK HOLE SURROUNDEDBY SUPERFLUID DARK MATTER

In this section we generalize the static and the spher-ical symmetric black hole solution to a spinning blackhole surrounded by dark matter halo applying theNewman-Janis method and adapting the approach in-troduced in [62 63] Applying the Newman-Janismethod we obtain the spinning black hole metric sur-rounded by dark matter halo as follows (see AppendixA)

ds212 = minus

(1minus 2Υ12(r)r

Σ

)dt2 +

Σ∆12

dr2 + Σdθ2

minus 2a sin2 θ2Υ12(r)r

Σdtdφ

+ sin2 θ

[(r2 + a2)2 minus a2∆12 sin2 θ

Σ

]dφ2 (45)

where we have introduced

Υ12(r) =r (1minusF12(r))

2 (46)

and identified F1 = F (r)exp and F2 = F (r)power re-spectively As a special case we find the Kerr BH limitY12(r) = M if a0 = MB = ρ0 = 0

In addition we can explore the shape of the ergore-gion of our black hole metric (45) that is we can plotthe shape of the ergoregion say in the xz-plane On theother hand the horizons of the black hole are found bysolving ∆12 = 0 te

r2F (r)12 + a2 = 0 (47)

while the inner and outer ergo-surfaces are obtainedfrom solving gtt = 0 ie

r2F (r)12 + a2 cos2 θ = 0 (48)

In Fig 2 we plot ∆12 as a function of r for differentvalues of a In the special case for some critical valuea = aE (blue line) the two horizons coincide in otherwords we have an extremal black hole with degeneratehorizons Beyond this critical value a gt aE there is noevent horizon and the solution corresponds to a nakedsingularity

6

00 05 10 15 20

-05

00

05

10

rM

Δ1

Exponential profile

00 05 10 15 20

-05

00

05

10

rM

Δ2

Power law profile

FIG 2 Left panel Variation of ∆1 as a function of r for BH-SFDM using the spherical exponential profile Right panelVariation of ∆2 as a function of r for BH-SFDM using the power profile law Note that a = 065 (black curve) a = 085 (red

curve) a = 1 (blue curve) respectively We have used m = 06 eV and Λ = 02 meV we find ρ0 = 002 times 10minus24 gcm3 or in eVunits ρ0 = 92 times 10minus8 eV4 For the Milky Way galaxy we can use MB = 6times 1010 M = 139 times 104 MBH and rc = 26

kpc= 124 times 1010 MBH

00 05 10 15 20-10

0

10

20

30

rM

Veff

Exponential profile

00 05 10 15 20-10

0

10

20

30

rM

Veff

Power law profile

FIG 3 Left panel The effective potential of photon moving for BH-SFDM using the exponsntial profile Right panel Theeffective potential of photon using thepower law profile We have chosen a = 05 (black color) a = 065 (red color) and a = 085

(blue color) respectively

A Geodesic equations

In order to find the contour of a black hole shadow ofthe rotating spacetime (45) first we need to find the nullgeodesic equations using the Hamilton-Jacobi methodgiven by

partSpartτ

= minus12

gmicroν partSpartxmicro

partSpartxν

(49)

where τ is the affine parameter and S is the Jacobi ac-tion Due to the spectime symmetries there are two con-served quantities namely the conserved energy E =minuspt and the conserved angular momentum L = pφ re-spectively

To find the separable solution of Eq (49) we need to

express the action in the following form

S =12

micro2τ minus Et + Jφ + Sr(r) + Sθ(θ) (50)

in which micro gives the mass of the test particle Of coursethis gives micro = 0 in the case of the photon That beingsaid it is straightforward to obtain the following equa-

7

-2 -1 0 1 2

-2

-1

0

1

2

rsin(θ)cos(φ)

rcos

(θ)

M=1 a=04

-2 -1 0 1 2

-2

-1

0

1

2

rsin(θ)cos(φ)

rcos

(θ)

M=1 a=065

-2 -1 0 1 2

-2

-1

0

1

2

rsin(θ)cos(φ)

rcos

(θ)

M=1 a=085

-2 -1 0 1 2

-2

-1

0

1

2

rsin(θ)cos(φ)

rcos

(θ)

M=1 a=098

FIG 4 The shape of the ergoregion and innerouter horizons for different values of a using the spherical exponential profileWe use M = 1 in units of the Sgr Alowast black hole mass given by MBH = 43 times 106 M and R = 158 kpc or R = 755 times 1010 MBH Furthermore for m = 06 eV and Λ = 02 meV we find ρ0 = 002 times 10minus24 gcm3 or in eV units ρ0 = 92 times 10minus8 eV4 For the

normal matter contribution of the Milky Way galaxy we can use MB = 6times 1010 M = 139 times 104 MBH and L = 26kpc= 124 times 1010 MBH

tions of motions from the Hamilton-Jacobi equation

Σdtdτ

=r2 + a2

∆12[E(r2 + a2)minus aL]minus a(aE sin2 θ minus L)

(51)

Σdrdτ

=radicR(r) (52)

Σdθ

dτ=radic

Θ(θ) (53)

Σdφ

dτ=

a∆12

[E(r2 + a2)minus aL]minus(

aEminus Lsin2 θ

) (54)

whereR(r) and Θ(θ) are given by

R(r) = [E(r2 + a2)minus aL]2 minus ∆12[m2r2 + (aEminus L)2 +K]

(55)

Θ(θ) = Kminus(

L2

sin2 θminus a2E2

)cos2 θ (56)

and K is the separation constant known as the Carterconstant

8

Kerr vacuum BH

BH-SFDM

-10 -5 0 5-6

-4

-2

0

2

4

6

α

β

a=02

Kerr vacuum BH

BH-SFDM

-10 -5 0 5-6

-4

-2

0

2

4

6

α

β

a=05

Kerr vacuum BH

BH-SFDM

-10 -5 0 5-6

-4

-2

0

2

4

6

α

β

a=085

Kerr vacuum BH

BH-SFDM

-10 -5 0 5-6

-4

-2

0

2

4

6

α

β

a=098

FIG 5 Variation in shape of shadow using the spherical exponential profile for baryonic matter We use M = 1 in units of theSgr Alowast black hole mass given by MBH = 43 times 106 M and R = 158 kpc or R = 755 times 1010 MBH Furthermore for m = 06 eV

and Λ = 02 meV we find ρ0 = 002 times 10minus24 gcm3 or in eV units ρ0 = 92 times 10minus8 eV4 For the Milky Way galaxy we can useMB = 6times 1010 M = 139 times 104 MBH and L = 26 kpc= 124 times 1010 MBH We observe that the dashed blue curve correspondingto Sgr Alowast with the above parameters is almost indistinguishable from the black dashed curve describing the Kerr vacuum BHThe solid blue curve corresponds to the case of increasing the baryonic mass by a factor of 102 and decrease of core radius by a

factor of 103

B Circular Orbits

Due to the strong gravity near the black hole it is thusexpected that the photons emitted near a black hole willeventually fall into the black hole or eventually scatteraway from it In this way the photons captured by theblack hole will form a dark region defining the contourof the shadow To elaborate the presence of unstablecircular orbits around the black hole we need to studythe radial geodesic by introducing the effective poten-tial Veff which can be written as follows

Σ2(

drdτ

)2+ V12

eff = 0 (57)

where V1eff = Vexp

eff and V2eff = Vpower

eff gives the expo-nential and the power law respectively At this point itis convenient to introduce two parameters ξ and η de-fined as

ξ = LE η = KE2 (58)

For the effective potential then we obtain the followingrelation

V12eff = ∆12((aminus ξ12)

2 + η12)minus (r2 + a2minus a ξ12)2 (59)

where we have replaced V12eff E2 by V12

eff For more de-tails in Figure (3) we plot the variation of the effectivepotential associated with the radial motion of photons

9

Kerr vacuum BH

BH-SFDM

-10 -5 0 5-6

-4

-2

0

2

4

6

α

β

a=02

Kerr vacuum BH

BH-SFDM

-10 -5 0 5-6

-4

-2

0

2

4

6

α

β

a=05

Kerr vacuum BH

BH-SFDM

-10 -5 0 5-6

-4

-2

0

2

4

6

α

β

a=085

Kerr vacuum BH

BH-SFDM

-10 -5 0 5-6

-4

-2

0

2

4

6

α

β

a=098

FIG 6 Variation in shape of shadow for different values of a using the power law profile We use M = 1 in units of the Sgr Alowast

black hole mass given by MBH = 43 times 106 M and R = 158 kpc or R = 755 times 1010 MBH Furthermore for m = 06 eV andΛ = 02 meV we find ρ0 = 002 times 10minus24 gcm3 or in eV units ρ0 = 92 times 10minus8 eV4 For the Milky Way galaxy we can use

MB = 6times 1010 M = 139 times 104 MBH and rc = 26 kpc= 124 times 1010 MBH The dashed red curve corresponds to Sgr Alowast usingthe above parameters which is almost indistinguishable from the black dashed curve discribing the Kerr vacuum BH The solid

red curve corresponds to the case of increasing the baryonic mass by a factor of 102 and decrease of core radius by a factor of 103

The circular photon orbits exists when at some constantr = rcir the conditions

V12eff (r) = 0

dV12eff (r)dr

= 0 (60)

Combining all these equations it is possible to showthat

ξ12 =(r2 + a2)(rF prime12(r) + 2F12(r))minus 4(r2F12(r) + a2)

a(rF prime12(r) + 2F12(r))

η12 =r3[8a2F prime12(r)minus r(rF prime12(r)minus 2F12(r))2]

a2(rF prime12(r) + 2F12(r))2 (61)

One can recover the Kerr vacuum case by letting a0 =MB = ρ0 = 0 yielding

ξ12 =r2(3Mminus r)minus a2(Mminus r)

a(rminusM) (62)

η12 =r3 (4Ma2 minus r(rminus 3M)2)

a2(rminusM)2 (63)

To obtain the shadow images of our black hole in thepresence of dark matter we assume that the observer islocated at the position with coordinates (ro θo) where roand θo represents the angular coordinate on observerrsquossky Furthermore we need to introduce two celestial co-ordinates α and β for the observer by using the follow-ing relations [29]

α = minusrop(φ)

p(t) β = ro

p(θ)

p(t) (64)

where (p(t) p(r) p(θ) p(φ)) are the tetrad componentsof the photon momentum with respect to locally non-rotating reference frame The observer bases emicro

(ν)can be

10

expanded as a form in the coordinate bases (see [64])

p(t) = minusemicro

(t)pmicro = E ζ12 minus γ12 L p(φ) = emicro

(φ)pmicro =

Lradicgφφ

p(θ) = emicro

(θ)pmicro =

pθradicgθθ

p(r) = emicro

(r)pmicro =prradicgrr

(65)

where the quantities E = minuspt and pφ = L are conserveddue to the associated Killing vectors If we use ξ = LEη = KE2 and pθ = plusmn

radicΘ(θ) we can rewrite these co-

ordinates in terms our parameters ξ and η as follows[43]

α12 = minusroξ12radicgφφ(ζ12 minus γ12ξ12)

|(ro θo)

β12 = plusmnro

radicη12 + a2 cos2 θ minus ξ2

12 cot2 θradic

gθθ(ζ12 minus γ12ξ12)|(ro θo) (66)

where

ζ12 =

radicgφφ

g2tφ minus gttgφφ

(67)

and

γ12 = minusgtφ

gφφζ12 (68)

Finally to simplify the problem further we are going toconsider that our observer is located in the equatorialplane (θ = π2) and very large but finite ro = D = 83kpc we find

α12 = minusradic

f12 ξ12

β12 = plusmnradic

f12radic

η12 (69)

Note that f12 are given by Eqs (32) and (34) respec-tively We see that due to the presence of SFDM our solu-tion is non-asymptotically flat In the special case whenSFDM is absent we recover f12 rarr 1 hence the aboverelations reduces to the asymptotically flat case Notethat for the observer located at the position with coor-dinates (ro θo) we have neglected the effect of rotationwhile ξ12 and η12 given by Eq (61) and are evaluatedat the circular photon orbits r = rcir

For more details in Figs 5 and 6 we plot the shape ofSgr Alowast black hole shadows Using the parameter valuesspecifying the SFDM and baryonic matter we obtain al-most no effect on the shadow images In fact one canobserve that the dashed curve (exponential profile) anddashed red curve (power law profile) are almost indis-tinguishable from the black dashed curve (Kerr vacuumBH) However we can observe that by increasing thebaryonic mass the shadow size increases considerably(solid bluered curve) The angular radius of the Sgr Alowastblack hole shadow can be estimated using the observ-able Rs as θs = Rs MD where M is the black hole mass

5196 5197 5198 5199 5200

5315

5316

5317

5318

Rs

θs [μas]

FIG 7 The expected value for the angular diameter in thecase of Sgr Alowast Note that we have adopted D = 83 kpc and

M = 43times 106M

and D is the distance between the black hole and theobserver The angular radius can be further expressedas θs = 987098times 10minus6Rs(MM)(1kpcD) microas In thecase of Sgr Alowast we have used M = 43 times 106M andD = 83 kpc [4] is the distance between the Earth andSgr Alowast center black hole In order to get more informa-tion about the effect of SFDM on physical observableswe shall estimate the effect of SFDM on the angular ra-dius for Sgr Alowast To simplify the problem let us considerour static and spherically symmetric black hole metricin a totally dominated dark matter galaxy described bythe function

F (r) = eminusX minus 2Mr

(70)

where we have introduced

X =ρ2

04Λ2m6 (71)

where we have used the approximation cos( π r2 R ) 1

since our solution is valid for r le R From the circularphoton orbit conditions one can show [41]

2minus rF prime(r)F (r) = 0 (72)

Now if we expand around X in the above function weobtain

F (r) = 1minus Xminus 2Mr

+ (73)

By solving this equation and considering only the lead-ing order terms one can determine the radius of the pho-ton sphere rps yielding

rps = 3M(1 + X) (74)

On the other hand one also can show the relation [41]

ξ2 + η =r2

ps

F (rps) (75)

11

The shadow radius Rs can be expressed in terms of thecelestial coordinates (α β) as follows

Rs =radic

α2 + β2 =radic

1minus XrpsradicF (rps)

(76)

Note that in the last equation we have used Eqs (69) and(75) Using the values m = 06 eV Λ = 02 meV andρ0 = 92 times 10minus8 eV4 in units of the Sgr Alowast black hole wefind Rs = 5196158313M Using this result we can findthat the angular diameter increases by δθs = 3times 10minus5

microas compared to the Schwarzschild vacuum This resultis consistent with the result reported in Ref [4] whereauthors estimated that the dark matter halo could influ-ence the shadow of Sgr Alowast at a level of order of magni-tude of 10minus3 microas and 10minus5 microas respectively They haveused the so called Cold Dark Matter and Scalar FieldDark Matter models to study the apparent shapes of theshadow A similar result has been obtained for the darkmatter effect on the M87 black hole (see [5]) using theBurkert dark matter profile

VI CONCLUSION

In this paper we have obtained a rotating black holesolution surrounded by superfluid dark matter alongwith baryonic matter To achieve this purpose in thepresent work we considered the superfluid dark mat-ter model and used two specific profiles to describe thebaryonic matter distribution namely the spherical ex-ponential profile and the power law profile followed bythe special case of a totally dominated dark matter caseUsing the current values for the baryonic mass centraldensity of the superfluid dark matter halo radius aswell as the radial scale length for the baryonic matterin our galaxy we found that the shadow size of Sgr Alowastblack hole remains almost unchanged compared to theKerr vacuum BH This result is consistent with a recentwork reported in [5] and also [4] For entirely dominateddark matter galaxies we find that the angular diame-ter increases by 10minus5microarcsec This result shows that itis very difficult to constrain the dark matter parametersusing the shadow images Such tiny effects on the angu-lar diameter are out of reach for the existing space tech-nology and it remains an open question if future astro-nomical observations can potentially detect such effectsThat being said the expected value for the angular di-ameter in the case of Sgr Alowast is of the order of θs 53 microasas predicted in GR

As an interesting observation we show that an in-crease of baryonic mass followed by a decrease of theradial scale length can increase the shadow size consid-erably This can be explained by the fact that the bary-onic matter is mostly located in the interior of the galaxyon the other hand dark matter is mostly located in theouter region of the galaxy In other words for the to-tally dominated dark matter galaxies we observe almost

no effect on black hole shadows but a more precise mea-surement of the Sgr Alowast shadow radius can play a signifi-cant role in determining the baryonic mass in our galaxy

Acknowledgements

TZ is supported by National Natural Science Founda-tion of China under the Grants No 11675143 the Zhe-jiang Provincial Natural Science Foundation of Chinaunder Grant No LY20A050002 and the FundamentalResearch Funds for the Provincial Universities of Zhe-jiang in China under Grants No RF-A2019015

Appendix A

Here we demonstrate the derivation of the spinningblack hole metric As a first step to this formalism wetransform Boyer-Lindquist (BL) coordinates (t r θ φ) toEddington-Finkelstein (EF) coordinates (u r θ φ) Thiscan be achieved by using the coordinate transformation

dt = du +drradic

F12(r)G12(r) (A1)

we obtain

ds212 = minusF12(r)du2minus 2

radicF12(r)G12(r)

dudr+ r2dθ2 + r2 sin2 θdφ2

(A2)This metric can be expressed in terms of null tetrads

as

gmicroν = minuslmicronν minus lνnmicro + mmicromν + mνmmicro (A3)

where the null tetrads are defined as

lmicro = δmicror (A4)

nmicro =

radicG12(r)F12(r)

δmicrou minus

12G12(r)δ

micror (A5)

mmicro =1radic2H

(δ

microθ +

ι

sin θδ

microφ

) (A6)

These null tetrads are constructed in such a way that lmicro

and nmicro while mmicro and mmicro are complex It is worth notingthatH = r2 and will be used later on It is obvious fromthe notation that mmicro is complex conjugate of mmicro Thesevectors further satisfy the conditions for normalizationorthogonality and isotropy as

lmicrolmicro = nmicronmicro = mmicrommicro = mmicrommicro = 0 (A7)lmicrommicro = lmicrommicro = nmicrommicro = nmicrommicro = 0 (A8)

minuslmicronmicro = mmicrommicro = 1 (A9)

12

Following the NewmanndashJanis prescription we write

xprimemicro = xmicro + ia(δmicror minus δ

microu ) cos θ rarr

uprime = uminus ia cos θrprime = r + ia cos θθprime = θφprime = φ

(A10)in which a stands for the rotation parameter Nextlet the null tetrad vectors Za undergo a transformationgiven by Zmicro = (partxmicropartxprimeν)Zprimeν following [62]

lprimemicro = δmicror (A11)

nprimemicro =

radicB12

A12δ

microu minus

12

B12δmicror

mprimemicro =1radic

2Σ12

[(δ

microu minus δ

micror )ιa sin θ + δ

microθ +

ι

sin θδ

microφ

]where we have assumed that (F12(r)G12(r)H(r))transform to (A12(a r θ) B12(a r θ) Σ12(a r θ)) Withthe help of the above equations new metric inEddington-Finkelste coordinate reads

ds212 = minusA12du2 minus 2

radicA12

B12dudr (A12)

+ 2a sin2 θ

(A12 minus

radicA12

B12

)dudφ

+ 2a

radicA12

B12sin2 θdrdφ + Σ12dθ2

+ sin2 θ

[Σ12 + a2 sin2 θ

(2

radicA12

B12minus A12

)]dφ2

Note that A12 is some function of r and θ as we al-ready pointed out Without going into details of the cal-culation one can revert the EF coordinates back to BLcoordinates by using the following transformation (see

for more details [62])

du = dtminus a2 + r2

∆12dr dφ = dφprime minus a

∆12dr (A13)

where in order to simplify the notation we introduce

∆(r)12 = r2 F12(r) + a2 (A14)

Making use of F12(r) = G12(r) one can obtain Σ1 =Σ2 = r2 + a2 cos2 θ herein we shall use just Σ Droppingthe primes in the φ coordinate and choosing

A12 =(F12H+ a2 cos2 θ)Σ12

(H+ a2 cos2 θ)2 (A15)

and

B12 =F12H+ a2 cos2 θ

Σ (A16)

we obtain the spinning black hole space-time metric ina SFDM halo

ds212 = minus

(1minus 2Υ12(r)r

Σ

)dt2 +

Σ∆12

dr2 + Σdθ2

minus 2a sin2 θ2Υ12(r)r

Σdtdφ

+ sin2 θ

[(r2 + a2)2 minus a2∆12 sin2 θ

Σ

]dφ2 (A17)

with

Υ12(r) =r (1minusF12(r))

2 (A18)

in which F1 = F (r)exp and F2 = F (r)power respec-tively

[1] The Event Horizon Telescope Collaboration Ap J Lett875 L1 (2019)

[2] Haroon S Jamil M Jusufi K Lin K Mann R B PhysRev D 99 no4 044015 (2019)

[3] Haroon S Jusufi K Jamil M Universe 6(2) 23 (2020)arXiv190400711[gr-qc]

[4] Hou X Xu Z Zhou M Wang J JCAP 1807 no07 015(2018)

[5] Jusufi K Jamil M Salucci P Zhu T Haroon S PhysRev D 100 044012 (2019)

[6] Xu Z Hou X Gong X Wang J JCAP 1809 no09 038(2018)

[7] Xu Z Hou X Wang J JCAP 1810 no10 046 (2018)[8] Berezhiani L Khoury J Phys Rev D 92 103510 (2015)[9] Synge J L Mon Not Roy Astron Soc 131 no 3 463

(1966)

[10] Luminet J P Astron Astrophys 75 228 (1979)[11] Bardeen J M in Black Holes (Proceedings Les Houches

France August 1972) edited by C DeWitt and B S De-Witt

[12] Abdujabbarov A Toshmatov B Stuchlik Z AhmedovB Int J Mod Phys D 26 no 06 1750051 (2016)

[13] Abdikamalov A B Abdujabbarov A A Ayzenberg DMalafarina D Bambi C Ahmedov B Phys Rev D 100(2019) no2 024014

[14] Abdujabbarov A Amir M Ahmedov B Ghosh S GPhys Rev D 93 no 10 104004 (2016)

[15] Amir M Jusufi K Banerjee A Hansraj S ClassQuant Grav 36 no21 215007 (2019)

[16] Amir M Ghosh S G Phys Rev D 94 no 2 024054(2016)

[17] Abdujabbarov A Atamurotov F Kucukakca Y Ahme-

3

The last equation determines the size of the condensate

R =

radicρ0

32πΛ2m6 ξ1 (13)

A simple analytical form that provides a good fit is

Ξ(ξ) = cos(

π

2ξ

ξ1

) (14)

The central density is related to the mass of the halo con-densate as follows

ρ0 =M

4πR3ξ1

|Ξprime(ξ1)| (15)

From the numerics we find Ξprime(ξ1) minus05 Substitut-ing (13) in Eq (15) we can solve for the central density

ρ0 (

MDM

1012M

)25 ( meV

)185(

ΛmeV

)6510minus24 gcm3

(16)Meanwhile the halo radius is

R (

MDM

1012M

)15 ( meV

)minus65(

ΛmeV

)minus2545 kpc

(17)Remarkably for m sim eV and Λ sim meV we obtain

DM halos of realistic size The mass profile of the darkmatter galactic halo is given by

MDM (r) = 4πint r

0ρDM

(rprime)

rprime2drprime (18)

To solve the last integral we use Eqs (10)-(14) and byrewriting the mass profile in terms of the new coordi-nate ξ In particular we obtain

MDM (r) = sin(πr

2R

) πr2

2Rρ2

08Λ2m6 r le R (19)

From the last equation one can find the tangential ve-locity v2

tg (r) = MDM(r)r of a test particle moving inthe dark halo in spherical symmetric space-time

v2tg (r) = sin(

πr2R

)πr2R

ρ20

8Λ2m6 (20)

In this section we derive the space-time geometry forpure dark matter To do so let us consider a static andspherically symmetric spacetime ansatz with pure darkmatter in Schwarzschild coordinates can be written asfollows

ds2 = minus f (r)dt2 +dr2

g(r)+ r2

(dθ2 + sin2 θdφ2

) (21)

in which f (r) = g(r) are known as the redshift andshape functions respectively

v2tg (r) =

d lnradic

f (r)d ln r

(22)

we find

f (r)DM = eminusρ2

0 cos( π r2 R )

4Λ2m6 (23)

with the constraint

limρ0rarr0

(eminus

ρ20 cos( π r

2 R )

4Λ2m6

)= 1 (24)

In the standard cold dark matter picture a halo ofmass MDM = 1012 M In addition for m = 06 eVand Λ = 02 meV we find ρ0 = 002 times 10minus24gcm3 =92 times 10minus8eV4 Note that here we have used the conver-sion 10minus19 gcm3 04 eV4

III EFFECT OF BARYONIC MATTER ANDSUPERFLUID DARK MATTER

In a realistic situation galaxies consist of a baryonic(normal) matter (consisting of stars of mass Mstar ion-ized gas of mass Mgas neutral hydrogen of mass MHIetc) the dark matter of mass MDM which we assumeto be in the form of a superfluid The total mass of thegalaxy is therefore MB = Mstar + Mgas + MHI as the to-tal baryonic mass in the galaxy But as we saw by intro-ducing the baryonic matter we must take into account aphonon-mediated force which describes the interactionbetween SFDM and baryonic matter

To calculate the radial acceleration on a test baryonicparticle within the superfluid core this is given by thesum of three contributions [52]

~a(r) =~aB(r) +~aDM(r) +~aphonon(r) (25)

where the first term is just the baryonic Newtonian ac-celeration aB = MB(r)r2 The second term is the grav-itational acceleration from the superfluid core aDM =MDM(r)r2 The third term is the phonon-mediated ac-celeration

~aphonon(r) = αΛ ~nablaψ (26)

The strength of this term is set by α and Λ or equiva-lently the critical acceleration

aphonon(r) =radic

a0aB (27)

which is nothing but the phonon force closely matchesthe deep-MOND acceleration a0 given by Eq(8) Thusin total the centrifugal acceleration of the particle of atest particle moving in the dark halo in spherical sym-metric space-time is given by

v2tg (r)

r=

MDM(r)r2 +

MBM(r)r2 +

radica0

MBM(r)r2 (28)

Outside the size of the superfluid rdquocorerdquo noted as Rthere should be a different phase for DM matter namely

4

it is surrounded by DM particles in the normal phasemost likely described by a Navarro-Frenk-White (NFW)profile or some other effective type dark matter profilemost probably described by the empirical Burkert pro-file (see for example [53ndash59])

However as was argued in [52] we do not expect asharp transition but instead there should be a transitionregion around at which the DM is nearly in equilibriumbut is incapable of maintaining long range coherence Inthe present work we are mostly interested to explore theeffect of the SFDM core and therefore we are going toneglect the outer DM region in the normal phase givenby the NFW profile

A Exponential profile

As a toy model for describing the baryonic distribu-tion we consider a spherical exponential profile of den-sity given by [52]

ρB(r) =MB

8πL3 eminusrL (29)

where the characteristic scale L plays the role of a radialscale length and MB is the baryonic mass In that casethe total mass can be written as a sum of masses of darkmatter and the disc The mass function of the stellar thindisk is written as

MB(r) =MB

[2L2 minus

(2L2 + 2rL + r2) eminus

rL

]2L2 (30)

We observe from these profiles that by going in theoutside region of core ie r gtgt L the mass MB(r) re-duces to a constant MB In that way the last term in(28) becomes independent of r and the empirical TullyndashFisher relation which describes the rotational curves ofgalaxies In our analyses we shall simplify the calcu-lations by considering a fixed baryonic mass MB in thelast term in (28) Let us introduce the quantity

v20 =

radicMBa0 (31)

and use the tangent velocity while assuming a spheri-cally symmetric solution resulting with

f (r)BM+DM asymp C r2 v20 eminus

ρ20 cos( π r

2 R )

4Λ2m6 (32)

times exp[minusMB

rL(2Lminus (2L + r)eminus

rL )

]

valid for r le R In the present work we shall focues onthe HSB type galaxyes such as the Milky Way galaxyWe can use MB = 5minus 7times 1010M and L = 2minus 3 kpc[60]

FIG 1 Schematic representation of the superfluid DM coreand baryonic matter surrounding a black hole at the center

Outside the superfluid core DM exists in a normal phaseprobably described by the NFW profile or Burkert profile

B Power law profile

As a second example we assume the baryonic matterto be concentrated into an inner core of radius rc andthat its mass profile MB(r) can be described by the sim-ple relation [61]

MB(r) = MB

(r

r + rc

)3β

(33)

in the present paper we shall be interested in the caseβ = 1 for high surface brightness galaxies (HSB) Mak-ing use of the tangent velocity and assuming a spheri-cally symmetric solution for the baryonic matter contri-bution in the case of HSB galaxies we find

f (r)BM+DM asymp C(πr

2R

)2 v20 eminusMB(2r+rc)

(r+rc)2 eminusρ2

0 cos( π r2 R )

4Λ2m6 (34)

Note that the above analyses holds only inside the su-perfluid core with r le R

C Totally dominated dark matter galaxies

There is one particular solution of interest which de-scribes a totally dominated dark matter galaxies (TD-DMG) or sometimes known as dark galaxies To do thiswe neglect the baryonic mass MBM rarr 0 and thereforea0 rarr 0 yielding

f (r)TDDMG = C eminusρ2

0 cos( π r2 R )

4Λ2m6 (35)

valid for r le R This result is obtained directly from Eqs(32) and (34) in the limit MB = 0 In other words this so-lution reduces to (23) as expected Note that the integra-tion constant C can be absorbed in the time coordinateusing dt2 rarr C dt2

5

IV BLACK HOLE METRIC SURROUNDED BYBARYONIC MATTER AND SUPERFLUID DARK

MATTER

Let us now consider a more interesting scenario byadding a black hole present in the SFDM halo Toachieve this aim we shall use the method introduced in[6] Namely we need to compute the spacetime metricby assuming the Einstein field equations given by [6]

Rνmicro minus

12

δνmicroR = κ2Tν

micro (36)

where the corresponding energy-momentum tensorsTν

micro = diag[minusρ pr p p] where ρ = ρDM + ρBM encodesthe total contribution coming from the surrounded darkmatter and baryonic matter respectively The space-time metric including black hole is thus given by [6]

ds2 = minus( f (r) + F1(r))dt2 +dr2

g(r) + F2(r)+ r2dΩ2 (37)

where dΩ2 = dθ2 + sin2 θdφ2 is the metric of a unitsphere Moreover we have introduced the followingquantities

F (r) = f (r) + F1(r) G(r) = g(r) + F2(r) (38)

In terms of these relations the space-time metric de-scribing a black hole in dark matter halo becomes [6]

ds2 = minus exp[int g(r)

g(r)minus 2Mr

(1r+

fprime(r)

f (r))drminus 1

rdr]dt2

+ (g(r)minus 2Mr

)minus1dr2 + r2dΩ2 (39)

In the absence of the dark matter halo ie f (r) =g(r) = 1 the indefinite integral reduces to a constant[6]

F1(r) + f (r) = exp[int g(r)

g(r) + F2(r)(

1r+

fprime(r)

f (r))minus 1

rdr]

= 1minus 2Mr

(40)

yielding F1(r) = minus2Mr In that way one can obtain theblack hole space-time metric with a surrounding matter

ds2 = minusF (r)dt2 +dr2

G(r) + r2dΩ2 (41)

Thus the general black hole solution surrounded bySFDM with F (r) = G(r) are given by

F (r)exp = C r2 v20 eminus

ρ20 cos( π r

2 R )

4Λ2m6 (42)

times exp[minusMB

rL(2Lminus (2L + r)eminus

rL )

]minus 2M

r

and

F (r)power = C(πr

2R

)2 v20 eminus

ρ20 cos( π r

2 R )

4Λ2m6 eminusMB(2r+rc)

(r+rc)2 minus 2Mr

(43)

As a special case we obtain a black hole in a totallydominated dark matter galaxy given by

F (r)TDDMG = C eminusρ2

0 cos( π r2 R )

4Λ2m6 minus 2Mr

(44)

valid for r le R Note that M is the black hole mass Inthe limit M = 0 our solution reduces to (23) as expected

V SHADOW OF SGR Alowast BLACK HOLE SURROUNDEDBY SUPERFLUID DARK MATTER

In this section we generalize the static and the spher-ical symmetric black hole solution to a spinning blackhole surrounded by dark matter halo applying theNewman-Janis method and adapting the approach in-troduced in [62 63] Applying the Newman-Janismethod we obtain the spinning black hole metric sur-rounded by dark matter halo as follows (see AppendixA)

ds212 = minus

(1minus 2Υ12(r)r

Σ

)dt2 +

Σ∆12

dr2 + Σdθ2

minus 2a sin2 θ2Υ12(r)r

Σdtdφ

+ sin2 θ

[(r2 + a2)2 minus a2∆12 sin2 θ

Σ

]dφ2 (45)

where we have introduced

Υ12(r) =r (1minusF12(r))

2 (46)

and identified F1 = F (r)exp and F2 = F (r)power re-spectively As a special case we find the Kerr BH limitY12(r) = M if a0 = MB = ρ0 = 0

In addition we can explore the shape of the ergore-gion of our black hole metric (45) that is we can plotthe shape of the ergoregion say in the xz-plane On theother hand the horizons of the black hole are found bysolving ∆12 = 0 te

r2F (r)12 + a2 = 0 (47)

while the inner and outer ergo-surfaces are obtainedfrom solving gtt = 0 ie

r2F (r)12 + a2 cos2 θ = 0 (48)

In Fig 2 we plot ∆12 as a function of r for differentvalues of a In the special case for some critical valuea = aE (blue line) the two horizons coincide in otherwords we have an extremal black hole with degeneratehorizons Beyond this critical value a gt aE there is noevent horizon and the solution corresponds to a nakedsingularity

6

00 05 10 15 20

-05

00

05

10

rM

Δ1

Exponential profile

00 05 10 15 20

-05

00

05

10

rM

Δ2

Power law profile

FIG 2 Left panel Variation of ∆1 as a function of r for BH-SFDM using the spherical exponential profile Right panelVariation of ∆2 as a function of r for BH-SFDM using the power profile law Note that a = 065 (black curve) a = 085 (red

curve) a = 1 (blue curve) respectively We have used m = 06 eV and Λ = 02 meV we find ρ0 = 002 times 10minus24 gcm3 or in eVunits ρ0 = 92 times 10minus8 eV4 For the Milky Way galaxy we can use MB = 6times 1010 M = 139 times 104 MBH and rc = 26

kpc= 124 times 1010 MBH

00 05 10 15 20-10

0

10

20

30

rM

Veff

Exponential profile

00 05 10 15 20-10

0

10

20

30

rM

Veff

Power law profile

FIG 3 Left panel The effective potential of photon moving for BH-SFDM using the exponsntial profile Right panel Theeffective potential of photon using thepower law profile We have chosen a = 05 (black color) a = 065 (red color) and a = 085

(blue color) respectively

A Geodesic equations

In order to find the contour of a black hole shadow ofthe rotating spacetime (45) first we need to find the nullgeodesic equations using the Hamilton-Jacobi methodgiven by

partSpartτ

= minus12

gmicroν partSpartxmicro

partSpartxν

(49)

where τ is the affine parameter and S is the Jacobi ac-tion Due to the spectime symmetries there are two con-served quantities namely the conserved energy E =minuspt and the conserved angular momentum L = pφ re-spectively

To find the separable solution of Eq (49) we need to

express the action in the following form

S =12

micro2τ minus Et + Jφ + Sr(r) + Sθ(θ) (50)

in which micro gives the mass of the test particle Of coursethis gives micro = 0 in the case of the photon That beingsaid it is straightforward to obtain the following equa-

7

-2 -1 0 1 2

-2

-1

0

1

2

rsin(θ)cos(φ)

rcos

(θ)

M=1 a=04

-2 -1 0 1 2

-2

-1

0

1

2

rsin(θ)cos(φ)

rcos

(θ)

M=1 a=065

-2 -1 0 1 2

-2

-1

0

1

2

rsin(θ)cos(φ)

rcos

(θ)

M=1 a=085

-2 -1 0 1 2

-2

-1

0

1

2

rsin(θ)cos(φ)

rcos

(θ)

M=1 a=098

FIG 4 The shape of the ergoregion and innerouter horizons for different values of a using the spherical exponential profileWe use M = 1 in units of the Sgr Alowast black hole mass given by MBH = 43 times 106 M and R = 158 kpc or R = 755 times 1010 MBH Furthermore for m = 06 eV and Λ = 02 meV we find ρ0 = 002 times 10minus24 gcm3 or in eV units ρ0 = 92 times 10minus8 eV4 For the

normal matter contribution of the Milky Way galaxy we can use MB = 6times 1010 M = 139 times 104 MBH and L = 26kpc= 124 times 1010 MBH

tions of motions from the Hamilton-Jacobi equation

Σdtdτ

=r2 + a2

∆12[E(r2 + a2)minus aL]minus a(aE sin2 θ minus L)

(51)

Σdrdτ

=radicR(r) (52)

Σdθ

dτ=radic

Θ(θ) (53)

Σdφ

dτ=

a∆12

[E(r2 + a2)minus aL]minus(

aEminus Lsin2 θ

) (54)

whereR(r) and Θ(θ) are given by

R(r) = [E(r2 + a2)minus aL]2 minus ∆12[m2r2 + (aEminus L)2 +K]

(55)

Θ(θ) = Kminus(

L2

sin2 θminus a2E2

)cos2 θ (56)

and K is the separation constant known as the Carterconstant

8

Kerr vacuum BH

BH-SFDM

-10 -5 0 5-6

-4

-2

0

2

4

6

α

β

a=02

Kerr vacuum BH

BH-SFDM

-10 -5 0 5-6

-4

-2

0

2

4

6

α

β

a=05

Kerr vacuum BH

BH-SFDM

-10 -5 0 5-6

-4

-2

0

2

4

6

α

β

a=085

Kerr vacuum BH

BH-SFDM

-10 -5 0 5-6

-4

-2

0

2

4

6

α

β

a=098

FIG 5 Variation in shape of shadow using the spherical exponential profile for baryonic matter We use M = 1 in units of theSgr Alowast black hole mass given by MBH = 43 times 106 M and R = 158 kpc or R = 755 times 1010 MBH Furthermore for m = 06 eV

and Λ = 02 meV we find ρ0 = 002 times 10minus24 gcm3 or in eV units ρ0 = 92 times 10minus8 eV4 For the Milky Way galaxy we can useMB = 6times 1010 M = 139 times 104 MBH and L = 26 kpc= 124 times 1010 MBH We observe that the dashed blue curve correspondingto Sgr Alowast with the above parameters is almost indistinguishable from the black dashed curve describing the Kerr vacuum BHThe solid blue curve corresponds to the case of increasing the baryonic mass by a factor of 102 and decrease of core radius by a

factor of 103

B Circular Orbits

Due to the strong gravity near the black hole it is thusexpected that the photons emitted near a black hole willeventually fall into the black hole or eventually scatteraway from it In this way the photons captured by theblack hole will form a dark region defining the contourof the shadow To elaborate the presence of unstablecircular orbits around the black hole we need to studythe radial geodesic by introducing the effective poten-tial Veff which can be written as follows

Σ2(

drdτ

)2+ V12

eff = 0 (57)

where V1eff = Vexp

eff and V2eff = Vpower

eff gives the expo-nential and the power law respectively At this point itis convenient to introduce two parameters ξ and η de-fined as

ξ = LE η = KE2 (58)

For the effective potential then we obtain the followingrelation

V12eff = ∆12((aminus ξ12)

2 + η12)minus (r2 + a2minus a ξ12)2 (59)

where we have replaced V12eff E2 by V12

eff For more de-tails in Figure (3) we plot the variation of the effectivepotential associated with the radial motion of photons

9

Kerr vacuum BH

BH-SFDM

-10 -5 0 5-6

-4

-2

0

2

4

6

α

β

a=02

Kerr vacuum BH

BH-SFDM

-10 -5 0 5-6

-4

-2

0

2

4

6

α

β

a=05

Kerr vacuum BH

BH-SFDM

-10 -5 0 5-6

-4

-2

0

2

4

6

α

β

a=085

Kerr vacuum BH

BH-SFDM

-10 -5 0 5-6

-4

-2

0

2

4

6

α

β

a=098

FIG 6 Variation in shape of shadow for different values of a using the power law profile We use M = 1 in units of the Sgr Alowast

black hole mass given by MBH = 43 times 106 M and R = 158 kpc or R = 755 times 1010 MBH Furthermore for m = 06 eV andΛ = 02 meV we find ρ0 = 002 times 10minus24 gcm3 or in eV units ρ0 = 92 times 10minus8 eV4 For the Milky Way galaxy we can use

MB = 6times 1010 M = 139 times 104 MBH and rc = 26 kpc= 124 times 1010 MBH The dashed red curve corresponds to Sgr Alowast usingthe above parameters which is almost indistinguishable from the black dashed curve discribing the Kerr vacuum BH The solid

red curve corresponds to the case of increasing the baryonic mass by a factor of 102 and decrease of core radius by a factor of 103

The circular photon orbits exists when at some constantr = rcir the conditions

V12eff (r) = 0

dV12eff (r)dr

= 0 (60)

Combining all these equations it is possible to showthat

ξ12 =(r2 + a2)(rF prime12(r) + 2F12(r))minus 4(r2F12(r) + a2)

a(rF prime12(r) + 2F12(r))

η12 =r3[8a2F prime12(r)minus r(rF prime12(r)minus 2F12(r))2]

a2(rF prime12(r) + 2F12(r))2 (61)

One can recover the Kerr vacuum case by letting a0 =MB = ρ0 = 0 yielding

ξ12 =r2(3Mminus r)minus a2(Mminus r)

a(rminusM) (62)

η12 =r3 (4Ma2 minus r(rminus 3M)2)

a2(rminusM)2 (63)

To obtain the shadow images of our black hole in thepresence of dark matter we assume that the observer islocated at the position with coordinates (ro θo) where roand θo represents the angular coordinate on observerrsquossky Furthermore we need to introduce two celestial co-ordinates α and β for the observer by using the follow-ing relations [29]

α = minusrop(φ)

p(t) β = ro

p(θ)

p(t) (64)

where (p(t) p(r) p(θ) p(φ)) are the tetrad componentsof the photon momentum with respect to locally non-rotating reference frame The observer bases emicro

(ν)can be

10

expanded as a form in the coordinate bases (see [64])

p(t) = minusemicro

(t)pmicro = E ζ12 minus γ12 L p(φ) = emicro

(φ)pmicro =

Lradicgφφ

p(θ) = emicro

(θ)pmicro =

pθradicgθθ

p(r) = emicro

(r)pmicro =prradicgrr

(65)

where the quantities E = minuspt and pφ = L are conserveddue to the associated Killing vectors If we use ξ = LEη = KE2 and pθ = plusmn

radicΘ(θ) we can rewrite these co-

ordinates in terms our parameters ξ and η as follows[43]

α12 = minusroξ12radicgφφ(ζ12 minus γ12ξ12)

|(ro θo)

β12 = plusmnro

radicη12 + a2 cos2 θ minus ξ2

12 cot2 θradic

gθθ(ζ12 minus γ12ξ12)|(ro θo) (66)

where

ζ12 =

radicgφφ

g2tφ minus gttgφφ

(67)

and

γ12 = minusgtφ

gφφζ12 (68)

Finally to simplify the problem further we are going toconsider that our observer is located in the equatorialplane (θ = π2) and very large but finite ro = D = 83kpc we find

α12 = minusradic

f12 ξ12

β12 = plusmnradic

f12radic

η12 (69)

Note that f12 are given by Eqs (32) and (34) respec-tively We see that due to the presence of SFDM our solu-tion is non-asymptotically flat In the special case whenSFDM is absent we recover f12 rarr 1 hence the aboverelations reduces to the asymptotically flat case Notethat for the observer located at the position with coor-dinates (ro θo) we have neglected the effect of rotationwhile ξ12 and η12 given by Eq (61) and are evaluatedat the circular photon orbits r = rcir

For more details in Figs 5 and 6 we plot the shape ofSgr Alowast black hole shadows Using the parameter valuesspecifying the SFDM and baryonic matter we obtain al-most no effect on the shadow images In fact one canobserve that the dashed curve (exponential profile) anddashed red curve (power law profile) are almost indis-tinguishable from the black dashed curve (Kerr vacuumBH) However we can observe that by increasing thebaryonic mass the shadow size increases considerably(solid bluered curve) The angular radius of the Sgr Alowastblack hole shadow can be estimated using the observ-able Rs as θs = Rs MD where M is the black hole mass

5196 5197 5198 5199 5200

5315

5316

5317

5318

Rs

θs [μas]

FIG 7 The expected value for the angular diameter in thecase of Sgr Alowast Note that we have adopted D = 83 kpc and

M = 43times 106M

and D is the distance between the black hole and theobserver The angular radius can be further expressedas θs = 987098times 10minus6Rs(MM)(1kpcD) microas In thecase of Sgr Alowast we have used M = 43 times 106M andD = 83 kpc [4] is the distance between the Earth andSgr Alowast center black hole In order to get more informa-tion about the effect of SFDM on physical observableswe shall estimate the effect of SFDM on the angular ra-dius for Sgr Alowast To simplify the problem let us considerour static and spherically symmetric black hole metricin a totally dominated dark matter galaxy described bythe function

F (r) = eminusX minus 2Mr

(70)

where we have introduced

X =ρ2

04Λ2m6 (71)

where we have used the approximation cos( π r2 R ) 1

since our solution is valid for r le R From the circularphoton orbit conditions one can show [41]

2minus rF prime(r)F (r) = 0 (72)

Now if we expand around X in the above function weobtain

F (r) = 1minus Xminus 2Mr

+ (73)

By solving this equation and considering only the lead-ing order terms one can determine the radius of the pho-ton sphere rps yielding

rps = 3M(1 + X) (74)

On the other hand one also can show the relation [41]

ξ2 + η =r2

ps

F (rps) (75)

11

The shadow radius Rs can be expressed in terms of thecelestial coordinates (α β) as follows

Rs =radic

α2 + β2 =radic

1minus XrpsradicF (rps)

(76)

Note that in the last equation we have used Eqs (69) and(75) Using the values m = 06 eV Λ = 02 meV andρ0 = 92 times 10minus8 eV4 in units of the Sgr Alowast black hole wefind Rs = 5196158313M Using this result we can findthat the angular diameter increases by δθs = 3times 10minus5

microas compared to the Schwarzschild vacuum This resultis consistent with the result reported in Ref [4] whereauthors estimated that the dark matter halo could influ-ence the shadow of Sgr Alowast at a level of order of magni-tude of 10minus3 microas and 10minus5 microas respectively They haveused the so called Cold Dark Matter and Scalar FieldDark Matter models to study the apparent shapes of theshadow A similar result has been obtained for the darkmatter effect on the M87 black hole (see [5]) using theBurkert dark matter profile

VI CONCLUSION

In this paper we have obtained a rotating black holesolution surrounded by superfluid dark matter alongwith baryonic matter To achieve this purpose in thepresent work we considered the superfluid dark mat-ter model and used two specific profiles to describe thebaryonic matter distribution namely the spherical ex-ponential profile and the power law profile followed bythe special case of a totally dominated dark matter caseUsing the current values for the baryonic mass centraldensity of the superfluid dark matter halo radius aswell as the radial scale length for the baryonic matterin our galaxy we found that the shadow size of Sgr Alowastblack hole remains almost unchanged compared to theKerr vacuum BH This result is consistent with a recentwork reported in [5] and also [4] For entirely dominateddark matter galaxies we find that the angular diame-ter increases by 10minus5microarcsec This result shows that itis very difficult to constrain the dark matter parametersusing the shadow images Such tiny effects on the angu-lar diameter are out of reach for the existing space tech-nology and it remains an open question if future astro-nomical observations can potentially detect such effectsThat being said the expected value for the angular di-ameter in the case of Sgr Alowast is of the order of θs 53 microasas predicted in GR

As an interesting observation we show that an in-crease of baryonic mass followed by a decrease of theradial scale length can increase the shadow size consid-erably This can be explained by the fact that the bary-onic matter is mostly located in the interior of the galaxyon the other hand dark matter is mostly located in theouter region of the galaxy In other words for the to-tally dominated dark matter galaxies we observe almost

no effect on black hole shadows but a more precise mea-surement of the Sgr Alowast shadow radius can play a signifi-cant role in determining the baryonic mass in our galaxy

Acknowledgements

TZ is supported by National Natural Science Founda-tion of China under the Grants No 11675143 the Zhe-jiang Provincial Natural Science Foundation of Chinaunder Grant No LY20A050002 and the FundamentalResearch Funds for the Provincial Universities of Zhe-jiang in China under Grants No RF-A2019015

Appendix A

Here we demonstrate the derivation of the spinningblack hole metric As a first step to this formalism wetransform Boyer-Lindquist (BL) coordinates (t r θ φ) toEddington-Finkelstein (EF) coordinates (u r θ φ) Thiscan be achieved by using the coordinate transformation

dt = du +drradic

F12(r)G12(r) (A1)

we obtain

ds212 = minusF12(r)du2minus 2

radicF12(r)G12(r)

dudr+ r2dθ2 + r2 sin2 θdφ2

(A2)This metric can be expressed in terms of null tetrads

as

gmicroν = minuslmicronν minus lνnmicro + mmicromν + mνmmicro (A3)

where the null tetrads are defined as

lmicro = δmicror (A4)

nmicro =

radicG12(r)F12(r)

δmicrou minus

12G12(r)δ

micror (A5)

mmicro =1radic2H

(δ

microθ +

ι

sin θδ

microφ

) (A6)

These null tetrads are constructed in such a way that lmicro

and nmicro while mmicro and mmicro are complex It is worth notingthatH = r2 and will be used later on It is obvious fromthe notation that mmicro is complex conjugate of mmicro Thesevectors further satisfy the conditions for normalizationorthogonality and isotropy as

lmicrolmicro = nmicronmicro = mmicrommicro = mmicrommicro = 0 (A7)lmicrommicro = lmicrommicro = nmicrommicro = nmicrommicro = 0 (A8)

minuslmicronmicro = mmicrommicro = 1 (A9)

12

Following the NewmanndashJanis prescription we write

xprimemicro = xmicro + ia(δmicror minus δ

microu ) cos θ rarr

uprime = uminus ia cos θrprime = r + ia cos θθprime = θφprime = φ

(A10)in which a stands for the rotation parameter Nextlet the null tetrad vectors Za undergo a transformationgiven by Zmicro = (partxmicropartxprimeν)Zprimeν following [62]

lprimemicro = δmicror (A11)

nprimemicro =

radicB12

A12δ

microu minus

12

B12δmicror

mprimemicro =1radic

2Σ12

[(δ

microu minus δ

micror )ιa sin θ + δ

microθ +

ι

sin θδ

microφ

]where we have assumed that (F12(r)G12(r)H(r))transform to (A12(a r θ) B12(a r θ) Σ12(a r θ)) Withthe help of the above equations new metric inEddington-Finkelste coordinate reads

ds212 = minusA12du2 minus 2

radicA12

B12dudr (A12)

+ 2a sin2 θ

(A12 minus

radicA12

B12

)dudφ

+ 2a

radicA12

B12sin2 θdrdφ + Σ12dθ2

+ sin2 θ

[Σ12 + a2 sin2 θ

(2

radicA12

B12minus A12

)]dφ2

Note that A12 is some function of r and θ as we al-ready pointed out Without going into details of the cal-culation one can revert the EF coordinates back to BLcoordinates by using the following transformation (see

for more details [62])

du = dtminus a2 + r2

∆12dr dφ = dφprime minus a

∆12dr (A13)

where in order to simplify the notation we introduce

∆(r)12 = r2 F12(r) + a2 (A14)

Making use of F12(r) = G12(r) one can obtain Σ1 =Σ2 = r2 + a2 cos2 θ herein we shall use just Σ Droppingthe primes in the φ coordinate and choosing

A12 =(F12H+ a2 cos2 θ)Σ12

(H+ a2 cos2 θ)2 (A15)

and

B12 =F12H+ a2 cos2 θ

Σ (A16)

we obtain the spinning black hole space-time metric ina SFDM halo

ds212 = minus

(1minus 2Υ12(r)r

Σ

)dt2 +

Σ∆12

dr2 + Σdθ2

minus 2a sin2 θ2Υ12(r)r

Σdtdφ

+ sin2 θ

[(r2 + a2)2 minus a2∆12 sin2 θ

Σ

]dφ2 (A17)

with

Υ12(r) =r (1minusF12(r))

2 (A18)

in which F1 = F (r)exp and F2 = F (r)power respec-tively

[1] The Event Horizon Telescope Collaboration Ap J Lett875 L1 (2019)

[2] Haroon S Jamil M Jusufi K Lin K Mann R B PhysRev D 99 no4 044015 (2019)

[3] Haroon S Jusufi K Jamil M Universe 6(2) 23 (2020)arXiv190400711[gr-qc]

[4] Hou X Xu Z Zhou M Wang J JCAP 1807 no07 015(2018)

[5] Jusufi K Jamil M Salucci P Zhu T Haroon S PhysRev D 100 044012 (2019)

[6] Xu Z Hou X Gong X Wang J JCAP 1809 no09 038(2018)

[7] Xu Z Hou X Wang J JCAP 1810 no10 046 (2018)[8] Berezhiani L Khoury J Phys Rev D 92 103510 (2015)[9] Synge J L Mon Not Roy Astron Soc 131 no 3 463

(1966)

[10] Luminet J P Astron Astrophys 75 228 (1979)[11] Bardeen J M in Black Holes (Proceedings Les Houches

France August 1972) edited by C DeWitt and B S De-Witt

[12] Abdujabbarov A Toshmatov B Stuchlik Z AhmedovB Int J Mod Phys D 26 no 06 1750051 (2016)

[13] Abdikamalov A B Abdujabbarov A A Ayzenberg DMalafarina D Bambi C Ahmedov B Phys Rev D 100(2019) no2 024014

[14] Abdujabbarov A Amir M Ahmedov B Ghosh S GPhys Rev D 93 no 10 104004 (2016)

[15] Amir M Jusufi K Banerjee A Hansraj S ClassQuant Grav 36 no21 215007 (2019)

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[17] Abdujabbarov A Atamurotov F Kucukakca Y Ahme-

4

it is surrounded by DM particles in the normal phasemost likely described by a Navarro-Frenk-White (NFW)profile or some other effective type dark matter profilemost probably described by the empirical Burkert pro-file (see for example [53ndash59])

However as was argued in [52] we do not expect asharp transition but instead there should be a transitionregion around at which the DM is nearly in equilibriumbut is incapable of maintaining long range coherence Inthe present work we are mostly interested to explore theeffect of the SFDM core and therefore we are going toneglect the outer DM region in the normal phase givenby the NFW profile

A Exponential profile

As a toy model for describing the baryonic distribu-tion we consider a spherical exponential profile of den-sity given by [52]

ρB(r) =MB

8πL3 eminusrL (29)

where the characteristic scale L plays the role of a radialscale length and MB is the baryonic mass In that casethe total mass can be written as a sum of masses of darkmatter and the disc The mass function of the stellar thindisk is written as

MB(r) =MB

[2L2 minus

(2L2 + 2rL + r2) eminus

rL

]2L2 (30)

We observe from these profiles that by going in theoutside region of core ie r gtgt L the mass MB(r) re-duces to a constant MB In that way the last term in(28) becomes independent of r and the empirical TullyndashFisher relation which describes the rotational curves ofgalaxies In our analyses we shall simplify the calcu-lations by considering a fixed baryonic mass MB in thelast term in (28) Let us introduce the quantity

v20 =

radicMBa0 (31)

and use the tangent velocity while assuming a spheri-cally symmetric solution resulting with

f (r)BM+DM asymp C r2 v20 eminus

ρ20 cos( π r

2 R )

4Λ2m6 (32)

times exp[minusMB

rL(2Lminus (2L + r)eminus

rL )

]

valid for r le R In the present work we shall focues onthe HSB type galaxyes such as the Milky Way galaxyWe can use MB = 5minus 7times 1010M and L = 2minus 3 kpc[60]

FIG 1 Schematic representation of the superfluid DM coreand baryonic matter surrounding a black hole at the center

Outside the superfluid core DM exists in a normal phaseprobably described by the NFW profile or Burkert profile

B Power law profile

As a second example we assume the baryonic matterto be concentrated into an inner core of radius rc andthat its mass profile MB(r) can be described by the sim-ple relation [61]

MB(r) = MB

(r

r + rc

)3β

(33)

in the present paper we shall be interested in the caseβ = 1 for high surface brightness galaxies (HSB) Mak-ing use of the tangent velocity and assuming a spheri-cally symmetric solution for the baryonic matter contri-bution in the case of HSB galaxies we find

f (r)BM+DM asymp C(πr

2R

)2 v20 eminusMB(2r+rc)

(r+rc)2 eminusρ2

0 cos( π r2 R )

4Λ2m6 (34)

Note that the above analyses holds only inside the su-perfluid core with r le R

C Totally dominated dark matter galaxies

There is one particular solution of interest which de-scribes a totally dominated dark matter galaxies (TD-DMG) or sometimes known as dark galaxies To do thiswe neglect the baryonic mass MBM rarr 0 and thereforea0 rarr 0 yielding

f (r)TDDMG = C eminusρ2

0 cos( π r2 R )

4Λ2m6 (35)

valid for r le R This result is obtained directly from Eqs(32) and (34) in the limit MB = 0 In other words this so-lution reduces to (23) as expected Note that the integra-tion constant C can be absorbed in the time coordinateusing dt2 rarr C dt2

5

IV BLACK HOLE METRIC SURROUNDED BYBARYONIC MATTER AND SUPERFLUID DARK

MATTER

Let us now consider a more interesting scenario byadding a black hole present in the SFDM halo Toachieve this aim we shall use the method introduced in[6] Namely we need to compute the spacetime metricby assuming the Einstein field equations given by [6]

Rνmicro minus

12

δνmicroR = κ2Tν

micro (36)

where the corresponding energy-momentum tensorsTν

micro = diag[minusρ pr p p] where ρ = ρDM + ρBM encodesthe total contribution coming from the surrounded darkmatter and baryonic matter respectively The space-time metric including black hole is thus given by [6]

ds2 = minus( f (r) + F1(r))dt2 +dr2

g(r) + F2(r)+ r2dΩ2 (37)

where dΩ2 = dθ2 + sin2 θdφ2 is the metric of a unitsphere Moreover we have introduced the followingquantities

F (r) = f (r) + F1(r) G(r) = g(r) + F2(r) (38)

In terms of these relations the space-time metric de-scribing a black hole in dark matter halo becomes [6]

ds2 = minus exp[int g(r)

g(r)minus 2Mr

(1r+

fprime(r)

f (r))drminus 1

rdr]dt2

+ (g(r)minus 2Mr

)minus1dr2 + r2dΩ2 (39)

In the absence of the dark matter halo ie f (r) =g(r) = 1 the indefinite integral reduces to a constant[6]

F1(r) + f (r) = exp[int g(r)

g(r) + F2(r)(

1r+

fprime(r)

f (r))minus 1

rdr]

= 1minus 2Mr

(40)

yielding F1(r) = minus2Mr In that way one can obtain theblack hole space-time metric with a surrounding matter

ds2 = minusF (r)dt2 +dr2

G(r) + r2dΩ2 (41)

Thus the general black hole solution surrounded bySFDM with F (r) = G(r) are given by

F (r)exp = C r2 v20 eminus

ρ20 cos( π r

2 R )

4Λ2m6 (42)

times exp[minusMB

rL(2Lminus (2L + r)eminus

rL )

]minus 2M

r

and

F (r)power = C(πr

2R

)2 v20 eminus

ρ20 cos( π r

2 R )

4Λ2m6 eminusMB(2r+rc)

(r+rc)2 minus 2Mr

(43)

As a special case we obtain a black hole in a totallydominated dark matter galaxy given by

F (r)TDDMG = C eminusρ2

0 cos( π r2 R )

4Λ2m6 minus 2Mr

(44)

valid for r le R Note that M is the black hole mass Inthe limit M = 0 our solution reduces to (23) as expected

V SHADOW OF SGR Alowast BLACK HOLE SURROUNDEDBY SUPERFLUID DARK MATTER

In this section we generalize the static and the spher-ical symmetric black hole solution to a spinning blackhole surrounded by dark matter halo applying theNewman-Janis method and adapting the approach in-troduced in [62 63] Applying the Newman-Janismethod we obtain the spinning black hole metric sur-rounded by dark matter halo as follows (see AppendixA)

ds212 = minus

(1minus 2Υ12(r)r

Σ

)dt2 +

Σ∆12

dr2 + Σdθ2

minus 2a sin2 θ2Υ12(r)r

Σdtdφ

+ sin2 θ

[(r2 + a2)2 minus a2∆12 sin2 θ

Σ

]dφ2 (45)

where we have introduced

Υ12(r) =r (1minusF12(r))

2 (46)

and identified F1 = F (r)exp and F2 = F (r)power re-spectively As a special case we find the Kerr BH limitY12(r) = M if a0 = MB = ρ0 = 0

In addition we can explore the shape of the ergore-gion of our black hole metric (45) that is we can plotthe shape of the ergoregion say in the xz-plane On theother hand the horizons of the black hole are found bysolving ∆12 = 0 te

r2F (r)12 + a2 = 0 (47)

while the inner and outer ergo-surfaces are obtainedfrom solving gtt = 0 ie

r2F (r)12 + a2 cos2 θ = 0 (48)

In Fig 2 we plot ∆12 as a function of r for differentvalues of a In the special case for some critical valuea = aE (blue line) the two horizons coincide in otherwords we have an extremal black hole with degeneratehorizons Beyond this critical value a gt aE there is noevent horizon and the solution corresponds to a nakedsingularity

6

00 05 10 15 20

-05

00

05

10

rM

Δ1

Exponential profile

00 05 10 15 20

-05

00

05

10

rM

Δ2

Power law profile

FIG 2 Left panel Variation of ∆1 as a function of r for BH-SFDM using the spherical exponential profile Right panelVariation of ∆2 as a function of r for BH-SFDM using the power profile law Note that a = 065 (black curve) a = 085 (red

curve) a = 1 (blue curve) respectively We have used m = 06 eV and Λ = 02 meV we find ρ0 = 002 times 10minus24 gcm3 or in eVunits ρ0 = 92 times 10minus8 eV4 For the Milky Way galaxy we can use MB = 6times 1010 M = 139 times 104 MBH and rc = 26

kpc= 124 times 1010 MBH

00 05 10 15 20-10

0

10

20

30

rM

Veff

Exponential profile

00 05 10 15 20-10

0

10

20

30

rM

Veff

Power law profile

FIG 3 Left panel The effective potential of photon moving for BH-SFDM using the exponsntial profile Right panel Theeffective potential of photon using thepower law profile We have chosen a = 05 (black color) a = 065 (red color) and a = 085

(blue color) respectively

A Geodesic equations

In order to find the contour of a black hole shadow ofthe rotating spacetime (45) first we need to find the nullgeodesic equations using the Hamilton-Jacobi methodgiven by

partSpartτ

= minus12

gmicroν partSpartxmicro

partSpartxν

(49)

where τ is the affine parameter and S is the Jacobi ac-tion Due to the spectime symmetries there are two con-served quantities namely the conserved energy E =minuspt and the conserved angular momentum L = pφ re-spectively

To find the separable solution of Eq (49) we need to

express the action in the following form

S =12

micro2τ minus Et + Jφ + Sr(r) + Sθ(θ) (50)

in which micro gives the mass of the test particle Of coursethis gives micro = 0 in the case of the photon That beingsaid it is straightforward to obtain the following equa-

7

-2 -1 0 1 2

-2

-1

0

1

2

rsin(θ)cos(φ)

rcos

(θ)

M=1 a=04

-2 -1 0 1 2

-2

-1

0

1

2

rsin(θ)cos(φ)

rcos

(θ)

M=1 a=065

-2 -1 0 1 2

-2

-1

0

1

2

rsin(θ)cos(φ)

rcos

(θ)

M=1 a=085

-2 -1 0 1 2

-2

-1

0

1

2

rsin(θ)cos(φ)

rcos

(θ)

M=1 a=098

FIG 4 The shape of the ergoregion and innerouter horizons for different values of a using the spherical exponential profileWe use M = 1 in units of the Sgr Alowast black hole mass given by MBH = 43 times 106 M and R = 158 kpc or R = 755 times 1010 MBH Furthermore for m = 06 eV and Λ = 02 meV we find ρ0 = 002 times 10minus24 gcm3 or in eV units ρ0 = 92 times 10minus8 eV4 For the

normal matter contribution of the Milky Way galaxy we can use MB = 6times 1010 M = 139 times 104 MBH and L = 26kpc= 124 times 1010 MBH

tions of motions from the Hamilton-Jacobi equation

Σdtdτ

=r2 + a2

∆12[E(r2 + a2)minus aL]minus a(aE sin2 θ minus L)

(51)

Σdrdτ

=radicR(r) (52)

Σdθ

dτ=radic

Θ(θ) (53)

Σdφ

dτ=

a∆12

[E(r2 + a2)minus aL]minus(

aEminus Lsin2 θ

) (54)

whereR(r) and Θ(θ) are given by

R(r) = [E(r2 + a2)minus aL]2 minus ∆12[m2r2 + (aEminus L)2 +K]

(55)

Θ(θ) = Kminus(

L2

sin2 θminus a2E2

)cos2 θ (56)

and K is the separation constant known as the Carterconstant

8

Kerr vacuum BH

BH-SFDM

-10 -5 0 5-6

-4

-2

0

2

4

6

α

β

a=02

Kerr vacuum BH

BH-SFDM

-10 -5 0 5-6

-4

-2

0

2

4

6

α

β

a=05

Kerr vacuum BH

BH-SFDM

-10 -5 0 5-6

-4

-2

0

2

4

6

α

β

a=085

Kerr vacuum BH

BH-SFDM

-10 -5 0 5-6

-4

-2

0

2

4

6

α

β

a=098

FIG 5 Variation in shape of shadow using the spherical exponential profile for baryonic matter We use M = 1 in units of theSgr Alowast black hole mass given by MBH = 43 times 106 M and R = 158 kpc or R = 755 times 1010 MBH Furthermore for m = 06 eV

and Λ = 02 meV we find ρ0 = 002 times 10minus24 gcm3 or in eV units ρ0 = 92 times 10minus8 eV4 For the Milky Way galaxy we can useMB = 6times 1010 M = 139 times 104 MBH and L = 26 kpc= 124 times 1010 MBH We observe that the dashed blue curve correspondingto Sgr Alowast with the above parameters is almost indistinguishable from the black dashed curve describing the Kerr vacuum BHThe solid blue curve corresponds to the case of increasing the baryonic mass by a factor of 102 and decrease of core radius by a

factor of 103

B Circular Orbits

Due to the strong gravity near the black hole it is thusexpected that the photons emitted near a black hole willeventually fall into the black hole or eventually scatteraway from it In this way the photons captured by theblack hole will form a dark region defining the contourof the shadow To elaborate the presence of unstablecircular orbits around the black hole we need to studythe radial geodesic by introducing the effective poten-tial Veff which can be written as follows

Σ2(

drdτ

)2+ V12

eff = 0 (57)

where V1eff = Vexp

eff and V2eff = Vpower

eff gives the expo-nential and the power law respectively At this point itis convenient to introduce two parameters ξ and η de-fined as

ξ = LE η = KE2 (58)

For the effective potential then we obtain the followingrelation

V12eff = ∆12((aminus ξ12)

2 + η12)minus (r2 + a2minus a ξ12)2 (59)

where we have replaced V12eff E2 by V12

eff For more de-tails in Figure (3) we plot the variation of the effectivepotential associated with the radial motion of photons

9

Kerr vacuum BH

BH-SFDM

-10 -5 0 5-6

-4

-2

0

2

4

6

α

β

a=02

Kerr vacuum BH

BH-SFDM

-10 -5 0 5-6

-4

-2

0

2

4

6

α

β

a=05

Kerr vacuum BH

BH-SFDM

-10 -5 0 5-6

-4

-2

0

2

4

6

α

β

a=085

Kerr vacuum BH

BH-SFDM

-10 -5 0 5-6

-4

-2

0

2

4

6

α

β

a=098

FIG 6 Variation in shape of shadow for different values of a using the power law profile We use M = 1 in units of the Sgr Alowast

black hole mass given by MBH = 43 times 106 M and R = 158 kpc or R = 755 times 1010 MBH Furthermore for m = 06 eV andΛ = 02 meV we find ρ0 = 002 times 10minus24 gcm3 or in eV units ρ0 = 92 times 10minus8 eV4 For the Milky Way galaxy we can use

MB = 6times 1010 M = 139 times 104 MBH and rc = 26 kpc= 124 times 1010 MBH The dashed red curve corresponds to Sgr Alowast usingthe above parameters which is almost indistinguishable from the black dashed curve discribing the Kerr vacuum BH The solid

red curve corresponds to the case of increasing the baryonic mass by a factor of 102 and decrease of core radius by a factor of 103

The circular photon orbits exists when at some constantr = rcir the conditions

V12eff (r) = 0

dV12eff (r)dr

= 0 (60)

Combining all these equations it is possible to showthat

ξ12 =(r2 + a2)(rF prime12(r) + 2F12(r))minus 4(r2F12(r) + a2)

a(rF prime12(r) + 2F12(r))

η12 =r3[8a2F prime12(r)minus r(rF prime12(r)minus 2F12(r))2]

a2(rF prime12(r) + 2F12(r))2 (61)

One can recover the Kerr vacuum case by letting a0 =MB = ρ0 = 0 yielding

ξ12 =r2(3Mminus r)minus a2(Mminus r)

a(rminusM) (62)

η12 =r3 (4Ma2 minus r(rminus 3M)2)

a2(rminusM)2 (63)

To obtain the shadow images of our black hole in thepresence of dark matter we assume that the observer islocated at the position with coordinates (ro θo) where roand θo represents the angular coordinate on observerrsquossky Furthermore we need to introduce two celestial co-ordinates α and β for the observer by using the follow-ing relations [29]

α = minusrop(φ)

p(t) β = ro

p(θ)

p(t) (64)

where (p(t) p(r) p(θ) p(φ)) are the tetrad componentsof the photon momentum with respect to locally non-rotating reference frame The observer bases emicro

(ν)can be

10

expanded as a form in the coordinate bases (see [64])

p(t) = minusemicro

(t)pmicro = E ζ12 minus γ12 L p(φ) = emicro

(φ)pmicro =

Lradicgφφ

p(θ) = emicro

(θ)pmicro =

pθradicgθθ

p(r) = emicro

(r)pmicro =prradicgrr

(65)

where the quantities E = minuspt and pφ = L are conserveddue to the associated Killing vectors If we use ξ = LEη = KE2 and pθ = plusmn

radicΘ(θ) we can rewrite these co-

ordinates in terms our parameters ξ and η as follows[43]

α12 = minusroξ12radicgφφ(ζ12 minus γ12ξ12)

|(ro θo)

β12 = plusmnro

radicη12 + a2 cos2 θ minus ξ2

12 cot2 θradic

gθθ(ζ12 minus γ12ξ12)|(ro θo) (66)

where

ζ12 =

radicgφφ

g2tφ minus gttgφφ

(67)

and

γ12 = minusgtφ

gφφζ12 (68)

Finally to simplify the problem further we are going toconsider that our observer is located in the equatorialplane (θ = π2) and very large but finite ro = D = 83kpc we find

α12 = minusradic

f12 ξ12

β12 = plusmnradic

f12radic

η12 (69)

Note that f12 are given by Eqs (32) and (34) respec-tively We see that due to the presence of SFDM our solu-tion is non-asymptotically flat In the special case whenSFDM is absent we recover f12 rarr 1 hence the aboverelations reduces to the asymptotically flat case Notethat for the observer located at the position with coor-dinates (ro θo) we have neglected the effect of rotationwhile ξ12 and η12 given by Eq (61) and are evaluatedat the circular photon orbits r = rcir

For more details in Figs 5 and 6 we plot the shape ofSgr Alowast black hole shadows Using the parameter valuesspecifying the SFDM and baryonic matter we obtain al-most no effect on the shadow images In fact one canobserve that the dashed curve (exponential profile) anddashed red curve (power law profile) are almost indis-tinguishable from the black dashed curve (Kerr vacuumBH) However we can observe that by increasing thebaryonic mass the shadow size increases considerably(solid bluered curve) The angular radius of the Sgr Alowastblack hole shadow can be estimated using the observ-able Rs as θs = Rs MD where M is the black hole mass

5196 5197 5198 5199 5200

5315

5316

5317

5318

Rs

θs [μas]

FIG 7 The expected value for the angular diameter in thecase of Sgr Alowast Note that we have adopted D = 83 kpc and

M = 43times 106M

and D is the distance between the black hole and theobserver The angular radius can be further expressedas θs = 987098times 10minus6Rs(MM)(1kpcD) microas In thecase of Sgr Alowast we have used M = 43 times 106M andD = 83 kpc [4] is the distance between the Earth andSgr Alowast center black hole In order to get more informa-tion about the effect of SFDM on physical observableswe shall estimate the effect of SFDM on the angular ra-dius for Sgr Alowast To simplify the problem let us considerour static and spherically symmetric black hole metricin a totally dominated dark matter galaxy described bythe function

F (r) = eminusX minus 2Mr

(70)

where we have introduced

X =ρ2

04Λ2m6 (71)

where we have used the approximation cos( π r2 R ) 1

since our solution is valid for r le R From the circularphoton orbit conditions one can show [41]

2minus rF prime(r)F (r) = 0 (72)

Now if we expand around X in the above function weobtain

F (r) = 1minus Xminus 2Mr

+ (73)

By solving this equation and considering only the lead-ing order terms one can determine the radius of the pho-ton sphere rps yielding

rps = 3M(1 + X) (74)

On the other hand one also can show the relation [41]

ξ2 + η =r2

ps

F (rps) (75)

11

The shadow radius Rs can be expressed in terms of thecelestial coordinates (α β) as follows

Rs =radic

α2 + β2 =radic

1minus XrpsradicF (rps)

(76)

Note that in the last equation we have used Eqs (69) and(75) Using the values m = 06 eV Λ = 02 meV andρ0 = 92 times 10minus8 eV4 in units of the Sgr Alowast black hole wefind Rs = 5196158313M Using this result we can findthat the angular diameter increases by δθs = 3times 10minus5

microas compared to the Schwarzschild vacuum This resultis consistent with the result reported in Ref [4] whereauthors estimated that the dark matter halo could influ-ence the shadow of Sgr Alowast at a level of order of magni-tude of 10minus3 microas and 10minus5 microas respectively They haveused the so called Cold Dark Matter and Scalar FieldDark Matter models to study the apparent shapes of theshadow A similar result has been obtained for the darkmatter effect on the M87 black hole (see [5]) using theBurkert dark matter profile

VI CONCLUSION

In this paper we have obtained a rotating black holesolution surrounded by superfluid dark matter alongwith baryonic matter To achieve this purpose in thepresent work we considered the superfluid dark mat-ter model and used two specific profiles to describe thebaryonic matter distribution namely the spherical ex-ponential profile and the power law profile followed bythe special case of a totally dominated dark matter caseUsing the current values for the baryonic mass centraldensity of the superfluid dark matter halo radius aswell as the radial scale length for the baryonic matterin our galaxy we found that the shadow size of Sgr Alowastblack hole remains almost unchanged compared to theKerr vacuum BH This result is consistent with a recentwork reported in [5] and also [4] For entirely dominateddark matter galaxies we find that the angular diame-ter increases by 10minus5microarcsec This result shows that itis very difficult to constrain the dark matter parametersusing the shadow images Such tiny effects on the angu-lar diameter are out of reach for the existing space tech-nology and it remains an open question if future astro-nomical observations can potentially detect such effectsThat being said the expected value for the angular di-ameter in the case of Sgr Alowast is of the order of θs 53 microasas predicted in GR

As an interesting observation we show that an in-crease of baryonic mass followed by a decrease of theradial scale length can increase the shadow size consid-erably This can be explained by the fact that the bary-onic matter is mostly located in the interior of the galaxyon the other hand dark matter is mostly located in theouter region of the galaxy In other words for the to-tally dominated dark matter galaxies we observe almost

no effect on black hole shadows but a more precise mea-surement of the Sgr Alowast shadow radius can play a signifi-cant role in determining the baryonic mass in our galaxy

Acknowledgements

TZ is supported by National Natural Science Founda-tion of China under the Grants No 11675143 the Zhe-jiang Provincial Natural Science Foundation of Chinaunder Grant No LY20A050002 and the FundamentalResearch Funds for the Provincial Universities of Zhe-jiang in China under Grants No RF-A2019015

Appendix A

Here we demonstrate the derivation of the spinningblack hole metric As a first step to this formalism wetransform Boyer-Lindquist (BL) coordinates (t r θ φ) toEddington-Finkelstein (EF) coordinates (u r θ φ) Thiscan be achieved by using the coordinate transformation

dt = du +drradic

F12(r)G12(r) (A1)

we obtain

ds212 = minusF12(r)du2minus 2

radicF12(r)G12(r)

dudr+ r2dθ2 + r2 sin2 θdφ2

(A2)This metric can be expressed in terms of null tetrads

as

gmicroν = minuslmicronν minus lνnmicro + mmicromν + mνmmicro (A3)

where the null tetrads are defined as

lmicro = δmicror (A4)

nmicro =

radicG12(r)F12(r)

δmicrou minus

12G12(r)δ

micror (A5)

mmicro =1radic2H

(δ

microθ +

ι

sin θδ

microφ

) (A6)

These null tetrads are constructed in such a way that lmicro

and nmicro while mmicro and mmicro are complex It is worth notingthatH = r2 and will be used later on It is obvious fromthe notation that mmicro is complex conjugate of mmicro Thesevectors further satisfy the conditions for normalizationorthogonality and isotropy as

lmicrolmicro = nmicronmicro = mmicrommicro = mmicrommicro = 0 (A7)lmicrommicro = lmicrommicro = nmicrommicro = nmicrommicro = 0 (A8)

minuslmicronmicro = mmicrommicro = 1 (A9)

12

Following the NewmanndashJanis prescription we write

xprimemicro = xmicro + ia(δmicror minus δ

microu ) cos θ rarr

uprime = uminus ia cos θrprime = r + ia cos θθprime = θφprime = φ

(A10)in which a stands for the rotation parameter Nextlet the null tetrad vectors Za undergo a transformationgiven by Zmicro = (partxmicropartxprimeν)Zprimeν following [62]

lprimemicro = δmicror (A11)

nprimemicro =

radicB12

A12δ

microu minus

12

B12δmicror

mprimemicro =1radic

2Σ12

[(δ

microu minus δ

micror )ιa sin θ + δ

microθ +

ι

sin θδ

microφ

]where we have assumed that (F12(r)G12(r)H(r))transform to (A12(a r θ) B12(a r θ) Σ12(a r θ)) Withthe help of the above equations new metric inEddington-Finkelste coordinate reads

ds212 = minusA12du2 minus 2

radicA12

B12dudr (A12)

+ 2a sin2 θ

(A12 minus

radicA12

B12

)dudφ

+ 2a

radicA12

B12sin2 θdrdφ + Σ12dθ2

+ sin2 θ

[Σ12 + a2 sin2 θ

(2

radicA12

B12minus A12

)]dφ2

Note that A12 is some function of r and θ as we al-ready pointed out Without going into details of the cal-culation one can revert the EF coordinates back to BLcoordinates by using the following transformation (see

for more details [62])

du = dtminus a2 + r2

∆12dr dφ = dφprime minus a

∆12dr (A13)

where in order to simplify the notation we introduce

∆(r)12 = r2 F12(r) + a2 (A14)

Making use of F12(r) = G12(r) one can obtain Σ1 =Σ2 = r2 + a2 cos2 θ herein we shall use just Σ Droppingthe primes in the φ coordinate and choosing

A12 =(F12H+ a2 cos2 θ)Σ12

(H+ a2 cos2 θ)2 (A15)

and

B12 =F12H+ a2 cos2 θ

Σ (A16)

we obtain the spinning black hole space-time metric ina SFDM halo

ds212 = minus

(1minus 2Υ12(r)r

Σ

)dt2 +

Σ∆12

dr2 + Σdθ2

minus 2a sin2 θ2Υ12(r)r

Σdtdφ

+ sin2 θ

[(r2 + a2)2 minus a2∆12 sin2 θ

Σ

]dφ2 (A17)

with

Υ12(r) =r (1minusF12(r))

2 (A18)

in which F1 = F (r)exp and F2 = F (r)power respec-tively

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[10] Luminet J P Astron Astrophys 75 228 (1979)[11] Bardeen J M in Black Holes (Proceedings Les Houches

France August 1972) edited by C DeWitt and B S De-Witt

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5

IV BLACK HOLE METRIC SURROUNDED BYBARYONIC MATTER AND SUPERFLUID DARK

MATTER

Let us now consider a more interesting scenario byadding a black hole present in the SFDM halo Toachieve this aim we shall use the method introduced in[6] Namely we need to compute the spacetime metricby assuming the Einstein field equations given by [6]

Rνmicro minus

12

δνmicroR = κ2Tν

micro (36)

where the corresponding energy-momentum tensorsTν

micro = diag[minusρ pr p p] where ρ = ρDM + ρBM encodesthe total contribution coming from the surrounded darkmatter and baryonic matter respectively The space-time metric including black hole is thus given by [6]

ds2 = minus( f (r) + F1(r))dt2 +dr2

g(r) + F2(r)+ r2dΩ2 (37)

where dΩ2 = dθ2 + sin2 θdφ2 is the metric of a unitsphere Moreover we have introduced the followingquantities

F (r) = f (r) + F1(r) G(r) = g(r) + F2(r) (38)

In terms of these relations the space-time metric de-scribing a black hole in dark matter halo becomes [6]

ds2 = minus exp[int g(r)

g(r)minus 2Mr

(1r+

fprime(r)

f (r))drminus 1

rdr]dt2

+ (g(r)minus 2Mr

)minus1dr2 + r2dΩ2 (39)

In the absence of the dark matter halo ie f (r) =g(r) = 1 the indefinite integral reduces to a constant[6]

F1(r) + f (r) = exp[int g(r)

g(r) + F2(r)(

1r+

fprime(r)

f (r))minus 1

rdr]

= 1minus 2Mr

(40)

yielding F1(r) = minus2Mr In that way one can obtain theblack hole space-time metric with a surrounding matter

ds2 = minusF (r)dt2 +dr2

G(r) + r2dΩ2 (41)

Thus the general black hole solution surrounded bySFDM with F (r) = G(r) are given by

F (r)exp = C r2 v20 eminus

ρ20 cos( π r

2 R )

4Λ2m6 (42)

times exp[minusMB

rL(2Lminus (2L + r)eminus

rL )

]minus 2M

r

and

F (r)power = C(πr

2R

)2 v20 eminus

ρ20 cos( π r

2 R )

4Λ2m6 eminusMB(2r+rc)

(r+rc)2 minus 2Mr

(43)

As a special case we obtain a black hole in a totallydominated dark matter galaxy given by

F (r)TDDMG = C eminusρ2

0 cos( π r2 R )

4Λ2m6 minus 2Mr

(44)

valid for r le R Note that M is the black hole mass Inthe limit M = 0 our solution reduces to (23) as expected

V SHADOW OF SGR Alowast BLACK HOLE SURROUNDEDBY SUPERFLUID DARK MATTER

In this section we generalize the static and the spher-ical symmetric black hole solution to a spinning blackhole surrounded by dark matter halo applying theNewman-Janis method and adapting the approach in-troduced in [62 63] Applying the Newman-Janismethod we obtain the spinning black hole metric sur-rounded by dark matter halo as follows (see AppendixA)

ds212 = minus

(1minus 2Υ12(r)r

Σ

)dt2 +

Σ∆12

dr2 + Σdθ2

minus 2a sin2 θ2Υ12(r)r

Σdtdφ

+ sin2 θ

[(r2 + a2)2 minus a2∆12 sin2 θ

Σ

]dφ2 (45)

where we have introduced

Υ12(r) =r (1minusF12(r))

2 (46)

and identified F1 = F (r)exp and F2 = F (r)power re-spectively As a special case we find the Kerr BH limitY12(r) = M if a0 = MB = ρ0 = 0

In addition we can explore the shape of the ergore-gion of our black hole metric (45) that is we can plotthe shape of the ergoregion say in the xz-plane On theother hand the horizons of the black hole are found bysolving ∆12 = 0 te

r2F (r)12 + a2 = 0 (47)

while the inner and outer ergo-surfaces are obtainedfrom solving gtt = 0 ie

r2F (r)12 + a2 cos2 θ = 0 (48)

In Fig 2 we plot ∆12 as a function of r for differentvalues of a In the special case for some critical valuea = aE (blue line) the two horizons coincide in otherwords we have an extremal black hole with degeneratehorizons Beyond this critical value a gt aE there is noevent horizon and the solution corresponds to a nakedsingularity

6

00 05 10 15 20

-05

00

05

10

rM

Δ1

Exponential profile

00 05 10 15 20

-05

00

05

10

rM

Δ2

Power law profile

FIG 2 Left panel Variation of ∆1 as a function of r for BH-SFDM using the spherical exponential profile Right panelVariation of ∆2 as a function of r for BH-SFDM using the power profile law Note that a = 065 (black curve) a = 085 (red

curve) a = 1 (blue curve) respectively We have used m = 06 eV and Λ = 02 meV we find ρ0 = 002 times 10minus24 gcm3 or in eVunits ρ0 = 92 times 10minus8 eV4 For the Milky Way galaxy we can use MB = 6times 1010 M = 139 times 104 MBH and rc = 26

kpc= 124 times 1010 MBH

00 05 10 15 20-10

0

10

20

30

rM

Veff

Exponential profile

00 05 10 15 20-10

0

10

20

30

rM

Veff

Power law profile

FIG 3 Left panel The effective potential of photon moving for BH-SFDM using the exponsntial profile Right panel Theeffective potential of photon using thepower law profile We have chosen a = 05 (black color) a = 065 (red color) and a = 085

(blue color) respectively

A Geodesic equations

In order to find the contour of a black hole shadow ofthe rotating spacetime (45) first we need to find the nullgeodesic equations using the Hamilton-Jacobi methodgiven by

partSpartτ

= minus12

gmicroν partSpartxmicro

partSpartxν

(49)

where τ is the affine parameter and S is the Jacobi ac-tion Due to the spectime symmetries there are two con-served quantities namely the conserved energy E =minuspt and the conserved angular momentum L = pφ re-spectively

To find the separable solution of Eq (49) we need to

express the action in the following form

S =12

micro2τ minus Et + Jφ + Sr(r) + Sθ(θ) (50)

in which micro gives the mass of the test particle Of coursethis gives micro = 0 in the case of the photon That beingsaid it is straightforward to obtain the following equa-

7

-2 -1 0 1 2

-2

-1

0

1

2

rsin(θ)cos(φ)

rcos

(θ)

M=1 a=04

-2 -1 0 1 2

-2

-1

0

1

2

rsin(θ)cos(φ)

rcos

(θ)

M=1 a=065

-2 -1 0 1 2

-2

-1

0

1

2

rsin(θ)cos(φ)

rcos

(θ)

M=1 a=085

-2 -1 0 1 2

-2

-1

0

1

2

rsin(θ)cos(φ)

rcos

(θ)

M=1 a=098

FIG 4 The shape of the ergoregion and innerouter horizons for different values of a using the spherical exponential profileWe use M = 1 in units of the Sgr Alowast black hole mass given by MBH = 43 times 106 M and R = 158 kpc or R = 755 times 1010 MBH Furthermore for m = 06 eV and Λ = 02 meV we find ρ0 = 002 times 10minus24 gcm3 or in eV units ρ0 = 92 times 10minus8 eV4 For the

normal matter contribution of the Milky Way galaxy we can use MB = 6times 1010 M = 139 times 104 MBH and L = 26kpc= 124 times 1010 MBH

tions of motions from the Hamilton-Jacobi equation

Σdtdτ

=r2 + a2

∆12[E(r2 + a2)minus aL]minus a(aE sin2 θ minus L)

(51)

Σdrdτ

=radicR(r) (52)

Σdθ

dτ=radic

Θ(θ) (53)

Σdφ

dτ=

a∆12

[E(r2 + a2)minus aL]minus(

aEminus Lsin2 θ

) (54)

whereR(r) and Θ(θ) are given by

R(r) = [E(r2 + a2)minus aL]2 minus ∆12[m2r2 + (aEminus L)2 +K]

(55)

Θ(θ) = Kminus(

L2

sin2 θminus a2E2

)cos2 θ (56)

and K is the separation constant known as the Carterconstant

8

Kerr vacuum BH

BH-SFDM

-10 -5 0 5-6

-4

-2

0

2

4

6

α

β

a=02

Kerr vacuum BH

BH-SFDM

-10 -5 0 5-6

-4

-2

0

2

4

6

α

β

a=05

Kerr vacuum BH

BH-SFDM

-10 -5 0 5-6

-4

-2

0

2

4

6

α

β

a=085

Kerr vacuum BH

BH-SFDM

-10 -5 0 5-6

-4

-2

0

2

4

6

α

β

a=098

FIG 5 Variation in shape of shadow using the spherical exponential profile for baryonic matter We use M = 1 in units of theSgr Alowast black hole mass given by MBH = 43 times 106 M and R = 158 kpc or R = 755 times 1010 MBH Furthermore for m = 06 eV

and Λ = 02 meV we find ρ0 = 002 times 10minus24 gcm3 or in eV units ρ0 = 92 times 10minus8 eV4 For the Milky Way galaxy we can useMB = 6times 1010 M = 139 times 104 MBH and L = 26 kpc= 124 times 1010 MBH We observe that the dashed blue curve correspondingto Sgr Alowast with the above parameters is almost indistinguishable from the black dashed curve describing the Kerr vacuum BHThe solid blue curve corresponds to the case of increasing the baryonic mass by a factor of 102 and decrease of core radius by a

factor of 103

B Circular Orbits

Due to the strong gravity near the black hole it is thusexpected that the photons emitted near a black hole willeventually fall into the black hole or eventually scatteraway from it In this way the photons captured by theblack hole will form a dark region defining the contourof the shadow To elaborate the presence of unstablecircular orbits around the black hole we need to studythe radial geodesic by introducing the effective poten-tial Veff which can be written as follows

Σ2(

drdτ

)2+ V12

eff = 0 (57)

where V1eff = Vexp

eff and V2eff = Vpower

eff gives the expo-nential and the power law respectively At this point itis convenient to introduce two parameters ξ and η de-fined as

ξ = LE η = KE2 (58)

For the effective potential then we obtain the followingrelation

V12eff = ∆12((aminus ξ12)

2 + η12)minus (r2 + a2minus a ξ12)2 (59)

where we have replaced V12eff E2 by V12

eff For more de-tails in Figure (3) we plot the variation of the effectivepotential associated with the radial motion of photons

9

Kerr vacuum BH

BH-SFDM

-10 -5 0 5-6

-4

-2

0

2

4

6

α

β

a=02

Kerr vacuum BH

BH-SFDM

-10 -5 0 5-6

-4

-2

0

2

4

6

α

β

a=05

Kerr vacuum BH

BH-SFDM

-10 -5 0 5-6

-4

-2

0

2

4

6

α

β

a=085

Kerr vacuum BH

BH-SFDM

-10 -5 0 5-6

-4

-2

0

2

4

6

α

β

a=098

FIG 6 Variation in shape of shadow for different values of a using the power law profile We use M = 1 in units of the Sgr Alowast

black hole mass given by MBH = 43 times 106 M and R = 158 kpc or R = 755 times 1010 MBH Furthermore for m = 06 eV andΛ = 02 meV we find ρ0 = 002 times 10minus24 gcm3 or in eV units ρ0 = 92 times 10minus8 eV4 For the Milky Way galaxy we can use

MB = 6times 1010 M = 139 times 104 MBH and rc = 26 kpc= 124 times 1010 MBH The dashed red curve corresponds to Sgr Alowast usingthe above parameters which is almost indistinguishable from the black dashed curve discribing the Kerr vacuum BH The solid

red curve corresponds to the case of increasing the baryonic mass by a factor of 102 and decrease of core radius by a factor of 103

The circular photon orbits exists when at some constantr = rcir the conditions

V12eff (r) = 0

dV12eff (r)dr

= 0 (60)

Combining all these equations it is possible to showthat

ξ12 =(r2 + a2)(rF prime12(r) + 2F12(r))minus 4(r2F12(r) + a2)

a(rF prime12(r) + 2F12(r))

η12 =r3[8a2F prime12(r)minus r(rF prime12(r)minus 2F12(r))2]

a2(rF prime12(r) + 2F12(r))2 (61)

One can recover the Kerr vacuum case by letting a0 =MB = ρ0 = 0 yielding

ξ12 =r2(3Mminus r)minus a2(Mminus r)

a(rminusM) (62)

η12 =r3 (4Ma2 minus r(rminus 3M)2)

a2(rminusM)2 (63)

To obtain the shadow images of our black hole in thepresence of dark matter we assume that the observer islocated at the position with coordinates (ro θo) where roand θo represents the angular coordinate on observerrsquossky Furthermore we need to introduce two celestial co-ordinates α and β for the observer by using the follow-ing relations [29]

α = minusrop(φ)

p(t) β = ro

p(θ)

p(t) (64)

where (p(t) p(r) p(θ) p(φ)) are the tetrad componentsof the photon momentum with respect to locally non-rotating reference frame The observer bases emicro

(ν)can be

10

expanded as a form in the coordinate bases (see [64])

p(t) = minusemicro

(t)pmicro = E ζ12 minus γ12 L p(φ) = emicro

(φ)pmicro =

Lradicgφφ

p(θ) = emicro

(θ)pmicro =

pθradicgθθ

p(r) = emicro

(r)pmicro =prradicgrr

(65)

where the quantities E = minuspt and pφ = L are conserveddue to the associated Killing vectors If we use ξ = LEη = KE2 and pθ = plusmn

radicΘ(θ) we can rewrite these co-

ordinates in terms our parameters ξ and η as follows[43]

α12 = minusroξ12radicgφφ(ζ12 minus γ12ξ12)

|(ro θo)

β12 = plusmnro

radicη12 + a2 cos2 θ minus ξ2

12 cot2 θradic

gθθ(ζ12 minus γ12ξ12)|(ro θo) (66)

where

ζ12 =

radicgφφ

g2tφ minus gttgφφ

(67)

and

γ12 = minusgtφ

gφφζ12 (68)

Finally to simplify the problem further we are going toconsider that our observer is located in the equatorialplane (θ = π2) and very large but finite ro = D = 83kpc we find

α12 = minusradic

f12 ξ12

β12 = plusmnradic

f12radic

η12 (69)

Note that f12 are given by Eqs (32) and (34) respec-tively We see that due to the presence of SFDM our solu-tion is non-asymptotically flat In the special case whenSFDM is absent we recover f12 rarr 1 hence the aboverelations reduces to the asymptotically flat case Notethat for the observer located at the position with coor-dinates (ro θo) we have neglected the effect of rotationwhile ξ12 and η12 given by Eq (61) and are evaluatedat the circular photon orbits r = rcir

For more details in Figs 5 and 6 we plot the shape ofSgr Alowast black hole shadows Using the parameter valuesspecifying the SFDM and baryonic matter we obtain al-most no effect on the shadow images In fact one canobserve that the dashed curve (exponential profile) anddashed red curve (power law profile) are almost indis-tinguishable from the black dashed curve (Kerr vacuumBH) However we can observe that by increasing thebaryonic mass the shadow size increases considerably(solid bluered curve) The angular radius of the Sgr Alowastblack hole shadow can be estimated using the observ-able Rs as θs = Rs MD where M is the black hole mass

5196 5197 5198 5199 5200

5315

5316

5317

5318

Rs

θs [μas]

FIG 7 The expected value for the angular diameter in thecase of Sgr Alowast Note that we have adopted D = 83 kpc and

M = 43times 106M

and D is the distance between the black hole and theobserver The angular radius can be further expressedas θs = 987098times 10minus6Rs(MM)(1kpcD) microas In thecase of Sgr Alowast we have used M = 43 times 106M andD = 83 kpc [4] is the distance between the Earth andSgr Alowast center black hole In order to get more informa-tion about the effect of SFDM on physical observableswe shall estimate the effect of SFDM on the angular ra-dius for Sgr Alowast To simplify the problem let us considerour static and spherically symmetric black hole metricin a totally dominated dark matter galaxy described bythe function

F (r) = eminusX minus 2Mr

(70)

where we have introduced

X =ρ2

04Λ2m6 (71)

where we have used the approximation cos( π r2 R ) 1

since our solution is valid for r le R From the circularphoton orbit conditions one can show [41]

2minus rF prime(r)F (r) = 0 (72)

Now if we expand around X in the above function weobtain

F (r) = 1minus Xminus 2Mr

+ (73)

By solving this equation and considering only the lead-ing order terms one can determine the radius of the pho-ton sphere rps yielding

rps = 3M(1 + X) (74)

On the other hand one also can show the relation [41]

ξ2 + η =r2

ps

F (rps) (75)

11

The shadow radius Rs can be expressed in terms of thecelestial coordinates (α β) as follows

Rs =radic

α2 + β2 =radic

1minus XrpsradicF (rps)

(76)

Note that in the last equation we have used Eqs (69) and(75) Using the values m = 06 eV Λ = 02 meV andρ0 = 92 times 10minus8 eV4 in units of the Sgr Alowast black hole wefind Rs = 5196158313M Using this result we can findthat the angular diameter increases by δθs = 3times 10minus5

microas compared to the Schwarzschild vacuum This resultis consistent with the result reported in Ref [4] whereauthors estimated that the dark matter halo could influ-ence the shadow of Sgr Alowast at a level of order of magni-tude of 10minus3 microas and 10minus5 microas respectively They haveused the so called Cold Dark Matter and Scalar FieldDark Matter models to study the apparent shapes of theshadow A similar result has been obtained for the darkmatter effect on the M87 black hole (see [5]) using theBurkert dark matter profile

VI CONCLUSION

In this paper we have obtained a rotating black holesolution surrounded by superfluid dark matter alongwith baryonic matter To achieve this purpose in thepresent work we considered the superfluid dark mat-ter model and used two specific profiles to describe thebaryonic matter distribution namely the spherical ex-ponential profile and the power law profile followed bythe special case of a totally dominated dark matter caseUsing the current values for the baryonic mass centraldensity of the superfluid dark matter halo radius aswell as the radial scale length for the baryonic matterin our galaxy we found that the shadow size of Sgr Alowastblack hole remains almost unchanged compared to theKerr vacuum BH This result is consistent with a recentwork reported in [5] and also [4] For entirely dominateddark matter galaxies we find that the angular diame-ter increases by 10minus5microarcsec This result shows that itis very difficult to constrain the dark matter parametersusing the shadow images Such tiny effects on the angu-lar diameter are out of reach for the existing space tech-nology and it remains an open question if future astro-nomical observations can potentially detect such effectsThat being said the expected value for the angular di-ameter in the case of Sgr Alowast is of the order of θs 53 microasas predicted in GR

As an interesting observation we show that an in-crease of baryonic mass followed by a decrease of theradial scale length can increase the shadow size consid-erably This can be explained by the fact that the bary-onic matter is mostly located in the interior of the galaxyon the other hand dark matter is mostly located in theouter region of the galaxy In other words for the to-tally dominated dark matter galaxies we observe almost

no effect on black hole shadows but a more precise mea-surement of the Sgr Alowast shadow radius can play a signifi-cant role in determining the baryonic mass in our galaxy

Acknowledgements

TZ is supported by National Natural Science Founda-tion of China under the Grants No 11675143 the Zhe-jiang Provincial Natural Science Foundation of Chinaunder Grant No LY20A050002 and the FundamentalResearch Funds for the Provincial Universities of Zhe-jiang in China under Grants No RF-A2019015

Appendix A

Here we demonstrate the derivation of the spinningblack hole metric As a first step to this formalism wetransform Boyer-Lindquist (BL) coordinates (t r θ φ) toEddington-Finkelstein (EF) coordinates (u r θ φ) Thiscan be achieved by using the coordinate transformation

dt = du +drradic

F12(r)G12(r) (A1)

we obtain

ds212 = minusF12(r)du2minus 2

radicF12(r)G12(r)

dudr+ r2dθ2 + r2 sin2 θdφ2

(A2)This metric can be expressed in terms of null tetrads

as

gmicroν = minuslmicronν minus lνnmicro + mmicromν + mνmmicro (A3)

where the null tetrads are defined as

lmicro = δmicror (A4)

nmicro =

radicG12(r)F12(r)

δmicrou minus

12G12(r)δ

micror (A5)

mmicro =1radic2H

(δ

microθ +

ι

sin θδ

microφ

) (A6)

These null tetrads are constructed in such a way that lmicro

and nmicro while mmicro and mmicro are complex It is worth notingthatH = r2 and will be used later on It is obvious fromthe notation that mmicro is complex conjugate of mmicro Thesevectors further satisfy the conditions for normalizationorthogonality and isotropy as

lmicrolmicro = nmicronmicro = mmicrommicro = mmicrommicro = 0 (A7)lmicrommicro = lmicrommicro = nmicrommicro = nmicrommicro = 0 (A8)

minuslmicronmicro = mmicrommicro = 1 (A9)

12

Following the NewmanndashJanis prescription we write

xprimemicro = xmicro + ia(δmicror minus δ

microu ) cos θ rarr

uprime = uminus ia cos θrprime = r + ia cos θθprime = θφprime = φ

(A10)in which a stands for the rotation parameter Nextlet the null tetrad vectors Za undergo a transformationgiven by Zmicro = (partxmicropartxprimeν)Zprimeν following [62]

lprimemicro = δmicror (A11)

nprimemicro =

radicB12

A12δ

microu minus

12

B12δmicror

mprimemicro =1radic

2Σ12

[(δ

microu minus δ

micror )ιa sin θ + δ

microθ +

ι

sin θδ

microφ

]where we have assumed that (F12(r)G12(r)H(r))transform to (A12(a r θ) B12(a r θ) Σ12(a r θ)) Withthe help of the above equations new metric inEddington-Finkelste coordinate reads

ds212 = minusA12du2 minus 2

radicA12

B12dudr (A12)

+ 2a sin2 θ

(A12 minus

radicA12

B12

)dudφ

+ 2a

radicA12

B12sin2 θdrdφ + Σ12dθ2

+ sin2 θ

[Σ12 + a2 sin2 θ

(2

radicA12

B12minus A12

)]dφ2

Note that A12 is some function of r and θ as we al-ready pointed out Without going into details of the cal-culation one can revert the EF coordinates back to BLcoordinates by using the following transformation (see

for more details [62])

du = dtminus a2 + r2

∆12dr dφ = dφprime minus a

∆12dr (A13)

where in order to simplify the notation we introduce

∆(r)12 = r2 F12(r) + a2 (A14)

Making use of F12(r) = G12(r) one can obtain Σ1 =Σ2 = r2 + a2 cos2 θ herein we shall use just Σ Droppingthe primes in the φ coordinate and choosing

A12 =(F12H+ a2 cos2 θ)Σ12

(H+ a2 cos2 θ)2 (A15)

and

B12 =F12H+ a2 cos2 θ

Σ (A16)

we obtain the spinning black hole space-time metric ina SFDM halo

ds212 = minus

(1minus 2Υ12(r)r

Σ

)dt2 +

Σ∆12

dr2 + Σdθ2

minus 2a sin2 θ2Υ12(r)r

Σdtdφ

+ sin2 θ

[(r2 + a2)2 minus a2∆12 sin2 θ

Σ

]dφ2 (A17)

with

Υ12(r) =r (1minusF12(r))

2 (A18)

in which F1 = F (r)exp and F2 = F (r)power respec-tively

[1] The Event Horizon Telescope Collaboration Ap J Lett875 L1 (2019)

[2] Haroon S Jamil M Jusufi K Lin K Mann R B PhysRev D 99 no4 044015 (2019)

[3] Haroon S Jusufi K Jamil M Universe 6(2) 23 (2020)arXiv190400711[gr-qc]

[4] Hou X Xu Z Zhou M Wang J JCAP 1807 no07 015(2018)

[5] Jusufi K Jamil M Salucci P Zhu T Haroon S PhysRev D 100 044012 (2019)

[6] Xu Z Hou X Gong X Wang J JCAP 1809 no09 038(2018)

[7] Xu Z Hou X Wang J JCAP 1810 no10 046 (2018)[8] Berezhiani L Khoury J Phys Rev D 92 103510 (2015)[9] Synge J L Mon Not Roy Astron Soc 131 no 3 463

(1966)

[10] Luminet J P Astron Astrophys 75 228 (1979)[11] Bardeen J M in Black Holes (Proceedings Les Houches

France August 1972) edited by C DeWitt and B S De-Witt

[12] Abdujabbarov A Toshmatov B Stuchlik Z AhmedovB Int J Mod Phys D 26 no 06 1750051 (2016)

[13] Abdikamalov A B Abdujabbarov A A Ayzenberg DMalafarina D Bambi C Ahmedov B Phys Rev D 100(2019) no2 024014

[14] Abdujabbarov A Amir M Ahmedov B Ghosh S GPhys Rev D 93 no 10 104004 (2016)

[15] Amir M Jusufi K Banerjee A Hansraj S ClassQuant Grav 36 no21 215007 (2019)

[16] Amir M Ghosh S G Phys Rev D 94 no 2 024054(2016)

[17] Abdujabbarov A Atamurotov F Kucukakca Y Ahme-

6

00 05 10 15 20

-05

00

05

10

rM

Δ1

Exponential profile

00 05 10 15 20

-05

00

05

10

rM

Δ2

Power law profile

FIG 2 Left panel Variation of ∆1 as a function of r for BH-SFDM using the spherical exponential profile Right panelVariation of ∆2 as a function of r for BH-SFDM using the power profile law Note that a = 065 (black curve) a = 085 (red

curve) a = 1 (blue curve) respectively We have used m = 06 eV and Λ = 02 meV we find ρ0 = 002 times 10minus24 gcm3 or in eVunits ρ0 = 92 times 10minus8 eV4 For the Milky Way galaxy we can use MB = 6times 1010 M = 139 times 104 MBH and rc = 26

kpc= 124 times 1010 MBH

00 05 10 15 20-10

0

10

20

30

rM

Veff

Exponential profile

00 05 10 15 20-10

0

10

20

30

rM

Veff

Power law profile

FIG 3 Left panel The effective potential of photon moving for BH-SFDM using the exponsntial profile Right panel Theeffective potential of photon using thepower law profile We have chosen a = 05 (black color) a = 065 (red color) and a = 085

(blue color) respectively

A Geodesic equations

In order to find the contour of a black hole shadow ofthe rotating spacetime (45) first we need to find the nullgeodesic equations using the Hamilton-Jacobi methodgiven by

partSpartτ

= minus12

gmicroν partSpartxmicro

partSpartxν

(49)

where τ is the affine parameter and S is the Jacobi ac-tion Due to the spectime symmetries there are two con-served quantities namely the conserved energy E =minuspt and the conserved angular momentum L = pφ re-spectively

To find the separable solution of Eq (49) we need to

express the action in the following form

S =12

micro2τ minus Et + Jφ + Sr(r) + Sθ(θ) (50)

in which micro gives the mass of the test particle Of coursethis gives micro = 0 in the case of the photon That beingsaid it is straightforward to obtain the following equa-

7

-2 -1 0 1 2

-2

-1

0

1

2

rsin(θ)cos(φ)

rcos

(θ)

M=1 a=04

-2 -1 0 1 2

-2

-1

0

1

2

rsin(θ)cos(φ)

rcos

(θ)

M=1 a=065

-2 -1 0 1 2

-2

-1

0

1

2

rsin(θ)cos(φ)

rcos

(θ)

M=1 a=085

-2 -1 0 1 2

-2

-1

0

1

2

rsin(θ)cos(φ)

rcos

(θ)

M=1 a=098

FIG 4 The shape of the ergoregion and innerouter horizons for different values of a using the spherical exponential profileWe use M = 1 in units of the Sgr Alowast black hole mass given by MBH = 43 times 106 M and R = 158 kpc or R = 755 times 1010 MBH Furthermore for m = 06 eV and Λ = 02 meV we find ρ0 = 002 times 10minus24 gcm3 or in eV units ρ0 = 92 times 10minus8 eV4 For the

normal matter contribution of the Milky Way galaxy we can use MB = 6times 1010 M = 139 times 104 MBH and L = 26kpc= 124 times 1010 MBH

tions of motions from the Hamilton-Jacobi equation

Σdtdτ

=r2 + a2

∆12[E(r2 + a2)minus aL]minus a(aE sin2 θ minus L)

(51)

Σdrdτ

=radicR(r) (52)

Σdθ

dτ=radic

Θ(θ) (53)

Σdφ

dτ=

a∆12

[E(r2 + a2)minus aL]minus(

aEminus Lsin2 θ

) (54)

whereR(r) and Θ(θ) are given by

R(r) = [E(r2 + a2)minus aL]2 minus ∆12[m2r2 + (aEminus L)2 +K]

(55)

Θ(θ) = Kminus(

L2

sin2 θminus a2E2

)cos2 θ (56)

and K is the separation constant known as the Carterconstant

8

Kerr vacuum BH

BH-SFDM

-10 -5 0 5-6

-4

-2

0

2

4

6

α

β

a=02

Kerr vacuum BH

BH-SFDM

-10 -5 0 5-6

-4

-2

0

2

4

6

α

β

a=05

Kerr vacuum BH

BH-SFDM

-10 -5 0 5-6

-4

-2

0

2

4

6

α

β

a=085

Kerr vacuum BH

BH-SFDM

-10 -5 0 5-6

-4

-2

0

2

4

6

α

β

a=098

FIG 5 Variation in shape of shadow using the spherical exponential profile for baryonic matter We use M = 1 in units of theSgr Alowast black hole mass given by MBH = 43 times 106 M and R = 158 kpc or R = 755 times 1010 MBH Furthermore for m = 06 eV

and Λ = 02 meV we find ρ0 = 002 times 10minus24 gcm3 or in eV units ρ0 = 92 times 10minus8 eV4 For the Milky Way galaxy we can useMB = 6times 1010 M = 139 times 104 MBH and L = 26 kpc= 124 times 1010 MBH We observe that the dashed blue curve correspondingto Sgr Alowast with the above parameters is almost indistinguishable from the black dashed curve describing the Kerr vacuum BHThe solid blue curve corresponds to the case of increasing the baryonic mass by a factor of 102 and decrease of core radius by a

factor of 103

B Circular Orbits

Due to the strong gravity near the black hole it is thusexpected that the photons emitted near a black hole willeventually fall into the black hole or eventually scatteraway from it In this way the photons captured by theblack hole will form a dark region defining the contourof the shadow To elaborate the presence of unstablecircular orbits around the black hole we need to studythe radial geodesic by introducing the effective poten-tial Veff which can be written as follows

Σ2(

drdτ

)2+ V12

eff = 0 (57)

where V1eff = Vexp

eff and V2eff = Vpower

eff gives the expo-nential and the power law respectively At this point itis convenient to introduce two parameters ξ and η de-fined as

ξ = LE η = KE2 (58)

For the effective potential then we obtain the followingrelation

V12eff = ∆12((aminus ξ12)

2 + η12)minus (r2 + a2minus a ξ12)2 (59)

where we have replaced V12eff E2 by V12

eff For more de-tails in Figure (3) we plot the variation of the effectivepotential associated with the radial motion of photons

9

Kerr vacuum BH

BH-SFDM

-10 -5 0 5-6

-4

-2

0

2

4

6

α

β

a=02

Kerr vacuum BH

BH-SFDM

-10 -5 0 5-6

-4

-2

0

2

4

6

α

β

a=05

Kerr vacuum BH

BH-SFDM

-10 -5 0 5-6

-4

-2

0

2

4

6

α

β

a=085

Kerr vacuum BH

BH-SFDM

-10 -5 0 5-6

-4

-2

0

2

4

6

α

β

a=098

FIG 6 Variation in shape of shadow for different values of a using the power law profile We use M = 1 in units of the Sgr Alowast

black hole mass given by MBH = 43 times 106 M and R = 158 kpc or R = 755 times 1010 MBH Furthermore for m = 06 eV andΛ = 02 meV we find ρ0 = 002 times 10minus24 gcm3 or in eV units ρ0 = 92 times 10minus8 eV4 For the Milky Way galaxy we can use

MB = 6times 1010 M = 139 times 104 MBH and rc = 26 kpc= 124 times 1010 MBH The dashed red curve corresponds to Sgr Alowast usingthe above parameters which is almost indistinguishable from the black dashed curve discribing the Kerr vacuum BH The solid

red curve corresponds to the case of increasing the baryonic mass by a factor of 102 and decrease of core radius by a factor of 103

The circular photon orbits exists when at some constantr = rcir the conditions

V12eff (r) = 0

dV12eff (r)dr

= 0 (60)

Combining all these equations it is possible to showthat

ξ12 =(r2 + a2)(rF prime12(r) + 2F12(r))minus 4(r2F12(r) + a2)

a(rF prime12(r) + 2F12(r))

η12 =r3[8a2F prime12(r)minus r(rF prime12(r)minus 2F12(r))2]

a2(rF prime12(r) + 2F12(r))2 (61)

One can recover the Kerr vacuum case by letting a0 =MB = ρ0 = 0 yielding

ξ12 =r2(3Mminus r)minus a2(Mminus r)

a(rminusM) (62)

η12 =r3 (4Ma2 minus r(rminus 3M)2)

a2(rminusM)2 (63)

To obtain the shadow images of our black hole in thepresence of dark matter we assume that the observer islocated at the position with coordinates (ro θo) where roand θo represents the angular coordinate on observerrsquossky Furthermore we need to introduce two celestial co-ordinates α and β for the observer by using the follow-ing relations [29]

α = minusrop(φ)

p(t) β = ro

p(θ)

p(t) (64)

where (p(t) p(r) p(θ) p(φ)) are the tetrad componentsof the photon momentum with respect to locally non-rotating reference frame The observer bases emicro

(ν)can be

10

expanded as a form in the coordinate bases (see [64])

p(t) = minusemicro

(t)pmicro = E ζ12 minus γ12 L p(φ) = emicro

(φ)pmicro =

Lradicgφφ

p(θ) = emicro

(θ)pmicro =

pθradicgθθ

p(r) = emicro

(r)pmicro =prradicgrr

(65)

where the quantities E = minuspt and pφ = L are conserveddue to the associated Killing vectors If we use ξ = LEη = KE2 and pθ = plusmn

radicΘ(θ) we can rewrite these co-

ordinates in terms our parameters ξ and η as follows[43]

α12 = minusroξ12radicgφφ(ζ12 minus γ12ξ12)

|(ro θo)

β12 = plusmnro

radicη12 + a2 cos2 θ minus ξ2

12 cot2 θradic

gθθ(ζ12 minus γ12ξ12)|(ro θo) (66)

where

ζ12 =

radicgφφ

g2tφ minus gttgφφ

(67)

and

γ12 = minusgtφ

gφφζ12 (68)

Finally to simplify the problem further we are going toconsider that our observer is located in the equatorialplane (θ = π2) and very large but finite ro = D = 83kpc we find

α12 = minusradic

f12 ξ12

β12 = plusmnradic

f12radic

η12 (69)

Note that f12 are given by Eqs (32) and (34) respec-tively We see that due to the presence of SFDM our solu-tion is non-asymptotically flat In the special case whenSFDM is absent we recover f12 rarr 1 hence the aboverelations reduces to the asymptotically flat case Notethat for the observer located at the position with coor-dinates (ro θo) we have neglected the effect of rotationwhile ξ12 and η12 given by Eq (61) and are evaluatedat the circular photon orbits r = rcir

For more details in Figs 5 and 6 we plot the shape ofSgr Alowast black hole shadows Using the parameter valuesspecifying the SFDM and baryonic matter we obtain al-most no effect on the shadow images In fact one canobserve that the dashed curve (exponential profile) anddashed red curve (power law profile) are almost indis-tinguishable from the black dashed curve (Kerr vacuumBH) However we can observe that by increasing thebaryonic mass the shadow size increases considerably(solid bluered curve) The angular radius of the Sgr Alowastblack hole shadow can be estimated using the observ-able Rs as θs = Rs MD where M is the black hole mass

5196 5197 5198 5199 5200

5315

5316

5317

5318

Rs

θs [μas]

FIG 7 The expected value for the angular diameter in thecase of Sgr Alowast Note that we have adopted D = 83 kpc and

M = 43times 106M

and D is the distance between the black hole and theobserver The angular radius can be further expressedas θs = 987098times 10minus6Rs(MM)(1kpcD) microas In thecase of Sgr Alowast we have used M = 43 times 106M andD = 83 kpc [4] is the distance between the Earth andSgr Alowast center black hole In order to get more informa-tion about the effect of SFDM on physical observableswe shall estimate the effect of SFDM on the angular ra-dius for Sgr Alowast To simplify the problem let us considerour static and spherically symmetric black hole metricin a totally dominated dark matter galaxy described bythe function

F (r) = eminusX minus 2Mr

(70)

where we have introduced

X =ρ2

04Λ2m6 (71)

where we have used the approximation cos( π r2 R ) 1

since our solution is valid for r le R From the circularphoton orbit conditions one can show [41]

2minus rF prime(r)F (r) = 0 (72)

Now if we expand around X in the above function weobtain

F (r) = 1minus Xminus 2Mr

+ (73)

By solving this equation and considering only the lead-ing order terms one can determine the radius of the pho-ton sphere rps yielding

rps = 3M(1 + X) (74)

On the other hand one also can show the relation [41]

ξ2 + η =r2

ps

F (rps) (75)

11

The shadow radius Rs can be expressed in terms of thecelestial coordinates (α β) as follows

Rs =radic

α2 + β2 =radic

1minus XrpsradicF (rps)

(76)

Note that in the last equation we have used Eqs (69) and(75) Using the values m = 06 eV Λ = 02 meV andρ0 = 92 times 10minus8 eV4 in units of the Sgr Alowast black hole wefind Rs = 5196158313M Using this result we can findthat the angular diameter increases by δθs = 3times 10minus5

microas compared to the Schwarzschild vacuum This resultis consistent with the result reported in Ref [4] whereauthors estimated that the dark matter halo could influ-ence the shadow of Sgr Alowast at a level of order of magni-tude of 10minus3 microas and 10minus5 microas respectively They haveused the so called Cold Dark Matter and Scalar FieldDark Matter models to study the apparent shapes of theshadow A similar result has been obtained for the darkmatter effect on the M87 black hole (see [5]) using theBurkert dark matter profile

VI CONCLUSION

In this paper we have obtained a rotating black holesolution surrounded by superfluid dark matter alongwith baryonic matter To achieve this purpose in thepresent work we considered the superfluid dark mat-ter model and used two specific profiles to describe thebaryonic matter distribution namely the spherical ex-ponential profile and the power law profile followed bythe special case of a totally dominated dark matter caseUsing the current values for the baryonic mass centraldensity of the superfluid dark matter halo radius aswell as the radial scale length for the baryonic matterin our galaxy we found that the shadow size of Sgr Alowastblack hole remains almost unchanged compared to theKerr vacuum BH This result is consistent with a recentwork reported in [5] and also [4] For entirely dominateddark matter galaxies we find that the angular diame-ter increases by 10minus5microarcsec This result shows that itis very difficult to constrain the dark matter parametersusing the shadow images Such tiny effects on the angu-lar diameter are out of reach for the existing space tech-nology and it remains an open question if future astro-nomical observations can potentially detect such effectsThat being said the expected value for the angular di-ameter in the case of Sgr Alowast is of the order of θs 53 microasas predicted in GR

As an interesting observation we show that an in-crease of baryonic mass followed by a decrease of theradial scale length can increase the shadow size consid-erably This can be explained by the fact that the bary-onic matter is mostly located in the interior of the galaxyon the other hand dark matter is mostly located in theouter region of the galaxy In other words for the to-tally dominated dark matter galaxies we observe almost

no effect on black hole shadows but a more precise mea-surement of the Sgr Alowast shadow radius can play a signifi-cant role in determining the baryonic mass in our galaxy

Acknowledgements

TZ is supported by National Natural Science Founda-tion of China under the Grants No 11675143 the Zhe-jiang Provincial Natural Science Foundation of Chinaunder Grant No LY20A050002 and the FundamentalResearch Funds for the Provincial Universities of Zhe-jiang in China under Grants No RF-A2019015

Appendix A

Here we demonstrate the derivation of the spinningblack hole metric As a first step to this formalism wetransform Boyer-Lindquist (BL) coordinates (t r θ φ) toEddington-Finkelstein (EF) coordinates (u r θ φ) Thiscan be achieved by using the coordinate transformation

dt = du +drradic

F12(r)G12(r) (A1)

we obtain

ds212 = minusF12(r)du2minus 2

radicF12(r)G12(r)

dudr+ r2dθ2 + r2 sin2 θdφ2

(A2)This metric can be expressed in terms of null tetrads

as

gmicroν = minuslmicronν minus lνnmicro + mmicromν + mνmmicro (A3)

where the null tetrads are defined as

lmicro = δmicror (A4)

nmicro =

radicG12(r)F12(r)

δmicrou minus

12G12(r)δ

micror (A5)

mmicro =1radic2H

(δ

microθ +

ι

sin θδ

microφ

) (A6)

These null tetrads are constructed in such a way that lmicro

and nmicro while mmicro and mmicro are complex It is worth notingthatH = r2 and will be used later on It is obvious fromthe notation that mmicro is complex conjugate of mmicro Thesevectors further satisfy the conditions for normalizationorthogonality and isotropy as

lmicrolmicro = nmicronmicro = mmicrommicro = mmicrommicro = 0 (A7)lmicrommicro = lmicrommicro = nmicrommicro = nmicrommicro = 0 (A8)

minuslmicronmicro = mmicrommicro = 1 (A9)

12

Following the NewmanndashJanis prescription we write

xprimemicro = xmicro + ia(δmicror minus δ

microu ) cos θ rarr

uprime = uminus ia cos θrprime = r + ia cos θθprime = θφprime = φ

(A10)in which a stands for the rotation parameter Nextlet the null tetrad vectors Za undergo a transformationgiven by Zmicro = (partxmicropartxprimeν)Zprimeν following [62]

lprimemicro = δmicror (A11)

nprimemicro =

radicB12

A12δ

microu minus

12

B12δmicror

mprimemicro =1radic

2Σ12

[(δ

microu minus δ

micror )ιa sin θ + δ

microθ +

ι

sin θδ

microφ

]where we have assumed that (F12(r)G12(r)H(r))transform to (A12(a r θ) B12(a r θ) Σ12(a r θ)) Withthe help of the above equations new metric inEddington-Finkelste coordinate reads

ds212 = minusA12du2 minus 2

radicA12

B12dudr (A12)

+ 2a sin2 θ

(A12 minus

radicA12

B12

)dudφ

+ 2a

radicA12

B12sin2 θdrdφ + Σ12dθ2

+ sin2 θ

[Σ12 + a2 sin2 θ

(2

radicA12

B12minus A12

)]dφ2

Note that A12 is some function of r and θ as we al-ready pointed out Without going into details of the cal-culation one can revert the EF coordinates back to BLcoordinates by using the following transformation (see

for more details [62])

du = dtminus a2 + r2

∆12dr dφ = dφprime minus a

∆12dr (A13)

where in order to simplify the notation we introduce

∆(r)12 = r2 F12(r) + a2 (A14)

Making use of F12(r) = G12(r) one can obtain Σ1 =Σ2 = r2 + a2 cos2 θ herein we shall use just Σ Droppingthe primes in the φ coordinate and choosing

A12 =(F12H+ a2 cos2 θ)Σ12

(H+ a2 cos2 θ)2 (A15)

and

B12 =F12H+ a2 cos2 θ

Σ (A16)

we obtain the spinning black hole space-time metric ina SFDM halo

ds212 = minus

(1minus 2Υ12(r)r

Σ

)dt2 +

Σ∆12

dr2 + Σdθ2

minus 2a sin2 θ2Υ12(r)r

Σdtdφ

+ sin2 θ

[(r2 + a2)2 minus a2∆12 sin2 θ

Σ

]dφ2 (A17)

with

Υ12(r) =r (1minusF12(r))

2 (A18)

in which F1 = F (r)exp and F2 = F (r)power respec-tively

[1] The Event Horizon Telescope Collaboration Ap J Lett875 L1 (2019)

[2] Haroon S Jamil M Jusufi K Lin K Mann R B PhysRev D 99 no4 044015 (2019)

[3] Haroon S Jusufi K Jamil M Universe 6(2) 23 (2020)arXiv190400711[gr-qc]

[4] Hou X Xu Z Zhou M Wang J JCAP 1807 no07 015(2018)

[5] Jusufi K Jamil M Salucci P Zhu T Haroon S PhysRev D 100 044012 (2019)

[6] Xu Z Hou X Gong X Wang J JCAP 1809 no09 038(2018)

[7] Xu Z Hou X Wang J JCAP 1810 no10 046 (2018)[8] Berezhiani L Khoury J Phys Rev D 92 103510 (2015)[9] Synge J L Mon Not Roy Astron Soc 131 no 3 463

(1966)

[10] Luminet J P Astron Astrophys 75 228 (1979)[11] Bardeen J M in Black Holes (Proceedings Les Houches

France August 1972) edited by C DeWitt and B S De-Witt

[12] Abdujabbarov A Toshmatov B Stuchlik Z AhmedovB Int J Mod Phys D 26 no 06 1750051 (2016)

[13] Abdikamalov A B Abdujabbarov A A Ayzenberg DMalafarina D Bambi C Ahmedov B Phys Rev D 100(2019) no2 024014

[14] Abdujabbarov A Amir M Ahmedov B Ghosh S GPhys Rev D 93 no 10 104004 (2016)

[15] Amir M Jusufi K Banerjee A Hansraj S ClassQuant Grav 36 no21 215007 (2019)

[16] Amir M Ghosh S G Phys Rev D 94 no 2 024054(2016)

[17] Abdujabbarov A Atamurotov F Kucukakca Y Ahme-

7

-2 -1 0 1 2

-2

-1

0

1

2

rsin(θ)cos(φ)

rcos

(θ)

M=1 a=04

-2 -1 0 1 2

-2

-1

0

1

2

rsin(θ)cos(φ)

rcos

(θ)

M=1 a=065

-2 -1 0 1 2

-2

-1

0

1

2

rsin(θ)cos(φ)

rcos

(θ)

M=1 a=085

-2 -1 0 1 2

-2

-1

0

1

2

rsin(θ)cos(φ)

rcos

(θ)

M=1 a=098

FIG 4 The shape of the ergoregion and innerouter horizons for different values of a using the spherical exponential profileWe use M = 1 in units of the Sgr Alowast black hole mass given by MBH = 43 times 106 M and R = 158 kpc or R = 755 times 1010 MBH Furthermore for m = 06 eV and Λ = 02 meV we find ρ0 = 002 times 10minus24 gcm3 or in eV units ρ0 = 92 times 10minus8 eV4 For the

normal matter contribution of the Milky Way galaxy we can use MB = 6times 1010 M = 139 times 104 MBH and L = 26kpc= 124 times 1010 MBH

tions of motions from the Hamilton-Jacobi equation

Σdtdτ

=r2 + a2

∆12[E(r2 + a2)minus aL]minus a(aE sin2 θ minus L)

(51)

Σdrdτ

=radicR(r) (52)

Σdθ

dτ=radic

Θ(θ) (53)

Σdφ

dτ=

a∆12

[E(r2 + a2)minus aL]minus(

aEminus Lsin2 θ

) (54)

whereR(r) and Θ(θ) are given by

R(r) = [E(r2 + a2)minus aL]2 minus ∆12[m2r2 + (aEminus L)2 +K]

(55)

Θ(θ) = Kminus(

L2

sin2 θminus a2E2

)cos2 θ (56)

and K is the separation constant known as the Carterconstant

8

Kerr vacuum BH

BH-SFDM

-10 -5 0 5-6

-4

-2

0

2

4

6

α

β

a=02

Kerr vacuum BH

BH-SFDM

-10 -5 0 5-6

-4

-2

0

2

4

6

α

β

a=05

Kerr vacuum BH

BH-SFDM

-10 -5 0 5-6

-4

-2

0

2

4

6

α

β

a=085

Kerr vacuum BH

BH-SFDM

-10 -5 0 5-6

-4

-2

0

2

4

6

α

β

a=098

FIG 5 Variation in shape of shadow using the spherical exponential profile for baryonic matter We use M = 1 in units of theSgr Alowast black hole mass given by MBH = 43 times 106 M and R = 158 kpc or R = 755 times 1010 MBH Furthermore for m = 06 eV

and Λ = 02 meV we find ρ0 = 002 times 10minus24 gcm3 or in eV units ρ0 = 92 times 10minus8 eV4 For the Milky Way galaxy we can useMB = 6times 1010 M = 139 times 104 MBH and L = 26 kpc= 124 times 1010 MBH We observe that the dashed blue curve correspondingto Sgr Alowast with the above parameters is almost indistinguishable from the black dashed curve describing the Kerr vacuum BHThe solid blue curve corresponds to the case of increasing the baryonic mass by a factor of 102 and decrease of core radius by a

factor of 103

B Circular Orbits

Due to the strong gravity near the black hole it is thusexpected that the photons emitted near a black hole willeventually fall into the black hole or eventually scatteraway from it In this way the photons captured by theblack hole will form a dark region defining the contourof the shadow To elaborate the presence of unstablecircular orbits around the black hole we need to studythe radial geodesic by introducing the effective poten-tial Veff which can be written as follows

Σ2(

drdτ

)2+ V12

eff = 0 (57)

where V1eff = Vexp

eff and V2eff = Vpower

eff gives the expo-nential and the power law respectively At this point itis convenient to introduce two parameters ξ and η de-fined as

ξ = LE η = KE2 (58)

For the effective potential then we obtain the followingrelation

V12eff = ∆12((aminus ξ12)

2 + η12)minus (r2 + a2minus a ξ12)2 (59)

where we have replaced V12eff E2 by V12

eff For more de-tails in Figure (3) we plot the variation of the effectivepotential associated with the radial motion of photons

9

Kerr vacuum BH

BH-SFDM

-10 -5 0 5-6

-4

-2

0

2

4

6

α

β

a=02

Kerr vacuum BH

BH-SFDM

-10 -5 0 5-6

-4

-2

0

2

4

6

α

β

a=05

Kerr vacuum BH

BH-SFDM

-10 -5 0 5-6

-4

-2

0

2

4

6

α

β

a=085

Kerr vacuum BH

BH-SFDM

-10 -5 0 5-6

-4

-2

0

2

4

6

α

β

a=098

FIG 6 Variation in shape of shadow for different values of a using the power law profile We use M = 1 in units of the Sgr Alowast

black hole mass given by MBH = 43 times 106 M and R = 158 kpc or R = 755 times 1010 MBH Furthermore for m = 06 eV andΛ = 02 meV we find ρ0 = 002 times 10minus24 gcm3 or in eV units ρ0 = 92 times 10minus8 eV4 For the Milky Way galaxy we can use

MB = 6times 1010 M = 139 times 104 MBH and rc = 26 kpc= 124 times 1010 MBH The dashed red curve corresponds to Sgr Alowast usingthe above parameters which is almost indistinguishable from the black dashed curve discribing the Kerr vacuum BH The solid

red curve corresponds to the case of increasing the baryonic mass by a factor of 102 and decrease of core radius by a factor of 103

The circular photon orbits exists when at some constantr = rcir the conditions

V12eff (r) = 0

dV12eff (r)dr

= 0 (60)

Combining all these equations it is possible to showthat

ξ12 =(r2 + a2)(rF prime12(r) + 2F12(r))minus 4(r2F12(r) + a2)

a(rF prime12(r) + 2F12(r))

η12 =r3[8a2F prime12(r)minus r(rF prime12(r)minus 2F12(r))2]

a2(rF prime12(r) + 2F12(r))2 (61)

One can recover the Kerr vacuum case by letting a0 =MB = ρ0 = 0 yielding

ξ12 =r2(3Mminus r)minus a2(Mminus r)

a(rminusM) (62)

η12 =r3 (4Ma2 minus r(rminus 3M)2)

a2(rminusM)2 (63)

To obtain the shadow images of our black hole in thepresence of dark matter we assume that the observer islocated at the position with coordinates (ro θo) where roand θo represents the angular coordinate on observerrsquossky Furthermore we need to introduce two celestial co-ordinates α and β for the observer by using the follow-ing relations [29]

α = minusrop(φ)

p(t) β = ro

p(θ)

p(t) (64)

where (p(t) p(r) p(θ) p(φ)) are the tetrad componentsof the photon momentum with respect to locally non-rotating reference frame The observer bases emicro

(ν)can be

10

expanded as a form in the coordinate bases (see [64])

p(t) = minusemicro

(t)pmicro = E ζ12 minus γ12 L p(φ) = emicro

(φ)pmicro =

Lradicgφφ

p(θ) = emicro

(θ)pmicro =

pθradicgθθ

p(r) = emicro

(r)pmicro =prradicgrr

(65)

where the quantities E = minuspt and pφ = L are conserveddue to the associated Killing vectors If we use ξ = LEη = KE2 and pθ = plusmn

radicΘ(θ) we can rewrite these co-

ordinates in terms our parameters ξ and η as follows[43]

α12 = minusroξ12radicgφφ(ζ12 minus γ12ξ12)

|(ro θo)

β12 = plusmnro

radicη12 + a2 cos2 θ minus ξ2

12 cot2 θradic

gθθ(ζ12 minus γ12ξ12)|(ro θo) (66)

where

ζ12 =

radicgφφ

g2tφ minus gttgφφ

(67)

and

γ12 = minusgtφ

gφφζ12 (68)

Finally to simplify the problem further we are going toconsider that our observer is located in the equatorialplane (θ = π2) and very large but finite ro = D = 83kpc we find

α12 = minusradic

f12 ξ12

β12 = plusmnradic

f12radic

η12 (69)

Note that f12 are given by Eqs (32) and (34) respec-tively We see that due to the presence of SFDM our solu-tion is non-asymptotically flat In the special case whenSFDM is absent we recover f12 rarr 1 hence the aboverelations reduces to the asymptotically flat case Notethat for the observer located at the position with coor-dinates (ro θo) we have neglected the effect of rotationwhile ξ12 and η12 given by Eq (61) and are evaluatedat the circular photon orbits r = rcir

For more details in Figs 5 and 6 we plot the shape ofSgr Alowast black hole shadows Using the parameter valuesspecifying the SFDM and baryonic matter we obtain al-most no effect on the shadow images In fact one canobserve that the dashed curve (exponential profile) anddashed red curve (power law profile) are almost indis-tinguishable from the black dashed curve (Kerr vacuumBH) However we can observe that by increasing thebaryonic mass the shadow size increases considerably(solid bluered curve) The angular radius of the Sgr Alowastblack hole shadow can be estimated using the observ-able Rs as θs = Rs MD where M is the black hole mass

5196 5197 5198 5199 5200

5315

5316

5317

5318

Rs

θs [μas]

FIG 7 The expected value for the angular diameter in thecase of Sgr Alowast Note that we have adopted D = 83 kpc and

M = 43times 106M

and D is the distance between the black hole and theobserver The angular radius can be further expressedas θs = 987098times 10minus6Rs(MM)(1kpcD) microas In thecase of Sgr Alowast we have used M = 43 times 106M andD = 83 kpc [4] is the distance between the Earth andSgr Alowast center black hole In order to get more informa-tion about the effect of SFDM on physical observableswe shall estimate the effect of SFDM on the angular ra-dius for Sgr Alowast To simplify the problem let us considerour static and spherically symmetric black hole metricin a totally dominated dark matter galaxy described bythe function

F (r) = eminusX minus 2Mr

(70)

where we have introduced

X =ρ2

04Λ2m6 (71)

where we have used the approximation cos( π r2 R ) 1

since our solution is valid for r le R From the circularphoton orbit conditions one can show [41]

2minus rF prime(r)F (r) = 0 (72)

Now if we expand around X in the above function weobtain

F (r) = 1minus Xminus 2Mr

+ (73)

By solving this equation and considering only the lead-ing order terms one can determine the radius of the pho-ton sphere rps yielding

rps = 3M(1 + X) (74)

On the other hand one also can show the relation [41]

ξ2 + η =r2

ps

F (rps) (75)

11

The shadow radius Rs can be expressed in terms of thecelestial coordinates (α β) as follows

Rs =radic

α2 + β2 =radic

1minus XrpsradicF (rps)

(76)

Note that in the last equation we have used Eqs (69) and(75) Using the values m = 06 eV Λ = 02 meV andρ0 = 92 times 10minus8 eV4 in units of the Sgr Alowast black hole wefind Rs = 5196158313M Using this result we can findthat the angular diameter increases by δθs = 3times 10minus5

microas compared to the Schwarzschild vacuum This resultis consistent with the result reported in Ref [4] whereauthors estimated that the dark matter halo could influ-ence the shadow of Sgr Alowast at a level of order of magni-tude of 10minus3 microas and 10minus5 microas respectively They haveused the so called Cold Dark Matter and Scalar FieldDark Matter models to study the apparent shapes of theshadow A similar result has been obtained for the darkmatter effect on the M87 black hole (see [5]) using theBurkert dark matter profile

VI CONCLUSION

In this paper we have obtained a rotating black holesolution surrounded by superfluid dark matter alongwith baryonic matter To achieve this purpose in thepresent work we considered the superfluid dark mat-ter model and used two specific profiles to describe thebaryonic matter distribution namely the spherical ex-ponential profile and the power law profile followed bythe special case of a totally dominated dark matter caseUsing the current values for the baryonic mass centraldensity of the superfluid dark matter halo radius aswell as the radial scale length for the baryonic matterin our galaxy we found that the shadow size of Sgr Alowastblack hole remains almost unchanged compared to theKerr vacuum BH This result is consistent with a recentwork reported in [5] and also [4] For entirely dominateddark matter galaxies we find that the angular diame-ter increases by 10minus5microarcsec This result shows that itis very difficult to constrain the dark matter parametersusing the shadow images Such tiny effects on the angu-lar diameter are out of reach for the existing space tech-nology and it remains an open question if future astro-nomical observations can potentially detect such effectsThat being said the expected value for the angular di-ameter in the case of Sgr Alowast is of the order of θs 53 microasas predicted in GR

As an interesting observation we show that an in-crease of baryonic mass followed by a decrease of theradial scale length can increase the shadow size consid-erably This can be explained by the fact that the bary-onic matter is mostly located in the interior of the galaxyon the other hand dark matter is mostly located in theouter region of the galaxy In other words for the to-tally dominated dark matter galaxies we observe almost

no effect on black hole shadows but a more precise mea-surement of the Sgr Alowast shadow radius can play a signifi-cant role in determining the baryonic mass in our galaxy

Acknowledgements

TZ is supported by National Natural Science Founda-tion of China under the Grants No 11675143 the Zhe-jiang Provincial Natural Science Foundation of Chinaunder Grant No LY20A050002 and the FundamentalResearch Funds for the Provincial Universities of Zhe-jiang in China under Grants No RF-A2019015

Appendix A

Here we demonstrate the derivation of the spinningblack hole metric As a first step to this formalism wetransform Boyer-Lindquist (BL) coordinates (t r θ φ) toEddington-Finkelstein (EF) coordinates (u r θ φ) Thiscan be achieved by using the coordinate transformation

dt = du +drradic

F12(r)G12(r) (A1)

we obtain

ds212 = minusF12(r)du2minus 2

radicF12(r)G12(r)

dudr+ r2dθ2 + r2 sin2 θdφ2

(A2)This metric can be expressed in terms of null tetrads

as

gmicroν = minuslmicronν minus lνnmicro + mmicromν + mνmmicro (A3)

where the null tetrads are defined as

lmicro = δmicror (A4)

nmicro =

radicG12(r)F12(r)

δmicrou minus

12G12(r)δ

micror (A5)

mmicro =1radic2H

(δ

microθ +

ι

sin θδ

microφ

) (A6)

These null tetrads are constructed in such a way that lmicro

and nmicro while mmicro and mmicro are complex It is worth notingthatH = r2 and will be used later on It is obvious fromthe notation that mmicro is complex conjugate of mmicro Thesevectors further satisfy the conditions for normalizationorthogonality and isotropy as

lmicrolmicro = nmicronmicro = mmicrommicro = mmicrommicro = 0 (A7)lmicrommicro = lmicrommicro = nmicrommicro = nmicrommicro = 0 (A8)

minuslmicronmicro = mmicrommicro = 1 (A9)

12

Following the NewmanndashJanis prescription we write

xprimemicro = xmicro + ia(δmicror minus δ

microu ) cos θ rarr

uprime = uminus ia cos θrprime = r + ia cos θθprime = θφprime = φ

(A10)in which a stands for the rotation parameter Nextlet the null tetrad vectors Za undergo a transformationgiven by Zmicro = (partxmicropartxprimeν)Zprimeν following [62]

lprimemicro = δmicror (A11)

nprimemicro =

radicB12

A12δ

microu minus

12

B12δmicror

mprimemicro =1radic

2Σ12

[(δ

microu minus δ

micror )ιa sin θ + δ

microθ +

ι

sin θδ

microφ

]where we have assumed that (F12(r)G12(r)H(r))transform to (A12(a r θ) B12(a r θ) Σ12(a r θ)) Withthe help of the above equations new metric inEddington-Finkelste coordinate reads

ds212 = minusA12du2 minus 2

radicA12

B12dudr (A12)

+ 2a sin2 θ

(A12 minus

radicA12

B12

)dudφ

+ 2a

radicA12

B12sin2 θdrdφ + Σ12dθ2

+ sin2 θ

[Σ12 + a2 sin2 θ

(2

radicA12

B12minus A12

)]dφ2

Note that A12 is some function of r and θ as we al-ready pointed out Without going into details of the cal-culation one can revert the EF coordinates back to BLcoordinates by using the following transformation (see

for more details [62])

du = dtminus a2 + r2

∆12dr dφ = dφprime minus a

∆12dr (A13)

where in order to simplify the notation we introduce

∆(r)12 = r2 F12(r) + a2 (A14)

Making use of F12(r) = G12(r) one can obtain Σ1 =Σ2 = r2 + a2 cos2 θ herein we shall use just Σ Droppingthe primes in the φ coordinate and choosing

A12 =(F12H+ a2 cos2 θ)Σ12

(H+ a2 cos2 θ)2 (A15)

and

B12 =F12H+ a2 cos2 θ

Σ (A16)

we obtain the spinning black hole space-time metric ina SFDM halo

ds212 = minus

(1minus 2Υ12(r)r

Σ

)dt2 +

Σ∆12

dr2 + Σdθ2

minus 2a sin2 θ2Υ12(r)r

Σdtdφ

+ sin2 θ

[(r2 + a2)2 minus a2∆12 sin2 θ

Σ

]dφ2 (A17)

with

Υ12(r) =r (1minusF12(r))

2 (A18)

in which F1 = F (r)exp and F2 = F (r)power respec-tively

[1] The Event Horizon Telescope Collaboration Ap J Lett875 L1 (2019)

[2] Haroon S Jamil M Jusufi K Lin K Mann R B PhysRev D 99 no4 044015 (2019)

[3] Haroon S Jusufi K Jamil M Universe 6(2) 23 (2020)arXiv190400711[gr-qc]

[4] Hou X Xu Z Zhou M Wang J JCAP 1807 no07 015(2018)

[5] Jusufi K Jamil M Salucci P Zhu T Haroon S PhysRev D 100 044012 (2019)

[6] Xu Z Hou X Gong X Wang J JCAP 1809 no09 038(2018)

[7] Xu Z Hou X Wang J JCAP 1810 no10 046 (2018)[8] Berezhiani L Khoury J Phys Rev D 92 103510 (2015)[9] Synge J L Mon Not Roy Astron Soc 131 no 3 463

(1966)

[10] Luminet J P Astron Astrophys 75 228 (1979)[11] Bardeen J M in Black Holes (Proceedings Les Houches

France August 1972) edited by C DeWitt and B S De-Witt

[12] Abdujabbarov A Toshmatov B Stuchlik Z AhmedovB Int J Mod Phys D 26 no 06 1750051 (2016)

[13] Abdikamalov A B Abdujabbarov A A Ayzenberg DMalafarina D Bambi C Ahmedov B Phys Rev D 100(2019) no2 024014

[14] Abdujabbarov A Amir M Ahmedov B Ghosh S GPhys Rev D 93 no 10 104004 (2016)

[15] Amir M Jusufi K Banerjee A Hansraj S ClassQuant Grav 36 no21 215007 (2019)

[16] Amir M Ghosh S G Phys Rev D 94 no 2 024054(2016)

[17] Abdujabbarov A Atamurotov F Kucukakca Y Ahme-

8

Kerr vacuum BH

BH-SFDM

-10 -5 0 5-6

-4

-2

0

2

4

6

α

β

a=02

Kerr vacuum BH

BH-SFDM

-10 -5 0 5-6

-4

-2

0

2

4

6

α

β

a=05

Kerr vacuum BH

BH-SFDM

-10 -5 0 5-6

-4

-2

0

2

4

6

α

β

a=085

Kerr vacuum BH

BH-SFDM

-10 -5 0 5-6

-4

-2

0

2

4

6

α

β

a=098

FIG 5 Variation in shape of shadow using the spherical exponential profile for baryonic matter We use M = 1 in units of theSgr Alowast black hole mass given by MBH = 43 times 106 M and R = 158 kpc or R = 755 times 1010 MBH Furthermore for m = 06 eV

and Λ = 02 meV we find ρ0 = 002 times 10minus24 gcm3 or in eV units ρ0 = 92 times 10minus8 eV4 For the Milky Way galaxy we can useMB = 6times 1010 M = 139 times 104 MBH and L = 26 kpc= 124 times 1010 MBH We observe that the dashed blue curve correspondingto Sgr Alowast with the above parameters is almost indistinguishable from the black dashed curve describing the Kerr vacuum BHThe solid blue curve corresponds to the case of increasing the baryonic mass by a factor of 102 and decrease of core radius by a

factor of 103

B Circular Orbits

Due to the strong gravity near the black hole it is thusexpected that the photons emitted near a black hole willeventually fall into the black hole or eventually scatteraway from it In this way the photons captured by theblack hole will form a dark region defining the contourof the shadow To elaborate the presence of unstablecircular orbits around the black hole we need to studythe radial geodesic by introducing the effective poten-tial Veff which can be written as follows

Σ2(

drdτ

)2+ V12

eff = 0 (57)

where V1eff = Vexp

eff and V2eff = Vpower

eff gives the expo-nential and the power law respectively At this point itis convenient to introduce two parameters ξ and η de-fined as

ξ = LE η = KE2 (58)

For the effective potential then we obtain the followingrelation

V12eff = ∆12((aminus ξ12)

2 + η12)minus (r2 + a2minus a ξ12)2 (59)

where we have replaced V12eff E2 by V12

eff For more de-tails in Figure (3) we plot the variation of the effectivepotential associated with the radial motion of photons

9

Kerr vacuum BH

BH-SFDM

-10 -5 0 5-6

-4

-2

0

2

4

6

α

β

a=02

Kerr vacuum BH

BH-SFDM

-10 -5 0 5-6

-4

-2

0

2

4

6

α

β

a=05

Kerr vacuum BH

BH-SFDM

-10 -5 0 5-6

-4

-2

0

2

4

6

α

β

a=085

Kerr vacuum BH

BH-SFDM

-10 -5 0 5-6

-4

-2

0

2

4

6

α

β

a=098

FIG 6 Variation in shape of shadow for different values of a using the power law profile We use M = 1 in units of the Sgr Alowast

black hole mass given by MBH = 43 times 106 M and R = 158 kpc or R = 755 times 1010 MBH Furthermore for m = 06 eV andΛ = 02 meV we find ρ0 = 002 times 10minus24 gcm3 or in eV units ρ0 = 92 times 10minus8 eV4 For the Milky Way galaxy we can use

MB = 6times 1010 M = 139 times 104 MBH and rc = 26 kpc= 124 times 1010 MBH The dashed red curve corresponds to Sgr Alowast usingthe above parameters which is almost indistinguishable from the black dashed curve discribing the Kerr vacuum BH The solid

red curve corresponds to the case of increasing the baryonic mass by a factor of 102 and decrease of core radius by a factor of 103

The circular photon orbits exists when at some constantr = rcir the conditions

V12eff (r) = 0

dV12eff (r)dr

= 0 (60)

Combining all these equations it is possible to showthat

ξ12 =(r2 + a2)(rF prime12(r) + 2F12(r))minus 4(r2F12(r) + a2)

a(rF prime12(r) + 2F12(r))

η12 =r3[8a2F prime12(r)minus r(rF prime12(r)minus 2F12(r))2]

a2(rF prime12(r) + 2F12(r))2 (61)

One can recover the Kerr vacuum case by letting a0 =MB = ρ0 = 0 yielding

ξ12 =r2(3Mminus r)minus a2(Mminus r)

a(rminusM) (62)

η12 =r3 (4Ma2 minus r(rminus 3M)2)

a2(rminusM)2 (63)

To obtain the shadow images of our black hole in thepresence of dark matter we assume that the observer islocated at the position with coordinates (ro θo) where roand θo represents the angular coordinate on observerrsquossky Furthermore we need to introduce two celestial co-ordinates α and β for the observer by using the follow-ing relations [29]

α = minusrop(φ)

p(t) β = ro

p(θ)

p(t) (64)

where (p(t) p(r) p(θ) p(φ)) are the tetrad componentsof the photon momentum with respect to locally non-rotating reference frame The observer bases emicro

(ν)can be

10

expanded as a form in the coordinate bases (see [64])

p(t) = minusemicro

(t)pmicro = E ζ12 minus γ12 L p(φ) = emicro

(φ)pmicro =

Lradicgφφ

p(θ) = emicro

(θ)pmicro =

pθradicgθθ

p(r) = emicro

(r)pmicro =prradicgrr

(65)

where the quantities E = minuspt and pφ = L are conserveddue to the associated Killing vectors If we use ξ = LEη = KE2 and pθ = plusmn

radicΘ(θ) we can rewrite these co-

ordinates in terms our parameters ξ and η as follows[43]

α12 = minusroξ12radicgφφ(ζ12 minus γ12ξ12)

|(ro θo)

β12 = plusmnro

radicη12 + a2 cos2 θ minus ξ2

12 cot2 θradic

gθθ(ζ12 minus γ12ξ12)|(ro θo) (66)

where

ζ12 =

radicgφφ

g2tφ minus gttgφφ

(67)

and

γ12 = minusgtφ

gφφζ12 (68)

Finally to simplify the problem further we are going toconsider that our observer is located in the equatorialplane (θ = π2) and very large but finite ro = D = 83kpc we find

α12 = minusradic

f12 ξ12

β12 = plusmnradic

f12radic

η12 (69)

Note that f12 are given by Eqs (32) and (34) respec-tively We see that due to the presence of SFDM our solu-tion is non-asymptotically flat In the special case whenSFDM is absent we recover f12 rarr 1 hence the aboverelations reduces to the asymptotically flat case Notethat for the observer located at the position with coor-dinates (ro θo) we have neglected the effect of rotationwhile ξ12 and η12 given by Eq (61) and are evaluatedat the circular photon orbits r = rcir

For more details in Figs 5 and 6 we plot the shape ofSgr Alowast black hole shadows Using the parameter valuesspecifying the SFDM and baryonic matter we obtain al-most no effect on the shadow images In fact one canobserve that the dashed curve (exponential profile) anddashed red curve (power law profile) are almost indis-tinguishable from the black dashed curve (Kerr vacuumBH) However we can observe that by increasing thebaryonic mass the shadow size increases considerably(solid bluered curve) The angular radius of the Sgr Alowastblack hole shadow can be estimated using the observ-able Rs as θs = Rs MD where M is the black hole mass

5196 5197 5198 5199 5200

5315

5316

5317

5318

Rs

θs [μas]

FIG 7 The expected value for the angular diameter in thecase of Sgr Alowast Note that we have adopted D = 83 kpc and

M = 43times 106M

and D is the distance between the black hole and theobserver The angular radius can be further expressedas θs = 987098times 10minus6Rs(MM)(1kpcD) microas In thecase of Sgr Alowast we have used M = 43 times 106M andD = 83 kpc [4] is the distance between the Earth andSgr Alowast center black hole In order to get more informa-tion about the effect of SFDM on physical observableswe shall estimate the effect of SFDM on the angular ra-dius for Sgr Alowast To simplify the problem let us considerour static and spherically symmetric black hole metricin a totally dominated dark matter galaxy described bythe function

F (r) = eminusX minus 2Mr

(70)

where we have introduced

X =ρ2

04Λ2m6 (71)

where we have used the approximation cos( π r2 R ) 1

since our solution is valid for r le R From the circularphoton orbit conditions one can show [41]

2minus rF prime(r)F (r) = 0 (72)

Now if we expand around X in the above function weobtain

F (r) = 1minus Xminus 2Mr

+ (73)

By solving this equation and considering only the lead-ing order terms one can determine the radius of the pho-ton sphere rps yielding

rps = 3M(1 + X) (74)

On the other hand one also can show the relation [41]

ξ2 + η =r2

ps

F (rps) (75)

11

The shadow radius Rs can be expressed in terms of thecelestial coordinates (α β) as follows

Rs =radic

α2 + β2 =radic

1minus XrpsradicF (rps)

(76)

Note that in the last equation we have used Eqs (69) and(75) Using the values m = 06 eV Λ = 02 meV andρ0 = 92 times 10minus8 eV4 in units of the Sgr Alowast black hole wefind Rs = 5196158313M Using this result we can findthat the angular diameter increases by δθs = 3times 10minus5

microas compared to the Schwarzschild vacuum This resultis consistent with the result reported in Ref [4] whereauthors estimated that the dark matter halo could influ-ence the shadow of Sgr Alowast at a level of order of magni-tude of 10minus3 microas and 10minus5 microas respectively They haveused the so called Cold Dark Matter and Scalar FieldDark Matter models to study the apparent shapes of theshadow A similar result has been obtained for the darkmatter effect on the M87 black hole (see [5]) using theBurkert dark matter profile

VI CONCLUSION

In this paper we have obtained a rotating black holesolution surrounded by superfluid dark matter alongwith baryonic matter To achieve this purpose in thepresent work we considered the superfluid dark mat-ter model and used two specific profiles to describe thebaryonic matter distribution namely the spherical ex-ponential profile and the power law profile followed bythe special case of a totally dominated dark matter caseUsing the current values for the baryonic mass centraldensity of the superfluid dark matter halo radius aswell as the radial scale length for the baryonic matterin our galaxy we found that the shadow size of Sgr Alowastblack hole remains almost unchanged compared to theKerr vacuum BH This result is consistent with a recentwork reported in [5] and also [4] For entirely dominateddark matter galaxies we find that the angular diame-ter increases by 10minus5microarcsec This result shows that itis very difficult to constrain the dark matter parametersusing the shadow images Such tiny effects on the angu-lar diameter are out of reach for the existing space tech-nology and it remains an open question if future astro-nomical observations can potentially detect such effectsThat being said the expected value for the angular di-ameter in the case of Sgr Alowast is of the order of θs 53 microasas predicted in GR

As an interesting observation we show that an in-crease of baryonic mass followed by a decrease of theradial scale length can increase the shadow size consid-erably This can be explained by the fact that the bary-onic matter is mostly located in the interior of the galaxyon the other hand dark matter is mostly located in theouter region of the galaxy In other words for the to-tally dominated dark matter galaxies we observe almost

no effect on black hole shadows but a more precise mea-surement of the Sgr Alowast shadow radius can play a signifi-cant role in determining the baryonic mass in our galaxy

Acknowledgements

TZ is supported by National Natural Science Founda-tion of China under the Grants No 11675143 the Zhe-jiang Provincial Natural Science Foundation of Chinaunder Grant No LY20A050002 and the FundamentalResearch Funds for the Provincial Universities of Zhe-jiang in China under Grants No RF-A2019015

Appendix A

Here we demonstrate the derivation of the spinningblack hole metric As a first step to this formalism wetransform Boyer-Lindquist (BL) coordinates (t r θ φ) toEddington-Finkelstein (EF) coordinates (u r θ φ) Thiscan be achieved by using the coordinate transformation

dt = du +drradic

F12(r)G12(r) (A1)

we obtain

ds212 = minusF12(r)du2minus 2

radicF12(r)G12(r)

dudr+ r2dθ2 + r2 sin2 θdφ2

(A2)This metric can be expressed in terms of null tetrads

as

gmicroν = minuslmicronν minus lνnmicro + mmicromν + mνmmicro (A3)

where the null tetrads are defined as

lmicro = δmicror (A4)

nmicro =

radicG12(r)F12(r)

δmicrou minus

12G12(r)δ

micror (A5)

mmicro =1radic2H

(δ

microθ +

ι

sin θδ

microφ

) (A6)

These null tetrads are constructed in such a way that lmicro

and nmicro while mmicro and mmicro are complex It is worth notingthatH = r2 and will be used later on It is obvious fromthe notation that mmicro is complex conjugate of mmicro Thesevectors further satisfy the conditions for normalizationorthogonality and isotropy as

lmicrolmicro = nmicronmicro = mmicrommicro = mmicrommicro = 0 (A7)lmicrommicro = lmicrommicro = nmicrommicro = nmicrommicro = 0 (A8)

minuslmicronmicro = mmicrommicro = 1 (A9)

12

Following the NewmanndashJanis prescription we write

xprimemicro = xmicro + ia(δmicror minus δ

microu ) cos θ rarr

uprime = uminus ia cos θrprime = r + ia cos θθprime = θφprime = φ

(A10)in which a stands for the rotation parameter Nextlet the null tetrad vectors Za undergo a transformationgiven by Zmicro = (partxmicropartxprimeν)Zprimeν following [62]

lprimemicro = δmicror (A11)

nprimemicro =

radicB12

A12δ

microu minus

12

B12δmicror

mprimemicro =1radic

2Σ12

[(δ

microu minus δ

micror )ιa sin θ + δ

microθ +

ι

sin θδ

microφ

]where we have assumed that (F12(r)G12(r)H(r))transform to (A12(a r θ) B12(a r θ) Σ12(a r θ)) Withthe help of the above equations new metric inEddington-Finkelste coordinate reads

ds212 = minusA12du2 minus 2

radicA12

B12dudr (A12)

+ 2a sin2 θ

(A12 minus

radicA12

B12

)dudφ

+ 2a

radicA12

B12sin2 θdrdφ + Σ12dθ2

+ sin2 θ

[Σ12 + a2 sin2 θ

(2

radicA12

B12minus A12

)]dφ2

Note that A12 is some function of r and θ as we al-ready pointed out Without going into details of the cal-culation one can revert the EF coordinates back to BLcoordinates by using the following transformation (see

for more details [62])

du = dtminus a2 + r2

∆12dr dφ = dφprime minus a

∆12dr (A13)

where in order to simplify the notation we introduce

∆(r)12 = r2 F12(r) + a2 (A14)

Making use of F12(r) = G12(r) one can obtain Σ1 =Σ2 = r2 + a2 cos2 θ herein we shall use just Σ Droppingthe primes in the φ coordinate and choosing

A12 =(F12H+ a2 cos2 θ)Σ12

(H+ a2 cos2 θ)2 (A15)

and

B12 =F12H+ a2 cos2 θ

Σ (A16)

we obtain the spinning black hole space-time metric ina SFDM halo

ds212 = minus

(1minus 2Υ12(r)r

Σ

)dt2 +

Σ∆12

dr2 + Σdθ2

minus 2a sin2 θ2Υ12(r)r

Σdtdφ

+ sin2 θ

[(r2 + a2)2 minus a2∆12 sin2 θ

Σ

]dφ2 (A17)

with

Υ12(r) =r (1minusF12(r))

2 (A18)

in which F1 = F (r)exp and F2 = F (r)power respec-tively

[1] The Event Horizon Telescope Collaboration Ap J Lett875 L1 (2019)

[2] Haroon S Jamil M Jusufi K Lin K Mann R B PhysRev D 99 no4 044015 (2019)

[3] Haroon S Jusufi K Jamil M Universe 6(2) 23 (2020)arXiv190400711[gr-qc]

[4] Hou X Xu Z Zhou M Wang J JCAP 1807 no07 015(2018)

[5] Jusufi K Jamil M Salucci P Zhu T Haroon S PhysRev D 100 044012 (2019)

[6] Xu Z Hou X Gong X Wang J JCAP 1809 no09 038(2018)

[7] Xu Z Hou X Wang J JCAP 1810 no10 046 (2018)[8] Berezhiani L Khoury J Phys Rev D 92 103510 (2015)[9] Synge J L Mon Not Roy Astron Soc 131 no 3 463

(1966)

[10] Luminet J P Astron Astrophys 75 228 (1979)[11] Bardeen J M in Black Holes (Proceedings Les Houches

France August 1972) edited by C DeWitt and B S De-Witt

[12] Abdujabbarov A Toshmatov B Stuchlik Z AhmedovB Int J Mod Phys D 26 no 06 1750051 (2016)

[13] Abdikamalov A B Abdujabbarov A A Ayzenberg DMalafarina D Bambi C Ahmedov B Phys Rev D 100(2019) no2 024014

[14] Abdujabbarov A Amir M Ahmedov B Ghosh S GPhys Rev D 93 no 10 104004 (2016)

[15] Amir M Jusufi K Banerjee A Hansraj S ClassQuant Grav 36 no21 215007 (2019)

[16] Amir M Ghosh S G Phys Rev D 94 no 2 024054(2016)

[17] Abdujabbarov A Atamurotov F Kucukakca Y Ahme-

9

Kerr vacuum BH

BH-SFDM

-10 -5 0 5-6

-4

-2

0

2

4

6

α

β

a=02

Kerr vacuum BH

BH-SFDM

-10 -5 0 5-6

-4

-2

0

2

4

6

α

β

a=05

Kerr vacuum BH

BH-SFDM

-10 -5 0 5-6

-4

-2

0

2

4

6

α

β

a=085

Kerr vacuum BH

BH-SFDM

-10 -5 0 5-6

-4

-2

0

2

4

6

α

β

a=098

FIG 6 Variation in shape of shadow for different values of a using the power law profile We use M = 1 in units of the Sgr Alowast

black hole mass given by MBH = 43 times 106 M and R = 158 kpc or R = 755 times 1010 MBH Furthermore for m = 06 eV andΛ = 02 meV we find ρ0 = 002 times 10minus24 gcm3 or in eV units ρ0 = 92 times 10minus8 eV4 For the Milky Way galaxy we can use

MB = 6times 1010 M = 139 times 104 MBH and rc = 26 kpc= 124 times 1010 MBH The dashed red curve corresponds to Sgr Alowast usingthe above parameters which is almost indistinguishable from the black dashed curve discribing the Kerr vacuum BH The solid

red curve corresponds to the case of increasing the baryonic mass by a factor of 102 and decrease of core radius by a factor of 103

The circular photon orbits exists when at some constantr = rcir the conditions

V12eff (r) = 0

dV12eff (r)dr

= 0 (60)

Combining all these equations it is possible to showthat

ξ12 =(r2 + a2)(rF prime12(r) + 2F12(r))minus 4(r2F12(r) + a2)

a(rF prime12(r) + 2F12(r))

η12 =r3[8a2F prime12(r)minus r(rF prime12(r)minus 2F12(r))2]

a2(rF prime12(r) + 2F12(r))2 (61)

One can recover the Kerr vacuum case by letting a0 =MB = ρ0 = 0 yielding

ξ12 =r2(3Mminus r)minus a2(Mminus r)

a(rminusM) (62)

η12 =r3 (4Ma2 minus r(rminus 3M)2)

a2(rminusM)2 (63)

To obtain the shadow images of our black hole in thepresence of dark matter we assume that the observer islocated at the position with coordinates (ro θo) where roand θo represents the angular coordinate on observerrsquossky Furthermore we need to introduce two celestial co-ordinates α and β for the observer by using the follow-ing relations [29]

α = minusrop(φ)

p(t) β = ro

p(θ)

p(t) (64)

where (p(t) p(r) p(θ) p(φ)) are the tetrad componentsof the photon momentum with respect to locally non-rotating reference frame The observer bases emicro

(ν)can be

10

expanded as a form in the coordinate bases (see [64])

p(t) = minusemicro

(t)pmicro = E ζ12 minus γ12 L p(φ) = emicro

(φ)pmicro =

Lradicgφφ

p(θ) = emicro

(θ)pmicro =

pθradicgθθ

p(r) = emicro

(r)pmicro =prradicgrr

(65)

where the quantities E = minuspt and pφ = L are conserveddue to the associated Killing vectors If we use ξ = LEη = KE2 and pθ = plusmn

radicΘ(θ) we can rewrite these co-

ordinates in terms our parameters ξ and η as follows[43]

α12 = minusroξ12radicgφφ(ζ12 minus γ12ξ12)

|(ro θo)

β12 = plusmnro

radicη12 + a2 cos2 θ minus ξ2

12 cot2 θradic

gθθ(ζ12 minus γ12ξ12)|(ro θo) (66)

where

ζ12 =

radicgφφ

g2tφ minus gttgφφ

(67)

and

γ12 = minusgtφ

gφφζ12 (68)

Finally to simplify the problem further we are going toconsider that our observer is located in the equatorialplane (θ = π2) and very large but finite ro = D = 83kpc we find

α12 = minusradic

f12 ξ12

β12 = plusmnradic

f12radic

η12 (69)

Note that f12 are given by Eqs (32) and (34) respec-tively We see that due to the presence of SFDM our solu-tion is non-asymptotically flat In the special case whenSFDM is absent we recover f12 rarr 1 hence the aboverelations reduces to the asymptotically flat case Notethat for the observer located at the position with coor-dinates (ro θo) we have neglected the effect of rotationwhile ξ12 and η12 given by Eq (61) and are evaluatedat the circular photon orbits r = rcir

For more details in Figs 5 and 6 we plot the shape ofSgr Alowast black hole shadows Using the parameter valuesspecifying the SFDM and baryonic matter we obtain al-most no effect on the shadow images In fact one canobserve that the dashed curve (exponential profile) anddashed red curve (power law profile) are almost indis-tinguishable from the black dashed curve (Kerr vacuumBH) However we can observe that by increasing thebaryonic mass the shadow size increases considerably(solid bluered curve) The angular radius of the Sgr Alowastblack hole shadow can be estimated using the observ-able Rs as θs = Rs MD where M is the black hole mass

5196 5197 5198 5199 5200

5315

5316

5317

5318

Rs

θs [μas]

FIG 7 The expected value for the angular diameter in thecase of Sgr Alowast Note that we have adopted D = 83 kpc and

M = 43times 106M

and D is the distance between the black hole and theobserver The angular radius can be further expressedas θs = 987098times 10minus6Rs(MM)(1kpcD) microas In thecase of Sgr Alowast we have used M = 43 times 106M andD = 83 kpc [4] is the distance between the Earth andSgr Alowast center black hole In order to get more informa-tion about the effect of SFDM on physical observableswe shall estimate the effect of SFDM on the angular ra-dius for Sgr Alowast To simplify the problem let us considerour static and spherically symmetric black hole metricin a totally dominated dark matter galaxy described bythe function

F (r) = eminusX minus 2Mr

(70)

where we have introduced

X =ρ2

04Λ2m6 (71)

where we have used the approximation cos( π r2 R ) 1

since our solution is valid for r le R From the circularphoton orbit conditions one can show [41]

2minus rF prime(r)F (r) = 0 (72)

Now if we expand around X in the above function weobtain

F (r) = 1minus Xminus 2Mr

+ (73)

By solving this equation and considering only the lead-ing order terms one can determine the radius of the pho-ton sphere rps yielding

rps = 3M(1 + X) (74)

On the other hand one also can show the relation [41]

ξ2 + η =r2

ps

F (rps) (75)

11

The shadow radius Rs can be expressed in terms of thecelestial coordinates (α β) as follows

Rs =radic

α2 + β2 =radic

1minus XrpsradicF (rps)

(76)

Note that in the last equation we have used Eqs (69) and(75) Using the values m = 06 eV Λ = 02 meV andρ0 = 92 times 10minus8 eV4 in units of the Sgr Alowast black hole wefind Rs = 5196158313M Using this result we can findthat the angular diameter increases by δθs = 3times 10minus5

microas compared to the Schwarzschild vacuum This resultis consistent with the result reported in Ref [4] whereauthors estimated that the dark matter halo could influ-ence the shadow of Sgr Alowast at a level of order of magni-tude of 10minus3 microas and 10minus5 microas respectively They haveused the so called Cold Dark Matter and Scalar FieldDark Matter models to study the apparent shapes of theshadow A similar result has been obtained for the darkmatter effect on the M87 black hole (see [5]) using theBurkert dark matter profile

VI CONCLUSION

In this paper we have obtained a rotating black holesolution surrounded by superfluid dark matter alongwith baryonic matter To achieve this purpose in thepresent work we considered the superfluid dark mat-ter model and used two specific profiles to describe thebaryonic matter distribution namely the spherical ex-ponential profile and the power law profile followed bythe special case of a totally dominated dark matter caseUsing the current values for the baryonic mass centraldensity of the superfluid dark matter halo radius aswell as the radial scale length for the baryonic matterin our galaxy we found that the shadow size of Sgr Alowastblack hole remains almost unchanged compared to theKerr vacuum BH This result is consistent with a recentwork reported in [5] and also [4] For entirely dominateddark matter galaxies we find that the angular diame-ter increases by 10minus5microarcsec This result shows that itis very difficult to constrain the dark matter parametersusing the shadow images Such tiny effects on the angu-lar diameter are out of reach for the existing space tech-nology and it remains an open question if future astro-nomical observations can potentially detect such effectsThat being said the expected value for the angular di-ameter in the case of Sgr Alowast is of the order of θs 53 microasas predicted in GR

As an interesting observation we show that an in-crease of baryonic mass followed by a decrease of theradial scale length can increase the shadow size consid-erably This can be explained by the fact that the bary-onic matter is mostly located in the interior of the galaxyon the other hand dark matter is mostly located in theouter region of the galaxy In other words for the to-tally dominated dark matter galaxies we observe almost

no effect on black hole shadows but a more precise mea-surement of the Sgr Alowast shadow radius can play a signifi-cant role in determining the baryonic mass in our galaxy

Acknowledgements

TZ is supported by National Natural Science Founda-tion of China under the Grants No 11675143 the Zhe-jiang Provincial Natural Science Foundation of Chinaunder Grant No LY20A050002 and the FundamentalResearch Funds for the Provincial Universities of Zhe-jiang in China under Grants No RF-A2019015

Appendix A

Here we demonstrate the derivation of the spinningblack hole metric As a first step to this formalism wetransform Boyer-Lindquist (BL) coordinates (t r θ φ) toEddington-Finkelstein (EF) coordinates (u r θ φ) Thiscan be achieved by using the coordinate transformation

dt = du +drradic

F12(r)G12(r) (A1)

we obtain

ds212 = minusF12(r)du2minus 2

radicF12(r)G12(r)

dudr+ r2dθ2 + r2 sin2 θdφ2

(A2)This metric can be expressed in terms of null tetrads

as

gmicroν = minuslmicronν minus lνnmicro + mmicromν + mνmmicro (A3)

where the null tetrads are defined as

lmicro = δmicror (A4)

nmicro =

radicG12(r)F12(r)

δmicrou minus

12G12(r)δ

micror (A5)

mmicro =1radic2H

(δ

microθ +

ι

sin θδ

microφ

) (A6)

These null tetrads are constructed in such a way that lmicro

and nmicro while mmicro and mmicro are complex It is worth notingthatH = r2 and will be used later on It is obvious fromthe notation that mmicro is complex conjugate of mmicro Thesevectors further satisfy the conditions for normalizationorthogonality and isotropy as

lmicrolmicro = nmicronmicro = mmicrommicro = mmicrommicro = 0 (A7)lmicrommicro = lmicrommicro = nmicrommicro = nmicrommicro = 0 (A8)

minuslmicronmicro = mmicrommicro = 1 (A9)

12

Following the NewmanndashJanis prescription we write

xprimemicro = xmicro + ia(δmicror minus δ

microu ) cos θ rarr

uprime = uminus ia cos θrprime = r + ia cos θθprime = θφprime = φ

(A10)in which a stands for the rotation parameter Nextlet the null tetrad vectors Za undergo a transformationgiven by Zmicro = (partxmicropartxprimeν)Zprimeν following [62]

lprimemicro = δmicror (A11)

nprimemicro =

radicB12

A12δ

microu minus

12

B12δmicror

mprimemicro =1radic

2Σ12

[(δ

microu minus δ

micror )ιa sin θ + δ

microθ +

ι

sin θδ

microφ

]where we have assumed that (F12(r)G12(r)H(r))transform to (A12(a r θ) B12(a r θ) Σ12(a r θ)) Withthe help of the above equations new metric inEddington-Finkelste coordinate reads

ds212 = minusA12du2 minus 2

radicA12

B12dudr (A12)

+ 2a sin2 θ

(A12 minus

radicA12

B12

)dudφ

+ 2a

radicA12

B12sin2 θdrdφ + Σ12dθ2

+ sin2 θ

[Σ12 + a2 sin2 θ

(2

radicA12

B12minus A12

)]dφ2

Note that A12 is some function of r and θ as we al-ready pointed out Without going into details of the cal-culation one can revert the EF coordinates back to BLcoordinates by using the following transformation (see

for more details [62])

du = dtminus a2 + r2

∆12dr dφ = dφprime minus a

∆12dr (A13)

where in order to simplify the notation we introduce

∆(r)12 = r2 F12(r) + a2 (A14)

Making use of F12(r) = G12(r) one can obtain Σ1 =Σ2 = r2 + a2 cos2 θ herein we shall use just Σ Droppingthe primes in the φ coordinate and choosing

A12 =(F12H+ a2 cos2 θ)Σ12

(H+ a2 cos2 θ)2 (A15)

and

B12 =F12H+ a2 cos2 θ

Σ (A16)

we obtain the spinning black hole space-time metric ina SFDM halo

ds212 = minus

(1minus 2Υ12(r)r

Σ

)dt2 +

Σ∆12

dr2 + Σdθ2

minus 2a sin2 θ2Υ12(r)r

Σdtdφ

+ sin2 θ

[(r2 + a2)2 minus a2∆12 sin2 θ

Σ

]dφ2 (A17)

with

Υ12(r) =r (1minusF12(r))

2 (A18)

in which F1 = F (r)exp and F2 = F (r)power respec-tively

[1] The Event Horizon Telescope Collaboration Ap J Lett875 L1 (2019)

[2] Haroon S Jamil M Jusufi K Lin K Mann R B PhysRev D 99 no4 044015 (2019)

[3] Haroon S Jusufi K Jamil M Universe 6(2) 23 (2020)arXiv190400711[gr-qc]

[4] Hou X Xu Z Zhou M Wang J JCAP 1807 no07 015(2018)

[5] Jusufi K Jamil M Salucci P Zhu T Haroon S PhysRev D 100 044012 (2019)

[6] Xu Z Hou X Gong X Wang J JCAP 1809 no09 038(2018)

[7] Xu Z Hou X Wang J JCAP 1810 no10 046 (2018)[8] Berezhiani L Khoury J Phys Rev D 92 103510 (2015)[9] Synge J L Mon Not Roy Astron Soc 131 no 3 463

(1966)

[10] Luminet J P Astron Astrophys 75 228 (1979)[11] Bardeen J M in Black Holes (Proceedings Les Houches

France August 1972) edited by C DeWitt and B S De-Witt

[12] Abdujabbarov A Toshmatov B Stuchlik Z AhmedovB Int J Mod Phys D 26 no 06 1750051 (2016)

[13] Abdikamalov A B Abdujabbarov A A Ayzenberg DMalafarina D Bambi C Ahmedov B Phys Rev D 100(2019) no2 024014

[14] Abdujabbarov A Amir M Ahmedov B Ghosh S GPhys Rev D 93 no 10 104004 (2016)

[15] Amir M Jusufi K Banerjee A Hansraj S ClassQuant Grav 36 no21 215007 (2019)

[16] Amir M Ghosh S G Phys Rev D 94 no 2 024054(2016)

[17] Abdujabbarov A Atamurotov F Kucukakca Y Ahme-

10

expanded as a form in the coordinate bases (see [64])

p(t) = minusemicro

(t)pmicro = E ζ12 minus γ12 L p(φ) = emicro

(φ)pmicro =

Lradicgφφ

p(θ) = emicro

(θ)pmicro =

pθradicgθθ

p(r) = emicro

(r)pmicro =prradicgrr

(65)

where the quantities E = minuspt and pφ = L are conserveddue to the associated Killing vectors If we use ξ = LEη = KE2 and pθ = plusmn

radicΘ(θ) we can rewrite these co-

ordinates in terms our parameters ξ and η as follows[43]

α12 = minusroξ12radicgφφ(ζ12 minus γ12ξ12)

|(ro θo)

β12 = plusmnro

radicη12 + a2 cos2 θ minus ξ2

12 cot2 θradic

gθθ(ζ12 minus γ12ξ12)|(ro θo) (66)

where

ζ12 =

radicgφφ

g2tφ minus gttgφφ

(67)

and

γ12 = minusgtφ

gφφζ12 (68)

Finally to simplify the problem further we are going toconsider that our observer is located in the equatorialplane (θ = π2) and very large but finite ro = D = 83kpc we find

α12 = minusradic

f12 ξ12

β12 = plusmnradic

f12radic

η12 (69)

Note that f12 are given by Eqs (32) and (34) respec-tively We see that due to the presence of SFDM our solu-tion is non-asymptotically flat In the special case whenSFDM is absent we recover f12 rarr 1 hence the aboverelations reduces to the asymptotically flat case Notethat for the observer located at the position with coor-dinates (ro θo) we have neglected the effect of rotationwhile ξ12 and η12 given by Eq (61) and are evaluatedat the circular photon orbits r = rcir

For more details in Figs 5 and 6 we plot the shape ofSgr Alowast black hole shadows Using the parameter valuesspecifying the SFDM and baryonic matter we obtain al-most no effect on the shadow images In fact one canobserve that the dashed curve (exponential profile) anddashed red curve (power law profile) are almost indis-tinguishable from the black dashed curve (Kerr vacuumBH) However we can observe that by increasing thebaryonic mass the shadow size increases considerably(solid bluered curve) The angular radius of the Sgr Alowastblack hole shadow can be estimated using the observ-able Rs as θs = Rs MD where M is the black hole mass

5196 5197 5198 5199 5200

5315

5316

5317

5318

Rs

θs [μas]

FIG 7 The expected value for the angular diameter in thecase of Sgr Alowast Note that we have adopted D = 83 kpc and

M = 43times 106M

and D is the distance between the black hole and theobserver The angular radius can be further expressedas θs = 987098times 10minus6Rs(MM)(1kpcD) microas In thecase of Sgr Alowast we have used M = 43 times 106M andD = 83 kpc [4] is the distance between the Earth andSgr Alowast center black hole In order to get more informa-tion about the effect of SFDM on physical observableswe shall estimate the effect of SFDM on the angular ra-dius for Sgr Alowast To simplify the problem let us considerour static and spherically symmetric black hole metricin a totally dominated dark matter galaxy described bythe function

F (r) = eminusX minus 2Mr

(70)

where we have introduced

X =ρ2

04Λ2m6 (71)

where we have used the approximation cos( π r2 R ) 1

since our solution is valid for r le R From the circularphoton orbit conditions one can show [41]

2minus rF prime(r)F (r) = 0 (72)

Now if we expand around X in the above function weobtain

F (r) = 1minus Xminus 2Mr

+ (73)

By solving this equation and considering only the lead-ing order terms one can determine the radius of the pho-ton sphere rps yielding

rps = 3M(1 + X) (74)

On the other hand one also can show the relation [41]

ξ2 + η =r2

ps

F (rps) (75)

11

The shadow radius Rs can be expressed in terms of thecelestial coordinates (α β) as follows

Rs =radic

α2 + β2 =radic

1minus XrpsradicF (rps)

(76)

Note that in the last equation we have used Eqs (69) and(75) Using the values m = 06 eV Λ = 02 meV andρ0 = 92 times 10minus8 eV4 in units of the Sgr Alowast black hole wefind Rs = 5196158313M Using this result we can findthat the angular diameter increases by δθs = 3times 10minus5

microas compared to the Schwarzschild vacuum This resultis consistent with the result reported in Ref [4] whereauthors estimated that the dark matter halo could influ-ence the shadow of Sgr Alowast at a level of order of magni-tude of 10minus3 microas and 10minus5 microas respectively They haveused the so called Cold Dark Matter and Scalar FieldDark Matter models to study the apparent shapes of theshadow A similar result has been obtained for the darkmatter effect on the M87 black hole (see [5]) using theBurkert dark matter profile

VI CONCLUSION

In this paper we have obtained a rotating black holesolution surrounded by superfluid dark matter alongwith baryonic matter To achieve this purpose in thepresent work we considered the superfluid dark mat-ter model and used two specific profiles to describe thebaryonic matter distribution namely the spherical ex-ponential profile and the power law profile followed bythe special case of a totally dominated dark matter caseUsing the current values for the baryonic mass centraldensity of the superfluid dark matter halo radius aswell as the radial scale length for the baryonic matterin our galaxy we found that the shadow size of Sgr Alowastblack hole remains almost unchanged compared to theKerr vacuum BH This result is consistent with a recentwork reported in [5] and also [4] For entirely dominateddark matter galaxies we find that the angular diame-ter increases by 10minus5microarcsec This result shows that itis very difficult to constrain the dark matter parametersusing the shadow images Such tiny effects on the angu-lar diameter are out of reach for the existing space tech-nology and it remains an open question if future astro-nomical observations can potentially detect such effectsThat being said the expected value for the angular di-ameter in the case of Sgr Alowast is of the order of θs 53 microasas predicted in GR

As an interesting observation we show that an in-crease of baryonic mass followed by a decrease of theradial scale length can increase the shadow size consid-erably This can be explained by the fact that the bary-onic matter is mostly located in the interior of the galaxyon the other hand dark matter is mostly located in theouter region of the galaxy In other words for the to-tally dominated dark matter galaxies we observe almost

no effect on black hole shadows but a more precise mea-surement of the Sgr Alowast shadow radius can play a signifi-cant role in determining the baryonic mass in our galaxy

Acknowledgements

TZ is supported by National Natural Science Founda-tion of China under the Grants No 11675143 the Zhe-jiang Provincial Natural Science Foundation of Chinaunder Grant No LY20A050002 and the FundamentalResearch Funds for the Provincial Universities of Zhe-jiang in China under Grants No RF-A2019015

Appendix A

Here we demonstrate the derivation of the spinningblack hole metric As a first step to this formalism wetransform Boyer-Lindquist (BL) coordinates (t r θ φ) toEddington-Finkelstein (EF) coordinates (u r θ φ) Thiscan be achieved by using the coordinate transformation

dt = du +drradic

F12(r)G12(r) (A1)

we obtain

ds212 = minusF12(r)du2minus 2

radicF12(r)G12(r)

dudr+ r2dθ2 + r2 sin2 θdφ2

(A2)This metric can be expressed in terms of null tetrads

as

gmicroν = minuslmicronν minus lνnmicro + mmicromν + mνmmicro (A3)

where the null tetrads are defined as

lmicro = δmicror (A4)

nmicro =

radicG12(r)F12(r)

δmicrou minus

12G12(r)δ

micror (A5)

mmicro =1radic2H

(δ

microθ +

ι

sin θδ

microφ

) (A6)

These null tetrads are constructed in such a way that lmicro

and nmicro while mmicro and mmicro are complex It is worth notingthatH = r2 and will be used later on It is obvious fromthe notation that mmicro is complex conjugate of mmicro Thesevectors further satisfy the conditions for normalizationorthogonality and isotropy as

lmicrolmicro = nmicronmicro = mmicrommicro = mmicrommicro = 0 (A7)lmicrommicro = lmicrommicro = nmicrommicro = nmicrommicro = 0 (A8)

minuslmicronmicro = mmicrommicro = 1 (A9)

12

Following the NewmanndashJanis prescription we write

xprimemicro = xmicro + ia(δmicror minus δ

microu ) cos θ rarr

uprime = uminus ia cos θrprime = r + ia cos θθprime = θφprime = φ

(A10)in which a stands for the rotation parameter Nextlet the null tetrad vectors Za undergo a transformationgiven by Zmicro = (partxmicropartxprimeν)Zprimeν following [62]

lprimemicro = δmicror (A11)

nprimemicro =

radicB12

A12δ

microu minus

12

B12δmicror

mprimemicro =1radic

2Σ12

[(δ

microu minus δ

micror )ιa sin θ + δ

microθ +

ι

sin θδ

microφ

]where we have assumed that (F12(r)G12(r)H(r))transform to (A12(a r θ) B12(a r θ) Σ12(a r θ)) Withthe help of the above equations new metric inEddington-Finkelste coordinate reads

ds212 = minusA12du2 minus 2

radicA12

B12dudr (A12)

+ 2a sin2 θ

(A12 minus

radicA12

B12

)dudφ

+ 2a

radicA12

B12sin2 θdrdφ + Σ12dθ2

+ sin2 θ

[Σ12 + a2 sin2 θ

(2

radicA12

B12minus A12

)]dφ2

Note that A12 is some function of r and θ as we al-ready pointed out Without going into details of the cal-culation one can revert the EF coordinates back to BLcoordinates by using the following transformation (see

for more details [62])

du = dtminus a2 + r2

∆12dr dφ = dφprime minus a

∆12dr (A13)

where in order to simplify the notation we introduce

∆(r)12 = r2 F12(r) + a2 (A14)

Making use of F12(r) = G12(r) one can obtain Σ1 =Σ2 = r2 + a2 cos2 θ herein we shall use just Σ Droppingthe primes in the φ coordinate and choosing

A12 =(F12H+ a2 cos2 θ)Σ12

(H+ a2 cos2 θ)2 (A15)

and

B12 =F12H+ a2 cos2 θ

Σ (A16)

we obtain the spinning black hole space-time metric ina SFDM halo

ds212 = minus

(1minus 2Υ12(r)r

Σ

)dt2 +

Σ∆12

dr2 + Σdθ2

minus 2a sin2 θ2Υ12(r)r

Σdtdφ

+ sin2 θ

[(r2 + a2)2 minus a2∆12 sin2 θ

Σ

]dφ2 (A17)

with

Υ12(r) =r (1minusF12(r))

2 (A18)

in which F1 = F (r)exp and F2 = F (r)power respec-tively

[1] The Event Horizon Telescope Collaboration Ap J Lett875 L1 (2019)

[2] Haroon S Jamil M Jusufi K Lin K Mann R B PhysRev D 99 no4 044015 (2019)

[3] Haroon S Jusufi K Jamil M Universe 6(2) 23 (2020)arXiv190400711[gr-qc]

[4] Hou X Xu Z Zhou M Wang J JCAP 1807 no07 015(2018)

[5] Jusufi K Jamil M Salucci P Zhu T Haroon S PhysRev D 100 044012 (2019)

[6] Xu Z Hou X Gong X Wang J JCAP 1809 no09 038(2018)

[7] Xu Z Hou X Wang J JCAP 1810 no10 046 (2018)[8] Berezhiani L Khoury J Phys Rev D 92 103510 (2015)[9] Synge J L Mon Not Roy Astron Soc 131 no 3 463

(1966)

[10] Luminet J P Astron Astrophys 75 228 (1979)[11] Bardeen J M in Black Holes (Proceedings Les Houches

France August 1972) edited by C DeWitt and B S De-Witt

[12] Abdujabbarov A Toshmatov B Stuchlik Z AhmedovB Int J Mod Phys D 26 no 06 1750051 (2016)

[13] Abdikamalov A B Abdujabbarov A A Ayzenberg DMalafarina D Bambi C Ahmedov B Phys Rev D 100(2019) no2 024014

[14] Abdujabbarov A Amir M Ahmedov B Ghosh S GPhys Rev D 93 no 10 104004 (2016)

[15] Amir M Jusufi K Banerjee A Hansraj S ClassQuant Grav 36 no21 215007 (2019)

[16] Amir M Ghosh S G Phys Rev D 94 no 2 024054(2016)

[17] Abdujabbarov A Atamurotov F Kucukakca Y Ahme-

11

The shadow radius Rs can be expressed in terms of thecelestial coordinates (α β) as follows

Rs =radic

α2 + β2 =radic

1minus XrpsradicF (rps)

(76)

Note that in the last equation we have used Eqs (69) and(75) Using the values m = 06 eV Λ = 02 meV andρ0 = 92 times 10minus8 eV4 in units of the Sgr Alowast black hole wefind Rs = 5196158313M Using this result we can findthat the angular diameter increases by δθs = 3times 10minus5

microas compared to the Schwarzschild vacuum This resultis consistent with the result reported in Ref [4] whereauthors estimated that the dark matter halo could influ-ence the shadow of Sgr Alowast at a level of order of magni-tude of 10minus3 microas and 10minus5 microas respectively They haveused the so called Cold Dark Matter and Scalar FieldDark Matter models to study the apparent shapes of theshadow A similar result has been obtained for the darkmatter effect on the M87 black hole (see [5]) using theBurkert dark matter profile

VI CONCLUSION

In this paper we have obtained a rotating black holesolution surrounded by superfluid dark matter alongwith baryonic matter To achieve this purpose in thepresent work we considered the superfluid dark mat-ter model and used two specific profiles to describe thebaryonic matter distribution namely the spherical ex-ponential profile and the power law profile followed bythe special case of a totally dominated dark matter caseUsing the current values for the baryonic mass centraldensity of the superfluid dark matter halo radius aswell as the radial scale length for the baryonic matterin our galaxy we found that the shadow size of Sgr Alowastblack hole remains almost unchanged compared to theKerr vacuum BH This result is consistent with a recentwork reported in [5] and also [4] For entirely dominateddark matter galaxies we find that the angular diame-ter increases by 10minus5microarcsec This result shows that itis very difficult to constrain the dark matter parametersusing the shadow images Such tiny effects on the angu-lar diameter are out of reach for the existing space tech-nology and it remains an open question if future astro-nomical observations can potentially detect such effectsThat being said the expected value for the angular di-ameter in the case of Sgr Alowast is of the order of θs 53 microasas predicted in GR

As an interesting observation we show that an in-crease of baryonic mass followed by a decrease of theradial scale length can increase the shadow size consid-erably This can be explained by the fact that the bary-onic matter is mostly located in the interior of the galaxyon the other hand dark matter is mostly located in theouter region of the galaxy In other words for the to-tally dominated dark matter galaxies we observe almost

no effect on black hole shadows but a more precise mea-surement of the Sgr Alowast shadow radius can play a signifi-cant role in determining the baryonic mass in our galaxy

Acknowledgements

TZ is supported by National Natural Science Founda-tion of China under the Grants No 11675143 the Zhe-jiang Provincial Natural Science Foundation of Chinaunder Grant No LY20A050002 and the FundamentalResearch Funds for the Provincial Universities of Zhe-jiang in China under Grants No RF-A2019015

Appendix A

Here we demonstrate the derivation of the spinningblack hole metric As a first step to this formalism wetransform Boyer-Lindquist (BL) coordinates (t r θ φ) toEddington-Finkelstein (EF) coordinates (u r θ φ) Thiscan be achieved by using the coordinate transformation

dt = du +drradic

F12(r)G12(r) (A1)

we obtain

ds212 = minusF12(r)du2minus 2

radicF12(r)G12(r)

dudr+ r2dθ2 + r2 sin2 θdφ2

(A2)This metric can be expressed in terms of null tetrads

as

gmicroν = minuslmicronν minus lνnmicro + mmicromν + mνmmicro (A3)

where the null tetrads are defined as

lmicro = δmicror (A4)

nmicro =

radicG12(r)F12(r)

δmicrou minus

12G12(r)δ

micror (A5)

mmicro =1radic2H

(δ

microθ +

ι

sin θδ

microφ

) (A6)

These null tetrads are constructed in such a way that lmicro

and nmicro while mmicro and mmicro are complex It is worth notingthatH = r2 and will be used later on It is obvious fromthe notation that mmicro is complex conjugate of mmicro Thesevectors further satisfy the conditions for normalizationorthogonality and isotropy as

lmicrolmicro = nmicronmicro = mmicrommicro = mmicrommicro = 0 (A7)lmicrommicro = lmicrommicro = nmicrommicro = nmicrommicro = 0 (A8)

minuslmicronmicro = mmicrommicro = 1 (A9)

12

Following the NewmanndashJanis prescription we write

xprimemicro = xmicro + ia(δmicror minus δ

microu ) cos θ rarr

uprime = uminus ia cos θrprime = r + ia cos θθprime = θφprime = φ

(A10)in which a stands for the rotation parameter Nextlet the null tetrad vectors Za undergo a transformationgiven by Zmicro = (partxmicropartxprimeν)Zprimeν following [62]

lprimemicro = δmicror (A11)

nprimemicro =

radicB12

A12δ

microu minus

12

B12δmicror

mprimemicro =1radic

2Σ12

[(δ

microu minus δ

micror )ιa sin θ + δ

microθ +

ι

sin θδ

microφ

]where we have assumed that (F12(r)G12(r)H(r))transform to (A12(a r θ) B12(a r θ) Σ12(a r θ)) Withthe help of the above equations new metric inEddington-Finkelste coordinate reads

ds212 = minusA12du2 minus 2

radicA12

B12dudr (A12)

+ 2a sin2 θ

(A12 minus

radicA12

B12

)dudφ

+ 2a

radicA12

B12sin2 θdrdφ + Σ12dθ2

+ sin2 θ

[Σ12 + a2 sin2 θ

(2

radicA12

B12minus A12

)]dφ2

Note that A12 is some function of r and θ as we al-ready pointed out Without going into details of the cal-culation one can revert the EF coordinates back to BLcoordinates by using the following transformation (see

for more details [62])

du = dtminus a2 + r2

∆12dr dφ = dφprime minus a

∆12dr (A13)

where in order to simplify the notation we introduce

∆(r)12 = r2 F12(r) + a2 (A14)

Making use of F12(r) = G12(r) one can obtain Σ1 =Σ2 = r2 + a2 cos2 θ herein we shall use just Σ Droppingthe primes in the φ coordinate and choosing

A12 =(F12H+ a2 cos2 θ)Σ12

(H+ a2 cos2 θ)2 (A15)

and

B12 =F12H+ a2 cos2 θ

Σ (A16)

we obtain the spinning black hole space-time metric ina SFDM halo

ds212 = minus

(1minus 2Υ12(r)r

Σ

)dt2 +

Σ∆12

dr2 + Σdθ2

minus 2a sin2 θ2Υ12(r)r

Σdtdφ

+ sin2 θ

[(r2 + a2)2 minus a2∆12 sin2 θ

Σ

]dφ2 (A17)

with

Υ12(r) =r (1minusF12(r))

2 (A18)

in which F1 = F (r)exp and F2 = F (r)power respec-tively

[1] The Event Horizon Telescope Collaboration Ap J Lett875 L1 (2019)

[2] Haroon S Jamil M Jusufi K Lin K Mann R B PhysRev D 99 no4 044015 (2019)

[3] Haroon S Jusufi K Jamil M Universe 6(2) 23 (2020)arXiv190400711[gr-qc]

[4] Hou X Xu Z Zhou M Wang J JCAP 1807 no07 015(2018)

[5] Jusufi K Jamil M Salucci P Zhu T Haroon S PhysRev D 100 044012 (2019)

[6] Xu Z Hou X Gong X Wang J JCAP 1809 no09 038(2018)

[7] Xu Z Hou X Wang J JCAP 1810 no10 046 (2018)[8] Berezhiani L Khoury J Phys Rev D 92 103510 (2015)[9] Synge J L Mon Not Roy Astron Soc 131 no 3 463

(1966)

[10] Luminet J P Astron Astrophys 75 228 (1979)[11] Bardeen J M in Black Holes (Proceedings Les Houches

France August 1972) edited by C DeWitt and B S De-Witt

[12] Abdujabbarov A Toshmatov B Stuchlik Z AhmedovB Int J Mod Phys D 26 no 06 1750051 (2016)

[13] Abdikamalov A B Abdujabbarov A A Ayzenberg DMalafarina D Bambi C Ahmedov B Phys Rev D 100(2019) no2 024014

[14] Abdujabbarov A Amir M Ahmedov B Ghosh S GPhys Rev D 93 no 10 104004 (2016)

[15] Amir M Jusufi K Banerjee A Hansraj S ClassQuant Grav 36 no21 215007 (2019)

[16] Amir M Ghosh S G Phys Rev D 94 no 2 024054(2016)

[17] Abdujabbarov A Atamurotov F Kucukakca Y Ahme-

12

Following the NewmanndashJanis prescription we write

xprimemicro = xmicro + ia(δmicror minus δ

microu ) cos θ rarr

uprime = uminus ia cos θrprime = r + ia cos θθprime = θφprime = φ

(A10)in which a stands for the rotation parameter Nextlet the null tetrad vectors Za undergo a transformationgiven by Zmicro = (partxmicropartxprimeν)Zprimeν following [62]

lprimemicro = δmicror (A11)

nprimemicro =

radicB12

A12δ

microu minus

12

B12δmicror

mprimemicro =1radic

2Σ12

[(δ

microu minus δ

micror )ιa sin θ + δ

microθ +

ι

sin θδ

microφ

]where we have assumed that (F12(r)G12(r)H(r))transform to (A12(a r θ) B12(a r θ) Σ12(a r θ)) Withthe help of the above equations new metric inEddington-Finkelste coordinate reads

ds212 = minusA12du2 minus 2

radicA12

B12dudr (A12)

+ 2a sin2 θ

(A12 minus

radicA12

B12

)dudφ

+ 2a

radicA12

B12sin2 θdrdφ + Σ12dθ2

+ sin2 θ

[Σ12 + a2 sin2 θ

(2

radicA12

B12minus A12

)]dφ2

Note that A12 is some function of r and θ as we al-ready pointed out Without going into details of the cal-culation one can revert the EF coordinates back to BLcoordinates by using the following transformation (see

for more details [62])

du = dtminus a2 + r2

∆12dr dφ = dφprime minus a

∆12dr (A13)

where in order to simplify the notation we introduce

∆(r)12 = r2 F12(r) + a2 (A14)

Making use of F12(r) = G12(r) one can obtain Σ1 =Σ2 = r2 + a2 cos2 θ herein we shall use just Σ Droppingthe primes in the φ coordinate and choosing

A12 =(F12H+ a2 cos2 θ)Σ12

(H+ a2 cos2 θ)2 (A15)

and

B12 =F12H+ a2 cos2 θ

Σ (A16)

we obtain the spinning black hole space-time metric ina SFDM halo

ds212 = minus

(1minus 2Υ12(r)r

Σ

)dt2 +

Σ∆12

dr2 + Σdθ2

minus 2a sin2 θ2Υ12(r)r

Σdtdφ

+ sin2 θ

[(r2 + a2)2 minus a2∆12 sin2 θ

Σ

]dφ2 (A17)

with

Υ12(r) =r (1minusF12(r))

2 (A18)

in which F1 = F (r)exp and F2 = F (r)power respec-tively

[1] The Event Horizon Telescope Collaboration Ap J Lett875 L1 (2019)

[2] Haroon S Jamil M Jusufi K Lin K Mann R B PhysRev D 99 no4 044015 (2019)

[3] Haroon S Jusufi K Jamil M Universe 6(2) 23 (2020)arXiv190400711[gr-qc]

[4] Hou X Xu Z Zhou M Wang J JCAP 1807 no07 015(2018)

[5] Jusufi K Jamil M Salucci P Zhu T Haroon S PhysRev D 100 044012 (2019)

[6] Xu Z Hou X Gong X Wang J JCAP 1809 no09 038(2018)

[7] Xu Z Hou X Wang J JCAP 1810 no10 046 (2018)[8] Berezhiani L Khoury J Phys Rev D 92 103510 (2015)[9] Synge J L Mon Not Roy Astron Soc 131 no 3 463

(1966)

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[15] Amir M Jusufi K Banerjee A Hansraj S ClassQuant Grav 36 no21 215007 (2019)

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