Eur. Phys. J. C (2020) 80:354 https://doi.org/10.1140/epjc/s10052-020-7899-5

Regular Article - Theoretical Physics

Shadows of Sgr A∗ black hole surrounded by superfluid dark matter halo

Kimet Jusufi1,2,a, Mubasher Jamil3,4,5,b ,TaoZhu3,5,c 1 Physics Department, State University of Tetovo, Ilinden Street nn, 1200 Tetovo, North Macedonia 2 Institute of Physics, Faculty of Natural Sciences and Mathematics, Ss. Cyril and Methodius University, Arhimedova 3, 1000 Skopje, North Macedonia 3 Institute for Theoretical Physics and Cosmology, Zhejiang University of Technology, Hangzhou 310023, China 4 Department of Mathematics, School of Natural Sciences (SNS), National University of Sciences and Technology (NUST), H-12, Islamabad 44000, Pakistan 5 United Center for Gravitational Wave Physics (UCGWP), Zhejiang University of Technology, Hangzhou 310023, China

Received: 7 January 2020 / Accepted: 2 April 2020 © The Author(s) 2020

Abstract In this paper we construct a black hole solution disruption of neighboring stars. From theoretical perspective, surrounded by superfluid dark matter and baryonic matter, BHs serve as a perfect lab to test gravity in strong field regime. and study their effects on the shadow images of the Sgr A∗ Recently, the Event Horizon Telescope (EHT) Collaboration black hole. To achieve this goal, we have considered two den- announced their first results concerning the detection of an sity profiles for the baryonic matter described by the spher- event horizon of a supermassive black hole at the center of ical exponential profile and the power law profile including a neighboring elliptical M87 galaxy [1].Duetothegrav- a special case describing a totally dominated dark matter itational lensing effect, the bright structure surrounding the galaxy. Using the present values for the parameters of the black hole also known as the accretion disk appears distorted. superfluid dark matter and baryonic density profiles for the Moreover, the region of accretion disk behind the black hole Sgr A∗ black hole, we find that the effects of the superfluid also gets visible due to the bending of light by black hole. dark matter and baryonic matter on the size of shadows are The shadow image captured by EHT is in good agreement almost negligible compared to the Kerr vacuum black hole. with the predictions of the spacetime geometry of black hole In addition, we find that by increasing the baryonic mass the described by the Kerr metric. Studies of the shadow images shadow size increases considerably. This result can be linked of various black holes, together with the current and future to the matter distribution in the galaxy, namely the baryonic observations, are expected to provide an important approach matter is mostly located in the galactic center and, therefore, to the geometric structure of black holes or small deviations increasing the baryonic matter can affect the size of black from Kerr metric in the strong field regime. hole shadow compared to the totally dominated dark matter On the other hand, it is believed that up to 90% of matter galaxy where we observe an increase of the angular diameter in the host galaxy of a central black hole consists of dark of the Sgr A∗ black hole of the magnitude 10−5 µarcsec. matter. Thus, it is natural to expect that the dark matter halo which surrounds the central black hole could lead to small deviations from the Kerr metric near the black hole hori- 1 Introduction zon. With this motivation, the black hole solutions with dark matter halo and their effects on the black hole shadow have It is widely believed that the region at the centre of many been studied in [2Ð7]. In this paper, we consider a scenario galaxies contain black holes (BHs). They are astrophysical of a BH solution surrounded by a halo, which contains both objects which perform manifestations of extremely strong the dark matter and the baryonic matter. For dark matter, we gravity such as formation of gigantic jets of particles and assume a superfluid dark matter model recently proposed in [8] and construct the corresponding black hole solution with a e-mail: kimet.jusufi@unite.edu.mk the halo of dark matter and baryonic matter. As a specific b example we are going to elaborate the shadow images of Sgr e-mail: [email protected] (corresponding author) ∗ c e-mail: [email protected] A to estimate its angular diameter with the effect of the halo.

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One aim here is to study the shadows of rotating black superfluid phonons are described by the modified Newtonian holes surrounded by SFDM and baryonic matter. As we men- dynamics (MOND) type Lagrangian tioned, shadows possess interesting observational signatures 3/2  of the black hole spacetime in the strong gravity regime 2(2m) LDM, T =0 = X |X| . (2) and, in the future we expect that shadow observations can 3 impose constraints on different gravity theories. It is interest- At first sight the fractional power of X may seem strange ing that the characteristic map of a shadow image depends on if (2) represents for example a scalar field; however, from the the details of the surrounding environment around the black theory of phonons one can argue that an exponent is crucial hole, while the shadow contour is determined only by the since it determines the superfluid equation of state. On the spacetime metric itself. In the light of this, there have been other hand, to obtain a mediator, a force between baryons made extensive efforts to investigate shadows cast by differ- and the DM phonons, one must have the coupling ent black hole and compact object spacetimes. The shadow of a Schwarzschild black hole was first studied in [9] and Lint = αψ ρB, (3) later in [10] and the shadow for a Kerr black hole was stud- ied by [11]. Since then various authors have studied shadows where α is a constant while ρB gives the baryonic density. of black holes in modified theories of gravity and wormholes Let us point out that at zero temperature, this superfluid geometries [12Ð52]. theory has three parameters, namely the particle mass m, The plan of the paper is as follows: in Sect. 2,weuse a self-interaction strength parameter , and finally the cou- the superfluid density profile to obtain the radial function pling constant α between phonons and baryons. Starting from of a spherically symmetric spacetime. In Sect. 3, we study the action composed of Eqs. (2) and (3) one can obtain a the effect of baryonic and SFDM in the spacetime metric, MOND type force law. In particular the equation of motion using the exponential and power law profile for the baryonic for phonons is the following [8]: matter. In Sect. 4, we find a spherically symmetric black hole ⎛ ⎞ metric surrounded by SFDM. And then in Sect. 5,wefirst (∇ ψ)2 − 2mμˆ αρ construct the spinning black hole surrounded by dark matter ∇· ⎝ ∇ ψ⎠ = b , (4) 2 halo based the spherically symmetric spacetime we obtained (∇ ψ)2 − 2mμˆ by applying the NewmanÐJanis method. With this spinning black hole, we then study null geodesics and circular orbits with μˆ ≡ μ − m. Using the limit (∇ ψ)2  2mμˆ and ∗ to explore the shadows of Sgr A black hole. In Sect. 6 we ignoring the curl term one has present the main conclusions of this paper. We shall use the   natural units G = c = h¯ = 1 throughout the paper. |∇ψ|∇ψ  α aB , (5)

with aB being the Newtonian acceleration due to baryons 2 Superfluid dark matter and spherically symmetric only. While the mediated acceleration resulting from (3)is dark matter spacetime  aphonon = α∇ψ. (6) 2.1 Superfluid dark matter In this way it can be shown that From the view of field theory, one can describe the superfluid  a = α32 a . (7) by the theory of a spontaneously broken global U(1) sym- phonon B metry, in a state of finite U(1) charge density. In particular, From this result it is easy to see the well-known MOND at low energy regime the relevant degree of freedom is the acceleration by identifying Goldstone boson for the broken symmetry of the phonon field ψ. On the other hand, the U(1) symmetry acts non-linearly 3 2 a0 = α  . (8) on ψ as a shift symmetry, ψ → ψ +c. In the non-relativistic μ regime, at finite chemical potential , the most general effec- From the galactic rotation curves for a0 it is found tive theory is described by [8] MOND  . × −8 / 2 . a0 1 2 10 cm s (9) LT =0 = P(X), (1) As we saw, the MOND type law obtained in the superfluid in which X = μ − m + ψ˙ − (∇ ψ)2/2m. Note that m is the model is not exact and only applies in the regime (∇ ψ)2  particle mass and  represents the Newtonian gravitational 2mμˆ . More importantly, the total acceleration experienced potential. The main idea put forward in [8]isthattheDM by a test particle is a contribution of aB, aphonon and aDM. 123 Eur. Phys. J. C (2020) 80:354 Page 3 of 13 354

2.2 Spacetime metric in the presence of superfluid dark To solve the last integral we use Eqs. (10)Ð(14) and rewrite matter the mass profile in terms of the new coordinate ξ. In particular we obtain The density profile can be given in terms of dimensionless π π 2 ρ2 r r 0 variables  and ξ, defined by [8] MDM (r) = sin , r ≤ R. (19) 2R 2R 82m6  ρ From the last equation one can find the tangential velocity 1/2 0 ρ(r) = ρ0 , r = ξ, (10) v2 (r) = M (r)/r of a test particle moving in the dark 32π2m6 tg DM halo in spherical symmetric space-time, ρ ≡ ρ( ) where 0 0 is the central density. From the LaneÐ πr πr ρ2 Emden equation, v2 ( ) = 0 . tg r sin (20) 2R 2R 82m6  1 d 2 d 1/2 ξ =− , (11) In this section, we derive the space-time geometry for ξ 2 dξ dξ pure dark matter. To do so, let us consider a static and spher- and using the boundary conditions (0) = 1 and  (0) = 0, ically symmetric spacetime ansatz with pure dark matter in the numerical solution yields Schwarzschild coordinates that can be written as follows: 2 2 2 dr 2 2 2 2 ξ1  2.75. (12) ds =−f (r)dt + + r dθ + sin θdφ , (21) g(r) The previous equation determines the size of the condensate in which f (r) = g(r) are known as the redshift and shape functions, respectively. We have  ρ0 √ R = ξ1 . (13) dln f (r) 32π2m6 v2 (r) = , (22) tg dlnr A simple analytical form that provides a good fit is and we find

π ξ ρ2 ( π r ) (ξ) = cos . (14) − 0 cos 2 R 2 6 2 ξ1 f (r)DM = e 4 m (23)

The central density is related to the mass of the halo conden- with the constraint sate as follows:   ρ2 ( π r ) − 0 cos 2 R M ξ1 lim e 42m6 = 1. (24) ρ = . ρ → 0 3 (15) 0 0 4π R | (ξ1)| In the standard cold dark matter picture, a halo of mass From the numerics we find  (ξ1) −0.5. Substituting (13) = 12 = .  = . in Eq. (15), we can solve for the central density, MDM 10 M . In addition, for m 0 6 eV and 0 2 −24 3 −8 4 meV we find ρ0 = 0.02 × 10 g/cm = 9.2 × 10 eV . − / / Note that here we have used the conversion 10 19 g/cm3  2 5 18/5  6 5 MDM m −24 3 . 4 ρ0  10 g/cm . 0 4eV . 1012 M eV meV (16) 3 Effect of baryonic matter and superfluid dark matter Meanwhile the halo radius is

1/5 − / −2/5 In a realistic situation, galaxies consist of a baryonic (normal) MDM m 6 5  R  45 kpc . (17) matter (consisting of stars of mass Mstar, ionized gas of mass 1012 M eV meV Mgas, neutral hydrogen of mass MHI etc.,), the dark matter of Remarkably, for m ∼ eV and  ∼ meV we obtain DM mass MDM, which we assume to be in the form of a superfluid. = + halos of realistic size. The mass profile of the dark matter The total mass of the galaxy is therefore MB Mstar + galactic halo is given by Mgas MHI as the total baryonic mass in the galaxy. But, as we saw by introducing the baryonic matter we must take

r   into account a phonon-mediated force which describes the ( ) = π ρ 2 . MDM r 4 DM r r dr (18) interaction between SFDM and baryonic matter. 0 123 354 Page 4 of 13 Eur. Phys. J. C (2020) 80:354

To calculate the radial acceleration on a test baryonic par- and the disc. The mass function of the stellar thin disk is ticle within the superfluid core this is given by the sum of written as three contributions [53]     2 2 2 − r MB 2L − 2L + 2rL + r e L a(r) =a (r) +a (r) +a (r), (25) MB(r) = . (30) B DM phonon 2L2 where the first term is just the baryonic Newtonian acceler- We observe from these profiles that by going in the out- = ( )/ 2 ation aB MB r r . The second term is the gravitational side region of core i.e. r >> L,themassMB(r) reduces to a = ( )/ 2 acceleration from the superfluid core, aDM MDM r r . constant MB. In that way the last term in (28) becomes inde- The third term is the phonon-mediated acceleration pendent of r and the empirical TullyÐFisher relation which describes the rotational curves of galaxies. In our analyses,  ( ) = α∇ ψ. aphonon r (26) we shall simplify the calculations by considering a fixed bary- onic mass MB in the last term in (28). Let us introduce the The strength of this term is set by α and , or equivalently quantity the critical acceleration  v2 = M a , (31) √ 0 B 0 a (r) = a a , phonon 0 B (27) and use the tangent velocity while assuming a spherically symmetric solution resulting with which is nothing but the phonon force closely matching the π ρ2 cos( r ) deep-MOND acceleration a0 given by Eq. (8). Thus, in total, v2 − 0 2 R ( ) ≈ 2 0 2m6 the centrifugal acceleration of the particle of a test particle f r BM+DM Cr e 4  MB − r moving in the dark halo in spherical symmetric space-time × exp − (2L − (2L + r)e L ) , (32) is given by rL  valid for r ≤ R. In the present work, we shall focus on HSB v2 (r) ( ) ( ) ( ) tg MDM r MBM r MBM r type galaxyes, such as the Milky Way. We can use MB = = + + a0 . (28) r r 2 r 2 r 2 5 − 7 × 1010 M and L = 2 − 3 kpc [61]. Outside the size of the superfluid “core” denoted as R, 3.2 Power law profile there should be a different phase for DM matter, namely it is surrounded by DM particles in the normal phase, most As a second example, we assume the baryonic matter to be likely described by a NavarroÐFrenkÐWhite (NFW) profile or concentrated into an inner core of radius r , and that its mass some other effective type dark matter profile, most probably c profile M (r) can be described by the simple relation [62] described by the empirical Burkert profile (see, for example, B [54Ð60]). β r 3 However, as was argued in [53] we do not expect a sharp M (r) = M , B B + (33) transition, but instead, there should be a transition region r rc around at which the DM is nearly in equilibrium but is inca- in the present paper we shall be interested in the case β = 1, pable of maintaining long range coherence. In the present for high surface brightness galaxies (HSB). Making use of work we are mostly interested to explore the effect of the the tangent velocity and assuming a spherically symmetric SFDM core and, therefore, we are going to neglect the outer solution for the baryonic matter contribution in the case of DM region in the normal phase given by the NFW profile. HSB galaxies we find

3.1 Exponential profile 2 π r v2 M (2r+rc) ρ cos( ) πr 2 0 − B − 0 2 R ( + )2 2 6 f (r)BM+DM ≈ C e r rc e 4 m . (34) As a toy model for describing the baryonic distribution, we 2R consider a spherical exponential profile of density given by Note that the above analysis holds only inside the superfluid [53] core with r ≤ R (Fig. 1).

MB − r ρB(r) = e L (29) 3.3 Totally dominated dark matter galaxies 8π L3 where the characteristic scale L plays the role of a radial There is one particular solution of interest which describes scale length and MB is the baryonic mass. In that case the a totally dominated dark matter galaxies (TDDMG), some- total mass can be written as a sum of masses of dark matter times known as dark galaxies. To address this, we neglect the 123 Eur. Phys. J. C (2020) 80:354 Page 5 of 13 354

where d2 = dθ 2 + sin2 θdφ2 is the metric of a unit sphere. Moreover, we have introduced the following quantities:

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

In terms of these relations, the space-time metric describing a black hole in dark matter halo becomes [6] ⎡ ⎤  

⎢ g(r) 1 f (r) 1 ⎥ ds2 =−exp ⎣ + dr − dr⎦ dt2 2M r f (r) r g(r) − r −1 2M 2 2 2 Fig. 1 Schematic representation of the superfluid DM core and bary- + g(r) − dr + r d . (39) onic matter surrounding a black hole at the center. Outside the superfluid r core DM exists in a normal phase probably described by the NFW pro- file or Burkert profile In the absence of the dark matter halo, i.e., f (r) = g(r) = 1, the indefinite integral reduces to a constant [6]     baryonic mass MBM → 0, and therefore a0 → 0, yielding g(r) 1 f (r) 1 F1(r)+ f (r) = exp + − dr g(r)+F2(r) r f (r) r ρ2 ( π r ) − 0 cos 2 R 2M f (r) = Ce 42m6 , (35) = 1 − , (40) TDDMG r yielding F (r) =−2M/r. In that way one can obtain the valid for r ≤ R. This result is obtained directly from Eqs. 1 black hole space-time metric with surrounding matter, (32)** and (34) in the limit MB = 0. In other words this solution reduces to (23), as expected. Note that the integra- dr 2 ds2 =−F(r)dt2 + + r 2d2. (41) tion constant C can be absorbed in the time coordinate using G(r) dt2 → C dt2. Thus the general black hole solution surrounded by SFDM with F(r) = G(r) is given by

π 4 Black hole metric surrounded by baryonic matter and ρ2 cos( r ) 2 v2 − 0 2 R F(r) = Cr 0 e 42m6 superfluid dark matter exp   MB − r 2M × exp − (2L − (2L + r)e L ) − Let us now consider a more interesting scenario by adding rL r a black hole present in the SFDM halo. To achieve this aim, (42) we shall use the method introduced in [6]. Namely we need to compute the spacetime metric by assuming the Einstein and field equations given by [6] 2 π r v2 ρ cos( ) M (2r+rc) πr 2 0 − 0 2 R − B 2M 2 6 ( + )2 F(r)power = C e 4 m e r rc − . ν 1 ν 2 ν 2R r R μ − δ μ R = κ T μ, (36) 2 (43)

ν where the corresponding energyÐmomentum tensors T μ = As a special case we obtain a black hole in a totally dom- diag[−ρ, pr , p, p], where ρ = ρDM + ρBM, encodes the inated dark matter galaxy given by total contribution coming from the surrounded dark mat- π ter and baryonic matter, respectively. The space-time metric ρ2 cos( r ) − 0 2 R 2M 2 6 including the black hole is thus given by [6] F(r)TDDMG = Ce 4 m − , (44) r

2 valid for r ≤ R. Note that M is the black hole mass. In the 2 =−( ( ) + ( )) 2 + dr + 2 2, ds f r F1 r dt r d (37) limit M = 0 our solution reduces to (23), as expected. g(r) + F2(r) 123 354 Page 6 of 13 Eur. Phys. J. C (2020) 80:354

5 Shadow of Sgr A∗ black hole surrounded by where τ is the affine parameter, and S is the Jacobi action. superfluid dark matter Due to the spectime symmetries there are two conserved quantities, namely the conserved energy E =−pt and the In this section we generalize the static and the spherical conserved angular momentum L = pφ, respectively. symmetric black hole solution to a spinning black hole sur- To find the separable solution of Eq. (49), we need to rounded by dark matter halo applying the NewmanÐJanis express the action in the following form: method and adapting the approach introduced in [63,64]. Applying the NewmanÐJanis method we obtain the spinning 1 2 S = μ τ − Et + Jφ + S (r) + Sθ (θ), (50) black hole metric surrounded by dark matter halo as follows 2 r (see Appendix A): in which μ gives the mass of the test particle. Of course, this 2ϒ , (r)r  s2 =− − 1 2 t2 + r 2 +  θ 2 gives μ = 0 in the case of the photon. That being said, it is d 1,2 1  d  d d 1,2 straightforward to obtain the following equations of motions 2ϒ , (r)r − a 2 θ 1 2 t φ from the HamiltonÐJacobi equation: 2 sin  d d   (r 2 + a2)2 − a2 , sin2 θ 2 + 2 + 2 θ 1 2 φ2  dt = r a [ ( 2 + 2) − ]− ( 2 θ − ), sin  d (45) E r a aL a aE sin L dτ 1,2 where we have introduced (51) dr   = R(r), (52) r (1 − F1,2(r)) τ ϒ , (r) = (46) d 1 2 2 dθ   = (θ), (53) dτ F = F( ) F = F( ) and identified 1 r exp and 2 r power, respec- dφ a L ( ) =  = [ ( 2 + 2) − ]− − , tively. As a special case we find the Kerr BH limit, Y1,2 r E r a aL aE 2 (54) dτ 1,2 sin θ M,ifa0 = MB = ρ0 = 0. In addition we can explore the shape of the ergoregion of where R(r) and (θ) are given by our black hole metric (45), that is, we can plot the shape of the ergoregion, say in the xz-plane. On the other hand the 2 2 2 2 2 2 R(r) =[E(r + a ) − aL] − 1,2[m r +(aE − L) +K], horizons of the black hole are found by solving 1,2 = 0, t.e., (55)

2 2 2 L 2 2 2 r F(r)1,2 + a = 0, (47) (θ) = K − − a E cos θ, (56) sin2 θ while the inner and outer ergo-surfaces are obtained from and K is the separation constant known as the Carter constant. solving gtt = 0, i.e.,

2 2 2 5.2 Circular orbits r F(r)1,2 + a cos θ = 0. (48) Due to the strong gravity near the black hole, it is thus In Fig. 2 we plot  , as a function of r for different values 1 2 expected that the photons emitted near a black hole will even- of a. In the special case for some critical value a = a (blue E tually fall into the black hole or eventually scatter away from line) the two horizons coincide, in other words we have an it. In this way, the photons captured by the black hole will extremal black hole with degenerate horizons. Beyond this form a dark region defining the contour of the shadow. To critical value, a > a , there is no event horizon and the E elaborate the presence of unstable circular orbits around the solution corresponds to a naked singularity. black hole we need to study the radial geodesic by intro- ducing the effective potential Veff which can be written as 5.1 Geodesic equations follows:

In order to find the contour of a black hole shadow of the 2 dr 1,2 rotating spacetime (45), first we need to find the null geodesic 2 + V = 0, (57) dτ eff equations using the HamiltonÐJacobi method given by

where V 1 = V exp and V 2 = V power give the exponential ∂S 1 μν ∂S ∂S eff eff eff eff =− g , (49) ∂τ 2 ∂xμ ∂xν and the power law, respectively. At this point, it is convenient 123 Eur. Phys. J. C (2020) 80:354 Page 7 of 13 354

Fig. 2 Left panel: variation of 1 as a function of r, for BH-SFDM respectively. We have used m = 0.6eVand = 0.2meVwefind −24 3 −8 4 using the spherical exponential profile. Right panel: variation of 2 as ρ0 = 0.02 × 10 g/cm , or in eV units, ρ0 = 9.2 × 10 eV .For 10 4 a function of r, for BH-SFDM using the power profile law. Note that the Milky Way we can use MB = 6 × 10 M =1.39 × 10 MBH and 10 a = 0.65 (black curve), a = 0.85 (red curve), a = 1 (blue curve), rc = 2.6 kpc = 1.24 × 10 MBH

Fig. 3 Left panel: the effective potential of photon moving for BH-SFDM using the exponsntial profile. Right panel: the effective potential of photon using thepower law profile. We have chosen a = 0.5 (black color), a = 0.65 (red color) and a = 0.85 (blue color), respectively

to introduce the two parameters ξ and η, defined as One can recover the Kerr vacuum case by letting a0 = MB = ρ0 = 0, yielding ξ = L/E,η= K/E2. (58)

2( − ) − 2( − ) For the effective potential then we obtain the following rela- ξ = r 3M r a M r , 1,2 ( − ) (62) tion:  a r M  r 3 4Ma2 − r(r − 3M)2 1,2 =  (( − ξ )2 + η ) − ( 2 + 2 − ξ )2, η1,2 = . (63) Veff 1,2 a 1,2 1,2 r a a 1,2 (59) a2(r − M)2 1,2/ 2 1,2 where we have replaced Veff E by Veff . For more details in Fig. 3 we plot the variation of the effective potential asso- To obtain the shadow images of our black hole in the pres- ciated with the radial motion of photons. The circular photon ence of dark matter we assume that the observer is located at ( ,θ ) θ orbits exist when at some constant r = rcir. the conditions the position with coordinates ro o , where ro and o repre- (Fig. 4) sents the angular coordinate on observer’s sky. Furthermore, we need to introduce two celestial coordinates, α and β,for 1,2 , dV (r) the observer by using the following relations [29]: V 1 2(r) = 0, eff = 0. (60) eff dr (φ) (θ) Combining all these equations it is possible to show that p p α =−r ,β= r , o (t) o (t) (64) p p (r2 + a2)(rF (r) + 2F , (r)) − 4(r2F , (r) + a2) ξ = 1,2 1 2 1 2 , 1,2 ( F ( ) + F ( )) a r 1,2 r 2 1,2 r 3[ 2F ( ) − ( F ( ) − F ( ))2] ( (t), (r), (θ), (φ)) r 8a 1,2 r r r 1,2 r 2 1,2 r where p p p p are the tetrad components of η , = . (61) 1 2 2( F ( ) + F ( ))2 a r 1,2 r 2 1,2 r the photon momentum with respect to locally non-rotating 123 354 Page 8 of 13 Eur. Phys. J. C (2020) 80:354

−24 3 Fig. 4 The shape of the ergoregion and inner/outer horizons for differ- eV and  = 0.2meVwefindρ0 = 0.02 × 10 g/cm ,orineV −8 4 ent values of a using the spherical exponential profile. We use M = 1 units, ρ0 = 9.2 × 10 eV . For the normal matter contribution of the ∗ 6 10 4 in units of the Sgr A black hole mass given by MBH = 4.3 × 10 M Milky Way we can use MB = 6 × 10 M =1.39 × 10 MBH and 10 10 and R = 158 kpc or R = 75.5 × 10 MBH. Furthermore for m = 0.6 L = 2.6 kpc = 1.24 × 10 MBH

μ reference frame. The observer basis e(ν) can be expanded in in terms our parameters ξ and η as follows [43]: the coordinate basis (see [65]) ξ α =− 1,2 | , 1,2 ro √ (ro,θo) gφφ(ζ1,2 − γ1,2ξ1,2) (t) μ (φ) μ L  p =−e( ) pμ = E ζ1,2 − γ1,2 L, p = e(φ) pμ = √ , t gφφ η + 2 2 θ − ξ 2 2 θ 1,2 a cos 1,2 cot (θ) μ pθ ( ) μ pr β =± √ | , = = √ , r = = √ , 1,2 ro (ro,θo) (66) p e(θ) pμ p e(r) pμ (65) θθ(ζ , − γ , ξ , ) gθθ grr g 1 2 1 2 1 2

where where the quantities E =−pt and pφ = L are conserved  due to the associated Killing vectors. If we use ξ = L/E, η = gφφ √ ζ1,2 = (67) 2 2 − K/E and pθ =± (θ) we can rewrite these coordinates gtφ gttgφφ 123 Eur. Phys. J. C (2020) 80:354 Page 9 of 13 354

10 Fig. 5 Variation in shape of shadow using the spherical exponential and L = 2.6 kpc = 1.24 × 10 MBH. We observe that the dashed blue profile for baryonic matter. We use M = 1 in units of the Sgr A∗ curve corresponding to Sgr A∗ with the above parameters is almost 6 black hole mass given by MBH = 4.3 × 10 M and R = 158 kpc or indistinguishable from the black dashed curve describing the Kerr vac- 10 R = 75.5 ×10 MBH. Furthermore for m = 0.6eVand = 0.2meV uum BH. The solid blue curve corresponds to the case of increasing the −24 3 −8 4 2 we find ρ0 = 0.02 ×10 g/cm , or in eV units, ρ0 = 9.2 ×10 eV . baryonic mass by a factor of 10 and decrease of core radius by a factor 10 4 3 For the Milky Way we can use MB = 6 × 10 M =1.39 × 10 MBH of 10

and effect on the shadow images. In fact one can observe that the dashed curve (exponential profile) and dashed red curve gtφ (power law profile) are almost indistinguishable from the γ1,2 =− ζ1,2. (68) gφφ black dashed curve (Kerr vacuum BH). However, we can observe that by increasing the baryonic mass the shadow Finally to simplify the problem further we are going to con- size increases considerably (solid blue/red curve). The angu- ∗ sider that our observer is located in the equatorial plane (θ = lar radius of the Sgr A black hole shadow can be estimated θ = / π/2) and that we have very large but finite ro = D = 8.3 using the observable Rs as s Rs M D, where M is the kpc, to find black hole mass and D is the distance between the black hole  and the observer. The angular radius can be further expressed −6 α1,2 =− f1,2 ξ1,2, as θs = 9.87098 × 10 Rs(M/M )(1 kpc/D) µas. In the  √ ∗ case of Sgr A ;wehaveusedM = 4.3×106M and D = 8.3 β1,2 =± f1,2 η1,2. (69) kpc [4] is the distance between the Earth and the Sgr A∗ cen-

Note that f1,2 are given by Eqs. (32) and (34), respectively. tral black hole. In order to get more information about the We see that due to the presence of SFDM our solution is non- effect of SFDM on the physical observables we shall esti- ∗ asymptotically flat. In the special case when SFDM is absent mate the effect of SFDM on the angular radius for Sgr A . we recover f1,2 → 1, hence the above relations reduces to To simplify the problem, let us consider our static and spher- the asymptotically flat case. Note that for the observer located ically symmetric black hole metric in a totally dominated at the position with coordinates (ro,θo) we have neglected dark matter galaxy described by the function (Fig. 7) the effect of rotation while ξ1,2 and η1,2 are given by Eq. (61) and are evaluated at the circular photon orbits r = r . − 2M cir F(r) = e X − , (70) For more details in Figs. 5 and 6 we plot the shape of Sgr r A∗ black hole shadows. Using the parameter values speci- fying the SFDM and baryonic matter we obtain almost no where we have introduced 123 354 Page 10 of 13 Eur. Phys. J. C (2020) 80:354

10 Fig. 6 Variation in shape of shadow for different values of a using and rc = 2.6 kpc = 1.24 ×10 MBH. The dashed red curve corresponds the power law profile. We use M = 1 in units of the- Sgr A∗ black to Sgr A∗ using the above parameters which is almost indistinguishable 6 hole mass given by MBH = 4.3 × 10 M and R = 158 kpc or from the black dashed curve discribing the Kerr vacuum BH. The solid 10 R = 75.5 ×10 MBH. Furthermore for m = 0.6eVand = 0.2meV red curve corresponds to the case of increasing the baryonic mass by a −24 3 −8 4 2 3 we find ρ0 = 0.02 ×10 g/cm , or in eV units, ρ0 = 9.2 ×10 eV . factor of 10 and decrease of core radius by a factor of 10 10 4 For the Milky Way we can use MB = 6 × 10 M =1.39 × 10 MBH

ρ2 X = 0 , (71) 42m6 ( π r )  where we have used the approximation cos 2 R 1 since our solution is valid for r ≤ R. From the circular photon orbit conditions one can show [41]

rF (r) 2 − = 0. (72) F(r)

Now if we expand around X in the above function we obtain Fig. 7 The expected value for the angular diameter in the case of Sgr ∗ A . Note that we have adopted D = 8.3 kpc and M = 4.3 × 106M 2M F(r) = 1 − X − +··· (73) r The shadow radius Rs can be expressed in terms of the celes- (α, β) By solving this equation and considering only the leading tial coordinates as follows: order terms one can determine the radius of the photon sphere  √ r 2 2 ps rps yielding Rs = α + β = 1 − X  . (76) F(rps) rps = 3M(1 + X). (74) Note that in the last equation we have used Eqs. (69) and On the other hand, one also can show the relation [41] (75). Using the values m = 0.6eV, = 0.2 meV and ρ0 = 9.2 × 10−8 eV4, in units of the Sgr A∗ black hole we find 2 rps Rs = 5.196158313M (i.e. Rs is measured in the units of M) ξ 2 + η = . (75) F(rps) (i.e. Rs is measured in the units of M). Using this result, we 123 Eur. Phys. J. C (2020) 80:354 Page 11 of 13 354

−5 can find that the angular diameter increases by δθs = 3×10 Data Availability Statement This manuscript has no associated data µas as compared to the Schwarzschild vacuum. This result or the data will not be deposited. [Authors’ comment: This paper deals is consistent with the result reported in Ref. [4] where the with a theoretical study of black holes. All the relevant data involved in this study is already presented in the research paper.] authors estimated that the dark matter halo could influence ∗ − the shadow of Sgr A at a level of order of magnitude of 10 3 Open Access This article is licensed under a Creative Commons Attri- µas and 10−5 µas, respectively. They have used the so called bution 4.0 International License, which permits use, sharing, adaptation, Cold Dark Matter and Scalar Field Dark Matter models to distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, pro- study the apparent shapes of the shadow. A similar result has vide a link to the Creative Commons licence, and indicate if changes been obtained for the dark matter effect on the M87 black were made. The images or other third party material in this article hole (see, [5]) using the Burkert dark matter profile. are included in the article’s Creative Commons licence, unless indi- cated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permit- ted use, you will need to obtain permission directly from the copy- 6 Conclusion right holder. To view a copy of this licence, visit http://creativecomm ons.org/licenses/by/4.0/. Funded by SCOAP3. In this paper we have obtained a rotating black hole solution surrounded by superfluid dark matter along with baryonic matter. To this aim we considered the superfluid dark matter Appendix A model and used two specific profiles to describe the baryonic matter distribution, namely the spherical exponential profile Here we demonstrate the derivation of the spinning black and the power law profile followed by the special case of a hole metric. As a first step to this formalism, we trans- totally dominated dark matter case. Using the current values form the BoyerÐLindquist (BL) coordinates (t, r,θ,φ) to for the baryonic mass, the central density of the superfluid EddingtonÐFinkelstein (EF) coordinates (u, r,θ,φ).This dark matter, halo radius, and the radial scale length for the can be achieved by using the coordinate transformation baryonic matter in our galaxy, we found that the shadow size ∗ dr of the Sgr A black hole remains almost unchanged compared dt = du +  , (A1) to the Kerr vacuum BH. This result is consistent with a recent F1,2(r)G1,2(r) work reported in [5] and also [4]. For entirely dominated dark we obtain matter galaxies, we find that the angular diameter increases  −5 µ F , (r) by 10 arcsec. This result shows that it is very difficult 2 =−F ( ) 2− 1 2 + 2 θ2+ 2 2 θ φ2. ds1,2 1,2 r du 2 dudr r d r sin d (A2) to constrain the dark matter parameters using the shadow G1,2(r) images. Such tiny effects on the angular diameter are out of reach for the existing space technology and it remains an open This metric can be expressed in terms of null tetrads as question if future astronomical observations can potentially μν μ ν ν μ μ ν ν μ g =−l n − l n + m m + m m , (A3) detect such effects. That being said, the expected value for the angular diameter in the case of Sgr A∗ is of the order of where the null tetrads are defined as θ  53 µas as predicted in GR. μ μ s l = δ , (A4) As an interesting observation, we show that an increase r G ( ) of baryonic mass followed by a decrease of the radial scale μ = 1,2 r δμ − 1G ( )δμ, n u 1,2 r r (A5) length can increase the shadow size considerably. This can F1,2(r) 2 be explained by the fact that the baryonic matter is mostly μ 1 μ ι˙ μ located in the interior of the galaxy; on the other hand, dark m = √ δ + δ . (A6) H θ sin θ φ matter is mostly located in the outer region of the galaxy. In 2 μ μ other words, for the totally dominated dark matter galaxies These null tetrads are constructed in such a way that l and n μ μ we observe almost no effect on black hole shadows, but a while m and m¯ are complex. It is worth noting that H = r 2 more precise measurement of the Sgr A∗ shadow radius can and this will be used later on. It is obvious from the notation μ μ play a significant role in determining the baryonic mass in that m¯ is the complex conjugate of m . These vectors further our galaxy. satisfy the conditions for normalization, orthogonality and isotropy: Acknowledgements TZ is supported by National Natural Science μ μ μ μ Foundation of China under the Grants No. 11675143, the Zhejiang l lμ = n nμ = m mμ =¯m m¯ μ = 0, (A7) Provincial Natural Science Foundation of China under Grant No. μ μ μ μ l mμ = l m¯ μ = n mμ = n m¯ μ = 0, (A8) LY20A050002, and the Fundamental Research Funds for the Provincial μ μ Universities of Zhejiang in China under Grants No. RF-A2019015. −l nμ = m m¯ μ = 1. (A9) 123 354 Page 12 of 13 Eur. Phys. J. C (2020) 80:354

Following the NewmanÐJanis prescription we write and

⎧ F H + 2 2 θ ⎪ u = u − iacos θ, = 1,2 a cos , ⎨⎪ B1,2 (A16) = + θ,  μ = μ + (δμ − δμ) θ → r r iacos x x ia r u cos ⎪ θ = θ, ⎩⎪ φ = φ. we obtain the spinning black hole space-time metric in a SFDM halo, (A10) 2ϒ , (r)r  s2 =− − 1 2 t2 + r 2 +  θ 2 d 1,2 1  d  d d in which a stands for the rotation parameter. Next, let the 1,2 null tetrad vectors Z a undergo a transformation given by 2ϒ1,2(r)r − a 2 θ t φ μ = (∂ μ/∂ ν) ν 2 sin  d d Z x x Z , following [63],   2 2 2 2 2 μ μ (r + a ) − a  , sin θ l = δ , + 2 θ 1 2 φ2 r sin  d (A17) μ = B1,2 δμ − 1 δμ, n u B1,2 r with A1,2 2   ( − F ( )) μ 1 μ μ μ ι˙ μ r 1 1,2 r m =  (δ − δ )ι˙a θ + δ + δ , ϒ1,2(r) = , (A18) u r sin θ θ φ 2 21,2 sin (A11) in which F1 = F(r)exp and F2 = F(r)power, respectively. wherewehaveassumedthat(F1,2(r), G1,2(r), H(r)) trans- form to (A1,2(a, r,θ),B1,2(a, r,θ),1,2(a, r,θ)). With the help of the above equations the new metric in EddingtonÐ References Finkelste coordinates reads  1. The Event Horizon Telescope Collaboration, Ap. J. Lett. 875,1 (2019). https://doi.org/10.3847/2041-8213/ab0ec7 2 =− 2 − A1,2 ds1,2 A1,2du 2 dudr 2. S. Haroon, M. Jamil, K. Jusufi, K. Lin, R.B. Mann, Phys. Rev. B1,2    D 99(4), 044015 (2019). https://doi.org/10.1103/PhysRevD.99. 044015 2 A1,2 +2a sin θ A1,2 − dudφ 3. S. Haroon, K. Jusufi, M. Jamil, Universe 6(2), 23 (2020). B1,2 arXiv:1904.00711 [gr-qc]  4. X. Hou, Z. Xu, M. Zhou, J. Wang, JCAP 1807(07), 015 (2018). + A1,2 2 θ φ +  θ 2 https://doi.org/10.1088/1475-7516/2018/07/015 2a sin drd 1,2d 5. K. Jusufi, M. Jamil, P.Salucci, T. Zhu, S. Haroon, Phys. Rev. D 100, B1,2     044012 (2019). https://doi.org/10.1103/PhysRevD.100.044012 6. Z. Xu, X. Hou, X. Gong, J. Wang, JCAP 1809(09), 038 (2018). 2 2 2 A1,2 2 + sin θ 1,2 + a sin θ 2 − A1,2 dφ . https://doi.org/10.1088/1475-7516/2018/09/038 B1,2 7. Z. Xu, X. Hou, J. Wang, JCAP 1810(10), 046 (2018). https://doi. (A12) org/10.1088/1475-7516/2018/10/046 8. L. Berezhiani, J. Khoury, Phys. Rev. D 92, 103510 (2015). https:// doi.org/10.1103/PhysRevD.92.103510 Note that A1,2 is some function of r and θ as we already 9. J.L. Synge, Mon. Not. R. Astron. Soc. 131, 3, 463 (1966). https:// pointed out. Without going into details of the calculation one academic.oup.com/mnras/article/131/3/463/2603317 can revert the EF coordinates back to BL coordinates by using 10. J.P. Luminet, Astron. Astrophys. 75, 228 (1979). http://articles. the following transformation (see for more details [63]): adsabs.harvard.edu/cgi-bin/nph-iarticle_query?bibcode=1979A 11. J.M. Bardeen, in Black Holes (Proceedings, Les Houches, France, 2 + 2 = − a r , φ = φ − a , August, 1972) edited by C. DeWitt and B. S. DeWitt du dt dr d d dr (A13) 12. A. Abdujabbarov, B. Toshmatov, Z. Stuchlik, B. Ahmedov, Int. 1,2 1,2 J. Mod. Phys. D 26(06), 1750051 (2016). https://doi.org/10.1142/ where in order to simplify the notation we introduce S0218271817500511 13. A.B. Abdikamalov, A.A. Abdujabbarov, D. Ayzenberg, D. Malafa- 2 2 rina,C.Bambi,B.Ahmedov,Phys.Rev.D100(2), 024014 (2019). (r) , = r F , (r) + a . (A14) 1 2 1 2 https://doi.org/10.1103/PhysRevD.100.024014 14. A. Abdujabbarov, M. Amir, B. Ahmedov, S.G. Ghosh, Phys. Rev. Making use of F1,2(r) = G1,2(r), one can obtain 1 = D 93(10), 104004 (2016). https://doi.org/10.1103/PhysRevD.93. 2 2 2 104004 2 = r + a cos θ; here we shall use just . Dropping the primes in the φ coordinate and choosing 15. M. Amir, K. Jusufi, A. Banerjee, S. Hansraj, Class. Quant. Gravit. 36(21), 215007 (2019). https://doi.org/10.1088/1361-6382/ ab42be/meta 2 2 (F1,2H + a cos θ)1,2 16. M. Amir, S.G. Ghosh, Phys. Rev. D 94(2), 024054 (2016). https:// A , = (A15) 1 2 (H + a2 cos2 θ)2 doi.org/10.1103/PhysRevD.94.024054 123 Eur. Phys. J. C (2020) 80:354 Page 13 of 13 354

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