Gravitational decoupling for axially symmetric systems and rotating black holes
GGravitational decoupling for axially symmetric systems and rotating black holes
E. Contreras ∗ Departamento de F´ısica, Colegio de Ciencias e Ingenier´ıa,Universidad San Francisco de Quito, Quito, Ecuador.
J. Ovalle † Research Centre of Theoretical Physics and Astrophysics, Institute of Physics,Silesian University in Opava, CZ-746 01 Opava, Czech Republic.
R. Casadio ‡ Dipartimento di Fisica e Astronomia, Alma Mater Universit`a di Bologna, 40126 Bologna, ItalyIstituto Nazionale di Fisica Nucleare, Sezione di Bologna, 40127 Bologna, Italy
We introduce a systematic and direct procedure to generate hairy rotating black holes by deform-ing a spherically symmetric seed solution. We develop our analysis in the context of the gravitationaldecoupling approach, without resorting to the Newman-Janis algorithm. As examples of possibleapplications, we investigate how the Kerr black hole solution is modified by a surrounding fluid withconserved energy-momentum tensor. We find non-trivial extensions of the Kerr and Kerr-Newmanblack holes with primary hair. We prove that a rotating and charged black hole can have the samehorizon as Kerr’s, Schwarzschild’s or Reissner-Nordstr¨om’s, thus showing possible observationaleffects of matter around black holes.
I. INTRODUCTION
Black holes (BHs) have been considered more thanmere exotic solutions of the Einstein equations for quitesome time now [1, 2]. Nonetheless, it is only very re-cently that their direct existence was detected, mainlydue to the spectacular results of both the LIGO [3] andEvent Horizon Telescope [4] collaborations. It is also fairto mention that some ultracompact stellar models couldact as “black-hole mimickers” [5, 6], although the exis-tence of such objects would not necessarily exclude theexistence of BHs [7], as it could be naively concluded.Starting with Kerr’s celebrated work [8], the interestin BHs has increased notably, and a large number of so-lutions have been found in various contexts (for somerecent notable works, see e.g. [9–15] ). Despite this di-versity, in four-dimensional space-time we can group allcases into two large groups: i) static spherically symmet-ric solutions and ii) stationary rotating solutions. (Notethat, if we include non-abelian matter fields, we can alsofind axisymmetric static BHs [16].) The study of theserotating and non-rotating BH metrics, and the shadowthey produce, have been extensively explored in recentyears [17–47], the Newman-Janis algorithm [48] and itsversion without complexification [49] being tremendouslyuseful tools to generate rotating systems.In all cases, it is well known that the presence of mat-ter around BHs could produce a significant distortionof the shadow in a highly model-dependent fashion (see e.g. [36, 50–52] and references therein). The resolution ∗ [email protected] † Corresponding author: [email protected] ‡ [email protected] of the first BH image is not enough to support or discardany of these models, hence it is important to study thisdistortion with a minimum set of assumptions. This isprecisely the topic of the present work. Namely, we willconsider a Kerr BH surrounded by an axially symmet-ric “tensor-vacuum” (analogous to the electro-vacuumand scalar-vacuum cases) represented by a conservedenergy-momentum tensor S µν which could account forone or more fundamental fields (scalar, vector or tensorfields representing any phenomenologically viable form ofmatter-energy, such as dark matter or dark energy). Theonly restriction we require is that S µν satisfies either thestrong (SEC) or the dominant energy condition (DEC)in the region outside the event horizon. Since the Grav-itational Decoupling approach (GD) [53, 54] is preciselydesigned for describing deformations of known solutionsof General Relativity induced by additional sources, wewill study this problem by first extending the GD to ax-ially symmetric systems.The GD is originally based on the so-called minimalgeometric deformation (MGD) [55, 56] (for some earlierworks on the MGD, see e.g. [57–65], and Refs. [66–84] forsome recent applications). The GD has been shown tobe particularly useful for at least three tasks [85–118]:a) to generate new and more complex solutions fromknown (seed) solutions of the Einstein field equations;b) to systematically reduce (decouple) a complex energy-momentum tensor T µν into simpler components; and c)to find solutions in gravitational theories beyond GeneralRelativity. Despite the above, one of the apparent limi-tations of the GD is that the decoupling of gravitational Indeed, a theoretical model with the minimum amount of as-sumptions is always desirable in any context a r X i v : . [ g r- q c ] J a n sources has only been achieved in the spherically symmet-ric case so far. One of the goals of this paper is to showthat indeed the GD can be implemented beyond spher-ical symmetry. In particular, we will show how the GDcan be obtained for axially symmetric systems, which isof particular importance for the study of rotating stellarsystems and BHs.The paper is organised as follows: in Section II, we firstreview the fundamentals of the GD approach for a spheri-cally symmetric system containing two sources, and thenwe show in detail how to extend the GD approach forthe axially symmetric case; in Section III, we apply ourresults to generate the axially symmetric version of twospherically symmetric hairy BH solutions, without imple-menting the Newman–Janis algorithm. The first solutioncontains a source satisfying the SEC and is an extensionof the Kerr metric, while the DEC holds for the source inthe second solution, which represents an extension of theKerr-Newman BH; finally, we summarize our conclusionsin Section IV. II. GRAVITATIONAL DECOUPLING
We start this section by briefly reviewing the key as-pects of the GD for spherically symmetric systems (de-scribed in detail in Ref. [54]). A particularly simple caseof GD is given by the MGD [55, 56], which will guide usto introduce a GD for the axially symmetric case.We start by considering the Einstein field equations ˜ G µν ≡ ˜ R µν −
12 ˜ R ˜ g µν = k ˜ T µν , (1)with a total energy-momentum tensor containing twocontributions, ˜ T µν = T µν + S µν , (2)where T µν is usually associated with a known solution ofGeneral Relativity, whereas S µν may contain new fieldsor a new gravitational sector. Since the Einstein tensor˜ G µν satisfies the Bianchi identity, the total source ˜ T µν must be covariantly conserved. A. Spherically symmetric case
For spherically symmetric and static systems, the met-ric ˜ g µν can be written as ds = e ν ( r ) dt − e λ ( r ) dr − r d Ω , (3)where ν = ν ( r ) and λ = λ ( r ) are functions of the arealradius r only and d Ω = dθ + sin θ dφ . The Einstein We use units with c = 1 and k = 8 π G N , where G N is Newton’sconstant. tensor in Eq. (1) then reads˜ G = 1 r − e − λ (cid:18) r − λ (cid:48) r (cid:19) (4)˜ G = 1 r − e − λ (cid:18) r + ν (cid:48) r (cid:19) (5)˜ G = − e − λ (cid:18) ν (cid:48)(cid:48) + ν (cid:48) − λ (cid:48) ν (cid:48) + 2 ν (cid:48) − λ (cid:48) r (cid:19) , (6)where f (cid:48) ≡ ∂ r f and ˜ T = ˜ T due to the spherical sym-metry. By simple inspection, we can identify an effectivedensity ˜ (cid:15) = T + S , (7)an effective radial pressure˜ p r = − T − S , (8)and an effective tangential pressure˜ p t = − T − S . (9)Since the anisotropy Π ≡ ˜ p t − ˜ p r usually does not van-ish, the system of Eqs. (4)-(6) may be viewed as ananisotropic fluid.We next consider a solution to the Eqs. (1) generatedby the seed source T µν alone [that is, for S µν = 0], whichwe write as ds = e ξ ( r ) dt − e µ ( r ) dr − r d Ω , (10)where e − µ ( r ) ≡ − k r (cid:90) r x T ( x ) dx = 1 − m ( r ) r (11)is the standard General Relativity expression containingthe Misner-Sharp mass function m = m ( r ). Adding thesource S µν can then be accounted for by the deformationof the seed metric (10) given by ξ → ν = ξ + α g (12) e − µ → e − λ = e − µ + α f , (13)where the parameter α is introduced to keep track ofthese deformations.By means of Eqs. (12) and (13), the Einstein equa-tions (1) split into the Einstein field equations for theseed metric (10), that is G νµ ( ξ, µ ) = k T νµ , (14)where G = 1 r − e − µ (cid:18) r − µ (cid:48) r (cid:19) , (15) G = 1 r − e − µ (cid:18) r + ξ (cid:48) r (cid:19) , (16) G = − e − µ (cid:18) ξ (cid:48)(cid:48) + ξ (cid:48) − µ (cid:48) ξ (cid:48) + 2 ξ (cid:48) − µ (cid:48) r (cid:19) , (17)and a second set containing the source S µν , which reads α G νµ ( ξ, µ ; f, g ) = k S νµ , (18)where G = − fr − f (cid:48) r , (19) G = − f (cid:18) r + ν (cid:48) r (cid:19) − Z (20) G = − f (cid:18) ν (cid:48)(cid:48) + ν (cid:48) + 2 ν (cid:48) r (cid:19) − f (cid:48) (cid:18) ν (cid:48) + 2 r (cid:19) − Z , (21)and Z = e − µ g (cid:48) r (22)4 Z = e − µ (cid:18) g (cid:48)(cid:48) + g (cid:48) + 2 g (cid:48) r + 2 ξ (cid:48) g (cid:48) − µ (cid:48) g (cid:48) (cid:19) . (23)One clearly sees that the tensor S µν must vanish whenthe metric deformations vanish ( α = 0). On assuming g = 0, we have Z = Z = 0 and Eq. (18) reduces to thesimpler “quasi-Einstein” system of the MGD of Refs. [55,56], in which the deformation f is only determined by thesource S µν and the seed metric (10).What makes the GD work is the fact that, underthe transformations (12) and (13), the Einstein tensorchanges as G σγ ( ξ, µ ) → G σγ ( ν, λ ) = G σγ ( ξ, µ ) + α G σγ ( ν, λ ) . (24)That is to say, Eqs. (12) and (13) yield a linear decom-position of the Einstein tensor in the parameter α , likethe two sources add linearly in the r.h.s. of Eq. (1). Wetherefore expect that a similar GD can be introduced forany given space-time, independently of its symmetries, ifwe can implement a linear decomposition for the Einsteintensor of the form in Eq. (24). A natural application isthen to consider axially symmetric systems. B. Axially symmetric case
Let us start with the simplest generic extension of theKerr metric, given by the Gurses-Gursey metric [119] ds = (cid:20) − r ˜ m ( r )˜ ρ (cid:21) dt + 4 ˜ a r ˜ m ( r ) sin θ ˜ ρ dt dφ − ˜ ρ ˜∆ dr − ˜ ρ dθ − ˜Σ sin θ ˜ ρ dφ , (25)with ˜ ρ = r + ˜ a cos θ (26)˜∆ = r − r ˜ m ( r ) + ˜ a (27)˜Σ = (cid:0) r + ˜ a (cid:1) − ˜ a ˜∆ sin θ (28)and ˜ a = ˜ J/ ˜ M , (29) where ˜ J is the angular momentum and ˜ M the total massof the system. Note that the line-element (25) reducesto the Kerr solution when the metric function ˜ m = ˜ M .Moreover, when ˜ a = 0, we obtain the Schwarzschild-likemetric in Eq. (3) with e ν = e − λ = 1 − m ( r ) r . (30)Hence, the correspondence between the metrics (3)and (25) is clear, with the latter being a rotational ver-sion of a Kerr-Schild spherically symmetric space-time(see e.g. Refs. [120–123]). Although Eq. (25) is not themost general axially symmetric line element, it can beused to describe rotating compact objects, like BHs andgravastars, among many others.The components of the Einstein tensor for the met-ric (25) read˜ G = 2 r + (cid:0) ρ − r (cid:1) + ˜ a (cid:0) r − ρ (cid:1) ρ ˜ m (cid:48) − r ˜ a sin θρ ˜ m (cid:48)(cid:48) (31)˜ G = 2 r ρ ˜ m (cid:48) , (32)˜ G = 2 ρ − r ρ ˜ m (cid:48) + rρ ˜ m (cid:48)(cid:48) (33)˜ G = 2 2 r (cid:0) ρ − r (cid:1) + ˜ a (cid:0) ρ − r (cid:1) ρ ˜ m (cid:48) + r (cid:0) ˜ a + r (cid:1) ρ ˜ m (cid:48)(cid:48) , (34)˜ G = 2 ˜ a (cid:0) r − ρ (cid:1) ρ ˜ m (cid:48) − ˜ a rρ ˜ m (cid:48)(cid:48) . (35)The key observation now is that this Einstein tensor is linear in derivatives of the mass function ˜ m ( r ), whereasthe rotational parameter ˜ a appears in a convoluted form.Any linear decomposition of the mass function,˜ m = m ( r ) + α m s ( r ) , (36)will therefore generate a linear decomposition of the Ein-stein tensor of the form in Eq. (24) with G σγ = G σγ ,provided the rotational parameter ˜ a is left unaffected,that is ˜ G σγ ( ˜ m, ˜ a ) = G σγ ( m, ˜ a ) + α G σγ ( m s , ˜ a ) . (37)Like for the spherically symmetric case in Section II A,we will assume that the mass functions m and m s aregenerated by the energy-momentum tensor T µν and S µν Of course, when ˜ m = M is constant, the Einstein tensor ˜ G µν =0, since Eq. (25) is the vacuum Kerr metric. in Eq. (2), respectively. It is convenient to introduce thetetrads [124]˜ e µt = (cid:0) r + ˜ a , , , ˜ a (cid:1)(cid:112) ρ ∆ , ˜ e µr = √ ∆ (0 , , , (cid:112) ρ ˜ e µθ = (0 , , , (cid:112) ρ , ˜ e µφ = − (cid:0) ˜ a sin θ, , , (cid:1)(cid:112) ρ sin θ , (38)so that the total source ˜ T µν generating the metric (25)can be written as˜ T µν = ˜ (cid:15) ˜ e µt ˜ e νt + ˜ p r ˜ e µr ˜ e νr + ˜ p θ ˜ e µθ ˜ e νθ + ˜ p φ ˜ e µφ ˜ e νφ , (39)where the energy density ˜ (cid:15) and the pressures ˜ p r , ˜ p θ and˜ p φ are given by˜ (cid:15) = − ˜ p r = 2 r ρ ˜ m (cid:48) (40)˜ p θ = ˜ p φ = − rρ ˜ m (cid:48)(cid:48) + 2 (cid:0) r − ρ (cid:1) ρ ˜ m (cid:48) , (41)which are also, consistently, linear in (derivatives of) themass function.We next consider a solution to the Eq. (1) for the seedsource T µν alone, which we write as ds = (cid:20) − r m ( r ) ρ (cid:21) dt + 4 a r m ( r ) sin θρ dt dφ − ρ ∆ dr − ρ dθ − Σ sin θρ dφ , (42)where the expressions for ρ , Σ and ∆ are the same asthose in Eqs. (26)-(28) but contain m and a instead of ˜ m and ˜ a . The addition of the second source S µν can thenbe accounted for by the GD of the seed metric (42) givenby m ( r ) → ˜ m = m ( r ) + α m s ( r ) , (43)with the parameter α introduced to keep track of thedeformation as usual. In order to achieve the decou-pling (37) of Eqs. (31)-(35), we must also demand˜ a = a = a s , (44)that is to say, the length-scales a and a s associated re-spectively with the sources T µν and S µν must be andremain equal. Finally, notice that the mass deforma-tion (43) corresponds to the particular metric deforma-tion f in Eq. (13) given by f ( r ) = − m s ( r ) r . (45)Unlike the general GD for the spherically symmetriccase, Eqs. (43) and (44) split the Einstein equations (1) intwo equal sets: A) one is given by Einstein field equationswith the energy-momentum tensor T µν , that is G νµ ( m, a ) = k T νµ . (46) whose solution is the seed metric (42); B) the second setcontains the source S µν and reads α G νµ ( m s , a ) = k S νµ . (47)whose solution has the same form as the one in Eq. (42)but with m ( r ) → α m s ( r ). In this case, we concludethat the two sources T µν and S µν can be decoupled bymeans of the metric deformations (43) and the length-scale invariant condition (44). Notice that the energyand pressures in Eqs. (40) and (41) can be written as [49]˜ (cid:15) = (cid:15) + α (cid:15) S (48)˜ p i = p i + α p Si ( i = r, θ, φ ) , (49)where (cid:15) S and p Si are the energy and pressures of thesource S µν . Finally, we see that for a = 0 the sets (46)and (47) reduce to those in (14) and (18) respectively.We want to emphasize once again that this procedure isexact and does not require a perturbative expansion inthe parameter α .
1. Strategy
We can now detail our scheme to generate new axiallysymmetric metrics from known solutions of the Einsteinfield equations:1. Consider the field equations ˜ G µν = k ˜ T µν whichdetermine the metric ˜ g µν ( ˜ m, ˜ a ) in Eq. (25), where˜ T µν = T µν + . . . + T nµν , and T iµν is the energy-momentum tensor of the i th gravitational source.2. Solve G iµν = k T iµν for each T iµν to find their re-spective g iµν ( m i , a i ) in Eq. (42) [of the same formas (25)], namely G µν = k T µν ⇒ g µν ( m , a )... G nµν = k T nµν ⇒ g nµν ( m n , a n ) .
3. The solution ˜ g µν ( ˜ m, ˜ a ) in Eq. (25) of the originalproblem ˜ G µν = k ˜ T µν is obtained by setting˜ m = m + . . . + m n ˜ a = a = ... = a n in the line-element (25).The previous scheme can be simplified even further bynoting that step 2. actually amounts to computing justthe mass functions m i = m i ( r ). We can thus do that for a i = 0 for each G iµν = k T iµν , i.e. solve the sphericallysymmetric cases G iµν = k T iµν ⇒ g iµν ( m i , a i = 0) (50)and generate the axially symmetric version by pluggingthe mass function ˜ m = m + ... + m n into Eq. (25) with theasymptotic angular momentum parameter ˜ a of choice.
2. Decoupling Einstein-Maxwell
With the aim of testing the consistency of our ap-proach, let us consider a well-known case, namely theEinstein-Maxwell system. In particular, we will considerthe axially symmetric electro-vacuum, for which the re-sult must be the well-known Kerr-Newman solution.Following our strategy, we start by identifying thesources T µν = T µν = 0 and T µν = S µν of relevance forthe case at hand, that is˜ T µν = (cid:0)(cid:0)(cid:18) T µν + S µν , (51)where S µν = 14 π (cid:18) F µα F αν + 14 g µν F αβ F αβ (cid:19) (52)is the Maxwell tensor.Next we solve the Einstein equations for each sourceseparately, in the particularly simple case a = a = 0.For the the vacuum T µν = 0, we find the Schwarzschildsolution with mass m = M . (53)For the source S µν , we find the Reissner-Nordstr¨om so-lution, whose mass function is given by m s ( r ) = A − Q r , (54)where A and Q are integration constants, with Q even-tually identified as the electric charge.The total mass function is given by m ( r ) = M + A − Q r ≡ M − Q r , (55)which, plugged into he metric (25), yields the well-knownKerr-Newman solution with˜∆ = r − r M + a + Q . (56)We see that the method is straightforward, and that wedo not need to use the Newman-Janis algorithm to mapthe spherically symmetric solution into the axially sym-metric one. III. ROTATING BLACK HOLE SOLUTIONS
In a recent paper [125], we developed a new method togenerate spherically symmetric hairy BHs by imposing aminimal set of requirements consisting of i) the existenceof a well defined event horizon, and ii) the SEC or DECfor the hair outside the horizon. In particular, we con-sidered a Schwarzschild BH surrounded by a sphericallysymmetric “tensor-vacuum” represented by a conservedenergy-momentum tensor S µν , which is dealt with as ex-plained in Section II A. We will here use those solutionsas seeds to generate axially symmetric systems accordingto our strategy in Section II B 1. A. Extended Kerr solution
When we demand that S µν satisfies the SEC in theregion outside the event horizon, we found the extendedSchwarzschild BH metric e ν = e − λ = 1 − r (2 M + (cid:96) ) + α e − r/M = 1 − M r + α e − r/ ( M− (cid:96) / , (57)where (cid:96) = α (cid:96) measures the increase of entropy from theminimum Schwarzschild value S = 4 π M caused by thehair, and must satisfy (cid:96) ≤ M ≡ (cid:96) K (58)in order to ensure asymptotic flatness.In order to extended this metric to the axially sym-metric case, we just need to identify the mass functionfrom Eq. (57), that is˜ m = M − α r e − r/ ( M− (cid:96) / , (59)which we then plug into the metric (25). This yields˜∆ = r + a − r M + α r e − r/ ( M− (cid:96) / . (60)The equation determining the horizon r = r H of the met-ric (25) is given by 0 = ˜ g rr ∼ ˜∆, which yields r + a − M r H + α r e − r H / ( M− (cid:96) / = 0 . (61)We see that the Kerr horizon r Kerr = M + (cid:112) M − a (62)is recovered for α = 0 and also when the inequality (58)is saturated.The function (60) is plotted in Fig. 1 for a few valuesof (cid:96) at fixed α and a . We see that the horizon shiftsto larger radii when (cid:96) increases, reaching a maximumvalue corresponding to the Kerr horizon for (cid:96) = (cid:96) K . Thesilhouette of the BH is also shown in Fig. 1 where σ and β are the usual celestial coordinates (see Appendix A).We conclude that the metric (25) with the mass func-tion (59) represents a family of rotating hairy BHs de-scribed by the parameters {M , a, (cid:96) } , where (cid:96) = α (cid:96) represents a charge associated with primary hair.Finally, the spherically symmetric metric with compo-nents (57) satisfies the SEC, that is (cid:15) S + p Sr + 2 p Sθ ≥ ρ S + p Sr ≥ ρ S + p Sθ ≥ . It can be checked straightforwardly that this property isinherited by the rotating solution. ℓ = r H < r Kerr ℓ = r H < r Kerr ℓ = ℓ k ; r H = r Kerr - - r Δ ( r ) - - - - σℳ β ℳ FIG. 1. Extended Kerr solution: function ˜∆ (left panel) and silhouette (right panel) of the shadow cast for different values of (cid:96) with α = 0 . a = 0 . M = 1. The Kerr horizon r Kerr here corresponds to the saturated case of Eq. (58).
B. Extended Kerr-Newman solution
The second case we will consider is the rotating versionof the spherically symmetric solution [125] e ν = e − λ = 1 − M + α (cid:96)r + Q r − α M e − r/M r = 1 − M r + Q r − αr (cid:18) M − α (cid:96) (cid:19) e − r/ ( M− α (cid:96)/ , (64)which extends a Reissner-Nordstr¨om-like metric toinclude a conserved source S µν satisfying the DEC.From (64) we read out the mass function˜ m = M − Q r + α (cid:18) M − (cid:96) (cid:19) e − r/ ( M− (cid:96) / , (65)which immediately yields the rotating version (25) with˜∆ = r + a + Q − r M − α r (cid:18)
M − (cid:96) (cid:19) e − r/ ( M− (cid:96) / , (66)where again the inequality (58) must hold to ensureasymptotic flatnessThe horizon is again determined by ˜∆ = 0 or r + a + Q − M r H = α r H (cid:18) M − (cid:96) (cid:19) e − r H / ( M− (cid:96) / . (67)The Kerr-Newman horizon r KN = M + (cid:112) M − a − Q (68) We remark that Q is not necessarily the electric charge, but couldbe a tidal charge of extra-dimensional origin or any other chargefor the tensor S µν . ℓ = r H > r KN ℓ = r H > r KN ℓ = ℓ KN ; r H = r KN - - r Δ ( r ) FIG. 2. Extended Kerr-Newman solution: function ˜∆ fordifferent values of (cid:96) with α = 0 . a = 0 . Q = 0 . M = 1. The Kerr-Newman horizon r KN here corresponds tothe saturated case of Eq. (58). is found for α = 0 (hence (cid:96) = 0) and also when theinequality in (58) is saturated.The metric function (66) is plotted in Fig. 2 for givenvalues of α , a and Q . For the range of (cid:96) shown there,the horizon shrinks to smaller radii when (cid:96) increases,reaching a minimum value corresponding to the Kerr-Newman horizon when Eq. (58) is saturated. We con-clude that the metric (25) with the mass function (65)represents rotating hairy BHs depending on the parame-ters {M , a, Q, α, (cid:96) } , where (cid:96) = α (cid:96) represents a chargeassociated with primary hair.Like with the SEC in Section III A, one can checkstraightforwardly that the rotating metrics inherit theDEC satisfied by the spherically symmetric metric func-tions in Eq. (64), ρ S ≥ ρ S ≥ | ˜ p Si | ( i = r, θ, φ ) . (70) - - - - σℳ β ℳ FIG. 3. Special extended Kerr-Newman solutions: BHshadow for (a) Schwarzschild horizon (black), (b) Kerr hori-zon (blue) and (c) Reissner-Nordstr¨om horizon (red), for α = 0 . (cid:96) = 0 . a = 0 . Q = 0 . M = 1. It is impossible to find analytical solutions to Eq. (67),except for some particular cases. Three of them areshown below.
Case 1
If the charge and angular momentum parameters sat-isfy the condition a + Q = α r H (cid:18) M − (cid:96) (cid:19) e − r H / ( M− (cid:96) / , (71)we have˜∆ = r − r M + α (cid:18) M − (cid:96) (cid:19) × (cid:104) r H e − r H / ( M− (cid:96) / − r e − r/ ( M− (cid:96) / (cid:105) , (72)and the event horizon is located at the Schwarzschild ra-dius r H = 2 M . This indicates that the source S µν fillingthe “electro-vacuum” produces a screening effect on thecharges a and Q in such a way that an external observerwill see a rotating space-time with the horizon apparentlygenerated by a non-rotating and neutral distribution. Case 2
Next, if the charge satisfies Q = α r H (cid:18) M − (cid:96) (cid:19) e − r H / ( M− (cid:96) / , (73) the metric function˜∆ = r + a − r M + α (cid:18) M − (cid:96) (cid:19) × (cid:104) r H e − r H / ( M− (cid:96) / − r e − r/ ( M− (cid:96) / (cid:105) , (74)and the event horizon is located at r H = r Kerr given inEq. (62), provided a ≤ M . This indicates a screeningeffect of the charge Q only, so that an external observerwill detect an horizon corresponding to a neutral distri-bution. Case 3
Lastly, if the angular momentum satisfies a = α r H (cid:18) M − (cid:96) (cid:19) e − r H / ( M− (cid:96) / , (75)which leads to∆ = r + Q − r M + α (cid:18) M − (cid:96) (cid:19) × (cid:104) r H e − r H / ( M− (cid:96) / − r e − r/ ( M− (cid:96) / (cid:105) , (76)the event horizon is given by Eq. (77) with a = 0, namelythe Reissner-Nordstr¨om horizon r RN = M + (cid:112) M − Q , (77)provided of course the charge Q ≤ M . In this case thescreening effect occurs on the rotational charge a . Anexternal observer will see a rotating BH with effectivehorizon corresponding to a non-rotating charged distri-bution.The three screening cases above can be described col-lectively by the metric function˜∆ = r + Z i − r M + αL (cid:16) r H e − r H /L − r e − r/L (cid:17) , (78)where L = M − (cid:96) /
2, and Z i = { Z S , Z K , Z RN } = { , a , Q } , (79)for the three effective horizons, namely, Schwarzschild,Kerr and Reissner-Nordstr¨om, respectively. Notice thatthe hair charge (cid:96) ≡ α (cid:96) in the expressions (71), (73)and (75) is related with the charges M , a and Q by theLambert W function as (cid:96) i = 2 M − r H i W (cid:16) α r i a + Q − Z i (cid:17) , (80)where the index i runs on the three cases in Eq. (79). Asan example, Fig. 3 shows the shadow cast in the threecases for a given choice of parameters. IV. CONCLUSIONS
Using the GD approach and the simplest extension ofthe Kerr metric (25), which could be generated by theNewman–Janis algorithm without complexification [49],we have proven that the decoupling of gravitationalsources in General Relativity is possible in the axiallysymmetric case, as long as the metric takes the form (25)and the asymptotic angular momentum parameter a inEq. (29) satisfies the critical condition (44). As a directconsequence, we provided a simple and systematic strat-egy to generate axially-symmetric BHs departing from aspherically symmetric seed solution, without implement-ing any variant of the Newman-Janis algorithm [126–129]. On a formal level, our results stem from observing thatthe Einstein tensor (31)-(35) for the metric (25) is linearin derivatives of the mass function. This property couldbe at the heart of other known methods to generate ax-ially symmetric solutions of the Einstein equations fromspherically symmetric solutions. Moreover, the GD couldhelp in investigating mass functions in non-sphericallysymmetric systems [130–135]. Following the aforementioned approach, we showedhow the Kerr BH, given by the metric (25) with massfunction equal to the total mass M , is modified whena fluid with conserved energy-momentum tensor S µν fills the axially-symmetric vacuum. We thus find non-trivial extensions of the Kerr BH, given by the massfunction (59), and Kerr-Newman BH, with mass func-tion (65). Both solutions can support a primary hair (cid:96) ≤ M , whose impact on the silhouette of the shadowis displayed in Figs. 1 and 3.Finally, in the case of the extended Kerr-Newman BH,whose horizon is found by solving Eq. (67), we identifyspecial cases describing a screening effect induced by thesource S µν on the charges a and Q , in such a way thatan external observer would see a rotating BH with effec-tive horizon corresponding to i) non-rotating and neutraldistribution, ii) neutral distribution, iii) non-rotating butcharged distribution. This clearly indicates that the mat-ter around BHs may have a significant observational im-pact and separating different models of BHs could remaina very hard task. In this respect, we notice that these re-sults do not contradict the conclusions of Ref. [136] aboutno hairs for BHs in astrophysical environments, since the(effective) fluid modifying the Kerr geometry overlaps theBH horizon in our case.Although it is not the main goal of the present work,we would like to conclude by mentioning that both ro-tating solutions [characterized by the metric functions Indeed, we can generate a new axially-symmetric solution by thedirect superposition of two different rotating solutions, as longas the critical condition (44) is satisfied. We plan to investigate these issues further in separate works. in Eqs. (60) and (66), respectively] could be investi-gated further, in particular, for possible observationalconstraints on the primary hairs (cid:96) and Q . However,this is beyond the purpose of the present work. Acknowledgments
R.C. is partially supported by the INFN grant FLAGand his work has also been carried out in the frame-work of activities of the National Group of MathematicalPhysics (GNFM, INdAM) and COST action
Cantata . Appendix A: Null geodesics around rotating BHs
We briefly review how to study null geodesics in a ro-tating space-time like the one in Eq. (25), and find thecelestial coordinates describing the BH shadow. Wavespropagate along characteristic curves described by theHamilton-Jacobi equation ∂S∂λ = 12 g µν ∂ µ S ∂ ν S , (A1)where λ is a parameter along the curve and S the Jacobiaction. Given the symmetries of the space-time (25),Eq. (A1) is separable and one has [137] S = − E t + Φ φ + S r ( r ) + S θ ( θ ) , (A2)with E and Φ being the conserved energy and angularmomentum, respectively. Replacing (A2) in Eq. (A1),we obtain S r = (cid:90) (cid:112) R ( r )∆ dr (A3) S θ = (cid:90) (cid:112) Θ( θ ) dθ , where R = (cid:2) ( r + a ) E − a Φ (cid:3) − ∆ (cid:2) Q + (Φ − a E ) (cid:3) (A4)Θ = Q − (Φ csc θ − a E ) cos θ , with Q the Carter constant.The (unstable) circular photon orbits are determinedby R = R (cid:48) = 0, namely (cid:0) a − a ξ + r (cid:1) − (cid:0) a + r F (cid:1) (cid:2) ( a − ξ ) + η (cid:3) = 0 (A5)4 (cid:0) a − a ξ + r (cid:1) − (cid:2) ( a − ξ ) + η (cid:3) ( r F (cid:48) + 2 F ) = 0 , where ξ = Φ /E and η = Q/E are the impact parame-ters. Accordingly, ξ = a + r a − (cid:0) a + r F (cid:1) a ( r F (cid:48) + 2 F ) ,η = r (cid:104) r + 2 a ( a − ξ ) − ( a − ξ ) F (cid:105) a + r F , (A6) F = 1 − mr , where r is the radius of the unstable photon orbit.The apparent shape of the shadow is finally describedby the celestial coordinates [138] σ ≡ lim r →∞ (cid:32) − r sin θ dφdr (cid:12)(cid:12)(cid:12)(cid:12) ( r ,θ ) (cid:33) = − ξ sin θ (A7) and β ≡ lim r →∞ (cid:32) r dθdr (cid:12)(cid:12)(cid:12)(cid:12) ( r ,θ ) (cid:33) = (cid:113) η − ξ cot θ + a cos θ , (A8)where ( r , θ ) are the coordinates of the observer. [1] S. Gillessen, F. Eisenhauer, S. Trippe, T. Alexander,R. Genzel, F. Martins, and T. Ott, Astrophys. J. ,1075 (2009), arXiv:0810.4674 [astro-ph].[2] A. Ghez et al. , Astrophys. J. , 1044 (2008),arXiv:0808.2870 [astro-ph].[3] B. P. Abbott et al. (LIGO Scientific, Virgo), Phys. Rev.Lett. , 061102 (2016), arXiv:1602.03837 [gr-qc].[4] K. Akiyama et al. (Event Horizon Telescope), Astro-phys. J. , L1 (2019).[5] P. O. Mazur and E. Mottola, Proc. Nat. Acad. Sci. ,9545 (2004), arXiv:gr-qc/0407075.[6] V. Cardoso and P. Pani, Living Rev. Rel. , 4 (2019),arXiv:1904.05363 [gr-qc].[7] J. Ovalle, C. Posada, and Z. Stuchl´ık, Class. Quant.Grav. , 205010 (2019), arXiv:1905.12452 [gr-qc].[8] R. P. Kerr, Phys. Rev. Lett. , 237 (1963).[9] Z.-Y. Fan and X. Wang, Phys. Rev. D , 124027(2016), arXiv:1610.02636 [gr-qc].[10] V. P. Frolov, Phys. Rev. D , 104056 (2016),arXiv:1609.01758 [gr-qc].[11] E. Babichev, C. Charmousis, and A. Leh´ebel, JCAP , 027 (2017), arXiv:1702.01938 [gr-qc].[12] D. D. Doneva and S. S. Yazadjiev, Phys. Rev. Lett. ,131103 (2018), arXiv:1711.01187 [gr-qc].[13] G. Antoniou, A. Bakopoulos, and P. Kanti, Phys. Rev.Lett. , 131102 (2018), arXiv:1711.03390 [hep-th].[14] C. A. Herdeiro, E. Radu, N. Sanchis-Gual, andJ. A. Font, Phys. Rev. Lett. , 101102 (2018),arXiv:1806.05190 [gr-qc].[15] H. O. Silva, J. Sakstein, L. Gualtieri, T. P. Sotiriou,and E. Berti, Phys. Rev. Lett. , 131104 (2018),arXiv:1711.02080 [gr-qc].[16] B. Kleihaus and J. Kunz, Phys. Rev. Lett. , 1595(1997), arXiv:gr-qc/9704060.[17] C. Bambi and K. Freese, Phys. Rev. D79 , 043002(2009), arXiv:0812.1328 [astro-ph].[18] C. Bambi and N. Yoshida, Class. Quant. Grav. ,205006 (2010), arXiv:1004.3149 [gr-qc]. [19] A. Abdujabbarov, F. Atamurotov, Y. Kucukakca,B. Ahmedov, and U. Camci, Astrophys. Space Sci. ,429 (2013), arXiv:1212.4949 [physics.gen-ph].[20] B. Toshmatov, B. Ahmedov, A. Abdujabbarov,and Z. Stuchlik, Phys. Rev. D , 104017 (2014),arXiv:1404.6443 [gr-qc].[21] P. V. P. Cunha, C. A. R. Herdeiro, E. Radu, andH. F. Runarsson, Phys. Rev. Lett. , 211102 (2015),arXiv:1509.00021 [gr-qc].[22] A. Abdujabbarov, B. Toshmatov, Z. Stuchl´ık, andB. Ahmedov, Int. J. Mod. Phys. D26 , 1750051 (2016),arXiv:1512.05206 [gr-qc].[23] A. Abdujabbarov, M. Amir, B. Ahmedov, andS. G. Ghosh, Phys. Rev. D , 104004 (2016),arXiv:1604.03809 [gr-qc].[24] Z. Younsi, A. Zhidenko, L. Rezzolla, R. Konoplya,and Y. Mizuno, Phys. Rev. D94 , 084025 (2016),arXiv:1607.05767 [gr-qc].[25] P. V. P. Cunha, C. A. R. Herdeiro, B. Kleihaus,J. Kunz, and E. Radu, Phys. Lett.
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