Black holes by gravitational decoupling
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Abstract
We investigate how a spherically symmetric fluid modifies the Schwarzschild vacuum solution when there is no exchange of energymomentum between the fluid and the central source of the Schwarzschild metric. This system is described by means of the gravitational decoupling realised via the minimal geometric deformation approach, which allows us to prove that the fluid must be anisotropic. Several cases are then explicitly shown.
1 Introduction
The study of black holes represents one of the most active areas of gravitational physics, from both a purely theoretical and the observational point of view. The interest black holes generate is due not only to their exotic nature, but also because they constitute ideal laboratories to study gravity in the strong field regime, and test general relativity therein. However, confronting theoretical predictions with observations is an arduous and complicated task. A formidable step in this direction is the recent direct observation of black holes through the detection of gravitational waves, which opens a new and promising era for gravitational physics [1, 2].
It is well known that general relativity predicts surprisingly simple solutions for black holes, characterised at most by three fundamental parameters, namely the mass M, angular momentum J and charge Q [3]. The original nohair conjecture states that these solutions should not carry any other charges [4]. Therefore, as the observations of systems containing black holes improve, the degree of consistency of these observations with the predictions determined according to the general relativistic solutions (with parameters M, J and Q) will result in a direct test of the validity of general relativity in the strong field regime. There could in fact exist other charges associated with inner gauge symmetries (and fields), and it is now known that black holes could have (soft) quantum hair [5]. The existence of new fundamental fields, which leave an imprint on the structure of the black hole, thus leading to hairy black hole solutions, is precisely the scenario under study in this paper.
Possible conditions for circumventing the nogo theorem have been investigated for a long time in different scenarios (see Refs. [6, 7, 8, 9, 10, 11, 12, 13, 14, 15] for some recent works and Refs. [16, 17, 18, 19, 20, 21, 22] for earlier works). In particular, a fundamental scalar field \(\phi \) has been considered with great interest (see Ref. [23] and references therein). In this work, we will take a different and more general approach than most of the investigations carried out so far and, instead of considering specific fundamental fields to generate hair in black hole solutions, we shall just assume the presence of an additional completely generic source described by a conserved energymomentum tensor \(\theta _{\mu \nu }\). Of course, this \(\theta _{\mu \nu }\) could account for one or more fundamental fields, but the crucial property is that it gravitates but does not interact directly with the matter that sources the (hairless) black hole solutions we start from. This feature may seem fanciful, but can be fully justified, for instance, in the context of the dark matter. Achieving this level of generality in the classical scheme represented by general relativity is a nontrivial task, and the gravitational decoupling by Minimal Geometric Deformation (MGDdecoupling, henceforth) is precisely the method that was developed for this purpose in Ref. [24].
 Extending simple solutions into more complex domains We can start from a simple spherically symmetric gravitational source with energymomentum tensor \({\hat{T}}_{\mu \nu }\) and add to it more and more complex gravitational sources, as long as the spherical symmetry is preserved. The starting source \({\hat{T}}_{\mu \nu }\) could be as simple as we wish, including the vacuum indeed, to which we can add a first new source, saywhere \(\alpha ^{(1)}\) is a constant that traces the effects of the new source \(T^{(1)}_{\mu \nu }\). We can then repeat the process with more sources, namely$$\begin{aligned} {\hat{T}}_{\mu \nu }\mapsto {\tilde{T}}^{(1)}_{\mu \nu }={\hat{T}}_{\mu \nu }+\alpha ^{(1)}\,T^{(1)}_{\mu \nu }, \end{aligned}$$(1.1)and so on. In this way, we can extend straightforward solutions of the Einstein equations associated with the simplest gravitational source \({\hat{T}}_{\mu \nu }\) into the domain of more intricate forms of gravitational sources \(T_{\mu \nu }={\tilde{T}}^{(n)}_{\mu \nu }\), step by step and systematically. We stress that this method works as long as the sources do not exchange energymomentum among them, namely$$\begin{aligned} {\tilde{T}}^{(1)}_{\mu \nu }\mapsto {\tilde{T}}^{(2)}_{\mu \nu }={\tilde{T}}^{(1)}_{\mu \nu }+\alpha ^{(2)}\,T^{(2)}_{\mu \nu }, \end{aligned}$$(1.2)which further clarifies that the constituents can only couple via gravity.$$\begin{aligned} \nabla _{\mu }{\hat{T}}^{\mu \nu } = \nabla _{\mu } T^{(1)\mu \nu } = \ldots = \nabla _{\mu } T^{(n)\mu \nu } = 0, \end{aligned}$$(1.3)

Deconstructing a complex gravitational source. The converse of the above also works. In order to find a solution to Einstein’s equations with a complex spherically symmetric energymomentum tensor \(T_{\mu \nu }\), we can split it into simpler components, say \({\hat{T}}_{\mu \nu }\) and \(T^{(i)}_{\mu \nu }\), provided they all satisfy Eq. (1.3), and solve Einstein’s equations for each one of these parts. Hence, we will have as many solutions as are the contributions \(T^{(i)}_{\mu \nu }\) in the original energymomentum tensor. Finally, by a straightforward combination of all these solutions, we will obtain the solution to the Einstein equations associated with the original energymomentum tensor \(T_{\mu \nu }\).
In analogy with the wellknown electrovacuum and scalarvacuum, in this paper we will consider a Schwarzschild black hole surrounded by a spherically symmetric “tensorvacuum”, represented by the aforementioned \(\theta _{\mu \nu }\). Following the MGDdecoupling, we will separate the Einstein field equations in (i) Einstein’s equations for the spherically symmetric vacuum and (ii) a “quasiEinstein” system for the spherically symmetric “tensorvacuum”. The MGD procedure will then allow us to merge the Schwarzschild solution for (i) with the solution for the “quasiEinstein” system (ii) into the solution for the complete system “Schwarzschild + tensorvacuum”. Like the case of the electrovacuum and (in some cases) scalarvacuum, new black hole solutions with additional parameters \(q_i\) besides the mass M can be obtained, each one associated with a particular equation of state for the “tensorvacuum”. Demanding the geometry is free of singularities and other pathologies, implies regularity conditions which show that not all of these parameters \(q_i\) can be independent.
The paper is organised as follows: in Sect. 2, we first review the fundamentals of the MGDdecoupling applied to a spherically symmetric system containing a perfect fluid and an additional source \(\theta _{\mu \nu }\); in Sect. 3, new hairy black holes solutions are found by assuming the perfect fluid has sufficiently small support so that only \(\theta _{\mu \nu }\) exists outside the horizon; finally, we summarise our conclusions in Sect. 4.
2 MGD decoupling for a perfect fluid
The Eqs. (2.6)–(2.8) contain seven unknown functions, namely: two physical variables, the density \(\rho (r)\) and pressure p(r); two geometric functions, the temporal metric function \(\nu (r)\) and the radial metric function \(\lambda (r)\); and three independent components of \(\theta _{\mu \nu }\). This system of equations is therefore indeterminate and we should emphasise that the spacetime geometry does not allow one to resolve for the gravitational source \(\{\rho , p, \theta _{\mu \nu }\}\) uniquely.
3 Black holes
3.1 Isotropic sector
3.2 Conformal sector
3.3 Barotropic equation of state
3.4 Linear equation of state
The final conclusion is thus that the linear equation of state (3.29) always produces black holes (with a Schwarzschild singularity at \(r=0\)) if \(2<b<4\) and \(a>1\), provided \(\alpha >0\) or \(\alpha <0\) and Eq. (3.46) holds.
3.5 A particular solution with no extra singularity
The reader can see that Eq. (3.29) leads to a system very rich in possibilities, whose generic solutions^{4} are given in Eqs. (3.34)–(3.37), and whose general analysis is detailed throughout Eqs. (3.38)–(3.46). The main feature of these solutions is that they do not satisfy the dominant energy condition. In this respect, let us recall that the energy conditions are a set of constraints which are usually imposed on the energymomentum tensor in order to avoid exotic matter sources, hence they can be viewed as sensible guides to avoid unphysical situations. However, it is wellknown that these energy conditions might fail for particular classical systems which are still reasonable [62]. In our case we are dealing with a gravitational source \(\theta _{\mu \nu }\) whose main characteristic is that it only interacts gravitationally with the matter that, by itself, would source the (hairless) black hole solution (2.27). Hence, one should not exclude a priori that such matter is a kind of exotic source (as indeed the conjectured dark matter is expected to be).
4 Conclusions
By making use of the MGDdecoupling approach, we have presented in detail how the Schwarzschild black hole is modified when the vacuum is filled by a generic spherically symmetric gravitational fluid, described by a “tensorvacuum” \(\theta _{\mu \nu }\), which does not exchange energymomentum with the central source. For this purpose, we have separated the Einstein field equations into (i) the Einstein equations for the spherically symmetric vacuum \(\rho =p=0\) and (ii) the “quasiEinstein” system in Eqs. (2.23)–(2.25) for the spherically symmetric “tensorvacuum” \(\theta _{\mu \nu }\). Following the MGD procedure, the superposition of the Schwarzschild solution found in (i) plus the solution for the “quasiEinstein” system in (ii), has led to the solution for the complete system “Schwarzschild + tensorvacuum.”
The quasiEinstein system (2.23)–(2.25) was solved by providing some physically motivated equations of state for the source \(\theta _{\mu \nu }\). In this respect, four different scenarios were considered, namely, (i) the isotropic \(\theta _1^{\,1}=\theta _2^{\,2}\); ii) the conformal \(\theta _\mu ^{\,\,\mu }=0\); iii) the polytropic \(\alpha \,\theta _1^{\,1}=K\,(\alpha \,\theta _0^{\,0})^\Gamma \) and iv) the generic linear equation of state in (3.29). In the isotropic case, we only found a metric which is not asymptotically flat for \(r\rightarrow \infty \), which means that the tensorvacuum for a black hole cannot be isotropic as long as its interaction with regular matter is purely gravitational. On the other hand, the conformal case leads to the hairy black hole solution in Eq. (3.12), whose primary hairs is represented by the length \(\ell \), which is constrained by the regularity condition (3.21). Among all polytropic equations of state, we have only considered the barotropic \(\Gamma =1\), which represents a tensorvacuum made of an isothermal selfgravitating sphere of gas. This leads to the exterior solution in Eq. (3.26) endowed with the parameters \(\{M,\alpha ,\ell _{p},K\}\). Since the Killing horizon \(r=r_{\mathrm{H}}=2\,M\) becomes a real singularity, this solution may represent the exterior of a selfgravitating system of mass M and radius \(R>r_{\mathrm{H}}\) but not a black hole solution.
Finally, we have analysed the generic linear equation of state in Eq. (3.29), which includes both the conformal and barotropic fluids as particular cases. This leads to the solution in Eq. (3.34), showing that even a simple linear equation of state may yield hairy black hole solutions with a rich geometry described by the parameters \(\{M,\alpha ,\ell ,a,b\}\), where \(\{\alpha ,\ell ,a,b\}\) represents a potential set of charges generating primary hairs. In this context, a particular black hole solution with primary hairs \(\{\alpha ,a\}\) was found in Eq. (3.47), whose main characteristic is the absence of other singularities in the region \(0<r<\infty \).
All the black holes solutions mentioned above have the horizon at \(r_\mathrm{H}=2\,M\) and primary hairs represented by a number of free parameters. However, these parameters can be restricted by demanding (i) the correct asymptotic behaviour and (ii) regularity conditions for black hole solutions free of pathologies. In this respect, there are always a potential singularity \(r_\mathrm{c}\) and a possible second horizon \(r_\mathrm{h}\) in our solutions. In order to have a proper black hole, it is necessary that \(r_\mathrm{c}\le r_{\mathrm{H}}\) to avoid a naked singularity, and \(r_\mathrm{h}=r_{\mathrm{H}}\) to have a metric with a proper signature. We emphasize that \(r_\mathrm{h}>r_{\mathrm{H}}\) yields both \(g_{tt}\) and \(g_{rr}\) positive inside the region \(r_{\mathrm{H}}<r<r_\mathrm{h}\). All these conditions yields restrictions on potential primary hairs. For instance, the linear equation of state (3.29) always produces black holes if \(2<b<4\) and \(a>1\), provided \(\alpha >0\) or \(\alpha <0\) and Eq. (3.46) holds.
We have shown that different characteristics of the gravitational source lead to different hairy black hole solutions. Therefore, the compatibility between some of these solutions and the observations could determine the main features of the tensorvacuum, and eventually the fundamental field(s) that constitute it. Finally, we would like to emphasize that the nonexistence of an isotropic tensorvacuum that does not exchange energymomentum with regular matter favours scenarios with Klein–Gordon type fields \(\phi \), which naturally induce anisotropy in the Einstein field equations. These scalar fields are found in a large number of alternative theories to general relativity.
Footnotes
 1.
This represents the simplest and so far the only known way to decoupling both gravitational sources in (1.1). An extension of the MGD approach (which represents the foundation of the MGDdecoupling) where both metric components are deformed, was developed in Ref. [28], but it works only in the vacuum and fails for regions where matter is present, since the Bianchi identities are no longer satisfied.
 2.
In fact, it approaches the radial component of the de Sitter metric for \(r\sim \ell _\mathrm{iso}\gg M\).
 3.
Notice that for the particular case \(B=2\), namely the barotropic fluid, the density does not change its sign. This remains an interesting exterior solution for a selfgravitating system of radius \(R>r_{\mathrm{H}}\).
 4.
 5.
Notes
Acknowledgements
J.O. and S.Z. have been supported by the Albert Einstein Centre for Gravitation and Astrophysics financed by the Czech Science Agency Grant No. 1437086G. R.C. is partially supported by the INFN grant FLAG and his work has been carried out in the framework of GNFM and INdAM and the COST action Cantata. R.dR. is grateful to CNPq (Grant No. 303293/20152), and to FAPESP (Grant No. 2017/188978) for partial financial support. A.S. is partially supported by Project Fondecyt 1161192, Chile.
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