Hairy black holes, scalar charges and extended thermodynamics

We explore the use of the recently defined scalar charge which satisfies a Gauss law in stationary spacetimes, in the context of theories with a scalar potential. We find new conditions that this potential has to satisfy in order to allow for static, asymptotically-flat black-hole solutions with regular horizons and non-trivial scalar field. These conditions are equivalent to some of the known ‘no-hair’ theorems (such as Bekenstein’s). We study the extended thermodynamics of these systems, deriving a first law and a Smarr formula. As an example, we study the Anabalón–Oliva hairy black hole.


Introduction
One of the most remarkable aspects of black holes is the fact that all their properties are determined by their conserved charges, irrespectively of their formation history.This fact is referred to as uniqueness (there is only one black hole for a given set of conserved charges) or absence of hair (there are no other parameters apart from the conserved charges characterizing different black holes) and it can be argued that it lies at the very heart of the thermodynamic interpretation of the dynamics of black holes.
This property has been tested in theories in which matter fields giving rise to different conserved charges are coupled to gravity.We can distinguish two broad types: 1. Matter fields with gauge symmetries, such as 1-forms in 4 dimensions.The conserved charges associated to these symmetries (electric and magnetic charges) are defined through surface integrals and are believed to be preserved by quantum gravity.

2.
Matter fields with global symmetries, such as scalar fields. 1 The conserved charges associated to these symmetries are defined through volume integrals and are believed not to be preserved by quantum gravity.Therefore, black holes should not be characterized by this kind of charge, and possible non-trivial fields of this kind in black-hole spacetimes would, then, be understood as "hair" violating uniqueness.
It goes without saying that it is the second type of matter and the possible violations of black-hole uniqueness that it may induce that has attracted most interest.It is also the subject of this work.
In order to discuss black holes with scalar hair we first have to characterize scalar hair more precisely.The most naive way to do it would be to use the conserved charge associated to the global symmetries of the theory that act on the scalars.There are several reasons why this is not possible, even though the contrary is sometimes assumed in the literature: 1.There may not be any global symmetry acting on the scalars at all and, therefore, there may not be an associated conserved charge.This is what actually happens in the theories considered in this paper in which the shift symmetry of a real scalar is broken by a scalar potential.In some works, the charge that would be conserved in absence of the potential is used, even though it is obviously not conserved and does not satisfy a Gauss law.The main problem with this kind of definitions comes from the next point, though.

2.
In static black-hole spacetimes the volume integral that gives the globally conserved charge usually vanishes when integrated over a spacelike hypersurface [1].
For these reasons, in most of the literature it has been customary to use a definition of scalar charge based on the asymptotic expansion of the scalar: the scalar charge would be given, up to normalization, by the coefficient of the 1/r term in that expansion (see, for instance, Ref. [2]).This definition can be used in simple settings but it is clear that a coordinate-dependent definition is necessary to study the properties of this charge and establish general results.
In Refs.[3,1] a covariant definition of scalar charge of a stationary black hole as the integral of an on-shell closed 2-form was proposed.In the cases considered so far, this definition gives the same value as the conventional definition based on the asymptotic expansion, but with the new definition one can go farther: the closedness of the 2-form charge implies that this scalar charge satisfies a Gauss law2 and the covariant definition can be used to recover the scalar term in the first law of black-hole mechanics found in Ref. [2].
One of the empirical properties of this scalar charge is that, in black-hole spacetimes, it is usually completely determined by the conserved charges and asymptotic values of the scalars.In the language of Ref. [4] this kind of scalar charge corresponds to "secondary hair" and the black hole is still completely determined by the values of its truly conserved charges (plus the asymptotic values of the scalars).Whenever there are solutions with the same conserved charges but the scalar charge does not have that value (a particular function of the conserved charges and asymptotic values of the scalars) but is a free parameter that describes "primary hair" in the language of Ref. [4], the solution does not have a regular horizon and does not describe a black hole.This is illustrated by the solutions in Refs.[5,6].The covariant definition of scalar charge of Refs.[3,1] can be used to determine the particular value of the scalar charge allowed in presence of a bifurcate black-hole horizon, which is, as a matter of fact, equivalent to a "no-hair theorem".
In this paper we want to study the extensions of the results obtained in Refs.[3,1] to the case in which a real scalar is coupled to itself via a scalar potential instead of being coupled to vector fields.This is a very simple case which has been very much studied in the past and several "no-hair theorems" have been proven for more or less general classes of scalar potentials in Refs.[7][8][9][10][11][12]. 3ne of the main assumptions in the proofs of these theorems is the positivity of the scalar potential, related to the energy conditions and, not surprisingly, asymptoticallyflat black-hole solutions with scalar hair have been found in theories whose scalar potential violates that condition [14][15][16][17][18].These solutions and their thermodynamics have not been studied from the point of view of their scalar charges 4 and our main goal is to do so, which, first of all requires a generalization of the definition of scalar charge of Refs.[3,1]: the theories considered in Refs.[3,1] have global symmetries and the scalar charges are related to them.As we are going to see, the covariant definition can be extended to the theories that we are going to consider, which have no global symmetries and it can be used to determine which values are allowed in the presence of a bifurcate black-hole horizon.As a byproduct we are going to see that the potentials that allow for asymptotically-flat black-hole solutions with well-defined scalar charges must satisfy a quite restrictive set of conditions previously found in Ref. [18].
This paper is organized as follows: in Section 2 we are going to describe the kind of theories that we are going to study.In Section 3 we are going to give a covariant definition of scalar charge for static solutions of these theories which satisfies a Gauss law and we are going to see which scalar potentials allow for well-defined scalar charges in the presence of a bifurcate horizon.In Section 4 we are going to derive a general Smarr formula for the black-hole solutions of these theories and in Section 5 we will derive the first law.in Section 6 we are going to test the general results obtained in the previous sections using the asymptotically-flat Anabalón-Oliva black hole [17].The results obtained are discussed in Section 7, which also contains pointers to further research.

The theory
The theory we are going to work with consists of gravity, described by the Vielbein e a = e a µ dx µ , coupled minimally to a real scalar field φ which couples to itself via a scalar potential V(φ).The action, in differential-form language, is simply given by5 where L is the Lagrangian 4-form.
Under a general variation of the fields, the action transforms as δS = {E a ∧ δe a + Eδφ + dΘ(e, φ, δe, δφ)} , ( where, ignoring the normalization factor (16πG N ) −1 for the time being (we will recover it when necessary), the Einstein equations E a and the scalar equation E are given by where V ′ ≡ dV/dφ, and where Θ(e, φ, δe, δφ) = − ⋆ (e a ∧ e b ) ∧ δω ab + ⋆dφδφ . (2.4) Several no-hair theorems for this system have been proven in the literature [8,9,12,10,11] with the positivity of the scalar potential as one of the main assumptions.Several asymptotically-flat black-hole solutions with regular horizon and scalar hair have been found in systems that violate this particular assumption [14][15][16][17][18] 6 .We will only study in detail the asymptotically-flat Anabalón-Oliva black hole Ref. [17].
Our goal in this paper is to study how the concept of scalar charge can be used to study these solutions and what allows them to exist at all.Thus, we first study the definition of scalar charge in this system.

Scalar charge
Following Refs.[3,1] we take the inner product of the Killing vector k with the scalar equation of motion, getting where we have defined The existence of the 2-form W k is (locally) guaranteed by the assumptions concerning the symmetry of the system: if the diffeomorphism generated by k leaves invariant all the fields of the configuration, Then, we define the scalar charge 2-form associated to the Killing vector k, We have just shown that Q φ [k] is closed on-shell or, in other words, that it satisfies a Gauss law on-shell.We are going to consider stationary black-hole spacetimes and k will be the timelike Killing vector that generates their Killing horizon.As we will show later, for spherically-symmetric, static, asymptotically-flat black holes, this choice gives the scalar charge Σ defined in Eq. (3.13) as the integral over any closed 2-dimensional surface Σ 2 enclosing the black hole horizon The on-shell closedness of Q φ [k] ensures that this definition does not depend on the integration surface chosen as long as they are homologically equivalent.
W k is defined up to closed forms.We can use that freedom to make it vanish at spatial infinity: Then, if we integrate Q φ [k] over the 2-sphere at spatial infinity, S 2 ∞ , we find that which recovers the conventional definition of scalar charge, as we are going to see.
If the black hole has a bifurcate horizon and we choose to integrate Q φ [k] over the bifurcation surface BH in which k = 0, we get which provides an interesting relation between the scalar potential and the scalar charge of a black hole with bifurcate horizon.If we use the boundary condition Eq. (3.6) and the definition of W k Eq. (3.2), applying Stokes' theorem we can rewrite the above formula in the form where Σ 3 is a hypersurface with boundaries at the bifurcation surface and spatial infinity.
Black holes with regular bifurcate horizons and scalar hair corresponding to the scalar charge Σ will only exist if the integral in the right-hand side is finite, which imposes strong conditions on the scalar potentials that allow for hairy black hole solutions.In order to find these conditions, we are going to focus on static, asymptoticallyflat, spherically-symmetric black holes with metrics of the form Since the integral of Q φ [k] over any 2-sphere of constant radius ρ should give the same result, Σ, where ω (2) is the volume 2-form of the unit sphere.
On the other hand, and, if φ behaves at spatial infinity as where φ ∞ is the constant value of the scalar at spatial infinity, we find that, in that limit, which implies that, in the same limit, which is consistent with the boundary condition we had chosen for W k , Eq. (3.6), and, in its turn, implies that Observe that, since λ must vanish on the horizon, ı k ⋆ dφ vanishes everywhere on the horizon and not just on the bifurcation surface.
Let us find an explicit expression for W k .For the metrics we are dealing with, where the function W k (ρ) is defined by From Eq. (3.11) and the definition of and, using Eqs.(3.12) and (3.17) we have If we expand asymptotically the right-hand side of the definition of W k (ρ), Eq. (3.18), assuming we get ) and, comparing with an asymptotic expansion of W k (r) that takes into account that we find that the potential and its first four derivatives must vanish at the asymptotic value of the scalar: where we have taken into account that we are considering asymptotically-flat black holes only and where we have assumed that Σ = 0.9 These conditions are satisfied by the potential of the theory of Ref. [17] for the asymptotic value of the asymptotically-flat (Anabalón-Oliva) black hole φ ∞ = 0. 10 They are not satisfied for a massive scalar, though, because V ′′ [φ ∞ ] = m 2 = 0. Therefore, the result that we have obtained, based on a definition of scalar charge that satisfies a Gauss law is equivalent to Bekenstein's no-hair theorem of Ref. [7] and also discards many other scalar potentials.
The Smarr formula follows from the integration of the generalized Komar charge [28] on the hypersurface Σ 3 that interpolates between the bifurcation sphere and the sphere at spatial infinity (its two boundaries).The generalized Komar charge is given by where Q[k] is the Noether-Wald charge associated to the Killing vector k and ω k is the 2-form implicitly defined by11 The (local) existence of ω k , as that of W k , is guaranteed by the assumption that k generates a symmetry of all the field of the solution: As explained in Ref. [29], in order to compute the Noether-Wald charge one must properly take into account the gauge freedoms of the fields of the theory.In this case, the only gauge symmetry of the theory (apart from diffeomorphisms), is the local Lorentz symmetry acting on the Vielbein and the right way to deal with it is to replace the Lie derivative of the Vielbein by the Lorentz-covariant (or Lie-Lorentz) derivative [30] (see also Ref. [31]), so that where P ξ ab , the Lorentz momentum map, is defined by the equation for Killing vectors k.This equation is satisfied by the Killing bivector Substituting the transformations Eqs.(4.4) into Eq.(2.2) and using the Noether identity associated to the invariance under local Lorentz transformations of the action and the Noether identity associated to the invariance under diffeomorphisms of the action we find that the variation of the action under the transformations Eqs.(4.4) is just a total derivative.Massaging this total derivative a bit, we arrive to Since the action is only invariant under a total derivative under these transformations, we arrive to the Noether-Wald charge of pure Einstein gravity [32], which is nothing but the Komar charge of pure Einstein gravity Now, in order to find ω k we need to evaluate the on-shell Lagrangian.We first take the trace of the Einstein equation Then, and, for a Killing vector k that leaves all the fields invariant Again, the (local) existence of the 2-form V k is guaranteed by the assumptions on the symmetry of the configurations.
The Komar charge of this theory is finally given by and it is not difficult to check that it is closed on-shell: recalling that, by assumption, ı k dφ = 0. Let us consider asymptotically-flat (V = 0 at spatial infinity), static black holes with bifurcate Killing horizons H associated to k and let us integrate dK[k] over a hypersurface Σ 3 whose boundaries are the bifurcation sphere BH where k = 0 and the 2-sphere at spatial infinity S 2 ∞ .Applying Stokes theorem At infinity, by assumption Over the bifurcation sphere Thus, using again Stokes' theorem and the definition of V k Eq. (4.14), we arrive at the Smarr formula This is not the form in which the Smarr formula is usually presented.The scalar potential must be proportional to one or several dimensionful coupling constants.Let α be that constant and let it have dimensions of inverse length squared so that Then, the Smarr formula can be written in the more standard form which, as proposed in Refs.[24,25,27], must be interpreted in terms of extended thermodynamics: α plays the role of a new thermodynamic variable and Φ α plays the role of its conjugate potential.The validity of this Smarr formula can be tested directly in the existing hairy solutions and we will do so for the Anabalón-Oliva black hole.
For the static, spherically symmetric metrics we are considering and For large values of ρ Asymptotic flatness implies V(φ ∞ ) = 0 and the convergence of the integral demands, again, Eqs.(3.24) to hold.

The first law of black hole mechanics
As shown in Ref. [27], in order to derive the first law of black-hole mechanics using Wald's formalism in theories with dimensionful parameters such as those that must necessarily occur in the scalar potential (α in the case discussed in Section 4), it is necessary to dualize those parameters into (d − 1)-form potentials which have a gauge symmetry generated by (d − 2)-form parameters and work with the dual formulation of the theory.In particular, we have to rederive the Noether-Wald charge, which will have an additional term associated to the new gauge symmetry.

The dual theory
We can directly draw from the results of Ref. [27] and write an action which contains two additional dynamical fields: the scalar ϑ and the 3-form C. ϑ is the square root of the dimensionful constant α discussed in Section 4, which in this setting is promoted to a scalar field, α = ϑ 2 , (5.1) so that 2) The 3-form C is the dual of ϑ and it is introduced in the action as a Lagrange multiplier enforcing the constraint dϑ = 0.The action is that in Eq. ( 2.1) supplemented by the Lagrange multiplier term, which is topological and does not modify the Einstein equations: Under a general variation of the fields, where the Einstein and scalar equations E a , E are identical to those of the original action Eqs.(2.3a) and (2.3b), and the equations of the 3-form C and of the dimensionful "constant" ϑ are given by ) where is the field strength of the 3-form C, invariant under the gauge transformations where χ is an arbitrary 2-form.
On-shell, the equation of motion of ϑ is a duality relation between the 3-form and ϑ [33] G = ⋆ ∂V ∂ϑ , (5.8) and, as expected, the equation of motion of C just says that ϑ is constant. 12inally, Θ contains a new term Θ(ϕ, δϕ) = − ⋆ (e a ∧ e b ) ∧ δω ab + ⋆dφδφ + Cδϑ . (5.9) Observe that the action Eq. ( 5.3) is only invariant under the gauge transformations of C up to a total derivative (defined itself up to total derivatives), which we will have to take into account: (5.10)

The Noether-Wald charge of the dual theory
We just have to use the general form of the variation of the dual action Eq. ( 5.4) in the particular case of the variations Eqs.(4.4) and where, following the general rules, we have defined the momentum map 2-form P ξ to satisfy the equation ı k G + dP k = 0 , (5.12) for a Killing vector k that leaves invariant all the fields of the theory.This assumption guarantees the existence of P k .The Noether-Wald and Komar charges can be simply read from Ref. [27] Q ⋆(e a ∧ e b )P ξ ab + ϑP ξ , (5.13a) ⋆(e a ∧ e b )P ξ ab − 1 2 ϑP ξ , (5.13b) but we have to take into account that they have been computed using the particular choice of total derivative Eq. (5.10) Before deriving the Smarr formula and the first law for the dual theory, it is necessary to consider the generalized, restricted, zeroth law [29].If k is the Killing vector that generates the horizon, in the bifurcation surface where k = 0 Eq.(5.12) implies that (5.15) In the static case in which we are interested here, this just means that where Φ G is a constant and ω (2) is the volume form of the unit 2-sphere.
Observe that, on-shell and using the homogeneity of V in ϑ, the equation of motion of C, the momentum map equation (5.12) and the equation of motion of ϑ we have the following relation: which means that V k defined in Eq. (4.14) is given by 1 Replacing this result in the Komar charge Eq. (5.13b) we recover that of the original theory Eq. (4.15), which means that we get the same Smarr formula Eq. (4.22).
Let us now consider the derivation of the first law in full detail, improving the derivations made in Refs.[29,35,36,26,27,37,1] in which either scalar charges had not been considered or the action was exactly invariant under gauge transformations (which is not the case here).
We start by defining the symplectic 3-form [38,32,39] where ϕ denotes collectively all the fields of the theory.In this case we have to choose in Eq. (5.19) δ 1 ϕ = δϕ, variations of the fields which satisfy the linearized equations of motion but which are, otherwise, arbitrary, and δ 2 ϕ = δ ξ ϕ, the transformations under diffeomorphisms given in Eqs.(4.4) and (5.11) which, we must recall, include induced gauge transformations δ σ ξ , δ χ ξ which we will denote, collectively, by δ Λ ξ , so that Under these transformations, using the general expression for the variation of the action, we get where Θ ′ is a combination of Θ and equations of motion, so that, on-shell, Θ = Θ ′ .On the other hand, varying directly the action we find that In the theory we are considering we can read X(Λ ξ ) from Eq. (5.14): Equating Eqs.(5.21) and (5.22) we arrive at Then, ( The last two terms must also be total derivatives: ) Let us consider ̟ ξ first.In the case at hands, there are two kinds of gauge transformations: 1. Local Lorentz transformations, δ σ .The parameter of the local Lorentz transformation induced by the diffeomorphism generated by ξ is (5.27) 2. Gauge transformations of the 3-form C, δ χ .The parameter of the gauge transformation induced by the diffeomorphism generated by ξ is We find that As for π ξ , we find (5.30) Thus, we find that with and we also find that for vector fields k that generate symmetries of all the fields .33)Integrating this identity over the same hypersurface Σ 3 we considered for the Smarr formula and using again the Stokes theorem we will derive the first law, but we must compute W[k] first.We find and, therefore, where we have used Eq.(5.18).As a non-trivial test of all the manipulations we have performed, it can be checked by an explicit and direct calculation that, indeed, W[k] is closed on-shell when the variations of the fields satisfy the linearized equations of motion and k leaves invariant the background solution.This calculation can be found in Appendix A.
Let us proceed to derive the first law: .36) where we have defined Observe that where Φ α is the potential that occurs in the Smarr formula Eq. (4.22).
In the above derivation we have used Eq.(3.7) and the vanishing of k over the bifurcation sphere.
(5.39) λ ≡ FΩ , and R 2 ≡ Ω , (6.6) the metric and scalar field of the solution take the form with the functions λ and R given by In this form the metric is asymptotically flat with the standard normalization for ρ → ∞: for large ρ λ ∼ 1 − 3η 2 + α 3η 3 ρ + O(1/ρ 2 ) , (6.9a) so that the ADM mass M, the scalar charge Σ and the asymptotic value of the scalar φ ∞ are given by M = 3η 2 + α 6η 3 , (6.10a) For this asymptotic value of the dilaton, the conditions Eqs. (3.24) are, indeed, satisfied.
Observe that, since α and ν are parameters of the theory, the solution contains only one free parameter, η, which determines the ADM mass.The scalar charge Σ can, then, be written in terms of the ADM mass and the parameters that define the theory.Therefore, it describes secondary hair.
It is not difficult to recover some known metrics: for |ν| = 1, the "hairless limit"14 the scalar vanishes identically and which correspond to the Schwarzschild metric with When α = 0 the potential vanishes and one recovers the Janis-Newman-Winicour solutions [5] R 2 (ρ) = W 1−1/ν ρ 2 , (6.13a) The parameter ν does not occur in the action and it is just an integration constant related to the mass and scalar charge by and the metric is singular except when the scalar charge vanishes, (Σ = 0, |ν| = 1) λ(ρ) vanishes for ρ = ν/η, but so does R 2 (ρ) (except for |ν| = 1, Schwarzschild), which indicates that there is a singularity there.It can be shown numerically that, for ν < 3 λ has another zero at some ρ h > ν/η which converges towards the singularity at ρ = ρ sing = ν/η.Finding an analytical expression for ρ h in terms of α, ν and η is too complicated, but we can check the Smarr formula and the results concerning the scalar charge that we have obtained in the previous sections using the property λ(ρ h ) = 0 and R 2 (ρ h ) = 0 only.In the calculations it is often convenient to use a coordinate 15x ≡ ν ηρ , Using these properties, we find the surface gravity and the area of the horizon We can, then, compute which is proportional to α.We have checked that the Smarr formula holds in this form.
We have also checked that Eq. (3.20) which follows from the coordinate-independent definition of scalar charge Eq. (3.5) holds.
Finally, checking the first law Eq.(5.39) in this solution is difficult.In order to test the term proportional to the scalar charge one must have a family of solutions in which the asymptotic value of the scalar is a free parameter, which is not the case here.

Discussion
In this paper we have derived a Smarr formula and a first law for the extended thermodynamics of the black-hole solutions of the theories described by the action Eq.(2.1) using Wald's formalism and the results of Refs.[29,27].Our results coincide with those of Ref. [19] except for the inclusion of the term proportional to the scalar charge and the variation of the asymptotic value of the scalar.This term is somewhat mysterious since there are no asymptotically flat black-hole solutions for asymptotic values of the scalar other than zero, but it may make sense in a wider class of not asymptotically-flat solutions.In any case, the term is clearly there since it arises exactly in the same way as in all the theories considered in Refs.[2,1].
Using an extension of the covariant definition of scalar charge given in Refs.[3,1] we have shown that, in the presence of a bifurcate horizon, the scalar charge is determined by the parameters of the theory, the particular scalar potential, and the value of the scalar on the horizon, and should be considered as "secondary hair".On the other hand, a well-defined scalar charge is possible in the presence of a bifurcate black-hole horizon only if the scalar potential satisfies a set of quite restrictive conditions given in Eq. (3.24), previously found in Ref. Ref. [18] following other considerations.These conditions are equivalent to a "no-hair theorem" for all the theories whose potentials do not satisfy them.Some scalar profiles such as those considered in Ref. [16] may evade these constraints, though.
Our results still leave some questions unanswered: are there hairy black-hole solutions in all the theories whose scalar potentials satisfy all the right conditions?What happens in theories with more scalar fields or in theories in which the scalars couple to curvature scalars?Clearly, much more work is necessary to find a general pattern of behaviour of scalar fields in black-hoe spacetimes and some work in this direction is already in progress [40].