Scalar hairy black holes and scalarons in the isolated horizons formalism

Alejandro Corichi,1,2,* Ulises Nucamendi,3,† and Marcelo Salgado1,‡

1Instituto de Ciencias Nucleares, Universidad Nacional Autónoma de México, A. Postal 70-543, México D.F. 04510, México2Instituto de Matemáticas, Universidad Nacional Autónoma de México, A. Postal 61-3, Morelia, Michoacán, 58090, México

3Instituto de Fı́sica y Matemáticas, Universidad Michoacana de San Nicolás de Hidalgo, Edif. C-3,

Ciudad Universitaria, Morelia, Michoacán, 58040, México(Received 27 April 2005; revised manuscript received 9 November 2005; published 3 April 2006)

The Isolated Horizons (IH) formalism, together with a simple phenomenological model for coloredblack holes has been used to predict nontrivial formulas that relate the ADM mass of the solitons and hairyBlack Holes of Gravity-Matter system on the one hand, and several horizon properties of the black holesin the other. In this article, the IH formalism is tested numerically for spherically symmetric solutions toan Einstein-Higgs system where hairy black holes were recently found to exist. It is shown that the massformulas still hold and that, by appropriately extending the current model, one can account for thebehavior of the horizon properties of these new solutions. An empirical formula that approximates theADM mass of hairy solutions is put forward, and some of its properties are analyzed.

DOI: 10.1103/PhysRevD.73.084002 PACS numbers: 04.70.Bw, 04.20.Cv, 04.40.Nr

I. INTRODUCTION

In recent years, the introduction of the Isolated Horizon(IH) formalism [1–3] has proved to be useful to gaininsight into the static sector of theories admitting ‘‘hair’’[4,5]. Firstly, it has been found that the Horizon Mass of theblack hole (BH), a notion constructed out of purely quasi-local quantities, is related in a simple way to the ADMmasses of both the colored black hole and the solitons ofthe theory [4]. Second, a simple model for colored blackholes as bound states of regular black holes and solitonshas allowed to provide heuristic explanations for the be-havior of horizon quantities of those black holes [5]. Third,the formalism is appropriate for the formulation of unique-ness conjectures for the existence of unique stationarysolutions in terms of horizon ‘‘charges’’ [4]. Finally, thecombination of the Mass formula, together with the factthat in theories such as Einstein-Yang-Mills-Higgs(EYMH) different ‘‘branches’’ of static solutions merge,has allowed to have a formula for the difference of solitonmasses in terms of black hole quantities [5–7]. Many ofthese predictions have been confirmed in more generalsituations and for other matter couplings [8,9]. For a recentreview on IH (including hair) see [10], and for a review ofhairy black holes see [11].

In this article, we explore further the consequences ofthe IH formalism in the static sector of the theory. Inparticular, we explore the behavior of a recently foundfamily of hairy static spherically symmetric (SSS) solu-tions to the Einstein-Higgs system [12], where the scalarpotential is allowed to be negative and therefore, the ex-isting no-hair theorems [13] do not apply. In the standardtreatment of stationary black holes with Killing horizons,

one is always restoring to several concepts that use asymp-totic information very strongly [14]. On the other hand, theIH formalism only uses quasilocal information defined onthe horizon, allowing it to prove very general results in-volving only these quasilocal quantities. The IH formalismhas proved to be generalizable, in the scalar sector, even tothe nonminimal coupling regime, where the energy con-ditions required for the consistency of the formalism aremuch weaker [15]. In the present paper we shall restrict ourattention to the minimally coupled case, and for a particu-lar form of the scalar potential for which static solutionsare known to exist [12]. We will study the one parameterfamily of solutions (that could be labeled by its geometricradius r�) and compare its properties with those of hairyblack holes in other theories, such as EYM, where thephenomenological predictions of the IH formalism havebeen shown to work very well [5,8]. As we will show, wefind that the mass formulas relating BH and soliton ADMmasses works also well, but the model of a hairy black holeas a bound state of a soliton and a bare black holes exhibitssome new unexpected features. As we shall see, one needsto slightly modify the model from its original formulationin Ref. [5]. Once this modification is made the model canagain explain all the qualitative behavior of the hairy BHsolutions.

The structure of the paper is as follows: In Sec. II wereview the consequences of the IH formalism for hairysolitons and BH solutions. In Sec. III we review the SSSfound recently in the Einstein-Higgs sector. Section IV isthe main section of the paper. In it, we show the numericalevidence for the mass formulas and the phenomenologicalpredictions of the model. Unlike the EYM case, there aresome unexpected features, such as the binding energybecoming positive. We then propose a modification ofthe formalism to deal with such situations. We show thatwith these modifications, the model can still account forthe geometrical phenomena found in several theories. In

*Electronic address: corichi@nucleares.unam.mx†Electronic address: ulises@itzel.ifm.umich.mx‡Electronic address: marcelo@nucleares.unam.mx

PHYSICAL REVIEW D 73, 084002 (2006)

1550-7998=2006=73(8)=084002(12)$23.00 084002-1 2006 The American Physical Society

Sec. V we explore the situation of the collapse of a hairyblack hole and use the model to put bound on the totalpossible energy to be radiated. These results should be ofsome relevance to full dynamical numerical evolutions ofsuch black holes. In Sec. VI we propose an empiricalformula for the horizon and ADM masses of scalar hairyblack holes that can also be applied to the EYM case.Finally, we end with a discussion in Sec. VII.

II. CONSEQUENCES OF THE ISOLATED

HORIZONS FORMALISM

In recent years, a new framework tailored to considersituations in which the black hole is in equilibrium (‘‘noth-ing falls in’’), but which allows for the exterior region to bedynamical, has been developed. This Isolated Horizons(IH) formalism is now in the position of serving as startingpoint for several applications. Notably, for the extraction ofphysical quantities in numerical relativity and also forquantum entropy calculations [1,10]. The basic idea is toconsider space-times with an interior boundary (to repre-sent the horizon), satisfying quasilocal boundary condi-tions ensuring that the horizon remains ‘‘isolated‘‘.Although the boundary conditions are motivated by geo-metric considerations, they lead to a well defined actionprinciple and Hamiltonian framework. Furthermore, theboundary conditions imply that certain ‘‘quasilocalcharges’’, defined at the horizon, remain constant ‘‘intime‘‘, and can thus be regarded as the analogous of theglobal charges defined at infinity in the asymptotically flatcontext. The isolated horizons Hamiltonian frameworkallows to define the notion of Horizon Mass M�, as afunction of the ‘‘horizon charges‘‘ (hereafter, the subscript‘‘�’’ stands for a quantity at the horizon).

In the Einstein-Maxwell and Einstein-Maxwell-Dilatonsystems considered originally [2], the horizon mass satis-fies a Smarr-type formula and a generalized first law interms of quantities defined exclusively at the horizon (i.e.without any reference to infinity). The introduction ofnonlinear matter fields like the Yang-Mills field hasbrought unexpected subtleties to the formalism [4].However, one still is in the position of defining a HorizonMass, and furthermore, this Horizon Mass satisfies a firstlaw.

An isolated horizon is a nonexpanding null surfacegenerated by a (null) vector field la. The IH boundaryconditions imply that the acceleration � of la (laralb ��lb) is constant on the horizon �. However, the precisevalue it takes on each point of phase space (PS) is notdetermined a-priori. On the other hand, it is known that foreach vector field tao on space-time, the induced vector fieldXto on phase space is Hamiltonian if and only if there exists

a function Eto such that �Eto � ���; Xto�, for any vectorfield � on PS. This condition can be rewritten as, �Eto ��to

8�G�a� � work terms. Thus, the first law arises as a nec-

essary and sufficient condition for the consistency of the

Hamiltonian formulation. Thus, the allowed vector fields ta

will be those for which the first law holds. Note that thereare as many ‘‘first laws‘‘ as allowed vector fields la b�ta onthe horizon. However, one would like to have a PhysicalFirst Law, where the Hamiltonian Eto be identified with the

‘‘physical mass‘‘ M� of the horizon. This amounts tofinding the ‘‘right �’’. This ‘‘normalization problem‘‘ canbe easily overcome in the EM system [2]. In this case, onechooses the function � � ��a�; Q�� as the correspondingfunction for the static solution with charges �a�; Q��.However, for the EYM system, this procedure is not asstraightforward. A consistent viewpoint is to abandon thenotion of a globally defined horizon mass on Phase Space,and to define, for each value of n � no (which labelsdifferent branches of the solutions), a canonical normal-

ization tano that yields the Horizon Mass M�no�� for the no

branch [3,4]. The horizon mass takes the form (from nowon we shall omit the n0 label),

M��r�� �1

2G0

Z r�0��r�dr; (1)

with r� the horizon radius. Here ��r�� is related to thesurface gravity as follows ��r�� � 2r���r��.

Furthermore, one can relate the horizon mass M� to theADM mass of static black holes. Recall first that generalHamiltonian considerations imply that the totalHamiltonian, consisting of a term at infinity, the ADMmass, and a term at the horizon, the Horizon Mass, isconstant on every connected component of static solutions(provided the evolution vector field ta0 agrees with the staticKilling field everywhere on this connected component)[2,3]. In the Einstein-Yang-Mills case, since theHamiltonian is constant on any branch, we can evaluateit at the solution with zero horizon area. This is just thesoliton, for which the horizon area a�, and the horizon

mass M� vanish. Hence we have that H�t0� � Msol. Thus,

we conclude [4]:

Msol � MADM �M�; (2)Thus, the ADM mass contains two contributions, oneattributed to the black hole horizon and the other to theoutside ‘‘hair‘‘, captured by the ‘‘solitonic residue’’. Theformula (2), together with some energetic considerations[5], lead to the model of a colored black hole as a boundstate of an ordinary, ‘‘bare’’, black hole and a ‘‘solitonicresidue’’, where the ADM mass of the colored black holeof radius r� is given by the ADM mass of the soliton plusthe horizon mass of the ‘‘bare‘‘ black hole plus the bindingenergy:

MADM � Msol �M� � M0� �Msol � Ebind; (3)with Ebind � M� �M0�. Simple considerations about thebehavior of the ADM masses of the colored black holesand the solitons, together with some expectations of thismodel (such as demanding for a nonpositive binding en-

CORICHI, NUCAMENDI, AND SALGADO PHYSICAL REVIEW D 73, 084002 (2006)

084002-2

ergy) give raise to several predictions about the behavior ofthe horizon parameters [5]. Among the predictions, wehave:

(i) The absolute value of the binding energy decreasesas r� increases.

(ii) ��r��, as a function of r�, is a positive function,bounded above by ��0��r�� � 1.

(iii) The curve��r�, as functions of r intersect the r � 0axis at distinct points between 0 and 1, and neverintersect. Finally,

(iv) The curve for �, for large value of its argument,becomes asymptotically tangential to the curve��0��r�� � 1.

One of the features of these solutions in, say, EYM isthat there is no limit for the size of the black hole. That is, ifwe plot the ADM mass of the BH as function of the radiusr� we get an infinite number of curves, each of the inter-secting the r� � 0 line at the value of the soliton mass, andnever intersecting each other.

The purpose of this paper is to test the mass formula (2),for the scalar hairy solutions found in Ref. [12] and also toconfront the predictions (i)–(iv) (obtained for the coloredEYM BH model [5]), with the corresponding properties forthe scalar hairy BH. In the next section we will review thescalar hairy solutions, and afterwards we shall study theirhorizon properties.

III. SCALAR SOLITONS AND BLACK HOLES IN

EINSTEIN-HIGGS THEORY

Let us consider the theory of a scalar -field minimallycoupled to gravity described by the total action:

Stot�g��; �� �Z ��������gp

�R

16���

1

2�r����r���

� V�����d4x (4)

(units where G0 � c � 1 are employed). The field equa-tions following from the variation of the action (4) are,

G�� � 8���r���r��� g��

�1

2�r����r��� � V���

��;

(5)

and,

�� � @V���@�

: (6)

It is well known that asymptotically flat static sphericallysymmetric solutions representing black holes solutions tothe Einstein-Higgs equations do not exist if the scalarmatter satisfies the weak energy condition (WEC) due tothe existence of the so called scalar no-hair theorems [13].Recently, numerical evidence for the existence of asymp-totically flat and static spherically symmetric solutionsrepresenting scalar hairy black holes (SHBH) and scalar

solitons (scalarons; hereafter SS) have been found in theo-ries represented by the action (4) and with a scalar potentialnon-positive-semidefinite [12] given by the asymmetricpotential,

V��� � �4

���� a�2 � 4�1 � 2�

3��� a� � 212

�

� ��� a�2; (7)

where �, i and a are constants. For this class of potentialone can see that, for 1 > 22 > 0, � � a corresponds tothe local minimum, � � a� 1 is a global minimum and� � a� 2 is a local maximum (see Fig. 1). The keypoint in the shape of the potential, V���, for the asymptoti-cally flat solutions to exist, is that the local minimumVlocalmin � V�a� is also a zero of V��� (see [12] for an de-tailed analysis for the existence of these solutions).Moreover, V��� is not positive definite (we assume � >0), which leads to a violation of the WEC and therefore thescalar no-hair theorems [13] can not be applied to this case.

In order to describe the asymptotically flat SHBH andSS, we use a standard parametrization for the metric andthe scalar field describing spherically symmetric and staticspacetimes

ds2 � ��1� 2m�r�

r

�e2��r�dt2 �

�1� 2m�r�

r

��1dr2

� r2d�2; (8)

� � ��r�; (9)

For SHBH we demand regularity on the event horizonr� which implies the conditions,

m� �r�2; ��r�� � ��; ��r�� � ��; (10)

FIG. 1. Qualitative shape of the scalar-field potential V��� asgiven by Eq. (7) used to construct the asymptotically flat blackhole and soliton solutions.

SCALAR HAIRY BLACK HOLES AND SCALARONS IN . . . PHYSICAL REVIEW D 73, 084002 (2006)

084002-3

�@r��� �r��@�V���1� 8�r2�V��

; �@rm�� � 4�r2�V�: (11)

For SS we impose regularity at the origin of coordinatesr � 0,m�0� � 0; ��0� � �0; ��0� ��0; �@r��0 � 0:

(12)

where �0 and �0 are to be found such as to obtain thedesired asymptotic conditions. In addition to the regularityconditions, we impose asymptotically flat conditions onthe spacetime for SHBH and SS:

m�1� � MADM; ��1� � 0; ��1� � �1: (13)Above, the value �1 corresponds to the local minimum ofV���. MADM is the ADM mass associated with a SHBH orSS configuration. For a given theory, the family of SHBHconfigurations is parametrized by the free parameter A�which specifies the area of the black hole horizon.Therefore for SHBH,MADM � MADM�A��, or equivalentlyMADM � MADM�r�� since in our coordinates the horizonarea A� � 4�r2�. The value �� � ���r�� is a shootingparameter rather than an arbitrary boundary value which isdetermined so that the asymptotic flat conditions are sat-isfied. On the other hand, for SS the value �0 is theshooting parameter, and the corresponding configurationis characterized by a unique MsolADM.

Finally, the surface gravity of a spherically symmetricstatic black hole can be calculated from the general ex-pression of spacetimes admitting a Killing horizon [14]:

� ��� 1

4r2

�1=2

r�r�; (14)

wherer2 stands for the Laplacian operator associated withthe stationary metric and � �@t; @t� is the norm of thetimelike (static) Killing field which is null at the horizon.

For the present case, � gtt � ��1� 2m�r�r �e2��r�.From the above formula, one can obtain the following

useful expression

� � limr!r�

�1

2

@rgtt������������gttgrrp

�: (15)

For the election of the parametrization of the metric (8) wehave

��r�� �1

2r�e��r���1� 2�@rm���: (16)

Introducing (11) in (16) we obtain the final expression forthe surface gravity of the SHBH

��r�� �1

2r�e��r���1� 8�r2�V��: (17)

In the next section, we shall analyze these solutions fromthe perspective of the IH formalism.

IV. MASS FORMULAS

Let us now turn to the straightforward application of theIH formalism mentioned in the Sec. II to the case of SHBHand SS in the Einstein-Higgs theory with action given by(4) and V��� by Eq. (7) [12]. As in Ref. [12], we shall takethe specific values 1 � 0:5, 2 � 0:1 and a � 0; all thequantities (e.g., MADM and r�) have been rescaled as

appropriate using 1=�����p

as a length-unit.The first consequence coming from the IH formalism is

that the horizon mass associated with the SHBH takes theform,1

M��r�� �1

2

Z r�0��r�dr; (18)

where ��r�� � 2r���r�� [the value of ��r�� is given by(17)]. We have dropped the subindex (n) in the expression(18) because in the Einstein-Higgs system considered herethere is only one branch of static spherically symmetricSHBH labeled by its horizon radius r� (the correspondingscalar configurations do not have nodes). Additionally,there is another branch of static spherically symmetricBH given by the family of Schwarzschild BH’s labeledby its corresponding horizon radius r� and with horizonmass

MSchwarz� �r�� � r�=2: (19)

The second consequence coming from the IH formalism isthat on the entire branch of SHBH we can expect thefollowing identity to be true

MADM � Msol �M�; (20)

where Msol is the ADM mass of the SS obtained taking thelimit r� ! 0 of the branch of SHBH and MADM is theADM mass corresponding to the SHBH with horizonradius r�, and the horizon mass is given by (18). Thus, ina similar way to the EYM theory, the total ADM mass ofthe solution contains two contributions, one attributed tothe horizon of the SHBH and the other to the outside‘‘hair‘‘, captured by the SS. We have performed numericalexplorations for SHBH for a large range of values of thehorizon radius (in normalized units) and have checked theidentity (20). We have found complete agreement withinthe numerical uncertainties. This can be seen in Fig. 2,where the identity was checked up to r� � 250.

Figure 3 depicts the behavior of ��r��. Unlike the EYMmodel, where � 1 for large r�, in this model � 1:24asymptotically.

Figure 4 shows an example of a BH solution with larger� (for a small BH see Fig. 2 of Ref. [12]).

1We remind the reader our choice of units G0 � c � 1.

CORICHI, NUCAMENDI, AND SALGADO PHYSICAL REVIEW D 73, 084002 (2006)

084002-4

A. A physical model of SHBH

The extrapolation of the model of a hairy black hole as abound state of an ordinary, ‘‘bare‘‘, black hole and a‘‘solitonic residue‘‘ (first applied successfully to the col-ored BH’s in the EYM theory) [5] does not apply directlyto the SHBH because the straightforward generalization ofthe formula (3) as

MADM � Msol �M� � Mschwarz� �Msol � Ebind; (21)

where Ebind � M� �Mschwarz� , to our case, has the problemthat the binding energy changes sign, becoming positivefor BH larger that r� 30, and then increasing in absolutevalue as r� gets larger. That is,

Ebind Br�; (22)

(where B is a constant whose value depends on the specificmodel) for r� � 1, which is contrary to the expectedfeature of a negative binding energy as in the EYM case(i.e. the prediction (i) mentioned above will not besatisfied).

In order to appreciate the origin of the failure of thisfeature, let us recall that we can writeMADM in terms of theSchwarzschild mass and the mass of the ‘‘hair‘‘ as,

MADM�r�� � Mschwarz� �r�� �Mhair�r��; (23)where

Mhair�r�� � �Z 1r�

r2Tttdr; (24)

By rescaling the r-coordinate in terms of r�, we canrewrite the mass of the hair. It becomes then,

Mhair�r�� � �r�Z 1

1x2 ~Tttdx; (25)

with

x � rr�; (26)

~T tt � r2�Ttt ; (27)

Ttt � ���

1� 2m�r�r

� �@r��22� V���

�: (28)

Now, for x� 1, the integral in Eq. (25) becomes almostindependent of r�, and in fact the numerical analysisprovides the following value

Mhair�r� � 1� Br�: (29)where B 0:12. Now, since Mschwarz� �r�� � r�=2 we havethen

MADM�r� � 1� Cr�: (30)where now C 0:62. Therefore we conjecture that C is aconstant that depends of the matter-theory involved. Forthe EYM case, one can easily show that the scaling prop-erties of the hair contribution of the energy makes theequivalent of the integral of Eq. (25) to behave like 1=r�rather than r�. Therefore, for the EYM case, C � 1=2. Aswe now show, this subtle difference in both theories makesthat the binding energy expression used in the EYM cannotbe used straightforwardly for the Einstein-Higgs theoryanalyzed here.

FIG. 3. ��r�� � 2�r� is plotted as function of r�. Note that itapproaches asymptotically the value 1:24. Here ��0� 0:324.

FIG. 2. The ADM mass (solid lines), the horizon BH mass (dash-dotted lines), and the mass of the Schwarzschild solution (dashed-lines) plotted as functions of r� (first panel). The second panel depicts similar quantities using logarithmic scales to appreciate bettertheir behavior for small r�. The soliton mass Msol 3:827.

SCALAR HAIRY BLACK HOLES AND SCALARONS IN . . . PHYSICAL REVIEW D 73, 084002 (2006)

084002-5

From (21) and (23) we find that

Ebind�r� � 1� Mhair�r� � 1� �Msol Br� �Msol;(31)

then Ebind scales as r� when r� � 1 (remember that Msolis a constant) [16]. It is clear that the sign of the bindingenergy as such defined changes sign and grows with thesize of the black hole.

To deal with this situation we now proceed to propose amodification of the model of a hairy black hole as a boundstate in order to adapt it to the more general case. Ourproposal consists in redefining the binding energy as,

Enewbind�r�� � Ebind�r�� � Br�� M��r�� �Mschwarz� �r�� � Br�; (32)

By this procedure we have ‘‘renormalized’’ the bindingenergy by subtracting the divergent term. That is, the newexpression for the binding energy is

Enewbind�r�� � M��r�� � Cr�: (33)

This definition shares now exactly the same properties asthe original expression for the EYM. Thus, it vanishes atr� � 0, and decreases monotonically to the negative value�Msol (see Fig. 5). It is interesting to note that this newdefinition reduces to the old one for the EYM case, since aswe previously remarked CEYM � 1=2.

The formula (21) can now be reformulated as,

MADM�r�� � Msol �M��r��� Mschwarz� �r�� �Msol � Enewbind�r�� � Br�� Cr� �Msol � Enewbin �r��; (34)

FIG. 5. Both binding energies are plotted. The ‘‘old’’ bindingenergy Ebind�r�� is shown to become positive and approachesasymptotically a straight line (dashed line). The ‘‘new’’ bindingenergy Enewbind�r�� is also plotted (solid line), showing the expectedbehavior.

FIG. 4. Large-black-hole configuration constructed with V��� as given by Eq. (7) with parameters 1 � 0:5, 2 � 0:1, a � 0, andr� � 150=

�����p

, �� 0:26111. The upper panels depict the scalar field and the mass function, respectively. The latter converges toMADM 93:096=

�����p

. The lower panels depicts the metric potentials (the first is a zoom of the second):�����������gttp (solid line), �������grrp

(dashed line), e� (dash-dotted line) and � (dotted line).

CORICHI, NUCAMENDI, AND SALGADO PHYSICAL REVIEW D 73, 084002 (2006)

084002-6

Thus, we would get the same structure for the ADM massof the hairy black hole where now the mass of the ‘‘bareblack hole‘‘ would be equal to ~M0� � �1=2� B�r� � Cr�.However it is not clear what the origin of this extra term(Br�) is, and we could very well have assigned it to thesoliton mass to form a new ‘solitonic residue with massMsol � Br� (whose interpretation however seems some-what obscure). We must explore more in order to decidewhich interpretation is best suited for our model. As westressed, the constant C depends on the theory considered;for the EYM and EYMH theories, C � 1=2. It would beinteresting to explore (in addition to the current analysis,see below) whether other theories admitting hair possesvalues of C different from 1=2.

To end this section, let us rewrite the form of the hairymass in terms of the old binding energy, in order to under-stand the behavior of the scalar system. First, let us noteusing Eqs. (21) and (23) that

Mhair�r�� � MADM�r�� �Mschwarz� �r�� � Msol � Ebind;(35)

so the binding energy is

Ebind � Mhair�r�� �Msol

� ��Z 1

r�

r2Tttdr�Z 1

0r2To

ttdr

�; (36)

where Tott is the stress-energy tensor of the solitonic

regular solution. This equation can be rewritten as,

Ebind � ��Z 1

r�

r2�Ttt � Tott�dr�Z r�

0r2To

ttdr

�: (37)

Here we can identify the first term as the difference be-tween the hair of the BH and the hair of the soliton. Ofcourse we are comparing the quantities (the integrands)that live on different manifolds, but the total integral is welldefined. What happens in the EYM case is that both stresstensors behave very much alike, for the exterior region(r > r�) and for large values of r�, and thus the onlyterm that contributes is the second one, that gives theADM mass of the soliton (recall that r� � 1, that is forBH’s much larger than the characteristic size of the soliton(of order one in this dimensionless units), so the integralcaptures most of the soliton mass). In the scalar-field case,the fact that the binding energy is proportional to r�, forlarge black holes, is captured by the fact that the BHcontribution to the first term in (37) is dominating. It wouldbe interesting to explore this issue in other gravity-mattersystems.

V. INSTABILITY AND FINAL STATE

The next question we want to consider has to do with thefollowing situation. Consider the case where a hairy blackhole of geometrical radius r� is slightly perturbed andtherefore it decays. The final state will be, one expects, a

black hole that in its near horizon geometry resembles theSchwarzschild solution, with the scalar field taking thevalue where the potential has a local minima and vanishes.This means that in this process the ‘‘scalar charge’’ at thehorizon, namely, the value �� must change. One can makean argument similar to the one in Ref. [5] to conclude that,in that situation, the horizon must grow in the process andtherefore, the available energy to be radiated can not all beradiated to infinity; part of it must fall into the black hole.Let us now recall the estimate for the upper bound of thetotal energy to be radiated.

The first step is to assume that the process illustrated inFig. 6 takes place. Then, we assume that in the initialsurface there was an isolated horizon �in and after theinitial unstable configuration has decayed, with part ofthe energy falling through the horizon and the rest radiatingaway to infinity, we are left with a horizon �fin of a hairlessblack hole (with rfin� > r

in�). If we denote by EI� the energy

radiated to future null infinity I�, and given that the ADMenergy does not change in the process, we have

MADM � M��rin�� �Msol � Mschwarz� �rfin� � � EI� ; (38)which can be rewritten as,

Msol � Einbind � Mschwarz� �rfin� � �Mschwarz� �rin�� � EI� :(39)

On the right-hand-side note that the first two terms can beidentified with �M0�, namely, the change in (bare) horizon

mass, while the second term corresponds to the radiatedenergy. Thus, it is natural to identify the quantity on the leftas the available energy Eavail on the system. We can thenwrite,

Eavail � MADM �Mschwarz� �rin�� � Msol � Ebind�rin��� Msol � Enewbind�rin�� � Brin� (40)

There are several comments regarding this quantity. First,we note that there is a qualitative change in the behavior ofEavail�rin�� as function of the initial horizon radius, as in theEYM case. Its functional dependence is very similar to the

∆

Σ

i

i

1

+

o

+

2Σ

∆

I

S1

S2

in

fin

FIG. 6. This figure illustrates a physical process where aninitial configuration with an isolated horizon �in is perturbedand the final state contains another isolated horizon �fin.

SCALAR HAIRY BLACK HOLES AND SCALARONS IN . . . PHYSICAL REVIEW D 73, 084002 (2006)

084002-7

binding energy since they differ only by the soliton mass.In the EYM case the available energy was equal to thesoliton mass when there was no initial black hole (there isno energy used in binding the BH), and decreases as theradius increases. For very large black holes, the availableenergy goes to zero. For the scalar case under considerationhere, we have a different behavior. The available energydecreases for small black holes but starts to increase andgrows linearly with r�. The fact that in the EYM case theavailable energy went to zero for large BHs was interpretedas meaning that those black holes were ‘‘less unstable‘‘.This expectation was confirmed by the fact that the fre-quencies of the linear perturbations was decreasing withthe radius of the initial BH [5,18]. It is natural then to askthe same question for the scalar black holes. We havecomputed the frequencies of the (single) unstable mode �t; r� � ��r�e�t present (where 2 turns to be alwaysnegative), as a function of the horizon radius and plotted itin Fig. 7 (where �t; r� represents a linear perturbation of��r�; see Ref. [12] for the details).

As can be seen from the Fig. 7, the frequencies stilldecrease as the black holes become larger, which is thesame behavior observed in the EYM case. It is convenientthen to reconsider the meaning of ‘‘less unstable’’. In thescalar-field case considered here, numerical investigationsof the dynamical evolution of the soliton as initial stateshow that the system is unstable [19]. The dynamicalevolution of the system depends on the sign of the initialperturbation on the extrinsic curvature. For one sign of theperturbation, the system collapses and forms a black holewith a final isolated horizon, while for the other sign thesystem expands as a domain wall and gets therefore radi-ated to infinity (for details see [19]). One should thenexpect that the dynamical evolution of slightly perturbedhairy black holes will show a similar qualitative behavior.

In that case, for one sign of the perturbation one mightexpect the situation considered before, namely, that thescalar field collapses and the BH grows. For the othersign, one can imagine that there could be, in some situ-ations, an expanding wall that radiates away while leavinga ‘‘naked black hole’’. The pressing question is whether, inthat case, this residue would be a Schwarzschild like or anAdS like black hole. This question arises since, for thesoliton collapse in the case of the expanding wall, theregion around the origin resembles an AdS spacetimewith an effective negative cosmological constant generatedby the (nonpositive) potential. One might need in that casea new interpretation of the formalism.

It is our belief that one needs to clarify what the criteriashould be for regarding the system as slightly unstable orvery unstable, other than the frequency of its perturbations.This and a full clarification of the nature of the resultingbare black hole could be achieved whenever full numericalsimulations of dynamical evolution staring from scalarhairy black holes become available.

Let us return our discussion to Eq. (40). The first thing tonote is that due to the characteristic behavior of thesesolutions, for horizons larger than r� 30, the horizonmass of the hairy black hole becomes larger than theSchwarzschild horizon mass of the same radius. This isalso the point at which the binding energy becomes posi-tive. One can thus speculate that the black holes of thisradius and larger will have more violent collapses with alarger fraction of the available energy radiated away.Finally, from Eq. (40) one could interpret that again, theterm (Br�) could be associated with the soliton mass toform a solitonic residue that, together with the new bindingenergy allows us to have the same qualitative features ofthe heuristic model of [5]. Again, a more detailed analysiswill have to wait for the numerical investigations of thefully dynamical process.

VI. AN EMPIRICAL FORMULA

The nonlinear behavior of the Einstein-Matter equationsand the nontrivial relations between the masses and thehorizon radius posses a challenge to obtain an analyticalformula for MADM�r��. One should expect that there is ingeneral no closed analytical formula for the masses ofhairy BH.

We have discovered that the following empirical for-mula reproduces the qualitative and quantitative features ofthe numerical analysis

MempADM�r�� �

��������������������������������������������������������������Msol �

D

2C

�2

� C2r2� �Dr�s

� D2C

� Msol� �������������������������������������

1� Cr�Msol

�2

� Fs

� 1�: (41)

where

FIG. 7. The unstable-mode frequency ! ������������2p

of the per-turbed BH and soliton (r� � 0) is plotted as a function of thehorizon radius r�. Note that the frequency decreases as thehorizon radius increases.

CORICHI, NUCAMENDI, AND SALGADO PHYSICAL REVIEW D 73, 084002 (2006)

084002-8

D � 2C�0Msol2C� �0

; Msol ��0Msol

2C� �0;

F ��2C

�0

�2

� 1(42)

�0 being the value of � at r� � 0.The formula (41) has the following nice properties:(1) M

empADM�0� � Msol.

(2) MempADM�r�� is a monotonically increasing function of

r�.(3) For large r�, M

empADM ! Cr�.

(4) The relative error between MempADM and the numerical

one is less that 10%. These errors become very smallfor small and large r�.

(5) One can define an empirical binding energy by usingE

empbin �r�� � M

empADM �Msol � Cr� [where the first

two terms provide the empirical horizon mass;here we are using Eq. (33)]. This formula reprodu-ces very well the numerical results.

(6) One can then obtain a fit for � as follows

�em�r�� � 2dM

empADM

dr�

� 2r�C2 �D����������������������������������������������������������

�Msol � D2C�2 � C2r2� �Dr�q

�2C�1� Cr�

Msol�

��������������������������������1� Cr�

Msol�2 � F

q : (43)

This formula reproduces the qualitative shape of thenumerical �, such as its exact value at the origin, itsmonotonically increasing behavior and its asymp-totic value 2C for large r�.

(7) For the Schwarzschild case (C � 1=2, Msol � 0),one obtains the expected results: M

empADM�r�� �

r�=2, �em�r�� � 1, Eempbind�r�� � 0.

Clearly by adding terms of the form r�� (1< �< 2)inside the square root of Eq. (41) one could improve thefit between the numerical results and the analytical for-mula. In Fig. 8 we compare between the empirical formulaand the numerical values of the hairy scalar black holes, forthe ADM mass, screened surface gravity � and the bindingenergy.

The empirical formula can be used also for the EYMcase with C � 1=2 and the corresponding values of Msoland D. Figure 9 compares the numerical values of theEYM n � 1-branch with those obtained from the empiricalformula.

We conjecture that the empirical formula can work alsofor different n, by using their corresponding values Mnsol,and �n0 . Moreover, we also speculate that such a formulamight hold for other theories admitting hair.

Now, we can further use the first law of thermodynamics�M � ��A�=�8�� for MempADM, and obtain the followingprediction

�

�M

empADM �

D

2C

�� C2 � D

2r�: (44)

Since the properties (1)–(7) show that the analytical resultsobtained from Eq. (41) work particularly well for large andsmall r�, the most reliable consequence of (44) is a re-markable simple relation between the surface gravity andthe ADM mass for sufficiently large hairy black holes:

�MADM C2: (45)

Note that Eq. (45) is consistent for the Schwarzschildcase (C � 1=2, D � 0, MADM � r�=2), where the identity�MADM � 1=4, holds exactly.

We have performed a nonexhaustive analysis of thesolutions with respect to variations of some of the parame-ters of the scalar potential Eq. (7). Notably, we havecomputed the effect of the variation of 2 on the globalquantities. It is to note that changing 2 modifies thepotential barrier between the global and the local mini-

FIG. 8. The three panels depict the ADM-mass, �, and binding energy Enewbind, respectively, as a function of the horizon radius. Thesolid lines correspond to the values obtained from a numerical analysis and the dashed lines were obtained from the empirical formulasdescribed in the main text. Note the good qualitative behavior of the empirical formulas. Remarkably good fits to the more precisenumerical values are obtained for small and large r�. The values for the empirical formula are C 0:62, Msol 3:827 and �0 0:324.

SCALAR HAIRY BLACK HOLES AND SCALARONS IN . . . PHYSICAL REVIEW D 73, 084002 (2006)

084002-9

mum. In fact, the closer the value 2 to 1=2, the lessnegative is V�a� 1� � �31�22 � 1�=12, and there-fore the potential approaches the conditions where theno-hair theorems apply. The details of the solutions thendepend in a nontrivial fashion between the interplay of thenegative global minimum (in order to avoid the applica-bility of the nonhair theorems) and the height of thepotential barrier.

Figure 10 depicts different global quantities as a func-tion of the horizon radius for five different values of 2.The soliton mass (r� � 0) as well as the ADM and horizonmasses (for large r�) tend to increase with2. Remarkably,the empirical formulas continue to provide reasonablegood results by changing the corresponding values of their

parameters ~P2 :� �Msol; C;�0�2 . The quality of the fit tothe numerical values can be appreciated by the dashedcurves of Fig. 10 which were computed with the empiricalformulas.

VII. DISCUSSION

Let us first summarize our results. By solving numeri-cally Einstein’s equations for static solutions of a self-gravitating scalar field, we have analyzed the behavior ofseveral spacetime quantities as functions of the black holehorizon radius. We have found that the ADM mass of thespacetimes exhibits two types of behavior: it is similar toother ‘‘hairy‘‘ theories for small black holes, but its behav-

FIG. 10. Panels 1– 4 depict the ADM-mass, horizon mass, � and the binding energy, respectively, as a function of r�. The solid lineswere obtained from the numerical analysis while the dashed lines were computed using the empirical formulas. The lines areassociated with the five different values used for 2 � 0:1, 0.11, 0.12, 0.13, 0.14 with 1 � 0:5 fixed. As seen from bottom to top (forlarge r�) the plots of panels 1–3 correspond to 2 in increasing order (in panel 4 the order is reversed). The values of the parameters~P2 � �Msol; C; �0�2 used in the empirical formulas are ~P0:1 �3:82; 0:62; 0:32�, ~P0:11 �5:22; 0:66; 0:24�, ~P0:12 �7:47; 0:72; 0:17�~P0:13 �11:78; 0:82; 0:11� ~P0:14 �23:51; 1:01; 0:06�.

FIG. 9. Same as Fig. 8, for the Einstein-Yang-Mills theory (n � 1 colored black holes). Here the values for the empirical formulasare C � 1=2, Msol 0:828 and �0 0:126.

CORICHI, NUCAMENDI, AND SALGADO PHYSICAL REVIEW D 73, 084002 (2006)

084002-10

ior changes dramatically for large black holes. In particularthe ADM mass of large BH scales not as r�=2 as in othertheories (EYM, EYMH, etc), but the proportionality con-stant (with respect to horizon radius) takes a different valuedepending on the form of the potential (C 0:63 for 2 �0:1). In this article we have analyzed the consequences ofthis fact for a model based on the isolated horizons formal-ism. In such a model, a hairy black hole is viewed as abound state of a soliton (which we have) and a ‘‘bare blackhole’’. The binding energy is found to be negative in EYMand EYMH, but in our case, for large BH, the bindingenergy becomes positive and grows linearly with r�.

2 Thisfact leads to several possibilities. We have seen that it ispossible to modify the original model by ‘‘renormalizing‘‘the binding energy in such a way that the newly definedenergy has the same qualitative behavior as in the EYMsystem. The price one has to pay is the need to reinterpreteither a new ‘‘solitonic residue’’, or a new bare black hole.As a first attempt towards giving a definite answer to thisquestion, we analyzed the frequency of the unstable modeof the linearized perturbation, and found that the behavioris the same as in EYM. This suggests that the properphysical interpretation is still unclear and that furthernumerical dynamical investigations are needed to fullysettle the question. In particular, the two different regimesof the theory might have some consequences in the dy-namical evolution of slightly perturbed BHs, where onecould conjecture a different qualitative behavior for smalland large black holes, regarding the endpoint of evolutionand the nature of the bare black hole to which the solution

settles. We have also conjectured that the constant thatfixed the proportionality between ADM mass and horizonradius for large BH’s is a theory-dependent constant, whichwould, in particular, imply that axi-symmetric nonspheri-cal BH solutions to the gravity-scalar-field system wouldhave the same asymptotic behavior, for each given poten-tial. It would be worth studying other gravity-matter sys-tems, such as nonminimally coupled scalars, to see whetherthey posses a different proportionality constant (work is inprogress in these directions).

We have shown also that a very simple heuristic analyticformula captures the essential qualitative behavior of theADM mass of the hairy scalar BH’s, specially for small andlarge values of the horizon radius. We have conjecturedthat such formula can also be useful for EYM and moregeneral hairy black holes. It remains a theoretical chal-lenge to fully understand the origin of such simple formula.

Perhaps the most important conclusion from the presentwork is the lesson that hairy black holes for different mattersystems exhibit new, and sometimes, unexpected behavior.This also point out to the need of a proper and deeperunderstanding of the reason why the heuristic hairy blackhole model works so well for the system that it does, andwhether the phenomenological modifications we have pro-posed here stand the test of full numerical investigations.

ACKNOWLEDGMENTS

We would like to thank A. Ashtekar and D. Sudarsky fordiscussions. This work was in part supported by grantsDGAPA-UNAM IN122002, and IN119005. U.N. acknowl-edges partial support from SNI, and Grants No. 4.8 CIC-UMSNH, No. PROMEP PTC-61 and No. CONACYT42949-F.

[1] A. Ashtekar, C. Beetle, O. Dreyer, S. Fairhurst, B.Krishnan, J. Lewandowski, and J. Wisniewski, Phys.Rev. Lett. 85, 3564 (2000).

[2] A. Ashtekar, C. Beetle, and S. Fairhurst, Class. Quant.Grav. 17, 253 (2000); A. Ashtekar and A. Corichi, Class.Quant. Grav. 17, 1317 (2000).

[3] A. Ashtekar, S. Fairhurst, and B. Krishnan, Phys. Rev. D62, 104025 (2000).

[4] A. Corichi and D. Sudarsky, Phys. Rev. D 61, 101501(R)(2000); A. Corichi, U. Nucamendi, and D. Sudarsky, Phys.Rev. D 62, 044046 (2000).

[5] A. Ashtekar, A. Corichi, and D. Sudarsky, Class. Quant.Grav. 18, 919 (2001).

[6] B. Kleihaus and J. Kunz, Phys. Lett. B 494, 130 (2000).[7] A. Corichi, U. Nucamendi, and D. Sudarsky, Phys. Rev. D

64, 107501 (2001).[8] B. Kleihaus, J. Kunz, A. Sood, and M. Wirschins, Phys.

Rev. D 65, 061502(R) (2002).[9] R. Ibadov, B. Kleihaus, J. Kunz, and M. Wirschins, Phys.

Lett. B 627, 180 (2005).[10] A. Ashtekar and B. Krishnan, gr-qc/0407042.[11] M. S. Volkov and D. V. Gal’tsov, Phys. Rep. 319, 1 (1999).[12] U. Nucamendi and M. Salgado, Phys. Rev. D 68, 044026

(2003).[13] M. Heusler, J. Math. Phys. (N.Y.) 33, 3497 (1992); D.

Sudarsky, Class. Quant. Grav. 12, 579 (1995); J. D.Bekenstein, Phys. Rev. D 51, R6608 (1995).

[14] M. Heusler, Black Hole Uniqueness Theorems (CambridgeUniversity Press, Cambridge 1996).

[15] A. Ashtekar, A. Corichi, and D. Sudarsky, Class. Quant.Grav. 20, 3413 (2003).

[16] Another example of hairy BH’s with the property that themasses Mhair scale also linearly with r� are the hairy BH’sof the Einstein-Skyrme system. To see this we take the

2Recently, another system in 5 dimensions was shown toposses a positive binding energy as well [20].

SCALAR HAIRY BLACK HOLES AND SCALARONS IN . . . PHYSICAL REVIEW D 73, 084002 (2006)

084002-11

equations of motion for the Einstein-Skyrme model asdescribed in the subsection (7.4) from Ref. [11]; specifi-cally, we take the equation (7.37):

dm�r�dr

� N�r2

2� sin2�

��d�

dr

�2

��r2 � sin

2�

2

�sin2�

r2

(46)

where the metric for static and spherically symmetricconfigurations is:

ds2 � �2�r�N�r�dt2 � 1N�r�dr

2 � r2d�2 (47)

with

N�r� � 1� 2m�r�=r; (48)and the Skyrmion field ��r� depending on the dimension-less coordinate r; to continuation we define a new coor-dinate as x � r=r� and after integration rewrite (46) asMADM�r���

r�2�Mhair�r��

� r�2�r�

Z 11

�Nx2

2

�d�

dx

�2

�sin2��dx

� 1r�

Z 11

�N

�d�

dx

�2

�sin2�

2x2

��sin2��dx (49)

The behavior for ��x� can see from left panel in the

Fig. 14. of such reference. From equation (7.41) we seethe asymptotic behavior for ��x� as x! 1 (with a aconstant): � ax�2. Although this analysis shows thatMADM�r�� grows linearly with r� for r� � 1, we can notapply the limit r� ! 1 to (49) because the existence ofSkyrme’s black holes is limited to configurations withhorizon radius r� & r

max� ��� (here � � 4�G0f2 is the

coupling constant of the theory and rmax� ��� is a maximalvalue depending of �) and � & �maxbh . The value of �

maxbh is

of the order of 0.0315. At the other hand, the existence ofsolitons is permitted for � & 0:0437. In the Einstein-Higgssystem at hand, r� seems to be limited by the precision ofthe shooting method. For large r�, the shooting parameter�� approaches a below limit ( 0:26 for the model ~P0:1of Fig. 10). It is unknown if this limit for the SHBHconfigurations is a fundamental limit or not. It turns theninteresting to construct a similar model based in theisolated-horizon formalism for the Einstein-Skyrme sys-tem to compare similarities and differences with the modelpresented here [17].

[17] A. Nielsen (to be published).[18] P. Bizoń and T. Chmaj, Phys. Rev. D 61, 067501 (2000).[19] M. Alcubierre, J. A. Gonzalez, and M. Salgado, Phys. Rev.

D 70, 064016 (2004).[20] Y. Brihaye and B. Hartmann, gr-qc/0503102.

CORICHI, NUCAMENDI, AND SALGADO PHYSICAL REVIEW D 73, 084002 (2006)

084002-12