Einstein-Maxwell-scalar black holes: classes of solutions, dyons and extremality
aa r X i v : . [ h e p - t h ] M a y Preprint typeset in JHEP style - HYPER VERSION
Einstein-Maxwell-scalar black holes:classes of solutions, dyons and extremality
D. Astefanesei † , C. Herdeiro ‡ , A. Pombo ⋆ and E. Radu ⋆ † Pontificia Universidad Cat´olica de Valpara´ıso, Instituto de F´ısica ,Av. Brasil 2950, Valpara´ıso, Chile ‡ CENTRA, Departamento de F´ısica, Instituto Superior T´ecnico - IST,Universidade de Lisboa - UL, Avenida Rovisco Pais 1, 1049 Lisboa, Portugal ⋆ Departamento de F´ısica da Universidade de Aveiro and CIDMA,Campus de Santiago, 3810-183 Aveiro, Portugal
Abstract:
Spherical black hole (BH) solutions in Einstein-Maxwell-scalar (EMS) modelswherein the scalar field is non-minimally coupled to the Maxwell invariant by some couplingfunction are discussed. We suggest a classification for these models into two classes, basedon the properties of the coupling function, which, in particular, allow, or not, the Reissner-Nordstr¨om (RN) BH solution of electrovacuum to solve a given model. Then, a comparativeanalysis of two illustrative families of solutions, one belonging to each class is performed: dilatonic versus scalarised
BHs. By including magnetic charge, that is considering dyons,we show that scalarised BHs can have a smooth extremal limit, unlike purely electric ormagnetic solutions. In particular, we study this extremal limit using the entropy functionformalism, which provides insight on why both charges are necessary for extremal solutionsto exist.
Keywords: black holes, numerical solutions, attractors. ontents
1. Introduction 22. The EMS model 4
3. Non-extremal black holes 9
4. Extremal BHs 16
5. Discussion 20A. Exact solutions with a linear coupling 22
A.1 Purely electric BHs 22A.2 Dyonic BHs 23A.2.1 α = 1 23A.2.2 α = √ . Introduction Einstein-Maxwell (EM) theory, a.k.a. electrovacuum, is the quintessential source-free, gravita-tional relativistic field theory. Its static physical black holes (BHs) belong to the 3-parameterReissner-Nordstr¨om (RN) family, described by mass M , electric Q and magnetic P charges.These BHs are perturbatively stable [1, 2], and, for given M , can only sustain charges ( P, Q )if p Q + P M . When the equality holds, the extremal limit is attained. Extremal RNBHs are special. They are non-singular spacetimes, on and outside a degenerate and C ∞ smooth event horizon, that: (i) have a vanishing Hawking temperature and are BPS statesthat possess Killing spinors when embedded in supergravity [3]; (ii) have a near horizon ge-ometry which is, itself, a solution of the EM theory [4] – the Robinson-Bertotti ( AdS × S )vacuum [5, 6]; and (iii) allow a no-force condition and a multi-BH generalisation, describedby the Majumdar-Papapetrou metrics [7–9].A simple and natural generalisation of the EM theory is to consider an additional dy-namical real scalar field, with a standard kinetic term. A variety of such EM-scalar (EMS)models are possible, depending on the way the scalar field couples to the Maxwell field. Remarkably, if the scalar field is minimally coupled to the Maxwell, no new charged BHsolutions are possible, beyond RN, even if the scalar field is allowed to have a non-negativeself-interactions potential [10]. In these conditions, charged BHs cannot have scalar hair .Quite different possibilities, however, arise if a non-minimal coupling between the scalar andMaxwell field is allowed. This is the case we shall be interested in this paper.The first such non-minimally coupled EMS model emerged in the pioneering unificationtheory of Kaluza [11] and Klein [12], soon after Einstein constructed General Relativity(GR) [13]. These EMS models turned out to be ubiquitous in the four dimensional descriptionof higher dimensional GR-inspired theories [14], as well as in supergravity, see e.g. [15]. In theformer, as well as in the latter with a higher dimensional origin, the scalar field describes howthe extra dimension(s) dilate along the four dimensional spacetime, being dubbed dilaton .The dilaton has a specific non-minimal coupling with the Maxwell term in the EMS action.This coupling prevents EM theory to be a consistent truncation of this class of EM-dilatonmodels. In particular, the RN solution of EM theory does not solve these EM-dilaton models.Instead, new charged BHs with a non-trivial scalar field profile exist [16,17], which are knownin closed analytic form and that present RN-unlike features. For instance, the BH charge tomass ratio can exceed unity (see, e.g. [18]). As another example, there are no extremal BHswith a regular horizon in the purely electric (or purely magnetic) case. These limiting solutionsbecome naked singularities , a sharp contrast with the physically interesting extremal RNBH. Nonetheless, dilatonic BHs provided an example of asymptotically flat charged BHs withscalar hair [22], albeit of secondary type [23]. Herein we shall always consider that the scalar field is minimally coupled to gravity. Asymptotically flat, purely electric BHs exist also in EMS models with a non-trivial scalar potential,explicit solutions being reported in [19, 20]. Since in this case there are two different terms that source thescalar field (a self-interaction potential and the term coming from the non-trivial coupling with the electricfield), a balance is possible, which allows for a well defined extremal limit [21]. – 2 –nce embedded in string theory, the dilaton φ controls the string coupling, which isrelated to the vacuum expectation of the asymptotic value of the dilaton, g s = e h φ ∞ i . There-fore, a consistent analysis of hairy BHs in string theory should consider a dynamical dilatonwhose asymptotic value can vary [24] (see, also, [25] for a resolution of the appearance of thescalar charges in the first law of thermodynamics). This need, however, is mitigated by theattractor mechanism [26, 27]: the near horizon data (particularly, the entropy) of extremalBHs is independent of the asymptotic values of the moduli. The mechanism is based ona simple physical intuition; when the temperature vanishes, there is a symmetry enhancednear horizon geometry: AdS × S . The infinite long throat of AdS yields the decouplingbetween the physics at the boundary from the physics at the extremal horizon [28]. A similardecoupling plays a central role in the AdS/CF T duality (see, e.g. [29, 30]).EMS models with more generic non-minimal couplings (than the dilatonic one) betweenthe scalar field and the Maxwell term are also of interest. For example, such models wereconsidered in cosmological inflationary scenarios [31,32]. In the context of BHs, it was recentlyrealised that a family of couplings can trigger a spontaneous scalarisation of the RN BH [33,34]. In this class of EMS models, unlike the aforementioned dilatonic models, the RN BHis a solution. For sufficiently large charge to mass ratio, however, the RN BH becomesunstable against scalar perturbations and dynamically grows a scalar field profile; it becomesenergetically favourable to scalarise. The hair growth stalls due to non-linear effects leading toa scalarised BH (to be distinguished from dilatonic BH). The fundamental scalarised chargedBHs, which are the ones formed dynamically are, moreover, perturbatively stable [38–40]and therefore represent the endpoint of the non-linear evolution of the unstable RN BHs.Consequently, these scalarised BHs are an example of dynamically grown scalar hair.The scalarised BHs studied up to now contain only electric charge. They possess noextremal limit. Rather, a critical solution is attained for the maximal charge a BH cansupport, which (numerical evidence suggests) is singular. This parallels the status of dilatonicBHs. For the latter, however, the introduction of an additional magnetic charge leads todyonic BHs with an extremal (non-singular) limit, which have been constructed for specificcouplings [41–43]. Given the importance of extremal solutions, it is interesting to inquirewhich are the properties of the family of dyonic scalarised BHs and, in particular, of theirextremal limit.In fact, the considerations above suggest a comparison between dilatonic and scalarisedBHs can be instructive. The purpose of this paper is to perform such a comparison, forthe canonical dilatonic coupling and the reference model of scalarised solutions introducedin [33]. Our results, within this comparative study, include: ( i ) the introduction of a generalframework to study EMS for any scalar non-minimal coupling; ( ii ) the first study of dyonicscalarised BHs; ( iii ) establishing that extremal scalarised BHs indeed exist (only) when bothelectric and magnetic charges are present; and ( iv ) the study of the corresponding near horizongeometries via the attractor and entropy function formalism of [44–46].This paper is organised as follows. In Section 2 we present the EMS models and proposea classification of the BH solutions, based on the behaviour of the coupling function. We also– 3 –erive the zero mode of the RN BHs for the models that allow BH scalarisation. Section 3contains a discussion of the non-extremal BHs for both two classes of solutions (dilatonicand scalarised). In Section 4, we study the extremal BHs toghether with the correspondingnear horizon geometries, using the attractors formalism. We conclude in Section 5 with adiscussion and some further remarks. The Appendix contains a brief review of the knownexact solutions, all of which occur for a dilatonic coupling.
2. The EMS model
The EMS family of models is defined by the following action (we set c = G = 4 πǫ = 1) S = 116 π Z d x √− g ( R − ∂ µ φ∂ µ φ − f ( φ ) F µν F µν ) , (2.1)where R is the Ricci scalar, F µν = ∂ µ A ν − ∂ ν A µ is the Maxwell field and φ is the scalar field.The coupling function f ( φ ) governs the non-minimal coupling of φ to the electromagnetic field.From the outset we are excluding an axion-type coupling of the scalar to the electromagneticfield, as well as any sort of self-interaction of the scalar field. The model may, of course, begeneralised in these directions.The field equations obtained by varying the above action principle with respect to thefield variables g µν , φ and A µ are R µν − Rg µν = 2 (cid:20) ∂ µ φ∂ ν φ − g µν ∂ ρ φ∂ ρ φ + f ( φ ) (cid:18) F µρ F ρν − g µν F ρσ F ρσ (cid:19)(cid:21) , (2.2)1 √− g ∂ µ ( √− g∂ µ φ ) = 14 df ( φ ) dφ F ρσ F ρσ , (2.3) ∂ µ ( √− gf ( φ ) F µν ) = 0 . (2.4)An ansatz suitable to address both the (generic) asymptotically flat solutions and theRobinson-Bertotti (near horizon) geometries reads ds = − a ( r ) dt + b ( r ) ( dθ + sin θdϕ ) + c ( r ) dr . (2.5)The gauge 4-potential ansatz compatible with the symmetries of (2.5) contains an electricpotential V ( r ) and a magnetic term, A = V ( r ) dt + P cos θdϕ , (2.6)where P =constant is the magnetic charge. The scalar field is a function of r only, φ ≡ φ ( r ).The Maxwell equation (2.4) yields a first integral V ′ = acb Qf ( φ ) , (2.7)– 4 –here Q =constant is the electric charge (measured at infinity), and henceforth a prime denotesa derivative w.r.t. the radial coordinate r .The equations of motion (2.2)-(2.4) are invariant under the electro-magnetic duality trans-formation { P → Q, Q → P } and f ( φ ) → /f ( φ ) . (2.8)In what follows, we shall assume, without any loss of generality, that both Q and P arepositive and that Q > P , (2.9)such that for scalarised BHs, the (electric) solutions in [33, 34] are recovered as P → The RN BH is a solution of (2.2)-(2.4) with f ( φ ) = 1, φ =constant and V ( r ) = − Qr , a ( r ) = 1 c ( r ) = 1 − Mr + Q + P r , b ( r ) = r , (2.10)where M is the BH ADM mass. For a more general f ( φ ) the RN BH may or may notsolve (2.2)-(2.4). This naturally leads to two classes of EMS models. (Note that, in thisclassification, we assume, without any loss of generality, that the scalar field vanishes asymp-totically, φ ( r ) r →∞ −→ Class I or dilatonic-type.
In this class of EMS models φ ( r ) = 0 does not solve the fieldequations. Thus RN is not a solution. Then, the scalar field equation (2.3) impliesthat f ,φ (0) ≡ df ( φ ) dφ (cid:12)(cid:12)(cid:12) φ =0 = 0 . (2.11)A representative example of coupling for this class is the standard dilatonic coupling f ( φ ) = e αφ , (2.12)in which case we refer to φ is a dilaton field. The arbitrary nonzero constant α is taken tobe positive without any loss of generality. Indeed, the solutions remain invariant underthe simultaneous sign change ( α, φ ) → − ( α, φ ). Thus, flipping the sign of α simplycorresponds to flipping the sign of φ . The coupling (2.12) appears naturally in Kaluza-Klein models and supergravity/low-energy string theory models. Three reference valuesfor the coupling constant α in (2.12) are: α = 0 (EM theory) α = 1 (low energy strings) α = √ . (2.13) There is an exceptional case: if Q = P , φ = 0 solves this class, so that the dyonic, equal charges RN BHis a solution. – 5 –ome exact, closed form BH solutions of (2.2)-(2.4) with (2.12) are known and pre-sented in Appendix A. Other exact solution examples in this class (with a non-dilatoniccoupling) are given in [47]. Class II or scalarised-type.
In this class of EMS models φ ( r ) = 0 solves the field equa-tions. Thus RN is a solution. This demands that f ,φ (0) ≡ df ( φ ) dφ (cid:12)(cid:12)(cid:12) φ =0 = 0 . (2.14)This condition is naturally implemented, for instance, if one requires the model to be Z -invariant under φ → − φ . The RN solution, however, is (in general) not unique.These EMS models may contain a second set of BH solutions, with a nontrivial scalarfield profile – the scalarised BHs . Below some conditions for this to occur are discussed.Such second set of BH solutions may, or may not, continuously connect with RN BHs.This leads to two subclasses. Subclass IIA or scalarised-connected-type.
In this subclass of EMS models, thescalarised BHs bifurcate from RN BHs, and reduce to the latter for φ = 0. Thisbifurcation moreover, may be associated to a tachyonic instability, against scalarperturbations, of the RN BH. Considering a small- φ expansion of the couplingfunction f ( φ ) = f (0) + 12 d f ( φ ) dφ (cid:12)(cid:12)(cid:12) φ =0 + . . . , (2.15)equation (2.3) linearised for small- φ reads:( (cid:3) − µ ) φ = 0 , where µ = F µν F µν d f ( φ ) dφ (cid:12)(cid:12)(cid:12) φ =0 . (2.16)The instability arises if µ <
0, which in particular requires f ,φφ (0) ≡ d f ( φ ) dφ (cid:12)(cid:12)(cid:12) φ =0 = 0 , (2.17)and with the opposite sign of F µν F µν . A reference example of a coupling functionin this subclass, which we consider in this work is [33] f ( φ ) = e αφ , (2.18)a case which is also relevant in cosmology [31,32]. Depending on the coupling, thissubclass could also contain another family of disconnected scalarised BHs, akin tothe ones of class IIB below. Subclass IIB or scalarised-disconnected-type.
In this subclass of EMS models,the scalarised BHs do not bifurcate from RN BHs, and do not reduce to the latter– 6 –or φ = 0. This is the case if there is no tachyonic instability of RN, for which asufficient (but not necessary) condition is that f ,φφ (0) ≡ d f ( φ ) dφ (cid:12)(cid:12)(cid:12) φ =0 = 0 . (2.19)We shall not address further this case in this paper (which, moreover, was notconsidered yet in the literature), but a representative coupling would be, say, f ( φ ) = 1 + αφ .Condition (2.14) guarantees RN is a solution. But it does not guarantee the existenceof scalarised BHs. In the case of purely electric (or magnetic) BHs, two Bekenstein-typeidentities can be derived, which put some constraints on f ( φ ) so that scalarised solutionsexist. These can be derived as follows.To derive the first identity, the scalar field equation (2.3) is multiplied by f ,φ and inte-grated over a spacetime volume. Integrating by parts and discarding the boundary terms, byvirtue of the horizon properties and asymptotic flatness, one obtains Z d x √− g f ,φφ ∂ µ φ∂ µ φ + f ,φ F ! = 0 . (2.20)The sign of neither term is fixed, in general, and specific considerations are required. Forinstance, a purely electric field has F <
0; this implies f ,φφ > , (2.21)must hold for some range of the radial coordinate r , otherwise the two terms of the integrandin (2.20) will have always the same sign, making the identity only possible for φ = 0.A second identity is found by multiplying (2.3) by φ , which results, via a similar proce-dure, in Z d x √− g (cid:18) ∂ µ φ∂ µ φ + φf ,φ F (cid:19) = 0 . (2.22)This imples that for a purely electric field the potential should satisfy the condition φf ,φ > , (2.23)for some range of r . Similar arguments hold for purely magnetic solutions, which implies f ,φφ < φf ,φ <
0, respectively. No such results can be established in the generic dyoniccase, since the sign of F = F µν F µν is not determined, a priori . Class IIA of EMS models is particularly interesting because it accommodates the dynamicalphenomenon of spontaneous scalarisation [33,34] (see also [35], [36], [37] for earlier discussions– 7 –f charged BHs scalarisation in different models). At the linear level this is manifest in thetachyonic instability (2.16). For a dyonic RN BHs (2.10), F µν F µν = − Q − P ) /r . Thus,under the assumption (2.9) a tachyonic instability requires f ,φφ (0) >
0. Let us study thisinstability, generalising the analysis in [33, 34] for the dyonic RN case.Assuming separation of variables, φ = Y ℓm ( θ, ϕ ) U ( r ) , (2.24)where Y ℓm are the real spherical harmonics and ℓ, m are the associated quantum numbers, i.e. ℓ = 0 , , . . . and − ℓ m ℓ , the equation for the radial amplitude U ( r ) reads (cid:18) r U ′ c ( r ) (cid:19) ′ = (cid:20) ℓ ( ℓ + 1) + ( P − Q )2 r f ,φφ (0) (cid:21) U , (2.25)where c ( r ) is given by (2.10). Observe that the term µ = ( P − Q ) f ,φφ (0) /r acts asthe effective mass for the perturbations and the condition µ < f ,φφ (0) >
0, asdiscussed above.For spherically symmetric perturbations ℓ = 0, and eq. (2.25) possesses an exact solutionwhich is regular on and outside the horizon and vanishes at infinity U ( r ) = P u (cid:18) Q − P )( r − r H ) r H + P − Q (cid:19) , where u = 12 (cid:18)q − f ,φφ (0) − (cid:19) , (2.26)where r H is the event horizon radial coordinate and P u is a Legendre function. This solutionis physical for f ,φφ (0) > /
2, a condition which, for the coupling function (2.18) implies α > / . (2.27)For generic parameters ( f ,φφ (0) , Q, P, r H ), the function U ( r ) approaches a constant nonzero value as r → ∞ , U ( r ) → U ∞ = F " − p − f ,φφ (0)2 , p − f ,φφ (0)2 , Q − P Q − P − r H + O (cid:18) r (cid:19) . (2.28)Thus finding the ℓ = 0 unstable mode of a RN BH with given P, Q, M reduces to a study ofthe zeros of the hypergeometric function F , so that U ∞ = 0.The value of U ∞ for the coupling function (2.18) and an illustrative value of α is shownin Fig. 1 (left panel). Therein, the integer n labels the number of nodes of the function U ( r )when the correct boundary condition at infinity is met: given a RN background, the solutionswith U ∞ = 0 are found for a discrete sequence α n , each one corresponding to a different nodenumber, cf. Fig. 1 (right panel). To simplify the picture, the results in Fig. 1 correspond to P = 0; a similar pattern holds also in the dyonic case. The limit P = 0 of this solution has been discussed in [34]. – 8 – U ∞ Q/M α =36n=0 n=1 n=2 l=0 -0.5 0 0.5 1 0 0.25 0.5 0.75 1 U (r) H /r P=0 Q/M=0.975n=0 ( α =0.75) n=1 ( α =3.32)n=2 ( α =7.92) Figure 1: (Left panel) The asymptotic value U ∞ of the zero-mode amplitude U for α = 36 as afunction of the charge to mass ratio of a RN BH. An infinite set of configurations with U ∞ = 0 exist,labelled by n , the number of nodes of U ( r ). (Right panel) The profiles of three zero mode amplitudes U ( r ) with a different node number, for a given RN background. The solution (2.26) yields a dyonic RN BH surrounded by a vanishingly small scalar field.The set of such RN BHs (varying α ) constitutes the existence line , the branching line betweenthe RN and scalarised BHs. The latter are the non-linear continuation of the (infinitesimallysmall) scalar clouds, i.e. the solutions of eq. (2.25). As remarked above, these clouds arelabelled by three integer numbers ( ℓ, m, n ). In what follows, however, we shall restrict ourstudy to the simplest case of nodeless, spherically symmetric configurations. More generalconfigurations in the purely electric case have been discussed in [33, 34].
3. Non-extremal black holes
Let us now construct, numerically, the non-linear BH solutions, for both class I and IIA,starting with the non-extremal BHs.
In the numerical study of the solutions, it is convenient to work in Schwarzschild-like co-ordinates, with a metric gauge choice b ( r ) = r , a ( r ) = e − δ ( r ) N ( r ) and c ( r ) = 1 /N ( r ).Then, (2.5) becomes: ds = − e − δ ( r ) N ( r ) dt + dr N ( r ) + r ( dθ + sin θdϕ ) , where N ( r ) ≡ − m ( r ) r . (3.1)The function m ( r ) corresponding to a local mass function, known as the Misner-Sharpmass [48]. – 9 –he equations of motion (2.2) and (2.3), together with the first integral (2.7) implies thatthe functions m , σ, φ solve the ordinary differential equations m ′ = 12 r N φ ′ + 12 r (cid:18) Q f ( φ ) + f ( φ ) P (cid:19) , (3.2) δ ′ + rφ ′ = 0 , (3.3)( e − δ r N φ ′ ) ′ + e − δ r f ( φ ) df ( φ ) dφ (cid:18) Q f ( φ ) − f ( φ ) P (cid:19) = 0 , (3.4)which can also be derived from the following effective action: S eff = Z dtdr (cid:20) e − δ m ′ − e − δ r N φ ′ + f ( φ )2 (cid:18) r e δ V ′ − e − δ r P (cid:19)(cid:21) , (3.5)while V ′ = e − δ Q/ ( r f ( φ )). The Einstein equations also yield a constraint equation,12 N ′′ − N δ ′′ + N ′ (cid:18) r − δ ′ (cid:19) + N δ ′ (cid:18) δ ′ − r (cid:19) + N φ ′ − r (cid:20) Q f ( φ ) + P f ( φ ) (cid:21) = 0 , (3.6)which can be shown to be a linear combination of equations (3.2)-(3.4) together with the firstderivatives of (3.2)-(3.3). It is also of interest to observe that equations (3.2)-(3.4) possessthe first integral e − δ r N " N δ ′ r − N (cid:18) N ′ N − δ ′ (cid:19) + 1 r (cid:18) Q f ( φ ) + P f ( φ ) (cid:19) = u , (3.7)where the constant u is fixed by the asymptotics.To assess possible singular behaviours we remark that the expression of the Ricci andKretschmann scalars for the line-element (3.1) read: R = N ′ r (3 rδ ′ −
4) + 2 r (cid:8) N (cid:2) r δ ′′ − (1 − rδ ′ ) (cid:3)(cid:9) − N ′′ , (3.8) K = 4 r (1 − N ) + 2 r (cid:2) N ′ + ( N ′ − N δ ′ ) (cid:3) + (cid:2) N ′′ − δ ′ N ′ + 2 N ( δ ′ − δ ′′ ) (cid:3) . To construct BH solutions, we assume the existence of a horizon located at r = r H >
0. In itsexterior neighbourhood, one finds the following approximate solution, valid for non-extremalBHs: m ( r ) = r H m ( r − r H ) + . . . , δ ( r ) = δ + δ ( r − r H ) + . . . , (3.9) φ ( r ) = φ + φ ( r − r H ) + . . . , V ( r ) = v ( r − r H ) + . . . , – 10 –here out of the six parameters, m , δ , δ , φ , φ , v , only two are essential, φ and δ , theremaining being determined in terms of these and the global charges as: m = 12 r H (cid:20) Q f ( φ ) + f ( φ ) P (cid:21) , φ = df ( φ ) dφ (cid:12)(cid:12)(cid:12)(cid:12) φ r H h Q f ( φ ) − f ( φ ) P ih Q f ( φ ) + f ( φ ) P − r H i ,δ = − r H φ , v = e − δ Qr H f ( φ ) . (3.10)Note that a similar result holds when considering higher order terms in the approximatesolution (3.9).For large r , one finds the following asymptotic expansions: m ( r ) = M − Q + P + Q s r + . . . , φ ( r ) = Q s r + . . . , V ( r ) = Φ e − Qr + . . . , δ ( r ) = Q s r + . . . . (3.11)The essential parameters introduced in the expansion at infinity (3.11) are the ADM mass M ,electric and magnetic charges Q, P , electrostatic potential at infinity Φ e and scalar ’charge’ Q s . The Ricci scalar (3.8) vanishes as r → r H , while the Kretschmann scalar (3.9) reads K = 12 r H ( − r H (cid:20) Q f ( φ ) + f ( φ ) P (cid:21) + 53 r H (cid:20) Q f ( φ ) + f ( φ ) P (cid:21) ) + O ( r − r H ) . (3.12) Two horizon physical quantities of interest are the Hawking temperature and horizon area T H = 14 π N ′ ( r H ) e − δ , A H = 4 πr H ; (3.13)these, together with the horizon scalar field value φ compose the relevant horizon data.The Smarr-like relation [49] for this family of models turns out to have no explicit imprintof the scalar hair, M = 12 T H A H + Φ e Q + Φ m P , (3.14)where we have defined a ‘magnetic’ potential as Φ m ≡ R ∞ r H dre − δ f ( φ ) P/r . One can thencompute a first law of BH thermodynamics for EMS BHs, that reads: dM = 14 T H dA H + Φ e dQ + Φ m dP . (3.15)A non-linear Smarr relation ( i.e. mass formula) can also be established for this family ofmodels, M + Q s = Q + P + 14 A H T H , (3.16)– 11 –hich is derived by evaluating the expression of the first integral (3.7) at the horizon and atinfinity, for the approximate form of the solutions (3.9) and (3.11), respectively.Finally, one can prove that the solutions satisfy the virial identity, which is obtained bya Derrick-type [50] scaling argument, see e.g. [23] Z ∞ r H dr (cid:26) e − δ φ ′ (cid:20) r H r (cid:16) mr − (cid:17)(cid:21)(cid:27) = Z ∞ r H dr (cid:26) e − δ (cid:18) − r H r (cid:19) r (cid:20) Q f ( φ ) + f ( φ ) P (cid:21)(cid:27) . (3.17)One can show that 1 + r H r ( mr − > i.e. the left hand side integrand, is strictly posi-tive. Thus, the virial identity shows that a nontrivial scalar field requires a nonzero elec-tric/magnetic charge so that the right hand side is nonzero.The model possesses the scaling symmetry r → λr , ( P, Q ) → λ ( P, Q ) , (3.18)where λ > e.g. , M → λM , while the coupling function f ( φ ) is unchanged. Thus, for a physicaldiscussion, we consider quantities which are invariant under the transformation (3.18). Con-sequently, we introduce the standard reduced quantities q ≡ p Q + P M , a H ≡ A H πM , t H ≡ πT H M . (3.19)For example, dyonic RN BHs have closed expressions for a H , t H : a (RN) H = 14 (1 + p − q ) , t (RN) H = 4 p − q (1 + p − q ) . (3.20)In Appendix A we exhibit the corresponding expressions for other dilatonic BHs known inclosed analytic form, which are class I solutions.The generic dilatonic dyonic solutions are not known in closed form, which hold also forall scalarised BHs. These solutions are found numerically, by matching the asymptotics (3.9),(3.11). Equations (3.2)-(3.4) are solved by using a standard Runge-Kutta ODE solver andimplementing a shooting method in terms of the parameters φ , δ . Let us start with our reference class I solutions. The behaviour of the dilatonic BHs with any α > α = 1 is a somewhat special point that separates the family intotwo subsets with respect to the behaviour of some physical quantities. This can be seen fromthe study of the exact solutions in Appendix A. For any given α , the branch of dilatonic BHsbifurcates from the Schwarzschild BH ( q = 0), rather than RN BHs, and ends in a criticalsolution which is approached for a certain maximal qq (D)max = p α , (3.21)– 12 – lass I (dilatonic) Class IIA (scalarised) a H q P=0 dilatonic model α =0 α =0.5 α =1 α =2 α =5 α =20 0 0.25 0.5 0.75 1 0 0.5 1 1.5 2 a H q α =0.375 α =2 α =5 α =20 α =77 α =288P=0 scalarised model RN BHs 0 0.5 1 1.5 2 0 0.5 1 1.5 2 t H q P=0 dilatonic model α =0 α =2 α =5 α =1 α =0.95 α =0.5 0 0.5 1 1.5 2 0 0.5 1 1.5 2 t H q P=0 scalarised model
RN BHs α =0.375 α =2 α =5 α =20 α =288 Figure 2:
Reduced area a H (top panels) and reduced temperature t H (bottom panels) vs. reducedcharge q for dilatonic (left panels) and scalarised solutions (right panels). All solutions have P = 0.The blue lines are the set of RN BHs ( φ = 0). The red lines are sequences of BHs with a nontrivialscalar field for a given α . Different sequences are presented, for a range of values of α . The black dotsindicate the RN solutions from which the scalarised BHs bifurcate. where the superscript ‘D’ refers to dilatonic. The critical solution has, for any α >
0, a singularhorizon, as one can see by evaluating the expression (3.12). The reduced temperature t H , onthe other hand, goes to zero for α < α >
1. The solutions with α = 1 have t H = 1. These features can be seen in Fig. 2 (left panels), where the behaviour of a H , t H vs.q are illustrated for several values of α .Let us now turn to our reference class IIA solutions. For the purely electric case thesehave been constructed in [33, 34]. Let us briefly review their basic properties, emphasizinga comparison with class I solutions. Given a value of the coupling constant α > /
8, thespherically symmetric scalarised BHs bifurcate from the corresponding RN BH, with a given q = q ( α ) = 0, as discussed above. Keeping constant the parameter α , this branch has a finiteextent, ending again in a critical configuration. This limiting solution possess a singularhorizon, as found when evaluating the Kretschmann scalar (3.12). The horizon area tendsto zero as the critical solution is approached and the temperature diverges, while the mass– 13 –nd scalar charge remain finite. This behaviour parallels that of the dilatonic solutions with α >
1. In the region of the parameter space wherein scalarised and RN BHs exist for thesame q , one always finds that the scalarised solutions are entropically favoured over the RNBHs, as it is manifest from the top right panel of Fig. 2.The domain of existence of the purely electrically charged BHs of both types will beexhibited below in Fig. 5, where we also compare it with the dyonic case that we shalldiscuss next. This domain of existence, in an ( α, q )-diagram is bounded by two curves. Inthe dilatonic case, these curves correspond to the Schwarzschild BH and the line of criticalsolutions. In the scalarised case, these curves correspond to the aforementioned existence line- the set of RN solutions from which the scalarised BHs bifurcate - and, again, the line ofcritical solutions.Finally, we remark that, for both dilatonic and scalarised solutions, along any branchwith fixed α , the ratio q = p Q + P /M increases and becomes larger than unity at somestage. In this sense, overcharged BHs are possible, in contrast with the RN family.
The purely electric solutions above, for both classes discussed, possess generalisations with anonzero magnetic charge. The profile functions of illustrative dyonic BHs are shown in Fig. 3,for both the dilatonic and scalarised cases. log r eve n t ho r i z on m(r) V(r) φ (r)500x δ (r) P=0.5 Q=1 α =4 r h =1.3 log r eve n t ho r i z on m(r)V(r) φ (r)100x δ (r) P=0.105 Q=0.182 α =66 r h Figure 3:
Examples of dyonic BHs radial profile functions for a dilatonic (left panel) and a scalarised(right panel) BH.
Dyonic BHs preserve some, but not all, of the qualitative characteristics of the purelyelectric solutions. In the dilatonic case, the branch of solutions with a given α starts againfrom the Schwarzschild limiting solution (which has a H = 1, t H = 1 and q = 0) and ends ina limiting configuration with a H > t H = 0 and q = q max > lass I (dilatonic) Class IIA (scalarised) a H q P/Q=0.1 α =0 α =1 α = √ α =3 dilatonic model a H q α =1 α =5 α =15 scalarised model P/Q=0.1RN BHsextremal SBHs 0 0.25 0.5 0.75 1 1.25 0 0.5 1 1.5 2 t H q P/Q=0.1 α =0 α =1 α = √ α =3 dilatonic model t H q scalarised model α =1 α =5 α =15RN BHsextremal SBHs P/Q=0.1 Figure 4:
Same as Fig. 1 but now for dyonic BHs. The ratio between magnetic and electric chargesis
P/Q = 0 . Unlike the dilatonic solutions, which exist for arbitrarily small q for any α , scalarisedBHs with a given α exist for q > q min only. They bifurcate from a RN BH (with q >
0) andform a branch ending again on an extremal solution with vanishing horizon temperature andnonzero horizon area - Fig. 4 (right panels). As for purely electric solutions, for the same globalcharges
M, P, Q , the scalarised solutions are entropically preferred over the corresponding RNsolution.The domain of existence of the dyonic BHs is shown in Fig. 5 for several values of the ratio
P/Q and for both dilatonic and scalarised BHs. In particular, observe that in both cases, themaximal allowed value of q for BHs with a given α decreases as the ratio P/Q increases. Inother words, the domain of existence shrinks, as the magnetic charge is increased, for fixed Q . – 15 – q α extremal linescritical line P/Q=0 P/Q=0.1P/Q=0.05 q α RN black holes existence lineextremal linescritical line P/Q=0 P/Q=0.05P/Q=0.1P/Q=1
Figure 5:
Domain of existence of dilatonic BHs (left panel) and scalarised BHs (right panel) forseveral values of the ratio
P/Q .
4. Extremal BHs
To address extremal BHs one needs to impose a different near-horizon expansion to thatin (3.9), which accounts for the degenerate horizon. The leading order terms of the appropriateexpansion are: N ( r ) = N ( r − r H ) + . . . , δ ( r ) = δ + δ ( r − r H ) k − + . . . ,φ ( r ) = φ + φ c ( r − r H ) k + . . . , V ( r ) = v ( r − r H ) + . . . . (4.1)One can show that the next to leading order term in the expression of N ( r ) is O ( r − r H ) .It is convenient to take r H and φ as essential parameters. Then the field equations imply Q = r H p f ( φ ) √ , P = r H p f ( φ ) , N = 1 r H . (4.2)Consequently, given an expression of the coupling function f ( φ ), one can express the value ofthe scalar field at the horizon φ as a function of P, Q , by solving the equation f ( φ ) = QP , while r H = p P Q . (4.3)The expansion (4.1) contains two free parameter φ c and δ which are fixed by numerics, while δ , v are fixed as δ = − r H φ c k k − , v = e − δ Qr H f ( φ ) . (4.4)The power k in (4.1) is given by k = 12 − s (cid:18) f ′ ( φ ) f ( φ ) (cid:19) > , (4.5)– 16 –hich, generically, takes non-integer values. However, a non-integer k implies that a suffi-ciently higher order derivative of the curvature tensor will diverge as r → r H . A minimalrequirement for smoothness is that the metric functions N, δ and their first and second deriva-tives are finite as r → r H ; this yields the condition k > / . (4.6)On the other hand, for analytic solutions on the horizon (as extremal RN), the power k in theabove near horizon expansion (4.1), (4.1) should be an integer. This imposes the condition f ′ ( φ ) f ( φ ) = ± p p ( p + 1) , with p = 1 , , . . . . (4.7)For the dilatonic case, condition (4.7) translates to [51, 52] (see also [53]) α = r p ( p + 1)2 , (4.8)again with an integer p . For scalarised solutions with the coupling function (2.18) the condi-tion (4.7) becomes α = p ( p + 1)4 log( QP ) . (4.9)The extremal solutions share the far field asymptotics (3.11) with the non-extremal ones;moreover, the relations (3.14)-(3.16) hold also for T H = 0.We have constructed extremal solutions by using the same numerical approach as in thegeneric non-extremal case. The profile of the various functions resulting from the integrationare not particularly enlightening, resembling those in the non-extremal case and shall notbe shown here. But we would like to point out a peculiar feature of the extremal scalarisedBHs. There exists a (presumably) infinite family of solutions with the same horizon dataas specified by ( φ , r H ) (or, equivalently, ( P, Q )), labelled by their node-number n . This isillustrated in Fig. 6: the scalar field always starts at the same horizon value; however, thebulk profile is different. As expected for excited states, the mass of these solutions increaseswith n . We remark no excited configurations were found in the dilatonic case, which alwayshas n = 0. The numerical construction of the extremal BHs is a difficult numerical task. Let us nowprovide a different argument for the existence of the EMS extremal dyonic BHs: the existenceof an exact solution describing a Robinson-Bertotti vacuum, namely an
AdS × S spacetime.As for extremal RN BHs, we expect that this solution describes the neighbourhood of theevent horizon of an extremal scalarised BH with nonzero magnetic and electric charges. Aswe shall see, both charges are mandatory for the Robinson-Bertotti vacuum to exist with anon-trivial scalar field. – 17 – φ (r) h /r eve n t ho r i z on n=0n=1 n=2 P=0.3 Q=0.35 α =5 Figure 6:
A sequence of scalar field profiles starting with the same horizon data in a scalarised model.Each solution possesses a different node number.
To search for the Robinson-Bertotti vacuum we consider (2.5) with a ( r ) = v r , b ( r ) = v , c ( r ) = v /r , that is the line element ds = v (cid:18) − r dt + dr r (cid:19) + v ( dθ + sin θdϕ ) , (4.10)and the matter fields ansatz A = erdt + P cos θdϕ , φ = φ . (4.11)The constant parameters { v , v ; e, P, φ } satisfy a set of algebraic relations which result fromthe EMS equations (2.2)-(2.4). However, instead of attempting to solve these, we shall, inwhat follows, determine these parameters by using the formalism proposed in [44–46], thusby extremizing an entropy function . This approach allows us also to compute the BH entropyand to show that the solutions exhibit an attractor-type behaviour.Let us consider the Lagrangian density of the model, as read off from (2.1), evaluated forthe ansatz (4.10)-(4.11) and integrated over the angular coordinates, L = 116 π Z dθdϕ √− g (cid:0) R − ∇ φ ) − f ( φ ) F (cid:1) = 12 (cid:20) v − v + f ( φ ) (cid:18) e v v − P v v (cid:19)(cid:21) . (4.12)Then, following [44–46], we define the entropy function E by taking the Legendre transformof the above integral with respect to a parameter Q , E = 2 π ( eQ − L ) , (4.13)– 18 –here Q = ∂ E /∂e is the electric charge of the solutions. It follows as a consequence of theequations of motion that the constants { v , v ; e, φ } are solutions of the equations ∂ E ∂φ = 0 , ∂ E ∂v i = 0 , ∂ E ∂e = 0 , (4.14)or, explicitly, ∂ E ∂v = 0 ⇒ − (cid:18) v v e + 1 v P (cid:19) f ( φ ) = 0 , (4.15) ∂ E ∂v = 0 ⇒ − (cid:18) v e + v v P (cid:19) f ( φ ) = 0 , (4.16) ∂ E ∂φ = 0 ⇒ (cid:0) P v − e v (cid:1) dfdφ = 0 , (4.17) ∂ E ∂e = 0 ⇒ Q = e v v f ( φ ) . (4.18)The sum of (4.15) and (4.16) leads to the generic relation v = v . (4.19)Thus, the ‘radius’ of the AdS is always equal with the one of S in the metric (4.10). Then,the equation (4.18) becomes Q = ef ( φ ) . (4.20)Consequently, eq. (4.17) implies the existence of two different families of solutions: a) eq. (4.17) is solved if the coupling function obeys df /dφ = 0. Then, e and P are inde-pendent quantities and, from (4.15), v = v = ( e + P ) f ( φ ) . (4.21)This family of solutions is only possible in the scalarised case. In this case, df /dφ = 0,with φ = 0. Therefore, one obtains the near horizon geometry of the extremal RN BH,with a vanishing scalar field. b) eq. (4.17) is also solved if e = P , (4.20) ⇒ Q = P f ( φ ) , (4.22)and, from (4.15), v = v = 2 P f ( φ ) . (4.23)This family of solutions is possible for both the scalarised and dilatonic cases anddemands both Q, P to be non-vanishing.– 19 –he scalarisation mechanism is encoded in the existence of two different types of attractorsolutions in the scalarised EMS models. This contrasts with the case of the dilatonic coupling,for which condition (4.22) is mandatory and only one type of solutions exists, that requiresboth electric and magnetic charges to be present.It is straighforward to check that in both cases the entropy function, E , evaluated at theattractor critical point is given by one-quarter of the area of angular sector in (4.10), S = πv . (4.24)Finally, we remark that the correspondence of the above parameters with the ones in thenear horizon expansion of the extremal BHs in Section 4.1 is straightforward: v = r H , v = 1 /N . (4.25)
5. Discussion
In this paper we have investigated the properties of static electromagnetically charged BHswith a non-trivial scalar field profile in EMS models, which are described by (2.1). A naturalclassification of these EMS models arises from the standard RN BH of electrovacuum being,or not, a solution. This divides EMS models into two classes. Class I, or dilatonic-type, doesnot admit RN as a solution. We have illustrated this class by a well-known family of dilatonic
BHs that naturally emerge in the low energy limit of string theory, as well as in Kaluza-Kleintheories. Class II, or scalarised-type, admits RN as a solution. The RN BH may, or may not,be continuously connected to the new BHs with a scalar field profile, naturally leading to twosubclasses. In class IIA RN is continuously connected to the new BHs. This class containsthe models wherein spontaneous scalarisation of the RN BH occurs [33], dynamically leadingto the new scalarised
BHs. We have illustrated this class by a particular choice of couplingfunction, introduced in [33] in this context. In class IIB, RN is not continously connected tothe new BHs and the RN BH is not unstable against scalarisation.One of the motivations for this work was to understand the effect of a magnetic chargein the EMS BHs. In the well known dilatonic case, dyonic BHs have a regular extremallimit, whereas purely electrically (or magnetically) charged ones do not; the latter becomesingular, approaching a critical solution when endowed with the maximal possible charge fora given mass. Given the special features of smooth extremal solutions, it is of interest tounderstand the status of these solutions in the generic EMS case, since for purely electicscalarised BHs maximal charge led to critical, rather than extremal, solutions [33, 34]. Herewe have shown that for scalarised BHs the conclusion is similar to dilatonic BHs (within acertain coupling regime) in this respect: dyonic BHs can have a regular extremal limit. Ouranalysis also allows constructing such dyonic extremal solutions for arbitrary coupling in thedilatonic case, since solutions where only known (in analytic closed form) for some particularvalues of the coupling. Morover, despite the defining difference in the two classes of solutions,– 20 –ig. 2 and 4 show that these two classes, for the illustrative families, present similar trendsin the behaviour of physical quantities.As evidence for the existence of dyonic extremal scalarised BHs, we have made use of thefact one expects such solutions to have a near-horizon geometry which is, itself, a solution ofthe field equations. Both for RN and Kerr extremal BHs (when T H = 0), the near horizongeometry has an enhanced symmetry that contains an AdS geometry (for Kerr, there exists anon-trivial fibration of an S on AdS in the near horizon geometry). It was proven in [44,45]that the existence of AdS factor is, in fact, at the basis of the attractor mechanism for ex-tremal BHs rather than supersymmetry [26, 27]. In string theory, the attractor mechanismprovides a non-renormalization theorem for the matching of statistical and thermodynamicentropies of extremal BHs [54] (see, also, section 5 of [55]). Here, the attractor mechanismprovides a clear and simple explanation of why the extremal limit is a naked singularity forsolutions with a single charge and a smooth geometry for dyonic BHs. Besides enabling apartial analytical understanding of the extremal solutions, analysing the near horizon geome-try provides an insight on how scalarisation leaves a trace at the level of attractors, allowingtwo families of near horizon geometries.Let us close considering some future research. It would be interesting to motivate classII models from a more fundamental viewpoint. In this respect, we remark that (2.1) maybe viewed as a member of a more general family of low energy string theory actions (see,e.g., [56]). For example, in four dimensions, the effective string theory can be described by N = 2 supergravity (and its deformations [57]), with the generic bosonic Lagrangian density L = − R h i ¯ ∂ µ z i ∂ µ ¯ z ¯ + 14 F ΛΣ ( z, ¯ z ) F Λ µν F Σ µν + 18 e D R ΛΣ ( z, ¯ z ) ǫ µνρσ F Λ µν F Σ ρσ − V ( z, ¯ z ) . (5.1)The model possesses n s complex scalars z i , i = 1 , . . . , n s , coupled to the vector fields F Λ µν in a non-minimal way through the real symmetric matrices F ΛΣ ( z, ¯ z ), R ΛΣ ( z, ¯ z ) and spana special K¨ahler manifold with the metric h i ¯ . The scalar potential V ( z, ¯ z ) originates fromelectric-magnetic Fayet-Illiopulos terms; a consistent truncation with only one scalar fieldand a concrete potential was presented in [58] (see, also, [59–61]). Despite the existence ofa potential for the scalar field, one can consider a situation when the effective cosmologicalconstant vanishes at the boundary even if the self-interaction in the bulk does not [62] –exact asymptotically flat hairy black hole solutions with a non-trivial dilaton potential wereobtained in [19]. We observe that the moduli metric and coupling with the gauge field canbe non-trivial. Depending of their form, the RN BH may, or may not, be obtained as asolution of the theory. For the dilatonic coupling (2.12) and a trivial moduli metric, the limit φ → not provide a consistent truncation and so RN BH is not a solution of the theory.However, this is not necessarily the case for any metric h i ¯ . Finding a concrete realisation ofclass II models in this context would be very promising.Amongst the several extension of the discussion herein, including in (2.1) an axion-typeterm could be interesting, due to the high energy physics motivation for axions. Another– 21 –bvious extension would be the consideration of solutions with less symmetry, either rotatingsolutions or solutions connected to zero modes with ℓ = 0.Finally, as a speculation, one can notice some analogy of the scalarised BHs and the AdSholographic duals of superconductors (the s − wave case) [63]. The general mechanism appearsto be the following: for both asymptotics, the RN BH remains a solution of the full model.However, for some range of the parameters, the non-trivial coupling of the scalar field withthe Maxwell field gives a tachyonic mass for the vacuum scalar perturbations around the RNBH, with the appearance of a scalar condensate. This implies the occurrence of a branch ofscalarised BHs, which are generically thermodynamically favoured over the RN configurations.It would be interesting to further pursue this apparent parallelism in the Minkowskian case,and to investigate the possible relevance of these aspects in providing ‘dual’ descriptions tophenomena observed in condensed matter physics. Acknowledgements
The work of D.A. has been funded by the Fondecyt Regular Grant 1161418. The work of C.H.,A.P., and E.R. is supported by the Funda¸c˜ao para a Ciˆencia e a Tecnologia (FCT) projectUID/MAT/04106/2019 (CIDMA), by CENTRA (FCT) project UID/FIS/00099/2013, by na-tional funds (OE), through FCT, I.P., in the scope of the framework contract foreseen in thenumbers 4, 5 and 6 of the article 23, of the Decree-Law 57/2016, of August 29, changed by Law57/2017, of July 19. We acknowledge support from the project PTDC/FIS-OUT/28407/2017.This work has also been supported by the European Union’s Horizon 2020 research and in-novation (RISE) programmes H2020-MSCA-RISE-2015 Grant No. StronGrHEP-690904 andH2020-MSCA-RISE-2017 Grant No. FunFiCO-777740. A. P. is supported by the FCT grantPD/BD/142842/2018. E. R. gratefully acknowledges the support of the Alexander von Hum-boldt Foundation. The authors would like to acknowledge networking support by the COSTAction CA16104.
A. Exact solutions with a linear coupling
A.1 Purely electric BHs
Purely electric dilatonic solutions of (2.1) with the dilatonic coupling (2.12) where first con-sidered by Gibbons and Maeda [16] and Garfinkle, Horowitz and Strominger [17]. The BHsolution has the line element (2.5) with a ( r ) = 1 c ( r ) = (cid:16) − r + r (cid:17) (cid:16) − r − r (cid:17) − α α , b ( r ) = r (cid:16) − r − r (cid:17) α α , (A.1)together with the Maxwell potential and dilaton field A = Qr dt , e φ = (cid:16) − r − r (cid:17) α α . (A.2) Following the conventions in the work, we fix φ ( ∞ ) = 0 for all solutions in the Appendix. – 22 –he two free parameters r + , r − (with r − < r + ) are related to the ADM mass, M , and (total)electric charge, Q , by M = 12 (cid:20) r + + (cid:18) − α α (cid:19) r − (cid:21) , Q = (cid:18) r − r + α (cid:19) . (A.3)For all α , the surface r = r H = r + is the location of the (outer) event horizon, with A H = 4 πr (cid:18) − r − r + (cid:19) α α , T H = 14 π r + − r − (cid:18) − r − r + (cid:19) α . (A.4)The extremal limit, which corresponds to the coincidence limit r − = r + , results in a singularsolution (as can be seen e.g. by evaluating the Kretschmann scalar). In this limit, the area ofthe event horizon goes to zero for α = 0. The Hawking temperature, however, only goes tozero in the extremal limit for α <
1, while for α = 1 it approaches a constant, and for α > q = 2 p (1 + α ) x α (1 − x ) + x , a H = (1 + α ) (1 − x ) α α (1 + α (1 − x ) + x ) , t H = (1 − x ) − α α (1 + α (1 − x ) + x )1 + α , where 0 x A.2 Dyonic BHsA.2.1 α = 1A dyonic dilatonic BH solution of (2.1), with the dilatonic coupling (2.12) and α = 1, wasfound in [43], and extensively discussed in the literature, since it can be embedded in N = 4supergravity. Taking the form (2.5), it has φ = 12 log ( r + Σ)( r − Σ) , a ( r ) = 1 c ( r ) = ( r − r + )( r − r − )( r − Σ ) , b ( r ) = r − Σ , (A.5)where r ± = M ± p M + Σ − Q − P , (A.6)and the outer horizon is at r H = r + , while M, Q, P are the mass and electric and magneticcharges. Σ corresponds to the scalar charge, which, however, is not an independent parameter(the hair is secondary): Σ = P − Q M . (A.7)The extremal limit of the above solution corresponds to r + = r − , in which case one finds tworelations between the charges0 = M + Σ − Q − P = ⇒ ( M + Σ) − P = 0 and ( M − Σ) − Q = 0 . (A.8)– 23 –he horizon area and Hawking temperature of the solutions are A H = 4 π (2 M r + − P − Q ) , T H = 12 π r + − M M r + − P − Q . (A.9)The expression of the reduced quantities is more involved in this case: a H = 14 (2 x − q ) , t H = 4( x − x − q , (A.10)with x a parameter expressed in terms of q as a solution of the equation (where k = PQ ) q − k ) (1 − k ) ( q + x ( x − . (A.11) A.2.2 α = √
3A dyonic dilatonic BH solution of (2.1), with the dilatonic coupling (2.12) and α = √
3, wasfound in [41,42]. This case arises from a suitable Kaluza-Klein reduction of a five-dimensionalvacuum BH. In the extremal limit, one obtains a non-BPS BH that can be embedded in N = 2supergravity.The generic solution can be written again in the form (2.5) with a ( r ) = 1 c ( r ) = ( r − r + )( r − r − ) √ AB , b ( r ) = √ AB and e φ ( r ) / √ = AB , (A.12)where A = ( r − r A + )( r − r A − ) , B = ( r − r B + )( r − r B − ) . (A.13)In the above relations one defines r ± = M ± p M + Σ − P − Q , (A.14)where, again, the outer horizon is at r H = r + , and r A ± = 1 √ ± P s − √ M , r B ± = − √ ± Q s √ M . (A.15)The solution possesses again three parameters
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