A new class of four-dimensional, hairy, stationary solutions of the
Einstein-Maxwell-Lambda system with a conformally coupled scalar field is
constructed in this paper. The metric belongs to the Plebanski-Demianski family
and hence its static limit has the form of the charged C-metric. It is shown
that, in the static case, a new family of hairy black holes arises. They turn
out to be cohomogeneity-two, with horizons that are neither Einstein nor
homogenous manifolds. The conical singularities in the C-metric can be removed
due to the back reaction of the scalar field providing a new kind of regular,
radiative spacetime. The scalar field carries a continuous parameter
proportional to the usual acceleration present in the C-metric. In the
zero-acceleration limit, the static solution reduces to the dyonic
Bocharova-Bronnikov-Melnikov-Bekenstein solution or the dyonic extension of the
Martinez-Troncoso-Zanelli black holes, depending on the value of the
cosmological constant.Comment: Published versio

Andrés Anabalón

Hideki Maeda

N. Bocharova

R. M. Wald

English

arXiv

A general class of four dimensional, stationary solutions of the Einstein-Maxwell system with a conformally coupled scalar field is constructed in this paper. The stationary case is presented and shown to belong to the Plebanski-Demianski family which implies that the static metric has the form of the C-metric. It is shown that in the static, AdS case, a new family of Black Holes arises. They turn out to be cohomogeneity two, with horizons that are not Einstein neither homogenous manifolds. The usual conical singularities present in the C-metric are automatically removed from the spacetime due to the backreaction of the scalar field. The scalar field carries a continuous parameter that resembles the usual acceleration present in the C-metric. When this parameter vanishes the static family it is shown to contain either to the dyonic Bocharova-Bronnikov-Melnikov-Bekenstein solution or the dyonic extension of the Martinez-Troncoso-Zanelli black holes, depending on the value of the cosmological constant

Anabalon, A.

Maeda, H.

MPG.PuRe

New Charged Black Holes with Conformal Scalar Hair

arXiv:0907.0219v1  [hep-th]  1 Jul 2009AEI-2009-059, CECS-PHY-09/05New Charged Black Holes with Conformal Scalar Hair.Andre´s Anabalo´n1,2∗ and Hideki Maeda2†1Max-Planck-Institut fu¨r Gravitationsphysik, Albert-Einstein-Institut,Am Mu¨hlenberg 1, D-14476 Golm, Germany.2Centro de Estudios Cientı´ficos (CECS),Arturo Prat 514, Valdivia, Chile(Dated: July 1, 2009)A general class of four dimensional, stationary solutions of the Einstein-Maxwell system with a conformallycoupled scalar field is constructed in this paper. The stationary case is presented and shown to belong to thePlebanski-Demianski family which implies that the static metric has the form of the C-metric. It is shown thatin the static, AdS case, a new family of Black Holes arises. They turn out to be cohomogeneity two, withhorizons that are not Einstein neither homogenous manifolds. The usual conical singularities present in theC-metric are automatically removed from the spacetime due to the backreaction of the scalar field. The scalarfield carries a continuous parameter that resembles the usual acceleration present in the C-metric. When thisparameter vanishes the static family it is shown to contain either to the dyonic Bocharova-Bronnikov-Melnikov-Bekenstein solution or the dyonic extension of the Martinez-Troncoso-Zanelli black holes, depending on thevalue of the cosmological constant.PACS numbers: 04.20.Jb 04.40.Nr 04.70.BwI. INTRODUCTION, DISCUSSION AND CONCLUSIONSThe scalar hair have played an important roˆle in our un-derstanding of four dimensional black holes as fundamen-tal objects, characterized by a small set of parameters (for ashort review see [1]). Within all the possible hairs the con-formal scalar hair is particularly interesting: it not only con-tains a well known family of U(1) charged, stationary, blackholes [2], [3], [5] , [6], [7], it also has the property that theasymptotically locally AdS solutions, when mapped to theEinstein frame, can be embedded in string theory [8] that theyare stable against linear perturbations [9] and provides a rele-vant arena for the gravitational description of superconductors[10].These interesting features are in contrast with the exigu-ous knowledge of exact solutions of this system. Moreover,the question on the existence of stationary axisymmetric so-lutions was already pointed out to be of relevance in one ofthe seminal papers of the subject [3] and, until now, there isno explicit construction of it. This is the first of a series ofpapers that will improve the situation on these regards. Thiswill be done taking advantage of the well known fact that, thetraceless of the energy momentum tensor for the conformallycoupled scalar field implies that any space of constant Ricciscalar could in principle support its backreaction. Being thePlebanski-Demianski family of solutions [11] the most gen-eral Petrov type D, Einstein-Maxwell spacetime, it providesthe natural starting point to further explore the space of so-lutions of the Einstein-Maxwell system and the conformallycoupled scalar hair.Thus, in section one the most general solution, withinthe Plebanski-Demianski family, of the Einstein-Maxwell-Conformally coupled scalar field is constructed. The addition∗Electronic address: anabalon-at-aei.mpg.de†Electronic address: hideki-at-cecs.clof a quartic self interaction of the scalar field is necessarywhen the cosmological constant is included and it is alsoconsidered. In order to show that all the known solutions areincluded within this new family, section two is devoted to thethe analysis of the static case. There is shown that the familyof metric presented here, besides reproducing the knownsolutions provides the dyonic extension of [6].The static solution, being of the form of the C-metric, isreanalyzed in the last section and a number of remarkable fea-tures are observed to occur. First, the conical singularities areremoved due to the back reaction of the scalar field, this is notdone at the expense of changing the asymptotic behavior ofthe spacetime (as opossite of the embedding of Ernst [12]);this seems to implies that the scalar field can be regarded asthe source of the acceleration in the classical interpretation ofthis spacetime [13]. Second, as was pointed out in [14], theAdS C-metric can be interpreted as a single black hole in acertain range of the parameters. This new AdS black hole isgeometrically characterized and it turns out to be a cohomo-geneity two black hole whose event horizon is not an Einsteinneither a homogenous manifold, resembling the five dimen-sional structure of the stationary black holes constructed in[15]. Third, this new family of static black holes are continu-ously connected with the maximally symmetric configuration,viz AdS. It turns out that in the limit where the spacetime isof constant curvature the scalar field develops a non-trivialvacuum expectation value: the energy momentum tensor van-ishes but there is a explicit dependence on the spacetime pointin the scalar field. These peculiar configurations were discov-ered in [16], however it was not known how they connect tonon conformally flat spacetimes.The elimination of conical singularities from the C-metric,due to the scalar field backreaction, is an interesting result anddeserves some comments. The conical defects associated tothe acceleration can be neatly described as follows: given thecharged C-metric2ds2 =1A2(q − p)2(dp2X(p)+X(p)dσ2 +dq2Y (q)− Y (q)dt2),X(p) = (1− p2)(1 +Ar+p)(1 +Ar−p), Y (q) = −X(q),r± = m±√m2 −Q2 (1.1)it is possible to compute the “surface gravity” (in the termi-nology of [17]), k = gµν∂µl2∂ν l24l2, of the angular killing vectorl = C∂σ = ∂χ (1.2)at the degeration surfaces p = ±1:kp=±1 = C2(A2Q2 + 1± 2Am)2 . (1.3)From the previous expression it seems to be impossible, keep-ing the acceleration, to have the same normalization at eachone of the degeneration surfaces (unless the mass vanishes inwhich case the charged C-metric metric represent a naked sin-gularity). This is equivalent to say that there is a conical de-fect (or excess) in, at least, one of the “poles” of the compact,spacelike section, defined at constant q.The situation completely changes in the presence of scalarfields. Slowly decaying scalar fields have a non-trivial con-tribution to the mass of the spacetime [18], [19]. Thus it ispossible to eliminate the m parameter from the metric func-tions, and thus the conical singularities, and still have a space-time with positive mass. Although this last point is not strictlyproved below, it is supported due to the existence of Killinghorizons of positive scalar curvature in the vanishing m limit.The present construction could have many applications.One of the most interesting, in our view, is when the metric islocally asymptotically AdS but, due to the acceleration hori-zon, not asymptotically static. The explicit, and intrinsic, timedependence in the asymptotic region would allow to study theelusive out of equilibrium phenomena in the dual condensematter system. In an accompanying paper we will further dis-cuss the rotating case, its thermodynamics and some of thephysical interpretations of these spacetimes [20].Our notations follows [21]. The conventions of curvaturetensors are [∇ρ,∇σ]V µ = RµνρσV ν and Rµν = Rρµρν . Themetric signature is taken to be mostly plus, greek letters arespacetime indices and we set c = 1.Note added in proof: An abstract of part of this work wassubmitted on June 19 to the “Fifth Aegean Summer School”organized by one of the authors (E.P.) of [23], who informs uson some overlap with their still unpublished work. Althoughthe line element in the static case coincide with the one of[23], their study is exclusively devoted to the case when theQ2 term of (1.1) is positive. What makes the inclusion of ascalar field special is that it allows for this Q2 term to not bethe square of the electric charge. Actually it can be negative,which is the case analyzed in the last section.II. THE STATIONARY SOLUTIONThe Einstein-Maxwell-conformally coupled scalar fieldwith quartic selfinteraction can be defined by the set of equa-tions:Rµν − 12gµνR+ Λgµν =κ4pi(FµλFλν −14gµνFτλFτλ) + κT φµν , R = 4Λ, F = dA, (2.1)T φµν = ∂µφ∂νφ−12gµνgαβ∂αφ∂βφ+16[gµν −∇µ∇ν +Gµν ]φ2 − αgµνφ4, (2.2)where κ := 8piG and Λ is a cosmological constant. The otherfield equations follows as consistency conditions of the previ-ous ones. The above form of the equations will be useful atthe time of solving them.Given the Pleabanski-Demianski ansatz:ds2 =1(1− qp)2[(p2 + q2)(dp2X(p)+dq2Y (q))+X(p)p2 + q2(dτ + q2dσ)2 − Y (q)p2 + q2(dτ − p2dσ)2], (2.3)it is possible to integrate the metric functions from the con-dition that the Ricci scalar is constant. Replacing it back inthe full set of field equations we found that the most generalsolution with a non-trivial scalar field is given by:3X(p) = a0 + a2p2 + a4p4, Y (q) = −a4 − Λ3− a2q2 −(a0 +Λ3)q4, (2.4)A =c1q + c2pq2 + p2dτ + pqc2q − c1pq2 + p2dσ, φ = ±√B1− pq1 + pq, (2.5)B =3(3κ(c21 + c22)+ 24pi (a4 + a0) + 8piΛ)4κpi (3a0 + 3a4 + Λ), α = − Λ6B. (2.6)In [20] it will be shown that the above spacetime has non-trivial angular momentum and that for certain values of theparameters it represent a black hole. In what follows the dis-cussion will be focussed on the static limit.III. CONTRACTIONS TO THE KNOWN SOLUTIONSIn this section an acceleration parameter β will be intro-duced. This will allow us to recover the known solutions.Furthermore, in order to have non-singular limit we found thatit is convenient to use another parametrization for the metricfunctions.With the above remarks in mind we just state the result; thestatic limit is given by:ds2 =1(q − βp)2(dq2Y (q)− Y (q)dτ2 + dp2X(p)+X(p)dσ2), A = c1qdτ + c2pdσ, (3.1)X(p) = a0 +a1βp+ a2p2 + βa3p3 + β2a4p4, Y (q) = −β2a0 − a1q − a2q2 − a3q3 − a4q4 − Λ3, (3.2)a1 = a3(4a2a4 − a23)8a24, φ = ±√Bβp− qβp+ q + a32a4B =3(κc21 + κc22 + 8pia4)4κpia4, α = − Λ6B. (3.3)The limit β → 0 makes sense if a3 = 0 or if a2 = a234a4.When a3 = 0 the limit implies a constant scalar field andthe discussion of the last section of [6] applies. In the moreinteresting case, a2 = a234a4the change of coordinates q = 1rand the parameters a4 = −M2G2, a3 = 2MG, c1 = e,c2 =g, a2 = −1, a0 = 1, provides the dyonic extension of theblack hole [6]:ds2 = −f(r)dτ2 + dr2f(r)+ r2(dp21− p2 +(1− p2) dφ2), A =erdτ + gpdσ, (3.4)φ = ∓√34pi√M2G− e2 − g2r −MG , f(r) =(1− MGr)2− Λ3r2,(e2 + g2)M2= G+2Λ9αG2. (3.5)In the same way is possible to prove that all the known so-lutions of the relevant system are contained within this staticfamily.IV. NEW STATIC, COHOMOGENEITY TWO ADS BLACKHOLESNow, let us study the above solution in all its generality. Tothis end we parameterize the solution in terms of the roots of4the metric functions, after a shift in the coordinates the staticmetric becomes 1:X(p) = b2(p2 − ξ21) (p2 − ξ22), (4.1)Y (q) = −b2 (q2 − ξ21) (q2 − ξ22)− Λ3 (4.2)Let us set ξ1 < ξ2. The manifold associated to the coordi-nates (p, σ) is Euclidean and compact if −ξ1 ≤ p ≤ ξ1. Thesame condition that follows from the field equations, namelythe fact that there are no linear neither cubic term in the metricfunctions implies that the spacetime is free of conical singu-larities. Indeed, from equation (1.2) we obtainkp=ξ1 = kp=−ξ1 = C2b4ξ21(ξ21 − ξ22)2. (4.3)Infinity is located at p = q and for vanishing cosmologicalconstant there is an event horizon at q = ξ2. When Λ 6= 0the horizon is located at the largest root, qH , of (4.2). In thecase of positive and vanishing cosmological constant there isa further horizon and the asymptotic region is no longer static.The asymptotic region is static for −Λ > 3b2ξ21ξ22 . It is alsointeresting to note that acceleration horizon is extremal when−Λ = 3b2ξ21ξ22 . There is a curvature singularity at q = ∞.From the above discussion it follows that the allowed rank forq is p < q <∞.Although the geometry is regular the scalar field divergesoutside the horizon. In this coordinates it is proportional toq − pp+ q, (4.4)so, although it goes to zero at infinity and is regular on thekilling horizon it is divergent on the surface p+ q = 0. From(3.3) it is possible to check that setting first a3 = c1 = c2 = 0and then a4 = 0, the space becomes of constant curvature andthe scalar field is nothing but an stealth field [16],[? ]. Indeed,although it is non trivial,φS = ±√6κq − pp+ q, (4.5)its energy momentum tensor vanishes Tµν (φ = φS) = 0 ifand only if the metric is of constant curvature.The horizon metric isds2H =1(qH − p)2(dp2X(p)+X(p)dσ2)(4.6)which is not an Einstein neither an homogeneous manifold. Inthe case Λ = 0 its scalar curvature isRH = 2b2(3p+ ξ2) (ξ2 − p)3 . (4.7)Note that suitable values of the roots can make the scalar cur-vature everywhere positive. To understand better the geometryof the horizon let us expand around the degeneration points,let us set a periodic coordinate χ ∈ [0, 2pi), related with σ asσ = Cχ, where C is obtained requiring (4.3) equals one, andp = −ξ1 + x212C, p = ξ1 − x222CUsing these coordinates it ispossible to show that the degeneration surface are smooth:ds2H∣∣x1=0−→ 1(qH + ξ1)2(dx21 + x21dχ2) (4.8)ds2H∣∣x2=0−→ 1(qH − ξ1)2(dx22 + x22dχ2) (4.9)The horizon looks locally as a two sphere, in close resem-blance with [15].The metric is asymptotically locally AdS in the sensethatRµν··λρ∣∣∣p=q=Λ3(δµλδνρ − δµρ δνλ) (4.10)Indeed, this is a local statement. In the AdS case exists ac-celeration horizons for certain values of the parameters. Thisallows to have a non-stationary behavior (because the explicittime dependence) of the asymptotic metric. In the case whenthe acceleration horizon is extremal (which only can occurwhen Λ < 0) it has been recently shown that the conformalstructure of the C-metric is topologically ℜ3 [22]. A furtherstudy of the conformal structure in the case presented here isnecessary to understand the properties of the theory inducedin the boundary. The dual, condense matter description ofthese spacetimes, as well as the thermodynamics propertiesare better understood in the Einstein frame. Thus, thisdiscussion will be postponed for a further work [20].AcknowledgmentsThe authors would like to thank D. Astefanesei, F. Ferrari,S. Hartnoll, G. Lavrelashvili, C. Martinez, J. Oliva, M. Ro-driguez, R. Troncoso, J. Podolsky´, O. Varela, E. Winstanleyand J. Zanelli for discussions and useful comments. A.A.wish to thank the precious support of Cecilia Gumucio whilewriting this manuscript. This work was partially funded bythe following Fondecyt grants: 3080024 (AA); and 1071125(HM). A.A is an Alexander von Humboldt fellow. The Cen-tro de Estudios Cientı´ficos (CECS) is funded by the ChileanGovernment through the Millennium Science Initiative andthe Centers of Excellence Base Financing Program of Coni-cyt. CECS is also supported by a group of private companieswhich at present includes Antofagasta Minerals, Arauco, Em-presas CMPC, Indura, Naviera Ultragas, and Telefo´nica delSur. CIN is funded by Conicyt and the Gobierno Regional deLos Rı´os.5[1] J. D. Bekenstein, arXiv:gr-qc/9605059.[2] N. Bocharova, K. Bronnikov, and V. Melnikov, Vestn. Mosk.Univ. Fiz. Astron. 6, 706 (1970).[3] J.D. Bekenstein, Ann. Phys. (N.Y.) 91, 75 (1975).[4] J. D. Bekenstein, Annals Phys. 82, 535 (1974); 91 (1975) 75.[5] K. S. Virbhadra and J. C. Parikh, Phys. Lett. B 331 (1994) 302[Erratum-ibid. B 340 (1994) 265] [arXiv:hep-th/9407121].[6] C. Martinez, R. Troncoso and J. Zanelli, Phys. Rev. D 67 (2003)024008 [arXiv:hep-th/0205319].[7] C. Martinez, J. P. Staforelli and R. Troncoso, Phys. 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D 71 (2005) 104037 [arXiv:hep-th/0505086].[17] M. Cvetic, H. Lu, D. N. Page and C. N. Pope, Phys. Rev. Lett.95 (2005) 071101 [arXiv:hep-th/0504225].[18] M. Henneaux, C. Martinez, R. Troncoso and J. Zanelli, Phys.Rev. D 70 (2004) 044034 [arXiv:hep-th/0404236].[19] M. Henneaux, C. Martinez, R. Troncoso and J. Zanelli, AnnalsPhys. 322 (2007) 824 [arXiv:hep-th/0603185].[20] A. Anabalon, H. Maeda. In preparation.[21] R. M. Wald, Chicago, Usa: Univ. Pr. ( 1984) 491p[22] R. Emparan and G. Milanesi, arXiv:0905.4590 [hep-th].[23] C. Charmousis, T. Kolyvaris and E. Papantonopoulos,arXiv:0906.5568 [gr-qc].

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New charged black holes with conformal scalar hair

New Charged Black Holes with Conformal Scalar Hair

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