A complex scalar field on a charged black hole in a cavity is known to experience a superradiant instability. We investigate possible final states of this instability. We find hairy black hole solutions of a fully coupled system of Einstein gravity and a charged scalar field. The black holes are surrounded by a reflecting mirror. We also investigate the stability of these black holes

Dolan, S.

Ponglertsakul, S.

Winstanley, E.

White Rose Research Online

This is a repository copy of Einstein-charged scalar field theory: black hole solutions and their stability.White Rose Research Online URL for this paper:http://eprints.whiterose.ac.uk/98079/Version: Submitted VersionArticle:Ponglertsakul, S., Dolan, S. and Winstanley, E. orcid.org/0000-0001-8964-8142 Einstein-charged scalar field theory: black hole solutions and their stability. (Unpublished) eprints@whiterose.ac.ukhttps://eprints.whiterose.ac.uk/Reuse Unless indicated otherwise, fulltext items are protected by copyright with all rights reserved. The copyright exception in section 29 of the Copyright, Designs and Patents Act 1988 allows the making of a single copy solely for the purpose of non-commercial research or private study within the limits of fair dealing. The publisher or other rights-holder may allow further reproduction and re-use of this version - refer to the White Rose Research Online record for this item. Where records identify the publisher as the copyright holder, users can verify any specific terms of use on the publisher’s website. Takedown If you consider content in White Rose Research Online to be in breach of UK law, please notify us by emailing eprints@whiterose.ac.uk including the URL of the record and the reason for the withdrawal request. arXiv:1511.05470v1  [gr-qc]  17 Nov 2015Einstein-Charged Scalar Field Theory: BlackHole Solutions and Their StabilitySupakchai Ponglertsakul, Sam Dolan and Elizabeth WinstanleyAbstract A complex scalar field on a charged black hole in a cavity is known toexperience a superradiant instability. We investigate possible final states of this in-stability. We find hairy black hole solutions of a fully coupled system of Einsteingravity and a charged scalar field. The black holes are surrounded by a reflectingmirror. We also investigate the stability of these black holes.1 IntroductionIn black hole physics, there is a mechanism where rotational(electromagnetic) en-ergy can be extracted from a rotating (charged) black hole. This is called superradi-ant scattering. More specifically, the amplitude of a scalarfield around a black holewill be amplified if the frequencyσ of the field satisfies [1]σ < mΩH+qΦH, wherem,q,ΩH andΦH are the azimuthal quantum number, scalar field charge, angulr ve-locity and electric potential at the outer horizon respectiv ly. By settingΩH = 0, thecharged version of the superradiant condition is obtained.One can create an instability of the spacetime background via a superradiantscattering process, if there is some mechanism to confine thebosonic field withinthe vicinity of the black hole. Then wave modes will be repeatedly scattered offthe black hole and their amplitude will be intensified. The back-reaction of the fieldSupakchai PonglertsakulSchool of Mathematics and Statistics, The University of Sheffield, Hicks Building, HounsfieldRoad, Sheffield, S3 7RH, United Kingdom e-mail:smp12sp@sheffield.ac.ukSam DolanSchool of Mathematics and Statistics, The University of Sheffield, Hicks Building, HounsfieldRoad, Sheffield, S3 7RH, United Kingdom e-mail:S.Dolan@sheffield.ac.ukElizabeth WinstanleySchool of Mathematics and Statistics, The University of Sheffield, Hicks Building, HounsfieldRoad, Sheffield, S3 7RH, United Kingdom, e-mail:E.Winstanley@sheffield.ac.uk12 Supakchai Ponglertsakul, Sam Dolan and Elizabeth Winstanleymodes on the background will eventually become significant.In the charged case,the trapping mechanism can be induced by either (i) a reflecting mirror [2] or (ii)anti-de Sitter boundary condition [4]. By having either of these with a charged blackhole, the superradiant instability can be triggered.An interesting question that one might ask is, what is the end-point of thischarged-scalar superradiant instability? To tackle this problem, a fully non-linearanalysis is required. Hence in this talk, we investigate possible end-points of the su-perradiant instability for charged black holes with a reflecting mirror. More specif-ically, a coupled system of gravity and a massless complex scalar field with amirror-like boundary condition is studied. Numerical black hole solutions with anon-trivial scalar field are obtained. By considering linearised perturbations of theseblack holes, a numerical analysis of the black hole’s stabili y is undertaken. Here wepresent a selection of plots to illustrate our numerical results. More details of thiswork can be found in Ref. [3].2 Linear Perturbations in ElectrovacuumIn this section, a massless complex scalar fieldφ on the Reissner-Nordström (RN)background in a cavity is considered. In the test-field limit, the dynamics of a scalarfield on RN spacetime is described by the Klein-Gordon (KG) equation. By substi-tuting the ansatzφ ∼ e−iσtR(r), whereσ andR(r) are respectively the frequencyand the radial part of the scalar field, into KG equation, we obtain a second orderdifferential equation in terms of the radial part. To solve this equation, we apply thefollowing boundary conditions: (i) an ingoing wave near thehorizonr → rh and (ii)at the mirror the scalar field vanishes, soR(rm) = 0, whererm is the location of themirror. Then a numerical technique called the shooting method is implemented. Wescan for corresponding frequenciesσ such that the perturbations satisfy the bound-ary conditions.With the black hole mass fixed to beM = 1, the example plot below (Fig. 1)illustrates the frequency as a function of the location of mirror. It is clear fromFig. 1(a) and 1(b) that this system experiences a superradiant instability as there areregions where Im(σ)> 0, indicating an unstable mode. One can learn the followingfrom these plots: when the location of mirrorrm is small (close to black hole), thefield mode decays exponentially in time; instability occurswhenrm reaches a certainvalue.We find that a massless charged scalar field on the RN background in a cavityexperiences a superradiant instability. These results arein agreement with with pre-vious work done by Herdeiro et al. [2], where they studied a massive complex scalarfield on a charged black hole with a mirror. To fully understand what could happen atthe end-point of this instability, a non-linear system of gravity and a charged scalarfield must be investigated. In the next section, a fully coupled Einstein-chargedscalar system will be considered.Einstein-Charged Scalar Field Theory: Black Hole Solutions a d Their Stability 35 10 15 20 25 30-0.0010.0000.0010.0020.003rmImHΣLQ=0.5Q=0.7Q=0.95 10 15 20 25 30-0.0010.0000.0010.0020.003rmImHΣLq=0.2q=0.4q=0.6(a) (b)Fig. 1 The imaginary part ofσ is plotted as a function of the location of the mirrorrm, (a) forfixed scalar chargeq = 0.5 and different values of the black hole chargeQ, (b) for fixedQ = 0.9and different values ofq. Taken from [3]3 Static Black HolesThe self-gravitating system of a charged scalar field is described by the followingactionS =∫d4x√−g[R16πG−14FabFab −12gabD∗(aφ∗Db)φ]. (1)In Eq. (1) Faraday tensor is defined byFab = ∇ aAb − ∇ bAa, whereAa is the elec-tromagnetic potential,Da = ∇ a − iqAa, q is the scalar field charge andX(ab) =12 (Xab +Xba). Varying Eq. (1), we obtain three equations of motionGab = 8πG(T Fab +Tφab), (2)∇ aFab =iq2(φ∗Dbφ−φ(Dbφ)∗), (3)DaDaφ = 0. (4)We consider a static spherically symmetry black hole spacetime with line elementds2 =− f (r)h(r)dt2+ f (r)−1dr2+ r2(dθ2+ sin2θ dϕ 2). (5)In addition, the electromagnetic vector potential isAa ≡ [A0(r),0,0,0] and the scalarfield depends onr only φ=φ(r). By inserting these ansatzes into Eqs. (2–4), we ob-tain three coupled ordinary differential equations. By imposing appropriate bound-ary conditions at the event horizon and at the mirror, these coupled equations can besolved numerically. We also require that the scalar field must vanish at the locationof the mirror.To obtain static solutions, three parameters must be specified: φh, the value ofthe scalar field on the horizon;Eh ≡ A′0(rh), the electric field on the horizon; andq. In Fig. 2, we show some example solutions where the black hole radius is fixedat rh = 1. The nontrivial structure of the scalar field can be seen. Because these4 Supakchai Ponglertsakul, Sam Dolan and Elizabeth Winstanleyscalar fields oscillate around zero therefore, one can put the mirror at any zero ofφ,however, in this work, we only consider the case where the mirror s located at thefirst zero, since these solutions are expected to be stable.5 10 15 20 25 30 35 40-0.050.000.050.100.150.200.250.30RadiusΦHrLq=0.4q=0.6q=0.810 20 30 40 50 60-0.20.00.20.40.60.81.0RadiusΦHrLEh=0.6Eh=0.8Eh=1.0(a) (b)Fig. 2 The scalar fieldφ(r) is plotted as a function of radius (a) for fixedφh = 0.3, Eh = 0.6 anddifferent values ofq, (b) for fixedq = 0.1 and different values ofEh. Taken from [3]By varying the three parametersφh,Eh and q, we obtain different hairy blackhole solutions. In Fig. 2(a), three distinct black hole soluti ns with three differentscalar charges are displayed. Note that these solutions possess three different mirrorradii. However, it is possible that different static solutions can share the same mirrorlocation as illustrated in Fig. 2(b).4 Stability of the Hairy Black HolesIn the previous section, we have shown that Einstein-charged scalar field theory ina cavity allows the existence of hairy black holes. Our next important question is,are these solutions stable or unstable? If they are shown to be stable, they couldrepresent a possible end-point of the superradiant instability for a massless chargedscalar perturbation on the RN background with a mirror.We consider linear spherically symmetric perturbations, where the four field vari-ables( f ,h,A0,φ) are rewritten as follows,f = f̄ (r)+δ f (t,r) and similarly for theother three quantities. In this notation,f̄ is the equilibrium quantity andδ f is theperturbed part. By linearising the field equations, we arrive at three coupled per-turbation equations in terms ofδA0 and the real and imaginary parts ofδφ. Theseperturbation equations are very complicated and lengthy, full details of these equa-tions can be found in [3]. The three perturbation equations cist of two dynamicalequations describing the real and imaginary parts of the scalar field respectively, theother one is a constraint equation. By substituting the ansatz δφ∼ e−iσtφ̃(r) (andsimilarly for the other perturbations) into the equations,we can integrate the per-turbation equations numerically. The perturbation modes ar required to satisfy theEinstein-Charged Scalar Field Theory: Black Hole Solutions a d Their Stability 5boundary conditions that̃φ(r) and other perturbation modes have an ingoing wave-like condition near the horizon, and at the mirror the scalarfield perturbations mustvanish(φ̃(rm) = 0).10 15 20 25 300.100.200.300.15rmlogHReHΣLLEh=0.5Eh=0.7Eh=0.910 15 20 25 300.00100.01000.00500.05000.00200.02000.00300.03000.00150.01500.00700.0700rmlogH ImHΣL¤L(a) (b)Fig. 3 The real (a) and imaginary parts (b) of the perturbation frequencyσ are plotted against thelocation of the mirrorrm with the scalar charge fixed to beq = 0.2, φh varying from 0.1 to 1.3 anddifferent values ofEh. We find that Im(σ)< 0. Taken from [3]0.2 0.4 0.6 0.8 1.00.050.100.150.200.250.30qReHΣL0.2 0.4 0.6 0.8 1.0-0.025-0.020-0.015-0.010-0.0050.000qImHΣLΦh=0.25,Eh=0.15Φh=0.10,Eh=0.20Φh=0.30,Eh=0.30(a) (b)Fig. 4 The real (a) and imaginary parts (b) of the perturbation frequencyσ are plotted against thescalar chargeq with a selection of various values ofφh andEh. Taken from [3]The numerical scheme is as follows. Firstly, static background parameters arespecified,q,φh andEh, then the equilibrium field equations are integrated. From thesolution, we locate the first zero of the equilibrium scalar field, setting this to bethe location of the mirrorm. Then the coupled perturbation equations are solved byscanning for frequenciesσ such that the perturbations satisfy the required boundaryconditions.In Fig. 3, the real and imaginary parts ofσ are plotted as functions of the mir-ror radiusrm for scalar field chargeq = 0.2. In each plot, the parameter describingstatic solution varies betweenφh = 0.1−1.3. Thus each point in this plot refers tothe perturbation frequency for one distinct hairy black hole. With the mirror at the6 Supakchai Ponglertsakul, Sam Dolan and Elizabeth Winstanleyfirst zero of the static scalar fieldφ, we find one value ofσ for which the perturba-tions satisfy the boundary conditions. Fig. 3(b) shows thate perturbation modesdecay exponentially in time since we find that Im(σ) < 0. In addition, the real andimaginary parts of the perturbation frequencyσ are plotted as functions of the scalarchargeq are displayed in Fig. 4. Here in this example, for each curve,a selection ofvalues of static black hole parametersφh andEh are fixed. Fig. 4(b) illustrates thatthe perturbation modes are exponentially decaying in time.W refer the reader to[3] where we find Im(σ)< 0 for all black hole solutions investigated.5 SummaryWe have studied a coupled system involving Einstein gravityand a complex chargedscalar field in the presence of a mirror-like boundary condition. Numerical hairyblack hole solutions were obtained by the shooting method. By putting a mirror atthe first node of the equilibrium scalar field, we showed theseblack hole solutionsare stable under spherically symmetric linear perturbations. Therefore, we concludethat these black holes could represent an end-point of the sup rradiant instability ofReissner-Nordstörm black holes to charged scalar field perturbations.Acknowledgements The work of SRD and EW is supported by the Lancaster-Manchester-Sheffield Consortium for Fundamental Physics under STFC grant ST/L000520/1. The work ofSRD is also supported by EPSRC grant EP/M025802/1. The work of SP is supported by the 90thAnniversary of Chulalongkorn University Fund (Ratchadaphiseksomphot Endowment Fund).References1. J. D. Bekenstein, Extraction of energy and charge from a black hole,Phys.Rev. D7 949 (1973).2. J. C. Degollado, C. A. R. Herdeiro and H. F. Rúnarsson, Rapid growth of superradiant insta-bilities for charged black holes in a cavity,Phys.Rev. D88 063003 (2013).3. S. Dolan, S. Ponglertsakul and E. Winstanley, Stability of black holes in Einstein-chargedscalar field theory in a cavity, arXiv:1507.02156 [gr-qc].4. N. Uchikata and S. Yoshida, Quasinormal modes of a massless charged scalar field on a smallReissner-Nordstrom-anti-de Sitter black hole,Phys.Rev. D83 064020 (2011).

Einstein-charged scalar field theory: black hole solutions and their stability

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Einstein-charged scalar field theory: black hole solutions and their stability

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