We investigate possible end-points of the superradiant instability for a charged black hole with a reflecting mirror. By considering a fully coupled system of gravity and a charged scalar field, hairy black hole solutions are obtained. The linear stability of these black hole solutions is studied

Dolan, S.

Ponglertsakul, S.

Winstanley, E.

White Rose Research Online

This is a repository copy of Black hole solutions in Einstein-charged scalar field theory.White Rose Research Online URL for this paper:http://eprints.whiterose.ac.uk/87951/Version: Submitted VersionProceedings Paper:Ponglertsakul, S., Dolan, S. and Winstanley, E. orcid.org/0000-0001-8964-8142 (Submitted: 2015) Black hole solutions in Einstein-charged scalar field theory. In: Proceedings of the Fourteenth Marcel Grossman Meeting on General Relativity. 14th Marcel Grossman Meeting . (Submitted) 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:1507.02462v1  [gr-qc]  9 Jul 2015July 10, 2015 0:18 WSPC Proceedings - 9.75in x 6.5in MG14 page 11Black hole solutions in Einstein-charged scalar field theorySUPAKCHAI PONGLERTSAKUL∗, SAM DOLAN and ELIZABETH WINSTANLEYConsortium for Fundamental Physics, School of Mathematics and Statistics,The University of Sheffield, Hicks Building, Hounsfield Road,Sheffield, S3 7RH, United Kingdom∗E-mail: smp12sp@sheffield.ac.ukWe investigate possible end-points of the superradiant instability for a charged blackhole with a reflecting mirror. By considering a fully coupled system of gravity and acharged scalar field, hairy black hole solutions are obtained. The linear stability of theseblack hole solutions is studied.Keywords: Superradiant instability; Charged black hole; Black hole stability.1. IntroductionIn classical general relativity, there is a process by which energy can be extractedfrom a black hole. This phenomenon is known as superradiant scattering. Morespecifically, the amplitude of a bosonic field around black hole will be amplified ifthe frequency ω and the charge q of the field satisfy1, ω < qΦH, where ΦH is electricpotential at the outer horizon. Note that this inequality is the charged version ofsuperradiant condition.Via superradiant scattering, an instability of the background spacetime canoccur if the bosonic field can be trapped within the proximity of the black hole.Wave modes are repeatedly scattered off and amplified by a black hole, so ultimatelytheir back-reaction on the background becomes non-negligible. For a charged scalarfield with non-zero mass, this instability can be triggered by surrounding a chargedblack hole with a reflecting mirror2.Superradiant instabilities for charged black holes have been extensively studiedin the linear regime. Although, the end-point of these instabilities remains un-known and a fully non-linear analysis is needed. Therefore in this talk, one possibleend-points of the superradiant instability for charged black holes in a cavity is in-vestigated. When a fully coupled system of gravity and a charged scalar field with amirror is considered, we find black hole solutions with scalar field hair. Consideringlinearised perturbations of these black holes, we perform a numerical analysis ofthe stability of these hairy black hole solutions. Here we present a summary of ourresults. More details of this work are to appear in Ref. 3.2. Massless charged scalar perturbation of Reissner-Nordströmspacetime with a mirrorAs a preliminary, we consider a massless charged scalar field φ on the Reissner-Nordström (RN) background with a mirror. In the test-field approximation, theusual Klein-Gordon (KG) equation in curved spacetime is obtained. This can besolved by using the ansatz φ ∼ e−iσtR(r) where σ and R(r) are respectively theJuly 10, 2015 0:18 WSPC Proceedings - 9.75in x 6.5in MG14 page 22frequency and the radial part of the scalar field. We impose an ingoing wave condi-tion near the horizon r → rh and at the mirror the scalar field vanishes R(rm) = 0,where rm is the location of the mirror. We integrate the KG equation numericallyand obtain the corresponding frequency. Figure 1 shows an example plot of thefrequency of the scalar field against the mirror radius when the black hole mass isfixed to be M = 1. The superradiant instability can be clearly seen from Figs. 1(a)and 1(b), since there are regions where Im(σ) > 0 indicating an unstable mode.5 10 15 20 25 30-0.0010.0000.0010.0020.003rmImHΣLQ=0.5Q=0.7Q=0.9(a)5 10 15 20 25 30-0.0010.0000.0010.0020.0030.004rmImHΣLq=0.2q=0.4q=0.6(b)Fig. 1. The imaginary part of σ is plotted as a function of the location of the mirror rm, (a) forfixed scalar charge q = 0.5 and different values of the black hole charge Q, (b) for fixed Q = 0.9and different values of q.Therefore from Fig. 1, a massless charged scalar field on the RN spacetime witha mirror experiences a superradiant instability. These results are in agreement withwith previous work done by Herdeiro et al.2, where a superradiant instability of amassive charged scalar field on a charged black hole in a cavity has been discussed.To proceed further, in the next section, we will investigate a fully coupled systemof gravity and a charged scalar field.3. Hairy black hole in Einstein-Maxwell-Klein-Gordon theory witha mirrorWe are interested in a fully coupled system of gravity and a charged scalar field.This system is described by the actionS =∫d4x√−g[R16πG− 14FabFab − 12gabD∗(aφ∗Db)φ], (1)where Fab = ∇aAb −∇bAa, Aa is the electromagnetic potential, Da = ∇a − iqAa,q is the scalar field charge and X(ab) =12 (Xab +Xba). The line element of a staticspherically symmetric spacetime is given byds2 = −f(r)h(r)dt2 + f(r)−1dr2 + r2(dθ2 + sin2 θ dϕ2), (2)the electromagnetic vector potential is Aa ≡ [A0(r), 0, 0, 0] and the scalar dependson r only φ = φ(r). Varying Eq. (1), three equations of motion are obtained, whichcan be solved numerically.July 10, 2015 0:18 WSPC Proceedings - 9.75in x 6.5in MG14 page 33In Fig. 2, the profile of the scalar field outside the black hole event horizon isillustrated, where the black hole radius is fixed at rh = 1. For each curve, there arethree parameters that need to be specified, namely, φh the value of the scalar fieldon the horizon, Eh ≡ A′0(rh) the electric field on the horizon and q. The oscillatorybehaviour of the scalar field allows us to put the mirror at any zero of φ. Here weconsider only the case for which the mirror is located at the first zero of φ. This isbecause these black hole solutions with the mirror at the first zero are expected tobe stable.10 20 30 40 50 60-0.050.000.050.100.150.200.250.30RadiusΦHrLq=0.4q=0.6q=0.8(a)10 20 30 40 50 60-0.20.00.20.40.60.81.0RadiusΦHrLEh=0.6Eh=0.8Eh=1.0(b)Fig. 2. The scalar field φ(r) is plotted as a function of radius (a) for fixed φh = 0.3, Eh = 0.6and different values of q, (b) for fixed q = 0.1 and different values of Eh.By changing one of these three parameters, we obtain different hairy black holesolutions. Figure 2(a) displays three distinct black hole solutions with varying scalarcharge and three different mirror radii. However, as shown in Fig. 2(b), it is possiblethat different solutions could share the same mirror location.4. Stability of the hairy black holesWe are now ready to perform a stability analysis of these equilibrium solutions.If these solutions are shown to be stable, they could represent the end-point ofthe superradiant instability for a massless charged scalar perturbation on the RNbackground with a mirror as discussed in Sec. 2.To investigate the stability of the equilibrium solutions, we perturb the four fieldvariables (f, h,A0, φ) in the theory as follows, f = f̄(r) + δf(t, r) and similarly forthe other three quantities. In this formalism, f̄ is the equilibrium quantity and δfis the perturbation. By linearising the field equations, three coupled perturbationequations for δA0 and the real and imaginary parts of δφ are obtained. The pertur-bation equations are solved by assuming δφ ∼ e−iσtφ̃(r), and similarly for the otherperturbations. These perturbation modes need to satisfy the following boundaryconditions, φ̃(r) and other perturbations modes follow an ingoing wave-like condi-tion near the horizon. The scalar field vanishes at the mirror (φ̃(rm) = 0).We numerically integrate the coupled perturbation equations by scanning fora frequency σ such that the boundary conditions are satisfied. An example plotJuly 10, 2015 0:18 WSPC Proceedings - 9.75in x 6.5in MG14 page 4420 25 30 350.080.100.120.140.160.180.20rmReHΣLEh=0.8Eh=1.0Eh=1.2(a)20 25 30 35-0.025-0.020-0.015-0.010-0.0050.000rmImHΣL(b)Fig. 3. The real (a) and imaginary part (b) of the perturbation frequency σ are plotted againstthe location of the mirror rm with the scalar charge is fixed at q = 0.1, φh varying from 0.1 to 1.4and different values of Eh.is shown in Fig. 3. In each plot, the charge of the scalar field is q = 0.1 andthe initial value of the scalar field varies between φh = 0.1 − 1.4. Each point inthis plot represents the perturbation frequency for one distinct hairy black holesolution. More precisely, with the mirror at the first zero of the static scalar fieldφ, we find one value of σ that satisfies the boundary conditions. The key resulthere is in Fig. 3(b) where Im(σ) < 0 indicates that the perturbation modes decayexponentially in time. Thus it is clearly shown that these black hole solutions arelinearly stable.5. SummaryWe have studied the possible end-point of the superradiant instability of a masslesscharged scalar field on the RN spacetime with a mirror. Numerical black holesolutions with charged scalar hair are obtained. Numerical study suggests thatthese hairy black hole solutions appear to be stable. Thus, they could represent theend-point of the superradiant instability.AcknowledgmentsThe work of EW and SRD is supported by the Lancaster-Manchester-Sheffield Con-sortium for Fundamental Physics under STFC grant ST/L000520/1. The work ofSRD is supported by EPSRC grant EP/M025802/1.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 super-radiant instabilities 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-charged scalar field theory in a cavity, arXiv:1507.02156 [gr-qc].

Black hole solutions in Einstein-charged scalar field theory

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Black hole solutions in Einstein-charged scalar field theory

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