Effect of scalar field mass on gravitating charged scalar solitons and black holes in a cavity
aa r X i v : . [ g r- q c ] N ov E ff ect of scalar field mass on gravitating charged scalar solitons and black holes in a cavity Supakchai Ponglertsakul, Elizabeth Winstanley
Consortium for Fundamental Physics, School of Mathematics and Statistics, University of She ffi eld,Hicks Building, Hounsfield Road, She ffi eld S3 7RH, United Kingdom Abstract
We study soliton and black hole solutions of Einstein charged scalar field theory in cavity. We examine the e ff ect of introducinga scalar field mass on static, spherically symmetric solutions of the field equations. We focus particularly on the spaces of soli-ton and black hole solutions, as well as studying their stability under linear, spherically symmetric perturbations of the metric,electromagnetic field, and scalar field. Keywords:
Einstein charged scalar field theory, black holes, solitons
PACS:
1. Introduction
In the phenomenon of charge superradiance, a classicalcharged scalar field wave incident on a Reissner-Nordstr¨omblack hole is scattered with a reflection coe ffi cient of greaterthan unity if the frequency, ω , of the wave satisfies the inequal-ity [1] 0 < ω < q Φ h , (1)where q is the charge of the scalar field and Φ h is the electro-static potential at the event horizon of the black hole. By thisprocess, the charged scalar field wave extracts some of the elec-trostatic energy of the black hole. If a charged scalar field wavesatisfying (1) is trapped near the event horizon by a reflectingmirror of radius r m , the wave can scatter repeatedly o ff the blackhole, and is amplified each time it is reflected. This can lead toan instability (the “charged black hole bomb”) where the am-plitude of the wave grows exponentially with time [2–5], pro-viding the scalar field charge q and mass µ satisfy the inequality[5] q µ > vt r m r − − r m r + − > , (2)where r + and r − are, respectively, the radius of the event horizonand inner horizon of the black hole. The inequality (2) ensuresthat the area of the event horizon increases as the scalar fieldevolves [2], and implies that for fixed q and µ , the mirror radius r m must be su ffi ciently large for an instability to occur. Physi-cally, the scalar field wave must extract more charge than massfrom the black hole, so that the black hole evolves away fromextremality.What is the ultimate fate of this charged black hole bombinstability? To answer this question, it is necessary to go be-yond the test-field limit and consider the back-reaction of the Email addresses: [email protected] (Supakchai Ponglertsakul),
[email protected] (Elizabeth Winstanley) charged scalar field on the black hole geometry. Recently, westudied static, spherically symmetric, black hole [6] and soli-ton [7] solutions of Einstein charged scalar field theory in acavity, in the case where the scalar field mass µ is set equalto zero. For both soliton and black hole solutions, the scalarfield vanishes on the mirror. We examined the stability of thesecharged-scalar solitons and black holes by considering linear,spherically symmetric, perturbations of the metric, electromag-netic field, and massless charged scalar field. In the black holecase [6], we found that if the scalar field has no zeros betweenthe event horizon and mirror, then the black holes appear to bestable. On the other hand, if the scalar field vanishes inside themirror then the system is unstable. The situation for solitons ismore complex [7]. Even if the scalar field has no zeros insidethe mirror, there are some solitons which are unstable. The un-stable solitons have small mirror radius and large values of theelectrostatic potential at the origin.In [6] we conjectured that the stable black holes with chargedscalar field hair could be possible end-points of the chargedblack hole bomb instability. This conjecture has been testedrecently [8, 9] by evolving the fully coupled, time-dependent,spherically symmetric, Einstein-Maxwell-Klein-Gordon equa-tions in a cavity. Starting from a Reissner-Nordstr¨om blackhole in a cavity with a small charged scalar field perturbation,the system evolved to a hairy black hole in which some of thecharge of the original black hole was transferred to the scalarfield.For a massless charged scalar field, the work of [9] confirmsour conjecture in [6] - the ultimate fate of the charged blackhole bomb is an equilibrium black hole with scalar field hair.However, in [8, 9] a massive charged scalar field is also consid-ered. In this paper we therefore study the e ff ect of introducinga scalar field mass on the soliton and black hole solutions foundin [6, 7]. Our aim is to examine whether the end-points of thecharged black hole bomb instability found in [8, 9] correspondto stable equilibrium solutions of the Einstein-Maxwell-Klein- Preprint submitted to Physics Letters B October 4, 2018 ordon equations.To this end, we begin in section 2 by introducing Einsteinmassive charged scalar field theory. We study numerical solitonand black hole solutions of the static, spherically symmetricfield equations in section 3, paying particular attention to thee ff ect of the scalar field mass on the phase space of solutions.The stability of the solutions is investigated in section 4, beforeour conclusions are presented in section 5.
2. Einstein massive charged scalar field theory
We consider a self-gravitating massive charged scalar fieldcoupled to gravity and an electromagnetic field, and describedby the action S = Z √− g d x " R − F ab F ab − g ab D ∗ ( a Φ ∗ D b ) Φ − µ Φ ∗ Φ (3)where g is the metric determinant, R the Ricci scalar, F ab = ∇ a A b − ∇ b A a is the electromagnetic field (with electromagneticpotential A a ), Φ is the complex scalar field, Φ ∗ its complexconjugate and D a = ∇ a − iqA a with ∇ a the usual space-timecovariant derivative. Round brackets in subscripts denote sym-metrization of tensor indices. The scalar field charge is q and µ is the scalar field mass. We use units in which 8 π G = = c andmetric signature ( − , + , + , + ).Varying the action (3) gives the Einstein-Maxwell-Klein-Gordon equations G ab = T Fab + T Φ ab , ∇ a F ab = J b , D a D a Φ − µ Φ = , (4)where the stress-energy tensor T ab = T Fab + T Φ ab is given by T Fab = F ac F bc − g ab F cd F cd , T Φ ab = D ∗ ( a Φ ∗ D b ) Φ − g ab h g cd D ∗ ( c Φ ∗ D d ) Φ + µ Φ ∗ Φ i , (5)and the current J a is J a = iq (cid:2) Φ ∗ D a Φ − Φ ( D a Φ ) ∗ (cid:3) . (6)We consider static, spherically symmetric, solitons and blackholes with metric ansatz ds = − f ( r ) h ( r ) dt + f − ( r ) dr + r h d θ + sin θ d ϕ i , (7)where the metric functions f and h depend only on the radialcoordinate r . It is useful to define an additional metric function m ( r ) by f ( r ) = − m ( r ) r . (8)By a suitable choice of gauge (see [6, 7] for details), we cantake the scalar field Φ = φ ( r ) to be real and depend only on r .The electromagnetic gauge potential has a single non-zero com-ponent which depends only on r , namely A µ = [ A ( r ) , , , E = A ′ , the static field equations (4) generalize those in [6, 7] to include a nonzero scalar field massand take the form h ′ = r (cid:16) qA φ f − (cid:17) + rh φ ′ , (9a) E + µ h φ = − r " f ′ h + f h ′ + hr ( f − , (9b)0 = f A ′′ + fr − f h ′ h ! A ′ − q φ A , (9c)0 = f φ ′′ + fr + f ′ + f h ′ h ! φ ′ + q A f h − µ φ. (9d)
3. Soliton and black hole solutions
We now consider soliton and black hole solutions of the staticfield equations (9). In both cases we have a mirror at radius r m ,on which the scalar field must vanish, so that φ ( r m ) =
0. As in[7], here we consider only solutions where the scalar field hasits first zero on the mirror, since it is shown in [6] that blackhole solutions for which the scalar field has its second zero onthe mirror are linearly unstable.
In order for all physical quantities to be regular at the origin,the field variables have the following expansions for small r : m = φ h a q + h µ i h r + O ( r ) , h = h + q a φ r + O ( r ) , A = a + a q φ r + O ( r ) ,φ = φ − φ h a q − h µ i h r + O ( r ) , (10)where φ , a and h are arbitrary constants. By rescaling thetime coordinate (see [7] for details), we can set h = q = .
1. For each value of the scalarfield mass µ , soliton solutions are then parameterized by the twoquantities a and φ .Scalar field profiles for some typical soliton solutions areshown in figure 1. From the expansions (10), it can be seenthat if the scalar field mass vanishes, µ =
0, and φ > µ >
0. For φ > h = | a | > µ/ q then the scalar field is decreasing close to the ori-gin, and, for the numerical solutions investigated, it monoton-ically decreases to zero on the mirror. If | a | < µ/ q then thescalar field is increasing close to the origin and must thereforehave a maximum before decreasing to zero on the mirror. Thisbehaviour can be seen in figure 1.2
20 40 60 80 100 1200.00.20.40.6
Radius Φ H r L a = Φ = a = Φ = a = Φ = Figure 1: Scalar field profiles for some typical soliton solutions with scalar fieldcharge q = . µ = . - - - - - a Φ Μ= - - - - - a Φ Μ= - - - - - a Φ Μ= Figure 2: Portions of the phase spaces of soliton solutions with scalar fieldcharge q = . µ . Shaded regionsindicate where solutions exist. The curves are contours at constant mirror radius r m =
20, 40, 60, 80, 100 and 300. The darkest regions have r m <
20; for thelightest regions, the mirror radius r m >
20 40 60 80 100 120 - Radius Φ H r L E h = Φ h = E h = Φ h = E h = Φ h = Figure 3: Scalar field profiles for some typical black hole solutions with eventhorizon radius r h =
1, scalar field charge q = . µ = . We find that the phase space of solitons depends on the scalarfield mass µ , see figure 2. As in the massless case [7], fornonzero µ there appears to be no upper bound on the value of | a | for which there are soliton solutions; accordingly only aportion of the phase space is shown in figure 2. When µ = | a | arbitrarily small (but nonzero).However, when µ >
0, we find that solitons exist only for | a | above some lower bound, which increases as µ increases. If φ > | a | is too small, then the scalar field is increasingsu ffi ciently rapidly close to the origin that it is unable to de-crease to zero before either the metric function f ( r ) has a zeroor the solution becomes singular.The other interesting feature in figure 2 is the existence ofsolitons with µ > q . For such values of the scalar field mass,there is no charged black hole bomb instability in the test-fieldlimit (2). We therefore now explore whether there are also blackhole solutions when µ > q . We consider black holes with event horizon radius r h , whichcan be set equal to unity using a length rescaling [7]. In a neigh-bourhood of the event horizon, the field variables have the ex-pansions m = r h + m ′ h ( r − r h ) + O ( r − r h ) , h = + h ′ h ( r − r h ) + O ( r − r h ) , A = E h ( r − r h ) + A ′′ h r − r h ) + O ( r − r h ) ,φ = φ h + φ ′ h ( r − r h ) + φ ′′ h r − r h ) + O ( r − r h ) , (11)where m ′ h = r h (cid:16) µ φ h + E h (cid:17) , h ′ h = r h φ h (cid:16) µ + q E h (cid:17)h − r h (cid:16) µ φ h + E h (cid:17)i ,φ ′ h = r h µ φ h − r h (cid:16) µ φ h + E h (cid:17) , (12)3 - - - E h Φ h q = Μ= - - - - E h Φ h q = Μ= - - - - E h Φ h q = Μ= - - - - E h Φ h q = Μ= - - - - E h Φ h q = Μ= - - - - E h Φ h q = Μ= - - - - E h Φ h q = Μ= - - - - E h Φ h q = Μ= - - - - E h Φ h q = Μ= - - - - E h Φ h q = Μ= - - - - E h Φ h q = Μ= - - - - E h Φ h q = Μ= Figure 4: Phase spaces of black hole solutions with event horizon radius r h = q and mass µ . Shaded regions indicatewhere solutions exist. The curves are contours at constant mirror radius r m =
20, 40, 60, 80, 100 and 300, except in the last two plots ( q = . µ = .
5, 0 .
98) wherethe outermost contour is r m =
5. The darkest regions have r m <
20; for the lightest regions, the mirror radius r m > q increases, theregion containing black holes with small r m (the darkest blue region) increases in size. .2 0.4 0.6 0.8 1.0 1.2 1.40.20.40.60.81.0 Μ q Figure 5: Phase space of black hole solutions with event horizon radius r h = µ and charge q forwhich we find hairy black holes. The red dashed line is q = µ . It is clear thatwe find solutions for which µ > q . and A ′′ h and φ ′′ h are given in terms of q , µ , r h , φ h and A ′ h = E h . For fixed µ and q , with r h =
1, black hole solutions areparameterized by φ h and E h . In order for the event horizon tobe nonextremal, we find that E h + µ φ h < r h =
1, whichrestricts the black hole phase space.Some typical scalar field profiles for black hole solutions areshown in figure 3. When the scalar field is massless, φ ′ h = φ ′′ h has the opposite sign to φ h [6]. Therefore, for µ = φ h >
0, the scalar field is decreasing close to the horizon.For a massive scalar field, from (12) we see that φ ′ h has thesame sign as φ h . Therefore, when φ h >
0, the scalar field isincreasing close to the event horizon and has a maximum be-tween the event horizon and mirror. This behaviour can be seenin the scalar field profiles shown in figure 3, and in the finalscalar field configurations resulting from the time-evolution ofthe charged black hole bomb instability [8, 9].The phase spaces of black hole solutions for various valuesof the scalar field charge q and mass µ are shown in figure 4.When µ >
0, we find that there is a minimum value of | E h | for which there are nontrivial black holes. This minimum isvery small when q is large and µ is small, when the gap in thephase space for small | E h | is not visible in figure 4. Below thisminimum, the scalar field does not have a zero before either f ( r ) has a second zero or the solution becomes singular.For each value of the scalar field charge q , we find a maxi-mum value of the scalar field mass µ for which there are hairyblack hole solutions. In figure 5 we plot the region of the ( q , µ )-plane (with event horizon radius r h = < q <
1) forwhich there are black hole solutions. It is clear that, for eachvalue of the scalar field charge q , the maximum scalar field massis always larger than q , in other words we find nontrivial blackholes with µ > q .
4. Stability analysis
We now examine the stability of the soliton and black holesolutions under linear, spherically symmetric, perturbations of the metric, electromagnetic field and scalar field. The method islargely unchanged from that employed in [6, 7] in the masslesscase. We therefore simply state the perturbation equations andbriefly discuss the numerical results, referring the reader to [6,7] for details of the derivation and numerical method used.
We begin by introducing two new field variables: γ = f h / , ψ = r φ, (13)where now γ , f , h , A , φ and ψ depend on the radial coordi-nate r and time t . We write the field variables as, for example, f ( t , r ) = ¯ f ( r ) + δ f ( t , r ) where barred variables are static equi-librium quantities and δ f (with similar notation for the othervariables) are time-dependent perturbations. All perturbationsare real, apart from the scalar field perturbation δψ , which wewrite in terms of its real and imaginary parts as [6]: δψ ( t , r ) = δ u ( t , r ) + i δ ˙ w ( t , r ) , (14)where δ u and δ w are real. The derivation of the linearized per-turbation equations is essentially the same as in the masslesscase [6, 7]. The metric perturbations can be eliminated to givethree perturbation equations for δ u , δ w and δ A . The final per-turbation equations are slightly modified by the inclusion of thescalar field mass µ , and take the form0 = δ ¨ u − ¯ γ δ u ′′ − ¯ γ ¯ γ ′ δ u ′ + q ¯ A + ¯ γ ¯ γ ′ r − ¯ f ¯ h ¯ ψ r ! ′ + ¯ f ¯ A ′ ¯ ψ r ! + ¯ ψ ′ − ¯ f ¯ ψ ¯ ψ ′ ¯ A ′ r + µ ¯ f ¯ h ( + ¯ ψ ¯ ψ r ! ′ + ¯ ψ ¯ ψ r ! ′ !) δ u + q ¯ A ¯ γ δ w ′′ + q ¯ f ¯ A p ¯ h ¯ γ ′ + − ¯ A ′ ¯ A A + ¯ hr + r ¯ A ′ ¯ ψ r ! ′ ¯ ψ − µ ¯ h ¯ ψ r + ¯ ψ ¯ ψ r ! ′ ! δ w ′ + q ¯ A " q ¯ A − γ ¯ γ ′ r + ¯ γ ¯ ψ ′ ¯ ψ r ! ′ ¯ γ ¯ A ′ ¯ A − ¯ γ ′ − ¯ γ r ! + µ ¯ f ¯ h (cid:0) − r + ¯ ψ ¯ ψ ′ (cid:1) r δ w , (15a)0 = δ ¨ w − ¯ γ δ w ′′ + − ¯ γ ¯ γ ′ + q ¯ A ¯ ψ r ¯ A ′ A δ w ′ + h − q ¯ A − q ¯ A ¯ ψ ¯ ψ ′ r ¯ A ′ A + ¯ γ ¯ γ ′ r + µ ¯ f ¯ h δ w − q ¯ A " + ¯ ψ ¯ ψ r ! ′ δ u + q ¯ A ¯ ψ ¯ A ′ δ A ′ − q ¯ ψδ A , (15b)5 .8 2.0 2.2 2.4 2.6 2.8 3.0 - a Σ Μ= Μ= Μ= Μ= Figure 6: Smallest eigenvalue σ for solitons with scalar field charge q = . µ . We have fixed φ = . = q ¯ ψ ¯ A ′ r A δ w ′′ + q ¯ ψ ¯ A r ¯ γ ′ ¯ A ¯ A ′ ¯ γ A − q ¯ ψ ¯ hr ¯ A ′ δ w ′ + q ¯ ψ ¯ A r A r ¯ A ¯ A ′ ¯ γ − ¯ γ ′ + rq ¯ A ¯ γ − µ r p ¯ h + q ¯ h ¯ ψ ¯ ψ ′ r ¯ A ′ δ w − ¯ ψ r ! ′ δ u ′ − " ¯ ψ r ! ′′ + r + ¯ γ ′ ¯ γ ! ¯ ψ r ! ′ − µ ¯ ψ r ¯ f δ u + δ A ′ ¯ A ′ ′ , (15c)where we have defined A = ¯ f ¯ h + r ¯ A ¯ A ′ . (16)At the mirror r = r m , the scalar field perturbations δ u and δ w must vanish; there is no restriction on the value of δ A there.The other boundary conditions depend on whether we are con-sidering equilibrium solitons or black holes. For soliton solutions, we consider time-periodic perturba-tions of the form [7] δ u ( t , r ) = Re h e − i σ t ˜ u ( r ) i , δ w ( t , r ) = Re h e − i σ t ˜ w ( r ) i ,δ A ( t , r ) = Re h e − i σ t ˜ A ( r ) i , (17)where ˜ u , ˜ w , ˜ A have the following expansions near the origin˜ u = r ∞ X j = u j r j , ˜ w = r ∞ X j = w j r j , ˜ A = ∞ X j = α j r j . (18)As in [7], we can use the residual gauge and di ff eomorphismfreedom to set w = = α and fix u since the perturbationequations (15) are linear. This leaves σ and w as free param-eters. We find that u , w , α all vanish and subsequent termsin the expansions (18) are determined by σ , w and u .In figure 6 we plot the smallest eigenvalue σ (which we findto be real) for some typical soliton solutions. The results arevery similar to those found in [7] when the scalar field mass µ =
0. Although including a scalar field mass µ does change thenumerical values of the eigenvalues σ , the qualitative results - - - - Φ h I m H Σ L Μ= Μ= Μ= Μ= Figure 7: Imaginary part of the perturbation frequency σ for black hole solu-tions with scalar field charge q = . E h = . r h = from [7] are unchanged. In particular, for larger values of themirror radius, all soliton solutions we investigated have σ > σ is real and the solutions arestable. However, if the mirror radius is su ffi ciently small, thenwe find that some solitons have eigenvalues σ <
0, giving apurely imaginary perturbation frequency. In this case there areperturbations which grow exponentially with time and hencethe solitons are unstable. When µ > q , we still find both stableand unstable solitons. Perturbations of black hole solutions have ingoing boundaryconditions at the event horizon, so we consider [6]: δ u ( t , r ) = Re h e − i σ ( t + r ∗ ) ˜ u ( r ) i , δ w ( t , r ) = Re h e − i σ ( t + r ∗ ) ˜ w ( r ) i ,δ A ( t , r ) = Re h e − i σ ( t + r ∗ ) ˜ A ( r ) i , (19)where the usual tortoise coordinate r ∗ is defined by dr ∗ dr = γ . (20)The quantities ˜ u , ˜ w and ˜ A have the following expansions nearthe horizon:˜ u = ˜ u + ˜ u ( r − r h ) + O ( r − r h ) , ˜ w = ˜ w + ˜ w ( r − r h ) + O ( r − r h ) , ˜ A = ˜ A ( r − r h ) + ˜ A ( r − r h ) + O ( r − r h ) . (21)Since the perturbation equations are linear, we can fix ˜ u with-out loss of generality, and then ˜ u , ˜ w , ˜ A and subsequent termsin the expansions (21) are determined by ˜ w and the eigenvalue σ .In contrast to the soliton case, for equilibrium black hole so-lutions the eigenvalue σ is, in general, complex. In figure 7we show the imaginary part of σ for some typical black holesolutions. Again our results are qualitatively similar to thoseobtained in [6] when µ =
0, although the numerical values of σ depend on the scalar field mass. In particular, for all the black6oles we investigated (including those with µ > q ), we find thatthe imaginary part of σ is negative, so the perturbations (19) areexponentially decaying with time and the black holes are stable.
5. Conclusions
We have studied the e ff ect of introducing a scalar field mass µ on static, spherically symmetric, charged scalar solitons andblack holes in a cavity, studied for µ = r = r m .The phase spaces of soliton and black hole solutions have anumber of interesting new features when µ is nonzero. For fixedscalar field charge q , for both solitons and black holes the phasespace shrinks as µ increases, with a nonzero lower bound on themagnitude of either the electrostatic potential at the origin (forsolitons) or the derivative of the electrostatic potential at thehorizon (for black holes). For black hole solutions, for fixed q there is a maximum value of the scalar field mass µ for whichwe find solutions.We have also studied the dynamical stability of our solutionsunder linear, spherically symmetric perturbations of the metric,scalar field and electromagnetic field. Recently, the thermody-namic stability of solitons and hairy black holes with a masslesscharged scalar field in a cavity has been studied [10]. A com-plex thermodynamic phase space emerges, in some regions ofwhich the solitons or the hairy black holes are the thermody-namically stable configuration. It would be interesting to inves-tigate the e ff ect of a scalar field mass µ on the thermodynamicphase space.Our work was motivated by the question of the end-point ofthe charged black hole bomb instability, which occurs in thetest-field limit if the scalar field mass µ and charge q satisfy theinequality q > µ [2, 5]. The hairy black holes we find with q > µ > | E h | for fixed µ and q for hairyblack hole solutions sets a limit on the amount of charge that thescalar field can extract from the black hole during the evolutionof the charged black hole bomb (see [8, 9] for detailed studiesof the extraction of charge and energy from the black hole asthe charged black hole bomb evolves).In this context our solutions with µ > q are particularly inter-esting. When µ > q , a linearized probe charged scalar field on aReissner-Nordstr¨om black hole background does not exhibit acharged black hole bomb instability [2, 5]. Since we find bothsoliton and black hole solutions with µ > q , we can nonethelessinterpret the hairy black holes as bound states of the solitons anda bald Reissner-Nordstr¨om black hole. We conjecture that the black holes in this case could form from the gravitational col-lapse of an unstable soliton with µ > q . To test this conjecture,a full nonlinear time-evolution of the Einstein-Maxwell-Klein-Gordon equations would be required, which we leave for futurework. Note added:
Very recently, the evolution of unstable soli-tons when the charged scalar field mass µ = µ . Acknowledgments
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