Black holes, stationary clouds and magnetic fields
BBlack holes, stationary clouds and magnetic fields
Nuno M. Santos , , ∗ and and Carlos A. R. Herdeiro , † Centro de Astrof´ısica e Gravitac¸ ˜ao — CENTRA, Departamento de F´ısica, Instituto Superior T´ecnico — IST,Universidade de Lisboa — UL, Av. Rovisco Pais 1, 1049-001 Lisboa, Portugal and Centre for Research and Development in Mathematics and Applications (CIDMA) andDepartamento de Matem´atica da Universidade de AveiroCampus de Santiago, 3810-183 Aveiro, Portugal (Dated: January 2021)As the electron in the hydrogen atom, a bosonic field can bind itself to a black hole occupying a discrete infiniteset of states. When (i) the spacetime is prone to superradiance and (ii) a confinement mechanism is present, someof such states are infinitely long–lived. These equilibrium configurations, known as stationary clouds, are states“synchronized” with a rotating black hole’s event horizon. For most, if not all, stationary clouds studied in theliterature so far, the requirements (i)–(ii) are independent of each other. However, this is not always the case. Thispaper shows that massless neutral scalar fields can form stationary clouds around a Reissner–Nordstr¨om blackhole when both are subject to a uniform magnetic field. The latter simultaneously enacts both requirements bycreating an ergoregion (thereby opening up the possibility of superradiance) and trapping the scalar field in theblack hole’s vicinity. This leads to some novel features, in particular, that only black holes with a subset of thepossible charge to mass ratios can support stationary clouds.
Keywords: black holes, magnetic fields, scalar fields, superradiance, stationary clouds
I. INTRODUCTION
Neutron stars and black holes in binary systems feed some ofthe most powerful astrophysical events in the Universe. Theirgravitational–wave luminosity can reach a peak of approxi-mately erg s − [1, 2], only comparable to the electromag-netic luminosity of the most luminous gamma–ray bursts [3].The Advanced LIGO/Virgo’s first and second observation runsreported the detection of gravitational waves from ten differ-ent binary black hole mergers and a single binary neutron starmerger. During the first half of the third observing run, a to-tal of 39 gravitational–wave candidate events were observed,three of which may have originated from neutron star–blackhole mergers [4]. Joint detections of gravitational and elec-tromagnetic waves from neutron star–black hole coalescencesare of particular interest for constraining the equation of stateof dense nuclear matter [5] and measuring the Hubble con-stant [6]. Furthermore, some neutron stars, known as mag-netars, are endowed with super–strong magnetic fields reach-ing – G [7]. For instance, the magnetar SGR J1745–2900, which orbits the supermassive black hole SagittariusA ∗ , has a surface dipolar magnetic field of G. Neutronstar–black hole binary systems are thus natural laboratories forprobing the intricate interaction of black holes with magneticfields.A magnetic field B permeating a black hole with mass M curves the spacetime in a non-negligible way beyond a thresh-old value set by M B ∼ [8], or reinstating familiar units B ≡ c G / M ∼ (cid:18) M (cid:12) M (cid:19) G , (1)where M (cid:12) is the solar mass. A magnetic field of order B or larger warps significantly spacetime in the vicinity of the ∗ [email protected] † [email protected] event horizon (without changing its topology). Since the fieldstrength of a magnetic dipole falls off as the cube of the dis-tance from it, it is unlikely that stellar–mass black holes or evensupermassive black holes are subject to magnetic fields of or-der B .Even if its strength is significantly smaller than B , the im-pact of a magnetic dipole on fields interacting with black holesmay be non–negligible, as they can acquire an effective massand be trapped in its vicinity. A massless field traversingthe black hole’s vicinity would then behave as if it had non–vanishing mass and its effective mass would depend on themagnetic field strength. In addition, if the field is bosonic,it can induce black–hole superradiance, i.e. the extraction ofenergy and angular momentum from rotating black holes (fora review, see [9]). Black–hole superradiance takes place whenthe phase angular velocity w of the bosonic field satisfies w < m Ω H , (2)where m is the azimuthal harmonic index and Ω H is theblack hole’s angular velocity. Together with a natural con-finement mechanism, black–hole superradiance is responsiblefor bosonic fields to form quasi–bound states. These are con-stinuously fed the extracted black hole’s energy and angularmomentum until Eq. (2) saturates, i.e. w = m Ω H , and theybecome bound states. The new equilibrium state is expectedto be a classical bosonic condensate in equilibrium with theslowed–down black hole, which for a complex bosonic field isa hairy black hole [10–14].The bosonic field remains trapped in the vicinity of the blackhole when it is massive. A non–vanishing intrinsic mass, how-ever, is not always mandatory. Trapping can be attained evenwhen the field is massless. For instance, a massless bosonicfield interacting with a black hole immersed in a magnetic fieldis likely to form bound states. The magnetic field creates a po-tential barrier, confining the field into the neighborhood of theblack hole. a r X i v : . [ g r- q c ] F e b An example that naturally embodies this idea is the interac-tion of a massless scalar field with a Reissner–Nordstr¨om blackhole embedded in a uniform axial magnetic field . The latteris described by the Reissner–Nordstr¨om–Melvin (RNM) so-lution [15, 16], obtained via a solution–generating techniqueknown as Harrison (or “magnetizing”) transformation. In-terestingly, the RNM solution is a stationary (rather than astatic) solution of the Einstein–Maxwell theory . The rotationis sourced by the coupling between the black hole’s electriccharge and the external magnetic field. Besides, the space-time features an ergoregion and, as a result, is prone to black–hole superradiance even for electrically neutral bosonic fields.This contrasts with the case of asymptotically–flat Reissner–Nordstr¨om black holes wherein (charged) superradiance ispossible but only for charged bosonic fields [17] and a superra-diant instability does not follow from a mass term; it requires,for instance, enclosing the black hole with a reflecting mirror– see, e.g. , [18–20].The present paper focuses on bound states between a mass-less scalar field and a RNM black hole (cf. [21]). These real–frequency states are characterized by the threshold of super-radiance w = m Ω H , hereafter referred to as synchronisationcondition , and were first reported in [22], in which the authornamed them stationary clouds . Much attention has been paidto such synchronized states since their discovery [14, 23–42],yet most works rely on intrinsically massive fields. For thecase under consideration here, the fields need not have a non-vanishing mass for stationary clouds to arise. A peculiar fea-ture of this model is that the scalar field’s effective mass is pro-portional to the black hole’s angular velocity, the proportion-ality constant being a function of the specific electric charge
Q/M alone, where M and Q are, respectively, the black hole’smass and electric charge. Curiously enough, the condition forthe existence of bound states is only met for values of Q/M ina subset of [ − , .The paper is organized as follows. First, the Einstein–Maxwell theory minimally coupled to a complex, ungaugedscalar field is introduced in section II. Together with a constantscalar field, the RNM solution is a particular case of the the-ory. Its main features are outlined in section II A, followed by alinear analysis of scalar field perturbations in section II B. Themain results on stationary clouds are presented in section III.A summary of the work can be found in section IV.Natural units ( G = c = 1 ) are consistently used through-out the text. Additionally, the metric signature ( − , + , + , +) isadopted. Although this is not a realistic astrophysical scenario, it suffices to sketchthe main argument of the paper. The same is true for AdS asymptotics – see, e.g. , [26].
II. FRAMEWORK
The action for the Einstein–Maxwell theory minimally cou-pled to a complex , ungauged scalar field Ψ is S = 14 π (cid:90) d x √− g (cid:20) R − F − ( ∇ µ Ψ ∗ )( ∇ µ Ψ) (cid:21) , (3)where F = d A is the electromagnetic tensor and A is elec-tromagnetic four–potential.The corresponding equations of motion read G µν = 2 (cid:104) T ( A ) µν + T (Ψ) µν (cid:105) , (cid:3) Ψ = 0 , ∇ µ F µν = 0 , (4)where (cid:3) ≡ ∇ µ ∇ µ is the d’Alembert operator and T ( A ) µν ≡ F µσ F νσ − g µν F σλ F σλ , (5) T (Ψ) µν ≡ ∂ ( µ Ψ ∗ ∂ ν ) Ψ − g µν ( ∂ λ Ψ ∗ )( ∂ λ Ψ) (6)are the stress–energy tensors of the electromagnetic and scalarfields, respectively. The action has a global U (1) invariancewith respect to the scalar field thanks to its complex character.This field theory admits all of the stationary solutions ofgeneral relativity. These are characterized by Ψ = Ψ , forsome constant Ψ . Linearizing the equations of motion around Ψ = Ψ , one obtains the ordinary Einstein–Maxwell equa-tions together with the Klein–Gordon equation for the scalarfield perturbation δ Ψ ≡ (Ψ − Ψ ) . This system describesthe linear or zero–backreaction limit of the theory: the limitin which the backreaction of both the gravitational and elec-tromagnetic fields to a non–constant scalar field is negligible.This first–order approximation suffices to capture potentiallyrelevant astrophysical phenomena such as superradiant scat-tering. The framework allows one to solve the Klein–Gordonequation (cid:3) ( δ Ψ) = 0 for a known solution { g , A } of theEinstein–Maxwell equations. A. Reissner–Nordstr¨om–Melvin black holes
This paper will focus on scalar field perturbations of RNMblack holes. These solutions belong to a family of elec-trovacuum type D solutions of the Einstein–Maxwell equa-tions which asymptotically resemble the magnetic Melvin uni-verse. The latter describes a non–singular, static, cylindri-cally symmetric spacetime representing a bundle of magneticflux lines in gravitational–magnetostatic equilibrium. It canbe loosely interpreted as Minkowski spacetime immersed ina uniform magnetic field; but it should be kept in mind thatsuch magnetic field, no matter how small, changes the globalstructure of the spacetime, in particular its asymptotics. Stationary clouds are not exclusive to complex scalar fields. A single realscalar field can equally form infinitely long–lived states at linear level –see [43].
Given an asymptotically–flat, stationary, axi–symmetric so-lution of Einstein–Maxwell equations, it is possible to embedit in a uniform magnetic field via a solution–generating tech-nique called Harrison transformation (also commonly knownas “magnetizing” transformation). This possibility, first real-ized by Harrison [44], was explored for the Schwarzschild andReissner-Nordstr¨om solutions [15] and for the Kerr and Kerr–Newman solutions [45].The RNM solution, which describes a Reissner–Nordstr¨omblack hole permeated by a uniform magnetic field, reads [16] g = | Λ | (cid:18) − ∆ r d t + r ∆ d r + r d θ (cid:19) + r sin ϑ | Λ | ( d ϕ − Ω d t ) , A = Φ d t + Φ ( d ϕ − Ω d t ) (7)where t ∈ ( −∞ , + ∞ ) , r ∈ (0 , + ∞ ) , ϑ ∈ [0 , π ] , ϕ ∈ [0 , π ) and ∆ = r − M r + Q , Λ = 1 + 14 B ( r sin ϑ + Q cos ϑ ) − iQB cos ϑ , Ω = − QB r + QB r (cid:18) r cos ϑ (cid:19) , Φ = − Qr + 34 QB r (cid:18) r cos ϑ (cid:19) , Φ = 2 B − | Λ | (cid:20) B + B (cid:0) r sin ϑ + 3 Q cos ϑ (cid:1)(cid:21) .B is the strength of the magnetic field, which is assumed to bemuch weaker than the threshold value (1), i.e M B (cid:28) M B =1 . When applied to the Reissner–Nordstr¨om solution, the Har-rison transformation produces a stationary (rather than a static)solution. The dragging potential Ω is directly proportional tothe coupling QB , which suggests that the interaction betweenthe charge Q and the magnetic field B serves as a source forrotation.The solution possesses two (commuting) Killing vectors, ξ = ∂ t and η = ∂ ϕ , associated to stationarity and axi–symmetry, respectively. The line element has coordinate sin-gularities at ∆ = 0 when Q ≤ M , which solves for r ± = M ± (cid:112) M − Q . The hypersurface r = r + ( r = r − )is the outer (inner) horizon. Besides, there is an ergo–regionthat extends to infinity along the axial direction, but not in theradial direction. Here, ergo–region means the regions outsidethe outer horizon wherein ξ is spacelike.The dragging potential Ω is constant (i.e. ϑ –independent)on r = r + , where it has the value Ω H ≡ − QB r + (cid:18) − r B (cid:19) . (8) Ω H is the angular velocity of the outer horizon. The Killingvector χ = ξ +Ω H η becomes null on the hypersurface r = r + and it is timelike outside it. B. Scalar field perturbations
In general, the Klein–Gordon equation (cid:3) ( δ Ψ) = 0 does notadmit a multiplicative separation of variables of the form δ Ψ( t, r ) = e − iwt R ( r ) S ( ϑ ) e + imϕ , (9)where w is the phase angular velocity, R and S are respectivelythe radial and angular fucntions and m ∈ Z is the azimuthalharmonic index. However, in the limit of sufficiently “weak”magnetic fields, i.e. neglecting terms of order higher than O ( B ) , the ansatz (9) actually reduces the problem to twodifferential equations in the coordinates r and ϑ . The radialand angular equations read [21] dd r (cid:18) ∆ d R d r (cid:19) + (cid:20) K ∆ − ( m B r + λ ) (cid:21) R = 0 , (10) ϑ dd ϑ (cid:18) sin ϑ d S d ϑ (cid:19) + (cid:18) λ − m sin ϑ − m Q B cot ϑ (cid:19) S = 0 , (11)respectively, where K = r w + 2 mQB r and λ is the sepa-ration constant. Equations (10)–(11) are both confluent Heunequations: the former (latter) has singular points at r = r ± ( ϑ = 0 , π ). They are coupled via the Killing eigenvalues { w, m } , B , Q and the separation constant λ and remaininvariant under the discrete transformation { w, mQB } →{− w, − mQB } . This guarantees that, without loss of gener-ality, one can take sgn( w ) = sgn( B ) = +1 . When mQB =0 , the angular equation reduces to the general Legendre equa-tion, whose canonical solutions are the associated Legendrepolynomials of degree (cid:96) and order m , P m(cid:96) ( ϑ ) , provided that λ = (cid:96) ( (cid:96) + 1) . Thus, if | mQB | (cid:28) , the angular dependenceof δ Ψ is approximately described by the scalar spherical har-monics of degree (cid:96) and order m , Y m(cid:96) ( ϑ, ϕ ) = P m(cid:96) ( ϑ ) e + imϕ .Equation (10) can be cast in Schr¨odinger–like form, yielding − d ρ d y + V eff ( y ) ρ = w ρ , (12)where ρ ≡ rR and y is the tortoise coordinate, defined by y ( r ) = r + r r + − r − log( r − r + ) − r − r + − r − log( r − r − ) , which maps the interval r ∈ [ r + , ∞ ) into r ∗ ∈ ( −∞ , + ∞ ) .The effective potential V eff , whose expression is omitted here,has the following limiting behavior: lim y →−∞ V eff ( y ) = w − ( w − m Ω H ) , (13) lim y → + ∞ V eff ( y ) = m B . (14) For a straightforward identification of the order of eachterm, it is convenient to introduce the dimensionless quanti-ties { tB , rB , MB , QB , w/B } so that all physical quantitiesare measured in units of the magnetic field strength. Note that the first fourquantities are of order O ( B ) , whereas the last is of order O ( B − ) . The last limit suggests that a non–vanishing external mag-netic field makes the scalar field acquire an effective mass µ eff = (cid:112) m B . It is important to remark, however, that theproblem at hand is not equivalent to that of a massive scalarfield perturbation on an asymptotically–flat stationary space-time, wherein the mass dominates the asymptotic behavior ofthe field. Besides providing the field an effective mass, themagnetic field also changes the asymptotic behavior at infinity(to be that of the Melvin magnetic universe), which has sim-ilarities with AdS asymptotics in the sense that it is naturallyconfining. FIG. 1. Effective potential for scalar field perturbations with (cid:96) = m = 1 and w = 0 . B of RNM black holes with MB = 0 . .(Inset) Zoom near rB ∼ to display the maximum of the effectivepotential. Figure 1 shows the effective potential as a function of theradial coordinate r for different (negative) specific electriccharges. In an asymptotically–Melvin spacetime, the magneticfield acts like a potential barrier at rB ∼ , whose maximum,about ten times larger than O ( B ) , approaches the outer hori-zon with decreasing Q/M (i.e. tending to extremality). More-over, there is a potential well for all positive specific electriccharges (not plotted in Figure 1) as well as for negative onesabove a certain threshold (away from extremality). The effec-tive potential resembles a mirror placed at rB ∼ and con-fines (low–frequency) scalar field perturbations in the blackhole’s vicinity [46, 47]. It is then natural to impose a Robin(or mixed) boundary condition at r = r as the outer bound-ary condition, tan( ζ ) = − R ( r ) R (cid:48) ( r ) , (15)where r is of order O ( B − ) , ζ ∈ [0 , π ) , with ζ = 0 ( ζ = π/ ) corresponding to a Dirichlet (Neumann) boundarycondition, and the prime denoting differentiation with respectto r .In realistic astrophysical scenarios, magnetic fields occur inaccretion disks around black holes. The “magnetic” potential barrier is then at a radial distance smaller than about the meanradius D of the disk, i.e. r (cid:46) D . Since the matter in theaccretion disk is expected to be close to the innermost stablecircular orbit, D ∼ M and it follows that M B (cid:38) . , whichclashes with the assumption M B (cid:28) (for a more completediscussion, see [47]). Despite this caveat, the main argumentof the paper holds at least from a purely theoretical perspective.Furthermore, physically meaningful solutions to the radialequation satisfy the inner boundary condition R | y →−∞ ∼ e − i ( ω − m Ω H ) y , (16)i.e. they behave as waves falling into (emanating from) theblack hole when w > m Ω H ( < w < m Ω H ). III. STATIONARY SCALAR CLOUDS
When the scalar field’s phase angular velocity is a naturalmultiple of the black hole’s angular velocity, i.e. w = m Ω H = − mQB r + + O ( B ) , (17)bound states, known as stationary clouds , are found. Equa-tion (17) is called synchronisation condition and does dependon the scalar field’s effective mass, µ eff = (cid:112) m B . The ra-tio | w/µ eff | = 2 | Q | /r + is independent of B and its absolutevalue is smaller than or equal to . Since it was assumed that sgn( w ) = sgn( B ) = +1 , the synchronisation condition dic-tates that the bound states satisfy sgn( mQ ) = − .The synchronisation occurs in one–dimensional subsets ofthe two–dimensional parameter space of Reissner–Nordst¨om–Melvin black holes, described by { M, Q } . These subsets –known as existence lines – are disjoint and can be labeled witha set of three “quantum” numbers: the number of nodes in theradial direction n , the orbital/total angular momentum (cid:96) andthe azimuthal harmonic index m . These states will be labeledwith | n, (cid:96), m (cid:105) .In the following, stationary scalar clouds around RNM blackholes are obtained both (semi–)analytically and numerically.The existence lines will be plotted in the ( M, Q ) –plane nor-malized to the magnetic field strength B . A. Analytical approach
The eigenvalue problem at hand can be solved using thematched asymptotic expansion method (see, e.g. , [48]), i.e.constructing approximations to the solutions of (10) that sep-arately satisfy the inner and outer boundary conditions. Theinterval r ∈ [ r + , r ] is thus split into two: (i) the inner re-gion, r − r + (cid:28) λ c , where λ c = µ − eff ≤ r + / ( m | Q | B ) isthe scalar field’s Compton wavelength; inspection shows that The number of nodes in the radial direction does not include the node at r = r when ζ = 0 (Dirichlet boundary condition). λ c (cid:29) M ; and (ii) the outer region, r − r + (cid:29) M . The in-ner and outer expansions are then matched in the overlap re-gion, where both conditions can hold simultaneously, definedby M (cid:28) r − r + (cid:28) λ c .
1. Outer region
The outer region is well–defined only if the outer bound-ary is sufficiently far from the black hole, i.e. as long as r (cid:29) M . Given that Q ≤ M , one can take ∆ ∼ r .Besides, if r (cid:29) | mQB /w | , then K ∼ wr . When thesyncrhonization condition (17) holds, the latter approximationis equivalent to r (cid:29) r + , which is consistent with r − r + (cid:29) M .The radial equation (10) then reduces to that of a masslessscalar field perturbation with phase angular velocity definedby (cid:36) ≡ w − µ eff = m B (4 Q /r − and angular mo-mentum (cid:96) in Minkowski spacetime , d d r ( rR + ) + (cid:20) (cid:36) − (cid:96) ( (cid:96) + 1) r (cid:21) ( rR + ) = 0 , (18)where R + ( r ) ≡ lim r → r R ( r ) . The general solution is R + ( r ) = α + j (cid:96) ( (cid:36)r ) + β + y (cid:96) ( (cid:36)r ) , (19)where j (cid:96) and y (cid:96) are the spherical Bessel functions of the firstand second kinds, respectively, and α + , β + ∈ C . For suf-ficiently large r , the spherical Bessel functions are a linearcombination of ingoing and outgoing waves if (cid:36) is real, i.e.if w > µ eff . The Robin boundary condition (28) fixes thequotient γ ≡ β + α + = (cid:20) − j (cid:96) ( (cid:36)r ) + tan( ζ ) j (cid:48) (cid:96) ( (cid:36)r ) y (cid:96) ( (cid:36)r ) + tan( ζ ) y (cid:48) (cid:96) ( (cid:36)r ) (cid:21)(cid:12)(cid:12)(cid:12)(cid:12) r = r . (20)The small– r behavior of the asymptotic solution (19) is R + ( r ) ∼ α + ( (cid:36)r ) (cid:96) (2 (cid:96) + 1)!! − β + (2 (cid:96) − (cid:36)r ) (cid:96) +1 . (21)
2. Inner region
Near the outer horizon, the radial equation (10) reduces to dd r (cid:18) ∆ d R − d r (cid:19) − (cid:96) ( (cid:96) + 1) R − = 0 , (22)where R − ( r ) ≡ lim r → r + R ( r ) . Introducing the radial co-ordinate z ≡ ( r − r + ) / ( r − r − ) and defining R − ( z ) =(1 − z ) (cid:96) +1 F ( z ) , one can bring the radial equation (22) intothe form z (1 − z ) d F d z + [ c − ( a + b + 1) z ] d F d z − abF = 0 , (23) Alternatively, one could say that Eq. (18) describes a scalar field with mass √ m B , phase angular velocity m Ω H and angular momentum (cid:96) . with a = b ≡ (cid:96) + 1 and c ≡ . Equation (23) is a Gaus-sian hypergeometric equation, which has three regular singularpoints: z = 0 , , ∞ . The most general solution is [11, 49] F ( z ) = α − F ( a, a ; 1; z )+ β − F ( a, a ; 1; z ) log z + 2 + ∞ (cid:88) j =1 f ( j ) z j , (24)where f ( j ) = (cid:20) ( a ) j j ! (cid:21) [ ψ ( a + j ) − ψ ( a ) − ψ ( j + 1) + ψ (1)] and ( a ) j = Γ( a + j ) / Γ( a ) and ψ is the digamma function.The second term in Eq. (24) diverges logarithmically as z → ( r → r + ). As the inner boundary condition must be regular,the constant β − must vanish. In terms of the radial function R − , the solution thus reads R − ( z ) = α − (2 (cid:96) + 1)!( (cid:96) !) (cid:20) ( − (cid:96) +1 R ( D ) ( z ) + R ( N ) ( z )( (cid:96) + 1) (cid:21) , where R ( D ) − ( z ) = (1 − z ) (cid:96) +1 F ( (cid:96) + 1 , (cid:96) + 1; 2 (cid:96) + 2; 1 − z ) ,R ( N ) − ( z ) = (1 − z ) − (cid:96) F ( − (cid:96), − (cid:96) ; − (cid:96) ; 1 − z ) . When r (cid:29) M , z ∼ and (1 − z ) ∼ ( r + − r − ) /r , meaningthat R ( D ) − ( z ) ∼ ( r + − r − ) (cid:96) +1 r − (cid:96) − ,R ( N ) − ( z ) ∼ ( r + − r − ) − (cid:96) r (cid:96) .
3. Matching
It is clear that the larger– r behavior of the asymptotic so-lution R − exhibits the same dependence on r as the small– r behavior of the asymptotic solution R + . Matching the two so-lutions, one gets γ = ( (cid:96) + 1) (2 (cid:96) + 1)!!(2 (cid:96) − (cid:36) ( r + − r − )] (cid:96) +1 . (25)Using Eq. (20), one finally obtains tan( ζ ) = − j (cid:96) ( (cid:36)r ) + γy (cid:96) ( (cid:36)r ) j (cid:48) (cid:96) ( (cid:36)r ) + γy (cid:48) (cid:96) ( (cid:36)r ) , (26)which establishes the existence condition for stationary scalarclouds around (non–extremal) RNM black holes. These existas long as the field perturbation has a radial oscillatory char-acter and therefore can satisfy a Robin boundary condition at r B ∼ . This requirement is met provided that (cid:36) is real,i.e. if w > µ eff ⇔ Q r > ⇒ Q M > , (27)or | Q/M | ∈ (0 . , . , where sgn( Q ) = ± for sgn( m ) = ∓ so that sgn( w ) = +1 . Note that this restriction on thespecific electric charge is a by–product of the proportionalitybetween w = m Ω H and µ eff . B. Numerical approach
Stationary clouds can also be found by solving numer-ically the coupled equations (10)–(11). For that purpose,it is convenient to replace the mass M by the outer hori-zon radius r + and work with the dimensionless quantities { r + B , QB , Ω H /B } . To impose the correct inner bound-ary condition the radial function may be written as a seriesexpansion around r = r + [50], R | r → r + ∼ + ∞ (cid:88) j =0 a ( j ) ( r − r + ) j . (28)The coefficients { a ( j ) } j> are obtained by plugging (28)into (10), writing the resulting equation in powers of ( r − r + ) and setting the coefficient of each power separately equal tozero. The resulting system of equations must then be solvedfor { a ( j ) } j> in terms of a (0) . The latter is set to with-out loss of generality. The coefficients { a ( j ) } j> depend onthe black hole’s parameters { r + , Q } , the Killing eigenvalue m and the separation constant λ . Instead of solving the an-gular equation (11), one approximates the latter by (cid:96) ( (cid:96) + 1) ,which is accurate enough if mQB (cid:28) . Since Q ≤ M and M B (cid:28) M B = 1 , the approximation is valid for moderatevalues of m .The parameters { r + , (cid:96), m } are assigned fixed values. Byvirtue of the regular singular point at r = r + , Eq. (10) mustbe integrated from r = r + (1 + δ ) , with δ (cid:28) , to r = r ,where r is the outer boundary radial coordinate. A simpleshooting method finds the Q –values for which the numericalsolutions satisfy a Robin boundary condition at r = r . C. Existence lines
Figure 2 displays the (numerical) existence lines for sta-tionary clouds | , , (cid:105) with r B ∈ { , , , } and ζ ∈{ , π , π } . The shaded bands represent the allowed regionsof the parameter space for the existence of bound states. Theupper boundary, defined by Q = M , corresponds to the extremal line . The RNM black holes in the lower boundarysatisfy Q = 0 . M , in accordance with the conclusion atthe end of section III A 3.The panels below the main plots show the absolute differ-ence σ between each existence line and that corresponding to r B = 4 and the absolute difference ε between the numeri-cal and analytical existence lines. As expected, given that theanalytical condition (26) is valid when M B (cid:28) , ε → as M B → .All existence lines lie within the shaded bands. Also, theyconverge to ( M, Q ) = (0 , , i.e. σ → as M B → ,which is in agreement with the expectation that scalar field per-turbations cannot attain stationary equilibrium with respect toasymptotically–Melvin black holes. Fixing M B , as the re-gion of influence of the magnetic field decreases, i.e. as r B decreases, the Coulomb energy of the black hole supportingthe stationary cloud increases. Vaster clouds thus require lowerangular velocities so that they do not collapse into the black FIG. 2. Stationary scalar clouds | n, (cid:96), m (cid:105) = | , , (cid:105) aroundReissner–Nordstr¨om black holes embedded in a uniform axial mag-netic field of strength B , for different Robin boundary conditions,parametrized by ζ , at the outer boundary r . Fixing MB , hole. Also, there is an overall decrease in the Coloumb energyas ζ varies continuously from (Dirichlet boundary condition)to π (Neumann boundary condition).The existence lines for the states | , (cid:96), m (cid:105) with (cid:96) = m =1 , . . . , , r B = 4 and ζ = 0 are plotted in Figure 3. Theseapproach the extremal line as (cid:96) = m decreases, a trend alreadynoticed in previous works (see, e.g. , [25]). FIG. 3. Stationary scalar clouds | n, (cid:96), m (cid:105) = | , (cid:96), (cid:96) (cid:105) aroundReissner–Nordstr¨om black holes embedded in a uniform axial mag-netic field of strength B and satisfying a Dirichlet boundary condi-tion ( ζ = 0 ) at r B = 4 . The impact of the orbital angular momentum (cid:96) is enlightnedin Figure 4, in which the existence lines for the states | , (cid:96), (cid:105) with (cid:96) = 1 , . . . , , r B = 6 and ζ = 0 are shown.As (cid:96) increases, so does | Q/M | , which suggests that stationaryclouds | , (cid:96), (cid:105) with (cid:96) > are more energetic than | , , (cid:105) . IV. CONCLUSION
The RNM black hole stands out as a toy model for a rotat-ing black hole immersed in an external axial magnetic field.In fact, it is the simplest stationary (but not static) solutionof Einstein–Maxwell equations asymptotically resembling themagnetic Melvin universe. Frequently overlooked due to itsastrophysical irrelevance, it is still worth studying as it may of-fer some insights into the interaction of black holes with mag-netic fields.The present paper aimed precisely to explore the interplaybetween bosonic fields and black holes when permeated by auniform magnetic field. It was shown in particular that RNMblack holes support synchronized scalar field configurationsknown as stationary clouds. They are somehow akin to atomicorbitals of the hydrogen atom in quantum mechanics in thatthey are both described by quantum number. In effect, sta-tionary clouds are characterized by the number of nodes in theradial direction, n , the orbital angular momentum, (cid:96) , and theazimuthal harmonic index, m , which labels the projection of FIG. 4. Stationary scalar clouds | n, (cid:96), m (cid:105) = | , (cid:96), (cid:105) , with (cid:96) =1 , . . . , , around Reissner–Nordstr¨om black holes embedded in a uni-form axial magnetic field of strength B and satisfying a Dirichletboundary condition ( ζ = 0 ) at r B = 6 . the orbital angular momentum along the direction of the mag-netic field.It is now well known that stationary equilibrium is possi-ble whenever a bosonic field at the threshold of superradiantinstabilities (i.e. obeying the so–called syncrhonization condi-tion) is confined in the black hole’s vicinity. The confinementmechanism (either natural or artificial) creates a potential bar-rier which may prevent the field from escaping to infinity. As aresult, infinitely long–lived configurations arise. For example,a massive bosonic field can form such stationary clouds aroundKerr black holes – with the field’s mass providing a naturalconfinement mechanism. So does a massless charged scalarfield in a cavity enclosing a Reissner–Nordstr¨om black hole– with the boundary of the cavity, a reflective mirror, sourc-ing an artificial confinement mechanism [18]. The propertiesof both equilibrium configurations are similar despite minorqualitative differences.Additionally worth mentioning is the fact that, in two pre-vious examples, the occurrence of superradiance does not relyon the existence of a confining environment; one could saythat the two ingredients are added separately. However, inthe setup under consideration, the magnetic field of the RNMblack hole is responsible not only for developing an ergoregionand hence trigger superradiant phenomena but also for mak-ing low–frequency fields acquire an effective mass and thusbe trapped, allowing the formation of stationary clouds. Inview of this, it does not come as a surprise that both the blackhole’s angular velocity Ω H and the field’s effective mass µ eff – synonyms for superradiance and confinement, respectively –depend on B .Lastly, a by–product of considering the RNM black hole wasthe realization that the quotient m Ω H /µ eff is a function of theblack hole’s specific electric charge Q/M only. Consequently,the condition for the existence of bound states constrains thevalues of
Q/M for which stationary clouds can exist.
V. ACKNOWLEDGEMENTS
This work has been supported by the Center for Astro-physics and Gravitation (CENTRA) and by the Center forResearch and Development in Mathematics and Applications(CIDMA) through the Portuguese Foundation for Science and Technology (FCT – Fundac¸ ˜ao para a Ciˆencia e a Tec-nologia), references UIDB/00099/2020, UIDB/04106/2020and UIDP/04106/2020. The authors acknowledge supportfrom the projects PTDC/FIS-OUT/28407/2017, CERN/FIS-PAR/0027/2019 and PTDC/FIS-AST/3041/2020. N. M. San-tos is supported by the FCT grant SFRH/BD/143407/2019.This work has further been supported by the EuropeanUnion’s Horizon 2020 research and innovation (RISE) pro-gram H2020-MSCA-RISE-2017 Grant No. FunFiCO-777740.The authors would like to acknowledge networking support bythe COST Action CA16104. [1] B. Abbott et al. (LIGO Scientific, Virgo), Phys. Rev. X ,031040 (2019), arXiv:1811.12907 [astro-ph.HE].[2] V. Cardoso, T. Ikeda, C. J. Moore, and C.-M. Yoo, Phys. 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