Infinitesimally thin static scalar shells surrounding charged Gauss-Bonnet black holes

We reveal the existence of a new form of spontaneously scalarized black-hole configurations. In particular, it is proved that Reissner-Nordstr\"om black holes in the highly charged regime $Q/M>(Q/M)_{\text{crit}}=\sqrt{21}/5$ can support {\it thin} matter shells that are made of massive scalar fields with a non-minimal coupling to the Gauss-Bonnet invariant of the curved spacetime. These static scalar shells, which become infinitesimally thin in the dimensionless large-mass $M\mu\gg1$ regime, hover a finite proper distance above the black-hole horizon [here $\{M,Q\}$ are respectively the mass and electric charge of the central supporting black hole, and $\mu$ is the proper mass of the supported scalar field]. In addition, we derive a remarkably compact analytical formula for the discrete resonance spectrum $\{\eta(Q/M,M\mu;n)\}_{n=0}^{n=\infty}$ of the non-trivial coupling parameter which characterizes the bound-state charged-black-hole-thin-massive-scalar-shell cloudy configurations of the composed Einstein-Maxwell-scalar field theory.

The main goal of the present paper is to explore the physical and mathematical properties of non-minimally coupled linearized massive scalar field configurations (massive scalar clouds) that are supported by charged Reissner-Nordström black holes with spatially regular horizons. In particular, using analytical techniques, we shall reveal the physically intriguing fact that the addition of a mass term to the supported non-minimally coupled scalar fields [see the action (1), which characterizes the composed Einstein-Maxwell-massive-scalar field theory] allows the existence of infinitesimally thin static scalar shells that hover a finite proper distance above the horizons of highly charged Gauss-Bonnet black holes.
In addition, below we shall derive a remarkably compact analytical resonance formula that describes, in the large-mass Mµ ≫ 1 regime, the functional dependenceη =η(Q/M, Mµ) of the critical existence line, which characterizes the composed charged-black-hole-thinmassive-scalar-shell bound-state cloudy configurations, on the electric charge Q of the central supporting black hole and on the dimensionless proper mass Mµ [33] of the non-minimally coupled scalar field.

II. DESCRIPTION OF THE SYSTEM
We study the physical and mathematical properties of 'cloudy' black-hole configurations which are made of central charged Reissner-Nordström black holes that support spatially regular bound-state static configurations of linearized massive scalar fields. The composed Einstein-Maxwell-Gauss-Bonnet-nonminimally-coupled-massive-scalar field theory is characterized by the action [21,22,34] where µ is the mass of the non-minimally coupled scalar field. As we shall explicitly prove below, the supported matter configurations owe their existence to the presence, in the action (1), of a direct (non-minimal) coupling between the massive scalar field φ and the Gauss- that characterizes the curved black-hole spacetime.
As discussed in [11,12,24], the leading-order functional behavior of the scalar coupling function guarantees that the bald Reissner-Nordström black-hole spacetime is a valid solution of the composed Einstein-Maxwell-scalar field equations in the weak-field φ → 0 limit [35]. The strength of the non-minimal coupling between the supported massive scalar field and the Gauss-Bonnet curvature invariant (2) is controlled by the physical parameter η [36].
The curved line element [37][38][39] characterizes the supporting Reissner-Nordström black hole of mass M and electric charge Q. The roots of the metric function (5) determine the radii of the (outer and inner) black-hole horizons.
The action (1) of the composed Einstein-Maxwell-massive-scalar field theory yields the Klein-Gordon differential equation [22] ∇ ν ∇ ν φ = µ 2 eff φ for the eigenfunctions of the supported scalar field configurations, where the effective mass term which depends on the Gauss-Bonnet curvature invariant of the Reissner-Nordström black-hole spacetime (4), reflects the non-trivial massive-scalarfield-Gauss-Bonnet coupling in the composed field theory (1).
Intriguingly, one finds that, depending on the relative magnitudes of the physical parameters {η, µ} of the composed field theory (1), the radially-dependent effective mass term (8) may become negative in the vicinity of the black-hole outer horizon. Below we shall use analytical techniques in order to prove that this property of the effective scalar-field-Gauss-Bonnet mass term (8) may allow the existence of infinitesimally thin non-minimally coupled massive scalar shells (thin massive scalar clouds) that hover a finite proper distance above the horizons of highly charged [see Eq. (23) below] Reissner-Nordström black holes.
Substituting the functional decomposition for the static non-minimally coupled massive scalar field into Eq. (7) [here Y lm (θ, ϕ) with l ≥ |m| are the familiar spherical harmonic functions] and using the curved black-hole line element (4), one obtains the radial differential equation [22,40] d dr which determines the spatial behavior of the supported massive scalar clouds in the curved black-hole spacetime (4).
The radial differential equation (11), supplemented by the physically motivated boundary conditions of spatially regular (bounded) functional behavior of the scalar field at the blackhole outer horizon [11,12,41], and an asymptotic exponential decay of the massive scalar eigenfunction at spatial infinity [11,12,41], In particular, we shall explicitly prove that the composed Reissner-Nordström-black-holenonminimally-coupled-massive-scalar-field system is amenable to an analytical treatment in the dimensionless large-mass regime (14), which corresponds to the dimensionless largecoupling regimeη To this end, it is convenient to define the radial scalar eigenfunction in terms of which the radial equation (11) can be expressed in the mathematically compact where the differential relation [42] determines the new radial coordinate y(r). The effective potential in the Schrödinger-like radial differential equation (17), which characterizes the composed black-hole-massive-scalarfield system, is given by the (rather cumbersome) functional expression The presence of the Gauss-Bonnet term in the effective interaction potential (19) is a direct consequence of the non-trivial (nonminimal) coupling between the Gauss-Bonnet curvature invariant (9) of the charged blackhole spacetime (4) and the supported massive scalar field.
We shall henceforth consider composed black-hole-massive-scalar-field cloudy configurations in the dimensionless large-mass regime (14) [or equivalently, in the dimensionless large-coupling regime (15)], in which case the effective black-hole-massive-field interaction potential (19) can be written in the form [43] V The Gauss-Bonnet term (20) has a peak whose charge-dependent radius is given by the simple functional relation Interestingly, one finds that, in the dimensionless charge-to-mass ratio regime the radial peak (22) is located outside the black-hole outer horizon [that is, r peak ≥ r + in the regime (23)]. Below we shall prove that this intriguing physical property of the nontrivial Gauss-Bonnet term (20) allows the existence of infinitesimally thin massive scalar shells that hover a finite proper distance above the outer horizons of central supporting Reissner-Nordström black holes in the highly charged regime (23) [44,45].
Substituting (22) into Eq. (20), one finds the functional relations [46] for the maximal value of the non-trivial Gauss-Bonnet term (20) in the exterior regions of charged Reissner-Nordström black holes.

A. Upper bound on the proper masses of non-minimally coupled scalar clouds
In the present subsection we shall derive a charge-dependent upper bound on the allowed proper masses of the non-minimally coupled scalar fields that can be supported by the central charged Reissner-Nordström black holes. To this end, we point out that the presence of a binding (attractive) potential well outside the black-hole outer horizon provides a necessary condition for the existence of static bound-state scalar field configurations (scalar clouds) that are supported in the curved black-hole spacetime.
In particular, the requirement [here {r t − , r t + } with r t − ≥ r + are the characteristic classical turning points of the effective curvature potential (19)] yields the series of inequalities [47] Taking cognizance of Eqs. (24) and (26), one finds the charge-dependent upper bound on the allowed proper masses of the supported non-minimally coupled scalar fields.

CLOUDS
In the present section we shall determine the effective widths of supported massive scalar field configurations in the charged Reissner-Nordström black-hole spacetime (4). In particular, we shall explicitly prove that, in the dimensionless large-mass Mµ ≫ 1 regime, the supported scalar clouds can be made arbitrarily thin.
The effective widths of the supported matter configurations in the charged black-hole spacetime are determined by the classically allowed radial region of the effective binding potential (21), which characterizes the composed black-hole-massivescalar-field system (1). Taking cognizance of Eqs. · 50 for the effective widths of the supported massive scalar field configurations, which in the large-mass regime (14) can be expressed in the form [see Eq. (39)] In the present section we shall analyze the spatial behavior of the non-minimally coupled massive scalar field configurations outside the narrow radial interval (40) [see also (43)] of the classically allowed region.
It is important to stress the fact that the scalar field is not strictly zero outside the classically allowed region (40). However, as we shall now demonstrate explicitly, the effective penetration depth of the scalar eigenfunction into the classically forbidden region becomes infinitesimally small in the large-mass Mµ ≫ 1 regime (14).
In particular, according to the standard WKB analysis [48][49][50], the radial function that characterizes the thin massive scalar field configurations is exponentially suppressed in the classically forbidden region outside the narrow radial interval (40). Interestingly, and most importantly for our analysis, one finds the relation [see Eqs. (21) and (39)] in the large-mass regime (14). Thus, the effective penetration depth of the WKB scalar eigenfunction into the classically forbidden region [that is, into the region outside the classically allowed narrow radial interval (40)] scales as 1/µ and therefore becomes infinitesimally small in the large mass Mµ ≫ 1 regime (14) that we consider in the present paper. note that this dimensionless ratio becomes even smaller for larger field masses. For example, for a supported massive scalar field with Mµ = 1000 one finds the characteristic small ratio ψ(r = r + )/ψ(r = r peak ) ≃ 8.7 × 10 −2190 ].
(3) It has been shown that the supported scalar clouds are characterized by the effective dimensionless widths [54] ∆r(Q/M, Mµ) Intriguingly, the analytically derived functional expression (47) implies that the supported scalar configurations, which are made of non-minimally coupled massive scalar fields, can be made arbitrarily thin in the dimensionless large-mass Mµ ≫ 1 regime.
Finally, it is worth emphasizing the fact that the analytically derived critical existenceline (48) marks, in the dimensionless large-mass regime (14), the sharp boundary between bald black-hole solutions of the Einstein-Maxwell-Gauss-Bonnet-massive-scalar field theory (1) and hairy (scalarized) black-hole spacetimes that characterize the composed physical system.