Onset of superradiant instabilities in rotating spacetimes of exotic compact objects

Exotic compact objects, horizonless spacetimes with reflective properties, have intriguingly been suggested by some quantum-gravity models as alternatives to classical black-hole spacetimes. A remarkable feature of spinning horizonless compact objects with reflective boundary conditions is the existence of a {\it discrete} set of critical surface radii, $\{r_{\text{c}}({\bar a};n)\}^{n=\infty}_{n=1}$, which can support spatially regular static ({\it marginally-stable}) scalar field configurations (here ${\bar a}\equiv J/M^2$ is the dimensionless angular momentum of the exotic compact object). Interestingly, the outermost critical radius $r^{\text{max}}_{\text{c}}\equiv \text{max}_n\{r_{\text{c}}({\bar a};n)\}$ marks the boundary between stable and unstable exotic compact objects: spinning objects whose reflecting surfaces are situated in the region $r_{\text{c}}>r^{\text{max}}_{\text{c}}({\bar a})$ are stable, whereas spinning objects whose reflecting surfaces are situated in the region $r_{\text{c}}<r^{\text{max}}_{\text{c}}({\bar a})$ are superradiantly unstable to scalar perturbation modes. In the present paper we use analytical techniques in order to explore the physical properties of the critical (marginally-stable) spinning exotic compact objects. In particular, we derive a remarkably compact {\it analytical} formula for the discrete spectrum $\{r^{\text{max}}_{\text{c}}({\bar a})\}$ of critical radii which characterize the marginally-stable exotic compact objects. We explicitly demonstrate that the analytically derived resonance spectrum agrees remarkably well with numerical results that recently appeared in the physics literature.

It is certainly of physical importance to explore the (in)stability properties of these horizonless exotic compact objects. In a very important work, Maggio, Pani, and Ferrari [13] have recently demonstrated numerically that, due to the physical mechanism of superradiant amplification [14,15], horizonless spinning compact objects with reflective boundary conditions and ergoregions may become unstable to scalar perturbation modes [16]. Intriguingly, the results presented in [13] have revealed the important fact that the ergoregion instability shuts down if the quantum reflective surface of the spinning exotic compact object is located far enough from the would-be classical black-hole horizon. This interesting and highly important numerical finding implies, in particular, that there exists a unique family of critical (marginally-stable) exotic compact objects which determine the boundary between stable and unstable horizonless spinning configurations.
In the present paper we shall use analytical techniques in order to explore the physical properties of the critical (marginally-stable) exotic compact objects. In particular, below we shall explicitly prove that horizonless spinning compact objects with reflective boundary conditions are characterized by the existence of a discrete set {r c (ā; n)} n=∞ n=1 of critical surface radii that can support spatially regular static (marginally-stable) massless scalar field configurations [17][18][19] [Hereā is the dimensionless angular momentum of the exotic compact object, see Eq. (12) below].
It should be emphasized that the physical significance of the outermost (largest) radius r max c (ā) ≡ max n {r c (ā; n)} stems from the fact that, for a given value of the dimensionless rotation parameterā, this unique critical radius marks the boundary between stable and unstable spinning exotic compact configurations. In particular, horizonless compact objects of rotation parameterā whose reflecting surfaces are characterized by the inequality r c < r max c (ā) are superradiantly unstable to scalar perturbation modes, whereas horizonless compact objects whose reflecting surfaces are characterized by the inequality r c > r max c (ā) are stable. The main goal of the present paper is to determine analytically the discrete spectrum {r c (ā; n)} n=∞ n=1 of radii which characterize the horizonless spinning exotic compact objects that can support the spatially regular static (marginallystable) scalar field configurations. In particular, below we shall derive a remarkably compact analytical formula for the critical (outermost) radii {r max c (ā)} of the exotic compact objects that mark the boundary between stable and unstable horizonless spinning configurations.

II. DESCRIPTION OF THE SYSTEM
We shall analyze the physical properties of horizonless spinning compact objects with reflective surfaces which are linearly coupled to massless scalar fields. Following the interesting work of Maggio, Pani, and Ferrari [13] (see also [1][2][3][4][5][6][7][8][9][10][11][12]), we shall study exotic compact objects which are characterized by curved geometries that modify the spinning Kerr spacetime only at some microscopic scale around the would-be classical horizon. In particular, we shall assume, as in [13], that the Kerr line element [20][21][22][23] with ∆ ≡ r 2 − 2M r + a 2 and ρ 2 ≡ r 2 + a 2 cos 2 θ, describes the external spacetime of the spinning exotic compact object of mass M , angular momentum J = M a, and radius r c . In addition, below we shall assume that the radius of the exotic compact object is characterized by the relation [24][25][26][27] where are the radii of the would-be classical horizons. The small-z c regime (2) corresponds to the physically interesting family of horizonless exotic compact objects whose quantum reflective surfaces are located a microscopic distance above the would-be classical black-hole horizons [13] (see also [1][2][3][4][5][6][7][8][9][10][11][12]). The dynamics of the linearized massless scalar field in the spinning spacetime of the exotic compact object is governed by the Klein-Gordon wave equation [28,29] ∇ ν ∇ ν Ψ = 0 .
Substituting into (4) the metric components of the curved line element (1), which characterizes the exterior spacetime of the spinning exotic compact object, and using the scalar field decomposition [28][29][30] one finds that the radial part R lm (r; M, a, ω) of the scalar eigenfunction is determined by the ordinary differential equation [28,29] ∆ d dr Here the frequency-dependent parameter K lm (aω) is the characteristic eigenvalue of the spatially regular angular eigenfunction S lm (θ; aω), which is determined by the familiar spheroidal differential equation [28,29,[31][32][33][34][35] For later purposes we note that, in the small frequency aω ≪ 1 regime, the angular eigenvalues of the spheroidal scalar eigenfunctions can be expanded in the from where the frequency-independent coefficients {c k (l, m)} are given in [33].
Following the physically interesting model of the spinning exotic objects studied numerically in [13], we shall assume that the horizonless compact objects are characterized by (Dirichlet or Neumann) reflecting boundary conditions [36]: Dirichlet B. C. ; dR(r = r c )/dr = 0 Neumann B. C. .
In addition, the marginally-stable (static) scalar modes that we shall analyze in the present paper are characterized by spatially regular (asymptotically decaying) eigenfunctions:

III. THE RESONANCE CONDITIONS OF THE MARGINALLY-STABLE SPINNING EXOTIC COMPACT OBJECTS
The composed horizonless-spinning-object-massless-scalar-field system is characterized by the existence of a unique family of marginally-stable (static) resonances which mark the onset of superradiant instabilities in the curved spinning spacetime. These physically interesting critical field modes are characterized by the simple property The set of equations (6)-(11) determines two discrete spectra of radii, {r Dirichlet c (ā, l, m; n)} and {r Neumann c (ā, l, m; n)}, which, for a given valueā of the dimensionless angular momentum parameter, characterize the critical (marginally-stable) spinning exotic compact objects. Interestingly, as we shall explicitly show in the present section, the characteristic radial equation (6) of the massless scalar fields in the curved spacetimes of the spinning exotic compact objects is amenable to an analytical treatment for the marginally-stable static modes. Substituting into Eq. (6) ω = 0 and where one obtains the characteristic radial equation [37] The general mathematical solution of the radial differential equation (15) is given by [33,38] where 2 F 1 (a, b; c; z) is the hypergeometric function and {A, B} are normalization constants. Substituting (16) into (13), one obtains the expression of the hypergeometric function, one finds from (17) of the radial eigenfunction. A physically acceptable (finite) solution at spatial infinity (r → ∞, or equivalently We therefore conclude that the marginally-stable (static) resonances of the massless scalar fields in the curved spacetimes of the horizonless spinning exotic compact objects are characterized by the radial eigenfunction Taking cognizance of the boundary conditions (9), which characterize the horizonless reflecting compact objects, one deduces that the compact resonance equations determine the discrete spectra of dimensionless critical radii {x c (ā, l, m; n)} which characterize the marginally-stable exotic compact objects.

IV. THE DISCRETE RESONANCE SPECTRA OF THE MARGINALLY-STABLE SPINNING EXOTIC COMPACT OBJECTS
The analytically derived resonance equations (22) and (23), which characterize the unique families of horizonless exotic compact objects that can support the static (marginally-stable) massless scalar field configurations, can easily be solved numerically. Interestingly, one finds that, for given physical parameters {ā, l, m} of the composed spinningexotic-compact-object-massless-scalar-field system, there exists a discrete set of critical radii, which can support the spatially regular static (marginally-stable) scalar field resonances.
In Table I we display, for various values of the dimensionless angular momentum parameterā, the largest (outermost) dimensionless radii [39] z max of the horizonless spinning compact objects that can support the marginally-stable massless scalar field configurations. From the data presented in Table I one learns that, for fixed values of the scalar angular harmonic indices, the critical radii z max c (ā) [and thus also r max c (ā)] which characterize the marginally-stable spinning exotic compact objects are a monotonically increasing function of the dimensionless angular momentum parameterā.
It is worth emphasizing again that the physical significance of the critical reflecting radius r max c (ā) stems from the fact that, for a given value of the dimensionless angular momentum parameterā, this supporting radius corresponds to a marginally-stable spinning object which marks the onset of superradiant instabilities in the composed exotic-compactobject-massless-scalar-field system. In particular, as nicely demonstrated numerically in [13], spinning compact objects of dimensionless angular momentumā whose reflecting surfaces are located in the region r c > r max c (ā) are stable, whereas spinning objects whose reflecting surfaces are located in the region r c < r max c (ā) are superradiantly unstable to scalar perturbation modes.   (24) and (25)] of the horizonless reflecting compact objects that can support the spatially regular static scalar field configurations with l = m = 1. One finds that the critical radii {z max c (ā)}, which characterize the marginally-stable spinning exotic compact objects, are a monotonically increasing function of the dimensionless angular momentum parameterā.
In Table II we display, for various equatorial (l = m) massless scalar field modes, the critical (largest) dimensionless radii z max c (l) [see Eq. (25)] of the horizonless spinning compact objects that can support the static (marginally-stable) scalar field resonances. From the data presented in Table II one (24) and (25)] of the horizonless reflecting compact objects with dimensionless angular momentum parameter a = 0.5 that can support static equatorial (l = m) scalar field configurations. One finds that the critical radii {z max c (l)}, which characterize the marginally-stable spinning exotic compact objects, are a monotonically increasing function of the scalar harmonic index l.

A. An analytical treatment
In the present section we shall explicitly prove that the resonance conditions (22) and (23), which respectively determine the dimensionless discrete radii {x Dirichlet c (ā, l, m; n)} and {x Neumann c (ā, l, m; n)} of the marginally-stable horizonless spinning exotic compact objects, can be solved analytically in the physically interesting regime As emphasized above, the small-x c regime (26) corresponds to the physically interesting family of highly compact exotic objects whose quantum reflective surfaces are located very near the would-be classical black-hole horizons. It is worth mentioning, in particular, that the strong inequality (26) characterizes the horizonless spinning exotic compact objects that were recently studied numerically in the interesting work of Maggio, Pani, and Ferrari [13] (see also [1][2][3][4][5][6][7][8][9][10][11][12] x iα 2 F 1 (l + 1 + 2iα, l + 1; 1 + 2iα; x) .
Taking cognizance of the characteristic asymptotic property (18) of the hypergeometric functions, one finds from (27) the small-x behavior of the radial scalar eigenfunction.

B. Numerical confirmation
It is physically important to verify the validity of the analytically derived resonance spectra (30) and (31) for the characteristic discrete radii of the highly compact (x c ≪ 1) spinning exotic objects that can support the static (marginally-stable) scalar field configurations. In Tables III and IV we display the dimensionless radii z analytical c (n) ≡ [r c (n) − r + ]/r + [39] of the spinning exotic compact objects with reflecting Dirichlet/Neumann boundary conditions as calculated from the analytically derived resonance spectra (30) and (31). For comparison, we also display the corresponding radii z numerical c (n) of the horizonless compact objects as obtained from a direct numerical solution of the exact (analytically derived) resonance conditions (22) and (23).
From the data presented in Tables III and IV one finds a very good agreement, especially in the physically interesting regime z c ≪ 1 of highly compact exotic objects [43] [see Eq. (2)], between the approximated radii of the horizonless compact objects that can support the static (marginally-stable) scalar resonances [as calculated from the analytical formulas (30) and (31)] and the corresponding exact radii of the spinning compact objects [as determined numerically from the characteristic resonance conditions (22) and (23)].  III: Spinning exotic compact objects with reflective Dirichlet boundary conditions. We present the analytically calculated discrete set of dimensionless radii z analytical c (n) which characterize the spinning compact objects that can support the static (marginally-stable) massless scalar field configurations. We also present the corresponding radii z numerical c (n) of the horizonless compact objects as obtained from a direct numerical solution of the characteristic resonance condition (22). The data presented is for horizonless compact objects with dimensionless rotation parameterā = 0.9 linearly coupled to a massless scalar field mode with l = m = 1. In the physically interesting regime zc ≪ 1 of highly compact exotic objects [43], one finds a remarkably good agreement between the approximated radii {z analytical c (n)} of the compact objects that can support the marginally-stable scalar resonances [as calculated from the analytical resonance spectrum (30)   IV: Spinning exotic compact objects with reflective Neumann boundary conditions. We present the analytically calculated discrete set of dimensionless radii z analytical c (n) which characterize the spinning compact objects that can support the static (marginally-stable) massless scalar field configurations. We also present the corresponding radii z numerical c (n) of the horizonless compact objects as obtained from a direct numerical solution of the characteristic resonance condition (23). The data presented is for horizonless compact objects with dimensionless rotation parameterā = 0.9 and a massless scalar field mode with l = m = 1. In the physically interesting zc ≪ 1 regime of highly compact exotic objects [43], one finds a remarkably good agreement between the approximated radii {z analytical c (n)} of the compact objects that can support the marginally-stable scalar resonances [as calculated from the analytical resonance formula (31)] and the corresponding exact radii {z numerical c (n)} of the compact exotic objects [as determined numerically from the resonance equation (23)].

VI. ANALYTICAL VS. FORMER NUMERICAL RESULTS
It is physically important to compare our analytical results for the marginally-stable exotic compact objects with the corresponding numerical data published recently in the very interesting work of Maggio, Pani, and Ferrari [13]. In Table V we display, for various values of the compactness parameter δ ≡ (r max c − r + )/M [44] introduced in [13], the dimensionless ratioā analytical (δ)/ā numerical (δ). Here {ā analytical (δ)} are the analytically derived dimensionless angular momenta which characterize the critical (marginally-stable) spinning exotic compact objects, and {ā numerical (δ)} are the corresponding numerically computed [13] values of the critical rotation parameter. Interestingly, from the data presented in Table V one finds a remarkably good agreement between our analytical formulas and the corresponding numerical data of [13].  , the dimensionless ratioā analytical (δ)/ā numerical (δ) between the analytically derived critical angular momentum which characterizes the marginally-stable spinning exotic compact object and the corresponding numerically computed [13] value of the critical rotation parameter. The data presented are for the fundamental (n = 1) resonances of the spinning exotic compact objects with reflecting Dirichlet boundary conditions and for a massless scalar field mode with l = m = 1. One finds a remarkably good agreement between our analytical results and the corresponding numerical data of [13].

VII. SUMMARY AND DISCUSSION
Horizonless exotic compact objects with reflective properties have recently attracted much attention from physicists as possible quantum-gravity alternatives to classical black-hole spacetimes (see [1][2][3][4][5][6][7][8][9][10][11][12][13] and references therein). In a physically important work, Maggio, Pani, and Ferrari [13] have recently studied numerically the stability properties of a family of horizonless spinning exotic compact objects which are characterized by curved spacetime geometries that modify the Kerr metric only at some microscopic scale around the would-be classical horizon.
The numerical analysis presented in [13] has demonstrated that horizonless spinning compact objects with reflective surfaces may become superradiantly unstable to scalar perturbation modes [16]. Intriguingly, however, the results presented in [13] have revealed the important fact that the ergoregion instability shuts down if the quantum reflective surface of the horizonless compact object is located far enough from the would-be classical black-hole horizon. This highly interesting numerical result implies, in particular, that there exist a unique family of marginally-stable spinning exotic compact objects which determine the critical boundary between stable and unstable horizonless spinning configurations.
In the present paper we have used analytical techniques in order to explore the physical properties of the critical (marginally-stable) spinning exotic compact objects. These horizonless reflecting objects are characterized by their ability to support spatially regular static configurations of massless scalar fields in their exterior spacetime regions. The physical significance of this unique family of critical (marginally-stable) exotic compact objects stems from the fact that it marks the boundary between stable and unstable composed spinning-exotic-compact-object-masslessscalar-field configurations.
The main results derived in the present paper and their physical implications are as follows: (1) We have proved that, for given angular harmonic indices (l, m) of the massless scalar field mode, there exist two discrete spectra of radii, {r Dirichlet c (ā, l, m; n)} n=∞ n=1 and {r Neumann c (ā, l, m; n)} n=∞ n=1 , which characterize the horizonless spinning compact objects with reflective boundary conditions that can support the spatially regular static (marginallystable) scalar field configurations [17][18][19]. In particular, we have shown that the analytically derived resonance equations [see Eqs. (22) and (23)] (32) determine the critical dimensionless radii of the marginally-stable spinning exotic compact objects.
for the discrete spectra of dimensionless radii which characterize the spinning compact objects that can support the static (marginally-stable) massless scalar field configurations. It is worth stressing the fact that the physical significance of the largest (outermost) radii r max c (ā) ≡ max n {r c (ā; n)} stems from the fact that these critical dimensionless radii mark the boundary between stable and unstable horizonless compact objects with reflecting boundary conditions. In particular, spinning exotic compact objects whose reflecting surfaces are located in the radial region r c < r max c (ā) are superradiantly unstable to scalar perturbation modes, whereas spinning exotic compact objects whose reflecting surfaces are located in the radial region r c > r max c (ā) are stable.
(3) It has been explicitly demonstrated (see Tables III and IV) that the analytically derived resonance spectra (30) and (31), which determine the dimensionless discrete radii of the horizonless exotic objects that can support the static (marginally-stable) scalar resonances, agree in the physically interesting regime x c ≪ 1 of highly-compact objects [43] with the corresponding exact radii of the critical exotic compact objects [as determined numerically directly from the resonance equations (22) and (23)].
(5) Finally, we have explicitly demonstrated that our analytical formulas agree remarkably well with the corresponding numerical data that recently appeared in the interesting work of Maggio, Pani, and Ferrari [13].