Ultra-spinning exotic compact objects supporting static massless scalar field configurations

Horizonless spacetimes describing highly compact exotic objects with reflecting (instead of absorbing) surfaces have recently attracted much attention from physicists and mathematicians as possible quantum-gravity alternatives to canonical classical black-hole spacetimes. Interestingly, it has recently been proved that spinning compact objects with angular momenta in the sub-critical regime ${\bar a}\equiv J/M^2\leq1$ are characterized by an infinite countable set of surface radii, $\{r_{\text{c}}({\bar a};n)\}^{n=\infty}_{n=1}$, that can support asymptotically flat static configurations made of massless scalar fields. In the present paper we study analytically the physical properties of ultra-spinning exotic compact objects with dimensionless angular momenta in the complementary regime ${\bar a}>1$. It is proved that ultra-spinning reflecting compact objects with dimensionless angular momenta in the super-critical regime $\sqrt{1-[{{m}/{(l+2)}}]^2}\leq|{\bar a}|^{-1}<1$ are characterized by a finite discrete family of surface radii, $\{r_{\text{c}}({\bar a};n)\}^{n=N_{\text{r}}}_{n=1}$, distributed symmetrically around $r=M$, that can support spatially regular static configurations of massless scalar fields (here the integers $\{l,m\}$ are the harmonic indices of the supported static scalar field modes). Interestingly, the largest supporting surface radius $r^{\text{max}}_{\text{c}}({\bar a})\equiv \text{max}_n\{r_{\text{c}}({\bar a};n)\}$ marks the onset of superradiant instabilities in the composed ultra-spinning-exotic-compact-object-massless-scalar-field system.

Horizonless spacetimes describing highly compact exotic objects with reflecting (instead of absorbing) surfaces have recently attracted much attention from physicists and mathematicians as possible quantum-gravity alternatives to canonical classical black-hole spacetimes. Interestingly, it has recently been proved that spinning compact objects with angular momenta in the sub-critical regimeā ≡ J/M 2 ≤ 1 are characterized by an infinite countable set of surface radii, {rc(ā; n)} n=∞ n=1 , that can support asymptotically flat static configurations made of massless scalar fields. In the present paper we study analytically the physical properties of ultra-spinning exotic compact objects with dimensionless angular momenta in the complementary regimeā > 1. It is proved that ultra-spinning reflecting compact objects with dimensionless angular momenta in the super-critical regime 1 − [m/(l + 2)] 2 ≤ |ā| −1 < 1 are characterized by a finite discrete family of surface radii, {rc(ā; n)} n=Nr n=1 , distributed symmetrically around r = M , that can support spatially regular static configurations of massless scalar fields (here the integers {l, m} are the harmonic indices of the supported static scalar field modes). Interestingly, the largest supporting surface radius r max c (ā) ≡ maxn{rc(ā; n)} marks the onset of superradiant instabilities in the composed ultra-spinning-exotic-compact-object-massless-scalar-field system.

I. INTRODUCTION
Curved black-hole spacetimes with absorbing event horizons are one of the most exciting predictions of the classical Einstein field equations. The physical and mathematical properties of classical black-hole spacetimes have been extensively explored during the last five decades [1,2], and it is widely believed that the recent detection of gravitational waves [3,4] provides compelling evidence for the existence of spinning astrophysical black holes of the Kerr family. Intriguingly, however, the physical properties of highly compact horizonless objects have recently been explored by many physicists (see [5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22] and references therein) in an attempt to determine whether these exotic curved spacetimes can serve as valid alternatives, possibly within the framework of a unified quantum theory of gravity, to canonical black-hole spacetimes.
In a very interesting work, Maggio, Pani, and Ferrari [17] have recently explored the complex resonance spectrum of massless scalar fields linearly coupled to horizonless spinning exotic compact objects. The numerical results presented in [17] have explicitly demonstrated the important physical fact that, for given values {l, m} of the scalar field harmonic indices, there is a critical compactness parameter characterizing the central reflecting objects, above which the massless scalar fields grow exponentially in time. This characteristic behavior of the fields in the horizonless spinning curved spacetimes indicates that the corresponding exotic objects may become unstable when coupled to bosonic (integer-spin) fields [23]. In particular, this superradiant instability [24][25][26][27][28] is attributed to the fact that the characteristic absorbing boundary conditions of classical black-hole spacetimes have been replaced in [17] by reflecting boundary conditions at the compact surfaces of the horizonless exotic objects.
The physical properties of marginally-stable spinning exotic compact objects were studied analytically in [19]. In particular, it was explicitly proved in [19] that reflecting compact objects with sub-critical angular momenta in the regime 0 <ā ≡ J/M 2 ≤ 1 [29, 30] are characterized by an infinite countable set of surface radii, {r c (ā; n)} n=∞ n=1 , which can support spatially regular static (marginally-stable) configurations made of massless scalar fields. The ability of spinning compact objects to support static scalar field configurations is physically interesting from the point of view of the no-hair theorems discussed in [31][32][33]. In particular, it was proved in [31,32] that spherically-symmetric (nonspinning) horizonless reflecting objects, like black holes with absorbing horizons [34][35][36], cannot support spatially regular nonlinear massless scalar field configurations [37][38][39].
Interestingly, the parameter space of the composed spinning-exotic-compact-object-massless-scalar-field system is divided by the outermost supporting radius, r max c (ā) ≡ max n {r c (ā; n)}, to stable and unstable configurations. In particular, horizonless reflecting objects whose surface radii lie in the regime r c > r max criticalā > 1 regime are characterized by a finite discrete family of surface radii, {r c (ā; n)} n=Nr n=1 [44], that can support the static (marginally-stable) scalar field configurations. This unique property of the ultra-spinning (ā > 1) reflecting compact objects should be contrasted with the previously proved fact [19] that sub-critical (ā < 1) spinning objects are characterized by an infinite countable family of surface radii, {r c (ā; n)} n=∞ n=1 , that can support spatially regular static scalar field configurations.
Using analytical techniques, we shall determine in this paper the characteristic critical (largest) surface radius, r max c (ā) ≡ max n {r c (ā; n)}, of the ultra-spinning reflecting objects that, for given value of the super-critical rotation parameterā, marks the boundary between stable and superradiantly unstable spinning configurations. In particular, below we shall derive a remarkably compact analytical formula for the discrete (and finite) family of supporting surface radii which characterizes exotic near-critical spinning horizonless compact objects in the physically interesting regime 0 <ā − 1 ≪ 1.

II. DESCRIPTION OF THE SYSTEM
We consider a spatially regular configuration made of a massless scalar field Ψ which is linearly coupled to an ultraspinning reflecting compact object of radius r c , mass M , and dimensionless angular momentum in the super-critical regimeā Following the interesting physical model of the exotic compact objects discussed by Maggio, Pani, and Ferrari [17] (see also [18][19][20]), we shall assume that the external spacetime geometry of the spinning compact object is described by the Kerr line element [1,2,29,[45][46][47][48][49][50] where the metric functions are given by ∆ ≡ r 2 − 2M r + a 2 and ρ 2 ≡ r 2 + a 2 cos 2 θ with a ≡ Mā.

III. THE RESONANCE CONDITION OF THE COMPOSED ULTRA-SPINNING-EXOTIC-COMPACT-OBJECT-MASSLESS-SCALAR-FIELD CONFIGURATIONS
In the present section we shall derive, for a given set of the dimensionless physical parameters {r c /M,ā, l, m}, the characteristic resonance condition for the existence of ultra-spinning reflecting exotic horizonless objects that support spatially regular static (marginally-stable) linearized scalar field configurations.
Substituting into the radial equation (5) the characteristic relation for the static scalar field configurations, one obtains the ordinary differential equation [19,60] x(1 − x) where and The physically acceptable solution of the characteristic radial scalar equation (10) which respects the asymptotic boundary condition (8) is given by [19,56,61,62] where A is a normalization constant and 2 F 1 (a, b; c; z) is the hypergeometric function. Substituting the radial solution (14) into the characteristic inner boundary condition (7) at the surface of the compact reflecting object, one obtains the remarkably compact resonance condition 2 F 1 (l + 1 − γ, l + 1; 2l + 2; 1 − x c ) = 0 (15) for the composed ultra-spinning-exotic-compact-object-massless-scalar-field configurations.
As we shall show below, the resonance equation (15) determines the discrete set of surface radii {r c = r c (ā, l, m; n)} which characterize the unique family of ultra-spinning exotic compact objects that can support the static spatially regular massless scalar field configurations.

IV. GENERIC PROPERTIES OF THE COMPOSED ULTRA-SPINNING-EXOTIC-COMPACT-OBJECT-MASSLESS-SCALAR-FIELD CONFIGURATIONS
In the present section we shall discuss two important features of the discrete resonance spectrum {r c (ā, l, m; n)} of surface radii that characterize the composed ultra-spinning-exotic-compact-object-massless-scalar-field configurations: (1) the distribution of the supporting radii, and (2) the (finite) number of supporting radii.
A. The resonance spectrum of surface radii is distributed symmetrically around r = M Interestingly, we shall now prove that the discrete set of supporting radii {r c (ā, l, m; n)}, which stems from the characteristic resonance equation (15), is distributed symmetrically around r = M . To this end, it is convenient to define the dimensionless symmetrical radial coordinate in terms of which the resonance equation (15) can be written in the form [63] Using the characteristic identity (see Eq. 15 of the hypergeometric function, one can express the resonance condition (17) in the symmetrical form The resonance equation (19) is obviously invariant under the reflection symmetry z c → −z c . We have therefore proved that if the dimensionless surface radius z c is a solution of the characteristic resonance equation (19), then −z c is also a valid resonance.
In addition, it is interesting to stress the fact that, for the static (ω = 0) scalar field modes, the radial scalar equation (5) is invariant under the reflection symmetries a → −a and m → −m [64]. One therefore deduces that if the dimensionless surface radius z c characterizes an ultra-spinning exotic compact object with ma > 0 that can support a spatially regular static (marginally-stable) scalar field configuration with harmonic indices {l, m}, then the same supporting radius also characterizes an ultra-spinning exotic compact object with ma < 0 that can support the same static scalar field configuration.
Taking cognizance of the three reflection symmetries, z c → −z c , a → −a, and m → −m, which characterize the composed ultra-spinning-exotic-compact-object-massless-scalar-field system, we shall henceforth assume, without loss of generality, that B. The number of discrete supporting radii is finite As emphasized above, it has recently been proved [19] that exotic compact objects in the sub-critical regimeā < 1 are characterized by an infinite set of surface radii, {r c (ā; n)} n=∞ n=1 , that can support static (marginally-stable) massless scalar field configurations.
On the other hand, we shall now show that super-critical (ā > 1) compact reflecting objects are characterized by a finite set of surface radii that can support the static massless scalar field configurations. In particular, one finds that, for positive integer values of the dimensionless physical parameter N (ā, l, m) ≡ γ − (l + 1), the resonance equation (15), which determines the characteristic spectrum of supporting radii of the ultra-spinning exotic compact objects, is a polynomial equation of degree N . Thus, in this case there is a finite number N of complex solutions {x c (ā, l, m; n)} n=N n=1 to the resonance condition (15) which in turn, using the relation (12), yield a finite discrete spectrum {r c (ā, l, m; n)} n=N n=1 of supporting surface radii. In addition, solving numerically the resonance equation (15) we find that, for positive non-integer values of the physical parameter N , the number of discrete surface radii that can support the static (marginally-stable) scalar field configurations is given by (see Tables I and II  To summarize, the (finite) number N r (ā, l, m) of discrete supporting radii that characterize the composed ultraspinning-exotic-compact-object-massless-scalar-field configurations is given by the simple relations [It is important to emphasize that cases 2 and 3 in (21) refer to non-integer values of the dimensionless composed parameter γ − (l + 1)].

V. THE REGIME OF EXISTENCE OF THE COMPOSED ULTRA-SPINNING-EXOTIC-COMPACT-OBJECT-MASSLESS-SCALAR-FIELD CONFIGURATIONS
In the present section we shall derive an upper bound on the characteristic surface radii {r c (ā, l, m; n)} n=Nr n=1 which characterize the ultra-spinning exotic compact objects that can support the static (marginally-stable) configurations of the massless scalar fields.
Substituting the scalar function into the characteristic radial equation (5), one obtains the ordinary differential equation for the static (ω = 0) scalar configurations.
Using the characteristic boundary conditions (7) and (8) of the spatially regular linearized scalar field configurations, which are supported in the asymptotically flat curved spacetime (2) of the exotic ultra-spinning reflecting compact object, one deduces that the radial scalar eigenfunction Φ(r) must have (at least) one extremum point, r = r peak , in the interval In particular, the simple functional relations characterize the spatial behavior of the radial scalar eigenfunction at this extremum point. Taking cognizance of Eqs. (23) and (25), one finds the simple relation The characteristic inequality (26) implies that r peak is bounded by the relations where Using Eqs. (16), (24), (27), and (28), one deduces that the composed ultra-spinning-exotic-compact-object-masslessscalar-field configurations are characterized by the simple dimensionless upper bound In particular, from the requirementā 2 [1 + l(l + 1) − m 2 ] − 1 ≤ l(l + 1) [see the r.h.s of (29)], one finds that the static (marginally-stable) massless scalar field configurations in the curved spacetimes of the ultra-spinning (ā > 1) exotic compact objects are characterized by the compact inequalities Interestingly, a stronger upper bound on the dimensionless angular momentum parameterā, which characterizes the unique family of ultra-spinning exotic compact objects that can support the spatially regular static (marginally-stable) massless scalar field configurations, can be obtained from the observations that [see Eq. (15)] [66] and 2 F 1 (−1, l + 1; 2l + 2; 2) = 2 F 1 (2l + 3, l + 1; 2l + 2; 2) = 0 .

VI. THE RESONANCE SPECTRUM OF THE COMPOSED ULTRA-SPINNING-EXOTIC-COMPACT-OBJECT-MASSLESS-SCALAR-FIELD CONFIGURATIONS
As mentioned above, the infinite countable spectrum of supporting surface radii {r c (ā, l, m; n)} n=∞ n=1 which characterizes the composed spinning-exotic-compact-object-massless-scalar-field configurations in the sub-critical regimē a < 1 has been determined in [19]. In the present section we shall explicitly show that ultra-spinning exotic compact objects in the complementary regimeā > 1 of super-critical angular momenta are characterized by a finite [see Eq. (21)] discrete set {r c (ā, l, m; n)} n=Nr n=1 of surface radii that can support the asymptotically flat static scalar field configurations.
The compact resonance equation (15) can be solved numerically, for a given set {ā, l, m} of the dimensionless physical parameters that characterize the composed compact-object-scalar-field system, to yield the discrete resonant spectrum {r c (ā, l, m; n)} n=Nr n=1 of supporting radii. In Table I we present, for various super-critical values of the dimensionless angular momentum parameterā, the smallest and largest dimensionless surface radii {z min c (ā, l, m), z max c (ā, l, m)} of the ultra-spinning exotic compact objects that can support the static spatially regular configurations of the massless scalar fields [67]. We also present in Table I  The data presented in Table I demonstrate the fact that, for given integer values {l, m} of the angular harmonic indices of the static (marginally-stable) massless scalar fields, the dimensionless supporting radius z max c (ā) of the ultra-spinning exotic compact objects is a monotonically decreasing function of the dimensionless physical parameter a. As a consistency check, it is worth noting that the numerically computed values {z max c (ā)} of the characteristic surface radii of the ultra-spinning reflecting compact objects, as displayed in Table I, conform to the analytically derived upper bound (29).
We would like to emphasize again that, for a given set of the physical parameters {ā, l, m}, the critical supporting radius r max c (ā) marks the boundary between stable and superradiantly unstable composed ultra-spinning-exoticcompact-object-massless-scalar-field configurations. In particular, the numerical results presented in the interesting work of Maggio, Pani, and Ferrari [17] indicate that ultra-spinning reflecting compact objects which are characterized by the inequality r c < r max c (ā) are superradiantly unstable to massless scalar perturbation modes, whereas ultraspinning exotic compact objects which are characterized by the relation r c > r max c (ā) are stable. In Table II we present, for various equatorial (l = m) modes of the supported static scalar fields, the smallest and largest dimensionless surface radii {z min c (ā, l, m), z max c (ā, l, m)} of the supporting marginally-stable ultra-spinning exotic compact objects [67]. Also displayed is the (finite) number [see Eq. (21)] of these unique supporting surface radii [68]. The data presented in Table II reveal the fact that, for a given value of the dimensionless physical parameter a, the critical (largest) supporting radius z max c (l) of the reflecting exotic compact objects is a monotonically increasing function of the harmonic index l which characterizes the static massless scalar field mode. It is worth noting that the numerically computed surface radii z max c (l) of the ultra-spinning marginally-stable exotic compact objects, as presented in Table II, conform to the analytically derived upper bound (29).

VII. THE RESONANCE SPECTRUM OF NEAR-CRITICAL ULTRA-SPINNING EXOTIC COMPACT OBJECTS
A. An analytical treatment Interestingly, as we shall explicitly show in the present section, the compact resonance equation (15), which determines the discrete family {x c (ā, l, m; n)} of dimensionless surface radii that characterize the marginally-stable  (16)], of the ultra-spinning exotic compact objects that can support the static (marginally-stable) massless scalar field configurations [67]. Also presented is the finite number [see Eq. (21)] of these unique supporting surface radii. The data presented is for the case l = m = 1. The critical supporting radii {z max c (ā)}, which characterize the marginally-stable ultra-spinning reflecting compact objects, are found to be a monotonically decreasing function of the dimensionless angular momentum parameterā. As a consistency check we note that the supporting radii of the ultra-spinning exotic compact objects conform to the analytically derived upper bound (29).   (16)] of the ultra-spinning exotic compact objects that can support the spatially regular static (marginally-stable) massless scalar field configurations [67]. We also present the finite number of these unique supporting radii. The data presented is for the case √ 1 −ā −2 = 1/4. The critical surface radii {z max c (l)}, which characterize the marginally-stable ultra-spinning exotic compact objects, are found to be a monotonically increasing function of the dimensionless harmonic index l of the supported static scalar field configurations. ultra-spinning exotic compact objects, is amenable to an analytical treatment in the physically interesting regime 0 <ā − 1 ≪ 1 (34) of near-critical horizonless spinning configurations. In particular, in the near-critical regime one may use the large-|b| asymptotic expansion [69] of the hypergeometric function in order to express the resonance condition (15) in the remarkably compact form [70] x m/ From the asymptotic relation (37) one finds the set of complex solutions [71] x c (n) = e −iπ(l+2n) which, taking cognizance of Eqs. (12) and (16), yields the discrete real resonance spectrum [63, 70,72] z c (n) = ā 2 − 1 · cot π(l + 2n) for 1 −ā −2 ≪ m l and π(l + 2n) ≫ 1 (39) for the dimensionless surface radii which characterize the near-critical (ā 1) exotic compact objects that can support the static massless scalar field configurations. Interestingly, the analytically derived resonance formula (39) can be further simplified in the π(l + 2n) √ 1 −ā −2 /2m ≪ 1 regime, in which case one finds the remarkably compact expression [72,73] z c (n) = 2mā π(l + 2n) for the characteristic radii of the ultra-spinning exotic compact objects that can support the static (marginally-stable) configurations of the massless scalar fields.

B. Numerical confirmation
It is of physical interest to verify the accuracy of the approximated (analytically derived) resonance spectrum (39) for the surface radii of the near-critical (0 <ā−1 ≪ 1) ultra-spinning exotic compact objects that can support the spatially regular static (marginally-stable) configurations of the massless scalar fields. In Table III we present the dimensionless discrete surface radii z analytical c (n) of the supporting near-critical ultra-spinning exotic reflecting objects as obtained from the analytically derived resonance spectrum (39). We also present in Table III the corresponding surface radii z numerical c (n) of the ultra-spinning exotic compact objects as computed numerically from the exact characteristic resonance equation (15).
The data presented in Table III nicely demonstrate the important fact that there is a good agreement between the approximated surface radii {z analytical c (n)} of the ultra-spinning exotic compact objects that can support the static massless scalar field configurations [as calculated from the compact analytically derived resonance formula (39)] and the corresponding exact surface radii {z numerical c (n)} of the reflecting compact objects [as determined numerically directly from the resonance equation (15)].  (n)} which characterize the ultra-spinning exotic compact objects that can support the static (marginallystable) spatially regular configurations of the massless scalar fields. Also displayed are the corresponding supporting radii {z numerical c (n)} of the near-critical exotic compact objects as obtained numerically directly from the characteristic resonance equation (15). The data presented is for static massless scalar field modes with l = m = 1 linearly coupled to near-critical ultra-spinning exotic compact objects with √ 1 −ā −2 = 10 −2 . The displayed data reveal a remarkably good agreement between the exact characteristic surface radii {z numerical c (n)} of the ultra-spinning exotic compact objects [as determined numerically from the resonance condition (15)] and the corresponding approximated radii {z analytical c (n)} of the near-critical ultra-spinning compact objects [as calculated analytically from the compact resonance formula (39)].

VIII. THE RESONANCE SPECTRUM OF ULTRA-SPINNING EXOTIC COMPACT OBJECTS WITH rc = M
Interestingly, as we shall now prove, the characteristic resonance equation (15) [or, equivalently, the symmetrical form (19) of the resonance condition] can also be solved analytically for the dimensionless angular momentum parameterā in the physically interesting case of horizonless ultra-spinning exotic compact objects of mass M whose compact reflecting surfaces coincide with the corresponding horizon radius r c = M of extremal Kerr black holes with the same mass parameter.
Interestingly, Maggio, Pani, and Ferrari [17] have recently provided compelling evidence that sub-critical (ā ≡ J/M 2 < 1) horizonless spinning spacetimes, in which the characteristic absorbing boundary conditions of classical black-hole spacetimes have been replaced by reflective boundary conditions at the surfaces of the exotic compact objects, may become superradiantly unstable [24][25][26][27][28] when linearly coupled to massless scalar (bosonic) field modes [23]. In particular, it has been explicitly proved in [19] that, in the sub-critical regimeā < 1 of the spinning reflecting objects and for given harmonic indices {l, m} of the massless scalar field, there exists an infinite countable set of surface radii, {r c (ā, l, m; n)} n=∞ n=1 , which can support spatially regular static (marginally-stable) massless scalar field configurations.
In the present paper we have explored the physical and mathematical properties of marginally-stable composed ultraspinning-exotic-compact-object-massless-scalar-field configurations which are characterized by super-critical (ā > 1) dimensionless rotation parameters. The following are the main results derived in this paper and their physical implications: (1) It has been explicitly proved that, for given dimensionless physical parameters {ā, l, m}, the unique discrete family {r c (ā, l, m; n)} of surface radii that characterize the ultra-spinning (ā > 1) exotic compact objects that can support the static (marginally-stable) massless scalar field configurations is determined by the resonance condition [see Eqs. (1), (16), and (19)] (2) We have shown that the composed ultra-spinning-exotic-compact-object-massless-scalar-field configurations, as determined by the resonance condition (44), are restricted to the physical regime [see Eq.
of the dimensionless super-critical rotation parameter a/M . In addition, it has been proved that, for a given set {ā, l, m} of the dimensionless physical parameters that characterize the composed compact-object-scalar-field system, the simple relation [see Eqs. (16) and (29)] provides an upper bound on the surface radii of the supporting ultra-spinning exotic compact objects.
(3) It has been pointed out that the analytically derived resonance condition in its symmetrical form (44) reveals the fact that, for ultra-spinning exotic compact objects, the discrete resonant spectrum of supporting surface radii is invariant under the reflection symmetries The symmetry transformations (47) imply, in particular, that if z c [see Eq. (16)] is a dimensionless supporting radius of a composed exotic-object-scalar-field system with dimensionless physical parameters {ā, l, m}, then: (1) −z c is also a valid supporting radius of the same composed physical system, and (2) z c is also a valid supporting radius of a composed exotic-object-scalar-field system with dimensionless parameters {±ā, l, ±m}.
(4) It has been shown that, for ultra-spinning exotic compact objects in the dimensionless physical regime (45), the finite number N r (ā, l, m) of surface radii that can support the spatially regular static (marginally-stable) scalar field configurations is given by [see Eqs. (13) and (21)] [65,75,76] if ⌊N ⌋ is a positive even integer ; where Interestingly, the fact that ultra-spinning (ā > 1) exotic compact objects are characterized by a finite discrete family {r c (ā, l, m; n)} n=Nr n=1 of surface radii that can support the static massless scalar field configurations should be contrasted with the complementary case of sub-critical spinning objects in theā < 1 regime which, as previously proved in [17,19], are characterized by an infinite countable family {r c (ā, l, m; n)} n=∞ n=1 of surface radii that can support the spatially regular static scalar fields [77].
(5) The ability of spinning objects to support spatially regular static scalar field configurations is physically intriguing from the point of view of the no-hair theorems that have recently been discussed in [31][32][33] for horizonless regular spacetimes. In particular, it has been proved in [31,32] that spherically-symmetric (non-spinning) horizonless reflecting stars cannot support nonlinear configurations made of massless scalar fields.
(6) It is important to stress the fact that, as shown in [17,19], the outermost (largest) surface radius r max c (ā) ≡ max n {r c (ā; n)} that can support the static scalar field configurations is of central physical importance since it marks the boundary between stable [r c > r max c (ā)] and unstable [r c < r max c (ā)] composed ultra-spinning-exotic-compactobject-massless-scalar-field configurations.
(7) Solving numerically the analytically derived resonance equation (15), we have demonstrated that the characteristic supporting radius r max c (ā, l, m) is a monotonically decreasing function of the dimensionless rotation parameter a of the ultra-spinning exotic compact objects (see Table I). Likewise, it has been demonstrated that the critical (outermost) supporting surface radius r max c (ā, l, m) is a monotonically increasing function of the harmonic parameter l which characterizes the massless scalar field modes (see Table II). (8) We have explicitly shown that the characteristic resonance equation (15) for the discrete family of supporting surface radii is amenable to an analytical treatment in the physically interesting regime 0 <ā − 1 ≪ 1 of near-critical horizonless spinning objects. In particular, the remarkably compact resonance formula [see Eqs. (16) and (40)] r c (n) = M + 2ma π(l + 2n) ; n ∈ Z (50) has been derived analytically for near-critical (ā 1) composed ultra-spinning-exotic-compact-object-massless-scalarfield configurations in the 1 ≪ π(l + 2n) ≪ 2m/ √ 1 −ā −2 regime. (9) We have verified that the predictions of the analytically derived resonance formula (50), which determines the unique family of surface radii of the near-critical ultra-spinning (0 <ā − 1 ≪ 1) compact reflecting objects that can support the static (marginally-stable) massless scalar field configurations, agree remarkably well (see Table III) with the corresponding exact values of the supporting surface radii as determined numerically from the characteristic resonance condition (15).
(10) Finally, it has been proved that the resonance equation (15) can be solved analytically in the physically interesting case of ultra-spinning (ā > 1) exotic compact objects whose reflecting surfaces coincide with the corresponding horizon radius r c = M of extremal (ā = 1) Kerr black holes with the same mass parameter. In particular, we have derived the remarkably compact discrete resonance spectrum [see Eq.
for the ultra-spinning compact configurations with r c = M [78]. Interestingly, one finds from the analytically derived resonance formula (51) that, in the l + 2n ≫ m regime, the dimensionless angular momenta {ā} n=∞ n=0 of the exotic ultra-spinning reflecting objects with physical parameters {M, r c = M } can be made arbitrarily close [79] to the corresponding dimensionless angular momentumā EK = 1 of an absorbing extremal Kerr black hole with the same mass and radius.