Onset of superradiant instabilities in rotating spacetimes of exotic compact objects

TL;DR

This paper analytically investigates the stability boundaries of rotating horizonless exotic compact objects with reflective surfaces, deriving a formula for the critical radii that determine superradiant instabilities, and confirms its accuracy against numerical results.

Contribution

The paper provides a new analytical formula for the critical radii of marginally-stable rotating exotic compact objects, advancing understanding of their superradiant instability boundaries.

Findings

Derived a compact analytical formula for critical radii.

The analytical spectrum matches well with numerical results.

Identified the stability boundary between stable and superradiantly unstable objects.

Abstract

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.

Tables

Table 1. Table 1: Marginally-stable spinning exotic compact objects. We display, for various values of the dimensionless angular momentum parameter a ¯ ¯ 𝑎 {\bar{a}} , the largest dimensionless radii z c max ​ ( a ¯ ) ≡ ( r c max − r + ) / r + subscript superscript 𝑧 max c ¯ 𝑎 subscript superscript 𝑟 max c subscript 𝑟 subscript 𝑟 z^{\text{max}}_{\text{c}}({\bar{a}})\equiv(r^{\text{max}}_{\text{c}}-r_{+})/r_{+} [see Eqs. ( 24 ) and ( 25 )] of the horizonless reflecting compact objects that can support the spatially regular static scalar field configurations with l = m = 1 𝑙 𝑚 1 l=m=1 . One finds that the critical radii { z c max ​ ( a ¯ ) } subscript superscript 𝑧 max c ¯ 𝑎 \{z^{\text{max}}_{\text{c}}({\bar{a}})\} , which characterize the marginally-stable spinning exotic compact objects, are a monotonically increasing function of the dimensionless angular momentum parameter a ¯ ¯ 𝑎 {\bar{a}} .

Boundary conditions	$z_{\text{c}}^{\text{max}}\left(\overline{a}=0.3\right)$	$z_{\text{c}}^{\text{max}}\left(\overline{a}=0.5\right)$	$z_{\text{c}}^{\text{max}}\left(\overline{a}=0.7\right)$	$z_{\text{c}}^{\text{max}}\left(\overline{a}=0.9\right)$	$z_{\text{c}}^{\text{max}}\left(\overline{a}=0.99\right)$	$z_{\text{c}}^{\text{max}}\left(\overline{a}=0.999\right)$
Dirichlet	$2.960\times {10}^{-10}$	$2.842\times {10}^{-6}$	$2.783\times {10}^{-4}$	$1.007\times {10}^{-2}$	$9.625\times {10}^{-2}$	$1.730\times {10}^{-1}$
Neumann	$6.455\times {10}^{-6}$	$6.662\times {10}^{-4}$	$7.417\times {10}^{-3}$	$5.432\times {10}^{-2}$	$2.105\times {10}^{-1}$	$3.078\times {10}^{-1}$

Table 2. Table 2: Marginally-stable spinning exotic compact objects. We present the largest dimensionless radii z c max ​ ( l ) ≡ ( r c max − r + ) / r + subscript superscript 𝑧 max c 𝑙 subscript superscript 𝑟 max c subscript 𝑟 subscript 𝑟 z^{\text{max}}_{\text{c}}(l)\equiv(r^{\text{max}}_{\text{c}}-r_{+})/r_{+} [see Eqs. ( 24 ) and ( 25 )] of the horizonless reflecting compact objects with dimensionless angular momentum parameter a ¯ = 0.5 ¯ 𝑎 0.5 {\bar{a}}=0.5 that can support static equatorial ( l = m 𝑙 𝑚 l=m ) scalar field configurations. One finds that the critical radii { z c max ​ ( l ) } subscript superscript 𝑧 max c 𝑙 \{z^{\text{max}}_{\text{c}}(l)\} , which characterize the marginally-stable spinning exotic compact objects, are a monotonically increasing function of the scalar harmonic index l 𝑙 l .

Boundary conditions	$z_{\text{c}}^{\text{max}}\left(l=1\right)$	$z_{\text{c}}^{\text{max}}\left(l=2\right)$	$z_{\text{c}}^{\text{max}}\left(l=3\right)$	$z_{\text{c}}^{\text{max}}\left(l=4\right)$	$z_{\text{c}}^{\text{max}}\left(l=5\right)$	$z_{\text{c}}^{\text{max}}\left(l=6\right)$
Dirichlet	$2.842\times {10}^{-6}$	$3.689\times {10}^{-4}$	$1.788\times {10}^{-3}$	$3.948\times {10}^{-3}$	$6.395\times {10}^{-3}$	$8.870\times {10}^{-3}$
Neumann	$6.662\times {10}^{-4}$	$6.141\times {10}^{-3}$	$1.268\times {10}^{-2}$	$1.832\times {10}^{-2}$	$2.297\times {10}^{-2}$	$2.680\times {10}^{-2}$

Table 3. Table 3: Spinning exotic compact objects with reflective Dirichlet boundary conditions. We present the analytically calculated discrete set of dimensionless radii z c analytical ​ ( n ) subscript superscript 𝑧 analytical c 𝑛 z^{\text{analytical}}_{\text{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 c numerical ​ ( n ) subscript superscript 𝑧 numerical c 𝑛 z^{\text{numerical}}_{\text{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 parameter a ¯ = 0.9 ¯ 𝑎 0.9 {\bar{a}}=0.9 linearly coupled to a massless scalar field mode with l = m = 1 𝑙 𝑚 1 l=m=1 . In the physically interesting regime z c ≪ 1 much-less-than subscript 𝑧 c 1 z_{\text{c}}\ll 1 of highly compact exotic objects Notephy , one finds a remarkably good agreement between the approximated radii { z c analytical ​ ( n ) } subscript superscript 𝑧 analytical c 𝑛 \{z^{\text{analytical}}_{\text{c}}(n)\} of the compact objects that can support the marginally-stable scalar resonances [as calculated from the analytical resonance spectrum ( 30 )] and the corresponding exact radii { z c numerical ​ ( n ) } subscript superscript 𝑧 numerical c 𝑛 \{z^{\text{numerical}}_{\text{c}}(n)\} of the compact exotic objects [as determined by a direct numerical solution of the resonance equation ( 22 )].

Formula	$z_{\text{c}}^{\text{Dir}}\left(n=1\right)$	$z_{\text{c}}^{\text{Dir}}\left(n=2\right)$	$z_{\text{c}}^{\text{Dir}}\left(n=3\right)$	$z_{\text{c}}^{\text{Dir}}\left(n=4\right)$	$z_{\text{c}}^{\text{Dir}}\left(n=5\right)$
Analytical [Eq. (30)]	$9.947\times {10}^{-3}$	$4.671\times {10}^{-4}$	$2.226\times {10}^{-5}$	$1.061\times {10}^{-6}$	$5.061\times {10}^{-8}$
Numerical [Eq. (22)]	$1.007\times {10}^{-2}$	$4.673\times {10}^{-4}$	$2.228\times {10}^{-5}$	$1.062\times {10}^{-6}$	$5.061\times {10}^{-8}$

Table 4. Table 4: Spinning exotic compact objects with reflective Neumann boundary conditions. We present the analytically calculated discrete set of dimensionless radii z c analytical ​ ( n ) subscript superscript 𝑧 analytical c 𝑛 z^{\text{analytical}}_{\text{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 c numerical ​ ( n ) subscript superscript 𝑧 numerical c 𝑛 z^{\text{numerical}}_{\text{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 parameter a ¯ = 0.9 ¯ 𝑎 0.9 {\bar{a}}=0.9 and a massless scalar field mode with l = m = 1 𝑙 𝑚 1 l=m=1 . In the physically interesting z c ≪ 1 much-less-than subscript 𝑧 c 1 z_{\text{c}}\ll 1 regime of highly compact exotic objects Notephy , one finds a remarkably good agreement between the approximated radii { z c analytical ​ ( n ) } subscript superscript 𝑧 analytical c 𝑛 \{z^{\text{analytical}}_{\text{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 c numerical ​ ( n ) } subscript superscript 𝑧 numerical c 𝑛 \{z^{\text{numerical}}_{\text{c}}(n)\} of the compact exotic objects [as determined numerically from the resonance equation ( 23 )].

Formula	$z_{\text{c}}^{\text{Neu}}\left(n=1\right)$	$z_{\text{c}}^{\text{Neu}}\left(n=2\right)$	$z_{\text{c}}^{\text{Neu}}\left(n=3\right)$	$z_{\text{c}}^{\text{Neu}}\left(n=4\right)$	$z_{\text{c}}^{\text{Neu}}\left(n=5\right)$
Analytical [Eq. (31)]	$4.839\times {10}^{-2}$	$2.145\times {10}^{-3}$	$1.019\times {10}^{-4}$	$4.860\times {10}^{-6}$	$2.318\times {10}^{-7}$
Numerical [Eq. (23)]	$5.432\times {10}^{-2}$	$2.153\times {10}^{-3}$	$1.020\times {10}^{-4}$	$4.860\times {10}^{-6}$	$2.318\times {10}^{-7}$

Table 5. Table 5: Marginally-stable spinning exotic compact objects. We display, for various values of the compactness parameter δ ≡ ( r c max − r + ) / M 𝛿 subscript superscript 𝑟 max c subscript 𝑟 𝑀 \delta\equiv(r^{\text{max}}_{\text{c}}-r_{+})/M Notedx introduced in Pan , the dimensionless ratio a ¯ analytical ​ ( δ ) / a ¯ numerical ​ ( δ ) superscript ¯ 𝑎 analytical 𝛿 superscript ¯ 𝑎 numerical 𝛿 {\bar{a}}^{\text{analytical}}(\delta)/{\bar{a}}^{\text{numerical}}(\delta) between the analytically derived critical angular momentum which characterizes the marginally-stable spinning exotic compact object and the corresponding numerically computed Pan value of the critical rotation parameter. The data presented are for the fundamental ( n = 1 𝑛 1 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 𝑙 𝑚 1 l=m=1 . One finds a remarkably good agreement between our analytical results and the corresponding numerical data of Pan .

$\delta \equiv \left(r_{\text{c}}-r_{+}\right)/M$	${10}^{-5}$	${10}^{-4}$	${10}^{-3}$	${10}^{-2}$	${10}^{-1}$
$\frac{{\overline{a}}^{\text{analytical}}}{{\overline{a}}^{\text{numerical}}}$	$0.999$	$0.999$	$1.001$	$1.000$	$1.001$

Equations

$$\displaystyle ds^{2}=-{{\Delta}\over{\rho^{2}}}(dt-a\sin^{2}\theta d\phi)^{2}+{{\rho^{2}}\over{\Delta}}dr^{2}+\rho^{2}d\theta^{2}+{{\sin^{2}\theta}\over{\rho^{2}}}\big{[}adt-(r^{2}+a^{2})d\phi\big{]}^{2}\ \ \ \text{for}\ \ \ \ r>r_{\text{c}}\ ,$$

$$z_{\text{c}}\equiv \frac{r_{\text{c}}-r_{+}}{r_{+}}\ll 1,$$

$$r_{\pm}=M+(M^{2}-a^{2})^{1/2}\$$

$${\nabla }^{\nu }{\nabla }_{\nu }\Psi =0.$$

$$\Psi =\sum _{l,m}{e}^{im\varphi }{S}_{lm}(\theta ;a\omega )R_{lm}(r;M,a,\omega )e^{-i\omega t},$$

$$\Delta{{d}\over{dr}}\Big{(}\Delta{{dR_{lm}}\over{dr}}\Big{)}+\Big{\{}[\omega(r^{2}+a^{2})-ma]^{2}+\Delta(2ma\omega-K_{lm})\Big{\}}R_{lm}=0\ .$$

$$\displaystyle{1\over{\sin\theta}}{{d}\over{\theta}}\Big{(}\sin\theta{{dS_{lm}}\over{d\theta}}\Big{)}+\Big{(}K_{lm}-a^{2}\omega^{2}+a^{2}\omega^{2}\cos^{2}\theta-{{m^{2}}\over{\sin^{2}\theta}}\Big{)}S_{lm}=0\ .$$

$$K_{lm}-a^2{\omega }^2=l(l+1)+\sum _{k=1}^{\infty }{c}_k(a\omega {)}^{2k},$$

$$\{\begin{array}{ll}R(r=r_{\text{c}})=0& \text{Dirichlet B. C.};\\ dR(r=r_{\text{c}})/dr=0& \text{Neumann B. C.}.\end{array}$$

$$R(r\to \infty )\to 0.$$

$$\omega =0.$$

$${\bar{a}}\equiv{{J}\over{M^{2}}}\$$

$$R(x)=x^{-i\alpha }(1-x{)}^{l+1}F(x),$$

$$x\equiv \frac{r-r_{+}}{r-r_{-}};\alpha \equiv \frac{ma}{r_{+}-r_{-}},$$

$$x(1-x)\frac{d^{2}F}{d{x}^2}+\{(1-2i\alpha )-[1+2(l+1)-2i\alpha ]x\}\frac{dF}{dx}-[(l+1{)}^2-2i\alpha (l+1)]F=0.$$

$$F(x)=A\cdot {}_2{F}_1(l+1-2i\alpha ,l+1;2l+2;1-x)+B\cdot (1-x{)}^{-2l-1}{}_2{F}_1(-l-2i\alpha ,-l;-2l;1-x),$$

$$\displaystyle R(x)=x^{-i\alpha}[A\cdot(1-x)^{l+1}{{}_{2}F_{1}}(l+1-2i\alpha,l+1;2l+2;1-x)+B\cdot(1-x)^{-l}{{}_{2}F_{1}}(-l-2i\alpha,-l;-2l;1-x)]\$$

$$\displaystyle{{}_{2}F_{1}}(a,b;c;z\to 0)\to 1\$$

$$\displaystyle R(x\to 1)=A\cdot(1-x)^{l+1}+B\cdot(1-x)^{-l}\$$

$$B=0.$$

$$R(x)=A\cdot x^{-i\alpha }(1-x{)}^{l+1}{}_2{F}_1(l+1-2i\alpha ,l+1;2l+2;1-x).$$

$${{}_{2}F_{1}}(l+1-2i\alpha,l+1;2l+2;1-x_{\text{c}})=0\ \ \ \ \ \text{for}\ \ \ \ \ \text{Dirichlet B. C.}\$$

$${{d}\over{dx}}[x^{-i\alpha}(1-x)^{l+1}{{}_{2}F_{1}}(l+1-2i\alpha,l+1;2l+2;1-x)]_{x=x_{\text{c}}}=0\ \ \ \ \ \text{for}\ \ \ \ \ \text{Neumann B. C.}\$$

$$\cdots r_{\text{c}}(n=3)<r_{\text{c}}(n=2)<r_{\text{c}}(n=1)\equiv r_{\text{c}}^{\text{max}}(\bar{a},l,m),$$

$$z^{\text{max}}_{\text{c}}({\bar{a}},l,m)\equiv{{r^{\text{max}}_{\text{c}}-r_{+}}\over{r_{+}}}\$$

$$x_{\text{c}}\ll 1.$$

$$R(x)$$

$$\displaystyle R(x\ll 1)=A{{(2l+1)!}\over{l!}}(1-x)^{l+1}\Big{[}{{\Gamma(2i\alpha)}\over{\Gamma(l+1+2i\alpha)}}x^{-i\alpha}+{{\Gamma(-2i\alpha)}\over{\Gamma(l+1-2i\alpha)}}x^{i\alpha}\Big{]}\cdot[1+O(x)]\$$

$$x^{2i\alpha }=\mp \frac{\Gamma (2i\alpha )\Gamma (l+1-2i\alpha )}{\Gamma (-2i\alpha )\Gamma (l+1+2i\alpha )},$$

$$x^{\text{Dirichlet}}_{\text{c}}(n)={e^{-\pi(n+{1\over 2})/\alpha}}\Big{[}{{\Gamma(2i\alpha)\Gamma(l+1-2i\alpha)}\over{\Gamma(-2i\alpha)\Gamma(l+1+2i\alpha)}}\Big{]}^{1/2i\alpha}\ \ \ ;\ \ \ n\in\mathbb{Z}$$

Full text

I Introduction

Black holes are certainly the most important prediction of general relativity, Einstein’s classical theory of gravity. These fundamental objects are characterized by curved spacetime geometries with absorbing boundary conditions (event horizons). Intriguingly, however, some researchers (see eco1 ; eco2 ; eco3 ; eco4 ; eco5 ; eco6 ; eco7 ; eco8 ; eco9 ; eco10 ; eco11 ; eco12 ; Pan and references therein) have recently argued that black-hole horizons may be quantum-mechanically unstable. Horizonless compact objects has therefore been proposed eco1 ; eco2 ; eco3 ; eco4 ; eco5 ; eco6 ; eco7 ; eco8 ; eco9 ; eco10 ; eco11 ; eco12 ; Pan as exotic quantum-gravity alternatives to the familiar (classical) black-hole spacetimes .

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 Pan have recently demonstrated numerically that, due to the physical mechanism of superradiant amplification Frid ; Bri , horizonless spinning compact objects with reflective boundary conditions and ergoregions may become unstable to scalar perturbation modes Notebi . Intriguingly, the results presented in Pan 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 $\{r_{\text{c}}({\bar{a}};n)\}^{n=\infty}_{n=1}$ of critical surface radii that can support spatially regular static (marginally-stable) massless scalar field configurations Notemsm ; Hodrc ; Herkr [Here ${\bar{a}}$ 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 $r^{\text{max}}_{\text{c}}({\bar{a}})\equiv\text{max}_{n}\{r_{\text{c}}({\bar{a}};n)\}$ stems from the fact that, for a given value of the dimensionless rotation parameter ${\bar{a}}$, this unique critical radius marks the boundary between stable and unstable spinning exotic compact configurations. In particular, horizonless compact objects of rotation parameter ${\bar{a}}$ whose reflecting surfaces are characterized by the inequality $r_{\text{c}}<r^{\text{max}}_{\text{c}}({\bar{a}})$ are superradiantly unstable to scalar perturbation modes, whereas horizonless compact objects whose reflecting surfaces are characterized by the inequality $r_{\text{c}}>r^{\text{max}}_{\text{c}}({\bar{a}})$ are stable.

The main goal of the present paper is to determine analytically the discrete spectrum $\{r_{\text{c}}({\bar{a}};n)\}^{n=\infty}_{n=1}$ of radii which characterize the horizonless spinning exotic compact objects that can support the spatially regular static (marginally-stable) scalar field configurations. In particular, below we shall derive a remarkably compact analytical formula for the critical (outermost) radii $\{r^{\text{max}}_{\text{c}}({\bar{a}})\}$ 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 Pan (see also eco1 ; eco2 ; eco3 ; eco4 ; eco5 ; eco6 ; eco7 ; eco8 ; eco9 ; eco10 ; eco11 ; eco12 ), 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 Pan , that the Kerr line element Chan ; Noteun ; Notebl ; Noteman

[TABLE]

with $\Delta\equiv r^{2}-2Mr+a^{2}$ and $\rho^{2}\equiv r^{2}+a^{2}\cos^{2}\theta$, describes the external spacetime of the spinning exotic compact object of mass $M$, angular momentum $J=Ma$, and radius $r_{\text{c}}$. In addition, below we shall assume that the radius of the exotic compact object is characterized by the relation Notesk ; kr1 ; kr2 ; kr3

[TABLE]

where

[TABLE]

are the radii of the would-be classical horizons. The small-$z_{\text{c}}$ 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 Pan (see also eco1 ; eco2 ; eco3 ; eco4 ; eco5 ; eco6 ; eco7 ; eco8 ; eco9 ; eco10 ; eco11 ; eco12 ).

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 Teuk ; Stro

[TABLE]

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 Teuk ; Stro ; Notedec

[TABLE]

one finds that the radial part $R_{lm}(r;M,a,\omega)$ of the scalar eigenfunction is determined by the ordinary differential equation Teuk ; Stro

[TABLE]

Here the frequency-dependent parameter $K_{lm}(a\omega)$ is the characteristic eigenvalue of the spatially regular angular eigenfunction $S_{lm}(\theta;a\omega)$, which is determined by the familiar spheroidal differential equation Heun ; Fiz1 ; Teuk ; Abram ; Stro ; Hodasy ; Hodpp

[TABLE]

For later purposes we note that, in the small frequency $a\omega\ll 1$ regime, the angular eigenvalues of the spheroidal scalar eigenfunctions can be expanded in the from

[TABLE]

where the frequency-independent coefficients $\{c_{k}(l,m)\}$ are given in Abram .

Following the physically interesting model of the spinning exotic objects studied numerically in Pan , we shall assume that the horizonless compact objects are characterized by (Dirichlet or Neumann) reflecting boundary conditions Noteby :

[TABLE]

In addition, the marginally-stable (static) scalar modes that we shall analyze in the present paper are characterized by spatially regular (asymptotically decaying) eigenfunctions:

[TABLE]

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

[TABLE]

The set of equations (6)-(11) determines two discrete spectra of radii, $\{r^{\text{Dirichlet}}_{\text{c}}({\bar{a}},l,m;n)\}$ and $\{r^{\text{Neumann}}_{\text{c}}({\bar{a}},l,m;n)\}$, which, for a given value

[TABLE]

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) $\omega=0$ and

[TABLE]

where

[TABLE]

one obtains the characteristic radial equation Noteklm0

[TABLE]

The general mathematical solution of the radial differential equation (15) is given by Abram ; Morse

[TABLE]

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

[TABLE]

for the radial scalar eigenfunction. Using the characteristic property (see Eq. 15.1.1 of Abram )

[TABLE]

of the hypergeometric function, one finds from (17) the asymptotic $r\to\infty$ [$x\to 1^{-}$, see Eq. (14)] behavior

[TABLE]

of the radial eigenfunction. A physically acceptable (finite) solution at spatial infinity ($r\to\infty$, or equivalently $x\to 1^{-}$) requires

[TABLE]

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

[TABLE]

Taking cognizance of the boundary conditions (9), which characterize the horizonless reflecting compact objects, one deduces that the compact resonance equations

[TABLE]

and

[TABLE]

determine the discrete spectra of dimensionless critical radii $\{x_{\text{c}}({\bar{a}},l,m;n)\}$ 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 $\{{\bar{a}},l,m\}$ of the composed spinning-exotic-compact-object-massless-scalar-field system, there exists a discrete set of critical radii,

[TABLE]

which can support the spatially regular static (marginally-stable) scalar field resonances.

In Table 1 we display, for various values of the dimensionless angular momentum parameter ${\bar{a}}$, the largest (outermost) dimensionless radii Notezx

[TABLE]

of the horizonless spinning compact objects that can support the marginally-stable massless scalar field configurations. From the data presented in Table 1 one learns that, for fixed values of the scalar angular harmonic indices, the critical radii $z^{\text{max}}_{\text{c}}({\bar{a}})$ [and thus also $r^{\text{max}}_{\text{c}}({\bar{a}})$] which characterize the marginally-stable spinning exotic compact objects are a monotonically increasing function of the dimensionless angular momentum parameter ${\bar{a}}$.

It is worth emphasizing again that the physical significance of the critical reflecting radius $r^{\text{max}}_{\text{c}}({\bar{a}})$ stems from the fact that, for a given value of the dimensionless angular momentum parameter ${\bar{a}}$, this supporting radius corresponds to a marginally-stable spinning object which marks the onset of superradiant instabilities in the composed exotic-compact-object-massless-scalar-field system. In particular, as nicely demonstrated numerically in Pan , spinning compact objects of dimensionless angular momentum ${\bar{a}}$ whose reflecting surfaces are located in the region $r_{\text{c}}>r^{\text{max}}_{\text{c}}({\bar{a}})$ are stable, whereas spinning objects whose reflecting surfaces are located in the region $r_{\text{c}}<r^{\text{max}}_{\text{c}}({\bar{a}})$ are superradiantly unstable to scalar perturbation modes.

In Table 2 we display, for various equatorial ($l=m$) massless scalar field modes, the critical (largest) dimensionless radii $z^{\text{max}}_{\text{c}}(l)$ [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 2 one learns that, for a fixed value of the dimensionless angular momentum parameter ${\bar{a}}$, the dimensionless critical radii $z^{\text{max}}_{\text{c}}(l)$ [and thus also $r^{\text{max}}_{\text{c}}(l)$] which characterize the marginally-stable spinning exotic compact objects are a monotonically increasing function of the scalar harmonic index $l$.

V Resonance spectra for highly-compact spinning exotic objects

V.1 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 $\{x^{\text{Dirichlet}}_{\text{c}}({\bar{a}},l,m;n)\}$ and $\{x^{\text{Neumann}}_{\text{c}}({\bar{a}},l,m;n)\}$ of the marginally-stable horizonless spinning exotic compact objects, can be solved analytically in the physically interesting regime

[TABLE]

As emphasized above, the small-$x_{\text{c}}$ 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 Pan (see also eco1 ; eco2 ; eco3 ; eco4 ; eco5 ; eco6 ; eco7 ; eco8 ; eco9 ; eco10 ; eco11 ; eco12 ).

Using Eq. 15.3.6 of Abram , one can express the characteristic radial scalar eigenfunction (21) in the form Notehg

[TABLE]

Taking cognizance of the characteristic asymptotic property (18) of the hypergeometric functions, one finds from (27) the small-$x$ behavior

[TABLE]

of the radial scalar eigenfunction.

Using the characteristic small-$x$ spatial behavior (28) of the radial scalar eigenfunction, one can express the Dirichlet and Neumann resonance equations (22) and (23) in the remarkably compact form Notehg

[TABLE]

where the upper/lower signs in (29) refer respectively to the reflecting Dirichlet/Neumann boundary conditions. From the resonance conditions (29) one finally finds the compact analytical formulas Notenn ; Notegam1

[TABLE]

and

[TABLE]

for the discrete families $\{x^{\text{Dirichlet}}_{\text{c}}(n),x^{\text{Neumann}}_{\text{c}}(n)\}$ of dimensionless critical radii which characterize the horizonless spinning compact objects that can support the spatially regular static (marginally-stable) massless scalar field resonances.

V.2 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 ($x_{\text{c}}\ll 1$) spinning exotic objects that can support the static (marginally-stable) scalar field configurations. In Tables 3 and 4 we display the dimensionless radii $z^{\text{analytical}}_{\text{c}}(n)\equiv[r_{\text{c}}(n)-r_{+}]/r_{+}$ Notezx 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 $z^{\text{numerical}}_{\text{c}}(n)$ 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 3 and 4 one finds a very good agreement, especially in the physically interesting regime $z_{\text{c}}\ll 1$ of highly compact exotic objects Notephy [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)].

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 Pan . In Table 5 we display, for various values of the compactness parameter $\delta\equiv(r^{\text{max}}_{\text{c}}-r_{+})/M$ Notedx introduced in Pan , the dimensionless ratio ${\bar{a}}^{\text{analytical}}(\delta)/{\bar{a}}^{\text{numerical}}(\delta)$. Here $\{{\bar{a}}^{\text{analytical}}(\delta)\}$ are the analytically derived dimensionless angular momenta which characterize the critical (marginally-stable) spinning exotic compact objects, and $\{{\bar{a}}^{\text{numerical}}(\delta)\}$ are the corresponding numerically computed Pan values of the critical rotation parameter. Interestingly, from the data presented in Table 5 one finds a remarkably good agreement between our analytical formulas and the corresponding numerical data of Pan .

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 eco1 ; eco2 ; eco3 ; eco4 ; eco5 ; eco6 ; eco7 ; eco8 ; eco9 ; eco10 ; eco11 ; eco12 ; Pan and references therein). In a physically important work, Maggio, Pani, and Ferrari Pan 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 Pan has demonstrated that horizonless spinning compact objects with reflective surfaces may become superradiantly unstable to scalar perturbation modes Notebi . Intriguingly, however, the results presented in Pan 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-massless-scalar-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, $\{r^{\text{Dirichlet}}_{\text{c}}({\bar{a}},l,m;n)\}^{n=\infty}_{n=1}$ and $\{r^{\text{Neumann}}_{\text{c}}({\bar{a}},l,m;n)\}^{n=\infty}_{n=1}$, which characterize the horizonless spinning compact objects with reflective boundary conditions that can support the spatially regular static (marginally-stable) scalar field configurations Notemsm ; Hodrc ; Herkr . In particular, we have shown that the analytically derived resonance equations [see Eqs. (22) and (23)]

[TABLE]

determine the critical dimensionless radii of the marginally-stable spinning exotic compact objects.

(2) We have shown that the physical properties of the critical (marginally-stable) horizonless spinning objects can be studied analytically in the physically interesting regime $x_{\text{c}}\ll 1$ which corresponds to highly compact exotic objects Notephy . In particular, using analytical techniques, we have derived the remarkably compact resonance formula [see Eqs. (30) and (31)]

[TABLE]

where

[TABLE]

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 $r^{\text{max}}_{\text{c}}({\bar{a}})\equiv\text{max}_{n}\{r_{\text{c}}({\bar{a}};n)\}$ 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 $r_{\text{c}}<r^{\text{max}}_{\text{c}}({\bar{a}})$ are superradiantly unstable to scalar perturbation modes, whereas spinning exotic compact objects whose reflecting surfaces are located in the radial region $r_{\text{c}}>r^{\text{max}}_{\text{c}}({\bar{a}})$ are stable.

(3) It has been explicitly demonstrated (see Tables 3 and 4) 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 $x_{\text{c}}\ll 1$ of highly-compact objects Notephy with the corresponding exact radii of the critical exotic compact objects [as determined numerically directly from the resonance equations (22) and (23)].

(4) It is worth pointing out that the compact resonance spectra (33), which characterize the marginally-stable horizonless configurations, can be further simplified in the slow rotation regime ${\bar{a}}\ll 1$. In particular, one finds from (33) and (34) the resonance spectra Notesma

[TABLE]

in the regime ${\bar{a}}\ll 1$ [or equivalently, $\alpha\simeq m{\bar{a}}/2\ll 1$, see Eq. (14)] of slowly-rotating exotic compact objects. It is worth noting that this analytical formula is especially useful since it is highly difficult to probe numerically Pan the small ${\bar{a}}\ll 1$ regime [which, as is evident from (35), corresponds to exponentially small values of the dimensionless radius $x_{\text{c}}({\bar{a}}\ll 1)$].

(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 Pan .

ACKNOWLEDGMENTS

This research is supported by the Carmel Science Foundation. I thank Yael Oren, Arbel M. Ongo, Ayelet B. Lata, and Alona B. Tea for stimulating discussions.