# 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.

## Key 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.