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.
Categories: hep-th astro-ph.HE gr-qc