# Stationary bound-state scalar configurations supported by rapidly-spinning exotic compact objects

## TL;DR

This paper derives an analytical formula for the discrete radii of rapidly-spinning exotic compact objects that can support stationary bound-state massive scalar fields, supported by numerical validation.

## Contribution

It provides the first explicit analytical expression for the radii of spinning exotic compact objects supporting scalar fields, extending previous numerical studies.

## Key findings

- Derived a compact analytical formula for supporting radii
- Confirmed analytical results with numerical computations
- Identified conditions for scalar field support on exotic objects

## Abstract

Some quantum-gravity theories suggest that the absorbing horizon of a classical black hole should be replaced by a reflective surface which is located a microscopic distance above the would-be classical horizon. Instead of an absorbing black hole, the resulting horizonless spacetime describes a reflective exotic compact object. Motivated by this intriguing prediction, in the present paper we explore the physical properties of exotic compact objects which are linearly coupled to stationary bound-state massive scalar field configurations. In particular, solving the Klein-Gordon wave equation for a stationary scalar field of proper mass $\mu$ and spheroidal harmonic indices $(l,m)$ in the background of a rapidly-rotating exotic compact object of mass $M$ and angular momentum $J=Ma$, we derive a compact analytical formula for the {\it discrete} radii $\{r_{\text{c}}(\mu,l,m,M,a;n)\}$ of the exotic compact objects which can support the stationary bound-state massive scalar field configurations. We confirm our analytical results by direct numerical computations.