Charged reflecting stars supporting charged massive scalar field configurations

TL;DR

This paper demonstrates that charged reflecting stars can support charged massive scalar fields outside their surface, challenging previous no-hair theorems, and provides analytical and numerical solutions for the supporting radii spectrum.

Contribution

It explicitly proves that charged reflecting stars can support charged scalar fields, providing analytical formulas for the supporting radii spectrum in specific regimes.

Findings

Charged reflecting stars can support charged scalar fields outside their surface.

The supporting radii are determined by zeros of confluent hypergeometric functions.

Analytical formulas for the star radii spectrum are validated by numerical computations.

Abstract

The recently published no-hair theorems of Hod, Bhattacharjee, and Sarkar have revealed the intriguing fact that horizonless compact reflecting stars {\it cannot} support spatially regular configurations made of scalar, vector and tensor fields. In the present paper we explicitly prove that the interesting no-hair behavior observed in these studies is not a generic feature of compact reflecting stars. In particular, we shall prove that charged reflecting stars {\it can} support {\it charged} massive scalar field configurations in their exterior spacetime regions. To this end, we solve analytically the characteristic Klein-Gordon wave equation for a linearized charged scalar field of mass $\mu$, charge coupling constant $q$, and spherical harmonic index $l$ in the background of a spherically symmetric compact reflecting star of mass $M$, electric charge $Q$, and radius $R_{\text{s}}\gg M,Q$. Interestingly, it is proved that the discrete set $\{R_{\text{s}}(M,Q,\mu,q,l;n)\}^{n=\infty}_{n=1}$ of star radii that can support the charged massive scalar field configurations is determined by the characteristic zeroes of the confluent hypergeometric function. Following this simple observation, we derive a remarkably compact analytical formula for the discrete spectrum of star radii in the intermediate regime $M\ll R_{\text{s}}\ll 1/\mu$. The analytically derived resonance spectrum is confirmed by direct numerical computations.