Charged reflecting stars supporting charged massive scalar field configurations

Abstract

Abstract The recently published no-hair theorems of Hod, Bhattacharjee, and Sarkar have revealed the intriguing fact that horizonless compact reflecting stars 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 can support 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 $$\mu $$ μ , 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 $$R_{\text {s}}\gg M,Q$$ Rs≫M,Q . Interestingly, it is proved that the discrete set $$\{R_{\text {s}}(M,Q,\mu ,q,l;n)\}^{n=\infty }_{n=1}$$ {Rs(M,Q,μ,q,l;n)}n=1n=∞ 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 $$M\ll R_{\text {s}}\ll 1/\mu $$ M≪Rs≪1/μ . The analytically derived resonance spectrum is confirmed by direct numerical computations.