Stationary bound-state massive scalar field configurations supported by spherically symmetric compact reflecting stars

TL;DR

This paper demonstrates that spherically symmetric compact reflecting stars can support stationary bound-state massive scalar fields outside their surface, providing analytical formulas for the resonant frequencies and verifying them numerically.

Contribution

It is the first to analytically prove the existence of stationary massive scalar fields around reflecting stars and derive explicit resonance frequency formulas.

Findings

Stationary scalar fields can exist outside reflecting stars.

Analytical resonance frequency formulas match numerical results.

Existence conditions depend on star compactness and scalar field parameters.

Abstract

It has recently been demonstrated that asymptotically flat neutral reflecting stars are characterized by an intriguing no-hair property. In particular, it has been proved that these {\it horizonless} compact objects cannot support spatially regular {\it static} matter configurations made of scalar (spin-0) fields, vector (spin-1) fields, and tensor (spin-2) fields. In the present paper we shall explicitly prove that spherically symmetric compact reflecting stars can support {\it stationary} (rather than static) bound-state massive scalar fields in their exterior spacetime regions. To this end, we solve analytically the Klein-Gordon wave equation for a linearized scalar field of mass $\mu$ and proper frequency $\omega$ in the curved background of a spherically symmetric compact reflecting star of mass $M$ and radius $R_{\text{s}}$. It is proved that the regime of existence of these stationary composed star-field configurations is characterized by the simple inequalities $1-2M/R_{\text{s}}<(\omega/\mu)^2<1$. Interestingly, in the regime $M/R_{\text{s}}\ll1$ of weakly self-gravitating stars we derive a remarkably compact {\it analytical} formula for the discrete spectrum $\{\omega(M,R_{\text{s}},\mu)\}^{n=\infty}_{n=1}$ of resonant oscillation frequencies which characterize the stationary composed compact-reflecting-star-linearized-massive-scalar-field configurations. Finally, we verify the accuracy of the analytically derived resonance formula of the composed star-field configurations with direct numerical computations.