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

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 horizonless compact objects cannot support spatially regular 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 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 μ and proper frequency ω in the curved background of a spherically symmetric compact reflecting star of mass M and radius R_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_s < (ω/μ)² < 1. Interestingly, in the regime M/R_s << 1 of weakly self-gravitating stars we derive a remarkably compact analytical equation for the discrete spectrum {ω(M,R_s, μ)}^n=∞_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. (orig.)