It has recently been revealed that, in curved black-hole spacetimes, non-minimally coupled massive Proca fields may be characterized by the existence of poles in their linearized perturbation equations and may therefore develop exponentially growing instabilities. Interestingly, recent numerical computations [H. W. Chiang, S. Garcia-Saenz, and A. Sang, arXiv:2504.04779] have provided compelling evidence that the onset of monopole instabilities in the composed black-hole-field system is controlled by the dimensionless physical parameter $\mu r_-$, where $\mu$ is the proper mass of the non-minimally coupled Proca field and $r_-\equiv (-2\alpha)^{ 1/3}r_{ \text{ H}}$ is the radial location of the pole [here $\alpha$ is the non-minimal coupling parameter of the Einstein-Proca theory and $r_{ \text{ H}}$ is the radius of the black-hole horizon]. In the present paper we use { \it analytical} techniques in order to explore the physical properties of critical (marginally-stable) composed Schwarzschild-black-hole-nonminimally-coupled-monopole-Proca-field configurations. In particular, we derive a remarkably compact analytical formula for the discrete spectrum $\{ \mu(r_{ \text{ H}},r_-;n) \}^{ n=\infty}_{ n=1}$ of Proca field masses which characterize the critical black-hole-monopole-Proca-field configurations in the dimensionless regime ${ { r_- -r_{ \text{ H}}}\over{ r_{ \text{ H}}}}\ll1$ of near-horizon poles. The physical significance of the analytically derived resonance spectrum stems from the fact that the critical field mass $\mu_{ \text{ c}}\equiv\mu(r_{ \text{ H}},r_-;n=1)$ marks the onset of instabilities in the Schwarzschild-black-hole-nonminimally-coupled-monopole-Proca-field system. In particular, composed black-hole-linearized-Proca-field configurations in the small-mass regime $\mu\leq\mu_{ \text{ c}}$ of the Proca field are stable.
Categories: gr-qc astro-ph.HE hep-th