It has recently been revealed that, in curved black-hole spacetimes, nonminimally 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 nonminimally coupled Proca field and $r_{-}\equiv (-2\alpha {)}^{1/3}{r}_{\mathrm{H}}$ is the radial location of the pole [here $\alpha$ is the nonminimal coupling parameter of the Einstein-Proca theory and $r_{\mathrm{H}}$ is the radius of the black-hole horizon]. In the present paper we use 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 $\{\mu (r_{\mathrm{H}},r_{-};n){\}}_{n=1}^{n=\infty }$ of Proca field masses which characterize the critical black-hole monopole-Proca-field configurations in the dimensionless regime $\frac{r_{-}-r_{\mathrm{H}}}{r_{\mathrm{H}}}\ll 1$ of near-horizon poles. The physical significance of the analytically derived resonance spectrum stems from the fact that the critical field mass ${\mu }_{\mathrm{c}}\equiv \mu (r_{\mathrm{H}},r_{-};n=1)$ 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 $\mu \le {\mu }_{\mathrm{c}}$ of the Proca field are stable.

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• General relativity
• Gravitation