Massive vector field perturbations in the Schwarzschild background: stability and quasinormal spectrum

We consider the perturbations of the massive vector field around Schwarzschild black hole, (generally, with non-vanishing $\Lambda$ - term). The monopole massive vector perturbation equations can be reduced to a single wave-like equation. We have proved the stability against these perturbations and investigated the quasinormal spectrum. The quasinormal behaviour for Schwarzschild black hole is quite unexpected: the fundamental mode and all higher overtones shows totally different dependence on the mass of the field $m$: as $m$ is increasing, the damping rate of the fundamental mode is decreasing, what results in appearing of the infinitely long living modes, while, on contrary, damping rate of all higher overtones are increasing, and their real oscillation frequencies gradually go to tiny values. Thereby, for all higher overtones, almost non-oscillatory, damping modes can exist. In the limit of asymptotically high damping, $Re \omega$ goes to $ln3/(8 \pi M)$, while imaginary part shows equidistant behaviour with spacing $Im \omega_{n+1}- Im \omega_{n}=i/4M$. In addition, we have found quasinormal spectrum of massive vector field for Schwarzschild-anti-de Sitter black hole.