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

We consider the perturbations of the massive vector field around Schwarzschild, Schwarzschild-de Sitter, and Schwarzschild-anti-de Sitter black holes. Equations for a spherically symmetric massive vector perturbation can be reduced to a single wavelike equation. We have proved the stability against these perturbations and investigated the quasinormal spectrum. The quasinormal behavior for Schwarzschild black hole is quite unexpected: the fundamental mode and higher overtones show 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 the 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 nonoscillatory, damping modes can exist. In the limit of asymptotically high damping, $\mathrm{Re}\omega$ goes to $ln3/(8\pi M)$, while imaginary part shows equidistant behavior with spacing $\mathrm{Im}{\omega }_{n+1}-\mathrm{Im}{\omega }_n=1/4M$. In addition, we have found quasinormal spectrum of massive vector field for Schwarzschild-anti-de Sitter black hole.