Late time tails of the massive vector field in a black hole background

Abstract

We investigate the late-time behavior of the massive vector field in the background of the Schwarzschild and Schwarzschild--de Sitter black holes. For Schwarzschild black hole, at intermediately late times the massive vector field is represented by three functions with different decay law ${\ensuremath{\Psi}}_{0}\ensuremath{\sim}{t}^{\ensuremath{-}(\ensuremath{\ell}+3/2)}\mathrm{sin}mt$, ${\ensuremath{\Psi}}_{1}\ensuremath{\sim}{t}^{\ensuremath{-}(\ensuremath{\ell}+5/2)}\mathrm{sin}mt$, ${\ensuremath{\Psi}}_{2}\ensuremath{\sim}{t}^{\ensuremath{-}(\ensuremath{\ell}+1/2)}\mathrm{sin}mt$, while at asymptotically late times the decay law $\ensuremath{\Psi}\ensuremath{\sim}{t}^{\ensuremath{-}5/6}\mathrm{sin}(mt)$ is universal and does not depend on the multipole number $\ensuremath{\ell}$. Together with a previous study of massive scalar and Dirac fields where the same asymptotically late-time decay law was found, it means that the asymptotically late-time decay law $\ensuremath{\sim}{t}^{\ensuremath{-}5/6}\mathrm{sin}(mt)$ does not depend also on the spin of the field under consideration. For Schwarzschild--de Sitter black holes it is observed in two different regimes in the late-time decay of perturbations: nonoscillatory exponential damping for small values of $m$ and oscillatory quasinormal mode decay for high enough $m$. Numerical and analytical results are found for these quasinormal frequencies.