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

TL;DR

This paper studies the late-time decay of massive vector fields around black holes, revealing universal decay laws and different regimes depending on the black hole type and field mass.

Contribution

It provides the first detailed analysis of late-time tails of massive vector fields in Schwarzschild and Schwarzschild-de Sitter backgrounds, highlighting universal decay behavior and regimes.

Findings

At intermediate times, decay laws depend on multipole number and oscillate with mass.

At late times, decay is universal with a t^{-5/6} law, independent of spin and multipole.

Different decay regimes are observed in Schwarzschild-de Sitter black holes, including exponential damping and quasinormal mode decay.

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 $\Psi_{0} \sim t^{-(\ell + 3/2)} \sin{m t}$, $\Psi_{1} \sim t^{-(\ell + 5/2)} \sin{m t}$, $\Psi_{2} \sim t^{-(\ell + 1/2)} \sin{m t}$, while at asymptotically late times the decay law $\Psi \sim t^{-5/6} \sin{(m t)}$ is universal, and does not depend on the multipole number $\ell$. Together with 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 $\sim t^{-5/6} \sin{(m t)}$ \emph{does not depend} also \emph{on the spin} of the field under consideration. For Schwarzschild-de Sitter black holes it is observed two different regimes in the late-time decay of perturbations: non-oscillatory 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.