Telling late-time tails for a massive scalar field in the background of brane-localized black holes

We examine perturbations of a massive scalar field around spherically symmetric, brane-localized black holes. Although the ringdown and asymptotic tails of various brane-world black holes have been extensively studied, there has been no analysis of the massive late-time tails for the simplest Schwarzschild-like, brane-localized black hole to date. We demonstrate that after the ringdown phase, two stages of oscillatory tails emerge —intermediate and asymptotic. The asymptotic decay law is distinct from those associated with Schwarzschild or Reissner-Nordström solutions. Specifically, during intermediate times, the signal decays as , while the asymptotic decay law is .