Asymptotic tails of massive scalar fields in Schwarzschild background

Abstract

We investigate the asymptotic tail behavior of massive scalar fields in Schwarzschild background. It is shown that the oscillatory tail of the scalar field has the decay rate of $t^{-5/6}$ at asymptotically late times, and the oscillation with the period $2\pi/m$ for the field mass $m$ is modulated by the long-term phase shift. These behaviors are qualitatively similar to those found in nearly extreme Reissner-Nordstr\"{o}m background, which are discussed in terms of a resonant backscattering due to the space-time curvature.