Asymptotic tails of massive scalar fields in a Schwarzschild background

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

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