High-order tail in Schwarzschild spacetime

Abstract

We present an analysis of the behavior at late times of linear field perturbations of a Schwarzschild black hole spacetime. In particular, we give explicit analytic expressions for the field perturbations (for a specific $\ensuremath{\ell}$-multipole) of general spin up to the first four orders at late times. These expressions are valid at arbitrary radius and include, apart from the well-known power-law tail decay at leading order ($\ensuremath{\sim}{t}^{\ensuremath{-}2\ensuremath{\ell}\ensuremath{-}3}$), a new logarithmic behavior at third leading order ($\ensuremath{\sim}{t}^{\ensuremath{-}2\ensuremath{\ell}\ensuremath{-}5}\mathrm{ln}t$). We obtain these late-time results by developing an analytical formalism initially formulated by Mano, Suzuki and Takasugi (MST) [Prog. Theor. Phys. 95, 1079 (1996); 96, 549 (1996)] formalism and by expanding the various MST Fourier-mode quantities for small frequency. While we give explicit expansions up to the first four leading orders (for small frequency for the Fourier modes, for late time for the field perturbation), we give a prescription for obtaining expressions to arbitrary order within a ``perturbative regime.''