High-order Tail in Schwarzschild Space-time

TL;DR

This paper analyzes late-time behavior of linear field perturbations in Schwarzschild space-time, revealing new logarithmic decay terms and providing explicit formulas up to the fourth order for various spins.

Contribution

It introduces explicit analytic expressions for late-time perturbations, including a novel logarithmic tail, using the MST formalism expanded for small frequencies.

Findings

Identified a new logarithmic decay tail at third order.

Derived explicit formulas for perturbations up to the fourth order.

Provided a method to obtain higher-order expressions within the perturbative regime.

Abstract

We present an analysis of the behaviour at late-times of linear field perturbations of a Schwarzschild black hole space-time. In particular, we give explicit analytic expressions for the field perturbations (for a specific 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 ($\sim t^{-2\ell-3}$), a new logarithmic behaviour at third leading order ($\sim t^{-2\ell-5}\ln t$). We obtain these late-time results by developing the so-called MST 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'.