Wave Propagation in Gravitational Systems: Late Time Behavior

TL;DR

A systematic treatment of the tail phenomenon for a broad class of models via a Green's function formalism and the Schwarzschild case with a power-law tail is exceptional among the class of the potentials having a logarithmic spatial dependence.

Abstract

It is well-known that the dominant late time behavior of waves propagating on a Schwarzschild spacetime is a power-law tail; tails for other spacetimes have also been studied. This paper presents a systematic treatment of the tail phenomenon for a broad class of models via a Green's function formalism and establishes the following. (i) The tail is governed by a cut of the frequency Green's function $\tilde G(ω)$ along the $-$~Im~$ω$ axis, generalizing the Schwarzschild result. (ii) The $ω$ dependence of the cut is determined by the asymptotic but not the local structure of space. In particular it is independent of the presence of a horizon, and has the same form for the case of a star as well. (iii) Depending on the spatial asymptotics, the late time decay is not necessarily a power law in time. The Schwarzschild case with a power-law tail is exceptional among the class of the potentials having a logarithmic spatial dependence. (iv) Both the amplitude and the time dependence of the tail for a broad class of models are obtained analytically. (v) The analytical results are in perfect agreement with numerical calculations.