Wave Propagation in Gravitational Systems: Late Time Behavior

TL;DR

This paper systematically analyzes wave tails in various gravitational systems using Green's function formalism, revealing that late-time decay behaviors depend on spatial asymptotics and are not always power-law, with analytical and numerical agreement.

Contribution

It generalizes the understanding of wave tails beyond Schwarzschild spacetime, showing decay depends on spatial structure and is often not a power law.

Findings

Late-time tails are governed by a frequency Green's function cut along the imaginary axis.

Decay behavior depends on spatial asymptotics, not just local structure or horizons.

Analytical results match numerical calculations across models.

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 $\tilde G(\omega)$ along the $-$~Im~$\omega$ axis, generalizing the Schwarzschild result. (ii) The $\omega$ 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.