Late-time Kerr tails: generic and non-generic initial data sets, "up" modes, and superposition

TL;DR

This paper investigates late-time scalar-field tails in Kerr spacetime, proposing a simplified formula for decay rates of multipole moments based on initial data, and clarifying the influence of initial conditions and symmetries.

Contribution

It generalizes the Barack-Ori decay formula to include contributions from any initial multipole, simplifying the understanding of tail decay rates in Kerr spacetime.

Findings

Decay rate depends only on initial and final multipoles, not on azimuthal number m.

Proposed decay index formula: n=ℓ'+ℓ+1 for ℓ<ℓ', and n=ℓ'+ℓ+3 for ℓ≥ℓ'.

Angular symmetry does not determine the decay rate.

Abstract

Three interrelated questions concerning Kerr spacetime late-time scalar-field tails are considered numerically, specifically the evolutions of generic and non-generic initial data sets, the excitation of "up" modes, and the resolution of an apparent paradox related to the superposition principle. We propose to generalize the Barack-Ori formula for the decay rate of any tail multipole given a generic initial data set, to the contribution of any initial multipole mode. Our proposal leads to a much simpler expression for the late-time power law index. Specifically, we propose that the late-time decay rate of the $Y_{\ell m}$ spherical harmonic multipole moment because of an initial $Y_{\ell' m}$ multipole is independent of the azimuthal number $m$, and is given by $t^{-n}$, where $n=\ell'+\ell+1$ for $\ell<\ell'$ and $n=\ell'+\ell+3$ for $\ell\ge\ell'$. We also show explicitly that the angular symmetry group of a multipole does not determine its late-time decay rate.