Transient dynamics of quasinormal mode sums

TL;DR

This paper investigates how non-normality in quasinormal mode spectra leads to long-lived energy packets near black hole horizons, with implications for black hole stability and holographic thermalization.

Contribution

It demonstrates the existence of arbitrarily long-lived sums of quasinormal modes due to spectral instabilities caused by non-normal operators.

Findings

Long-lived sums of quasinormal modes scale as log(M).

Provided closed-form examples for scalar fields in de Sitter and BTZ black holes.

Numerical evidence for scalar and gravitational perturbations in various spacetimes.

Abstract

Quasinormal modes of spacetimes with event horizons are typically governed by a non-normal operator. This gives rise to spectral instabilities, a topic of recent interest in the black hole pseudospectrum programme. In this work we show that non-normality leads to the existence of arbitrarily long-lived sums of short-lived quasinormal modes, corresponding to localising packets of energy near the future horizon. There exist sums of $M$ quasinormal modes whose lifetimes scale as $\log{M}$. This transient behaviour results from large cancellations between non-orthogonal quasinormal modes. We provide simple closed-form examples for a massive scalar field in the static patch of dS$_{d+1}$ and the BTZ black hole. We also provide numerical examples for scalar perturbations of Schwarzschild-AdS$_{d+1}$, and gravitational perturbations of Schwarzschild in asymptotically flat spacetime, using hyperboloidal foliations. The existence of these perturbations is linked to certain properties of black hole pseudospectra. We comment on implications for thermalisation times in holographic plasmas.