Transient dynamics of quasinormal mode sums

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.

42 pages, 12 figures, 1 table. Added some comments and references. Matches published version

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• Pseudospectra of Quasinormal Modes and Holography
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• Perturbing a quantum black hole
• Pseudospectrum and time-domain analysis of the EFT corrected black holes
• Transients in black hole perturbation theory
• QNM orthogonality relations for AdS black holes
• Relaxation time approximation revisited and non-analytical structure in retarded correlators
• Spectral instability of horizonless compact objects within astrophysical environments
• Shear transport in far-from-equilibrium isotropization of supersymmetric Yang-Mills plasma
• Pseudospectrum and (in)stability of black hole total transmission modes