# QSimFP QBH Seminar: Pseudospectrum and black hole quasinormal mode instability: an ultraviolet universality conjecture

Abstract:

Can we measure the ‘effective regularity’ of spacetime from the perturbation of quasi-normal mode (QNM) overtones? Black hole (BH) QNMs encode the resonant response of black holes under linear perturbations, their associated complex frequencies providing an invariant probe into the background spacetime geometry. In the late nineties, Nollert and Price found evidence of a BH QNM instability phenomenon, according to which perturbed QNMs of Schwarzschild spacetime migrate to new perturbed branches of different qualitative behaviour and asymptotics. Here we revisit this BH QNM instability issue by adopting a pseudospectrum approach. Specifically, we cast the QNM problem as an eigenvalue problem for a non-selfadjoint operator by adopting a hyperboloidal formulation of spacetime. Non-selfadjoint (more generally non-normal) operators suffer potentially of spectral instabilities, the notion of pseudospectrum providing a tool suitable for their study. We find evidence that perturbed Nollert & Price BH QNMs track the pseudospectrum contour lines, therefore probing the analytic structure of the resolvent, showing the following (in)stability behaviour: i) the slowest decaying (fundamental) mode is stable, whereas ii) (all) QNM overtones are ultraviolet unstable (for sufficiently high frequency). Building on recent work characterizing Burnett’s conjecture as a low-regularity problem in general relativity, we conjecture that (in the infinite-frequency limit) generic ultraviolet spacetime perturbations make BH QNMs migrate to ‘Regge QNM branches’ with a precise universal logarithmic pattern. This is a classical general relativity (effective) low-regularity phenomenon, agnostic to possible detailed (quantum) descriptions of gravity at higher-energies and potentially observationally accessible.

Short bio:

Jose Luis Jaramillo works at the Institut de Mathématiques de Bourgogne (IMB) in Dijon, in the Mathematical Physics group. He did his Ph.D at the Instituto de Astrofísica de Andalucía (IAA-CSIC), followed by postdoctoral stays at the Observatoire de Paris-Meudon, the Albert Einstein Institut (Max-Planck Institut for Gravitational Physics) in Golm and the Laboratoire de Physique Océanographique (LPO) in Brest (France). Since 2015 he is professor in the Mathematics Department at the Université de Bourgogne in Dijon.

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