The overtone level spacing of a black hole quasinormal frequencies: a fingerprint of a local $SL(2,\mathbb{R})$ symmetry

TL;DR

This paper demonstrates that the evenly spaced overtone level spacing of black hole quasinormal frequencies reflects a hidden local $SL(2, ext{R})$ symmetry near the horizon and photon sphere regions, unifying understanding across damping regimes.

Contribution

It reveals that the overtone level spacing in black hole quasinormal modes is a signature of a hidden local $SL(2, ext{R})$ symmetry in different spacetime regions.

Findings

Overtone spacing is linked to a hidden $SL(2, ext{R})$ symmetry.

Symmetry exists near the horizon and photon sphere regions.

Both highly damped and weakly damped modes exhibit this spacing.

Abstract

The imaginary part of the quasinormal frequencies spectrum for a static and spherically symmetric black hole is analytically known to be equally spaced, both for the highly damped and the weakly damped families of quasinormal modes. Some interesting attempts have been made in the last twenty years to understand in simple ways this level spacing for the only case of highly damped quasinormal frequencies. Here, we show that the overtone level spacing, for both the highly damped and weakly damped families of quasinormal modes, can simply be understood as a fingerprint of a hidden local $SL(2,\mathbb{R})$ symmetry, near different regions of the black hole spacetime, i.e. the near-horizon and the near-photon sphere regions.