Hidden Conformal Symmetry and Quasi-normal Modes

TL;DR

This paper introduces an algebraic method leveraging hidden conformal symmetry to efficiently compute quasi-normal modes of black holes, unifying scalar, vector, and tensor cases through Lie algebra techniques.

Contribution

It presents a novel algebraic approach to derive quasi-normal modes using hidden conformal symmetry, applicable to various field types and black hole dimensions.

Findings

Constructed quasi-normal modes algebraically from highest-weight modes.

Rewrote scalar Laplacian in terms of $SL(2,R)$ Casimir.

Achieved agreement with previous numerical results.

Abstract

We provide an algebraic way to calculate the quasi-normal modes of a black hole, which possesses a hidden conformal symmetry. We construct an infinite tower of quasi-normal modes from the highest-weight mode, in a simple and elegant way. For the scalar, the hidden conformal symmetry manifest itself in the fact that the scalar Laplacian could be rewritten in terms of the $SL(2,R)$ quadratic Casimir. For the vector and the tensor, the hidden conformal symmetry acts on them through Lie derivatives. We show that for three-dimensional black holes, with appropriate combination of the components the radial equations of the vector and the tensor could be written in terms of the Lie-induced quadratic Casimir. This allows the algebraic construction of the quasi-normal modes feasible. Our results are in good agreement with the previous study.