$SL(2,R)$ symmetry and quasi-normal modes in the BTZ black hole

TL;DR

This paper reveals an intrinsic $SL(2,R)$ symmetry in the equations of motion for fields in the BTZ black hole and uses it to derive quasi-normal modes, confirming previous exact solutions.

Contribution

It introduces new tensor fields linked to the $SL(2,R)$ Casimir and applies an algebraic method to determine quasi-normal modes without coordinate dependence.

Findings

Quasi-normal modes form an infinite tower of descendants.

The $SL(2,R)$ symmetry is intrinsic and coordinate-independent.

Results agree with previous exact solutions.

Abstract

With the help of two new intrinsic tensor fields associated with the $SL(2,R)$ quadratic Casimir of Killing fields, we uncover the $SL(2,R)$ symmetry satisfied by the solutions to the equations of motion for various fields in the BTZ black hole in a uniform way by performing tensor and spinor analysis without resorting to any specific coordinate system. Then with the standard algebraic method developed recently, we determine the quasi-normal modes for various fields in the BTZ black hole. As a result, the quasi-normal modes are given by the infinite tower of descendants of the chiral highest weight mode, which is in good agreement with the previous analytic result obtained by exactly solving equations of motion instead.