Conformal structure of the Schwarzschild black hole

TL;DR

This paper uncovers a hidden SL(2,R) symmetry in the low-frequency scalar wave equation of Schwarzschild black holes, linking it to AdS_2 symmetries and using it to algebraically derive quasinormal frequencies.

Contribution

It reveals a hidden SL(2,R) invariance in Schwarzschild black holes and connects it to AdS_2 symmetries, providing a new algebraic method to compute quasinormal modes.

Findings

Hidden SL(2,R) symmetry in Schwarzschild scalar wave equation

Connection between Schwarzschild symmetry and AdS_2 isometries

Algebraic derivation of quasinormal frequencies for large damping

Abstract

We show that the scalar wave equation at low frequencies in the Schwarzschild geometry enjoys a hidden SL(2,R) invariance, which is not inherited from an underlying symmetry of the spacetime itself. Contrary to what happens for Kerr black holes, the vector fields generating the SL(2,R) are globally defined. Furthermore, it turns out that under an SU(2,1) Kinnersley transformation, which maps the Schwarzschild solution into the near horizon limit AdS_2 x S^2 of the extremal Reissner-Nordstr"om black hole (with the same entropy), the Schwarzschild hidden symmetry generators become exactly the isometries of the AdS_2 factor. Finally, we use the SL(2,R) symmetry to determine algebraically the quasinormal frequencies of the Schwarzschild black hole, and show that this yields the correct leading behaviour for large damping.