Conformal structure of the Schwarzschild black hole

TL;DR

A hidden $SL(2,R)$ conformal structure underpins the low-frequency, near-region scalar wave equation in Schwarzschild spacetime, providing a chiral symmetry with globally defined generators. Through a Kinnersley SU(2,1) transformation, Schwarzschild is mapped to the near-horizon AdS2 × S^2 geometry, revealing that the hidden generators become AdS2 isometries and the dynamics reduce to AdS2 × S^2 in the appropriate limit. The authors use the $SL(2,R)$ algebra to construct Schwarzschild quasinormal modes algebraically as descendants of a lowest-weight state, recovering the correct leading large-n damping. These results support a potential two-dimensional conformal (holographic) description of Schwarzschild and connect horizon symmetries to AdS/CFT-inspired structures, with extensions to higher dimensions discussed in the appendix.

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.