Conformal structure of the Schwarzschild black hole

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 × S 2 of the extremal ReissnerNordström 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.

Quasinormal modes and hidden conformal symmetry in the Reissner–Nordström black hole

“…Finally, we obtain (26) by employing the lowenergy approximation [9]. In the near-region and low-energy approximations which are necessary to develop the hidden conformal symmetry, the first two terms of the ω-dependent potential (20) disappear, leading to the last term only.…”

“…Recently, the authors [9] have shown that the scalar wave equation in the near-region and low-energy limits enjoys a hidden SL(2,R) invariance in the Schwarzschild geometry. They have used the SL(2,R) symmetry to determine algebraically the quasinormal frequencies (QNFs) of the Schwarzschild black hole, and also shown that this yields the purely imaginary QNFs describing large damping.…”

“…It seems difficult to derive QNFs of a scalar propagating on the RN black hole by using a hidden conformal symmetry solely. The reason is that quasinormal modes do not satisfy the outgoing wave-boundary condition because these modes were developed using the near-region and low-energy limits [9] where the frequency ω should satisfy inequalities of ω ≪ 1/r and ω ≪ 1/r + . Importantly, the geometry modified in this way (the subtracted geometry) shows that it has the same near-horizon properties as the original RN black hole, but different asymptotes [12].…”

On extremal limits and duality orbits of stationary black holes

“…In fact this approach has provided a valuable tool for their classification [7][8][9][10][11][12][13][14][15][16][17] and consists in describing this kind of solutions as solutions to an effective D = 3 Euclidean sigma-model which is formally obtained by reducing the D = 4 theory along the time direction and dualizing the vector fields into scalars. The action of the global symmetry group (duality group) G of this Euclidean model has been extensively used in the literature as a solutiongenerating technique to construct non-extremal, rotating, electrically charged black hole solutions coupled to scalar fields [7,8] and, more recently, found application in the context of subtracted geometry [18][19][20][21][22].…”

A Kaluza–Klein subtractor

“…4 See [19,20,9] for previous discussions on Harrison transformations in the subtracted geometry context. and the energy constraint from Einstein equations is…”