Hidden conformal symmetry of the Kerr black hole

Abstract

Extreme and very-near-extreme spin $J$ Kerr black holes have been conjectured to be holographically dual to two-dimensional (2D) conformal field theories (CFTs) with left and right central charges ${c}_{L}={c}_{R}=12J$. In this paper it is observed that the 2D conformal symmetry of the scalar wave equation at low frequencies persists for generic nonextreme values of the mass $M\ensuremath{\ne}\sqrt{J}$. Interestingly, this conformal symmetry is not derived from a conformal symmetry of the spacetime geometry except in the extreme limit. The $2\ensuremath{\pi}$ periodic identification of the azimuthal angle $\ensuremath{\phi}$ is shown to correspond to a spontaneous breaking of the conformal symmetry by left and right temperatures ${T}_{L}={M}^{2}/2\ensuremath{\pi}J$ and ${T}_{R}=\sqrt{{M}^{4}\ensuremath{-}{J}^{2}}/2\ensuremath{\pi}J$. The well-known low-frequency scalar-Kerr scattering amplitudes coincide with correlators of a 2D CFT at these temperatures. Moreover, the CFT microstate degeneracy inferred from the Cardy formula agrees exactly with the Bekenstein-Hawking area law for all $M$ and $J$. These observations provide evidence for the conjecture that the Kerr black hole is dual to a ${c}_{L}={c}_{R}=12J$ 2D CFT at temperatures (${T}_{L},{T}_{R}$) for every value of $M$ and $J$.