The Kerr/CFT correspondence

Abstract

Quantum gravity in the region very near the horizon of an extreme Kerr black hole (whose angular momentum and mass are related by $J=G{M}^{2}$) is considered. It is shown that consistent boundary conditions exist, for which the asymptotic symmetry generators form one copy of the Virasoro algebra with central charge ${c}_{L}=\frac{12J}{\ensuremath{\hbar}}$. This implies that the near-horizon quantum states can be identified with those of (a chiral half of) a two-dimensional conformal field theory (CFT). Moreover, in the extreme limit, the Frolov-Thorne vacuum state reduces to a thermal density matrix with dimensionless temperature ${T}_{L}=\frac{1}{2\ensuremath{\pi}}$ and conjugate energy given by the zero mode generator, ${L}_{0}$, of the Virasoro algebra. Assuming unitarity, the Cardy formula then gives a microscopic entropy ${S}_{\mathrm{micro}}=\frac{2\ensuremath{\pi}J}{\ensuremath{\hbar}}$ for the CFT, which reproduces the macroscopic Bekenstein-Hawking entropy ${S}_{\mathrm{macro}}=\frac{\mathrm{Area}}{4\ensuremath{\hbar}G}$. The results apply to any consistent unitary quantum theory of gravity with a Kerr solution. We accordingly conjecture that extreme Kerr black holes are holographically dual to a chiral two-dimensional conformal field theory with central charge ${c}_{L}=\frac{12J}{\ensuremath{\hbar}}$, and, in particular, that the near-extreme black hole GRS 1915+105 is approximately dual to a CFT with ${c}_{L}\ensuremath{\sim}2\ifmmode\times\else\texttimes\fi{}{10}^{79}$.