SL(2,R) families of Kerr black holes

The stationary, axisymmetric sector of vacuum general relativity (with zero cosmological constant) enjoys an SL(2,R) symmetry called the Matzner-Misner group. We study the action of the Matzner-Misner group on the Kerr black hole. We show that the group acts naturally on a three parameter generalization of the usual two parameter Kerr solution. The new parameter represents a large diffeomorphism which gives the spacetime an asymptotic angular velocity. We explain how the SL(2,R) symmetry organizes the space of three parameter Kerr solutions into the classical analogue of principal series representations. We show that the SL(2,R) Casimir operator is the Bekenstein-Hawking entropy. The Matzner-Misner group sits inside a much larger Kac-Moody symmetry called the Geroch group. We show that the Kac-Moody level of the Kerr black hole is the Bekenstein-Hawking entropy.