$\mathrm{SL}(2,\mathbb{R})$ families of Kerr black holes

The stationary, axisymmetric sector of vacuum general relativity (with zero cosmological constant) enjoys an $\mathrm{SL}(2,\mathbb{R})$ 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 $\mathrm{SL}(2,\mathbb{R})$ symmetry organizes the space of three parameter Kerr solutions into the classical analogue of principal series representations. We show that the $\mathrm{SL}(2,\mathbb{R})$ 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.