Abstract: Gravitational perturbations of a black hole illustrate the invaluable synergy between theory, experiment, and numerical simulations in general relativity. A recent development in the theory side has been the identification of the relevant degrees of freedom describing the low energy physics driving a black hole away from extremality.  For simple cases, this low energy sector determines important aspects of the gravitational backreaction, and several properties that are key to our microscopic (quantum) understanding of black hole physics.

In this talk I will discuss these developments in the context of the (near-)extreme Kerr black hole. In particular, I will revisit the spectrum of linear axisymmetric gravitational perturbations of this black hole. The aim is to characterise those perturbations that are responsible for the deviations away from extremality, and to contrast them with the linearized perturbations treated in the Newman-Penrose formalism. I will show that for Kerr the low-lying mode sector is subtle and intricate—features that their charged spherical symmetric cousins do not display. This unveils new clues on how to decode a microscopic, and holographic, understanding of the Kerr black hole.