Critical Exponents of Extremal Kerr Perturbations

TL;DR

This paper demonstrates that perturbations of extremal Kerr black holes exhibit self-similarity and critical behavior, with computed decay rates, explaining the Aretakis instability through emergent symmetry and showing scalar quantities decay despite derivative growth.

Contribution

It introduces the concept of critical exponents for extremal Kerr perturbations and links the Aretakis instability to emergent near-horizon symmetry.

Findings

Perturbations are asymptotically self-similar under near-horizon scaling.

Critical exponents determine decay rates of perturbations.

Scalar quantities decay to zero despite derivative growth.

Abstract

We show that scalar, electromagnetic, and gravitational perturbations of extremal Kerr black holes are asymptotically self-similar under the near-horizon, late-time scaling symmetry of the background metric. This accounts for the Aretakis instability (growth of transverse derivatives) as a critical phenomenon associated with the emergent symmetry. We compute the critical exponent of each mode, which is equivalent to its decay rate. It follows from symmetry arguments that, despite the growth of transverse derivatives, all generally covariant scalar quantities decay to zero.