No Dynamics in the Extremal Kerr Throat

TL;DR

This work analyzes the near-horizon region of extremal Kerr black holes (NHEK) to determine whether dynamics can occur beyond the NHEK geometry under asymptotic NHEK conditions. By developing linearization-stability constraints and studying both a scalar toy model and linearized gravity via Teukolsky formalisms, the authors show that all physically admissible perturbations must have vanishing $SL(2,\mathbb{R}) \times U(1)$ charges, severely restricting possible excitations. A stationary axisymmetric uniqueness theorem demonstrates that asymptotically-NHEK vacua with smooth horizons are diffeomorphic to NHEK (or finite-temperature variants), implying no nontrivial nonlinear backreaction can yield a distinct solution within these asymptotics. Collectively, the results support the interpretation of NHEK as a ground state in Kerr/CFT, with dynamics constrained by the specified boundary conditions and linearization-stability requirements.

Abstract

Motivated by the Kerr/CFT conjecture, we explore solutions of vacuum general relativity whose asymptotic behavior agrees with that of the extremal Kerr throat, sometimes called the Near-Horizon Extreme Kerr (NHEK) geometry. We argue that all such solutions are diffeomorphic to the NHEK geometry itself. The logic proceeds in two steps. We first argue that certain charges must vanish at all times for any solution with NHEK asymptotics. We then analyze these charges in detail for linearized solutions. Though one can choose the relevant charges to vanish at any initial time, these charges are not conserved. As a result, requiring the charges to vanish at all times is a much stronger condition. We argue that all solutions satisfying this condition are diffeomorphic to the NHEK metric.

Figures (2)

• Figure 1: Penrose diagrams of AdS${}_2$ showing coordinate patches covered by (a) global coordinates, (b) Poincaré coordinates, (c) finite temperature coordinates. In each case the associated time translation and any horizons are shown.
• Figure 2: Typical plots of ${{\frak{Im}}}[\tilde{A}/\tilde{B}]$ for power law modes (left) and $|\tilde{A}/\tilde{B}| - 1$ for oscillatory modes (right) over the complex $\omega$ plane. The shading denotes positive/negative values. Frequencies that satisfy boundary conditions which conserve Klein-Gordon flux at the $y=+\infty$ boundary lie on the boundaries. For each plot, $A/B$ has already been fixed to conserve flux at the other boundary ($y=-\infty$).