No Dynamics in the Extremal Kerr Throat

TL;DR

This paper demonstrates that all vacuum solutions with extremal Kerr throat asymptotics are diffeomorphic to the NHEK geometry, showing no nontrivial dynamics exist in this near-horizon limit.

Contribution

It proves that solutions with NHEK asymptotics are diffeomorphic to the NHEK geometry, indicating a lack of dynamics in this extremal Kerr throat setting.

Findings

All solutions with NHEK asymptotics are diffeomorphic to NHEK.

Charges associated with these solutions are not conserved over time.

No nontrivial dynamical solutions exist with extremal Kerr throat asymptotics.

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.