Horizon Instability of the Extremal BTZ Black Hole

TL;DR

This paper analyzes the Aretakis instability of a scalar field on extremal BTZ black holes, revealing how certain derivatives grow over time and linking this to null geodesic behavior near the horizon.

Contribution

It provides a new time-domain Green function expression and connects the instability to null geodesics orbiting near the horizon, enhancing understanding of horizon dynamics.

Findings

Field derivatives grow with time on the horizon.

Instability linked to null geodesics orbiting near the horizon.

Consistent results obtained in both time and frequency domains.

Abstract

We study real-time propagation of a massive scalar field on the extremal BTZ black hole spacetime, focusing on the Aretakis instability of the event horizon. We obtain a simple time-domain expression for the $\textrm{AdS}_3$ retarded Green function with Dirichlet boundary conditions and construct the corresponding time-domain BTZ retarded Green function using the method of images. The field decays at different rates on and off the horizon, indicating that transverse derivatives grow with time on the horizon (Aretakis instability). We solve the null geodesic equation in full generality and show that the instability is associated with a class of null geodesics that orbit near the event horizon arbitrarily many times before falling in. In an appendix we also treat the problem in the frequency domain, finding consistency between the methods.