Horizon Instability of the Extremal BTZ Black Hole

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 ${\text{AdS}}_3$ 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.

arXiv comment23 pages, 6 figures

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Cited by in corpus (4)

• From regular black holes to horizonless objects: quasi-normal modes, instabilities and spectroscopy
• Slow scrambling in extremal BTZ and microstate geometries
• The Aretakis constants and instability in general spherically symmetric extremal black hole spacetimes: higher multipole modes, late-time tails, and geometrical meanings
• Revisiting the Aretakis constants and instability in two-dimensional Anti-de Sitter spacetimes