Horizon instability of the extremal BTZ black hole
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Springer
Received: December 12, Revised: April 16, Accepted: May 3, Published: May 20,
2019 2020 2020 2020
Samuel E. Gralla,a Arun Ravishankara,1 and Peter Zimmermanb a
Department of Physics, University of Arizona, 1118 E. Fourth Street, Tucson, Arizona, 85721, U.S.A b Max Planck Institute for Gravitational Physics (Albert Einstein Institute), Am M¨ uhlenberg 1, 14476 Potsdam, Germany
E-mail: [email protected], [email protected], [email protected] 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 AdS3 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. Keywords: Black Holes, Black Holes in String Theory, AdS-CFT Correspondence ArXiv ePrint: 1911.11164
1
Corresponding author.
c The Authors. Open Access, Article funded by SCOAP3 .
https://doi.org/10.1007/JHEP05(2020)094
JHEP05(2020)094
Horizon instability of the extremal BTZ black hole
Contents 1
2 Metric 2.1 Global AdS3 2.2 Poincar´e coordinates and patch 2.3 Extremal coordinates and patch 2.4 Horizon coordinates 2.5 Extremal Killing field and BTZ
3 3 3 4 5 6
3 Scalar field 3.1 AdS3 Green function 3.2 Extremal BTZ Green function
6 6 8
4 Late-time 4.1 AdS3 4.2 BTZ 4.2.1 4.2.2 4.2.3
decay
Both points outside the horizon Field point on the horizon and source point outside Both points on the horizon
9 10 10 11 12 13
5 Null geodesics
14
A AdS3 Green function
18
B Massless axisymmetric perturbations: Aretakis’ method
20
C Mode approach C.1 Preliminaries C.2 Modes at late times near the horizon C.3 Non-periodic limit — AdS3
21 21 23 24
1
Introduction
In the decade since Aretakis’ initial study of massless scalar fields in the extremal ReissnerN¨ordstrom spacetime [1, 2], it has become clear that extremal horizons generically exhibit weak derivative instabilities [3–5]. Independent of the type of extremal black hole (and even of the type of field that perturbs it!), the evidence suggests that sufficiently highorder transverse derivatives always grow at least polynomially in advanced time along the event horizon. This has the physical consequence that infalling observers experience large field gradients [4, 6, 7].
–1–
JHEP05(2020)094
1 Introduction
1
We demonstrate this fact numerically and give heuristic analytical arguments for the behavior, but do not provide a rigorous proof.
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