Abstract
In this article, building on our recent investigations and motivated by the fuzzball-paradigm, we explore normal modes of a probe massless scalar field in the rotating BTZ-geometry in an asymptotically AdS spacetime and correspondingly obtain the Spectral Form Factor (SFF) of the scalar field. In particular, we analyze the SFF obtained from the single-particle partition function. We observe that, a non-trivial Dip-Ramp-Plateau (DRP) structure, with a Ramp of slope one (within numerical precision) exists in the SFF which is obtained from the grand-canonical partition function. This behaviour is observed to remain stable close to extremality as well. However, at exact extremality, we observe a loss of the DRP-structure in the corresponding SFF. Technically, we have used two methods to obtain our results: (i) An explicit and direct numerical solution of the boundary conditions to obtain the normal modes, (ii) A WKB-approximation, which yields analytic, semi-analytic and efficient numerical solutions for the modes in various regimes. We further re-visit the non-rotating case and elucidate the effectiveness of the WKB-approximation in this case, which allows for an analytic expression of the normal modes in the regime where a level-repulsion exists. This regime corresponds to the lower end of the spectrum as a function of the scalar angular momentum, while the higher end of this spectrum tends to become flat. By analyzing the classical stress-tensor of the probe sector, we further demonstrate that the back-reaction of the scalar field grows fast as the angular momenta of the scalar modes increase in the large angular momenta regime, while the back-reaction remains controllably small in the regime where the spectrum has non-trivial level correlations. This further justifies cutting the spectrum off at a suitable value of the scalar angular momenta, beyond which the scalar back-reaction significantly modifies the background geometry.
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Acknowledgments
We thank Roberto Emparan, Sumit Garg, Masanori Hanada, Shota Komatsu, Chethan Krishnan, Kyriakos Papadodimas, Nicholas Warner for useful discussions, conversations and feedbacks on this work. AK is partially supported by CEFIPRA 6304-3, DAE-BRNS 58/14/12/2021-BRNS and CRG/2021/004539 of Govt. of India. AK further thanks the warm hospitality of the Department of Theoretical Physics, CERN where a large part of this work was carried out.
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Das, S., Kundu, A. Brickwall in rotating BTZ: a dip-ramp-plateau story. J. High Energ. Phys. 2024, 49 (2024). https://doi.org/10.1007/JHEP02(2024)049
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DOI: https://doi.org/10.1007/JHEP02(2024)049