Brickwall in Rotating BTZ: A Dip-Ramp-Plateau Story

TL;DR

This paper investigates the spectral form factor of a scalar field in rotating BTZ black hole spacetime, revealing a dip-ramp-plateau structure that persists near extremality and analyzing the effects of back-reaction and approximation methods.

Contribution

It introduces a detailed analysis of the spectral form factor in rotating BTZ geometry, employing numerical and WKB methods, and explores the effects of extremality and back-reaction on the spectral features.

Findings

Dip-Ramp-Plateau structure observed in SFF from grand-canonical partition function

Structure remains stable near extremality but is lost at exact extremality

WKB approximation effectively captures normal modes and level-repulsion regimes

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 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. 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.