Slow scrambling in extremal BTZ and microstate geometries

Abstract

Out-of-time-order correlators (OTOCs) that capture maximally chaotic properties of a black hole are determined by scattering processes near the horizon. This prompts the question to what extent OTOCs display chaotic behaviour in horizonless microstate geometries. This question is complicated by the fact that Lyapunov growth of OTOCs requires nonzero temperature, whereas constructions of microstate geometries have been mostly restricted to extremal black holes.In this paper, we compute OTOCs for a class of extremal black holes, namely maximally rotating BTZ black holes, and show that on average they display “slow scrambling”, characterized by cubic (rather than exponential) growth. Superposed on this average power-law growth is a sawtooth pattern, whose steep parts correspond to brief periods of Lyapunov growth associated to the nonzero temperature of the right-moving degrees of freedom in a dual conformal field theory.Next we study the extent to which these OTOCs are modified in certain “superstrata”, horizonless microstate geometries corresponding to these black holes. Rather than an infinite throat ending on a horizon, these geometries have a very deep but finite throat ending in a cap. We find that the superstrata display the same slow scrambling as maximally rotating BTZ black holes, except that for large enough time intervals the growth of the OTOC is cut off by effects related to the cap region, some of which we evaluate explicitly.

Reparametrization modes in 2d CFT and the effective theory of stress tensor exchanges

“…Agreement was indeed found between this effective description and more conventional CFT methods [1]. Although this is not the primary focus of the present paper, the strong connection between maximal chaos and the AdS/CFT correspondence is worth mentioning as it appears that holographic CFTs are maximally chaotic [15][16][17][18][19][20][21][22].…”

Chaos in CFT dual to rotating BTZ

“…In this way we reproduce the gravitational results found in [1]. We also study the extremal and zero-temperature limits, which require backing off from some of the approximations used in the previous result, and reproduce the gravitational results found in [2].…”

“…With this result for boosted ensembles on the line in hand, we next analyse the case of rotating ensembles on the circle. Gravitational computations of the OTOC have found that in the Lyapunov regime the instantaneous Lyapunov exponent exhibits a sawtooth pattern which when averaged over a full period gives an averaged Lyapunov exponent controlled by the temperature [1,2]. We show that in the Lyapunov regime, where the global conformal block of the stress tensor dominates, the OTOC on the circle can be obtained by that on the line from a reasoning that involves the relation between global conformal blocks and geodesic Witten diagrams.…”

“…In [1], it was shown that the OTOC in a rotating BTZ black hole exhibits an average exponential decay on which is superposed a sawtooth pattern describing alternating exponential behaviours determined by the left and right moving temperatures of the dual CFT. A sawtooth pattern was also observed in the extremal limit of the rotating BTZ black hole [2]. Thus it is natural to wonder how this behaviour emerges in the CFT holographically dual to rotating BTZ.…”

Tidal excitation as mixing in thermal CFT

“…Indeed, tidal stresses have been found to be significant also in horizonless microstate geometries [10][11][12][13] that, seen from a distance, resemble black holes. These geometries are also associated with scrambling [14], another probe of strong gravity. Finally, evanescent modes are related to the strong gravity inside the innermost stable orbits, but the relation to mixing is less clear.…”