Slow scrambling in extremal BTZ and microstate geometries

TL;DR

This paper investigates how out-of-time-order correlators behave in extremal BTZ black holes and microstate geometries, revealing slow, power-law scrambling patterns and the effects of horizonless caps on chaos indicators.

Contribution

It demonstrates that extremal BTZ black holes and their microstate geometries exhibit slow, power-law scrambling of OTOCs, contrasting with exponential growth in non-extremal cases.

Findings

OTOCs show cubic growth in extremal BTZ black holes.

Superstrata microstate geometries display similar slow scrambling.

Cap effects in microstates cut off OTOC growth at large times.

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.