Abstract
The growth of the "size" of operators is an important diagnostic of quantum chaos. In arXiv:1802.01198 [hep-th] it was conjectured that the holographic dual of the size is proportional to the average radial component of the momentum of the particle created by the operator. Thus the growth of operators in the background of a black hole corresponds to the acceleration of the particle as it falls toward the horizon. In this note we will use the momentum-size correspondence as a tool to study scrambling in the field of a near-extremal charged black hole. The agreement with previous work provides a non-trivial test of the momentum-size relation, as well as an explanation of a paradoxical feature of scrambling previously discovered by Leichenauer [arXiv:1405.7365 [hep-th]]. Naively Leichenauer's result says that only the non-extremal entropy participates in scrambling. The same feature is also present in the SYK model. In this paper we find a quite different interpretation of Leichenauer's result which does not have to do with any decoupling of the extremal degrees of freedom. Instead it has to do with the buildup of momentum as a particle accelerates through the long throat of the Reissner-Nordstrom geometry.
Highlights
TWO PUZZLESAll horizons are locally the same; namely they are Rindler-like.1 one might expect their properties as scramblers and complexifiers to be universal
The growth of the “size” of operators is an important diagnostic of quantum chaos
The growth of operators in the background of a black hole corresponds to the acceleration of the particle as it falls toward the horizon
Summary
All horizons are locally the same; namely they are Rindler-like. one might expect their properties as scramblers and complexifiers to be universal. DC ∼ S ; dt Rs ð1:1Þ where S and Rs are the entropy and Schwarzschild radius of the black hole [1]. Rs is the area radius of the horizon (otherwise known as rþ) and S0 is the entropy of the extremal black hole with the same charge. A simple explanation would be that the extremal degrees of freedom (d.o.f.) are somehow decoupled from the chaotic behavior, leaving only the nonextremal component to actively “compute.” But given the fact that all horizons are Rindler-like, it is difficult to understand why this should be so. (1.3) and (1.4) that has nothing to do with any decoupling of extremal d.o.f. Horizons do have universal computational properties in which all S d.o.f. actively compute. Neutral and charged black hole horizons compute in exactly the same way. Our results would apply to black holes of small or intermediate size in AdS
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