Falling toward charged black holes

The growth of the ``size'' of operators is an important diagnostic of quantum chaos. Susskind 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 paper 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 nontrivial test of the momentum-size relation, as well as an explanation of a paradoxical feature of scrambling previously discovered by Leichenauer. Naively Leichenauer's result says that only the nonextremal entropy participates in scrambling. The same feature is also present in the Sachdev-Ye-Kitaev (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-Nordstr\"om geometry. We also conjecture that the proportionality factor in size-momentum relation varies through the throat. This result agrees with direct calculations in SYK.