# A note on size-momentum correspondence and chaos

## TL;DR

This paper investigates the relationship between operator size and bulk momentum in chaotic theories within various gravity models, revealing how parameters like coupling constants and charges influence chaos and operator growth.

## Contribution

It extends Susskind's size-momentum correspondence to Gauss-Bonnet and Lifshitz-hyperscaling violating theories, showing how these parameters affect operator size and chaos.

## Key findings

- Operator size is suppressed with increasing Gauss-Bonnet coupling.
- Operator size increases with Lifshitz parameter z and hyperscaling parameter θ.
- Chaos suppression observed for charged particles falling into black holes.

## Abstract

The aim of this note is to explore Susskind's proposal [arXiv:1802.01198] on the connection between operator size in chaotic theories and the bulk momentum of a particle falling into black holes (see also [arXiv:1804.04156, 1806.05574, 1904.12819, 1912.05996, 2006.03019] for more recent generalizations), in a broad class of models involving Gauss-Bonnet(GB) and Lifshitz-Hyperscaling violating theories in AdS. For Gauss-Bonnet black holes, the operator size is seen to be suppressed as the coupling constant $\lambda$ is increased. For the Lifshitz-hyperscaling violating theories characterised by the parameters $z$ and $\theta$, the operator size is higher as compared to case $z=1,\theta=0$ (Reissner-Nordstrom AdS black holes). In the case of operators with global charge corresponding to charged particles falling into black holes, suppression of chaos is seen in general theories of gravity, in conformity with the original proposal [arXiv:1802.01198] and earlier findings [arXiv:1806.05574].