Chaotic LLM billiards

TL;DR

This paper investigates the chaotic behavior of null geodesics confined to the LLM plane in ten-dimensional geometries, revealing complex dynamics akin to billiards and implications for black hole physics.

Contribution

It demonstrates that in-plane null geodesics in LLM geometries exhibit chaos and introduces a billiard-like model to analyze their dynamics and black hole analogies.

Findings

In-plane null geodesics are chaotic and confined by obstacles.

Geodesic dynamics resemble billiard problems with fixed obstacles.

Long trapping times suggest diffusion-like escape mechanisms.

Abstract

We study null geodesics of the ten-dimensional LLM geometries. In particular, we show that there are a subset of these null geodesics that are confined to the LLM plane. The effective dynamics of these in-plane geodesics is that of a Hamiltonian system with two degrees of freedom (a phase space of dimension 4). We show that these are chaotic. In the two-coloring of the LLM plane, if they start in the empty region, they cannot penetrate the filled region and viceversa. The dynamical problem is therefore very similar to that of a billiards problem with fixed obstacles. We study to what extent LLM geometries with many droplets may be treated as an incipient black hole and draw analogies with the fuzzball proposal. We argue that for in-plane null geodesics deep in the interior of a region with a lot of droplets, in order to exit towards the $AdS$ boundary they will need to undergo a process that resembles diffusion. This mechanism can account for signals getting lost in the putative black hole for a very long time.