Holographic Complexity from the Crofton's Formula in Lorentzian AdS$_3$

TL;DR

This paper explores a new holographic representation of complexity in Lorentzian AdS$_3$ using Crofton's formula, linking surface area to geodesic flux and discussing implications for information theory.

Contribution

It introduces a novel geometric approach to holographic complexity via Crofton's formula in Lorentzian AdS$_3$, connecting surface area with geodesic flux and complexity.

Findings

Surface area of spacelike surfaces equals flux of spacelike geodesics.

Holographic complexity can be represented by counting geodesics.

Potential links between geometric flux and information-theoretic measures.

Abstract

We study the Crofton's formula in the Lorentzian AdS$_3$ and find that the area of a generic space-like two dimensional surface is given by the flux of space-like geodesics. The "complexity=volume" conjecture then implies a new holographic representation of the complexity in terms of the number of geodesics. Finally, we explore the possible explanation of this result from the standpoint of information theory.