Holographic Complexity and de Sitter Space

TL;DR

This paper investigates the behavior of spacelike geodesics in certain two-dimensional geometries with de Sitter or black hole horizons, exploring implications for holographic complexity and the nature of de Sitter space.

Contribution

It analyzes geodesic lengths in flow geometries with de Sitter horizons, contrasting their behavior with black hole horizons to inform holographic complexity theories.

Findings

Geodesic lengths grow linearly in black hole geometries at late times.

Finite geodesic lengths exist only for short times in de Sitter geometries.

De Sitter horizons do not support linear growth of complexity as black holes do.

Abstract

We compute the length of spacelike geodesics anchored at opposite sides of certain double-sided flow geometries in two dimensions. These geometries are asymptotically anti-de Sitter but they admit either a de Sitter or a black hole event horizon in the interior. While in the geometries with black hole horizons, the geodesic length always exhibit linear growth at late times, in the flow geometries with de Sitter horizons, geodesics with finite length only exist for short times of the order of the inverse temperature and they do not exhibit linear growth. We comment on the implications of these results towards understanding the holographic proposal for quantum complexity and the holographic nature of the de Sitter horizon.