Notes on Euclidean de Sitter space

TL;DR

This paper explores the topology of Euclidean de Sitter space, revealing different structures in (2+1) dimensions depending on the approach, and discusses potential generalizations to higher dimensions.

Contribution

It clarifies the Euclidean topology of de Sitter space in (2+1) dimensions and proposes a connection between the solid torus and the three-hemisphere topologies.

Findings

Euclidean de Sitter space in (2+1) dimensions can be a three-hemisphere or a solid torus.

The topology transitions from a solid torus to a three-hemisphere when moving from the stretched horizon to the horizon.

Implications for higher-dimensional generalizations are discussed.

Abstract

We discuss issues relating to the topology of Euclidean de Sitter space. We show that in (2+1) dimensions, the Euclidean continuation of the`causal diamond', i.e the region of spacetime accessible to a timelike observer, is a three-hemisphere. However, when de Sitter entropy is computed in a `stretched horizon' picture, then we argue that the correct Euclidean topology is a solid torus. The solid torus shrinks and degenerates into a three-hemisphere as one goes from the `stretched horizon' to the horizon, giving the Euclidean continuation of the causal diamond. We finally comment on the generalisation of these results to higher dimensions.