Ideally embedded space-times

TL;DR

This paper extends the concept of ideal embeddings from Riemannian to indefinite spaces, proving inequalities and demonstrating specific space-times like de Sitter and Robertson-Walker can be ideally embedded in five-dimensional pseudo-Euclidean space.

Contribution

It introduces the notion of ideal embeddings in indefinite spaces and shows certain space-times can be embedded with minimal tension in higher-dimensional spaces.

Findings

De Sitter spaces can be ideally embedded in 5D pseudo-Euclidean space.

A curvature inequality relating intrinsic invariants and mean curvature is established.

Some anisotropic perfect fluid metrics also admit ideal embeddings.

Abstract

Due to the growing interest in embeddings of space-time in higher-dimensional spaces we consider a specific type of embedding. After proving an inequality between intrinsically defined curvature invariants and the squared mean curvature, we extend the notion of ideal embeddings from Riemannian geometry to the indefinite case. Ideal embeddings are such that the embedded manifold receives the least amount of tension from the surrounding space. Then it is shown that the de Sitter spaces, a Robertson-Walker space-time and some anisotropic perfect fluid metrics can be ideally embedded in a five-dimensional pseudo-Euclidean space.