Embeddings in Non-Vacuum Spacetimes

TL;DR

This paper presents a method for embedding n-dimensional Riemannian manifolds into (n+1)-dimensional Einstein spaces, with criteria for embeddings in scalar field solutions, supported by illustrative examples.

Contribution

It introduces a new scheme for embedding manifolds into Einstein spaces and establishes criteria for embeddings in scalar field spacetimes, expanding the understanding of geometric embeddings in general relativity.

Findings

Developed a scheme for embedding manifolds in Einstein spaces.

Established criteria for embeddings in scalar field solutions.

Provided illustrative examples of the embedding procedures.

Abstract

A scheme is discussed for embedding n-dimensional, Riemannian manifolds in an (n+1)-dimensional Einstein space. Criteria for embedding a given manifold in a spacetime that represents a solution to Einstein's equations sourced by a massless scalar field are also discussed. The embedding procedures are illustrated with a number of examples.