On Applications of Campbell's Embedding Theorem

TL;DR

This paper explores Campbell's embedding theorem to demonstrate local embeddings of various Einstein and Ricci-flat spacetimes into higher-dimensional spaces, highlighting the theorem's applications and limitations.

Contribution

It extends Campbell's theorem to include embeddings of n-dimensional Einstein spaces and discusses the local nature of these embeddings.

Findings

Embedded 4D spacetimes in 5D Ricci-flat spaces.

Extended embedding results to n-dimensional Einstein spaces.

Highlighted the local nature of Campbell's theorem.

Abstract

A little known theorem due to Campbell is employed to establish the local embedding of a wide class of 4-dimensional spacetimes in 5-dimensional Ricci-flat spaces. An embedding for the class of n-dimensional Einstein spaces is also found. The local nature of Campbell's theorem is highlighted by studying the embedding of some lower-dimensional spaces.