The embedding of the spacetime in five dimensions: an extension of Campbell-Magaard theorem

TL;DR

This paper extends the Campbell-Magaard theorem to show that any n-dimensional semi-Riemannian manifold can be locally embedded in an (n+1)-dimensional Einstein space, with implications for higher-dimensional spacetime theories.

Contribution

The paper proves a generalized embedding theorem for semi-Riemannian manifolds into Einstein spaces, expanding the mathematical foundation for higher-dimensional gravity models.

Findings

Any n-dimensional semi-Riemannian manifold can be embedded in an (n+1)-dimensional Einstein space.

Provides examples demonstrating the application of the extended theorem.

Discusses relevance to modern higher-dimensional spacetime theories.

Abstract

We extend Campbell-Magaard embedding theorem by proving that any n-dimensional semi-Riemannian manifold can be locally embedded in an (n+1)-dimensional Einstein space. We work out some examples of application of the theorem and discuss its relevance in the context of modern higher-dimensional spacetime theories.