Application of the Campbell-Magaard theorem to higher-dimensional physics

TL;DR

This paper explores the Campbell-Magaard theorem's application to higher-dimensional physics, analyzing how 4D spacetimes can be embedded in 5D manifolds and implications for theories like STM and braneworld models.

Contribution

It extends the Campbell-Magaard theorem to n dimensions and examines its relevance to modern higher-dimensional theories, including observer confinement and specific cosmological embeddings.

Findings

Arbitrary spacetimes can be embedded in 5D models under certain conditions.

Observers can be confined to the 4D hypersurface if energy conditions are met.

A specific embedding of radiation-dominated cosmology in 5D vacuum is demonstrated.

Abstract

Stated succinctly, the original version of the Campbell-Magaard theorem says that it is always possible to locally embed any solution of 4-dimensional general relativity in a 5-dimensional Ricci-flat manifold. We discuss the proof of this theorem (and its variants) in n dimensions, and its application to current theories that postulate that our universe is a 4-dimensional hypersurface Sigma_0 within a 5-dimensional manifold, such as Space-Time-Matter (STM) theory and the Randall & Sundrum (RS) braneworld scenario. In particular, we determine whether or not arbitrary spacetimes may be embedded in such theories, and demonstrate how these seemingly disparate models are interconnected. Special attention is given to the motion of test observers in 5 dimensions, and the circumstances under which they are confined to Sigma_0. For each 5-dimensional scenario considered, the requirement that observers be confined to the embedded spacetime places restrictions on the 4-geometry. For example, we find that observers in the thin braneworld scenario can be localized around the brane if its total stress-energy tensor obeys the 5-dimensional strong energy condition. As a concrete example of some of our technical results, we discuss a Z_2 symmetric embedding of the standard radiation-dominated cosmology in a 5-dimensional vacuum.