Dynamically generated embeddings of spacetime

TL;DR

This paper explores how embeddings of spacetime, guaranteed by the Campbell-Magaard theorem, can be interpreted as dynamical evolutions of initial data, establishing a link between initial conditions and higher-dimensional embeddings.

Contribution

It demonstrates that any analytic spacetime can be embedded in a five-dimensional vacuum space via initial data, providing a physical interpretation of the embeddings.

Findings

Embeddings can be viewed as dynamical evolution of initial data.

Any analytic spacetime can be embedded in a 5D vacuum space.

Embedded spacetimes are Cauchy stable with respect to initial data.

Abstract

We discuss how embeddings in connection with the Campbell-Magaard (CM) theorem can have a physical interpretation. We show that any embedding whose local existence is guaranteed by the CM theorem can be viewed as a result of the dynamical evolution of initial data given in a four-dimensional spacelike hypersurface. By using the CM theorem, we establish that for any analytic spacetime, there exist appropriate initial data whose Cauchy development is a five-dimensional vacuum space into which the spacetime is locally embedded. We shall see also that the spacetime embedded is Cauchy stable with respect these the initial data.