Harmonic maps and isometric embeddings of the spacetime

TL;DR

This paper demonstrates that any n-dimensional Lorentzian manifold can be harmonically and isometrically embedded into a higher-dimensional Ricci-flat space, extending to Einstein spaces and including applications to cosmological models.

Contribution

It provides a general scheme for minimally embedding vacuum solutions of general relativity into Ricci-flat spaces using harmonic and isometric maps.

Findings

Any n-dimensional Lorentzian manifold can be embedded in a (n+1)-dimensional Ricci-flat space.

Extension of embeddings to Einstein target spaces.

Explicit embedding of Friedmann-Robertson-Walker spacetime in five-dimensional Ricci-flat space.

Abstract

We investigate harmonic maps in the context of isometric embeddings when the target space is Ricci-flat and has codimension one. With the help of the Campbell-Magaard theorem we show that any $n$-dimensional ($n\geqslant 3$) Lorentzian manifold can be isometrically and harmonically embedded in a (n+1)-dimensional semi-Riemannian Ricci-flat space. We then extend our analysis to the case when the target space is an Einstein space. Finally, as an example, we work out the harmonic and isometric embedding of a Friedmann-Robertson-Walker spacetime in a five-dimensional Ricci-flat space and proceed to obtain a general scheme to minimally embed any vacuum solution of general relativity in Ricci-flat spaces with codimension one.