Harmonic maps and isometric embeddings of the spacetime

Details Harmonic maps and isometric embeddings of the spacetime Harmonic maps and isometric embeddings of the spacetime

2003-12-03

arxiv.org/abs/gr-qc/0312022v1

Phys.Lett. A326 (2004) 171-177

Astronomy and Astrophysics

Astrophysics

General Relativity and Quantum Cosmology

17 pages/ RevTex

Scientific paper

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.

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