Astron. Astrophys. 364, 894-900 (2000)

The Friedmann-Lemaître models in perspective

Embeddings of the Friedmann-Lemaître models in flat 5-dimensional space

M. Lachièze-Rey

CNRS URA-2052 Service d'Astrophysique, C.E. Saclay 91191 Gif sur Yvette cedex, France (marclr@cea.fr)

Received 20 July 2000 / Accepted 31 August 2000

Abstract

I show that all FRW models (four dimensional pseudo-Riemannian manifolds with maximally symmetric space) can be embedded in a flat Minkowski manifold with 5 dimensions. The pseudo Riemannian metric of space-time is induced by the flat metric. This generalizes the usual embedding widely used for the de Sitter models. I give the coordinate transformations for the embedding. Taking into account the spatial isotropy, one can reduce space-time to a two-dimensional surface, embedded in a three-dimensional Minkowski space. This allows to give exact graphic representations of the FRW models, and in particular of their curvature.

Key words: cosmology: miscellaneous cosmology: theory

This article contains no SIMBAD objects.

Contents

• 1. Introduction
• 2. Cosmology in 5 dimensions
  • 2.1. Negative curvature:
    • 2.1.1. The metric
  • 2.2. Positive curvature:
    • 2.2.1. Metric
  • 2.3. Zero spatial curvature:
    • 2.3.1. Metric
  • 2.4. The de Sitter case
    • 2.4.1. Radial light rays
• 3. Friedmann equations
  • 3.1. The spatially flat case
    • 3.1.1. The matter dominated universes
• 4. Conclusion
• References

© European Southern Observatory (ESO) 2000

Online publication: January 29, 2001
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