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

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4. Conclusion

These calculations generalize, to arbitrary Friedmann-Robertson-Walker models, the embedding usually used for the de Sitter models. The fact that the embedding space is flat offers a very good convenience to illustrate in an intuitive way the geometrical properties of these models. For instance, time durations, or lengths between events could be obtained by measuring (Lorentzian) lengths of the corresponding curves with a ruler in [FORMULA] or [FORMULA]. Also, the curvature coefficients would be those obtained for the hypersurface in [FORMULA]. Care must be taken, however, in this case, that the signature of the embedding space is Lorentzian. Many text books have illustrated this fact for the de Sitter case. Here we observe the curious fact that a flat space appears as a parabola (or a paraboloid in more dimensions). However, a correct measure of the curvature would confirm the flatness of the corresponding surface.

Beside their pedagogical interest, these representations could be of great help for various calculations. I mention for instance the calculation of cosmic distances or time intervals (generalizing those of Triay et al. (1996) for the case of spatial distances). This would be also of great help to gain intuition in any theory with more than five dimensions.

Among other speculative ideas, it would be tempting to consider dynamics (here cosmic dynamics) as a geometrical effect in a manifold with 5 (or more) dimensions, which is flat (like here) or Ricci flat (this track is being explored by Wesson 1994, and references therein).

This suggest prolongations of the present work in the spirit of the old Kaluza-Klein attempts: consider other solutions of general relativity (Wesson & Liu 2000), consider solutions of gravitation theories other than general relativity, consider embeddings in Ricci-flat (rather than flat) manifolds, embeddings in manifolds with more dimensions, etc. For instance, Darabi et al. (2000, see also references therein) suggest that this may offer a starting point for quantum cosmology. This may also offer an angle of attack for quantization in curved space time, following the work already done in de Sitter space-time. This is motivated by recent work (see, e.g., Bertola et al. 2000 and references therein) which have shown interesting relations between quantum field theories in different dimensions (for instance, they suggest the idea that "a thermal effect on a curved manifold can be looked at as an Unruh effect in a higher dimensional flat spacetime").

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© European Southern Observatory (ESO) 2000

Online publication: January 29, 2001