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*Astron. Astrophys. 364, 894-900 (2000)*
## 1. Introduction
Although immediate experience indicates that our
space-time has four dimensions, modern physics evocates
additional dimensions in various occasions. Gauge theories involve
(principal) fiber bundles where the fibers may be seen as additional
(internal) dimensions where the gauge fields live, usually not
considered as physical, since they do not mix with the
space-time dimensions. However, the simplest gauge theory, namely
the electromagnetism, has been tentatively described by a five
dimensional theory (Kaluza 1921; Klein 1926,1927; Thiry 1947). It is
not clear, in this case, that the 5 th dimension may be seen as a
physical one, but Souriau (1963) has proposed a genuine 5 dimensional
theory of this type.
More recently, string theories, M-theory, branes are formulated in
a multidimensional space-time. Although most often compactified, the
additional dimensions are considered as physical, in the sense that
some interactions are able to propagate through them.
An appealing property of the Kaluza-Klein theories is the fact
that the five-dimensional space-time, in which the Einstein equations
are solved, is *Ricci flat* (and thus devoid of matter), although
the embedded 4 dimensional manifold corresponding to space-time,
our world, is curved according to the four-dimensional Einstein
equations with sources.
In this paper, I show that all the
Friedmann-Robertson-Walker cosmological models can be embedded in
a *flat* (Minkowskian) five-dimensional
space-time . Such an embedding is
known for a long time for the de Sitter space-time, which appears so
as an hyperboloid in
. This embedding is widely used,
mainly for pedagogical and illustrative purposes (see, e.g., Hawking
& Ellis 1973), and presents interesting properties for
cosmological calculations. Recently, it has for instance been used to
explore the quantification on
de Sitter space-time (Bertola et al. 2000). Also it is
well known that a three dimensional space with maximal symmetry can be
embedded in a flat Euclidean or Lorentzian manifold, also allowing
interesting possibilities for calculations (see, e.g., Triay et al.
1996).
This work can be seen as a generalization of such embeddings to
space-time s with less symmetries (in fact with maximal *spatial*
symmetry only). All embeddings are in a flat five-dimensional space
with Lorentzian signature (because of
this signature, a flat space does not appear as a plane, as can be
seen below). This generalizes also some work made by Wesson (1994) for
some peculiar big bang models. The potential applications are the same
as for the de Sitter case. First, this allows to visualize the
arbitrary and varying curvature of space-time, in the same way as
de Sitter space-time is visualized under the form of a
hyperboloid embedded in . I emphasize
that this makes the space-time curvature visible, not only its
spatial part (which is very simple in all cases, of the three well
known types, flat, spherical or hyperbolic), the temporal part being
*not* given by the curve . In
Sect. 2, I give explicitly the embedding formulae for an
arbitrary space-time with maximal spatial curvature,
distinguishing three cases according to its sign. In Sect. 3, I
consider the cosmic dynamics which, by the Friedmann equations,
restricts the geometrical possibilities. I consider in more details
some cosmological models.
© European Southern Observatory (ESO) 2000
Online publication: January 29, 2001
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