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

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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 [FORMULA]. Such an embedding is known for a long time for the de Sitter space-time, which appears so as an hyperboloid [FORMULA] in [FORMULA]. 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 [FORMULA] 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 [FORMULA]. 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 [FORMULA]. 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.

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© European Southern Observatory (ESO) 2000

Online publication: January 29, 2001