## 2. Cosmology in 5 dimensionsI recall the metric of a Friedmann-Robertson-Walker model , where is the metric of a maximally symmetric 3-d space with curvature , where , , . The function is given by the dynamics (see 3). In this section, it will remain arbitrary, so that the geometrical embedding appears very general. I call the Minkowski space-time in five dimensions with the flat metric I will show that every Friedmann-Lema&^inodot;tre model can be seen as a four dimensional submanifold (hypersurface) of . In the Friedmann-Lema&^inodot;tre models, space has maximal symmetry and is, in particular, isotropic. This isotropy is expressed by the action of the group SO(3) of the spatial rotations , which coincides with the subgroup of rotations in the three-dimensional subspace of described by the coordinates . Thus, any point of may be written so that and Thus, in the following, I will simply consider the three dimensional flat manifold as embedding space, with the three coordinates and (putting ), since the others can be trivially reconstructed by action of the spatial rotations. Thus, any point in represents a two-sphere in . Space and time are measured in arbitrary identical units (I impose ). In the next sections, I will use , the Hubble time, as a common unit. ## 2.1. Negative curvature:I consider defined parametrically in through the equations In this case, I have written explicitly the five equations to give insights to the geometry. In , they reduce to which can be inverted as and
where
and
are the inverse functions of
The whole space-time is obtained by the action of the hyperbolic rotations around the axis, with angle (followed by the spherical rotations , as indicated above), on the world line . The latter illustrates the temporal part of the curvature. It is defined by its equations ## 2.1.1. The metricDifferentiation of the equations (2) leads to Inserting in (1) leads to the metric induced onto the surface i.e., that of a Friedmann-Lema&^inodot;tre model. ## 2.2. Positive curvature:I consider defined parametrically through the equations Their inversion leads to and The whole space-time is obtained by the action of the spherical rotations , around the axis, with angle [followed by the spherical rotations ], on the world line , which has the parametric equations ## 2.2.1. MetricDifferentiation of Eq. (8) leads to ## 2.3. Zero spatial curvature:I consider defined parametrically through the equations Their inversion gives and . ## 2.3.1. MetricDifferentiation of Eq. (12) leads to It may be easily verified that, on the four-dimensional hypersurface , this leads to It is advantageous to introduce the new system of coordinates: and . The rotation preserves the value
of the coordinate In this representation, (flat) space is represented in
by the parabola of parametric
Eqs. (14), where It may appear curious that a ## 2.4. The de Sitter caseA peculiar case is the de Sitter space-time, with the topology . Space-time is the hyperboloid in but, as it is well known, different cosmological models may be adjusted to it, depending on how the time coordinate is chosen. This gives the opportunity to illustrate the previous cases (all these formulae are standard and may be found, for instance, in Hawking & Ellis 1973). -
Negative spatial curvature: . -
Positive spatial curvature: . -
Zero spatial curvature: . Also, for this case,Inversion gives and These coordinates cover half of the hyperboloid (). The metric takes the form , that of a static universe.
Only in the second case (negative spatial curvature), the space-time corresponds to the whole hyperboloid, that I consider now. ## 2.4.1. Radial light raysLight rays are null geodesics with respect to the metric of
. For a point After some algebra, this leads to the relations and They describe a straight line in , which proves that the light rays of the (ruled) hyperboloid are straight lines in . In particular, the light rays through the origin are described by and . A similar treatment shows that, in the general (non de Sitter) case, the light rays are not, in general, straight lines in . © European Southern Observatory (ESO) 2000 Online publication: January 29, 2001 |