2. Cosmology in 5 dimensions
I 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 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
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:
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 R and , respectively.
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 metric
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:
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
2.3. Zero spatial curvature:
Their inversion gives and .
It may be easily verified that, on the four-dimensional hypersurface , this leads to
The rotation preserves the value of the coordinate v, transforms to and to
In this representation, (flat) space is represented in by the parabola of parametric Eqs. (14), where t remains fixed, or a paraboloid in : Fig. 2 shows this flat space (reduced to two dimensions), in the subspace of described by the coordinates . This (flat) hypersurface at constant time appears as the revolution paraboloid obtained by the action of the (spherical) rotation around w, of angle , of the parabolic section seen above.
It may appear curious that a flat space is represented by a parabola (or a paraboloid), rather than by a straight line (or an hyperplane). This is due to the Lorentzian (rather than Euclidean) nature of the embedding space (or ). Because of the signature of the metric, any curve in , with parametric equations , represents a flat space, with f an arbitrary function, and A and B two arbitrary constants. In other words, this curve lies in the plane of equation , inclined by 45 with respect to the "vertical" axis. The flat character is expressed by the fact that an arc of such a curve corresponding to a range of the coordinate y has precisely for length: the contributions due to the other coordinates cancel exactly. However, for the Friedmann-Lema&^inodot;tre models, the form (13) is the unique one which gives the complete Robertson-Walker metric.
2.4. The de Sitter case
A 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).
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 rays
Light rays are null geodesics with respect to the metric of . For a point y describing a light ray passing through a point x, we have , and the constraints that both x and y belong to M. For a de Sitter universe, this implies
After some algebra, this leads to the relations
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