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

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2. Cosmology in 5 dimensions

I recall the metric of a Friedmann-Robertson-Walker model [FORMULA],

[EQUATION]

where [FORMULA] is the metric of a maximally symmetric 3-d space with curvature [FORMULA],

[EQUATION]

where [FORMULA], [FORMULA], [FORMULA]. The function [FORMULA] is given by the dynamics (see 3). In this section, it will remain arbitrary, so that the geometrical embedding appears very general.

I call [FORMULA] the Minkowski space-time in five dimensions with the flat metric

[EQUATION]

I will show that every Friedmann-Lema&^inodot;tre model can be seen as a four dimensional submanifold (hypersurface) of [FORMULA].

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 [FORMULA], which coincides with the subgroup [FORMULA] of rotations in the three-dimensional subspace [FORMULA] of [FORMULA] described by the coordinates [FORMULA]. Thus, any point of [FORMULA] may be written

[EQUATION]

so that

[EQUATION]

[EQUATION]

and

[EQUATION]

Thus, in the following, I will simply consider the three dimensional flat manifold [FORMULA] as embedding space, with the three coordinates [FORMULA] and [FORMULA] (putting [FORMULA]), since the others can be trivially reconstructed by action of the spatial rotations. Thus, any point in [FORMULA] represents a two-sphere in [FORMULA]. Space and time are measured in arbitrary identical units (I impose [FORMULA]). In the next sections, I will use [FORMULA], the Hubble time, as a common unit.

2.1. Negative curvature:

I consider [FORMULA] defined parametrically in [FORMULA] through the equations

[EQUATION]

[EQUATION]

[EQUATION]

[EQUATION]

[EQUATION]

In this case, I have written explicitly the five equations to give insights to the geometry. In [FORMULA], they reduce to

[EQUATION]

[EQUATION]

[EQUATION]

which can be inverted as [FORMULA] and [FORMULA] where [FORMULA] and [FORMULA] are the inverse functions of R and [FORMULA], respectively.

The whole space-time [FORMULA] is obtained by the action of the hyperbolic rotations [FORMULA] around the [FORMULA] axis, with angle [FORMULA] (followed by the [FORMULA] spherical rotations [FORMULA], as indicated above), on the world line [FORMULA]. The latter illustrates the temporal part of the curvature. It is defined by its equations

[EQUATION]

[EQUATION]

[EQUATION]

2.1.1. The metric

Differentiation of the equations (2) leads to

[EQUATION]

Inserting in (1) leads to the metric induced onto the surface

[EQUATION]

i.e., that of a [FORMULA] Friedmann-Lema&^inodot;tre model.

2.2. Positive curvature:

I consider [FORMULA] defined parametrically through the equations

[EQUATION]

[EQUATION]

[EQUATION]

Their inversion leads to [FORMULA] and [FORMULA]

The whole space-time [FORMULA] is obtained by the action of the spherical rotations [FORMULA], around the [FORMULA] axis, with angle [FORMULA] [followed by the [FORMULA] spherical rotations [FORMULA]], on the world line [FORMULA], which has the parametric equations

[EQUATION]

[EQUATION]

[EQUATION]

2.2.1. Metric

Differentiation of Eq. (8) leads to

[EQUATION]

2.3. Zero spatial curvature:

I consider [FORMULA] defined parametrically through the equations

[EQUATION]

[EQUATION]

[EQUATION]

Their inversion gives [FORMULA] and [FORMULA].

2.3.1. Metric

Differentiation of Eq. (12) leads to

[EQUATION]

It may be easily verified that, on the four-dimensional hypersurface [FORMULA], this leads to

[EQUATION]

It is advantageous to introduce the new system of coordinates:

[EQUATION]

and [FORMULA].
This makes apparent the fact that the whole space-time [FORMULA] is obtained by the action of parabolic rotations [FORMULA], of angle r, around the [FORMULA] axis, of the world-line [FORMULA] (followed by the spatial SO(3) rotations). The latter [see an illustration in Fig. 4] is defined parametrically by

[EQUATION]

[EQUATION]

[EQUATION]

The rotation [FORMULA] preserves the value of the coordinate v, transforms [FORMULA] to [FORMULA] and [FORMULA] to [FORMULA]

In this representation, (flat) space is represented in [FORMULA] by the parabola of parametric Eqs. (14), where t remains fixed, or a paraboloid in [FORMULA]: Fig. 2 shows this flat space (reduced to two dimensions), in the subspace of [FORMULA] described by the coordinates [FORMULA]. This (flat) hypersurface at constant time appears as the revolution paraboloid obtained by the action of the (spherical) rotation around w, of angle [FORMULA], 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 [FORMULA] (or [FORMULA]). Because of the signature of the metric, any curve in [FORMULA], with parametric equations [FORMULA], 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 [FORMULA], inclined by 45[FORMULA] 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 [FORMULA] of the coordinate y has precisely [FORMULA] 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 [FORMULA]. Space-time is the hyperboloid [FORMULA] in [FORMULA] 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: [FORMULA].

    [EQUATION]

    [EQUATION]

    [EQUATION]

  • Positive spatial curvature: [FORMULA].

    [EQUATION]

    [EQUATION]

    [EQUATION]

  • Zero spatial curvature: [FORMULA].

    [EQUATION]

    [EQUATION]

    [EQUATION]

    Also, for this case,

    [EQUATION]

    [EQUATION]

    [EQUATION]

    Inversion gives

    [EQUATION]

    and

    [EQUATION]

    These coordinates cover half of the hyperboloid ([FORMULA]). The metric takes the form [FORMULA], 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 rays

Light rays are null geodesics with respect to the metric of [FORMULA]. For a point y describing a light ray passing through a point x, we have [FORMULA], and the constraints that both x and y belong to M. For a de Sitter universe, this implies

[EQUATION]

[EQUATION]

After some algebra, this leads to the relations

[EQUATION]

and

[EQUATION]

[EQUATION]

They describe a straight line in [FORMULA], which proves that the light rays of the (ruled) hyperboloid are straight lines in [FORMULA]. In particular, the light rays through the origin [FORMULA] are described by [FORMULA] and [FORMULA]. A similar treatment shows that, in the general (non de Sitter) case, the light rays are not, in general, straight lines in [FORMULA].

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Online publication: January 29, 2001
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