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

## 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 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 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

Differentiation 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

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. Metric

Differentiation of Eq. (8) leads to

### 2.3. Zero spatial curvature:

I consider defined parametrically through the equations

Their inversion gives and .

#### 2.3.1. Metric

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

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).

• 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.