The Universe Of Friedmann

The theory of relativity allows me to complete my description of space with a variable radius by introducing new concepts.

This space is in the interior of a sphere, the center can be choosen arbitrarily.

This sphere is not the boundary of the system. It is the edge of my map.

It is the place at which the two opposite, half straight lines are soldered into a closed straight line.

Cosmic repulsion is manifested as a force proportional to the distance to the center of the diagram.

In terms of the gravity in this sphere:

• the regions farther away from the center than the point being considered have no influence on its motion.
• the interior points act as though they were concentrated at the center.

The homogeneity of the distribution of matter gives a constant density.

• This makes the force of attraction proportional to a distance, just as is cosmic repulsion.

Therefore, a certain density exists as ’the density of equilibrium’ or ’the cosmic density’, for which the two forces will be in equilibrium.

These elementary considerations permit recognition, in a result which calculation gives and which is contained in Friedmann’s equation:

( dR)2 2M R2
dt =-i+ if+;P'

The last term represents cosmic repulsion (it is double the function of the forces of this repulsion).

T is a constant depending on the value ofthe cosmological constant and able to replace this. The next-to-iast term is double the potential of attraction due to the interior mass.

The radius of space R is the distance from the origin of a point of angular distance (] =i. if one multiplied the equation by ~, one would have the<. corresponding equation for a point at any distance.

That which is remarkable in Friedmann’s equation is the first term -i. The elementary considerations which we have just advanced would allow us to assign it a value which is more or less constant; it is the constant of . energy in the motion which takes place under the action of two forces.

The complete theory determines this constant and thus links the geometric properties to the dynamic properties.

Einstein’s Equilibrium

The radius R is set constant and so the state of the universe in equilibrium, or Einstein’s universe, is obtained.

The conditions of the universe in equilibrium are easily deduced from Friedmann’s equation:

T 31 T RE = fj; PE = 4'1T T2; M = fj'

In these formulas, the distances are calculated in light-time, which amounts to taking the velocity of light c as equal to unity, but, in addition, the unit of mass is chosen in such a way that the constant-of gravitation may also be equal to unity.

It is easy to pass on to the numerical values of c.g.s. by re-establishing in the formulas the constants c and G in such a manner as to satisfy the equations of dimension.

In particular, if one takes T as being equal to 2 Xi09 years, as we shall suppose in a moment, one finds that the density PE is equal toi0-27 gram per cubic centimeter.

These considerations can be extended to a region in which the distribution is no longer homogeneous and where even the spherical symmetry is no longer verified, provided that the region under consideration be of small dimension.

In a small region, Newtonian mechanics is always a good approximation.

Naturally, it is necessary, in applying Newtonian mechanics, to take account of cosmic repulsion but, aside from this easy modification, it is perfectly legitimate to utilize the intuition acquired by the practice of classic mechanics and its application to systems which are more or less complicated.

This equilibrium is unstable and can even be disturbed in one sense, in one place, and in the opposite sense in another region.

Friedmann’s equation is only rigorously exact if the mass M remains constant.

While one takes account of the radiation which circulates in space and also of the characteristic velocities of the particles which cross one another in the manner of molecules in a gas and, as in a gas, give rise to pressure, it is necessary to consider the work of this pressure during the expansion of space, in the evaluation ofthe mass or the energy.

But it is apparent that such an effect is generally negligible, as detailed researchers elsewhere have shown.

The Significance Of Clusters Of Nebulae

What followed the disintegration of the primeval atom?

In a first period of rapid expansion, gaseous clouds must have been formed, animated by great, proper velocities.

We are now going to suppose that the mass M is slightly larger than fj'

The second member of Friedmann’s equation will thus be able to become smaller, but it will not be able to vanish.

Thus, the expansion has 3 phases.

The first rapid expansion will be followed by a period of deceleration. In this period, attraction and repulsion will bring themselves into equilibrium.

Finally, repulsion will definitely prevail over attraction. The universe will enter into the third phase, that of the resumption of expansion under the dominant action of cosmic repulsion.

The phase of slow expansion

The gaseous clouds are not distributed uniformly.

Let us consider in a region sufficiently small, and that only from the point of view of classical mechanics, the conflict between the forces of repulsion and attraction which almost produces equilibrium.

As a result of local fluctuations of density, there will be regions where attraction will finally prevail over repulsion, in spite of the fact that we have supposed that, for the universe in its entirety, it is the contrary which takes place.

These regions in which attraction has prevailed will thus fall back on themselves, while the universe will be entering on a period of renewed expansion.

This produces a universe formed of regions of condensations which are separated from one another.

It will be possible for large regions where the density or the speed of expansion differ slightly from the average to hesitate between expansion and contraction, and remain in equilibrium, while the universe has resumed expansion.

Could these regions not be identified with the clus~ers of nebulae, which are made up of several hundred nebulae located at relative distances from one another, which are a dozen times smaller than those of isolated nebulae?

According to this interpretation, these clusters are made up of nebulae which are retarded in the phase of equilibrium; they represent a sample of the distribution of matter, as it existed everywhere, when the radius of space was a dozen times smaller than it is at present, when the universe was passing through equilibrium.