Formation Of Clouds

We picture the primeval atom as filling space which has a very small radius (astronomically speaking).

Therefore, there is no place for superficial electrons, the primeval atom being nearly an isotope of a neutron. This atom is conceived as having existed for an instant only, in fact, it was unstable and, as soon as it came into being, it was broken into pieces which were again broken, in their turn.

Among these pieces electrons, protons, alpha particles, etc., rushed out.

An increase in volume resulted, the disintegration of the atom was thus accompanied by a rapid increase in the radius of space which the fragments of the primeval atom filled, always uniformly.

When these pieces became too small, they ceased to break up; certain ones, like uranium, are slowly disintegrating now, with an average life of four billion years, leaving us a meager sample of the universal disintegration of the past. "

In this first phase of the expansion of space, starting asymptotically with a radius practically zero, we have particles of enormous velocities (as a result of recoil at the time of the emission of rays) which are immersed in radiation, the total energy of which is, without doubt, a notable fraction of the mass energy of the atoms.

The effect of the rapid expansion of space is the attenuation of this radiation and also the diminution of the relative velocities of the atoms.

Let us imagine that an atom has, along the radius of the sphere in which we are representing closed space, a radial velocity which is greater than the velocity normal to the region in which it is found. Then this atom will depart faster from the center than the ideal material particle which has normal velocity.

Thus the atom will reach, progressively, regions where its velocity is less abnormal, and its proper velocity, that is, its excess over normal velocity, will diminish. Calculation shows that proper velocity varies in this way in inverse ratio to the radius of space.

We must therefore look for a notable attenuation of the relative velocities of atoms in the first period of expansion.

From time to time as a result of favorable chances, the collisions between atoms will become sufficiently moderate so as not to give rise to atomic transformations or emissions of radiation.

These collisions will be elastic collisions, controlled by superficial electrons, so considered in the theory of gases.

Thus we shall obtain, at least locally, a beginning of statistical equilibrium, that is, the formation of gaseous clouds.

These gaseous clouds will still have considerable velocities, in relation to one another, and they will be mixed with radiations that are themselves attenuated by expansion.

It is these radiations which will endure until our time in the form of cosmic rays, while the gaseous clouds will have given place to stars and to nebulae by a process which remains to be explained.

Cosmic Repulsion

For that explanation, we must say a few words about the theory of relativity.

When Einstein established General Relativity, he accepted the principle of equivalence, that the ideas of special relativity were approximately valid in a sufficiently small domain.

In the special theory, the differential element of space-time measurements had for its square a quadratic form with four coordinates, the coefficients of which had special constant values.

In the generalization, this element will still be the square root of a quadratic form, but the coefficients, designated collectively by the name of the metric tensor, will vary from place to place.

The geometry of space-time is then the general geometry of Riemann at three plus one dimensions.

The spaces with variable radii are a particular case in this general geometry, since the hypothesis of spatial homogeneity or of the equivalence of observers is introduced here.

This geometry differs only apparently from that of special relativity. This is what happens when the quadratic form can be transformed, by a simple change of coordinates, into a form having constant coefficients.

Then one says with Riemann that the corresponding variety (that is, space-time) is flat or Euclidian.

For that, it is necessary that certain expressions, expressed by components of a tensor with four indices called Riemann’s tensor, vanish completely at all points. When it is not so, the tensor of Riemann expresses the departure from flatness.

Riemann’s tensor is calculated by the average of second derivatives of the metric tensor. Starting with Riemann’s tensor with four indices, it is easy to obtain a tensor which has only two indices like the metric tensor; it is called the contracted Riemannian tensor.

One can also obtain a scalar, the totally contracted Riemannian tensor.

In special relativity, a free point describes a straight line with uniform motion, that is the principle of inertia.

One can also say that, in an equivalent manner, it describes a geodesic of space-time. in the generalization, it is again presumed that a free point describes a geodesic.

These geodesies are no longer representable by a uniform, rectilinear motion, they now represent a motion of a point under the action of the forces of gravitation.

Since the field of gravitation is caused by the presence of matter, it is necessary that there be a relation between the density of the distribution of matter and Riemann’s tensor which expresses the departure from flatness.

The density is, in itself, considered as the principal component of a tensor with two indices called the material tensor.

Thus one obtains as a possible expression of the material tensor Tii as a function of the metric tensor gli and of the two tensors of Riemann, contracted to Rii and totally contracted to R,

where a, b, and c are three constants.

Certain identities must exist between the components of the material tensor and its derivatives.

These identities can be interpreted, for a convenient choice:( of coordinates, a choice which corresponds, moreover, to the practical conditions of observations, as expressing the principles of conservation, that of energy and that of momentum.

in order that such identities may be satisfied, it is no longer possible to choose arbitrarily the values of the three constants. b must be taken as equal to-a/2.

Theory cannot predict either their magnitude or their sign. it is only observation which can determine them.

The constant a is linked to the constant of gravitation. in fact, when theory is applied to conditions which are met in the applications (in particular, the fact that astronomical velocities are small in comparison to the speed of light) and when one profits from these conditions by introducing coordinates which facilitate comparison with experiment, one finds that the geodesics differ from rectilinear motion by an acceleration which can be interpreted as an attraction in inverse ratio to the square of the distances, and which is exercised by the masses represented by the material tensor.

This is simply the principal effect foreseen by the theory; this theory predicts small departures which, in favorable cases, have been confirmed by observation.

A good agreement with planetary observations is obtained by leaving out the term in c. That does not prove that this term may not have experimental consequence. in fact, in the conditions which were employed to obtain Newton’s law as an approximation of the theory, the term in c would furnish a force varying, not in the inverse square ratio of the distance, but proportionally to this distance. This force could therefore have a marked action at very great distances although, for the distances of the planets, its action would be negligible. Also, the relation c/ a, designated customarily by the letter A, lambda, is called the cosmological constant. When A is positive, the additional force proportional to the distance is called cosmic repulsion.

The theory of relativity has thus unified the theory of Newton. in Newton’s theory, there were two principles posed independently of one another: universal attraction and the conservation of mass. In the theory of relativity, these principles take a slightly modified form, while being practically identical to those of Newton in the case where these have been confronted with the facts. But universal attraction is now a result of the conservation of mass. The size of the force, the constant of gravitation, is determined experimentally.

The theory again indicates that the constancy of mass has, as a result, besides the Newtonian force of gravitation, a repulsion proportional to the distance of which the size and even the sign can only be determined by observation and by observation requiring great distances.

Cosmic repulsion is not a special hypothesis, introduced to avoid the difficulties which are presented in the study of the universe. if Einstein has re-introduced it in his work on cosmology, it is because he remembered having arbitrarily dropped it when he had established the equations of gravitation. To suppress it amounts to determining it arbitrarily by giving it a particular value: zero.