Annex 10 - Consideration on the universe

There is a second fundamental problem with the classical view of the universe. If we think about how the universe as a whole should be seen the first answers could be:
As regard space and time, the universe is infinite.
There are stars everywhere so that the density of matter -although variable locally- is, on average, the same everywhere.
In other words, however far we travel in the space we should find everywhere fixed stars of approximately the same size and density.
This does not agree with Newton's theory that requires that the universe should have a kind of centre where the star density is the highest. Moving away front this centre the star density should decrease until, far away, we reach an infinite region of emptiness. This could also be expressed by saying that, according to Newton's law, the stellar universe is a finite island in the middle of infinite empty space.

However this concept is not satisfactory because the light emitted by the stars are sent without possibility of returning into the empty space and without ever interacting with any object in nature. As a consequence the finite universe would cool down. To solve this problem the Newton law was modified by assuming that for great distances the attraction between two masses decreases more rapidly than the inverse square distance would suggest. In this way it would be possible for the mean density of the matter to be the same everywhere, even to infinity and without infinitely large gravitational fields being created. This avoids the necessity for the universe to have a centre but this require modifying Newton's law without theoretical justification.

The introduction of non-Euclidean geometry put some doubt about the idea of an infinite space.

Let us first consider a two-dimensional space, in other words a plane. Flat rigid measuring rods are free to move on this plane and nothing exists outside it. This two-dimensional space extends to infinity and there is room for an infinite number of identical squares made up of rods. This infinite universe is "plane" and this means that one can construct plane Euclidean geometry with the rods, the individual rods always represent the same distance, wherever they are on the plane.

Now we will consider another two-dimensional space but on a spherical surface instead of a plane. Again we can imagine a large number of measuring rods set on the surface of the sphere. In this case it is not possible to consider the spherical universe as plane geometry and the measuring rods do not represent the same distance anymore. If we try to draw a straight line on the sphere we get a curve that we describe as a great circle, a self-contained line of finite length that can be measured using measuring rods. This spherical universe has also a finite area and yet it has no limits and it is not Euclidean.

Starting from a point let us draw "straight lines" (they are circle in our usual three-dimensional space) of equal length in all directions. For a plane surface the ration of its circumference to its diameter, both measured with the same unit rod, is in the Euclidean geometry of the plane equal to and this value is a constant that does not depend of the diameter of the circle. On the surface of the sphere one would find that for the same ratio the value * sin(r/R)/(r/R) and this is smaller than . This value is smaller the greater is the radius r of the circle in relation to the radius R of the sphere.

By means of this relation one can calculate the radius of the spherical universe when a small but all the same sufficient part of the sphere is available to make the measurements.

According to the general theory of relativity the properties of space are not independent, but are determined by matter. As that for a well-chosen set of co-ordinates, the velocity of the stars is small compared with the velocity of light. We can get a first approximation of the nature of the whole universe if we assume that it is at rest.

Measuring-rods and clocks, we know, are influenced by gravitational fields that is by the distribution of matter. From this we deduce that the Euclidean geometry is not exactly valid in the universe but it differs only in a small way from a Euclidean one. From this we can assume that our universe behave like a surface that is irregularly curved locally but that, in the whole, is very close to a plane. Such a universe may be called a quasi-Euclidean universe and its space is infinite. Calculation shows that in such a quasi-Euclidean space the average density of matter would be zero and this is contrary to our experience. As a result if we, as we know it is true, that the universe contains a small average density of matter, then this universe cannot be quasi-Euclidean. Calculation shows that if matter is uniformly distributed in the space, then the universe must be spherical or elliptical. But the matter is not distributed uniformly in the universe, it will deviate in some parts from a pure sphere. In conclusion our universe is quasi-spherical (or elliptical) but, in any case, it is finite.