The Structure Of Space According to General Relativity
According General Relativity (GR), the geometrical properties of space are determined by matter.Thus, we can draw conclusions about the universe’s geometrical structure based on matter.
The velocities of the stars are small as compared with the speed of light. From this, we can conclude the nature of the universe.
The behaviour of measuring-rods and clocks is influenced by gravitational fields caused by the distribution of matter. This eliminates the need for the exact validity of Euclidean geometry in our universe.
But our universe differs only slightly from a Euclidean one. This is proven by the outer space of our solar system being influenced only to an exceedingly small extent by masses even as large as our sun.
We might imagine that our universe is like the rippled surface of a lake, as a “quasi-Euclidean universe” which has infinite space. But such a quasi-Euclidean universe would have an average density of matter as nil. Thus such a universe could not be inhabited by matter everywhere.
On the contrary, if matter were distributed uniformly, the universe would necessarily be spherical (or elliptical).
Since the distribution of matter is not uniform, the real universe will not be totally spherical. It will be quasi-spherical, but finite. In fact, GR connects the space-expanse of the universe and the average density of matter in it.
For the radius
R of the universe we obtain the equation
R^2 = 2 / κρ
The use of the C.G.S. system in this equation gives
2 / κ = = 1.08.10^27 ;
ρ is the average density of the matter.