XXXII. The Structure of Space According to the General Theory of Relativity
Relativity: The Special and General TheoryAlbert Einstein
According to the general theory of relativity, the geometrical properties of space are not independent, but they are determined by matter. Thus we can draw conclusions about the geometrical structure of the universe only if we base our considerations on the state of the matter as being something that is known. We know from experience that, for a suitably chosen coordinate system, the velocities of the stars are small as compared with the velocity of transmission of light. We can thus as a rough approximation arrive at a conclusion as to the nature of the universe as a whole, if we treat the matter as being at rest.
We already know from our previous discussion that the behaviour of measuringrods and clocks is influenced by gravitational fields, i.e. by the distribution of matter. This in itself is sufficient to exclude the possibility of the exact validity of Euclidean geometry in our universe. But it is conceivable that our universe differs only slightly from a Euclidean one, and this notion seems all the more probable, since calculations show that the metrics of surrounding space is influenced only to an exceedingly small extent by masses even of the magnitude of our sun. We might imagine that, as regards geometry, our universe behaves analogously to a surface which is irregularly curved in its individual parts, but which nowhere departs appreciably from a plane: something like the rippled surface of a lake. Such a universe might fittingly be called a quasiEuclidean universe. As regards its space it would be infinite. But calculation shows that in a quasiEuclidean universe the average density of matter would necessarily be nil. Thus such a universe could not be inhabited by matter everywhere; it would present to us that unsatisfactory picture which we portrayed in Section XXX.
If we are to have in the universe an average density of matter which differs from zero, however small may be that difference, then the universe cannot be quasiEuclidean. On the contrary, the results of calculation indicate that if matter be distributed uniformly, the universe would necessarily be spherical (or elliptical). Since in reality the detailed distribution of matter is not uniform, the real universe will deviate in individual parts from the spherical, {\it i.e.} the universe will be quasispherical. But it will be necessarily finite. In fact, the theory supplies us with a simple connection^{1} between the spaceexpanse of the universe and the average density of matter in it.
Appendix I. Simple Derivation of the Lorentz Transformation [Supplementary to Section XI]

For the radius R of the universe we obtain the equation
$R^2=\frac{2}{\kappa p}$The use of the C.G.S. system in this equation gives $2/k = 1^.08 \cdot 10^{27}$; $p$ is the average density of the matter and $k$ is a constant connected with the Newtonian constant of gravitation.
↩
Subscribe to The Empty Robot
Get the latest posts delivered right to your inbox
Spread the word: