Page:The Meaning of Relativity - Albert Einstein (1922).djvu/130

118 From this follows {{MathForm2|(123)| $$ \left. \begin{align} p & = -\frac{\sigma}{2} \\ a & = \sqrt{\frac{2}{\kappa\sigma}} \end{align} \right\}. $$}}

If the universe is quasi-Euclidean, and its radius of curvature therefore infinite, then $$\sigma$$ would vanish. But it is improbable that the mean density of matter in the universe is actually zero; this is our third argument against the assumption that the universe is quasi-Euclidean. Nor does it seem possible that our hypothetical pressure can vanish; the physical nature of this pressure can be appreciated only after we have a better theoretical knowledge of the electromagnetic field. According to the second of equations (123) the radius, $$a$$, of the universe is determined in terms of the total mass, $$M$$, of matter, by the equation

The complete dependence of the geometrical upon the physical properties becomes clearly apparent by means of this equation.

Thus we may present the following arguments against the conception of a space-infinite, and for the conception of a space-bounded, universe:—

1. From the standpoint of the theory of relativity, the condition for a closed surface is very much simpler than the corresponding boundary condition at infinity of the quasi-Euclidean structure of the universe.