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

From this follows

$\left.{\begin{aligned}p&=-{\frac {\sigma }{2}}\\a&={\sqrt {\frac {2}{\kappa \sigma }}}\end{aligned}}\right\}.$

(123)

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

$a={\frac {M\kappa }{4\pi ^{2}}}.$

(124)

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.