Page:The Meaning of Relativity - Albert Einstein (1922).djvu/126

able choice of co-ordinates, $g_{44}$ is nearly constant in the universe, at least, in that part of the universe in which there is matter. The assumption appears natural, moreover, that there are stars in all parts of the universe, so that we may well assume that the inconstancy of $g_{44}$ depends only upon the circumstance that matter is not distributed continuously, but is concentrated in single celestial bodies and systems of bodies. If we are willing to ignore these more local non-uniformities of the density of matter and of the $g_{\mu \nu }$-field, in order to learn something of the geometrical properties of the universe as a whole, it appears natural to substitute for the actual distribution of masses a continuous distribution, and furthermore to assign to this distribution a uniform density $\sigma$. In this imagined universe all points with space directions will be geometrically equivalent; with respect to its space extension it will have a constant curvature, and will be cylindrical with respect to its $x_{4}$-co-ordinate. The possibility seems to be particularly satisfying that the universe is spatially bounded and thus, in accordance with our assumption of the constancy of $\sigma$, is of constant curvature, being either spherical or elliptical; for then the boundary conditions at infinity which are so inconvenient from the standpoint of the general theory of relativity, may be replaced by the much more natural conditions for a closed surface.

According to what has been said, we are to put

$ds^{2}=dx_{4}^{2}-\gamma _{\mu \nu }dx_{\mu }dx_{\nu }$

(119)

in which the indices $\mu$ and $\nu$ run from 1 to 3 only. The