Schwarzschild Metrics, Quasi-Universes and Wormholes

Abstract

It is well known [1] that the three-dimensional space

$$ {d^{(3)}}{8^2} = {{d{r^2}} \over {1 - {{{r^2}} \over {{R^2}}}}} + {r^2}(d{\vartheta ^2} + {\sin ^2}\vartheta d{\varphi ^2}) $$

(1)

of some models of closed homogeneous and isotropic universes has an especially simple geometry which can be seen best introducing a angular coordinate 0 ≤ ϰ ≤ π via r = R sin ϰ and transforming the line element (1) into the form

$$ {d^{(3)}}{8^2} = {R^2}(d{\chi ^2} + {\sin ^2}\chi d{\Omega ^2}) $$

(2)

where

$$ d{\Omega ^2} = d{\vartheta ^2} + {\sin ^2}\vartheta d{\varphi ^2} $$

(3)

The metric (2) is that of a three-dimensional hypersurface of radius R which can be represented in a flat, four-dimensional Euclidean embedding space.