[105] Of course even from a mathematical point of view the attempt would have been impossible from the start. See [Appendix IV].
[106] De Sitter’s universe is truly spherical only when we argue in terms of imaginary time it. In this case, for any given observer, both time and space close round on themselves. When, however, we use real time t, as indeed we should, we find that de Sitter’s universe yields a three-dimensional spherical Riemann extension, for the space of a given observer, but that real time no longer curls round on itself. This universe can be represented on the surface of a hyperboloid of one sheet, open at both ends in the time direction, and there is no fear of a return of time with the past becoming the future. It is easy to see why time is differentiated in its curvature from space in de Sitter’s universe. All we have to do is to notice that in
, since the three space-
’s (
) are always positive and
, the time-