Note 106 — Chapter XXXIV · THE FINITENESS OF THE UNIVERSE

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-, is always negative, or vice versa (except when they all vanish), we always have a difference in sign between the curvatures , , , on the one hand, and , on the other. Were we to appeal to imaginary time, would also be positive, so that would be of the same sign as the other ’s. Analogously, in the special theory, by putting it in place of , we obtained in place of , giving us in place of .