. As for
, its magnitude is proportional to the universal curvature. As a matter of fact, the appellation hyperbolical universe would be more correct, for it is space-time which is curved spherically, not space and time. Here we must remember that in space-time, space is real and time imaginary; and if we wish to express the shape of the universe in terms of real time instead of imaginary time, the hyperbolical form is the correct one.[106]
As we have explained in the note, de Sitter’s universe is open at both ends in the time direction, so that time does not come back as it would in a spherical universe. When we consider points farther and farther removed from where we stand, all the
’s decrease and finally vanish, as was demanded by the requirements discussed previously. In particular, this decrease and this ultimate vanishing of the potential
indicate, respectively, the gradual slowing down and the complete arrest of time in the distant regions of the universe. This arrest of time, however, is fictitious, for if we transported ourselves to the point considered, we should find that things went on as usual. It would now be where we had stood formerly that the arrest of time would appear complete. Thus, for every observer there would exist a locus of distant points where nothing would appear to change or move. A ray of light could never circle round the universe, but would gradually slow down and ultimately be arrested when this passive horizon of the observer was reached.
We also see that the nearer an incandescent atom is situated to this passive horizon, the slower would its vibrations appear; hence, in the case of a sodium atom (emitting a yellow light under normal conditions), the redder would be its light, till finally the light would cease completely when the passive horizon was reached by the atom. Now it happens that among the most distant luminous bodies known, namely, the spiral nebulæ, the great majority actually do appear to emit slowed-down light vibrations. Of course this effect might be attributed to the general tendency of all spiral nebulæ to recede from the solar system. But as no satisfactory reason can be advanced to account for this general recession, Eddington considers that de Sitter’s hypothesis of the hyperbolical universe has much to commend it.
A few further remarks must be made with reference to de Sitter’s universe. We remember that in Einstein’s theory the perfect flatness of space-time entailed the complete absence of gravitational or inertial forces (so far as a Galilean observer is concerned). Therefore, were it not for the curvature that matter itself imposes upon space-time around it, two parcels of matter would never attract each other. In de Sitter’s universe, however, there exists a residual hyperbolical curvature of space-time at every point, regardless of the additional curvature that would be superimposed by matter. This universal curvature, as can be shown by calculation, would produce a mutual repulsion between bodies. Unless these bodies were of minute proportions, this universal repulsion would be counterbalanced and overcome by the more important curvature generated by the bodies themselves; so that, in spite of all, gravitational attraction would be accounted for. Nevertheless, we should be led to a most displeasing duality in our conception of gravitational action, since we should be in the presence of two separate phenomena: