The Radial Velocities of Globular Clusters, and de Sitter's Cosmology

Abstract

IF R be the curvature radius of the world, an invariant of the manifold, de Sitter's empty space-time is characterised, in the two-dimensional case which interests us here, by the corresponding elliptic space being covered by the interval σ=o to ½π. With the observer's station O as origin, let a star be coerced to remain at a fixed distance r. Then, the permanence of atoms as natural clocks being granted, the observer should perceive a wave-length increment of a line of the stellar, as compared with his local spectrum, the latter for small r/R. This is de Sitter's own result (1917), a second-order effect due to mere distance, based upon the artificial assumption that the lightsource is at rest relatively to the observer. Its characteristic feature is that it gives always a positive or red-shift, as would be the common Doppler effect for a receding star. Since the B-stars were known to show a systematic displacement corresponding to about +4.5 km./sec., de Sitter welcomed them as so many supporters of his formula. Ascribing of the effect to their own gravitation, he represented the remaining by that formula, which, with r=3.10⁷ as the average distance of these stars¹, gave him R=7.10⁹. This would, nowadays, seem rather too small, being just below Shapley's estimate, 9.10⁹, of the semi-diameter of our galaxy. Such considerations, however, need not detain us.