On the Coordination of the Riemannian and Kinematic Techniques in Theoretical Cosmology, with Particular Reference to the Shift-Distance Law

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Abstract
This paper is concerned with homogeneous and isotropic model-universes. The two main alternative descriptions of such universes, (i) by means of a Riemannian space-time map characterised by the Robertson–Walker metric and (ii) as a kinematic equivalence, are compared. The generating function ψ(t) and signal-parameter α of the equivalence are related to the scale-function R(t) and radial coordinate r of the corresponding metric. Milne's distance is related to the other distances commonly used in cosmology, for which expansions in terms of proper distance are given. The empirical determination from nebular red-shift data of the functions R(t) and ψ(t) appropriate to the observable universe is discussed with reference to Whitrow's second-order shift-distance criterion and in the light of recently published observations. Also, new general shift-ratio formulae for kinematic equivalences are derived.

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