Absolute parallelism and metric in the expanding universe theory

In the existing theories of the expanding universe, the idea of metric or, alternatively, that of distance between two non-neighbouring points is taken as the starting-point. Thus in general relatively metric is the basis. When the metric is found and it is required to compare the predictions of theory with observation, further definitions, giving rise to different kinds of distance, must be made. On the other hand, Milne takes a definition of distance as the basis of his theory. This definition involves two assumptions. Firstly, it is supposed that observers only use the time-measures of their clocks by which to define distance, and secondly they define it is an special relativity. In each relativistic model there is therefore some ambiguity as to what is meant by distance, whilst Milne’s theory is open to the objection that there is no reason why observers should, in fact, adopt his definition of distance. The question therefore arises: up to what point is it possible to construct a theory of the mechanics of the expanding universe without using the ideas of metric and distance at all? We attempt to provide an answer by employing the theory of absolute or distant parallelism, so that we substitute the idea ofdirection for that ofmetric (or of distance) as the fundamental notion. The resulting theory is similar to that of Milne and its generalization recently proposed, but differs from them in the use of non-metrical equations. When metric and distance are eventually introduced, their function is merely to interpret, and not to obtain, formulae already found by non-metrical methods. A physical picture of the situation we wish to deal with is as follows. Consider a hydrodynamical fluid consisting of non-interacting particles in a state of continuous flux. We suppose that in this fluid there is a set of observers moving with it and making measurements of the state of the fluid around them. These observers make similar kinds of measurements. Each observer finds that the fluid recedes from him with a velocity proportional to distance from himself, as in the system of the spiral nebulae regarded as forming a “fluid”.