The principle of least action in Milne's kinematical relativty

In a recent paper, Professor Milne has obtained a gravitational field with non-zero density of matter in flat space-time, the field in question being appropriate to the whole cosmos. It was obtained, without recourse to a formal theory of gravitation, by constructing a system of particles in motion satisfying Einstein’s cosmological principle applied to a set of fundamental observers in uniform relative motion: the result was a set of motions and a particle-density distribution which would be described in the same way by each one of the fundamental observers. This method of obtaining a gravitational field is fundamentally different from Einstein’s, the applicability being dependent upon explicit recognition of the priniciple that “an observer can either ( a ) select any one of the spaces of pure geometry presented to him by the mathematician, use it in order to describe the phenomena in the space; or alternatively ( b ) posit beforehand the laws of nature he wishes to see obeyed and then determine the space in which, in consequence, he must embed the phenomena he describes.” Einstein’s theory of gravitation consists essentially in obtaining a metric ds 2 such that a free particle obeys the law of nature δ ∫ ds = 0, and is an example of alternative ( b ). The gravitational field for the system described above begins by selecting Euclidean space and Newtonian time for any one observer, the different observer’s space and times being connected by Lorentz transformations, and then determines the laws of motion in this space; it is an example of alternative ( a ). The laws of motion were obtained as formulæ for the components of acceleration of a free particle as functions of the seven variables, t, x, y, z, u, v, w , reckoned from defined zeros. The alternative procedures have been recently stated very clearly by Milner. He wrote: ”Two courses are open to us. (1) We can modify the geometry assumed in the relation ds 2 = ∑ i=1 4 ( dx i ) 2 so that a mathematically straight track ( i. e ., its length obeys a stationery principle still continues to represent the non-uniform motion of a particle; this is the method of general relativity. (2) We can retain the four-fold with unaltered geometry and specify a curved track which represents the observed motion by weighting each element of its length so that the integral weighted length between two points is stationery; this is the method of ‘least action’. Both these methods of describing the motions of bodies must be considered equally logical when one remembers that a manifold (even when it is called ‘space-time’) is not the actual world, but a mental concept, in which real phenomena are represented symbolically.”