Milne's Kinematic Relativity

Scientific Paper
Title	Milne's Kinematic Relativity
Read in full	Link to paper
Author(s)	Mogens True Wegener
Keywords	Cosmology, kinematic relativity, cosmic time, steady state, continued creation
Published	2000
No. of pages	19
Pages	255-274

Read the full paper here

Abstract

Revised version. # Cosmology, a Science?

Milne's 'Kinematic' Relativity
Walker's Analysis of Milne's Ideas
Revolt Against Prejudice in Science!
Theories of Continued Creation
A Program for Synthesis

In our time, Cosmology is generally acknowledged to be the science of the universe. But what is the universe? Is it being, entity, or substance? Is it nature itself, ultimate reality? How do we overcome the desperate difficulties of speaking sensibly of everything at once? And in what sense can such an elusive subject be the object of anything like a real science? Can we avoid the danger of assuming either too little or too much even before we begin? Finally, the universe is one, or unique: how can it then give rise to a legitimate science at all such questions cry for their rational answers ... Hoping for better progress later, we shall start by making our language a bit more precise.

Overview

The full title of this piece is "Milne's Kinematic Relativity — Ideas of Cosmology: A Philosopher's Synthesis". It is the 2011 revised version of a paper printed in Duffy and Wegener, eds., Recent Advances in Relativity Theory, vol. 1 (Hadronic Press, 2000). Mogens True Wegener writes as a philosopher of science rather than a working cosmologist, and the essay does two things at once: it reconstructs and defends E. A. Milne's kinematic relativity of the 1930s, largely forgotten after its rejection by the mainstream, and it uses that reconstruction to build a new steady-state model of continued creation.

The dissident thesis is that cosmology has been miscast. Wegener argues for cosmology as a deductive science — "the art of designing world-models" — against the prevailing inductive practice of generalising local laws outward, which he calls "piecemeal engineering" producing "a patchwork of conflicting elements." The concrete casualty of the standard approach, on his account, is time: models representing reality as a timeless multi-dimensional continuum, "like a piece of crystalline mineral," are incompatible with the passage of time, which he takes to be a presupposition of science rather than something science can derive. Where Einstein's programme aimed at "a geometrization of physics", Milne's aimed at "an arithmetization of physics", and where Einstein sought to reduce inertia to gravitation in the spirit of Mach's Principle, Milne sought to reduce gravitation to inertia in opposition to it.

The argument

Cosmology as a science

Wegener opens with a Kantian preliminary. He distinguishes a practical metaphysics — the working assumptions that keep us alive — from a theoretical ontology which he calls "redundant, and often odious," and proposes a "critical idea of the universe" presenting it as "an unknown X, an unique, absolute and ultimate referent ('thing-in-itself') devoid of any specifiable properties." Following Harrison's convention he writes Universe (capitalised) for a metaphysical idea and universe for the propertyless referent. The point is to sustain scientific realism while conceding that theories can be falsified but never verified, so that "it would be rash to hypostasize an ontology by ascribing definite properties to the universe."

Milne's two problems of time-keeping

Milne's world is built from particle-observers or monads — an abstract image of human consciousness, whose only property is a capacity for continual signal exchange. Wegener notes, following Merleau-Ponty, that this is essentially Leibnizian monadology in mathematical form. Milne is a conventionalist in Poincaré's spirit: any device associating events with an increasing scale of numbers is a clock.

The first problem asks whether observer Q can set up a clock congruent with P's, irrespective of relative motion, when the universe of discourse contains nothing but P, Q and their clocks. Signals are exchanged in an unbroken zig-zag, each carrying the clock reading of the last reflection, so each observer can plot his readings against the other's, generating signal-functions θ and φ. Congruence is achieved if Q can regraduate his clock so that θ ≡ φ, which reduces to finding the square root of the functional product θφ — a construction Milne and Whitrow showed is always possible. Two observers are equivalent exactly when their signal-functions are fully symmetric.

The second problem extends this to three or more observers. For observers collinear in fixed order, transitivity of the signal-functions implies that they commute, and the general commutative identity has exactly two solutions, taken to define a linear equivalence. Wegener's picture for it is a set of discrete particles at equal intervals along a homogeneous elastic string: if one particle's motion is uniform relative to a given observer, so is every other's, and their distances are proportional to their velocities. Generalised to three dimensions, the linear equivalence becomes the universal substratum. Wegener adds an observation of his own: if the substratum is discrete rather than a continuous fluid, its particles cannot be equidistant from neighbours in all directions, so "small perturbations should be expected to occur which might imitate the newly observed minute ripples in the cosmic background radiation" (see Cosmic Microwave Background).

Two time-scales, and the derivation of force

With distances defined by radar signals the rigid rod becomes redundant, and the ratio of radar distance to elapsed time serves as an alternative to Einstein's light principle. Assuming the linear equivalence in uniform expansion on the t-scale (which Milne later identified with atomic time), Milne derived the Lorentz–Poincaré transformations. The double solution to the commutativity problem then permits a second scale related logarithmically to the first, τ = t₀ log(t/t₀). Regraduating every clock from t to τ converts "a uniformly expanding substratum in flat space into a stationary substratum in hyperbolic space" — at the cost that all atoms must shrink continuously on the τ-scale. Wegener quotes Eddington's remark that "the theory of the expanding universe might also be called the theory of the shrinking atom."

This double scale carries Milne's most striking result. Distinguishing fundamental particles of the substratum from accidental particles superposed on it, a free accidental particle moves inertially in τ-time; mapped onto t-time the same motion appears accelerated. Milne thus proposed "to explain the nature of 'forces' by local deviations from global symmetry." To extend this from one test particle to a whole ensemble he inverted the Boltzmann equation, deriving accelerations from a position–velocity distribution rather than the reverse, and obtained an induced acceleration with two components: one toward the fundamental particle relative to which the test particle is momentarily at rest, the other toward the local centre of the accidental particles. Milne also built a new electrodynamics from "superpotentials" and treated cosmic rays, galactic structure and atomic structure. Wegener notes that Milne died before resolving the relation between t-mappings and τ-mappings for optical phenomena, "so his cosmology remains partially incomplete."

Walker's analysis

A. G. Walker, whom Wegener regards as the one genuine follower, dropped Milne's assumption that the Lorentz formulae connect observations in t-time and asked what follows from the cosmological principle plus the principle of symmetry alone — later proving the first contained in the second. The result was the Robertson–Walker metric, whose derivation Wegener stresses came out of Milne's kinematic programme, with the additional generality that free paths need not be geodesics. Walker's key passage, quoted at length, establishes τ as "a cosmic time" serving as a public coordinate of the four-dimensional manifold, before assuming t ≡ τ. Walker's objection to General Relativity is reported as this: it lets the form of space depend on the distribution of matter, but no system can exist in isolation from the universe, so world-structure must come first.

Wegener presses a philosophical point of his own here. There is no difficulty in a fundamental observer's master-clock appearing retarded relative to another observer's slave-clocks; but the claim that one fundamental master-clock is retarded relative to another "involves a formal contradiction," since non-congruent master-clocks mean at least one of the two observers is not fundamental. That the Robertson–Walker metric "not merely involves, but presupposes, a Cosmic Time" is offered as evidence that "a very widespread interpretation of relativity is wrong."

Non-standard time coordinates

Section 4 attacks the standard reading of Special Relativity directly. "Laws of nature being distinguished by invariance, how can the dilatation of time ever be a law?" The Lorentz transformations follow inevitably from Einstein's coordinate definitions, but "definitions are conventions" — and the light principle itself, Wegener argues, should be read as constraining only the two-way or round-trip velocity of light. It is generally agreed that no experiment fixes the one-way velocity without a conventional definition of distant Simultaneity; round-trip tests carry no such circularity. Once the one-way velocity is admitted as a theoretical variable, alternatives to the Einsteinian time coordinate open up. Wegener sketches one, relating fundamental observers pairwise through the frame of the particle midway between them, and arriving at τ as a common proper time shown by both master-clocks. On this "alternative relativity", clock retardation relative to the substratum survives but "is not just an effect of inertial motion... it can always be interpreted as a gravitational effect of the substratum", and the deviation of observed time from true time measures the object's departure from fundamentality — "its motional anisotropy." He also endorses Weyl's compass of inertia, identifying it with the substratum rather than, as in Gödel, with some further reality.

A new steady state

Wegener denies that Big Bang and steady-state models are mutually exclusive: one can build a universe originating in a singularity that later approaches a stationary state. He rejects the original Bondi–Gold metric on two grounds — a horizon at t = t₀ ln 2 beyond which "an actual infinity of particles" would exist unobservably, reducing the visible universe to "a finite drop in an infinite ocean of 'reality'", which he calls gross speculation; and the empirical failure of its number–distance statistics for galaxies. Milne's uniform-expansion model, by contrast, has no horizon. The proposal is to transfer that property to a steady state by taking Milne's metric adapted to hyperbolic 3-space, with clock congruence postulated transitive for all fundamental observers rather than derived from commuting signal-functions.

The resulting model dissolves several familiar objections. Fundamentality becomes a matter of degree — "the less the atoms regulating the master-clock of a particle-observer deviate from the ideal standard of cosmic time, the more fundamental he is" — and cosmic time joins the perfect vacuum and absolute zero as a regulative ideal. The old charge that continued creation violates energy conservation is met by asking what volume the conservation law is stated over: "in a pure 'big bang' model energy conservation is defined relative to constant coordinate space, whereas it is defined with respect to constant proper space in a pure 'steady state' model", and Wegener sees no reason to privilege coordinate over proper space. The universe becomes "a pseudosphere in flat space, of finite radius and finite mass, containing an infinite (yet apparently ever growing) number of atoms which apparently shrink with their distance from an observer, reaching zero value precisely as they pass its periphery" — nothing escapes, so it "appears to be the perfect 'black hole'". He introduces a third time-scale t on which recession looks exponential, making the analogy with Bondi's model explicit, and compares the picture to Escher's Circle Limit IV. The redshift law is 1 + z = e^r, which he reports Prokhovnik found matches the number–redshift counts used to reject Bondi's steady state — but which, Wegener argues, supports his own model rather than Prokhovnik's, because only a steady state makes Milne's world-map and world-view structurally identical. Finally, since gravity does not brake the expansion in Milne's theory, the internal pressure of a cosmic gas is floated as the mechanism of "universal dissipation, hence also gravity."

Assessment

The essay's strongest feature is its recovery of a genuinely neglected line of work, together with a fair account of why it was neglected. Wegener does not conceal that "at least some of Milne's more inflated claims were untenable," quotes North's judgement that Milne "seems always to have hovered on the verge of a perfectly reasonable hypothetico-deductive account", and reproduces McCrea's deflationary reply — that deriving law-like properties from general hypotheses is unsurprising, because "we can start with physical laws and construct model-universes; if desired, we can reverse the mathematics." That is the central objection to Milne's programme, and it is put in the paper by the paper's own author. The historical point about the Robertson–Walker metric emerging from Milne's kinematic method is accurate and worth making, as is the observation that the metric presupposes a cosmic time. The conventionality-of-simultaneity argument is likewise standard rather than heterodox: the impossibility of measuring the one-way speed of light without a synchrony convention is widely accepted, and building an alternative coordinatisation on it is legitimate.

The difficulties begin where the philosophy stops and the model starts. The new steady-state construction is presented through metrics and regraduations but never confronted with the measurements that decide between cosmologies. The essay offers no calculation of primordial abundances, no account of the blackbody spectrum of the microwave background — the deficiency that killed the original steady state — and no treatment of its acoustic peak structure. The only observational test invoked is a number–redshift relation of the form 1 + z = e^r, taken at second hand from Prokhovnik with no data reproduced. Set against the modern supernova magnitude–redshift measurements and, decisively, the (1 + z) stretching of Type Ia supernova light curves, an empty kinematic expansion with shrinking atoms owes an explicit prediction; none is given. Where evidence against the Big Bang is cited it is thin and dated — Lerner's claim that certain galactic structures "seem to be as old as a hundred billion yrs" is repeated without qualification, though it rests on assumptions about structure-formation timescales that are exactly what is in dispute.

Several central steps are asserted rather than derived. The suggestion that discreteness of the substratum produces perturbations "which might imitate the newly observed minute ripples in the cosmic background radiation" is offered without an amplitude or an angular spectrum, and so cannot be compared with the measured 10⁻⁵ anisotropies. The final proposal that the internal pressure of a cosmic gas "might explain universal dissipation, hence also gravity" is a single speculative sentence carrying the weight of the paper's boldest claim. The formal-contradiction argument against mutual clock retardation is the weakest link in the philosophical chain: it is valid only given Milne's definition of fundamental observers as members of a congruence-preserving substratum, so it does not refute the relativistic account but restates the premise that a preferred cosmic frame exists. Reciprocal time dilation between observers in relative motion is directly measured — in the decay rates of moving muons and in the Ives–Stilwell and Doppler-shift experiments — and any "alternative relativity" must reproduce those results, which the essay assumes rather than demonstrates.

Judged as what it announces itself to be — a philosopher's synthesis, a reconstruction of a discarded programme with a sketch of where it might lead — the paper is thoughtful, well-read and unusually honest about the objections to its own hero. Judged as a cosmological proposal, it is a metric in search of a confrontation with data.