New Axioms for Cosmology
| Scientific Paper | |
|---|---|
| Title | New Axioms for Cosmology |
| Read in full | Link to paper |
| Author(s) | Mogens True Wegener |
| Keywords | Cosmology, simultaneity |
| Published | 2011 |
| No. of pages | 19 |
Read the full paper here
Abstract
In the present paper it is shown how it is possible by means of a time-based concept of equidistance to construct a spatial geometry for relativistic cosmology. In analogy to a sphere defined as the geometrical site for all those points which are equidistant from a given point, we construct a plane as the site for all those points that are equidistant from two points, and a line as the site for all those points equidistant from three points. Having defined parallellity and perpendicularity, we proceed to define the Cosmic Substrate in a way analogous to the cosmological principle of the Cusan. Assuming that we can always construe the center point for any three non-collinear members of this Substrate, we can prove simultaneity to be universally transitive for all members of the Substrate if the simultaneity is defined indirectly by means of equidistance.
Overview
This is a revised (2011) version of a paper Wegener presented at the 2009 PIRT conference "Mathematics, Physics and Philosophy in the Interpretations of Relativity Theory" at Loránd Eötvös University in Budapest. It is an axiomatisation of relativistic cosmology built on a single primitive: the exchange of signals ("photons") between particles. Space is not assumed; it is constructed. Distance is defined as light-time, equidistance is defined by a signal reflected off two targets and returning together, and from that "temporal pair of compasses" Wegener derives spheres, circles, planes, straight lines, parallelism and perpendicularity — dispensing, as he puts it, with the ruler and keeping only the compasses, in the spirit of Georg Mohr's Euclides Danicus (1672).
The purpose of this machinery is polemical as well as formal. Wegener wants to recover an absolute cosmic time and a universally transitive simultaneity, and he says so plainly: he regards the "spatialization of time" as "a great mistake on behalf of science, indeed the ultimate catastrophe," and the block universe "wherein nothing happens" as foreign to experience. Against the standard reading of Special Relativity he argues that if simultaneity at a distance is conventional, then so is the one-way Speed of Light, and therefore so is the opening angle of the light cone — which in his view "puts the geometric light-cone enterprise completely in jeopardy." His constructive alternative revives E. A. Milne's Kinematic Relativity (1948), with its substratum of equivalent fundamental particles, and asks physics to derive gravity from asymmetry rather than assume it.
The axiomatisation
Signals as the primitive
The paper is sketched in first-order predicate logic with basic set theory. The Universe is the set of all particle-observers, each itself a set of observational events. Ax.1 (Causation by Signals) states that for any absorption event in a particle P there was exactly one prior emission event in some other particle P′ that caused it; Ax.2 makes causal succession transitive. Wegener notes explicitly that he prefers "backwards explanation" — inferring a present effect's past cause — to "forwards prediction," which involves probabilities.
Distance is then wholly temporal. Two particles momentarily coincide when the light-time between them is zero; they are identical when that distance is always zero. Momentary equidistance (Df.5) holds when a single emission event in P produces two reflections, at P′ and P″, whose returns are absorbed in the same event in P. Wegener stresses the limits of this: it presupposes only that photons travel with invariant two-way speed in all directions. "We do not assume a constant one-way light speed."
Geometry from equidistance
From that definition the geometrical objects follow as sets:
- a sphere is the set of points equidistant from a given point;
- a plane is the set of points equidistant from two given points;
- a straight line is the set of points equidistant from three given points that are not in optical line;
- a circle is the intersection of two similar spheres, and equivalently the intersection of a sphere with a plane.
The defining points themselves do not belong to the object they define. Theorems 1–9 establish that the plane between P and P′ is identical with the plane between P′ and P, that a line is indifferent to the order of its three defining points, that a line is the intersection of two planes, and that a plane is the set of all the lines it contains. Parallelism of planes is defined by their defining points lying on a common line, parallelism of lines by their defining points lying in a common plane, and perpendicularity by the corresponding cross-membership condition.
The Substrate
Df.23 introduces the Substrate: a set of particles such that if two members ever keep the same distance from a third they always do, and if three members ever form an optical line they always do. Members of the Substrate are fundamental particles and are said to be at rest in it; everything else is an accidental particle and is in motion. Axioms 3–5 make the Substrate non-empty, dense (any two fundamental particles have a fundamental midway particle) and infinite. Ax.6 has the striking consequence that if any two members of the Substrate ever coincide, all of them do — from which Wegener draws the comment that "an expanding and accelerating universe need not stem from an initial singularity."
Ax.7 imposes cosmic isotropy on every fundamental particle's observational perspective, which Wegener attributes to Nicholas of Cusa (De Docta Ignorantia, ca. 1450): the machine of the world has its centre everywhere and its circumference nowhere. Ax.8 fixes three dimensions by requiring that any three mutually equidistant fundamental particles admit exactly two further equidistant ones — the two tetrahedra being, he notes, the origin of handedness. Ax.9 makes the Substrate serve "as an Aether, i.e. a medium for the transmission of photons." Flatness (Df.28) is left open: a regular hexagon of seven Substrate particles either fits its plane, leaves a gap (hyperbolic), or overlaps (spherical).
Simultaneity and the two clocks
P-simultaneity is defined by equidistance relative to an observer P (Df.29). Theorems 11–13 prove it reciprocal and transitive among points of any sphere centred on P, and transitive along equidistant points of a straight line by repeated midway-point bisection. Wegener concedes that this procedure covers only finitely many points of a line, so that securing simultaneity everywhere forces a choice between atoms of time and postulating simultaneity absolute; he takes the latter as Ax.10, adding that absolute simultaneity does not prevent moving clocks from being retarded, "as evidenced by experiment in agreement with standard relativity."
The final move distinguishes cosmic clocks — Langevin clocks built from zig-zag signals between two fundamental particles, keeping cosmic time T — from atomic clocks keeping atomic time t. A universe is stationary if t/T is constant and expanding if t/T is a function of time. Quoting Eddington's remark that "the theory of the expanding cosmos is equivalent to the theory of the shrinking atom," Wegener concludes that only the comparison of the two clock types gives the question of expansion any content. He assumes inertia holds relative to cosmic time, so that free motion referred to atomic time shows spontaneous accelerations — "this might be a key to explaining gravitation." Gravity would then vindicate Mach's Principle the other way round: gravity explained by Inertia, not inertia by gravity. The Einstein and Friedmann field equations are rejected.
Assessment
The distinctive achievement here is economy of primitives. Almost every axiomatisation of spacetime helps itself to a metric or to a coordinate manifold; Wegener starts with nothing but emission, absorption and causal order, and gets planes and straight lines out of a purely temporal equidistance relation. The refusal to assume a one-way Speed of Light is intellectually honest and is exactly the point at which his construction differs from light-cone axiomatics — his objection to the cone's opening angle is a real one, and rests on the conventionality results of Winnie (1970) and Øhrstrøm (2000) that he cites. The Milne-style separation of cosmic time from atomic time is likewise a genuine physical question, not a verbal one, and Wegener is right that it is a question standard cosmology tends to hide rather than answer.
The difficulties are of two kinds. The first is that the paper is, by its author's own admission, "exceedingly sketchy": he says he preferred "to convey an impression of the main ideas rather than to offer an impeccable presentation." Several of the proofs are one-line set identities that restate the definitions, and the crucial theorems 11–13 establish transitivity of simultaneity only on spheres and on equidistant points of a line; the extension to the whole Substrate is not proved but postulated in Ax.10. The Scholium concedes exactly this. So the abstract's claim that simultaneity is "proved" universally transitive is stronger than what the text delivers.
The second is that the most interesting physical claims are gestured at rather than derived. The identification of gravity with the change of description between cosmic and atomic time scales is stated in the preamble and again in the conclusion — an accidental particle's kinetic energy as judged by one fundamental observer must appear as potential energy to the observer relative to which it is at rest, "so, what to the first observer appears as inertia, to the second one must appear as gravity" — but no field equation, no force law, and no quantitative prediction is offered. Nothing in the paper is confronted with measurement: not the perihelion precession, not the Shapiro delay, not light deflection, all of which the Einstein equations Wegener rejects reproduce to high precision, and all of which Walker's variational alternative (which Wegener mentions) only "almost" reproduced. Whether the Substrate can be identified observationally — the obvious candidate being the frame in which the Cosmic Microwave Background dipole vanishes — is not discussed, and without such an identification "fundamental particle" remains a formal category.
Taken for what it is, the paper is a philosophical axiomatics rather than a physical theory: a careful demonstration that a Milne-type cosmology with absolute simultaneity is internally constructible from signal exchange alone. That is a real result, and it is worth more than the loose claim that Einstein's obliteration of Milne was a matter of shopping-bag fame. But the step from "constructible" to "true of this universe" is the step the paper does not take.