The Arrow of Time in the Dynamic Theory

Scientific Paper
Title	The Arrow of Time in the Dynamic Theory
Read in full	Link to paper
Author(s)	Pharis E Williams
Keywords	Entropy, Arrow of Time, Quasars
Published	1981
No. of pages	13

Read the full paper here

Abstract

A newly proposed, and as yet unverified, theory provides new answers to the old questions concerning the symmetry of time in nature. The theory requires an asymmetry in time for systems whose Newtonian or relativistic description is symmetrical. This is accompanied with the prediction that the universe must forever grow older and continually expand and provides new insight on the extreme red shift of quasars.

Overview

This paper is a short expository report — Los Alamos Scientific Laboratory report LA-8690-MS, issued February 1981 — in which Pharis E Williams, then a Lieutenant Commander in the US Navy, applies his "Dynamic Theory" to the oldest embarrassment in theoretical physics: the fact that the laws of thermodynamics single out a direction of time while the Newton-Einstein laws of motion do not. Williams opens by contrasting the irreversibility "embodied in the laws of thermodynamics" with its absence "in the Newton-Einstein laws of motion", and quotes Omar Khayyám's moving finger as the common human experience the equations fail to capture.

His proposal is not to add an arrow of time to mechanics from outside, but to derive mechanics from a footing that already contains the arrow. The Dynamic Theory "adopts generalizations of the classical laws of thermodynamics as the basis for a new view of all physical phenomena." Williams concedes this "may at first seem preposterous", since mechanical systems obey laws of motion while the thermodynamic laws have never yielded equivalent equations for thermodynamic systems. But he argues the implication runs the other way: if equations of motion can be obtained from generalized thermodynamic laws, then "the irreversibility so prominent in thermodynamics would also be embodied within these equations." The arrow of time would then be a structural feature of mechanics, not an anomaly to be explained away by initial conditions or coarse-graining. The departure from the mainstream account is therefore quite radical — where the standard view treats macroscopic irreversibility as statistical and the underlying dynamics as time-symmetric, Williams makes irreversibility fundamental and time-symmetric dynamics the special (reversible) limiting case.

The argument

The three generalized laws

Williams begins from a generalized first law, written as a differential energy balance δE = dU − F_idxⁱ, with i = 1, 2, …, n, where E represents "any energy transferred between the system and its surroundings by any means other than expressible by work terms", U is the system energy, and n is fixed by the number of independent work terms needed to describe the system. He stresses that such a law is path-dependent and therefore "could not give rise to equations that could specify a path. Another law is needed."

For the second law he rejects the temperature-and-heat-flow statements as hard to generalize and adopts instead Carathéodory's abstract form: "In the neighborhood (however close) of any equilibrium state of a system of any number of dynamic coordinates, there exists states that cannot be reached by reversible E-conservative (δE = 0) processes." This, Williams says, "is the law that houses the notion of irreversibility."

The third law is generalized as: "The generalized entropy of the system, when the integrating factor, φ, becomes infinite, is a constant and may be taken to be zero."

Mechanical entropy and the speed of light

The second law, applied to a system with any number of independent variables, guarantees an integrating factor for the first law, and that factor "is independent of the system and hence is applicable to all systems." Its existence in turn guarantees a differential form dS = δE/φ, where S is a generalized entropy. Restricting attention to the mechanical properties of a system yields what Williams calls a "mechanical entropy" — the central novelty of the paper.

Restricting to mechanical systems produces a second immediate result: the integrating factor φ is "strictly a function of velocity", so there exists a unique velocity that drives it to infinity. Williams identifies this as exactly analogous to absolute zero temperature in thermodynamics, and states that the unique velocity can be shown to be the speed of light c, so that "Einstein's postulate concerning the constancy of the speed of light follows immediately." Imposing the third law alongside the other two then gives the result that a system moving below that unique velocity may never exceed it.

Geometry from the stability conditions

Newton assumed Euclidean geometry; Einstein assumed Riemannian. Williams asks whether the geometry can instead be dictated rather than assumed. He argues it can: the quadratic form arising from maximizing or minimizing a function of several variables "becomes a natural metric describing distances", and the second law supplies just such a quadratic form through its stability conditions. Several candidate forms appear, depending on the choice of independent variables, but imposing the mechanical entropy principle selects one. The arc length of the resulting metric is the mechanical entropy, (dq°)² = h_ijdxⁱdx^j with i, j = 0, 1, 2, 3 and x⁰ = ct.

A second, related metric also appears, coupled to the first by a gauge function f: (dq°)² = f(g_ijdxⁱdx^j) = f(dσ)². Williams reports that the entropy space must be Riemannian while the "sigma" space is a Weyl space. He treats the gauge function coupling the two as "the analog, in differential geometry, to the integrating factor coupling the differential change in entropy to the first law."

Recovering the established theories

With the entropy principle imposed, Williams claims the conditions Weyl set up in his unified theory are completed, so that Weyl's own results may be invoked: variations of the gauge function produce Maxwellian electromagnetism (see Maxwell's Equations), variations of the metric coefficients g_ij produce General Relativity, and letting the coefficients become constants produces Special Relativity. Quantum effects are sketched in a paragraph: since the mechanical entropy may never decrease, either it increases (irreversible process) or it stays fixed (reversible), and a fixed entropy makes the equations of motion null trajectories in the entropy space — a condition London showed in 1927 to produce quantization within the coupled metrics, with the fundamental quantum number specifying the "order" of the null trajectory "just as poles and zeros have order."

The arrow itself, and its cosmology

The pivot of the paper is a single identification: the mechanical entropy is related to relativistic proper time by dq° = c d τ. Because the entropy principle requires dS ≥ 0 for an isolated (δE = 0) system, proper time can never run backwards, and "any system that is the least bit irreversible must forever be growing older."

Applied to the universe as a whole — taken as an isolated system — this gives a set of consequences Williams lists rather than derives in detail:

Since mechanical entropy is a measure of distance in the entropy space and must forever increase, "the scale of the universe must be forever increasing", ruling out any model in which the universe later contracts.
Universal expansion does not oblige every source to show a red shift. The entropy principle applies only to isolated systems, so an individual star interacting with the rest of the universe could in principle show a blue shift — though Williams judges this improbable given irreversibility.
If red shift is partly a measure of irreversibility, added to the gravitational red shift, then the very large quasar red shifts "may be viewed in an entirely new light": they would indicate that "the processes going on within quasars are more irreversible than the processes within other stars."
Local ages need not track cosmic age. Parts of the universe interacting with other parts "may age faster or slower than the universe as a whole", so a geological date exceeding the supposed age of the universe "would not necessarily be an inconsistency."
Heat death is not compulsory. Increased generalized entropy may be taken up as increased mechanical entropy — that is, as expansion — "thus removing the necessity of increasing thermodynamic entropy and a death by fire."

Williams also offers a light-hearted corollary: a man isolated from the rest of the universe must grow older, but "there may be a particular set of interactions between the man and the rest of the universe that would allow time for the man to slow down; perhaps even reverse", by analogy with lowering a thermodynamic system's entropy through an appropriate transfer of heat.

Assessment

What is genuinely attractive here is the economy of the starting point. Williams asks for three laws — a generalized first law, Carathéodory's second law, and a generalized third law — and claims to get back the constancy of c, the light-speed barrier, the geometry of spacetime, electrodynamics, gravitation and a quantization condition, without separately postulating any of them. Deriving the geometry rather than assuming it is a real ambition, and one that few unified programmes even attempt; Newton and Einstein both had to put the geometry in by hand, and Williams is right that a theory which fixes it earns something. The identification dq° = cdτ is also a clean and testable-sounding move: it converts the second law directly into a statement about proper time, so that the arrow of time becomes a theorem rather than a boundary condition. And the entropy-space/sigma-space pair, with a gauge function playing the role the integrating factor plays in thermodynamics, is an elegant structural analogy.

The difficulties are equally plain, and most of them are difficulties of what the report does not do. Almost every load-bearing step is asserted rather than derived. That the integrating factor is "strictly a function of velocity" is stated without argument; that the velocity at which it diverges "may be shown" to be c is stated without the showing; that the entropy space "MUST be a Riemannian space" while the sigma space is Weyl is announced as a result. Williams repeatedly says a demonstration "would lead us too far from the present theme" and refers the reader to his longer LA-8370-MS report. As an expository summary that is a legitimate choice, but it means the paper cannot be assessed on its own; nothing in it can be checked. There are no numbers at all — no computed red shift, no predicted expansion rate, no quasar figure — so there is nothing to compare against measurement.

The physical claims that are made are weakly specified. The proposal that quasar red shifts include an irreversibility term additional to the gravitational red shift is offered qualitatively, with no rule fixing how much irreversibility yields how much shift, and so it makes no prediction that could distinguish it from the ordinary cosmological interpretation. It also sits awkwardly beside the measured (1+z) stretching of Type Ia supernova light curves, which ties red shift to an actual time-dilation factor with a definite coefficient — any additional non-kinematic contribution to z has to leave that relation intact, and the paper does not say how it would. Similarly, the suggestion that a geological date may legitimately exceed the age of the universe is presented as an attractive freedom, but with no mechanism specifying how large the discrepancy may be it is unfalsifiable rather than explanatory; and the reciprocal claim, that the late-time behaviour must be perpetual expansion, is exactly the sort of conclusion for which observational discrimination requires a quantitative model.

There is also a conceptual tension the paper does not confront. Williams recovers Special Relativity as the constant-coefficient limit, which brings with it the ordinary symmetry of the equations of motion under time reversal. He then wants irreversibility to be present in those same equations by virtue of their entropy-space origin. Whether the arrow survives the limit that produces the reversible theory, or is quietly lost in it, is precisely the question at issue, and the report does not address it. Finally, the treatment of the universe as an isolated system — required for dS ≥ 0 to apply — is assumed rather than justified, and the paper's own argument that subsystems can age backwards makes clear how much work that assumption is doing.

The honest summary is that this is a programme statement rather than a result. Read as such, it is clear, candid about its own unverified status (Williams says so in the first sentence of the abstract), and unusually free of overclaiming for its genre. Read as a physics paper, it offers no calculation that can be confirmed or refuted.