Reference-Frame Independent Dynamics, Or How to Get Off Einstein's Train
| Scientific Paper | |
|---|---|
| Title | Reference-Frame Independent Dynamics, Or How to Get Off Einstein's Train |
| Read in full | Link to paper |
| Author(s) | Greg Volk |
| Keywords | Hertzian Electromagnetism, convective derivative, reference frame, Mach's Principle, universal instant, Maxwell's equations |
| Published | 2009 |
| No. of pages | 12 |
Read the full paper here
Abstract
Einstein spent his life trying to develop a system of dynamics independent of reference frame. But this lofty goal demands a hard look at the very meaning of reference frame, a starting point for this paper. Einstein's fruitless attempts were based on his own observer-based theory of relativity, in turn based on Lorentzian covariant paradigms. Covariant mathematics admits transformations between coordinate systems and reference frames, but does not suggest a path towards invariance. However, the del operator provides a means of expressing spatial derivatives independent of coordinate system. Likewise, if time is independent of space, as this paper suggests, the total time derivative operator, d/dt, is also independent of reference frame. Therefore, physical equations that can be expressed with the del operator and the total time derivative, as opposed to reference-frame dependent partial derivatives, are naturally independent of reference frame. In particular, Maxwell's equations can be correct only if expressible by the del and d/dt operators, as proposed by Hertz and more recently advocated by Phipps. From fluid dynamics, developed by Bernoulli, Euler, Langrange, and many others, over a hundred years before Maxwell, we have the convective or Lagrangian time derivative d/dt = d/dt + v • del. Both sides of this equality are independent of observer, but the two right-hand terms differ in weight depending on how an observer moves with respect to the quantity being differentiated. One observer might see a buildup of material, while another sees an altered flow; one observer might see a changing field, another an acceleration. But the total change, the sum, remains invariant. With the convective derivative, we can readily derive Maxwell's famous "displacement current" term in Ampere's Law and clear up mysteries surrounding Faraday's Law, particularily relating to unipolar induction and the Sagnac experiment. Invariant Hertzian dynamics provides a means to finally get off Einstein's covariant train.
Overview
Greg Volk's paper is an argument in two halves that are meant to lock together. The first half is conceptual: it takes apart the notion of "reference frame", distinguishes it sharply from "coordinate system", and argues that the ordinary idea of an inertial frame smuggles in unexamined assumptions about space. The second half is mathematical: it revives the form of Maxwell's Equations proposed by Heinrich Hertz, in which every partial time derivative is replaced by the total (convective) derivative d/dt = ∂/∂t + v · ∇, and shows what happens when the advective term is expanded with vector identities. The bridge between the halves is a single claim: an equation written entirely in ∇ and d/dt is automatically frame-independent, because both operators are "facts of nature" rather than artefacts of a coordinate choice.
The position taken throughout is Machian rather than Einsteinian or absolutist. Volk agrees with Einstein that empty space provides no privileged frame, but rejects the conclusion that motion is therefore observer-relative; instead motion, and with it energy, is determined with respect to matter. He accordingly redefines "at rest" as "experiencing no net force or torque" rather than "not moving with respect to the observer", and re-reads the constant c not as the speed of light — a property of space — but as the "speed of charge", the speed at which the electrostatic repulsion between like elements of matter is balanced by the Amperian attraction between parallel currents. If c is a property of matter, he argues, "the concept of 'space-time' becomes ridiculous", and simultaneity can be restored. The paper acknowledges Thomas E Phipps as the modern champion of the Hertzian formulation and cites Peter Erickson for absolute time.
The argument
The Gedanken-experiment and the universal instant
The paper opens with a deliberately trivial exercise: imagine a point P at time t; now imagine the same point at t '. Volk's point is that step two cannot be taken without smuggling in a reference frame, since "same point" can only mean same with respect to some body of matter. Thirteen numbered conclusions are drawn, the load-bearing ones being that a point in space is meaningful only with respect to matter; that the serial flow of time demands the existence of the "instant", a universal simultaneous snapshot; that the existence of the instant is equivalent to the independence of space and time; that reference frames are spatial coordinate systems that slide with matter over time; and that in a single instant a coordinate system is meaningful but a reference frame is not. Two further items — that we never measure space or time themselves, only compare distances with standard rulers and cycles with standard clocks — return later.
Against the inertial frame
Seven hidden assumptions are listed as concealed in the phrase "inertial frame", among them that an object can be at rest with respect to space itself, that reference frames are linear, and that parallel paths exist. The critique is carried by two examples. A fly on the windshield of a car at 60 mph and the driver behind it are, to an Einsteinian, both "at rest" in the car's frame; but the fly feels the wind and the driver does not, and on the Machian definition that difference is a fact independent of any observer. Conversely a stargazer and a geostationary satellite maintain a constant separation, yet the stargazer feels weight and the satellite is in free fall — so it is the satellite, not the observer, that is "at rest".
From this the paper argues that only orbital or rotational frames are physically realisable, since nothing anywhere is permanently at rest in a linear frame: the Earth orbits, the Sun orbits the galaxy, and so on. Parallel paths, it adds, presuppose infinite or bounded space; a nine-step argument is sketched for finite unbounded space, resting on conservation implying finite content and on the claim that a boundary would define a preferred frame in violation of Mach's Principle. The globe analogy follows: on a sphere there is no qualitative difference between the "translational" path of the equator and the "rotational" path at 89.99° latitude, so in closed space translation and rotation are the same thing seen differently. Newton's first law is then restated twice, first as "objects at rest in some frame stay at rest in that frame" and finally as "objects in stable orbits stay in orbit".
Partial derivatives and the convective derivative
The mathematical section rests on a claim about partial derivatives: since a coordinate such as x is not a property of nature, the total derivative with respect to x is meaningless, while partials are meaningful only in concert — "any isolated partial derivative is physically meaningless". Time, however, is a fact of nature if it is independent of space, so d/dt is meaningful. Euler's convective (Lagrangian, hydrodynamic) derivative
df/dt = ∂f/∂t + (v · ∇)f
is presented as the reconciliation: each term on the right is observer-dependent, their sum is not. In the fly example the friend at the roadside sees all the change in the ∂/∂t term, the driver sees all of it in the advective term, and both agree on the total.
Hertz's Maxwell equations and the advective expansion
Maxwell's equations are then written with script fields and total time derivatives — ∇·D = ρ, ∇×H = dD/dt, ∇·B = 0, ∇×E = −dB/dt — the script fields being declared the true invariant ones, from which the observer-dependent D, H, B, E are to be derived. Standard vector identities for ∇×(A×v) and ∇(A·v) give two inequivalent expansions of the advective term; applying one to Ampère's and Faraday's laws and setting J ≡ ρv recovers the familiar textbook forms once ∇·v and the dyad ∇v are set to zero, together with the low-velocity field relations D = D script − (v×H)/c2 and B = B script + (v×E)/c2, with terms of order β2 discarded.
Volk's emphasis falls on what has to be thrown away. Setting ∇·v and ∇v to zero, he argues, is the assumption of an inertial frame: it asserts that v is the same everywhere, whereas on a matter-based view v is measured against a proper frame that differs from point to point — the Earth for the fly, the car for the driver. Those discarded terms are where he expects the interesting physics to be: ∇v near the surface of a moving solid, ∇·v in vortex expansion and compression, and both in the Sagnac and Michelson–Gale experiments, "where the mirrors do not all reside in the same gauge". A closing section reads Ampère's and Faraday's laws as statements that motion determined through time alone (the dD/dt side) must equal motion determined through space alone in the universal instant (the ∇×H side), and as expressing the correspondence between translation (D, E) and rotation (H, B).
Assessment
The paper is unusually clear about what it is doing, and its central methodological observation is sound and worth having: the convective derivative genuinely does partition a single invariant rate of change into two observer-dependent pieces, and the habit of writing ∂/∂t in field equations does presuppose a choice about what is held fixed. The distinction between coordinate system and reference frame — the latter being a coordinate system that "slides with matter over time" — is a useful one that many textbooks blur. The fly-and-driver and stargazer-and-satellite examples make the Machian definition of rest vivid, and the point that free fall, not coordinate stillness, is the physically distinguished condition is one that general relativity itself endorses, though the paper does not say so. Reviving Hertz's formulation is a legitimate research programme with a real history behind it, and Volk represents Phipps's version of it accurately.
The difficulties are of two kinds. The first is that a load-bearing identification in the electrodynamic section is simply mislabelled. What "spills out" of the advective term when [5]–[8] are expanded is the term (∇·D)v = ρv ≡ J — a convection current density, the flow of charge. That is not Maxwell's displacement current, which is the ∂D/∂t term and which was already present in the equation before the expansion began. The abstract's promise to "derive Maxwell's famous 'displacement current' term" is therefore not delivered by the calculation shown; what is exhibited is the well-known fact that a static charge distribution seen from a moving frame looks like a current. Since this is offered as the concrete pay-off of the whole apparatus, the confusion matters.
The second is that almost every specific claim is deferred. Unipolar induction and the Sagnac experiment appear in the abstract as things the method will "clear up", but neither is calculated anywhere in the paper; the ∇v and ∇·v terms are said to "account for many 'anomalies'" without a single worked example; the assertion that "all the so-called relativistic solutions of electrodynamics are derivable from Hertzian total time derivatives" is postponed to "a subsequent paper". This leaves the historic objection to Hertz's formulation untouched. Because the total derivative convects the field with the matter that defines the proper frame, Hertzian electrodynamics implies complete entrainment of light by a moving medium; that prediction is contradicted by Fizeau's 1851 measurement of the drag coefficient in flowing water (1 − 1/n2, not 1) and by Airy's water-filled-telescope test, in which the aberration angle of starlight is unchanged by filling the tube. Phipps has proposed answers by reinterpreting v as a detector velocity; Volk gestures at the same move by making the proper frame local and matter-defined, but offers no quantitative treatment of either experiment, and without one the reinterpretation cannot be assessed.
Two smaller lapses are worth noting because they are checkable. The argument from Fermat's principle — that light "follows only one path, the path of least time", so we "will never see light from the same source arriving via different paths" — is false as stated: Fermat's principle selects stationary, not uniquely minimal, paths, and multiple imaging of a single source is directly observed in gravitational lensing, beginning with the twin quasar QSO 0957+561 in 1979 and now catalogued in hundreds of systems, several with measured time delays between the images. And the closing appeal to Peter Woit's Not Even Wrong is decorative: that book is a critique of string theory and says nothing about inertial frames. The redefinition of c as a balance between electrostatic repulsion and Amperian attraction is likewise asserted without derivation, though it is the hinge on which the rejection of spacetime turns. As a programmatic essay setting out why one might want a matter-referred, Hertzian electrodynamics, the paper succeeds; as a demonstration that such a theory works, it defers all the load to work not shown.