Principles of a Frame Indifferent Classical Electromagnetic Field Theory
| Scientific Paper | |
|---|---|
| Title | Principles of a Frame Indifferent Classical Electromagnetic Field Theory |
| Read in full | Link to paper |
| Author(s) | Burak Polat |
| Keywords | electromagnetic field, Maxwell equations, frame indifference, comoving time derivative, continuum mechanics, moving media |
| Published | 2011 |
| No. of pages | 21 |
Read the full paper here
Abstract
In this three part investigation we provide the mathematical foundations and principles of a frame indifferent classical electromagnetic field theory (FIEFT) for arbitrarily moving material media with arbitrary constitution based on convective and comoving time derivative operators. Part 1 is devoted to the mathematical tools utilized in establishing the field theory. It starts with the description of material points in arbitrary Euclidean motion, which is a characteristic of rigid (non-deforming) bodies and incompressible inhomogeneous fluids in continuum mechanics. Next we establish the mathematical link between spatial and time derivatives of vector fields between Eulerian and Lagrangian frames via coordinate transformations in Euclidean space. Regarding the images of time derivatives of field quantities, we necessarily invoke the convective and comoving time derivatives. We also provide a proof of the representation of the comoving time derivative for scalar and vector density fields along with its certain differential, commutative and integral properties. In Part 2 we provide the axiomatic structure of our field theory where the frame indifferent electromagnetic field equations are obtained directly as images of Maxwell equations of stationary media under Euclidean (aka observer) transformations. The commutative properties derived between spatial differential and comoving time derivative operators help us derive progressive wave equations for the two standard (translational and rotational) types of Euclidean motion. In Part 3 we describe the general formulation of a boundary value problem for an arbitrarily moving object and investigate three canonical problems of practical interest to demonstrate the predictions of FIEFT.
Overview
Burak Polat, an electrical engineer at Trakya University in Edirne, Turkey, sets out to build a complete classical electrodynamics of moving bodies without using the Lorentz Transformation. His starting point is the principle of "material frame indifference" (MFI) from rational continuum mechanics: the structural form and physical content of a physical law, subjected to arbitrary coordinate transformations, should not depend on quantities that merely define the geometry of the underlying space-time manifold. Polat notes explicitly that this programme "clearly contradicts" the worldview of Special Relativity: where relativity favours general covariance, form invariance, Minkowski space-time and Lorentz transformations, frame indifference in the continuum-mechanical tradition favours general invariance, frame indifference, Newtonian space-time and Euclidean (observer) transformations.
The resulting frame indifferent electromagnetic field theory (FIEFT) is constructed from five postulates. The decisive one is Postulate 1: between a laboratory ("Eulerian", E-) frame and a frame comoving with the material ("Lagrangian", L-), "no time dilation or length contraction is assumed" for measurements by ideal devices. Time is treated as a non-physical parameter, t = t' , and the Euclidean metric is the same in both frames. Everything relativistic is thus removed from the kinematics at the outset; the whole burden of describing motion is shifted onto the time derivative operator. The paper's central claim is that if one replaces the partial time derivative of the stationary-media Maxwell equations by the comoving (Oldroyd) time derivative, the resulting equations are the correct field equations for a medium in arbitrary motion, and they retain the exact form of the stationary ones. The work is presented as an extension to arbitrary constitutive media of a vacuum formulation given by C. I. Christov, whose "meta-continuum" programme treats the electromagnetic field as an incompressible viscoelastic material continuum.
The argument
Part 1: kinematics and the two progressive derivatives
The mathematical apparatus rests on two operators. The convective (substantial) derivative is the familiar
Dg/Dt = ∂g/∂t + v · grad g
established by a Taylor expansion of g(r(t+Δt); t+Δt) about r(t). The comoving derivative ∂g/∂t (written with a special symbol in the paper) is defined by a limit taken on the deformed medium, and Polat shows it is precisely the image of the L-frame partial time derivative under the coordinate map. For a scalar density field this gives
∂g/∂t = Dg/Dt + g div v
and for a vector density field
∂A/∂t = ∂A/∂t + v·grad A − A·grad v + A div v.
The proof is elegant and is the technical core of Part 1: instead of the usual limit argument, Polat imposes conservation of a volume integral, ∫g dΩ = ∫g 'dΩ', expands the Jacobian of the deformation gradient as J = 1 + Δt div v + o(Δt), and reads off the transformation rule. He identifies the operator with Oldroyd's upper-convected derivative from rheology, and remarks that it is the only member of the family of invariant time derivatives that "correctly postulates field equations" in both continuum mechanics and the electromagnetism of moving bodies. Ten differential properties (O1–O10), surface analogues, and the Reynolds and Helmholtz transport theorems are then derived as consequences.
Part 2: the field equations
Two commutation theorems do the real work. Divergence commutes cleanly with the comoving derivative; curl does not, in general — Theorem 8 produces a "sophisticated structure" that Polat concedes is "impractical in obtaining FIEFE in the most general case". Restricting to Euclidean motion (rigid-body translation and rotation, and incompressible inhomogeneous fluids) restores a simple commutation rule, and this restriction is what makes the theory workable.
With Postulate 4 — "the laws of macroscopic electromagnetism are frame indifferent" — the stationary Maxwell equations map directly into the E-frame with the partial time derivatives replaced by comoving ones: curl E + ∂B/∂t = 0, curl H − ∂D/∂t = JC, div D = ρf, div B = 0, together with the continuity relation. The constitutive relations carry over unchanged, and the convective current JV = ρfv appears in the E-frame total free current, as it must. Integral forms of the electromotive and magnetomotive force follow, reproducing the motional terms.
For pure translation the comoving derivative reduces to the convective one and the wave operator becomes LD = lap − με D2/Dt2 − μσ D/Dt. Polat notes these are "the Hertz equations", the point Heinrich Hertz had reached in his 1890 paper before his death in 1894.
The velocity discriminant
In one spatial dimension the operator becomes
(μεv2 − 1)∂2/∂x12 + 2μεv ∂2/∂t∂x1 − με ∂2/∂t2 + ...
whose discriminant is Δ = 4με > 0, the same value as in the stationary frame. Two conclusions are drawn: the operator stays hyperbolic at any material speed, and v2 = 1/με — the speed of light in the medium — is the critical value at which wave propagation breaks down. Polat then observes that nothing in the construction forbids μεv2 − 1 from being negative, "which might address a possibility of speeds of material points faster than the speed of light in the same simple medium".
Part 3: three canonical scattering problems
Boundary-value problems are solved by "frame hopping": map the incident field E→L, solve the ordinary stationary problem, map the scattered field L→E. Postulate 5 (after İdemen, 1973) asserts that the Maxwell equations hold in the sense of Schwartz–Sobolev distributions, which supplies the jump and edge conditions.
- Uniform motion of a dielectric half-space under TM plane-wave incidence. The mapping yields cinc' = c cosθ / (cosθ − G/c) and ωinc' = ωinc(1 − βcosθ), with reflected and transmitted frequencies ωsc = ωinc(1 − 2βcosθ) and ωtr = ωinc(1 − βndcosθ) — "the famous Doppler effect" in a first-order, non-relativistic form. The condition β < cosθ appears as "a physical limit on G for the realization of scattering phenomenon".
- Harmonic motion of the same half-space, v = Gcos(Ωt)x1. Using the Bessel identity eiαsinΩt = ΣJm(α)eimΩt, the incident wave decomposes into an infinite comb of sidebands of amplitude Jm(α) and frequency ωinc + mΩ, with α = Gkcosθ/Ω.
- A rotating PEC cylinder under TE incidence, solved by separation of variables and Bessel functions. The notable result is negative: the total scattered field is monochromatic at the incident frequency and "independent of the frequency of rotation, coinciding with the result for the stationary case".
Assessment
The paper's real strength is its mathematical hygiene. Polat does not hand-wave about "the ether frame"; he states his postulates, proves his operator identities, and shows exactly where his construction fails to generalise. The admission that curl and the comoving derivative do not commute for arbitrary velocity fields — and the consequent restriction of the theory to Euclidean motion — is stated openly rather than buried. The unified proof of Theorems 4 and 5 via Jacobian conservation is clean and independently useful, as is the demonstration that the wave operator's discriminant is a frame invariant. The tutorial register, aimed at electrical engineers with no continuum-mechanics background, is a genuine service, and the boundary-value problems are worked to closed form rather than gestured at.
The central conceptual claim is nevertheless asserted, not established. Postulate 1 simply declares that there is no time dilation or length contraction; the paper offers no argument for this beyond its convenience within the continuum-mechanical framework, and no engagement with the measurements that motivate the relativistic kinematics — the muon lifetime dilation in cosmic-ray and storage-ring experiments, the Ives–Stilwell transverse Doppler measurements, or the Hafele–Keating and GPS clock-rate results. Likewise, Postulate 4 asserts frame indifference of electromagnetism rather than deriving or testing it. The theory is therefore best read as an exploration of what follows if relativistic kinematics is set aside, not as a refutation of it — and Polat's own conclusion, which proposes comparing FIEFT's answers with relativity's "both conceptually and numerically" for the same boundary-value problems, concedes exactly this: the comparison has not yet been made.
Where such a comparison would bite is precisely in the results of Part 3. The Doppler shifts obtained are first-order in β with no transverse (second-order) term; special relativity predicts a measured second-order shift, and that term has been confirmed since Ives and Stilwell in 1938. The paper does not address the discrepancy, or note that its own expressions differ from the relativistic ones at order β2. The condition β < cosθ, presented as "a physical limit ... for the realization of scattering", is a limit of the formalism rather than a derived physical bound, and it sits awkwardly beside the later suggestion that superluminal material speeds may be admissible. That suggestion is itself made in passing, on the strength of a sign in a discriminant, without any discussion of what a hyperbolic operator with negative leading coefficient would mean physically.
Finally, the parts do not carry equal weight. Part 1 is a rigorous piece of applied mathematics that would stand on its own in a continuum-mechanics setting. Parts 2 and 3 depend on it, but their physical content rests entirely on the four postulates, and the third canonical problem — the rotating cylinder — returns the stationary answer, so it discriminates between FIEFT and any rival theory not at all. Within its own terms, however, the construction is consistent, and its debt to Christov's material-continuum programme is properly acknowledged.