Radiacion Electromagnetica

Scientific Paper
Title	Radiacion Electromagnetica
Read in full	Link to paper
Author(s)	Zbigniew Oziewicz
Keywords	Maxwell theory, special relativity, electric, magnetic, and electromagnetic fields
Published	2007
No. of pages	33

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Abstract

Presented at International Conference on Applied Analysis, Querétaro, 2007. Segundo Congreso Cientifico Tecnologico, Cuautitlan 2007. A solution of the Maxwell differential linear equations, the electric and magnetic fields, is said to be the electromagnetic radiation, if and only if there is a transport of energy, i.e. if the Poynting vector does not vanishes in no-one reference system. It is known that this is the case if and only if holds the two non-linear algebraic conditions, E.B = 0, and, E² = B². It is proposed to solve first the non-linear algebraic equations, and after look for solutions of the linear differential Maxwell's equations. In this it is shown that each electromagnetic radiation needs no more than two scalar fields introduced by Robert Yamaleev in 2005. These scalar fields are conceptually different from introduced by Edward Whittaker in 1904, and are distinct from Peter Debye potentials

Overview

Written in Spanish and presented in 2007 at the International Conference on Applied Analysis in Querétaro and at the Segundo Congreso Científico Tecnológico in Cuautitlán, this paper by Zbigniew Oziewicz (UNAM) makes a single sharp point about what electromagnetic radiation is. Textbooks, he argues, define radiation as a solution of the wave equation far from its sources — he cites Landau and Lifshitz and Jauch and Rohrlich by name — and this misses the property that matters: radiation must carry energy. A field carries energy only if the Poynting vector fails to vanish in every reference frame, and that is not a differential condition but an algebraic one.

The two conditions, E·B = 0 and E² = B², are quadratic and therefore non-linear. Oziewicz calls them "the exterior Plücker equations" and attributes their recognition to Oliver Heaviside (1893), with later appearances in Lightman et al. (1975) and Choquet-Bruhat et al. The methodological proposal that gives the paper its structure is to invert the usual order: solve the non-linear algebraic Plücker conditions first, determining the most general absolute field satisfying them, and only then impose the linear differential Maxwell equations. Carried through, this shows that any electromagnetic radiation is described by exactly two scalar fields — the potentials guessed by Robert Yamaleev in 2005, which Oziewicz here motivates and derives rather than guesses, and which he is careful to distinguish from Whittaker's 1904 pair and from Debye potentials.

A second, pedagogical purpose runs alongside. The paper is addressed in part to engineering teachers and students, who are told that relativity may be ignored because one frame — the Earth — suffices. Oziewicz's counter-argument is that relativity does not complicate electromagnetism but simplifies it: the mathematical structure, the understanding of the four Maxwell laws, and the theory of radiation all become clearer, and "electromagnetism memorised artificially can be understood with clear reasons." Eight appendices (A–H) supply the Grassmann algebra, metric tensor, orientation and pseudo-tensors, Hodge star, and coordinate-free differential operators the argument needs.

The argument

Electric and magnetic fields are relative

The paper opens by crediting Oliver Heaviside with showing in 1888 that E and B depend on the choice of reference frame — a result Oziewicz places before Einstein 1905 and Minkowski 1908, the latter of which he calls "less known, but clearer and deeper." With two observers, Rosa (R) and Pedro (P), moving perpendicular to the observed fields, Heaviside's formulas read

γ_v = 1/√(1 − v²/c²), E(P) = γ_v{E(R) + (v/c) × B(R)}, B(P) = γ_v{B(R) − (v/c) × E(R)},

and Oziewicz stresses that they hold strictly only when v·E = v·B = 0. An observer at rest with a charge sees E ≠ 0 and B = 0; a moving observer sees both. The moral he draws is stark: "without a choice of reference frame, without an observer, we do not have electric and magnetic fields."

That raises the question the rest of the section answers: if there are no observers and no laboratories, is there no electromagnetism? Oziewicz postulates an absolute electromagnetic field, independent of observers and dependent only on its sources — the bivector Minkowski introduced in 1908, universally miscalled "the Faraday tensor" F although "Michael Faraday did not introduce it." Following his own programme he calls the resulting theory a relativity groupoid rather than a group, to distinguish it from Einstein's Lorentz-group relativity in which F is Lorentz-covariant and therefore frame-dependent (he cites Landau and Lifshitz's postulate that the potential A, with F = dA, is Lorentz-covariant). Fortunately, he notes, this metaphysical choice does not matter for radiation: Heaviside's theorem (5) is derivable in both theories, and the predictions differ only when v·E ≠ 0 or v·B ≠ 0.

Minkowski's definitions then read, for a unit timelike observer field with (Obs)² = −1: E(F, Obs) ≡ (Obs)·F and B(F, Obs) ≡ ⋆{(Obs)∧F}, with F² ≃ E² − B² and F·(⋆F) ≃ 2B·E. Oziewicz deliberately writes ≃ rather than =, because the left side is postulated observer-independent while the right side is explicitly observer-dependent. Following Rainich, Pleban'ski and Synge, F is electric if F² > 0, null if F² = 0, magnetic if F² < 0; pure (decomposable) if F∧F = 0.

When the Poynting vector cannot be made to vanish

The technical core is a small, clean calculation. Ask: given a field seen by Rosa, can some other observer Pedro see zero Poynting vector? Any relative velocity orthogonal to E, B and the Poynting vector must have the form v = f·(E×B). Substituting Heaviside's transformation into Pedro's Poynting vector gives a quadratic in the scalar f,

f²·(E∧B)² = 1 + f·(E² + B²), Δ ≡ (E² − B²)² + 4(E·B)²,

whose solution is v²/c² = (E² + B² ± √Δ)/(E² + B² ∓ √Δ). The frame that would kill the energy flux therefore requires |v| = c precisely when Δ = 0. Since no material observer can move at c, Δ = 0 — that is, E·B = 0 together with E² = B² — is necessary and sufficient for the Poynting vector to be non-zero in every frame. In absolute language this is the Plücker pair F∧⋆F = 0 and F∧F = 0: the field is pure and null. A pure null bivector factorises as F = k∧l with k² = 0 = k·l, where k is the wave form (a Killing vector of the radiation's symmetry) and l, together with the m in ⋆F = k∧m, describes polarisation. Rosa's electric field of radiation is E(R) = (k·R)l − (l·R)k, with k·R the frequency she measures.

The consequence Oziewicz emphasises repeatedly is that radiation is a non-linear phenomenon even though Maxwell's equations are linear: "the superposition of two radiations is not radiation."

Maxwell's four laws, and the Lorenz gauge

Two reorganisations are proposed. First, the four laws collapse into two absolute statements: magnetic Gauss plus Faraday, each frame-dependent, are together equivalent to F being irrotational, dF = 0 ⟺ F = dA; electric Gauss plus Ampère-Oersted are two consequences of the absolute conservation of charge-current, δJ = 0, with δF = 0 in the source-free case. Second, and more polemically, the wave equation is not a consequence of Maxwell's equations alone. The d'Alembert operator is the square of a Dirac operator, △ ≡ (grad + div)², and Maxwell gives only "half" of it; to obtain a genuine wave equation for the potential one must add Ludwig Lorenz's 1867 condition. Oziewicz's verdict is blunt: the Lorenz gauge, "like any other, for example the Coulomb gauge, is not a law of physics", and it is a source of trouble in the treatment of photon radiation.

The two scalar fields

The final step invokes Darboux's classification of one-forms in four dimensions. Of the four Darboux classes, only classes 2 and 3 satisfy F∧F = 0; combined with dF = 0 this yields (Darboux 1887; Stachel 1969) the existence of two scalar fields with

F = dφ ∧ dψ.

Radiation is then completely characterised by the differential equation δ(dφ∧dψ) = −(△φ)dψ + (△ψ)dφ + [dφ, dψ] = 0 together with the algebraic null condition {(dφ)·(dψ)}² = (dφ)²(dψ)². Relative to an observer field R, the measured fields are

E(R) = (Rφ) grad ψ − (Rψ) grad φ, B(R) = (grad ψ) ×_R (grad φ),

which are exactly Yamaleev's 2005 expressions — but Yamaleev, Oziewicz notes, guessed them for the inertial observer R = ∂_t, whereas here they are derived and hold for a general observer field. Oziewicz adds a caveat with teeth: the familiar magnetic Gauss law divB(R) = 0 holds strictly only for inertial (holonomic, integrable) observers, div{B(R)} = 0 ⟺ dg_R = 0. He also observes that Maxwell's equation (34) makes the two-dimensional distribution gradφ∧gradψ involutive, so by Frobenius' theorem an integral two-surface exists.

Assessment

The central observation is correct, elegant, and better known to relativists than to the electrical engineers the paper addresses. That the null and pure conditions on the electromagnetic bivector are algebraic and select radiation from among all wave-equation solutions is standard in the Rainich-Misner-Wheeler and Petrov-classification literature, and Oziewicz's derivation of it — computing the velocity that would annihilate Pedro's Poynting vector and finding it forced to c exactly when Δ = 0 — is a genuinely instructive route to a result usually stated as a definition. The consequence he draws, that superposing two radiations does not in general give radiation, is a real and underemphasised point: the linearity of Maxwell's equations does not descend to the subset of solutions that transport energy in every frame. Reducing that subset to two scalar potentials via Darboux's classification is clean, and supplying a motivation and proof for what Yamaleev had guessed, while generalising it off the inertial observer, is a definite contribution. The historical corrections are also well made and well documented: Heaviside's 1888 priority on field relativity, and the fact that the "Faraday tensor" is Minkowski's 1908 bivector.

The reservations are mostly about scope and framing. The paper is a conference presentation with roughly half its length given to appendices on Grassmann algebra, the Hodge star and coordinate-free operators, so the new material is compressed; some steps are stated as exercises with answers rather than worked, and the reader is sent to Cruz Guzmán and Oziewicz (2003) for the reformulation of Maxwell's laws without the curl operator. The "relativity groupoid" postulate — that F is absolute rather than Lorentz-covariant — is introduced with some fanfare and then explicitly set aside as irrelevant to radiation, since Heaviside's transformation follows in either theory and the two differ only when v·E ≠ 0 or v·B ≠ 0. That is honest, but it means the paper's most contentious claim is never tested here; and the case where the predictions do differ is exactly the case Heaviside's formulas as quoted do not cover, so no comparison with measurement is offered. Given that the Lorentz transformation of fields for arbitrary v is confirmed daily in accelerator and synchrotron practice — the relativistic beaming and the E = γ(E + v×B) scaling underlying every storage-ring design — a theory whose predictions diverge from it for non-perpendicular velocities carries a substantial empirical burden that the paper does not take up.

The attack on the Lorenz gauge is likewise half an argument. It is quite true that a gauge condition is a mathematical convenience and not a physical law, and true that Maxwell's equations by themselves give (div∘grad)A = 0 rather than a full wave equation. But gauge freedom is precisely why this is unproblematic: the physical content lies in F, which the paper itself treats as primary, and no measurable prediction depends on the choice. Saying the Lorenz gauge "is a source of problems for the treatment of photon radiation" without naming a problem leaves the charge unsupported. Finally, the paper is entirely formal — there is no new prediction, and none is claimed. Judged as what it is, a clarification of a definition plus a derivation of a two-potential representation, it is sound and worth reading; the polemical framing around groupoids and gauges is separable from that result and does not affect it.

See also

• Zbigniew Oziewicz
• Oliver Heaviside
• Hermann Minkowski
• Maxwell's Equations
• James Clerk Maxwell
• Electromagnetism
• Special Relativity
• Lorentz Transformation
• Light
• Albert Einstein