Electric and Magnetic Fields: Do They Need Lorentz Covariance?
| Scientific Paper | |
|---|---|
| Title | Electric and Magnetic Fields: Do They Need Lorentz Covariance? |
| Read in full | Link to paper |
| Author(s) | Zbigniew Oziewicz |
| Keywords | Lorentz covariance, magnetic fields, Relativistic Dynamics, Maxwell equations, groupoid category |
| Published | 2011 |
| Volume | 330 |
| Number | 012012 |
| No. of pages | 29 |
Read the full paper here
Abstract
7th Biennial Conf. on Classical and Quantum Relativistic Dynamics of Particles and Fields. Electric and magnetic fields are relative. They depend not only on a choice of electromagnetic sources via Maxwell equations, but also on a choice of observer, a choice of material reference-system. In 1908 Minkowski defined electric and magnetic fields on a four-dimensional spacetime, as tensorial concomitants of observer. Minkowski defined Lorentz-group-covariance of concomitant tensor field as group-action that commute with contractions. Present-day textbooks interpret Lorentz-group-covariance of concomitant tensor differently than Minkowski in 1908. In 2003-2005 Tomislav Ivezić re-invented Minkowski's group-covariance. Different interpretations of group-covariance, lead to different relativity transformations of electric and magnetic fields.
An objective of present article is to explore third possibility, implicit in [Minkowski 1908, §11.6], where a set of all relativity transformations of all material observers forms a groupoid category, which is not a group.
Overview
This is a mathematical-physics paper, published in the Journal of Physics: Conference Series for the 7th Biennial Conference on Classical and Quantum Relativistic Dynamics of Particles and Fields, and it is unusually technical for this archive. Zbigniew Oziewicz argues that the transformation law for electric and magnetic fields between two moving observers is not settled physics but depends on a choice about what "Lorentz covariance" means — and that three inequivalent answers are available, all of which agree at first order in u/c and differ only at second order.
The paper's historical claim is that Hermann Minkowski's last paper of 1908 contains a definition of electric and magnetic fields as concomitants of an observer — one-forms built from an absolute electromagnetic biform F and a time-like observer field — together with a definition of group-covariance as a group action commuting with contractions, and that both were lost when Max Born rewrote the material in 1910 with the emphasis placed on Lorentz-group covariance. Oziewicz credits Tomislav Ivezić with rediscovering Minkowski's notion of covariance in 2003–2005, and then argues for a third possibility that he says is implicit in Minkowski §11.6: that the set of all transformations between material reference systems is not a group at all but a groupoid category. His departure from the mainstream is therefore structural rather than empirical — the claim is that the Lorentz group was mistakenly imported into a role (permuting material observers) that it was never shown to occupy, and that it should be replaced by a weaker algebraic structure in which relative velocity is unique and reciprocity fails.
The argument
The symmetry group of Maxwell's equations is not the Lorentz group
Oziewicz opens with a historical point. In 1904 Lorentz and Poincaré identified the symmetry group of Maxwell's equations with the isometry group of the metric tensor. In 1909–1910 Bateman and Cunningham independently showed that this is only a subgroup: the actual symmetry group of the massless Maxwell equations is the 15-dimensional conformal group. Yet the conformal group has never been considered a group permuting material reference systems. Oziewicz draws the moral that the identification of the isometry group with the group of relativity transformations was an assumption, not a derivation.
He then argues that the domains differ. The Lorentz isometry must act on all vectors, including light-like ones; but light-like massless radiation cannot be a reference system, and so must lie outside the domain of transformations between material observers. It follows that "a set of all transformations among all material reference systems do not need to be postulated a priori to be the Lorentz isometry-group."
What a groupoid is
A group has a unique neutral element; a groupoid category need not. In Lorentz relativity, the zero velocity of Earth relative to Earth and that of Sun relative to Sun are identified as the same neutral boost, whereas in groupoid relativity 0Sun ≠ 0Earth. Second, group composition is global: any two elements compose. In a groupoid, arrows P → Q and R → S need not be composable when Q ≠ R and S ≠ P. Oziewicz notes that Einstein's second postulate — light speed independent of source motion — follows equally from groupoid relativity, so it cannot be used to infer the Lorentz transformations: "Einstein postulated group structure." Equivalently, group structure follows from postulating reciprocity of relative velocity, that the inverse of v is −v, and reciprocity is exactly what fails in the groupoid.
Observer as monad, not tetrad
A long methodological section insists that physics is basis-free and coordinate-free. Oziewicz rejects the vocabulary of "3D" and "4D" quantities outright: a vector does not have a dimension, only a manifold or vector space does, and a time-dependent field E(r,t) is a vector field on four-dimensional spacetime however many of its components happen to vanish in some basis. What is really being said when E is called three-dimensional is that it is observer-constrained, E·P = 0 — and, he remarks, "nobody call E to be 'nonrelativistic' because exists another vector P orthogonal to E."
This underwrites the paper's central contrast between two models of an observer. Einstein in 1905 identified a physical reference system with a coordinate basis, a tetrad. Oziewicz objects that neither coordinates nor bases involve the physical concept of a material body: there is no inertial mass in a frame. The alternative, traced to Euler's fluid of 1754 and reinvented by Minkowski in 1908 (and later by Eckart, Ehlers and Zelmanov), is the monad — a normalized time-like vector field, P2 = R2 = −1, carrying a mass density. Space is then not a fibre but a quotient, Space ≡ Spacetime/material body, and Time ≡ Spacetime/convention of simultaneity. He also insists that a groupoid observer need not be inertial: the condition ∇P = 0 is a differential condition that, he notes acidly, is hard to find actually imposed in the special-relativity literature.
Minkowski's definition and the cross product in four dimensions
Minkowski's 1908 definition, which Oziewicz numbers 7.1, is
- E(F, Paul) ≡ P·F, B(F, Paul) ≡ ⋆(P ∧ F)
so that F is absolute and observer-free while E and B are derived, observer-relative concomitants. The textbook matrix "definition" of Fαβ in terms of E and B — which he traces through Sommerfeld, Landau and Lifshitz, Møller, Fock, Tonnelat, Jackson and Barut — reverses the dependence, making the absolute object a function of the relative ones. Minkowski's own decomposition F = E ∧ P − iP⋆B is a theorem, he stresses, not a definition, and the observer hidden inside the matrix is the source of much confusion, including, he suggests, Dirac's magnetic monopole reading of Maxwell's equations.
A necessary technical ingredient is that the Gibbs cross product does not exist as a binary operation in four dimensions. Oziewicz defines A ×P B ≡ ⋆(A ∧ P ∧ B), a ternary operation depending on an auxiliary vector field, and argues that this observer-dependence of the cross product is "either not realized or thoughtlessly suppressed" in textbook presentations of both the field transformations and the ponderomotive force.
Three inequivalent transformation laws
The heart of the paper is the exhibition of three different answers to a single question: how are the fields measured by Paul (monad P) and Rose (monad R = γ(P + u/c)) related?
(i) Ivezić's Lorentz-covariant transformation. If both F and the observer are Lorentz-covariant, one obtains
- EI = E + γ(E·u/c){P + γ/(γ+1) u/c}
with the corresponding magnetic expression. Oziewicz derives it from a Cartan bivector-generated isometry and checks it against Ivezić's Clifford-rotor derivation. Its corollaries are that (EI)2 = E2, that EI·BI = E·B, and — the diagnostic feature — that EI is no longer orthogonal to the first observer, P·EI = −γE·u/c.
(ii) Fixed-observer ("Æther") transformation. If instead the observer is held fixed and only F transforms, one recovers exactly the familiar textbook law of Pauli, Sommerfeld, Fock, Jackson and Landau–Lifshitz:
- EL = γ{E + (u/c) ×P B} − [γ2/(γ+1)]{(u/c)·E} u/c
Oziewicz insists this "must not be called the relativity transformation", since only one reference system is present and the two fields are different electromagnetic fields due to different sources. The transformed field is necessarily orthogonal to the fixed observer, and "absolute Æther-observer P and his proper-time are intact." He also notes that John Field's 2006 group-free derivation for a moving charge gives the coefficient γ rather than γ2/(γ+1).
(iii) Groupoid transformation. Taking F as absolute and letting only the observer change by the groupoid arrow P → R = γ(P + u/c), Minkowski's own equations (47–48, 51–52) give
- EM = γ{E + (u/c) ×P B} + γ{(u/c)·E} P
The only visible difference from (ii) is that the (u·E) term is time-like rather than space-like. Oziewicz shows directly that this map is not an isometry: for two monads with P·u = Q·u = 0, the product P·Q goes to γ2P·Q + γ2 − 1 ≠ P·Q.
The predicted differences
The three laws are then compared numerically. For the projection of the transformed field on the original, with β ≡ u/c,
- E(u)·E − E2 = ½ × { β2E2 − (E·β)2 for fixed Æther; (E·β)2 for Ivezić; β2E2 for groupoid }
so the three differ at order β2. A second prediction concerns convection current. Transforming the charge and spin densities gives ρ(R) = γ{ρ(P) + (u/c)·s(P)} — which agrees with the Lorentz result — but the current density transforms differently, and when s(P) = 0 the magnitude of the pure convection current is (γ2 − 1)ρ in the groupoid but √(γ2 − 1)ρ under the Lorentz group. Since for small u/c the former is ≈ |u/c|2 and the latter ≈ |u/c|, groupoid relativity predicts a substantially weaker convection current, the two agreeing only at u = 0 and u/c = 1/√2. Oziewicz also derives an observer-free ponderomotive form with three terms, J·F = −ρE + s·B + (Es)P, the third arising because the magnetic field does not interact with scalar charge density while the electric field interacts with both charge and magnetic spin.
Assessment
The paper is careful, technically literate, and unusual in this archive for making a difference that is in principle measurable rather than merely interpretive. Several of its criticisms of standard exposition are correct and worth stating. The observation that the Bateman–Cunningham conformal group, not the Lorentz group, is the symmetry group of the source-free Maxwell equations is simply true and is genuinely under-emphasised in textbooks. The insistence that E and B are observer-dependent quantities extracted from an absolute F, rather than primitive objects from which F is assembled, is Minkowski's own position and is also the modern differential-geometric one; the matrix "definition" of Fαβ really does invert the logical order. The point that a binary cross product does not exist in four dimensions, so that the familiar u × B silently carries an observer, is correct and rarely made. And the tetrad/monad distinction — that a coordinate frame contains no mass and so is a poor model of a material body — is a substantive conceptual point with a real pedigree in Euler, Eckart, Ehlers and Zelmanov.
The central weakness is that the paper's own key theorem is not proved. Oziewicz says so himself, in unusually frank terms: of Theorem 11.3 he writes, "Our 'proof' is not conceptually correct, because we use covariance as Grassmann algebra map ... and we only hope that such transformation of differential biform of electromagnetic field F ↦ FL (without transforming vectors) can be deduced from transformation of electromagnetic sources." That hope is doing load-bearing work: the fixed-observer case is precisely where the paper claims to recover the textbook law, and if the recovery rests on an unjustified step then the three-way comparison at the end is not on firm ground. Similarly, the groupoid law (iii) is presented as Minkowski's own equations (47–48, 51–52) rather than derived independently, so the argument that Minkowski's transformation "is not the Lorentz transformation" depends entirely on a reading of a 1908 text that the paper's own acknowledgments record as contested — Ivezić, the collaborator whose work motivated the paper, "vigorously disagree[s] that Minkowski introduced the modern concept of observer as a time-like vector field."
The physical stakes are also understated. Abandoning reciprocity of relative velocity is not a small matter: it means that Rose's velocity relative to Paul is not the negative of Paul's velocity relative to Rose, which conflicts with the operational symmetry demonstrated in every reciprocal time-of-flight and two-way Doppler measurement. Oziewicz cites Świerk and Matolcsi for the failure of reciprocity but does not confront the experimental situation. More seriously, the convection-current prediction is where the paper is most vulnerable and least tested. A current density scaling as (γ2 − 1)ρ rather than √(γ2 − 1)ρ is not an exotic second-order correction: at the low velocities of ordinary laboratory currents it differs from the standard result by a factor of order u/c, which is to say the groupoid prediction for the magnetic field of a slowly moving charge is smaller by that factor. The magnetic field of convection currents has been measured since Rowland's 1876 experiment, and modern electrodynamics — including the operation of every electric motor and the whole of accelerator beam optics — is built on the standard scaling. Oziewicz says the consequence "could eventually be tested experimentally" without noting that a large body of existing measurement already bears on it, and without computing the size of the discrepancy in any concrete apparatus. That omission is the paper's most substantial gap: it names a testable difference and then does not test it, or say where it has already been tested.
Finally, much of the paper is polemic about notation rather than physics. The extended complaint that calculus textbooks fail to say that every derivative is directional, the insistence that "the most frequent Calculus-books problem 'Calculate derivative of 2x + 1' is meaningless", and the repeated attacks on the "3D"/"4D" vocabulary, are defensible pedagogical positions but consume a considerable fraction of a conference paper whose novel content is the groupoid proposal. Readers wanting that proposal should go directly to §§ 5, 12 and 12.3. Within its own terms, the mathematics of the groupoid category is sound — a groupoid is a perfectly respectable structure and Oziewicz's demonstration that his transformation is not an isometry is a two-line calculation that checks out. Whether physics should adopt it is a question the paper poses well and answers only by promise.