Electric and Magnetic Fields According to Hermann Minkowski
| Scientific Paper | |
|---|---|
| Title | Electric and Magnetic Fields According to Hermann Minkowski |
| Read in full | Link to paper |
| Author(s) | Cynthia Kolb Whitney, Zbigniew Oziewicz |
| Keywords | history of of relativity theory, group-covariance, groupoid category, relativity groupoid, reference frame, observer, tetrad versus monad, electric and magnetic fields |
| Published | 2008 |
| Journal | Proceedings of the NPA |
| Volume | 5 |
| Number | 2 |
| No. of pages | 28 |
| Pages | 183-194 |
Read the full paper here
Abstract
Since publications by Oliver Heaviside in 1888-1889, it is known that the electric and magnetic fields are relative, i.e. that these fields depend on a choice of an observer, a choice of a reference system, and not only on a choice of electromagnetic sources via the differential Maxwell equations. Einstein in 1905 re-derived the Heaviside relativity transformation by applying the Lorentz-isometry group to a system of four differential Maxwell equations. Minkowski in 1908 defined the electric and magnetic fields on a four-dimensional spacetime, as the tensorial concomitants of an observer-monad, and explains the meaning of the Lorentz-group-covariance of a concomitant tensor field. Present textbooks interpret the Lorentz-group-covariance of concomitant tensor differently than Minkowski in 1908. Tomislav Ivezić in 2001 re-invented the Minkowski definition of the Lorentz-group-covariance. Different interpretations of the group-covariance, lead to different relativity transformations of the electric and magnetic fields on spacetime. The main objective of the present note is the third possibility, implicit in [Minkowski 1908, §11.6], where a set of all relativity transformations of all massive observers-monads, is a groupoid category, that is not a group [Oziewicz since 2005]. Heaviside in 1888 derived the relativity of the electric and magnetic fields without the concept of a relativity group. The Heaviside relativity transformation actually could be a groupoid transformation, and not a Lorentz-group transformation.
Overview
Written by Zbigniew Oziewicz of UNAM for the centenary of Minkowski's 1908 memoir, and published in the Proceedings of the NPA, this is a mathematically technical paper about a historical and conceptual question: what, exactly, does it mean to say that the electric and magnetic fields are "Lorentz covariant"? Oziewicz's contention is that three logically distinct answers have circulated under that one name, that they yield three different transformation laws for E and B, and that the version in modern textbooks is not the one Minkowski defined.
The historical hinge is Oliver Heaviside. In 1888–89 — seventeen years before Einstein and twenty before Minkowski — Heaviside derived the relativity of the fields, E(R) = γ{E(P) + (u/c) × B(P)} and B(R) = γ{B(P) − (u/c) × E(P)}, with no concept of a relativity group whatever. Oziewicz asks how that was possible, and answers that Heaviside was implicitly using not a group but a groupoid: a category in which every arrow is invertible but in which arrows between different objects need not be composable. In his reading, the set of relativity transformations relating massive observers is intrinsically a groupoid, not the Lorentz isometry group, because massive observers exhaust neither the domain nor the composability structure a group requires. The paper is thus simultaneously a defence of Minkowski's original definitions, a criticism of textbook electromagnetism, and an advertisement for the author's own relativity-groupoid programme.
The argument
Terminology: no "3D quantities"
Oziewicz opens with an insistence on vocabulary. Minkowski's "space vector" means what is now called a space-like vector and his "space-time vector" simply a vector; the modern habit of contrasting "3D" against "4D" quantities he regards as a misreading that "grows into" a false ontology. All tensor fields in the paper — E, B, the electromagnetic biform F, the observer field — live on the four-dimensional spacetime manifold and nowhere else. That a vector field has three non-vanishing components in some particular adapted basis is a fact about the basis, not about the field: "the basis-dependent classification into '4D', '3D', '2D' and '1D' physical quantities, is totally irrelevant for mathematics and for physics."
Against the Jammer reading of Heaviside
Jammer (1961) interpreted Heaviside's derivation as a coordinate change turning the d'Alembert wave operator into a Poisson equation. Oziewicz shows this cannot be right: the d'Alembertian is coordinate-free, so no change of chart can convert it into a different operator. Carrying out the substitution {x,t} → {x − ut,t} explicitly — and being careful that ∂t is a different vector field in each chart, a distinction he notes thermodynamics handles better with its (∂T)P ≠ (∂T)V notation — he obtains an operator with a surviving cross term, not a Poisson operator. What actually does the work, he shows, is the condition Hajra and Ghosh use: the Lie derivative of the electromagnetic potential along the observer's time-like field vanishes, LRA = 0. Combined with the wave equation, that does give Poisson's equation, on four-dimensional spacetime and with no coordinate change at all. Oziewicz proposes elevating the Lie-symmetry condition to an axiom relating the potential to the reference system, and remarks in passing that Hajra and Ghosh's own route through dx = u dt is defective because then dx ∧ dt = 0 and {x,t} ceases to be a chart.
Tetrad versus monad
Two models of "observer" are contrasted. Einstein in 1905 identified a physical reference system with a coordinate basis (a tetrad); on that view all tensors are absolute and only their scalar components are observer-dependent. Minkowski in 1908 identified the observer instead with a normalized time-like vector field — a monad. Oziewicz sides with the monad: bases are mathematical, not physical, and a monad is necessarily massive since a time-like field cannot describe massless radiation. He records that monads were independently reinvented by Eckart (1940), Ehlers (1961) and Zel'manov (1944), "the Minkowski first invention in 1908 goes to oblivion," and suggests Heaviside was tacitly a monad theorist too.
The cross product is ternary in spacetime
A technical section establishes that the Gibbs cross product, which requires the Hodge star (invented, Oziewicz notes, by Grassmann in 1862 under the name Ergänzung), cannot be binary in four dimensions. He defines A ×C B ≡ ⋆(A ∧ C ∧ B), which depends on an auxiliary vector field C — in practice the observer. His judgement on the three-dimensional cross product in electromagnetism is blunt: it renders Maxwell's equations, the Lorentz force and Heaviside's transformation "a thoughtlessness mechanical set of strange formulas and this is mortal for the spirit of electromagnetism."
Three transformation laws
Minkowski's definitions are E(F,Paul) ≡ P · F and B(F,Paul) ≡ ⋆(P ∧ F): the fields are concomitants of two arguments, the source-determined biform and the observer. The question the paper calls "the heart of this note" is how many distinct routes lead from these definitions to Heaviside's transformation. Three are identified.
- The textbook (Jackson/Fock) law, labelled J. The Lorentz boost acts on F while the observer-monad Paul is held fixed and absolute. This yields EJ = γ{E + (u/c) ×P B} − [γ2/(γ+1)]{(u/c)·E}(u/c), with the invariant E2 − B2. Oziewicz's verdict, following Ivezić's criticism from 2001, is that this is not covariance at all: it is "meaningless mathematically" to Lorentz-transform F while treating the observer's own time-like vector as invariant. He calls it "Lorentz-not-covariance."
- The Minkowski–Ivezić law, labelled I. If a Lorentz transformation is an isomorphism of the vector space, then the entire tensor algebra must transform, including the observer. Then EI = E − γ(E·u/c){P − [γ/(γ+1)]u/c}, and the magnitudes are preserved: (EI)2 = E2, (BI)2 = B2, EI·BI = E·B. Oziewicz notes that "Minkowski never used his definition of group-covariance in practise."
- The groupoid law, labelled M, which is the paper's own proposal. Here F is postulated absolute — observer-free, in the spirit of Gregori's "principle of absolute reality" — and only the monads transform, P ↦ R = γ(P + u/c). The induced law is EM = γ{E + (u/c) ×P B} + γ{(u/c)·E}P, which Oziewicz identifies with Minkowski's own equations (47–48) and (51–52).
Why the groupoid is not a group
The relativity groupoid takes massive bodies as objects and unique binary relative velocities, u(P ↦ R)/c ≡ −(P − R)/(P·R), as arrows. Since arrows P → R and Q → S with R ≠ Q are not composable, this is a genuine groupoid rather than a one-object group. Oziewicz then shows directly that the monad transformation is not an isometry: for P·u = Q·u = 0, the boost sends P·Q to γ2P·Q + γ2 − 1 ≠ P·Q. The three laws are then distinguished by a clean diagnostic: the difference terms in J are proportional to the space-like velocity u, while in the groupoid they are time-like, along P. When u × B = 0, EM·E − γE2 = 0 exactly, whereas EJ·E − γE2 = −[γ2/(γ+1)](u·E/c)2 and EI·E − E2 carries the same term with opposite sign. All three coincide with Heaviside's 1888 result in the special case where the relative velocity is spacetime-orthogonal to both fields, u·E = u·B = 0 — which is why, Oziewicz argues, the discrepancy has gone unnoticed.
Assessment
The paper's strongest contribution is diagnostic and it is largely correct on its own ground. The observation that "Lorentz covariance" is used in the literature for at least two inequivalent operations — transforming the field while freezing the observer, versus transforming the whole tensor algebra including the observer — is a real logical point, and Ivezić has pressed it in refereed venues (Foundations of Physics, 2001 and 2003) without it being refuted so much as ignored. Oziewicz's demonstration that all three candidate laws collapse to the same familiar formula when u·E = u·B = 0 is genuinely illuminating: it explains at once why the ambiguity is invisible in the standard textbook worked examples and why it has never bitten anyone experimentally. The correction of Jammer's account of Heaviside is careful and, as far as the algebra goes, decisive: the d'Alembertian does not become a Poisson operator under a chart change, and the actual work is done by a Lie-symmetry condition. The recovery of Heaviside's 1888–89 priority, and of Minkowski's monad as against Einstein's tetrad, is also useful history that is not widely known.
The difficulties are of two kinds. First, the paper never confronts the fact that its three rival laws must be experimentally distinguishable if the distinction is physical, and it identifies no experiment. The difference terms are the longitudinal components u·E and u·B — precisely the configuration realized in, for example, a charge moving parallel to a field — yet no measurement is proposed, no existing datum is cited as favouring one law, and no bound is estimated. Since the standard law is embedded in the design of every accelerator, every synchrotron light source and the relativistic beam optics on which those machines depend, a rival law differing at order γ2(u·E/c)2 carries a substantial burden of showing where it does not disagree with practice. The paper does not take that burden up.
Second, the central positive claim — that the electromagnetic biform F is absolute rather than Lorentz-covariant — is a postulate, adopted from Gregori's "principle of absolute reality," and it is what generates the groupoid law. But it is precisely the negation of the Ivezić position the paper otherwise endorses, and Oziewicz says so plainly without arguing for the choice on physical grounds. Given that his objection to the textbook law is that it is internally inconsistent to freeze one tensor while transforming another, adopting a scheme in which F is frozen and the monads transform is at least an ironic resolution and at most a symmetric instance of the same complaint. Relatedly, if the relativity transformations form a groupoid rather than a group, then relative velocities do not compose associatively across three observers, and the resulting non-transitivity of "relative rest" is a considerable physical claim about which the paper says little beyond the categorical formalism.
The presentation also works against the argument. The rhetoric is sharp where precision would serve better — the three-dimensional cross product is "mortal for the spirit of electromagnetism," the standard boost is "strange," textbook treatments are "misunderstood" — and the paper assumes fluency in exterior algebra, Hodge duality and category theory while offering no worked physical example. Read as what it is, a centenary note arguing that a specific definition of Minkowski's has been silently replaced by a different one, it makes its case. Read as a proposal to replace the Lorentz group in physics, it is a programme statement whose empirical content remains to be supplied.