Do We Understand the Field Transformations in Classical Electrodynamics?
| Scientific Paper | |
|---|---|
| Title | Do We Understand the Field Transformations in Classical Electrodynamics? |
| Read in full | Link to paper |
| Author(s) | Alexander L Kholmetskii |
| Keywords | Field transformations, special relativity, covariant ether theories |
| Published | 2004 |
| Journal | Apeiron |
| Volume | 11 |
| Number | 1 |
| No. of pages | 22 |
Read the full paper here
Abstract
This paper considers a number of physical problems, dealing with transformation of the non-radiating electric and magnetic fields between different inertial frames, and analyses an origin of these fields through their sources and the laws of electrostatics and magnetostatics. It has been found that, in all problems of classical electrodynamics dealing either with single space-time transformations or with successive spacetime transformations with collinear velocities, a relationship between the fields and their sources in terms of the electrostatic and magnetostatic laws can be established. In problems dealing with successive field transformations with non-collinear relative velocities, relativity theory fails to indicate an origin of the fields obtained formally via such transformations. This can be done in covariant ether theories, and some physical inferences from the obtained results are discussed.
Overview
Alexander Kholmetskii, of Belarus State University, asks a question that is rarely pressed in textbook treatments: when the field transformations of special relativity tell an observer in one frame that an electric field is present where another observer sees only a magnetic one, can that new field always be traced back to its sources — to a definite distribution of charge density ρ and current density j obeying the ordinary laws of electrostatics and magnetostatics? He restricts attention to non-radiating fields, which remain attached to their sources, so that the question is well posed.
His answer is a partition. For any single Lorentz transformation, for successive transformations with collinear velocities, and for problems with circular currents, the source account can always be constructed, and the classic wire-and-charge example works out exactly. For successive transformations with non-collinear velocities it cannot: he presents a case in which the field transformations require a source configuration whose magnitude is twice, and whose sign is opposite to, what relativistic kinematics independently predicts. His proposed resolution is that the missing physics is real, absolute contraction — a genuine deformation of moving bodies producing genuine mechanical stresses — which is available in covariant ether theories (CETs) but not in special relativity, where contraction is purely kinematic.
The paper's stance is unusual among dissident treatments: Kholmetskii does not claim the transformations give wrong numbers. He grants that the predicted forces agree across frames. His claim is that special relativity supplies no physical account of why those fields are there, and that an account exists in a theory with a preferred frame.
The argument
The standard field transformations
The paper begins from the antisymmetric field tensor Fik = ∂Ak/∂xi − ∂Ai/∂xk and the familiar transformation rules for relative velocity along x: Ex = E′x, Ey = (E′y + vB′z)/√(1−v2/c2), and so on. Since ρ and j transform too, understanding the physics of the field transformations means finding the fields' origin in the transformed sources.
Where the source account succeeds
The worked example is Feynman's: a long straight wire carrying current I along x, and a positive charge Q moving parallel to it at velocity v. In the wire's rest frame the answer is a Lorentz force Fy = QvB. In the particle's rest frame there is no relevant magnetic field, and the force must be electric. It is: the ion skeleton moves at v and contracts by √(1−v2/c2), while the conduction electrons move at the composed velocity u′ = (u+v)/(1+uv/c2) and contract by a different factor. The mismatch leaves the wire net-charged in that frame, giving
- Ey = vB√(1 − v2/c2)
which is precisely what the field transformation gives. Kholmetskii accepts this completely: "for the problem in Fig. 1 we fully understand the transformations (2) for the electric and magnetic fields, proceeding from analysis of their sources."
He then classifies the cases where such an account can always be built: all single-transformation problems (his example is a moving parallel-plate capacitor, where the enhanced field follows from increased surface charge density and the induced magnetic field from the motion of the charged plates); all successive transformations with collinear velocities, of which the wire example is one; and all circular-current problems, such as a moving solenoid, where the electron velocities u⊕v(φ) vary with angular position so that the surface charge density varies harmonically with φ, and electrostatics then gives the constant interior electric field along y.
The underlying reason is a theorem he attributes to covariant ether theories: for single Lorentz transformations, or successive ones with collinear velocities, special relativity and an infinite set of ether theories satisfying the general relativity principle give identical results for every inertial observer, because in all of them the measured coordinates obey the Lorentz transformations and collinear boosts commute. The same commutativity holds for the field transformations.
The failure case: perpendicular motion
The crucial problem changes one thing. The same wire carries current I along x, but the probe charge now moves along y at velocity −v. In the laboratory frame the Lorentz force is Fx = QvB, directed along x. Transforming to the particle frame with the boost along y, the only surviving electric component is
- E′x = vBz√(1 − v2/c2)
and the forces agree between frames. The difficulty is the origin of Ex. From electrostatics, a field along the wire's own axis can arise only if the line of positive ions and the filament of conduction electrons are not parallel but lie at some small angle α. Kholmetskii estimates that angle by comparing Ex with the electrons' own field Ey = −λ−/2πε0r, obtaining to order c−2
- α ≈ Ex/Ey ≈ uv/c2
Now, relativistic kinematics does predict a tilt here, through the Thomas–Wigner rotation: for the successive transformations KQ → Ki → Ke with non-collinear velocities, the axes of the electron frame and the particle frame are rotated relative to one another by
- γ ≈ uv/2c2
Kholmetskii's objection has two parts. The magnitudes disagree by a factor of two, γ = α/2. And the signs disagree: the kinematic rotation implies Ex in the negative x-direction, while the field transformations give it positive. To this he adds that the premise is physically absurd in any case, since the ion lattice and the conduction electrons "belong to the same solid body" and cannot sensibly be described as lying at a non-vanishing angle. Deriving Ex as −∂A/∂t from the vector potential is, he says, "acceptable for purely mathematical, but not physical, theory."
The covariant-ether-theory resolution
CETs, as Kholmetskii sets them out, keep space-time homogeneity, space isotropy and causality but replace the Einstein relativity principle with the general relativity principle. This admits a preferred frame K0 with Galilean metric; in any moving frame the metric is no longer Galilean, and physical four-vectors xph and measured four-vectors xm differ, transforming by separate rules. The measured coordinates still obey Lorentz transformations, so Maxwell's equations and the field transformations carry over unchanged. The decisive difference is that "Nature does not 'know' a direct relative velocity of two arbitrary inertial frames": velocities are always composed as v1⊕v2 from absolute velocities in K0. Choosing the transformation matrix A = G (Galilean) recovers Lorentz ether theory, with absolute time dilation and absolute contraction along the direction of absolute velocity.
Applied to the perpendicular problem with Q at rest in K0: the electron filament moves at absolute velocity V = u⊕v. Absolute contraction along V rotates the x axis of Ke counter-clockwise by γ and the y axis clockwise by the same angle, so the axes make an angle π/2 − 2γ. Were the filament mechanically free it would rotate counter-clockwise by γ, and E clockwise by γ — giving a positive projection Ex, the correct sign, but only half the required magnitude.
The second step supplies the other half, and is the paper's real proposal. The electrons are not free: they move inside the wire. Because in CETs the contraction is a real deformation rather than a kinematic appearance, it produces real mechanical stresses. The filament's tendency to rotate through γ calls forth a reactive force from the ion lattice that holds it parallel to the x axis — an additional clockwise rotation by γ, carried along by the electric field vector. The result is E at 2γ = α to the y axis, exactly the angle demanded by the field transformation. Kholmetskii notes that assuming KQ to be the absolute frame is a simplification and that the same Ex follows for arbitrary absolute velocity, and he defers the underlying theory of deformation-induced forces in solids to a promised separate paper.
Charge conservation in a rotating frame
The final problem, taken from Avramenko and colleagues, sharpens the point. A superconducting ring carries current I and rests in the laboratory with a charge Q at distance r > R; the ring is neutral and no force acts. The ring is then spun up to angular velocity ω. Transforming the current-density four-vector from an inertial to a rotating frame gives a non-zero charge density spread uniformly around the perimeter,
- ρ = ωRj/√(1 − ω2R2/c2)
and the same result follows by dividing the ring into short straight segments and applying the wire analysis of the first problem to each. The rotating ring would then be charged, produce a radial electric field, and attract Q.
But the ring is a closed system connected to no charge source, so a net charge appearing on it violates charge conservation. The authors of the original reference were prepared to contemplate a violation of that law; Kholmetskii is not. He argues instead that the transformation of the current-density four-vector fails as a "mathematical law" applied without regard to material reality: the electron filament rotates inside the ion skeleton, absolute contraction of the filament induces reactive forces from the lattice, and those forces keep the filament undeformed, so the total charge density stays zero and no force acts on Q. He cites his own analysis of the Faraday induction law as an earlier instance where "the failure of formal mathematics of classical EM theory to describe real physical situations" appeared.
He closes with an asymmetry: if instead the ring rests and the charged particle is made to rotate at −ω, the particle does experience a radial Lorentz force QωrB(r). Rotating ring gives zero force, rotating charge gives a force — which he takes to indicate "a violation of relativity of rotational motion, as revealed in the Barnett experiment."
Assessment
The paper's virtues are real and worth stating plainly. Kholmetskii asks a legitimate and under-examined question — whether the transformed fields can always be grounded in transformed sources — and he answers the easy part of it honestly and correctly, working through the wire, capacitor and solenoid cases and conceding that special relativity handles them fully. He does not overclaim: nowhere does he assert that relativity predicts a wrong force. The problems are well chosen, the calculations are clean, and the rotating-ring paradox is a genuinely instructive one that most textbooks avoid. His instinct that a non-vanishing net charge on an isolated superconducting ring cannot be right, and that the fault lies in applying a transformation law rather than in charge conservation, is sound. It is also to his credit that he identifies the exact boundary of the theorem he relies on — collinearity and commutativity — rather than making a blanket claim.
The central argument, however, rests on a misidentification. The Thomas–Wigner rotation is not a physical tilting of the wire's material lines; it is a rotation of the coordinate axes relating two frames reached by non-collinear boosts. To compare γ with an angle α extracted from an electrostatic model in which the ion line and the electron filament are geometrically inclined is to compare a frame rotation with a material configuration, and the factor of two and the sign discrepancy are what one should expect from such a comparison rather than evidence of a defect. More fundamentally, the premise that Ex requires a tilt is not correct. In the particle's frame the wire is in motion, so the situation is not electrostatic at all: relativity of simultaneity means the charge distribution along the wire is evaluated at different lab times at different positions, which produces a longitudinal field gradient with no geometric inclination whatever. Kholmetskii's own parenthetical hedge — that field lines of moving and resting charges differ in general, "however, this is not the case when the relative velocity is orthogonal to the axis of the wire" — is where the argument turns, and it is asserted rather than demonstrated. The step is the load-bearing one in the whole paper, and it is the least defended.
The CET resolution then has the character of a construction fitted to a known answer. The first rotation by γ supplies the right sign but half the magnitude; a second rotation by γ, attributed to lattice reactive forces, is introduced to supply the remainder. Nothing in the paper derives that second rotation from a stress calculation. The theory that would be needed — "a theory of forces in solid bodies, resulting from the absolute contraction effect" — is explicitly deferred to a future publication, with the present results used because they "seem to be obvious at qualitative level". A mechanism that happens to double an angle, with no independent determination of its magnitude, cannot be counted as an explanation of the factor of two; it restates it. Nor is it explained why the same lattice forces, invoked here to rotate the field vector, do not disturb the collinear cases where relativity's account is granted to be complete.
The rotating-ring section conflates two distinct things. The failure of the naive transformation is real, but its cause is well understood and does not require absolute contraction: a rotating frame is not inertial, the transformation to it is not a Lorentz transformation, and the resulting coordinate system is not globally synchronisable — the Ehrenfest and Sagnac problems are the standard consequences. Applying the four-current transformation as if it were a boost is exactly the error, and Kholmetskii's diagnosis that "it is necessary to take into account another effects" is right in spirit while pointing at the wrong effect. His conclusion that no force acts on the stationary charge is almost certainly correct; the argument that gets him there is not needed. The closing claim of a "violation of relativity of rotational motion" is weaker still, since relativity does not assert an equivalence between rotating a ring and rotating a charge — rotation is absolute in the sense of being locally detectable, as Foucault's pendulum and ring-laser gyroscopes show, and no relativity principle is breached.
One further point is worth noting for a reader assessing the wider programme. Kholmetskii is quite explicit that CETs and special relativity give identical predictions for all single and collinear-successive transformations, and that the theories differ observationally only through the Thomas–Wigner rotation regime. That is an unusually honest statement of the empirical stakes, and it also shows how narrow they are. If the two frameworks agree on every measured force in the problems considered, then the choice between them here is a choice between explanations, not a decidable experimental question — and the paper does not propose an experiment that would decide it.
The paper is best read, then, as a careful and well-motivated set of puzzles about the physical interpretation of the field transformations, whose diagnostic step does not hold up but whose examples repay working through.