Scandal in Electrodynamics (continued): Mr Savarkar and the Case Rindler
| Scientific Paper | |
|---|---|
| Title | Scandal in Electrodynamics (continued): Mr. Savarkar and the Case Rindler |
| Read in full | Link to paper |
| Author(s) | Sadanand S. Savarkar |
| Keywords | torque paradox, Trouton-Noble experiment, lever paradox, hidden momentum, third law of motion, central force, Ampere force law |
| Published | 2007 |
| No. of pages | 10 |
Read the full paper here
Abstract
Attention to prior semblance of the Electrodynamic Torque Paradox has been brought forward by one Sadanand Savarkar, physics author and researcher. The work of Professor Wolfgang Rindler is cited in Mr. Savarkar's general NPA message attachment of 13 July 2006.
Overview
This is a polemical reply, the second instalment of a series titled "Scandal in Electrodynamics". Its subject is the Electrodynamic Torque Paradox — the old observation that two charges carried along parallel lines but not abreast (a charged rod moving obliquely to its own length) appear, when the Lorentz Force is applied in the laboratory frame, to experience a couple that has no counterpart in the rod's rest frame. The paper answers two people at once: Wolfgang Rindler, whose Introduction to Special Relativity sets the situation as an exercise, and Sadanand S. Savarkar, who circulated the Rindler passage as a settled refutation and then supplied his own treatment in Work and Making Relativity Work.
The claim advanced here is that neither answer is a resolution. Both, the author says, rescue Special relativity by invoking a "non-material momentum density" — an energy current said to flow along the rod in the moving frame only — and this device, whatever it does for the bookkeeping of Angular Momentum, violates Newton's Third Law the moment the applied forces are switched off. The mainstream account treats the same energy flux (today usually called hidden momentum, and derived from the stress-energy tensor rather than postulated) as the standard and correct resolution of the Trouton-Noble problem; the paper's departure is to reject it as "phantom physics" and to insist that a relativistic Electrodynamics can only be built on a central force law of the Ampère-Weber type.
The argument
What Rindler actually says
The paper quotes Savarkar quoting Rindler's Exercise VI.12 and Exercise VII.9 in full, and then makes a point about status rather than physics: Rindler does not resolve a paradox, he sets a problem, and instructs the student to prove that although the forces constitute a couple the rod does not turn. The author's reading is that Rindler "sees no paradox at all", and that his aside — that the Trouton-Noble null result "contributed to the later acceptance of relativity" — inverts the lesson, since on this account relativity "requires the rod both to acquire angular momentum and not acquire angular momentum".
A second objection separates the two cases Savarkar had assimilated. The Lewis and Tolman lever (1909) is purely mechanical and has a reaction mass — a pivot bolted to a platform — against which any torque can react. The electrodynamic case has no such body: strip the platform away and "the charged rod exerts a torque upon itself". The author also insists that the electrostatic tensions between the end charges act strictly along the rod in both frames and cancel exactly, so that the residual magnetic moments are the whole of the effect.
Savarkar's derivation, as reproduced
The paper reprints Savarkar's own analysis at length. A rod AB of length l* lies at angle θ* in a rest system Σ*, pulled at its ends by equal and opposite forces at angle φ*; equilibrium requires φ* = θ*. Set Σ* moving with velocity α along x. If the length, angle and forces suffer "velocity-modifications", the torque in Σ becomes G = Fl (sin φ cos θ − cos φ sin θ), which need not vanish.
Savarkar then applies Planck's 1908 theorem that momentum is proportional to energy flux, p = κEv, with κ a universal constant. The force at A does work at the rate αF cos φ and the force at B does equal negative work, so a steady energy current traverses the rod; if its elastic propagation speed is w, the stored energy is E = (l/w)Fα cos φ, and the moment of the associated momentum about the origin grows as dL/dt = −wE sin θ. Setting this equal to G gives the stability condition
- tan φ = tan θ (1 − κα2)
Savarkar then notes that Lorentz-Einstein electrodynamics gives force components Fx ∝ l cos θ and Fy ∝ l sin θ (1 − α2/c2), hence tan φ = (1 − α2/c2) tan θ, so that the condition is met if and only if κ = 1/c2. His "momentous" conclusion is that κ is a mechanical parameter and c an electromagnetic one, that their equality is no logical necessity, and that no relativistic electrodynamics based on a central force law can survive the test.
The rebuttal
Against this the author argues that no energy flows at all: "the forces are constant and the stresses are entirely static in a non-rotating rod", so the force at A merely "strives to do work". The translational work is cancelled by the counter-force at B; the only work that could act on the rod would be rotational; and the whole construction rests on conflating the torque F × l with the work function F · l. Since Planck's theorem presupposes mass-energy equivalence, which the author holds refuted in his companion paper Mass Disparities from Relativistic Electrodynamics, the appeal is doubly illegitimate. Even granting the energy current, he argues, the platform must acquire a real rotation that Σ* does not show, and when the horses stop pulling the platform's accumulated angular momentum is left uncompensated — "a recurrent violation of the third law of motion". He therefore concludes that φ = θ of logical necessity, and that only a central force law is compatible with relativity — the exact inverse of Savarkar's corollary — dismissing the preference for the Grassmann/Biot-Savart form as "purely mathematical convenience".
The lever paradox by simultaneity
The one resolution the paper does allow is built on the relativity of simultaneity. Weld a second lever, rotated 180°, to the pivot to make a symmetric crossbar ACED; replace the steady forces with instantaneous impulses from identical particles launched perpendicular to the four ends, simultaneous in S′. In S the arm along the motion is length-contracted but the launches are not simultaneous: the impulse at D occurs at −γvΔx/c2 and that at C at +γvΔx/c2, each displaced by γΔx′ from the origin. The moments in S are therefore greater, not smaller, and balance requires the transverse and longitudinal momenta to stand in exact inverse proportion to the corresponding moments. That balances the torques — but yields a ratio of longitudinal to transverse mass which, the author observes, is not the γ2 that relativistic dynamics elsewhere requires, so the paradox is traded for a fresh inconsistency.
Assessment
The paper is unusually careful about one thing that is often fudged in this literature: it reproduces its opponent's argument in full before attacking it, so a reader can check the target. And its central complaint has a legitimate core. The "hidden momentum" resolution genuinely does require that a static, non-rotating rod carry momentum, and that is counter-intuitive enough to have needed a century of clarification.
The arithmetic it reprints is sound. Savarkar's condition tan φ = tan θ (1 − κα2) does reduce to the Lorentz-force ratio when κ = 1/c2, and the algebra is consistent. But the "independent parameters" argument does not deliver what is claimed for it. κ = 1/c2 is not a numerical coincidence awaiting mechanical confirmation: Planck's relation p = Ev/c2 is mass-energy equivalence written for a flux, and the inertia of energy has been measured directly — in nuclear mass defects, in pair production thresholds, and in the Pound-Rebka frequency shift. Savarkar's premise that κ "could be determined from purely mechanical collision experiments" and might come out otherwise is the weakest step in the section the author chose to attack, and he does not attack it.
The rebuttal's own load-bearing step is asserted rather than derived. "The forces are constant and the stresses are entirely static, therefore no energy flows" is a rest-frame statement smuggled into the moving frame. Power is F · v, and in Σ the ends move with velocity α while the forces have unequal-signed x components — so the work is not zero at each end, only in the sum, which is exactly the condition for a steady current with no accumulation. Calling this "striving to do work" renames the problem. The "when the forces quit" objection likewise omits the transient: switching the forces off is a time-dependent process during which the flux itself carries angular momentum out, and no conservation law is broken once that interval is included. Momentum conservation in relativistic mechanics is a statement about the divergence of the stress-energy tensor, not about pairs of point forces, and Newton's Third Law in its instantaneous form is not the right ledger for a system with fields in it.
The final claim — that only a central force law can be relativistic — sits badly with the fact the paper itself relies on. Ampère's force law between current elements and the Grassmann/Biot-Savart form are known to give identical results for any closed circuit; they differ only for isolated elements, which is why no experiment has yet separated them. The paper offers no measurement that does. And the Trouton-Noble experiment, invoked here as "proof of just such a law", is a null result: it is equally proof that the couple does not appear, which is what every account under discussion, including Savarkar's, already predicts. Finally, the longitudinal/transverse mass discrepancy in the last section is a contradiction only inside the obsolete practice of writing F = ma with two masses; modern relativistic dynamics uses p = γmv with one invariant Mass, and the γ2 ratio is an artefact of the older bookkeeping rather than a separate physical requirement.