The Special Theory of Relativity: Conditions of Performance of Laws of Preservation Impulse and Energy (Reduced Variant of Article)
| Scientific Paper | |
|---|---|
| Title | The Special Theory of Relativity: Conditions of Performance of Laws of Preservation Impulse and Energy (Reduced Variant of Article) |
| Read in full | Link to paper |
| Author(s) | Victor Nikolayevich Cochetkov |
| Keywords | special theory of a relativity, invariancy of a velocity of light, conservation of energy, conservation of an impulse |
| Published | 2011 |
| No. of pages | 16 |
Read the full paper here
Abstract
In article it is shown that use of laws of preservation of an impulse and energy of the closed mechanical system presumes to check up justice of the special theory of relativity theoretically.
Overview
Kochetkov (the paper is signed V. N. Kochetkov; the wiki index spells the surname Cochetkov) sets out to test special relativity not against an experiment but against two conservation laws. His method is to construct the simplest possible closed rotating system, transform it into a moving inertial frame using the Lorentz transformations, evaluate its total momentum and energy at two different instants of the moving frame's time, and ask whether the two evaluations agree.
They do not. The paper's conclusion is that the total momentum and the total energy of the system, as computed in the moving frame, come out different at the two instants, and that the discrepancy can be made to vanish only by setting 1/c2 = 0 — that is, by taking the speed of light to be infinite. Since it is not, Kochetkov concludes that "application of the special theory of relativity at the description of movement of the closed mechanical system of the bodies considered in the given example, leads to default of the law of preservation of an impulse," and likewise of energy. He adds one alternative escape, offered in a single sentence: "It is possible that the made assumption that a constant c in Lorentz's transformations is a velocity of light, not truly."
The paper is a machine-assisted translation from Russian, and its vocabulary needs decoding: "system of readout" means reference frame, "impulse" means momentum, "preservation" means conservation, "weight" means mass, and "default" means failure. It is a shortened version of a longer article; a fuller treatment appeared in the Journal of Vectorial Relativity in 2011.
The construction
The system
Two point bodies, 1 and 2, each of rest mass M0, are joined by a thread 3 whose mass "because of its small size can be neglected." They rotate counterclockwise with angular velocity ω at radius R about their common centre of mass Oc, which is placed at the origin of a rest frame Oxyz. At t = 0 body 1 sits on the positive x axis and body 2 on the negative. Their coordinates and velocity components are the elementary ones: x1 = R cos ωt1, y1 = R sin ωt1, v1x = −ωR sin ωt1, v1y = ωR cos ωt1, with the signs reversed for body 2.
A second frame O′x′y′z′ moves with constant speed V along Ox, axes parallel, origins coincident at t = t′ = 0. Kochetkov writes out the Lorentz transformations for the coordinates, the velocity-addition formulae for both components, and from them the relativistic momentum P′ = M0v′/√(1 − v′2/c2) and kinetic energy E′ = M0c2[1/√(1 − v′2/c2) − 1] for each body.
The simultaneity condition
The pivot of the whole argument is his equation (27). To speak of "the momentum of the system at time t′," the two bodies must be evaluated at the same primed time, and because they sit at different x coordinates, that single primed time corresponds to two different unprimed times t1 and t2:
- t′ = [t1 − (VR/c2) cos ωt1] / √(1 − V2/c2) = [t2 + (VR/c2) cos ωt2] / √(1 − V2/c2).
Two instants are then selected. At t′p, body 1 is taken at ωt1p = π/2, so that it lies on the y axis; equation (27) then reduces to an equation whose root is ωt2p = π/2 as well, so both bodies lie on a line parallel to O′y′ and both have x = 0. At this instant the simultaneity offset vanishes and the momenta come out symmetrically: P′x1p = −M0(V + ωR)/√[(1 − V2/c2)(1 − ω2R2/c2)], P′x2p = +M0(ωR − V)/(same), and both y components zero.
At the second instant t′h, body 1 is taken at t1h = 0, on the x axis. Now equation (27) gives, for body 2, the transcendental relation c2ωt2h/(VRω) = −1 − cos ωt2h, whose solution is negative: body 2 is not on the x axis at that instant, but lags behind it by a small angle of order VRω/c2. Kochetkov leaves its velocity components as unknowns vx2h, vy2h, subject only to v2h2 = ω2R2.
The two checks
Imposing conservation of the x momentum between t′p and t′h, the common factors cancel and the condition collapses to −(V + ωR) + (ωR − V) = −V + (vx2h − V), from which vx2h = 0. Imposing conservation of the y momentum gives vy2h = −ωR. Together these say that body 2 must be exactly diametrically opposite body 1 on the x axis, i.e. t2h = 0. But substituting t1h = t2h = 0 back into the simultaneity condition (27) yields 0 = (VR/c2)(1 + 1), which Kochetkov writes as the requirement 0 = 1/c2. Since c is finite, the requirement cannot be met, and momentum conservation "is executed cannot be."
The energy check runs the same way. Assuming — and he states this as an assumption — that if no change of potential energy occurs in one inertial frame then none occurs in any other, he equates the summed kinetic energies at the two instants. The relativistic factors again cancel, leaving (1 + VωR/c2) + (1 − VωR/c2) = 1 + (1 − Vvx2h/c2), hence once more vx2h = 0 and the same contradiction. The conclusion is that in this example special relativity fails both conservation laws.
Assessment
The paper's virtue is that it is short, concrete and checkable. Kochetkov does not hand-wave; he writes out every transformation, picks two explicit instants, and lets the algebra decide. The core observation he isolates is real and is worth stating plainly: for a spatially extended rotating body, a plane of constant time in a moving frame cuts the two bodies' worldlines at different proper times, so the "instantaneous configuration" of a rigid rotor is frame-dependent, and the two masses are not diametrically opposite in the moving frame even though they always are in the rest frame. That is a genuine consequence of the relativity of simultaneity, correctly derived, and it is the kind of thing worth making students confront.
The conclusion drawn from it, however, does not follow, and the reason is visible in the paper's own setup. Kochetkov discards the thread's contribution on the ground that its mass is negligible. But what matters here is not the thread's mass; it is the thread's tension. The thread must supply the centripetal force holding each body in its orbit, so it carries a stress of order M0ω2Rγ along its length, and in relativity stress is a component of the stress-energy tensor exactly as mass-energy is. When a stressed body is viewed from a moving frame, the stress terms contribute to the total momentum a piece of order (V/c2) × (tension × length) — which is precisely the order VωR/c2 of the residual that Kochetkov is left with, and which he has thrown away by construction. A system of two point particles plus a stressed connector is not described by the sum of the two particle momenta alone. Its total four-momentum is the integral of Tμν over a simultaneity slice, and the particle terms are only part of that integral.
This is not a new difficulty invented to rescue the theory. It is the century-old problem of the four-momentum of extended stressed systems, worked out by Max von Laue in 1911 in response to exactly this family of puzzles — the right-angle lever, the Trouton–Noble apparatus, and the electromagnetic mass of the electron. Laue's result is that for a system held together by internal stresses, the naive sum of parts is not a four-vector and is not conserved, while the full integral of the stress-energy tensor is both. The relevant experimental check is also on record: the Trouton–Noble experiment of 1903 looked for precisely the kind of unbalanced torque that a failure of this bookkeeping would produce in a charged capacitor moving through the aether, and found none, at a sensitivity that has since been improved by many orders of magnitude in modern Lorentz-invariance tests. Kochetkov cites neither the problem's history nor any of these measurements; his reference list contains one earlier paper of his own and two Russian handbooks.
Two further details tell against the argument on its own terms. First, the condition he reaches at equation (61) is 0 = (VR/c2)(1 + 1), which is satisfied not only by 1/c2 = 0 but also by V = 0 or R = 0. He reports only the first. That the alleged contradiction disappears when the relative velocity vanishes, or when the system shrinks to a point, is the signature of a simultaneity effect in an extended body rather than of an inconsistency in the theory — a genuine contradiction would not care whether the observer was moving. Second, his energy check rests on an assumption he states but does not defend: that a system with no change of potential energy in one frame has none in any other. For a system with internal stress this is exactly the assumption that fails, since the work done by the internal stresses is distributed differently across different simultaneity slices. The assumption is doing the work that the conclusion is credited to.
There are also small blemishes. Equation (63) as printed sets E′1p + E′2p equal to E′1h + P′x2h, adding an energy to a momentum component; the following line uses E′2h, so this is a typographical slip rather than a substantive one. The text refers to "the condition (83)" where it means equation (62), the paper having no equation (83). And the transcendental equation for t2p is said to be solved "using a graphic method," when ωt2p = π/2 is an exact root by inspection, both sides vanishing there.
The closing suggestion — that c in the Lorentz transformations may not be the speed of light — does not help, since the algebra is identical for any finite constant, and it is the finiteness alone that Kochetkov's contradiction turns on. Read as a demonstration that special relativity is internally inconsistent, the paper does not succeed; the omitted term is the resolution, and it was identified a century before the paper was written. Read as a clean, self-contained worked illustration of how badly the naive picture of a rigid rotor fails when the simultaneity slice is tilted, it is genuinely instructive, and the arithmetic is sound as far as it is taken.