The Special Theory of Relativity: Conditions of Performance of Laws of Preservation Impulse and Energy: Difference between revisions
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In article attempt to show becomes 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. | In article attempt to show becomes 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== | |||
Victor Nikolayevich Cochetkov (Kochetkov) proposes that the conservation laws for momentum ("impulse") and energy can be used as a theoretical test of special relativity, without any new experiment. His procedure is to take a very simple closed mechanical system — two equal point masses joined by a massless thread, rotating about their common centre of mass — describe it in a frame in which the centre of mass is at rest, then transform the whole description into a second inertial frame moving uniformly past it, and ask whether the total momentum and total energy of the system come out constant in that second frame. | |||
His answer is that they do not. Evaluating the system at two different instants of the moving frame's time, he finds that momentum conservation and energy conservation each force conditions on the state of one of the bodies, and that those conditions can be satisfied only if 1/''c''<sup>2</sup> = 0. Since the speed of light is finite, he 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, by a parallel calculation, of the law of conservation of energy. He adds one alternative escape, offered without elaboration: "it is possible that the made assumption that a constant ''с'' in Lorentz's transformations is a velocity of light, not truly." The paper is a continuation of the author's earlier study in the ''Journal of Vectorial Relativity'' (2011) on the time-dependence of the momentum of a closed system of bodies. | |||
==The argument== | |||
===The rotating pair=== | |||
Two point bodies 1 and 2, each of rest mass ''M''<sub>0</sub>, are connected by a thread 3 whose mass is neglected. They rotate with constant angular velocity ω at radius ''R'' about the common centre of mass ''O''<sub>c</sub>. The frame ''Oxyz'' is chosen so that ''O''<sub>c</sub> is fixed at the origin and rotation is anticlockwise in the ''Oxy'' plane, with body 1 on the positive ''x'' axis and body 2 on the negative ''x'' axis at ''t'' = 0. In this frame the positions and velocity components are elementary: | |||
: ''x''<sub>1</sub> = ''R'' cos ω''t''<sub>1</sub>, ''y''<sub>1</sub> = ''R'' sin ω''t''<sub>1</sub>, ''v''<sub>1''x''</sub> = −ω''R'' sin ω''t''<sub>1</sub>, ''v''<sub>1''y''</sub> = ω''R'' cos ω''t''<sub>1</sub> | |||
with body 2 the same quantities reversed in sign. Each body therefore moves at constant speed ω''R'' in the rest frame of the centre of mass. | |||
A second inertial frame ''O'x'y'z''' moves with constant velocity ''V'' along ''Ox'', axes parallel and origins coincident at ''t'' = ''t' '' = 0. The [[Lorentz transformation]] and the relativistic velocity-addition formulae then give the primed coordinates, times and velocity components of each body. The transformed speeds are | |||
: ''v'<sub>x</sub>''<sub>1</sub> = (''v''<sub>1''x''</sub> − ''V'')/(1 − ''Vv''<sub>1''x''</sub>/''c''<sup>2</sup>), ''v'<sub>y</sub>''<sub>1</sub> = ''v''<sub>1''y''</sub>√(1 − ''V''<sup>2</sup>/''c''<sup>2</sup>)/(1 − ''Vv''<sub>1''x''</sub>/''c''<sup>2</sup>) | |||
and the momenta and kinetic energies follow from ''P'' = ''M''<sub>0</sub>''v'' /√(1 − ''v''<sup>2</sup>/''c''<sup>2</sup>) and ''E'' = ''M''<sub>0</sub>''c''<sup>2</sup>[1/√(1 − ''v''<sup>2</sup>/''c''<sup>2</sup>) − 1]. | |||
===The relativity of simultaneity as the engine of the paper=== | |||
The whole argument turns on Cochetkov's insistence that, to speak of "the momentum of the system at time ''t' ''" in the moving frame, the two bodies must be sampled at the ''same'' primed instant. Because the two bodies sit at different ''x'' positions, the same ''t' '' corresponds to two ''different'' unprimed times ''t''<sub>1</sub> and ''t''<sub>2</sub>, tied together by his equation (29): | |||
: (''t''<sub>1</sub> − ''VR'' cos ω''t''<sub>1</sub>/''c''<sup>2</sup>) = (''t''<sub>2</sub> + ''VR'' cos ω''t''<sub>2</sub>/''c''<sup>2</sup>) | |||
This is [[Simultaneity|the relativity of simultaneity]] applied to a rotating pair: in the moving frame the two bodies are, in general, caught at different phases of their orbit. | |||
===Two chosen instants=== | |||
'''Instant ''t'<sub>p</sub>'''''. He picks ''t''<sub>1''p''</sub> = π/(2ω), so body 1 is on the ''+y'' axis. Equation (29) then reduces to ''c''<sup>2</sup>(π/2 − ω''t''<sub>2''p''</sub>)/(''VR''ω) = cos ω''t''<sub>2''p''</sub>, which he solves "using a graphic method" to obtain ''t''<sub>2''p''</sub> = π/(2ω) as well. At this instant both bodies lie on a line parallel to ''O'y' '', both have zero ''y'' momentum in the primed frame, and | |||
: ''P'<sub>x</sub>''<sub>1''p''</sub> = −''M''<sub>0</sub>(''V'' + ω''R'')/√[(1 − ''V''<sup>2</sup>/''c''<sup>2</sup>)(1 − ω<sup>2</sup>''R''<sup>2</sup>/''c''<sup>2</sup>)], ''P'<sub>x</sub>''<sub>2''p''</sub> = ''M''<sub>0</sub>(ω''R'' − ''V'')/√[same] | |||
'''Instant ''t'<sub>h</sub>'''''. He picks ''t''<sub>1''h''</sub> = 0, so body 1 is on the ''x'' axis moving purely in ''+y''. Equation (29) now reads ''c''<sup>2</sup>ω''t''<sub>2''h''</sub>/(''VR''ω) = −1 − cos ω''t''<sub>2''h''</sub>, and he observes that "value of the moment of time ''t''<sub>2''h''</sub> should be less than 0" — body 2 cannot be on the ''x'' axis at this primed instant. | |||
===The contradiction=== | |||
Leaving body 2's velocity components ''v''<sub>''x''2''h''</sub>, ''v''<sub>''y''2''h''</sub> unknown, he imposes conservation of the total momentum between ''t'<sub>p</sub>'' and ''t'<sub>h</sub>''. The ''x'' equation collapses to | |||
: −(''V'' + ω''R'') + (ω''R'' − ''V'') = −''V'' + (''v''<sub>''x''2''h''</sub> − ''V'') | |||
giving ''v''<sub>''x''2''h''</sub> = 0, and the ''y'' equation gives ''v''<sub>''y''2''h''</sub> = −ω''R''. But those two values, substituted back into the unprimed velocity formulas (7) and (8), require ''t''<sub>2''h''</sub> = 0 — exactly what equation (53) had just excluded. Feeding ''t''<sub>1''h''</sub> = ''t''<sub>2''h''</sub> = 0 into the simultaneity relation (29) yields 0 = (''VR''/''c''<sup>2</sup>)(1 + 1), from which he divides out ''VR'' to obtain his final condition | |||
: 0 = 1/''c''<sup>2</sup> | |||
"But since the size of a velocity of light ''c'' is not equal to infinity, therefore the condition (83) is not feasible." | |||
The energy check runs in parallel and is shorter. Equating ''E'<sub>1p</sub>'' + ''E'<sub>2p</sub>'' with the total at ''t'<sub>h</sub>'', the common radicals cancel and the equation reduces to | |||
: (1 + ''V''ω''R''/''c''<sup>2</sup>) + (1 − ''V''ω''R''/''c''<sup>2</sup>) = 1 + (1 − ''Vv''<sub>''x''2''h''</sub>/''c''<sup>2</sup>) | |||
which again forces ''v''<sub>''x''2''h''</sub> = 0 and so the same impossible condition. Cochetkov's conclusion: "use of the special theory of relativity by consideration of separate examples can lead to default of laws of preservation of an impulse and energy of the closed mechanical system in inertial systems of readout." | |||
==Assessment== | |||
What is attractive here is the discipline of the construction. Cochetkov does not appeal to paradoxical thought experiments or to interpretation; he sets up the simplest possible bound rotating system, applies the standard [[Lorentz transformation]] and the standard relativistic expressions for momentum and kinetic energy without modification, and does the algebra explicitly enough that every step can be checked. He is also right about the feature that makes the problem interesting and that is often glossed over in textbooks: because simultaneity is frame-dependent, an extended rotating body sampled at a single instant of a moving frame is caught at different orbital phases at different points, so its total momentum in that frame is not simply the boosted total of its rest-frame momentum. His equation (29) is a correct statement of that fact, and his observation that ''t''<sub>2''h''</sub> < 0 at the second instant is a genuine and correctly derived consequence. | |||
The decisive difficulty is that the system he analyses is not the system his conservation laws apply to. Bodies 1 and 2 are held in their circular paths by a thread under tension; the paper explicitly neglects the thread's mass and then omits it from every subsequent equation. In relativity a stressed body's energy–momentum is not the sum of its particle terms: the stress in the connecting member carries momentum density in any frame in which it moves, and this is precisely the effect identified by von Laue in 1911 in resolving the Trouton–Noble and right-angle-lever problems, and known more generally as hidden momentum in stressed or current-carrying systems. A rotating pair on a taut thread is the textbook case in which the omitted term is of the same order as the terms retained. Cochetkov's ''P' ''<sub>total</sub> is therefore not the total momentum of a closed system, and its failure to be constant is not a failure of the conservation law. That the missing piece is of order ''V''ω''R''/''c''<sup>2</sup> — exactly the size of the discrepancy that survives to give equation (83) — is a strong indication that this, and not relativity, is what the calculation has found. The energy check inherits the same omission: the thread's elastic potential energy is not merely assumed unchanged between frames, as the paper states, but its transformation is where the compensating term lives. | |||
There are also internal problems. The final step from 0 = (''VR''/''c''<sup>2</sup>)(1 + 1) to 0 = 1/''c''<sup>2</sup> divides by ''V'' and ''R'', which is legitimate only for ''V'' ≠ 0 and ''R'' ≠ 0; what the equation actually says is ''VR''/''c''<sup>2</sup> = 0, i.e. that no contradiction arises in the non-relativistic or point-particle limits — which is what one would expect if the neglected relativistic stress term were the source. Equation (84) as printed sets ''E'<sub>1p</sub>'' + ''E'<sub>2p</sub>'' equal to ''E'<sub>1h</sub>'' + ''P'<sub>x2h</sub>'', adding a momentum to an energy; the subsequent working shows this is a typographical slip for ''E'<sub>2h</sub>'', but it is uncorrected. The solution ''t''<sub>2''p''</sub> = π/(2ω) is obtained by an unspecified "graphic method"; it does satisfy the equation, but no argument is given that it is the only root, and the whole contradiction depends on the pairing of instants being unique. Finally, the paper's alternative escape — that the constant ''c'' in the Lorentz transformation might not be the speed of light — is raised in a single sentence and never developed, so it does no work. | |||
Taken on its own terms the calculation is carefully executed and reproducible, which is more than can be said for many arguments of this kind. Its conclusion does not follow, because the closed system it names is not the closed system it computes. | |||
==See also== | |||
* [[Victor Nikolayevich Cochetkov]] | |||
* [[Lorentz transformation]] | |||
* [[Simultaneity]] | |||
* [[Time Dilation]] | |||
* [[Length Contraction]] | |||
* [[Mass]] | |||
* [[Speed of Light]] | |||
[[Category:Scientific Paper|special theory relativity conditions performance laws preservation impulse energy]] | [[Category:Scientific Paper|special theory relativity conditions performance laws preservation impulse energy]] | ||
Latest revision as of 11:47, 21 July 2026
| Scientific Paper | |
|---|---|
| Title | The Special Theory of Relativity: Conditions of Performance of Laws of Preservation Impulse and Energy |
| Read in full | Link to paper |
| Author(s) | Victor Nikolayevich Cochetkov |
| Keywords | The special theory of a relativity, principle of invariancy of a velocity of light, the law of conservation of energy, the law of preservation of an impulse |
| Published | 2011 |
| No. of pages | 19 |
Read the full paper here
Abstract
In article attempt to show becomes 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
Victor Nikolayevich Cochetkov (Kochetkov) proposes that the conservation laws for momentum ("impulse") and energy can be used as a theoretical test of special relativity, without any new experiment. His procedure is to take a very simple closed mechanical system — two equal point masses joined by a massless thread, rotating about their common centre of mass — describe it in a frame in which the centre of mass is at rest, then transform the whole description into a second inertial frame moving uniformly past it, and ask whether the total momentum and total energy of the system come out constant in that second frame.
His answer is that they do not. Evaluating the system at two different instants of the moving frame's time, he finds that momentum conservation and energy conservation each force conditions on the state of one of the bodies, and that those conditions can be satisfied only if 1/c2 = 0. Since the speed of light is finite, he 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, by a parallel calculation, of the law of conservation of energy. He adds one alternative escape, offered without elaboration: "it is possible that the made assumption that a constant с in Lorentz's transformations is a velocity of light, not truly." The paper is a continuation of the author's earlier study in the Journal of Vectorial Relativity (2011) on the time-dependence of the momentum of a closed system of bodies.
The argument
The rotating pair
Two point bodies 1 and 2, each of rest mass M0, are connected by a thread 3 whose mass is neglected. They rotate with constant angular velocity ω at radius R about the common centre of mass Oc. The frame Oxyz is chosen so that Oc is fixed at the origin and rotation is anticlockwise in the Oxy plane, with body 1 on the positive x axis and body 2 on the negative x axis at t = 0. In this frame the positions and velocity components are elementary:
- x1 = R cos ωt1, y1 = R sin ωt1, v1x = −ωR sin ωt1, v1y = ωR cos ωt1
with body 2 the same quantities reversed in sign. Each body therefore moves at constant speed ωR in the rest frame of the centre of mass.
A second inertial frame O'x'y'z' moves with constant velocity V along Ox, axes parallel and origins coincident at t = t' = 0. The Lorentz transformation and the relativistic velocity-addition formulae then give the primed coordinates, times and velocity components of each body. The transformed speeds are
- v'x1 = (v1x − V)/(1 − Vv1x/c2), v'y1 = v1y√(1 − V2/c2)/(1 − Vv1x/c2)
and the momenta and kinetic energies follow from P = M0v /√(1 − v2/c2) and E = M0c2[1/√(1 − v2/c2) − 1].
The relativity of simultaneity as the engine of the paper
The whole argument turns on Cochetkov's insistence that, to speak of "the momentum of the system at time t' " in the moving frame, the two bodies must be sampled at the same primed instant. Because the two bodies sit at different x positions, the same t' corresponds to two different unprimed times t1 and t2, tied together by his equation (29):
- (t1 − VR cos ωt1/c2) = (t2 + VR cos ωt2/c2)
This is the relativity of simultaneity applied to a rotating pair: in the moving frame the two bodies are, in general, caught at different phases of their orbit.
Two chosen instants
Instant t'p. He picks t1p = π/(2ω), so body 1 is on the +y axis. Equation (29) then reduces to c2(π/2 − ωt2p)/(VRω) = cos ωt2p, which he solves "using a graphic method" to obtain t2p = π/(2ω) as well. At this instant both bodies lie on a line parallel to O'y' , both have zero y momentum in the primed frame, and
- P'x1p = −M0(V + ωR)/√[(1 − V2/c2)(1 − ω2R2/c2)], P'x2p = M0(ωR − V)/√[same]
Instant t'h. He picks t1h = 0, so body 1 is on the x axis moving purely in +y. Equation (29) now reads c2ωt2h/(VRω) = −1 − cos ωt2h, and he observes that "value of the moment of time t2h should be less than 0" — body 2 cannot be on the x axis at this primed instant.
The contradiction
Leaving body 2's velocity components vx2h, vy2h unknown, he imposes conservation of the total momentum between t'p and t'h. The x equation collapses to
- −(V + ωR) + (ωR − V) = −V + (vx2h − V)
giving vx2h = 0, and the y equation gives vy2h = −ωR. But those two values, substituted back into the unprimed velocity formulas (7) and (8), require t2h = 0 — exactly what equation (53) had just excluded. Feeding t1h = t2h = 0 into the simultaneity relation (29) yields 0 = (VR/c2)(1 + 1), from which he divides out VR to obtain his final condition
- 0 = 1/c2
"But since the size of a velocity of light c is not equal to infinity, therefore the condition (83) is not feasible."
The energy check runs in parallel and is shorter. Equating E'1p + E'2p with the total at t'h, the common radicals cancel and the equation reduces to
- (1 + VωR/c2) + (1 − VωR/c2) = 1 + (1 − Vvx2h/c2)
which again forces vx2h = 0 and so the same impossible condition. Cochetkov's conclusion: "use of the special theory of relativity by consideration of separate examples can lead to default of laws of preservation of an impulse and energy of the closed mechanical system in inertial systems of readout."
Assessment
What is attractive here is the discipline of the construction. Cochetkov does not appeal to paradoxical thought experiments or to interpretation; he sets up the simplest possible bound rotating system, applies the standard Lorentz transformation and the standard relativistic expressions for momentum and kinetic energy without modification, and does the algebra explicitly enough that every step can be checked. He is also right about the feature that makes the problem interesting and that is often glossed over in textbooks: because simultaneity is frame-dependent, an extended rotating body sampled at a single instant of a moving frame is caught at different orbital phases at different points, so its total momentum in that frame is not simply the boosted total of its rest-frame momentum. His equation (29) is a correct statement of that fact, and his observation that t2h < 0 at the second instant is a genuine and correctly derived consequence.
The decisive difficulty is that the system he analyses is not the system his conservation laws apply to. Bodies 1 and 2 are held in their circular paths by a thread under tension; the paper explicitly neglects the thread's mass and then omits it from every subsequent equation. In relativity a stressed body's energy–momentum is not the sum of its particle terms: the stress in the connecting member carries momentum density in any frame in which it moves, and this is precisely the effect identified by von Laue in 1911 in resolving the Trouton–Noble and right-angle-lever problems, and known more generally as hidden momentum in stressed or current-carrying systems. A rotating pair on a taut thread is the textbook case in which the omitted term is of the same order as the terms retained. Cochetkov's P' total is therefore not the total momentum of a closed system, and its failure to be constant is not a failure of the conservation law. That the missing piece is of order VωR/c2 — exactly the size of the discrepancy that survives to give equation (83) — is a strong indication that this, and not relativity, is what the calculation has found. The energy check inherits the same omission: the thread's elastic potential energy is not merely assumed unchanged between frames, as the paper states, but its transformation is where the compensating term lives.
There are also internal problems. The final step from 0 = (VR/c2)(1 + 1) to 0 = 1/c2 divides by V and R, which is legitimate only for V ≠ 0 and R ≠ 0; what the equation actually says is VR/c2 = 0, i.e. that no contradiction arises in the non-relativistic or point-particle limits — which is what one would expect if the neglected relativistic stress term were the source. Equation (84) as printed sets E'1p + E'2p equal to E'1h + P'x2h, adding a momentum to an energy; the subsequent working shows this is a typographical slip for E'2h, but it is uncorrected. The solution t2p = π/(2ω) is obtained by an unspecified "graphic method"; it does satisfy the equation, but no argument is given that it is the only root, and the whole contradiction depends on the pairing of instants being unique. Finally, the paper's alternative escape — that the constant c in the Lorentz transformation might not be the speed of light — is raised in a single sentence and never developed, so it does no work.
Taken on its own terms the calculation is carefully executed and reproducible, which is more than can be said for many arguments of this kind. Its conclusion does not follow, because the closed system it names is not the closed system it computes.