Connection between coordinates and time in pseudo-inertial systems of readout
| Scientific Paper | |
|---|---|
| Title | Connection between coordinates and time in pseudo-inertial systems of readout |
| Read in full | Link to paper |
| Author(s) | Victor Nikolayevich Cochetkov |
| Keywords | The special theory of a relativity, Lorentz's transformation, inertial systems of readout, pseudo-inertial systems of readout, velocity of light, constancy of a course of time |
| Published | 2011 |
| No. of pages | 11 |
Read the full paper here
Abstract
In article attempt to establish connection between coordinates and time in pseudo-inertial systems of readout becomes.
Overview
This eleven-page paper by Victor Nikolayevich Cochetkov, a specialist at the Russian FSUE "TSENKI" centre for space ground infrastructure, is a direct sequel to his earlier study "The special theory of relativity: linear example of infringement of laws of preservation of an impulse" (The General Science Journal, 2011). The earlier paper argued that applying special relativity to a closed mechanical system whose parts interact continuously leads to a violation of momentum conservation in ordinary inertial frames, and that the violation disappears only if the constant c in the Lorentz transformation is taken to be infinite. The present paper carries the same construction over to what the author calls "pseudo-inertial systems of readout" — frames laid out along a closed circular line rather than an infinite straight line.
The claim is that the same contradiction reappears, and reappears in a form the author regards as harder to evade: in one-dimensional and two-dimensional pseudo-inertial frames the course of time must be identical, so events simultaneous in one frame are simultaneous in all. Since relativity of simultaneity is the very feature that distinguishes special relativity from Galilean kinematics, the paper concludes that special relativity is contradicted and that time is absolute. The text is a machine translation from Russian, and its vocabulary is idiosyncratic throughout — "systems of readout" for reference frames, "impulse" for momentum, "weight" for mass, "constancy of size P1" for conservation of momentum.
The argument
One-dimensional frames and the torus construction
Cochetkov begins by discarding the usual three-dimensional frame in favour of a one-dimensional frame O1x1: a single axis, pictured as "the space concluded inside of infinitely long tube which internal radius is infinitesimal size", along which bodies may move but not sideways. Space and time along the tube are assumed homogeneous. A second frame O2x2 slides along the same line at constant speed V, with origins coinciding at t1 = t2 = 0.
For this pair he writes the standard one-dimensional Lorentz transformation, his equations (1), (2), (5) and (6), together with the velocity-addition rules (3) and (4) — all quoted from the Russian physics handbook of Yavorsky and Detlaf. Nothing here departs from textbook relativity.
The novelty is the next step. "If this tube to bend and connect its ends from a tube it is possible to receive a torus." The infinite straight tube is closed into a ring of radius R, assumed "absolutely rigid and its weight infinitely big". The two frames become O1l1 and O2l2, with curvilinear arc-length coordinates l1 and l2 running around the circle, and O2l2 circulating at constant speed V. Bodies inside the torus feel no resistance, so a free body keeps a constant speed. Cochetkov then asserts, "by analogy to Lorentz's transformations", that equations (7)–(10) hold with l substituted for x — the same transformation in the same algebraic form.
The two-body oscillator
Into frame O2l2 he places the closed mechanical system from his earlier paper: two point bodies of equal rest mass M0 joined by a perfectly elastic spring whose own mass is infinitesimal. The bodies oscillate symmetrically about their common centre of mass S, which is placed at rest at the origin O2. At t2 = 0 the spring is fully compressed and both bodies are momentarily at rest; thereafter the velocities V21 and V22 are always equal in magnitude and opposite in direction.
Viewed from O1l1, the velocities V11 and V12 follow from the velocity-addition formulas, and the momenta are written in the usual relativistic form, his equations (11) and (12),
P11 = M0V11 / √(1 − V112/c2), P12 = M0V12 / √(1 − V122/c2).
He defines the quantity P1 as the sum of the magnitudes of P11 and P12 when body 2 moves forwards, and their difference when it moves backwards — that is, the net momentum of the pair of bodies in O1l1. His figure 6 shows this quantity oscillating in time t1 between a minimum and the value 2M0V / √(1 − V2/c2). Since the torus is rigid, infinitely massive and frictionless, and space and time in the frame are homogeneous, he argues that P1 ought to be constant; relativity makes it a function of t1; therefore relativity fails.
The condition for constancy, and c = ∞
Section 3 tests this by comparing two instants: t1 = 0 (spring compressed) and t1 = t1t, the moment the spring is fully released and body 1 has acquired exactly the frame speed V. Constancy of P1 then requires his equations (13) and (14), V11t = V12t = V and V21t = V22t = 0, and these in turn require equation (15), t21t = t22t — the two bodies must reach their turning points at the same moment of frame-2 time as well as of frame-1 time. Given the transformation (10), two events with the same t1 but different l1 share the same t2 only if V l1/c2 vanishes, so Cochetkov arrives at equation (16), c = ∞.
He closes by noting that the argument is independent of R, so it holds for every one-dimensional ring; and since a two-dimensional pseudo-inertial frame can be decomposed into infinitely many such rings, the course of time is identical there too. The conclusion: in all pseudo-inertial and all inertial frames time runs alike, "that contradicts the special theory of a relativity according to which should be non-simultaneity of the events occurring in different systems of readout."
Assessment
The construction is unusually clean for a paper of this kind. Cochetkov does not appeal to paradoxes of light propagation or to any disputed experiment; he takes a system whose internal dynamics are completely specified, transforms it, and asks whether a conserved quantity stays conserved. That is a legitimate way to probe a theory, and the arithmetic of his equations (1)–(12) is textbook-correct as written. The torus is also a genuinely interesting choice, because a closed spatial loop really does raise questions that an infinite line does not.
The decisive difficulty, however, is that the momentum he tracks is not the momentum of the closed system. The system consists of two bodies and a spring. The paper dismisses the spring on the grounds that its mass is "infinitesimal in comparison with weights of bodies 1 and 2" — but in relativity what carries momentum is energy, not rest mass, and the compressed spring stores an energy U that is not negligible at all. It is exactly the kinetic energy the bodies do not have at that instant. In the centre-of-mass frame O2l2 the total energy E0 is constant and the total momentum is zero; boosting to O1l1 gives total momentum γVE0/c2, which is constant. Working the two instants of section 3 explicitly: at full compression the bodies alone carry 2M0Vγ while the spring holds U; at full release the spring holds nothing and the bodies alone carry 2M0Vγγu, where γu refers to their speed in O2l2. The difference between the two is precisely γVU/c2, since U = 2M0c2(γu − 1). The oscillation drawn in his figure 6 is therefore not a violation of conservation; it is the spring's stored energy, moving back and forth between the field and the particles, and it cancels exactly. This is a standard result, and the paper's central claim is an artefact of leaving one term out of the ledger.
A second difficulty is that step (15) assumes what is to be proved. Requiring the two turning-point events, which are simultaneous in O1l1 and spatially separated, to be simultaneous also in O2l2 is precisely the denial of relativity of simultaneity. Feeding that requirement into equation (10) can only return c = ∞; the derivation is a restatement of the premise rather than an independent consequence. Equation (17), t21 < t22, is then correctly identified as what relativity actually predicts.
Third, the torus is asserted, not derived. Equations (7)–(10) are written down "by analogy" with the flat case, but a frame circulating around a closed ring is not inertial and no global Lorentz transformation connects it to the ring frame. The interesting physics of a closed loop is the opposite of the paper's conclusion: attempting to synchronise clocks continuously all the way around the ring in the moving frame fails to close, leaving a gap of order γVL/c2 with L = 2πR. That is the Sagnac effect, measured routinely in ring-laser gyroscopes and corrected for in GPS timing, and it says that on a closed path time is less globally definable than in flat space, not more. A paper that closes a line into a circle in order to recover absolute time needs to confront the one experiment that tests exactly that geometry, and this one does not.
Finally, the conclusion stands against direct measurement of the time dilation the paper's c = ∞ would abolish: the lifetimes of circulating muons in storage rings, and the rates of atomic clocks flown around the Earth, agree with relativistic prediction to fractions of a percent. Setting c = ∞ in equations (5) and (6) restores the Galilean transformation and with it a velocity-addition law that light does not obey.