Einstein's Lorentz Transformation is a Mathematical Game

Scientific Paper
Title	Einstein's Lorentz Transformation is a Mathematical Game
Read in full	Link to paper
Author(s)	Qing Zeng
Keywords	Einstein?s Lorentz transformation, positive transformation, inverse transformation, the geniture of infinite relativity theories
Published	2010
No. of pages	10

Read the full paper here

Abstract

This paper indicates that the calculation of time dilation in relativity theory is as the indirect calculation of t' from 'positive transformation'. While, the calculation of the length contraction is different. X is gotten from the 'inverse transformation', and then x' is resolved. From the mathematical point, if we reverse the calculation methods, it will be time contraction and length dilation... In addition, this paper adopts Einstein's methods and gets W relativity theory. When W is given infinite value, there will be infinite relativity theories. From these, we can conclude that Einstein's Lorentz transformation is a mathematical magic, and it is not only without any mathematical logic, but also without any physical significance.

Overview

The author (bylined in the PDF as Zeng Qingping, of the Air Force Radar Academy, Wuhan) attacks not the postulates of special relativity but its bookkeeping. His observation is that the textbook derivations of the theory's two signature results draw on different halves of the same transformation pair: time dilation is read off what he calls the "positive transformation" (S′ → S), length contraction off the "inverse transformation" (S → S′). Nothing in the mathematics, he argues, dictates that choice. Swap the two and one obtains time contraction and length dilation; use the inverse for both and one obtains contraction of both; use the positive for both and one obtains dilation of both. If four mutually contradictory results follow from four equally available routes through the same algebra, then the algebra carries no physical content and "Lorentz transformation is a pure mathematical game."

The second half of the paper turns the same method into a construction. Instead of postulating that light travels at c₀ in both frames, the author postulates c₀ in S′ and an arbitrary value w in S — deliberately abandoning light-speed invariance — and runs Einstein's algebra unchanged. The result, "w relativity theory", has a length-contraction formula identical to Einstein's and a time formula differing only by a constant factor. Since w is free, there are infinitely many such theories, and the author takes this to vindicate Lorentz's own remark that local time is "just a mathematical hypothesis without real physical meaning."

The argument

Deriving the transformation

The paper first reproduces the standard derivation to fix notation. From the light-sphere conditions x′² + y′² + z′² = (c₀t′)² and x² + y² + z² = (c₀t)², with y = y′, z = z′ and the linear ansatz x = ax′ + bt′, t = ex′ + ft′, matching coefficients gives a² − c₀²e² = 1, c₀²f² − b² = c₀² and c₀²ef = ab; the condition that x = 0 corresponds to x′ = −vt′ gives b = av. Solving yields a = f = γ = 1/√(1 − β²), and the familiar pair, which the author labels the positive transformation (10) and, on inversion, the inverse transformation (11).

The four routes

For a clock at rest at x′ in S′, the positive transformation gives Δt = γΔt′ — dilation. For a rod at rest in S′ measured by an S observer, the inverse transformation gives l = l′/γ — contraction. The author then performs what he calls the "[Imitation]": applying the inverse transformation to the clock problem gives Δt′ = γΔt, "time compression"; applying the positive transformation to the rod problem gives x₂ − x₁ = γl′, "length dilation". His conclusion: "the trick of Einstein is: using Lorentz transformation (10) to get time dilation result and then using Lorentz reverse transformation (11) to get length contraction result."

The light-sphere objection

A subsidiary complaint, illustrated by three figures, concerns which frame the flash belongs to. If the source is in the moving system it is in motion; if in the static system it is at rest; if the two origins spark on coincidence there are "two light sources at the same time", of the same frequency and status, and hence two spherical wavefronts. "How did Einstein join two spherical waves?" The author's answer is that the joining is really a statement about the space positions of the wavefront, not about light speed at all — and that one could equally well impose x² + y² + z² = (wt)² in S and still obtain a transformation of the same shape.

w relativity theory

That is the construction. With the S light-sphere written as (wt)², the same coefficient-matching gives

x = γ(x′ + vt′),    t = γ(c₀t′ + βx′)/w

with inverse x′ = γ(x − vw/c₀ · t), t′ = γ(wt − βx)/c₀. The length contraction that follows is l = l′√(1 − β²), which the author notes is "fully equal to that of Einstein", while the time relation carries the extra factor c₀/w, which he describes as a "high-order infinite small amount of difference". Since w may be assigned any value, "when given infinite values, there will be infinite relativity theories" — and a length contraction derived from a variable light speed reproduces exactly the one Einstein derived from an invariant light speed.

Assessment

Two of the paper's technical results are correct, and one of its two main conclusions does not follow from them.

The w-transformation is algebraically sound. Repeating the coefficient matching independently — a² − w²e² = 1, w²f² − b² = c₀², ab = w²ef, b = av — one does get a²(c₀² − v²) = c₀², so a = γ with β = v/c₀, together with f = γc₀/w and e = βγ/w. The author's Eqs. (10)′ and (11)′ are exactly right, and a direct check confirms that light does propagate isotropically at speed w in S under them: for x′ = ±c₀t′, x/t = ±w. His statement that the spatial factor is untouched is likewise correct, since a comes out independent of w.

But that last fact is the diagnosis, not the mystery. Comparing the two transformations term by term, x_w = x_Einstein exactly and t_w = (c₀/w)·t_Einstein. The "w relativity theory" is Einstein's Lorentz transformation with the S-frame time coordinate multiplied by a constant — that is, with the S second redefined. Nothing else is different, which is precisely why the length formula is unchanged and why the time formula differs by a pure constant. Test the construction against itself: since x is untouched and t is rescaled, every velocity in S, not only that of light, is multiplied by w/c₀. If w ≠ c₀, a train that moves at 100 km/h in Einstein's description moves at 100·(w/c₀) km/h in the author's. Either that is a real prediction, in which case it is a claim about ordinary mechanics and is wrong, or the S clock has simply been recalibrated, in which case nothing has been constructed. The "infinity of relativity theories" is an infinity of choices of the second. A genuinely different light-speed postulate would have altered the spatial coefficient too; that it did not is the tell.

The paper's own numerical characterisation also slips here. It writes the discrepancy as c₀/w = c₀/(c₀ + v) and calls it a "high-order infinite small amount". It is not: c₀/(c₀ + v) ≈ 1 − β, a difference of first order in v/c. For the Earth's orbital speed that is one part in 10⁴ in every time interval measured in S — the same order as the first-order ether-drift effects that terrestrial optics ruled out in the nineteenth century, and far above what modern frequency standards would tolerate. A first-order term described as higher-order is an error of arithmetic, not of interpretation.

The "four routes" argument fails for a plainer reason: each of the four calculations omits its own physical side condition, and the omitted conditions are what make the two standard results non-arbitrary. A clock measurement fixes x′ (the clock sits at one place in S′); a length measurement fixes t (both ends are marked at one instant in S). Once those conditions are written down, the choice of which transformation to substitute into is forced, not free. Inspect the author's own "[Imitation]" formulae and the substitution he has actually made is visible in them. In his "time compression" derivation he holds x fixed across both events — which means the clock is now at rest in S, not in S′. His Δt′ = γΔt is therefore not a contradiction of Δt = γΔt′; it is the same result with the frames exchanged, and its existence is the reciprocity that special relativity insists on. In his "length dilation" derivation he holds t′ fixed across the two ends of the rod. The two events so selected are not simultaneous in S (they differ by γβΔx′/c), so their spatial separation in S is not the length of anything measured in S. The paper has changed the experiment and reported the change as an inconsistency in the theory. That the relativity of simultaneity is never mentioned in the section is not incidental; it is the whole of what has gone missing.

The same gap accounts for the light-sphere objection. There are not two flashes but one event at the coincidence of the origins, and one light cone. The two "spheres" of Figs. 2–4 are two different simultaneity slices through that single cone, which is why both descriptions can be spherical without either being a second wave to be "joined". Finally, the historical premise is only half right: Lorentz did treat local time as an auxiliary quantity, but he proposed the contraction of moving bodies as a physical effect of motion on intermolecular forces, not as a mathematical convenience, so he cannot be enlisted as a witness that the transformation "is without physical meaning".

What is worth keeping is the observation that started the paper. It is true that the standard textbook presentation moves between the two transformations without always saying why, and a reader who is not told to track the side conditions can easily be left thinking the choice is discretionary. The paper is a clear demonstration of what happens if it is treated that way. The English is a translation and the author says so directly at the end of the text — "Above translat possible hove problems" — and some passages, particularly the discussion of the figures, are hard to follow as a result.

See also

• Qing Zeng
• Lorentz Transformation
• Special relativity
• Simultaneity
• Length Contraction
• Time Dilation
• Speed of Light
• Hendrik Lorentz
• Albert Einstein
• Hermann Minkowski
• Galilean transformation
• Michelson–Morley experiment