The Twin Paradox in Special Relativity and in Lorentz Ether Theory
| Scientific Paper | |
|---|---|
| Title | The Twin Paradox in Special Relativity and in Lorentz Ether Theory |
| Read in full | Link to paper |
| Author(s) | Alexander L Kholmetskii |
| Keywords | twin paradox, special theory of relativity, Lorentz |
| Published | 2003 |
| Journal | Apeiron |
| Volume | 10 |
| Number | 3 |
| No. of pages | 27 |
Read the full paper here
Abstract
The present paper analyses the twin paradox as presented by Van Flandern [1] and confirms the results obtained by standard relativistic calculations. A variety of the twin paradox with symmetrical causal chains of events is considered, where the motional trajectories of two twins can be realized with some probability. It seems that such a probabilistic presentation of the twin paradox destroys a conception about equivalence of all inertial observers in SRT. The twin problem has been considered within Lorentz ether theory (LET). For better understanding of the observations of both twins, a mathematical apparatus of LET has been developed. It has been proved that LET postulates represent a direct consequence of the Galilean transformations in physical space-time under limited speed of light.
Overview
Alexander L. Kholmetskii's paper, published in Apeiron in July 2003, is a direct reply to Tom Van Flandern's article in the same journal on what the Global Positioning System tells us about the twin paradox. It has two distinct halves. The first works through Van Flandern's setup using the standard formalism of special relativity (SRT) and confirms his arithmetic, then presses on a point Kholmetskii thinks the standard treatment cannot absorb. The second constructs the same problem inside ether theory and argues that the paradox dissolves there.
The departure from the mainstream account is not a claim that relativity's predictions are wrong. Kholmetskii explicitly "confirms the results obtained by standard relativistic calculations," and his final theorem is that the measured intervals in a wide class of ether theories obey the Lorentz transformation exactly. The break is interpretational and, he argues, at one point empirical: the paper claims that the accelerating twin's inferred jumps in remote Earth time must be counted as apparent rather than real, that special relativity has no room for apparent phenomena in flat spacetime, and that a probabilistic variant of the experiment produces a contradiction with causality that the equivalence of all inertial observers cannot survive.
The argument
Restating Van Flandern's scenario
Two twins begin together. Twin 0 remains on Earth; twin 1 travels at v = 0.99c toward a star, with γ = 7. Each frame carries an infinite lattice of clocks, Einstein-synchronised within its own frame. The key effect, which Kholmetskii calls after Van Flandern the time slippage effect, is that while twin 0 sees each individual travelling clock run slow by γ, the succession of travelling clocks streaming past a fixed Earth point advances γ times faster than Earth time. He derives this straightforwardly: clocks synchronised in frame 1 are not synchronised in frame 0, and two clocks separated by Δxi differ in frame 0 by γvΔxi/c2, giving the reading γt for the clock passing at the moment the traveller's own clock reads t/γ.
With t0 = 49 months for the outward leg, the table is symmetric: the Earth twin's clock reads 49 months, the passing spacecraft clock 343 months, and he infers 7 months aboard the spacecraft; the spacecraft twin's clock reads 7 months, he sees 49 months on the star-side Earth clock, and infers 1 month on Earth itself.
What acceleration does
Van Flandern's turnaround has twin 1 orbit the star. Kholmetskii replaces the orbit with a uniform relativistic deceleration from v to 0 and re-acceleration to −v, and reproduces the jump. During deceleration the traveller infers a uniform gravitational field, so clocks at smaller x tick faster; when his momentary velocity is zero the Earth frame is instantaneously co-moving, so all clocks synchronised in frame 0 are synchronised in frame 1 too, and the Earth clock must read what the star-side clock reads — 4 years. Before the deceleration it read one month. The jump is Δta = γvxn/c2, and he confirms it in the low-velocity limit from the standard formula t(x) ≈ t(0)(1 − ax/c2). He stresses that "a short-time deceleration process 'causes' a large time change Δta, which does not depend on deceleration time," and that after the second stage the inferred Earth time is 8 years.
So far he agrees entirely with the relativistic account: "there is no essential change of local time under acceleration of a twin; sudden changes occur only with remote clocks."
The objection
The objection follows immediately. There is no causal relation between the spacecraft's acceleration and the readings of clocks on Earth. If the traveller orbits the star repeatedly, inferred Earth time drops back to one month each time he heads outward and jumps to 8 years each time he heads inward. "The Earth time cannot physically change by 8 years and oscillate within this range during continuous orbiting of the spacecraft." Such changes must therefore be apparent.
But, Kholmetskii argues, SRT cannot say that. Einstein's postulates make the geometry of empty spacetime pseudo-Euclidean with a Galilean metric in Cartesian coordinates in every inertial frame, and "this and only this geometry has an exclusive property: the measured space and time coordinates give their physical magnitudes directly." A theory in which measurement reads off physical magnitude has no category of the merely apparent. He concedes the escape: because the Earth and star events are not causally connected, no mathematically inconsistent result can actually be derived.
The butterflies paradox
To close that escape he constructs a variant with causal consequences. Twin 0 is at rest in a global frame G; twin 1 moves at v = 0.99c. At x = ±L sit two identical barriers, B1 at rest in G and B0 moving toward twin 0 at v. Each barrier has the property that a spacecraft passes through with probability ½ and collides — losing its velocity — with probability ½. L/v = 7 hours; each spacecraft carries a butterfly that lives 20 hours. The question is whether each butterfly is alive when its spacecraft meets its barrier.
In frame G the answer is unambiguous. Both meetings occur at t = 7 hours; twin 0's elapsed time is 7 hours and twin 1's is L/vγ = 1 hour, so both butterflies live, whatever the outcome of the encounters.
Now take twin 0's frame. If he passes through B0, he reports a live butterfly aboard 1 as expected. If instead he collides and is suddenly accelerated to velocity v in G, his own local time is unaffected — but he now knows that his frame was accelerated while frame 1 was inertial throughout, so by the standard result his clock ran slow relative to twin 1's. His clock reads 7 hours; twin 1's must therefore read 49 hours, well beyond the butterfly's 20-hour lifespan, and he concludes butterfly 1 died hours ago.
Kholmetskii draws the sting: "just before the collision, twin 0 does not know what to think: in fact, he rolls the dice in order to decide whether the butterfly 1 is alive or dead," and one of the two verdicts contradicts the factual observation in G. The natural resolution — accept the G-frame answer because no absolute events occur in it, so that G is preferred ("and 'G' can be transcribed as God") — is precisely what relativity forbids.
Light clocks in Lorentz ether theory
The second half rebuilds the problem in Lorentz ether theory, whose modern postulates Kholmetskii lists: an absolute frame K0 in which light speed is isotropic and equal to c; in a frame K moving at v, a light velocity c′ = c − v; contraction of scales by √(1 − v2/c2) along v; and dilation of time by the same factor.
Two identical light clocks are analysed. In K0 the resting clock has period 2L/c and the moving clock 2L/c√(1 − v2/c2). Transferring to twin 1's frame and using the Galilean addition law, the light in clock Cl0 has components v and c, modulus √(c2 + v2), and its round-trip time is again 2L/c; for Cl1 the vectors combine to √(c2 − v2) along y, giving the same slowed period. Both twins therefore agree on the true rates of both clocks: time dilation is absolute.
The relativity of the effect is then recovered as an artefact of measurement. Twin 1's rulers are contracted, his standard clocks dilated — neither detectable locally — and, crucially, Einstein synchronisation of separated clocks in K1 leaves a residual offset Δts = Lv/γ(c2 − v2) because of light-speed anisotropy. Carrying this error through the measurement of Cl0's rate gives Δtex = 2L/c · √(1 − v2/c2) — "exactly equal to the value derived from the Lorentz transformations." Meanwhile twin 1's own absolute dilation is invisible because his measuring clock is dilated by the same factor. So he concludes his clock runs fast and Cl0 slow, though the physical rate of Cl1 is γ times slower. This is what Kholmetskii calls "the illusory relativity of the time dilation effect."
The general theorem
The last section generalises. He introduces physical four-vectors xph and measured four-vectors xex, together with formal Minkowskian four-vectors xL subject to the Lorentz transformation, and writes xph = B(v)xL with B → 1 at v = 0, the metric in a moving frame being oblique-angled. Working out the Einstein-synchronisation error in an oblique-angled spacetime and substituting, he obtains
- (xex)0 = (xL)0
— the measured time intervals always obey the Lorentz transformation, for any' ether theory adopting a Galilean metric of absolute space. Adding the reciprocity principle A−1(−v) = A(v) forces Bα0 = 0 and extends the identity to spatial intervals, so that "an observer in any inertial frame moving in the absolute space sees the world as in SRT, for an infinite set of ether space-time theories."
He then singles out LET by making the simplest choice, A = G, the Galilean matrix, and shows that substituting it recovers, one by one, the second, third and fourth LET postulates. The conclusion he draws is his main positive result: "the postulates of LET, which appeared artificial to many physicists for a century, now represent a direct consequence of the Galilean transforms under a natural assumption about Galilean metrics in an absolute space," and this simplicity "assigns an exclusive place to LET." He notes in a footnote that LET differs from Newtonian physics in the limited velocity of light, and cites his own earlier proposal for a Mössbauer synchrotron test as evidence that SRT and ether theories are not in principle indistinguishable.
Assessment
This is a careful and technically competent paper, and the second half is its strongest part. The demonstration that measured intervals obey the Lorentz transformation for an entire class of ether theories with Galilean absolute-space metrics plus reciprocity is a genuine result, correctly derived, and it is stated with the right modesty — Kholmetskii does not claim to have refuted relativity's predictions but to have shown they can be reached from a different ontology. The observation that LET's four postulates fall out of the single choice A = G is elegant and answers a real historical complaint, that Lorentz's contraction and local time were bolt-on hypotheses. The light-clock treatment is transparent, and the separation of "true" from "measured" intervals is exactly the right conceptual apparatus for making the LET position precise. He is also scrupulous about the limits of his argument, conceding that the non-causal version of the paradox yields no mathematical inconsistency.
The interpretational objection in the first half is weaker than it is presented as being. That the traveller's inferred Earth time oscillates as he orbits is not a claim about Earth; it is a statement about which spacelike hypersurface he calls "now," and the choice of hypersurface changes when his velocity changes. Kholmetskii's argument that SRT cannot admit apparent phenomena because a Galilean metric makes measured coordinates equal physical magnitudes conflates a frame-dependent labelling of distant events with a local measurement. Nothing in relativity claims that a distant simultaneity assignment is a measurement; the invariant content is the proper time along each worldline, which no orbiting changes. The word "apparent" is doing more work in the argument than the physics licenses.
The butterflies paradox is the paper's most interesting move and also, on inspection, its clearest error. The step that generates the contradiction is twin 0's inference, after his own collision, that "his clock ticked slower than the clock in spacecraft 1" and therefore reads 7 hours against twin 1's 49. That inverts the relation. The elapsed proper time along a worldline is an invariant; twin 1 has been moving at 0.99c in G throughout, so his elapsed time to the encounter is L/vγ = 1 hour, exactly as computed in G and in frame 0 before the collision. Twin 0's later acceleration cannot retroactively alter twin 1's already-accumulated proper time, and the correct post-collision analysis has him recompute, not reverse, the ratio. What the collision changes is his simultaneity convention going forward — the same time-slippage effect Kholmetskii has already derived — not a past invariant. Once the factor of γ is applied in the correct direction the G-frame verdict is reproduced in every frame, both butterflies live, and the contradiction with causality does not arise. The paper offers no independent derivation of the 49-hour figure, remarking only that "one can show that this result can be directly derived from Eqs. (2)–(5)"; that assertion is the load-bearing step and it is not carried out.
There is also a gap between the two halves. Having concluded that measured intervals in LET obey the Lorentz transformation identically, Kholmetskii has shown that LET and SRT make the same predictions for the twins — including for the butterflies. If the probabilistic variant really produced a causal contradiction, LET would inherit it, since the observable content is identical by his own theorem. The paper does not confront this, and the claim that in LET "the twin problem loses its paradoxical nature" is therefore a claim about interpretive comfort rather than about physics. That is a defensible position, and many working physicists hold something like it; but it is not the destruction of the equivalence of inertial observers that the abstract advertises.
Finally, the empirical question is deferred rather than answered. Kholmetskii points to his own 2000 Mössbauer synchrotron proposal for a possible discriminating test but explicitly places it outside the paper's scope, so nothing here bears on whether a preferred frame exists. Against that, the accumulated null results — the Kennedy–Thorndike and modern Michelson–Morley resonator experiments, and the GPS system's own consistency under the Sagnac correction — constrain any anisotropy to parts in 1017 or better, which the theory accommodates by construction but does not predict. Read as what it demonstrably is — a rigorous account of how far a Lorentzian ether can be pushed and why its postulates are less arbitrary than they look — the paper is worth study. Read as a refutation of special relativity, it rests on one arithmetic step going the wrong way.