What the Global Positioning System Tells Us about the Twin's Paradox
| Scientific Paper | |
|---|---|
| Title | What the Global Positioning System Tells Us about the Twin's Paradox |
| Read in full | Link to paper |
| Author(s) | Tom Van Flandern |
| Keywords | GPS, atomic clocks, twin paradox |
| Published | 2003 |
| Journal | Apeiron |
| Volume | 10 |
| Number | 1 |
| No. of pages | 18 |
| Pages | 18 |
Read the full paper here
Abstract
In the GPS, all atomic clocks in all reference frames (in orbit and on the ground) are set once and stay synchronized. We can use this same trick to place a GPS-type clock aboard the spacecraft of a traveling twin. That clock will stay synchronized with Earth clocks, allowing a clear resolution of the twin's paradox in special relativity — why the traveller expects to come back younger, and why the stay-at-home twin is not entitled to the same expectation.
Overview
This is the sequel to Van Flandern's earlier What the Global Positioning System Tells Us about Relativity (in Selleri's Open Questions in Relativistic Physics, 1998). The earlier paper argued that the GPS favours Lorentzian relativity (LR) over Einstein's special relativity (SR) on grounds of simplicity; this one takes the same engineering fact and turns it into a thought experiment. Since every GPS clock is rate-adjusted once before launch and thereafter stays synchronized in epoch and rate with every ground clock, Van Flandern proposes putting such a clock aboard the spacecraft of the travelling twin. The traveller then carries two clocks: an unadjusted "natural" clock, and a GPS clock that by construction always reads the same as the nearest Earth-frame clock outside the window.
The departure from the textbook account is twofold. First, Van Flandern rejects the standard resolution of the twin's paradox — that the traveller's acceleration breaks the symmetry — as "illusory," on the grounds that accelerations can in principle be instantaneous and that cyclotron experiments show accelerations as large as 1019 g have no effect on clock rates. Second, he insists that SR and LR are not cosmetically different: they differ over whether material bodies can exceed the speed of light in forward time, and the GPS as actually built uses Lorentz synchronization to a preferred frame, not Einstein synchronization. He is careful to concede that SR is internally consistent and that no mathematical contradiction can be produced from it; his argument is about credulity and practicality, not about a formal error.
The argument
The GPS as a realization of Lorentz's "universal time"
Van Flandern's factual starting point is that after a one-time pre-launch rate adjustment, all GPS satellite clocks in all orbits remain synchronized with each other and with ground clocks, needing no further relativity corrections apart from a small term for orbital non-circularity. This, he says, is a practical realization of Lorentz's universal time. The conventional account achieves the same thing by synchronizing each clock to an imaginary clock in the Earth-centred inertial (ECI) frame instantaneously co-located with it and assumed to sit at the sea-level potential at Earth's poles — and he notes the "coincidence" that this device makes use of precisely the Lorentzian preferred frame, the local gravity field.
He then poses the converse. Had the clocks not been rate-adjusted pre-launch, Einstein synchronization would require corrections unique to each observer's frame and changing from moment to moment, since both clocks are accelerating. The gain would be constancy of the measured speed of light in all inertial frames; but since all clocks are in fact synchronized to the ECI frame alone, invariance in other frames "is of no practical value." His sharpest illustration is a closure argument: if a ring of satellites A, B, C, …, Z circled the Earth in a common orbit and each Einstein-synchronized with the next, then when Z closed the circuit back to A the required corrections would give A a different reading from its starting one, making closure impossible.
He also disputes the claim that SR's two postulates are experimentally confirmed. None of the eleven independent experiments testing SR verifies either postulate or tests frame reciprocity; and following Erlichson he holds that no experiment can do so in principle, since the postulates become true by convention under Einstein synchronization and false under Lorentz's. In a footnote he records that de Sitter, Sagnac, Michelson and Ives each concluded from their own experiments that SR was falsified in favour of the Lorentz theory, and that "subsequent re-interpretation of SR allowed that theory to survive these objections."
Setting up the twins
Two twins separate; one travels to Alpha Centauri, four light-years away, at 0.99c, giving γ ≈ 7 — a deliberately large value so the effects are "large and obvious, not subtle." The round trip takes 98 months of Earth time; the traveller returns 14 months old. To remove acceleration from the argument entirely, the traveller may simply be replaced at turn-around by an inbound traveller of identical biological age passing the other way. The paradox is then stated in its sharpest form: why is the traveller not entitled to regard the spacecraft as at rest and the Earth as having made the round trip?
Time slippage: the term that does the work
The technical core is the clock-epoch or "time slippage" term vX/c2 in the Lorentz transformation
- t = γ(T − vX/c2), x = γ(X − vT)
together with its reciprocal inverse. Van Flandern evaluates it twice. Looking at the single spacecraft clock, X = vT, the transformation gives x = 0 and t = T/γ: the familiar result that the moving clock ticks seven times slower. But looking instead at the fixed Earth position X = 0, one gets t = γT: the Earth observer sees a succession of spacecraft-frame clocks parading past, and time on that succession runs seven times faster than Earth time.
This is the point he wants the reader to dwell on: "in any inertial frame with a relative motion, all individual clocks tick slower but overall frame time moves forward at a faster rate," and in general the slippage effect dominates the rate effect. Crucially, the effect survives even if the GPS trick is used to eliminate all rate differences, since slippage is a function of location, not of rate. Because SR is reciprocal, exactly the same two results hold from the spacecraft frame looking back — which is SR's answer to the symmetry challenge.
Working the journey through
At arrival at Alpha Centauri, 49 months of Earth time have elapsed and seven months of spacecraft time. But the traveller infers that only one month has passed on Earth, because a single remote clock is affected by rate slowing alone; he simultaneously agrees that 49 months have elapsed at Alpha Centauri, because the journey began with a 48-month slippage for AC plus one month accrued en route. In calendar terms: a journey commencing 2000 January has the on-board GPS clock reading 2004 February on arrival, while the traveller infers Earth clocks read 2000 February and an AC resident infers 2004 February — "and both are correct for their respective frames."
The turn-around changes nothing locally but reverses the sign of the slippage, so the traveller now infers Earth time to be 2008 February. One more inferred month elapses on the return, and all parties agree Earth reads 2008 March at reunion. LR gives the same arrival ages by a different route: the traveller simply had a high speed relative to the preferred frame, and there never was any symmetry.
Van Flandern then removes the turn-around altogether. Let the spacecraft continue past AC to Beta Centauri, eight light-years out. There is no turn-around event and no acceleration, yet the traveller is still 14 months old on arrival while twins born on Beta Centauri simultaneously with the Earth twin are 98 months old — and the traveller infers those twins began the journey already 96 months old and aged two months during it. Neither acceleration nor turn-around is essential to the result. A footnote adds that if SR expects a frame change to affect remote clocks, that would itself constitute instantaneous action at a distance.
The orbiting variant
The paper's rhetorical climax replaces the instantaneous turn-around with several orbits around Alpha Centauri. Each time the traveller heads outbound, inferred Earth time drops back to 2000; each time inbound, it jumps to 2008; intermediate orbital positions give intermediate years. Van Flandern draws the consequence bluntly: on each cycle to 2008 many people on Earth will have died and others been born, and on each reversion to 2000 "some of the dead will be resurrected and some living young children in 2008 will cease to exist." All the while the on-board GPS clock, representing LR's universal time, insists it is 2004 on Earth, at AC and aboard ship alike. He acknowledges that SR shields these effects from observation because they lie outside the observer's light cone, but stresses that SR nonetheless regards them as real physical time, not illusion.
The concluding contrast is that LR treats a changed clock rate as a change in the rate of a physical process — "just as we do not assume that time has been affected when the temperature rises and causes a pendulum clock to slow down" — leaving space and time themselves untouched, preserving a universal "now," and removing c as a speed limit. He closes by pointing to his separate work on the speed of gravity, arguing that anything propagating faster than light in forward time would falsify SR outright, and that physics "may have no speed limit when the driving forces are gravitational or electrodynamic rather than electromagnetic in nature."
Assessment
The paper is unusually well constructed for a critique of relativity, and its central pedagogical point is genuinely valuable. Most textbook treatments of the twin's paradox do lean on acceleration to break the symmetry, and Van Flandern is right that this is not where the asymmetry lives: the Beta Centauri variant, in which nobody accelerates and no turn-around occurs, makes the point cleanly, and the relativity-of-simultaneity term vX/c2 really is what does the work. The observation that the traveller's inference about "now on Earth" jumps discontinuously across the turn-around, and can be made to oscillate by orbiting, is a correct consequence of the transformation and is stated more vividly here than in most textbooks. His concession that SR is internally consistent, and that his objection is to its demands on credulity rather than to its mathematics, is honest and is what separates this paper from cruder attacks. The GPS-clock device itself is a clean way to make an operational preferred frame visible inside a thought experiment.
The difficulties begin with the operative word "infers." The oscillating Earth-year — 2000, 2008, 2000, 2008 — depends on identifying "Earth time now" with the time coordinate of the traveller's instantaneous inertial frame under Einstein synchronization. That coordinate is a labelling convention, and no measurement, signal or causal influence tracks it; nothing on Earth changes when the traveller turns. The people being "resurrected" are an artefact of re-labelling a coordinate axis, not a physical prediction, and SR's claim that both frames are "correct" is precisely the claim that the label carries no ontological weight. Van Flandern's insistence that SR regards these inferences as "real, physical time, and not an illusion" is an interpretive gloss he supplies; the relativist reply is that a coordinate assignment is neither real nor illusory but conventional. Because the whole polemical force of the orbiting variant rests on that gloss, the argument does not reach the theory it is aimed at.
Several other steps are asserted rather than shown. That the GPS is "designed to use Lorentz synchronization" is true as an engineering description of the ECI-referenced timescale, but the ECI frame is used because Earth's mass defines a convenient locally non-rotating frame, not because it is dynamically preferred — and the standard analysis derives the same rate offsets from general relativity without any preferred frame. The ring-of-satellites closure argument is really the Sagnac effect, which GPS handles explicitly with a rotation correction; that a rotating chain of clocks cannot be consistently Einstein-synchronized is a known and understood result rather than a difficulty for SR. The claim that none of eleven experiments tests frame reciprocity is a strong one, and the paper supports it only by reference to Erlichson.
The paper is also silent on the measurements that most directly constrain a preferred frame. Modern Michelson–Morley and Kennedy–Thorndike experiments using cryogenic optical resonators, and Hughes–Drever clock-comparison tests, bound frame-dependent anisotropies at the 10−17 level, and any LR variant must reproduce those nulls exactly — which it does, at the cost of the observational equivalence Van Flandern himself concedes. Since he grants that SR and LR "explain all existing electromagnetic-based experiments," the choice between them cannot be settled by the GPS, and the paper's real case reduces to a simplicity argument plus the separate and much more contentious claim about the speed of gravity. That claim is asserted here and argued elsewhere; a reader should note that the standard rebuttal — that the near-cancellation of gravitational aberration in general relativity arises from velocity-dependent terms in the field, not from instantaneous propagation, and that the (1+z) time dilation of Type Ia supernova light curves and binary-pulsar orbital decay both track the retarded-field prediction — is not engaged. Within its stated aim, however, the paper does what it sets out to do: it exhibits, with correct arithmetic, exactly which term in the Lorentz transformation resolves the twin's paradox, and why acceleration is a red herring.