Twin Paradoxes

Scientific Paper
Title	Twin Paradoxes
Read in full	Link to paper
Author(s)	Thomas E Phipps
Keywords	twin problem, special relativity, paradoxical
Published	2007
Journal	Apeiron
Volume	14
Number	3
No. of pages	20
Pages	300-319

Read the full paper here

Abstract

Several paradoxical aspects of the twin problem of special relativity theory are reviewed, with the conclusion that none has been permanently resolved.

Overview

Phipps' paper is a critical review, not a new theory of motion. Its target is not the fact of differential ageing — he accepts that flatly, citing "CERN and GPS data" as putting the asymmetry beyond doubt — but the claim, repeated in a century of textbooks, that special relativity theory (SRT) has resolved the twin paradox. Phipps argues that the standard resolutions succeed only by quietly using quantities that the theory itself has declared meaningless, and that when SRT is held strictly to its own rules it "defaults on the topic."

The paper is deliberately plural in its title: Phipps separates out several distinct paradoxes tangled together in the usual discussion. The primary paradox is the logical one — how a theory whose central transformation is symmetric between inertial observers can deduce an asymmetric result. The second paradox concerns clock rates as against elapsed times: the traveller is told his twin's clock runs slow throughout the journey, yet on return finds that it must have run fast throughout. A third concerns the notion of "meaninglessness" itself as a device for protecting theory from experience, and a fourth concerns the concept of spacetime.

Phipps' own positive suggestion is modest and appears only near the end. He proposes returning the relativity principle to "its original form — that the laws of nature are the same in all inertial systems" — and allowing clock rates to be objectively, physically altered by the energy-imparting actions that change a body's state of motion. On this view the ageing asymmetry has a physical cause rather than a kinematic one, and the transformation relating the twins' times is the simple reciprocal pair t ′ = tγ with inverse t = t ′/γ, which he says accords with the GPS evidence. He acknowledges this is not developed here, referring the reader to his book Old Physics for New.

The argument

The setup

Phipps first restates the problem in its standard forms: Einstein's original closed polygonal path traversed at uniform speed v, with the traveller's clock reading carried over unchanged at each vertex; and the later simplified out-and-back trip, either with a single accelerated traveller or with three inertial observers (stay-at-home, outgoer, incomer). In both cases SRT predicts that a journey of duration T by the stay-at-home's clock corresponds to elapsed proper time T/γ for the traveller, where γ = 1/√(1 − v²/c²).

He stresses at once that the asymmetry of ageing is "not inherently paradoxical." The traveller has undergone energy-state changes the stay-at-home has not, and such changes may well affect clock rates — "just as gravity potential energy state changes (are known from other theory and observation to) affect such clocks." The paradox is not the asymmetry; it is deriving that asymmetry from a postulate of symmetry.

The primary paradox: symmetry versus asymmetry

The Lorentz transformation makes clock-rate measurements strictly symmetric: each inertial observer measures the other's clock as slow. How can symmetric rates yield asymmetric elapsed times? The standard answer invokes the relativity of simultaneity. Phipps works the standard calculation through. With D the one-way distance in the stay-at-home's frame, at the turnaround the outgoing observer assigns the distant stay-at-home's clock the reading

T₁(out) = (D/v)(1 − v²/c²)

while the incoming observer at the same event and place assigns

T₁(in) = (D/v)(1 + v²/c²).

He notes in passing that the average of these two, [T₁(out) + T₁(in)]/2 = D/v, agrees with the stay-at-home's own figure, "so in this instance the relativity of simultaneity is removable by averaging."

Taking T₁(in) and adding the incomer's account of the return leg gives total stay-at-home ageing 2D/v — the right answer, larger than the traveller's 2(D/v)/γ by a factor γ. But taking T₁(out) instead gives 2(D/v)(1 − v²/c²) — the wrong answer.

Here is Phipps' pivot. Relativists standardly declare distant clock readings on which inertial observers disagree to be "meaningless" or "irrelevant" — he cites W. H. McCrea's reply to Dingle. But if disagreement is the criterion of meaninglessness, then T₁(in) and T₁(out) are equally meaningless, and neither may be used. "But, deprived of both these quantities... the theory cannot be used by either the traveler or the stay-at-home to make any legitimate inferences whatever about total agings during the journey." Since total ageing between co-located departure and return events is plainly an absolute physical fact, "paradox is conserved." Asymmetry enters accepted analyses, Phipps says, "only through the asymmetrical introduction of the meaningless number T₁(in)."

He is careful to concede what is not in dispute: a Minkowski diagram does display the asymmetry, and experiment leaves no doubt of it. "Nature is unambiguous; she has simply not been meaningfully described."

The second paradox: clock rates versus elapsed times

If the traveller's own clock runs at rate R, SRT tells him the stay-at-home's runs at R/γ throughout; observation on return tells him it must have run at Rγ. Phipps examines the usual reconciliation — the account of Taylor and Wheeler, in which a "clock phase-jump" at turnaround replaces the outgoer's notion of distant simultaneity by the incomer's. He objects that this resetting "corresponds to no operation performed by any observer. Nobody can actually do this resetting... Its locus is the head of the traveler."

He then runs a bookkeeping argument. Let S be the number of ticks of the stay-at-home's clock "measured" during each leg, and J = 2Dv/c² the jump. The traveller attributes 2S + J ticks in total, but only 2S are "measured", so J ticks appear to vanish — which is absurd, as signal-exchange analyses show. Phipps grants that SRT has an answer: the 2S figure is obtained not by one observer but by a spatially extended set of co-moving observers, so the Lorentz coupling of space to time accounts for the discrepancy. "Such an explanation will satisfy most relativists." His reply is that the same procedural symmetry applies to both twins, so no explanation of the observed asymmetry survives without "the moral equivalent of the one-sided J-jump."

He then puts the case in human terms: an ordinary astronaut, not a mathematician, "has never encountered a biological process that undergoes anything analogous to a clock resetting, time skip, or phase jump." He could believe a genuine steady rate asymmetry — that would contradict none of his experience. A discontinuous jump in his twin's age challenges his credulity, and the twin's own testimony. Phipps' summary is blunt: the theory "obtains its right answers by concatenating manifestly false stories," compensating falsehood #1 (the stay-at-home's clock appears slow) with falsehood #2 (the stay-at-home's clock appears to jump). The "simple truth, thoroughly obscured by these artifacts, is that the stay-at-home's clock runs steadily faster than the traveler's."

The meaning of meaninglessness

Phipps devotes a section to "meaninglessness" as a general device. It is not, he says, a concept internal to any physical theory but one "introduced from outside to hide (by a master stroke of verbal obfuscation) discrepancies between theory and experience." He gives precedents: imaginary velocities in Newtonian mechanics, and above all the advanced solutions of Maxwell's field theory, which are mathematically as good as the retarded ones and are discarded solely because they conflict with experience. The choice of T₁(in) over T₁(out) is, he argues, exactly analogous — a choice made on grounds no stronger than plausibility, and made because one option "gives the answer we are determined to get."

Worse, he says, plausibility here favours the third party: only the stay-at-home's account exhibits the timekeeping symmetry between the two legs that matches their spatial and speed symmetry, and only his account is free of J altogether. In an aside characteristic of the paper's tone, "the incoming inertial observer is more equivalent than the outgoing one."

Spacetime as an incidental paradox

The final section attacks the identification of time with a spatial dimension. A space coordinate, Phipps argues, is single-valued — a measure of linear extension marked by one number. Time is bi-valued: the parameter t serves both to label a moment (a date or epoch) and to represent flow (a rate). "Relatively to our laboratory, a point of space can be revisited, a point of time cannot." He notes that the twin problem itself forces this very distinction into the open, since it is the "date" aspect that undergoes the jump J while the "flow" aspect is what clock rates measure — the sinusoid analogy being phase versus frequency. It is paradoxical, he concludes, that a theory built on undifferentiated "spacetime" must invoke time's bi-valuedness, absent in space, to handle the very first problem of non-uniform motion it meets.

Acknowledgment

Phipps notes that the paper grew out of discussions with C. Roy Keys and Stephane Baune, and that "the analysis on which it is based was done by Dr. Baune, who, however, does not subscribe to my lucubrations."

Assessment

The paper's real strength is that it separates questions usually run together. Phipps concedes the empirical facts outright — differential ageing is real, CERN and GPS confirm it — so nothing here depends on denying data. What he isolates is a genuine tension in the standard exposition: the criterion by which distant simultaneity judgements are dismissed as "meaningless" is applied selectively, since the incoming observer's T₁(in) is used to obtain the accepted answer while the outgoer's T₁(out), equally the product of an inertial frame, is not. That is a fair point about the rhetoric of the resolution, and it is sharpened by the observation that the stay-at-home's account alone is jump-free and leg-symmetric. His distinction between clock rates and elapsed times, and between time's "flow" and "date" aspects, is also a clarifying one that survives independently of his conclusions.

The difficulties are equally clear. First, the argument is largely one about interpretation rather than prediction: Phipps nowhere disputes that SRT computes the elapsed times correctly, and he says so. A defender will reply that a theory whose observable predictions are confirmed is not refuted by discomfort with the language used to explain it, and that the "meaninglessness" of a coordinate-dependent quantity is a statement about frame-dependence, not a suppression of data. Second, Phipps himself acknowledges the standard rebuttal of his tick-counting argument — that the 2S figure belongs to an extended set of co-moving observers rather than a single detector — and grants it "will satisfy most relativists"; his counter, that the same applies symmetrically to both twins, asserts rather than demonstrates that no asymmetric account can then be given, since the traveller's frame changes and the stay-at-home's does not.

Third, and most substantially, the alternative is only gestured at. The transformation t ′ = tγ with inverse t = t ′/γ is stated in a bracketed remark, not derived, and the claim that it accords with GPS evidence is referred to Old Physics for New rather than argued here. Whether such a reciprocal, absolute-rate scheme can reproduce the full body of results ordinarily credited to the Lorentz group — including the transverse Doppler and particle-lifetime measurements — is not addressed in this paper. Finally, the closing pages shift from analysis to polemic about editorial policies and the "mob psychology" of the profession; readers unpersuaded by the technical case are unlikely to be moved by that, and it makes the paper easier to dismiss than its better arguments deserve.