The clock that does not feel the turn

The twin who travels comes back younger, and the standard account attributes the asymmetry to the turnaround: one twin accelerated and the other did not.

That account is correct about which twin and unhelpfully vague about why. It is easy to read it as saying the acceleration causes the difference — that a clock is slowed by being pushed — and to expect a harder turn to produce a larger effect.

The opposite is true, and it is worth seeing quantitatively.

Rounding the corner #

Replace the instantaneous turnaround with a hyperbolic one: the ship decelerates at a constant proper acceleration, comes to rest, and accelerates back, so the corner becomes a smooth arc. Fix the far point of the journey and let the acceleration vary.

The turnaround’s own duration is $2\varphi c/a$ in proper time, with $\varphi$ the rapidity, so at a tenth of a gravity it lasts more than thirteen years and at a thousand gravities half a day. Its contribution to the age difference falls in the same way, and by a thousand gravities it is four decimal places down.

The total difference does not follow it down. It tends to $2.67$ years for the trip drawn, which is exactly $2(D/\beta)(1 - 1/\gamma)$ — the instantaneous-turnaround formula, obtained by taking the limit of a calculation in which the turnaround was fully accounted for.

So the acceleration can be made to contribute as little as anyone likes and the twins still differ. What the turn supplies is not the difference but the bending: it is what makes one twin’s worldline the one that is not straight, and a straight worldline between two events has the longest proper time — which is the reverse of the Euclidean statement and is what the signature of the interval buys. The acceleration is the reason there is an asymmetry to speak of, and the legs are where the asymmetry’s size is.

The two worldlines are the whole geometrical content of the twin problem, and the surprising part is a sign. In Euclidean geometry the straight line between two points is the shortest; in spacetime the straight worldline between two events has the longest proper time. The traveller who turns round takes a bent path and arrives younger, and nothing about the turn itself is doing the ageing — the bending is what makes the path shorter in the only sense that matters.

Where the difference actually accumulates #

It is worth being precise about the accounting, because “the legs” is not much more informative than “the acceleration”.

On each inertial leg the traveller’s clock runs slow by $\gamma$ relative to coordinate time in the stay-at-home’s frame. The leg lasts a coordinate time $D/\beta c$ each way, so the difference accumulated over both is $2(D/\beta c)(1 - 1/\gamma)$ — proportional to the distance travelled and independent of anything about the turn.

During the turnaround the traveller’s speed passes through zero, so the instantaneous rate difference passes through zero too: at the moment of rest the two clocks are running at the same rate. The turn therefore contributes less difference per unit time than the legs do, and it contributes nothing at all at its own midpoint.

That is the reason the limit works. Making the turn shorter removes a stretch of journey during which the twins were ageing at nearly the same rate, and replacing it with a stretch at full speed increases the difference slightly — which is why the total in the figure rises as the acceleration rises, rather than falling. It approaches the instantaneous idealisation from below, and the idealisation is the maximum rather than an approximation to something larger.

The assumption underneath #

The calculation above used a formula that has not been justified: that the proper time along any worldline is

$$\tau = \int \frac{\mathrm{d}t}{\gamma(v(t))}.$$

That expression contains the speed and nothing else. It does not contain the acceleration, or the jerk, or how long the acceleration has been going on. Writing it down is asserting that a clock’s rate depends on its instantaneous velocity alone — the clock hypothesis — and that assertion is not in the postulates of special relativity.

The postulates are about inertial frames: the laws are the same in all of them, and light has the same speed in all of them. They say nothing whatever about what happens to an object that is not in one. Getting from them to the integral above requires the additional step of saying that an accelerating clock, at each instant, ticks like an inertial clock moving at the same speed — that it is momentarily comoving.

Einstein made the step in 1905, in the paper’s discussion of two clocks brought together by different routes, and made it without comment. Everybody has made it since, and it deserves the name it now has, because it is exactly the sort of quiet assumption that is worth isolating and testing.

The light clock derives time dilation from a specific mechanism, and it shows both why the clock hypothesis is plausible and why it is not free. An accelerating light clock has its mirrors pushed, and whether the light bouncing between them notices depends on how small the clock is: a sufficiently small clock experiences a sufficiently uniform push over a sufficiently short flight time. The hypothesis is that this limit is reached, and it is an assumption about clocks rather than a theorem about spacetime.

Why it is plausible #

The argument for it is a statement about size.

An ideal clock is one small enough that the region it occupies is, over the time of one tick, indistinguishable from an inertial one. Acceleration in general relativity’s sense is not detectable locally — that is the equivalence principle — so a sufficiently small clock in free fall or under thrust cannot tell what it is doing, and must tick at its proper rate.

What is detectable locally is the variation of acceleration across the clock: a tidal effect, which scales with the clock’s size. A clock of size $\ell$ under acceleration $a$ has an internal disparity of order $a\ell/c^2$ in rate between its ends, and the clock hypothesis is the statement that this can be made negligible by making $\ell$ small.

That framing is useful because it says what would break it. A clock that is not small compared with $c^2/a$ is not an ideal clock, and there is no expectation whatever that it will obey the integral. A pendulum clock in an accelerating rocket runs at a completely different rate for entirely mundane reasons; so does a mechanical watch. The hypothesis is about clocks whose internal physics is unaffected, and asserting it for an actual object is asserting something about that object.

The hypothesis in two other forms #

The same assumption is made routinely under other names, and recognising it is a good check on how much is being taken for granted.

A rigid rod carried by an accelerating observer is assumed to have the length its instantaneous rest frame would give it — the length hypothesis, which does the same job for space that the clock hypothesis does for time and which is why a rigid body cannot really exist in relativity.

A local inertial frame in general relativity is assumed to exist at every event, with special relativity holding within it to whatever accuracy the region is small. That is the same statement again: physics at a point does not know about the acceleration of the observer who is passing through it.

And a particle’s decay rate is assumed to be a property of its proper time. That is what makes the muon experiment a test of the hypothesis rather than a test of muons: the reasoning is that if a muon’s internal clock were acceleration-dependent, its lifetime would be, and the two are the same assumption looked at from either end.

What has been measured #

The natural test is a clock that is being accelerated enormously and whose rate can be read against the speed-only prediction.

Muons in a storage ring supply it. At CERN in 1977 muons were held on a circular orbit at $\gamma = 29.3$, and their decay was timed. The dilation of their lifetime agreed with $\gamma$ — the speed-only prediction — to about a part in a thousand.

The interesting number is the acceleration those muons were under: $\gamma^2v^2/r$ works out at roughly $10^{18}$ times the acceleration due to gravity. If a clock’s rate contained a term proportional to the acceleration, that experiment would have seen it unless the coefficient were smaller than $10^{-3}$ at $10^{18}$ g — which is to say the term is bounded at a level that makes it irrelevant to anything anybody will ever build.

An earlier and quite different experiment did the same job at lower acceleration and much higher precision. A Mössbauer source on the rim of a rotating disc, with an absorber at the centre, measures the rim’s time dilation through the resonance’s frequency shift; the accelerations are of order $10^5$ g and the agreement with the speed-only formula is at the level of a per cent of the effect.

Both experiments share the feature that makes them convincing: the acceleration and the speed are varied independently. A measurement at one radius and one speed cannot separate them, and the storage ring’s value comes from comparing rings, geometries and speeds.

What has been measured is a decay. A muon is a clock whose ticking is a decay probability, and an unstable population thins exponentially at a rate its own physics fixes — so comparing a stored muon’s lifetime against a stationary one’s is comparing two clock readings. At CERN the stored muons were held at $10^{18}$ times the acceleration of gravity and their lifetimes matched the prediction from speed alone to two parts in a thousand.

What would change if it failed #

Suppose the hypothesis were false and the rate contained a term in the acceleration. Three things would follow, and listing them shows how much rests on it.

Proper time would stop being a property of a path. The integral above is a functional of the worldline alone, and that is what makes proper time a geometrical quantity — the “length” of a curve in spacetime, computed from the metric with no extra input. An acceleration-dependent term would make the elapsed time depend on the parametrisation as well, which is to say on how the journey was made rather than merely on which journey it was.

The twin result would depend on the turnaround. The whole content of the first section is that it does not, and that conclusion used the hypothesis. With an acceleration term the age difference would depend on how hard the turn was, and the limit as the turn became infinitely hard would diverge rather than converge.

And general relativity would lose its footing. The theory identifies proper time with the metric interval along a worldline, which is exactly the clock hypothesis promoted to a definition. Every use of a clock as a probe of geometry — the global positioning system, gravitational redshift measurements, pulsar timing — assumes that what a clock reads is a path length and not a history.

How the muon experiment separates the two effects #

The storage-ring measurement is worth describing in a little detail, because a single measurement at a single speed and radius would prove nothing at all.

Muons were injected into a ring of $7$ metres radius and circulated at $\gamma = 29.3$, decaying as they went; the decay electrons were counted against time. The measured lifetime came out as $\gamma$ times the lifetime of muons at rest, which is what the speed-only prediction says.

By itself that is one number and could be fitted by any number of formulae in which the speed and the acceleration both appear, because in a ring the two are locked together: $a = \gamma^2 v^2/r$, so changing the speed changes the acceleration.

What separates them is comparing experiments. The muon ring, an earlier CERN measurement at a different $\gamma$ and a different radius, the Mössbauer rotor at $10^5$ g, and ordinary linear time-dilation measurements at essentially zero acceleration all report the same relation between speed and rate. Fitting a hypothetical term $\xi(a/a_0)^n$ across that whole set is what produces the bound, and the bound is a statement about the family of experiments rather than about any one of them.

That is the ordinary structure of a null test and it is worth naming, because a single spectacular number — $10^{18}$ gravities — reads like a single decisive experiment and is not one.

The thing acceleration does do #

It would be wrong to leave the impression that acceleration has no effects, because it has several and they are dramatic.

An observer accelerating for ever has a horizon behind them: events beyond a certain surface can never send them a signal. That is the wall of silence behind a rocket, and its distance is $c^2/a$ — entirely a function of the acceleration.

More strikingly, such an observer finds the vacuum to be thermally populated at a temperature proportional to the acceleration. The Unruh temperature is $\hbar a/2\pi c k_B$, which is $4\times10^{-21}$ K at one gravity and about a kelvin at $10^{20}$ m/s².

Neither of those contradicts the clock hypothesis, and keeping them separate is the point. The hypothesis says an ideal clock’s rate is set by its speed. It does not say that an accelerating observer’s circumstances are the same as an inertial one’s, and they emphatically are not — the accelerating observer is in a hot vacuum with a horizon behind them, keeping perfectly good time.

The thing acceleration does do is set which events a worldline can reach. A uniformly accelerated observer has a horizon behind them — a boundary beyond which no signal will ever catch up — and that is a real consequence of accelerating, unrelated to any effect on the rate of a carried clock. The clock hypothesis says acceleration does not affect the ticking; it says nothing about what the accelerating observer can see.

Where it stops #

The bound is a bound and not a proof. Experiments have excluded an acceleration-dependent term above a certain size at a certain acceleration; they cannot exclude one that is smaller, or one with a different functional form, or one that appears only above $10^{18}$ g. What has been established is that the hypothesis is an excellent approximation over an enormous range, which is the most any assumption of this kind can have.

Real clocks are not ideal. Everything above concerns clocks small enough not to notice. A caesium fountain accelerated at $10^{18}$ g would not run slow; it would be destroyed, and the question of its rate would not arise. The experiments use clocks that are elementary particles precisely because those come closest to the idealisation.

A bound of this kind is model-dependent. Saying that any acceleration term is smaller than $10^{-3}$ at $10^{18}$ g presumes a form for the term — here a term linear in the acceleration, normalised at that scale. A term quadratic in the acceleration, or one that switches on at a threshold, or one involving the acceleration’s rate of change, would be constrained differently and in some cases not at all by the same data. The honest summary is that the simplest departures are excluded over a huge range and that the space of possible departures is not exhausted, which is true of every null result in physics and is worth saying rather than assuming.

And “acceleration” here means proper acceleration. The quantity that appears in the bound is what an accelerometer carried with the clock would read, not the coordinate acceleration in any particular frame. Those differ substantially at high speed, and quoting a bound in the wrong one changes the number by powers of $\gamma$.

Where it stops is where the clock stops being small. The hypothesis is a statement about ideal point clocks, and a real clock has a size — so an acceleration steep enough to vary appreciably across it will affect it, not because of any failure of relativity but because the two ends are doing different things. That is a tidal effect rather than an acceleration effect, and it is the honest boundary of the assumption.

What the figures cannot settle #

Two limitations of the drawings are worth stating, since both concern the very thing being argued about.

The first figure computes the age difference using the clock hypothesis. It has to: there is no other formula available for the proper time along an accelerated worldline. So it demonstrates that the difference is insensitive to the turnaround within the theory, which answers the question people usually mean and does not test the hypothesis at all. Only an experiment can do that, and the second figure is a summary of experiments rather than a computation.

The second is that the flat line in that figure is flat by construction. Plotting $1/\gamma$ against acceleration at fixed speed produces a horizontal line whatever the acceleration is, because $\gamma$ contains no acceleration — that is the assumption, drawn, and the drawing cannot be evidence for it. What makes the figure worth having is the other curve: the shape an acceleration-dependent term of the smallest size not yet excluded would produce, which shows how far from the data such a term would have to sit.

A figure that argues for an assumption by displaying its consequences is a figure that argues for nothing. The pair here is honest only because the second curve says what would have been seen if the assumption were false.

The ladder from here #

Later rungs on this anchor: the Rindler coordinates in which an accelerating observer’s own description is written, with the position-dependent rate that appears in them and its relation to gravitational redshift; the clock hypothesis in general relativity, where it is promoted from an assumption to the definition of proper time and where its failure would be a failure of the theory’s foundations; tests of local position invariance, which ask the parallel question about a clock’s rate depending on where it is rather than on how it is moving; and the Unruh effect in detail, where the acceleration-dependent physics that does exist is derived.

The neighbouring ladders are the twin who comes back younger, which is the calculation this makes legitimate; the ship that never arrives at c, which is the hyperbolic motion used for the turn; and the temperature of an acceleration, which is what acceleration does instead of slowing clocks.