The longest way round is the shortest clock

The twin who comes back younger settles a particular case: one twin goes out and comes back, the other stays, and the traveller’s clock reads less. The usual explanation invokes the turnaround, which is where the asymmetry between the two lives.

That explanation is not wrong and it is answering a smaller question than the one worth asking. The general statement covers every pair of worldlines between every pair of events, needs no turnaround, and has a form that survives into general relativity unchanged.

The extremal principle, with the sign that surprises #

Of all worldlines joining two timelike-separated events, the inertial one carries the most proper time.

That is the whole statement, and the only surprising thing in it is the word most. Euclidean intuition says a straight line is the shortest path, and every physical variational principle taught before this one — least time, least action, least energy — is a minimum. Here the straight route is a maximum.

The reason is one sign. The interval is $c^2\mathrm{d}t^2 - \mathrm{d}x^2$, with the space term subtracted rather than added, so moving sideways reduces the interval instead of adding to it. Every metre of wandering is subtracted from the accumulated time, and the route that wanders least keeps the most.

The quantity nobody argues about establishes that the interval is what every observer agrees on, so this is a frame-independent statement about frame-independent quantities. Which route carries more time is not a matter of who is measuring.

Flat at the top, which is the useful part #

The curve’s flatness at its peak is worth more than its height.

A quantity that is stationary at a configuration can be turned into a variational principle: the condition “the first-order change vanishes” picks out the configuration, and that condition is a differential equation. Applied here, “proper time is stationary along the actual worldline” gives the geodesic equation, and in flat spacetime the geodesic equation says the acceleration is zero.

So the principle reproduces Newton’s first law rather than assuming it, and it does so in a form that will still work when the spacetime is curved and there are no straight lines to appeal to.

The flatness also settles a common misreading of the twin problem. Because the falloff is second order in the wandering, a route that differs only slightly from the straight one loses almost nothing — which is why the effect is invisible for anything moving slowly, and why the deficit is not attributable to any single moment of the trip. It is an integral over the whole route, and the turnaround is only where the route is forced to differ, not where the time is lost. That is the same point the clock that does not feel the turn makes by rounding the corner off and watching the answer not change.

More excursions, and the same verdict #

Changing the shape of the detour is a useful control, because a result about one family of curves is a result about one family of curves.

Two swings in the same interval need twice the speed for the same amplitude, so the drawn excursions have to be halved to keep the routes physical — which is itself a reminder that a worldline is not free to be any shape at all. The generator refuses an amplitude that would need light speed, and the refusal is part of the figure rather than a guard around it.

What does not change is the ordering. The straight route still carries the most, and the deficit still grows with the wandering. That is the general theorem doing its work: it says nothing about the shape of the detour, only that there is one.

A deficit that does not care about the shape is a deficit that comes from the metric, and this is the practical way to tell that apart from a deficit that comes from a mechanism. A mechanism — a force applied at the turnaround, a stress in the traveller’s body — would care a great deal which shape the route had.

Two measures on one drawing #

Putting both measures on one plot makes the sign concrete rather than rhetorical.

The curves are literally the same curves. Nothing about the drawing changed between the two measurements; the only difference is whether the square of the horizontal displacement is added to the square of the vertical one or subtracted from it. One choice makes the straight route the shortest and the other makes it the longest.

That is worth sitting with, because it is the single structural difference between Euclidean geometry and the geometry of spacetime, and almost everything else follows from it. The light cone exists because a subtracted square can cancel; the interval can be zero between distinct events for the same reason; and boosts are hyperbolic rather than circular rotations because a hyperbola is what a difference of squares holds constant.

There is a familiar object that behaves the same way for the same reason, and it is worth naming because it makes the reversal less exotic. A hyperbola holds a difference of squares constant, and among the points on one branch there is no nearest one to the centre in the Euclidean sense and there is a furthest in the hyperbolic sense. Every calibration curve in a spacetime diagram is such a branch, and every statement about proper time is a statement about where a route sits relative to those curves.

A minus sign is not a small change to a geometry. It is a different geometry, and the reversal of an extremal principle is one of its plainer consequences.

The ball that maximises its own clock #

The principle survives gravity, and there it stops being a restatement of Newton’s first law and becomes a replacement for his second.

A clock higher in a gravitational field runs faster, by $gh/c^2$ per unit time. A clock moving runs slower, by $v^2/2c^2$. A thrown ball trades one against the other: it climbs, gaining from the height term, and it has to move to get there, losing from the speed term.

The figure fixes the two endpoints — the ball leaves and returns four seconds later — and scans a family of paths joining them, scaling the real parabola up and down. The proper time is greatest at the real trajectory, and the search locates it to better than a part in five hundred.

So the answer to why a thrown ball follows a parabola is that a parabola is the path with the most time on it. No force is needed in the statement. That is the equivalence principle’s version of gravity, and this figure is the simplest quantitative demonstration of it that fits on a page.

The numbers are worth noticing too. Over a four-second flight the whole effect is a few hundredths of a nanosecond, which is smaller than the ball’s own thermal expansion in any meaningful sense — and it is enough, because it is a maximum and the path is determined by where the maximum is rather than by how large it is.

What the principle refuses to explain #

A principle this clean invites a question it cannot answer, and being clear about that is part of using it.

It does not say why a free particle takes the extremal route. In Newtonian terms the first law is an assumption and here it is a theorem about a functional, which moves the assumption rather than removing it: something has to be said about why proper time is the thing extremised. Quantum mechanics supplies an answer — every route contributes an amplitude, the phases cancel except near the stationary one, and the classical path is where the cancellation fails — and that answer belongs to a different subject.

It also does not privilege any route as “what really happened” in a case where several are extremal. Two geodesics between the same pair of events are equally good answers, and in gravitational lensing both are taken, by different photons, arriving at different times. A principle that selects a stationary point selects all of them.

And it says nothing about routes that are not free. A rocket burning fuel follows a route with less proper time on it than the free route, and the principle does not forbid that or make it costly in any sense the rocket notices. What it does is identify which route needs no explanation, which is a narrower and more useful job than it first appears.

An extremal principle answers “which one” and never “why that one”, and reading it as an explanation of the second kind is how variational principles acquire a mystical reputation they do not deserve.

Where the two terms come from #

It is worth separating the two contributions in that calculation, because they have different origins and are usually taught in different chapters.

The speed term is special relativity: a moving clock runs slow, and the effect is $v^2/2c^2$ to first order. It has been measured in aircraft, in storage rings, and by comparing atomic clocks moved at walking pace.

The height term is general relativity, or rather is the one piece of it that follows from the equivalence principle alone: a clock higher in a field runs fast, by $gh/c^2$. It has been measured over a tower, over a metre, and now over a centimetre with optical clocks.

In free fall the two are of comparable size and opposite sign, and their competition is the whole content of the trajectory. Near the Earth’s surface a satellite in low orbit has the speed term winning; a satellite in geostationary orbit has the height term winning; and the navigation satellites sit where the two nearly cancel, which is why their clock corrections are a small residue of two much larger numbers.

Gravity as geometry is not a metaphor here. The trajectory is being computed by extremising a length in a curved geometry, and the answer is Newton’s, and the calculation used nothing that Newton would recognise as a force.

The same principle in three currencies #

The statement has three standard forms and it is worth putting them side by side, because each is the one somebody was taught and none of them looks like the others.

As a maximum of proper time, which is this essay: among routes between two events, the free one carries the most time on its own clock.

As a stationary action. The action of a free relativistic particle is minus its mass times its proper time, so maximising the time is minimising the action, and the machinery of Lagrangian mechanics applies unchanged. The minus sign that turns one into the other is a convention, and it is the reason the free-particle Lagrangian is written with a square root and a leading minus that looks arbitrary until this is said.

As a geodesic, in the sense the clock that has to slow sets up. The stationarity condition, written out, is the geodesic equation: the four-velocity is parallel-transported along the worldline. That is the form that generalises, because it never mentions comparing routes and works locally.

The three are the same statement and they have different reaches. Proper time is the most physical and needs the two endpoints. The action connects to the rest of mechanics and to quantum mechanics, where the phase of a free particle’s amplitude is the action over $\hbar$ and the classical path is where the phases stop cancelling. The geodesic equation is the one that survives into a spacetime where “route between two events” is no longer a well-posed comparison.

Which form is the right one depends on what is being asked, and a subject that offers three equivalent statements of the same principle is usually offering three different generalisations of it.

Where the model stops #

The maximum is local. In curved spacetime two events can be joined by more than one geodesic — light bending around a massive object is exactly that — and then no single one is the global maximum. The principle is properly “stationary”, and the global statement holds only within a region small enough for the geodesic to be unique.

Only timelike routes are compared. The result says nothing about spacelike separations, where the interval is a distance and behaves the other way round, and nothing about null routes, whose proper time is zero and for which the principle has to be restated entirely.

The gravitational calculation is to first order in $1/c^2$. Both terms are the leading corrections, and the Schwarzschild treatment differs at the next order. Nothing about the argument changes; the coefficients do, and for a ball on Earth the difference is beyond anything measurable.

And the family of trial paths is one-parameter. Scanning a scaled parabola shows the real trajectory is a maximum within that family. Proving it against all paths is the calculus of variations, which the figure gestures at and does not perform — although the family was chosen so that the maximum is at exactly one, which is a test the figure could have failed.

And nothing here is a statement about ageing in any biological sense. Proper time is what a clock carried along the route accumulates, and the claim that a traveller ages by that amount is a separate assumption — that every physical process, chemical and biological alike, runs on proper time and on nothing else. It is a well-tested assumption and it is not part of the geometry, and the essays that speak of twins are borrowing it. The clock that does not feel the turn is the same borrowing examined directly, for clocks rather than for people.

The comparison also assumes both routes are available. Two events joined by a straight worldline are joined by infinitely many wandering ones, and every one of them requires a rocket that can supply the accelerations. Nothing in the principle asks whether the detour is achievable, which is why a statement about the maximum is a statement about geometry rather than about what any traveller could do.

What the pictures cannot show #

The route figures draw worldlines in one frame and the proper times attached to them are frame-independent, so the drawing carries two kinds of quantity at once and looks like it carries one. Redrawing in another frame would move every curve on the page and change none of the numbers in the caption.

The gravitational figure plots proper time against a family parameter, and that parameter is a choice. A different family through the same maximum would give a differently shaped curve with the peak in the same place, and nothing in the picture indicates which features are the physics and which are the parameterisation.

A third omission is scale. Every figure here is drawn at excursions of a light-second or more, because nothing smaller is visible; the routes an aircraft or a satellite actually takes differ from the straight one by parts in $10^{10}$, and drawn honestly they would be indistinguishable lines. The pictures are therefore portraits of a regime that has never been visited, illustrating a principle whose measured consequences are all in the regime the pictures cannot draw.

Where the ladder goes next #

The time-dilation ladder began with the clock that has to slow, passed through the twin problem and the clock that is wrong in two directions, where the speed and height terms first compete, and reached the two clocks that flew in opposite directions. This rung replaces the case-by-case comparisons with a principle that decides all of them. The rungs after it: the geodesic equation as the differential form of this statement; the action for a free particle, which is the proper time times the mass and is where the connection to Lagrangian mechanics is made; and the geodesics of curved spacetime, where the same principle produces orbits.

The habit worth carrying away is that an extremal principle is a statement about a geometry. The sign of the extremum is the signature of the metric, and a subject in which the straight path is the longest is announcing something about its geometry before any calculation is done.