The twin who never turns round

The twin who comes back younger resolves the most famous paradox in relativity with a single observation. If motion slows clocks, and motion is relative, each twin should see the other’s clock run slow, and it is hard to see how, when they meet again, one of them can be younger. The answer is that the twins are not symmetric: one of them turned round. Their worldlines join the same two events, the one who stayed at home has a straight worldline, the traveller’s has a kink at the turn, and the straight worldline between two events carries the most time. Acceleration is not the cause of the ageing, but the turn is where the asymmetry is visible.

That answer leans on something taken for granted: that the only way for the traveller to come back is to turn. In ordinary space it is. But space need not be ordinary. General relativity says how space is curved; it says nothing about its shape on the largest scales — its topology — and a flat space can close on itself. The simplest example is a universe whose space is a circle: travel far enough in a straight line in one direction and arrive back where the journey began. In such a universe the traveller can come home without ever turning, and the standard resolution has nothing to point at.

Two straight worldlines that meet twice #

The drawing uses a device that turns a closed space into an ordinary one, in the spirit of the diagrams that bring infinity onto a page. Space is a circle ten light-years round, and the diagram unrolls it into a line, repeating it every ten light-years: the shaded strip is all of space, and the strips beside it are copies. A point in one strip and the corresponding point in the next are the same place. So twin A, at rest, appears as a solid vertical line and as dashed copies of that line every ten light-years. Twin B leaves A at 0.6 of the speed of light, keeps going and, after 16.67 years by A’s clock, reaches the first copy of A — which is A. They have met again.

Neither twin ever feels a force. Both are in free fall for the whole journey, and both worldlines are straight lines on the diagram. Nothing in B’s experience distinguishes it from A’s: each carries an accelerometer that reads zero throughout. Yet when they meet, A’s clock reads 16.67 years and B’s 13.33. B is younger by exactly the time-dilation factor for 0.6c, as though B had been the moving one all along and A the one at rest.

This case was taken seriously in the physics literature from 1973, when Carl Brans and D. R. Stewart published it as the unaccelerated returning twin, and later analyses — John Barrow and Janna Levin’s and Jeffrey Weeks’s around 2001, among others — agree on the resolution. It is worth following the drawings to see why, because the resolution teaches more about relativity than the turn does.

Many straight lines between the same two events #

In ordinary space, two events are joined by exactly one straight worldline. In a closed space they are joined by many — one for each number of times the worldline winds round space before arriving. The drawing takes two events forty years apart on A’s worldline and counts them: staying put, going round once in either direction at a quarter of the speed of light, twice at half, three times at three quarters. Each of those seven worldlines is straight; each is a possible history for a free particle; and they carry different amounts of proper time.

The rule that a straight worldline carries the most time between its ends survives, but it has to be restated. It holds among worldlines that can be deformed into one another without leaving spacetime — among those that wind round space the same number of times. Worldlines with different winding numbers belong to different families, and the rule compares within families, not across them. A’s worldline carries the most time of all of them. B’s is the longest worldline of its own family, and it is shorter than A’s.

The same structure appears in a place with no closed space at all. The orbit that ages less than a throw compares a clock in orbit with a clock thrown straight up, both in free fall, both leaving and returning to the same events, and finds the thrown clock older. There the geodesics are in curved spacetime around the Earth, and a geodesic that goes round the planet is a different family from one that goes up and down. Winding round something — a planet, or space itself — is what lets two free-falling clocks disagree.

Light sent both ways round the universe #

Which twin is special is not a matter of opinion, and there is an experiment that decides it without the twins having to meet. At the moment of parting, each sends a flash of light both ways round the universe and waits for the flashes to come back from behind. For A, the two flashes travel ten light-years each and return together after ten years. For B the situation is not symmetric. The flash sent forward travels round the universe chasing B, and has to make up the distance B covers in the meantime; the flash sent backward meets B coming towards it. By B’s own clock the two return fifteen years apart.

This is the whole resolution. The twins’ local physics is identical — every laboratory experiment either could do in a small room gives the same result for both, exactly as special relativity requires. But a closed space is not a small room. It gives every observer a global experiment, light sent round the whole universe, and that experiment picks out one state of motion: the one for which the two flashes return together. That observer’s frame is the one in which the universe’s circumference is measured at a single instant — the frame in which space closes up simultaneously — and it is special in the way the frame of a turntable is special, not in any way that local physics can see.

A speedometer that needs no window #

The gap between the two returns is $2\gamma v L$ by the sender’s own clock, where $v$ is the sender’s speed in the special frame and $L$ the universe’s circumference. It is zero for one frame and grows with speed, so it is a speedometer that works without looking at anything outside — a measurement of absolute motion, in a universe where absolute motion has a meaning.

The formula is familiar from another place. The ring where the two beams disagree sends light both ways round a loop on a turntable and finds the two beams return at different times, by an amount that depends only on the rotation and the area of the loop. The Sagnac effect detects rotation, which is absolute in relativity — a rotating observer can tell, by a closed-loop light experiment, that they are rotating. A closed universe makes translation absolute in the same way. In both cases the experiment works because the light path closes, and in both cases it detects motion relative to the one frame in which the closed path’s clocks can be synchronised all the way round.

The traveller’s now does not close #

The last drawing shows the same asymmetry in the language of simultaneity. Now is a choice of slicing: each observer’s set of events “happening at the same time” is a line through spacetime tilted according to their motion. For A the line through the parting event is horizontal, and followed once round the universe it comes back to A at the same moment. A’s “now” is a closed loop round space.

B’s line is tilted. Followed once round the universe it reaches B’s copy — which is B — not at the parting but seven and a half years later by B’s clock. B cannot consistently say what is happening “now” all the way round space: going round once, B’s now arrives in B’s own future. This is exactly the failure a rotating observer meets on a disc that cannot be spun, where synchronising clocks round the rim fails to close, and exactly what how big now is describes as the limit of a local patch of simultaneity. Here it is produced by the topology of space, with nothing rotating anywhere.

The asymmetry between the twins is therefore not in their motion, their clocks or their accelerations. It is in whether their notion of simultaneity closes up round the universe. The twin whose slices close is the one who ages most, and the one whose slices do not — every other observer — comes home younger.

The traveller’s own reckoning, and where the missing years are #

It is worth doing the calculation from B’s side, because B’s reasoning is where the paradox lives. B says: in my frame A is the one moving at 0.6c, so A’s clock runs slow, so A should be the younger. The first two clauses are correct, and the drawings do not contradict them. What B has left out is the shape of space in B’s own coordinates.

Transform A’s description of the universe into B’s frame and two things change. The circumference, measured along B’s lines of simultaneity, is longer — 12.5 light-years rather than ten, stretched by γ. And the gluing that joins one end of space to the other is no longer between simultaneous events: going once round space in B’s coordinates, the calendar jumps back by γvL, seven and a half years, which is the non-closing slice of the last drawing seen from B’s side. Space in B’s frame is a circle with a time step in it.

Now B can compute. A moves round B’s 12.5-light-year circle at 0.6c, which takes 20.83 years of B’s time; during that time A’s clock runs slow by γ and advances 16.67 years, just as B said it would. But when A has gone once round, the gluing moves the meeting back by 7.5 years of B’s time, so B has aged only 20.83 − 7.5 = 13.33 years when they meet. Both twins’ calculations agree on both clocks. B was right that A’s clock runs slow in B’s frame and wrong to conclude A would be younger, because B’s coordinates contain a seam at which B’s own time jumps, and A’s journey crosses it.

This is precisely the structure of the ordinary twin paradox, with the seam in a different place. There, the travelling twin’s lines of simultaneity swing round at the turn, and the stay-at-home’s clock appears to jump forward during it; the jump is where the “missing” years are. Here nothing turns, and the jump is built into space. In both cases time dilation is symmetric and the asymmetry is in simultaneity.

What this does and does not say about relativity #

It is tempting to read the closed universe as a refutation of the principle of relativity, and it is not one. The principle says that the laws of physics are the same in every inertial frame, and here they are: every local experiment by A or B gives identical results, and the equations of motion have the same form for both. What differs is a boundary condition — the way space is joined up far away — and boundary conditions are allowed to single out frames. A laboratory next to a large wall has a special frame too, the wall’s, and nobody thinks the wall refutes relativity.

There is one convention the closed universe does break. The speed that cannot be measured one way explains that every measurement of light’s speed sends it out and brings it back, because timing a one-way trip needs two distant clocks synchronised by a rule that is itself a choice. In a closed universe light sent round the universe comes back to the same clock without being reflected — it has made a one-way trip, round the whole of space, and a single clock times it. A finds the trip takes the same time in both directions and so may take light’s one-way speed to be the same both ways; B finds the two trips unequal, and must either say that light is faster one way round, or say that B is moving. The synchronisation convention that is free in an open universe is fixed, in a closed one, by the universe.

The closed universe simply makes that boundary condition inescapable, because it is everywhere at once. Relativity’s symmetry is a local symmetry of the laws. Whether the universe as a whole respects it is a separate question about the universe, answered by its contents and its shape. How big now is notes a third view of simultaneity, in which the cosmic microwave background picks out a preferred frame; a closed topology would pick out one too, and in a closed universe filled with radiation the two would plausibly coincide.

Is space closed? #

The question is not idle. Einstein’s equations allow flat, spherical and hyperbolic spaces, each in many topologies, and observations constrain the curvature much more tightly than the topology. If space closed on itself on a scale smaller than the distance light has travelled since the microwave background was released, the same distant region would be seen in more than one direction, and the pattern of the background would contain matching circles of temperature on opposite sides of the sky. Searches through the maps made by the WMAP and Planck satellites have found no such circles, which rules out a closed space smaller than about the diameter of the observable universe in the simplest topologies. Space could still close on a larger scale, and nothing forbids it.

In a universe that did, the special frame would almost certainly be the frame of the cosmic background itself, and the Earth, moving at about 370 kilometres per second relative to that background, would be a twin B. A flash sent round a universe a hundred billion light-years round would return with its two halves separated by roughly a quarter of a billion years — which is to say that the experiment is possible in principle and useless in practice. Its interest is what it shows about which parts of relativity are local laws and which are facts about the universe.

Where the flat cylinder stops #

Flat spacetime. The drawings use special relativity on a space that is flat but closed — a cylinder, with one closed direction. A real closed universe would also expand and, in general, curve; the twins’ clocks would then also be affected by the expansion, and the special frame would be the frame that sees the expansion as uniform.

One closed dimension. Space here is closed in one direction and infinite in the others. Three-dimensional closed spaces — a three-torus, or the more complicated spaces that a positively curved universe can have — have several closed directions, and each supplies its own light-round-the-universe experiment; the special frame is the one in which all of them close simultaneously.

Ideal clocks and signals. The flashes are assumed to travel round the universe through empty space. Across a real universe, light would be delayed by the matter it passes, and would have to travel far longer than any civilisation has existed.

What the diagrams cannot show #

The unrolled diagram is a trick for drawing a closed space on flat paper, and it works because the space is flat. What it cannot show is the experience of living in such a universe: looking in any direction along the closed dimension and seeing an image of oneself, far away, in the past — once in each direction, and again further away, and again, receding without end. Each image is light that has gone round the universe once, twice, three times. For a twin at rest the images in the two directions are identical in age; for a moving twin they are not, and their difference is the speedometer of the fourth drawing, read with a telescope instead of a clock.

Still open: what shape space is #

Whether the universe’s space is closed, and if so how, is not known. The absence of matching circles in the microwave background rules out small closed universes; larger ones leave subtler statistical traces — slight correlations between opposite sides of the sky at the largest angles, a suppression of the largest-scale fluctuations — and the largest-scale features of the background do show some anomalies whose significance is argued over, including a lower-than-expected strength on the very largest scales. None of them is established as a sign of topology, and some analyses find a closed space of about the right size could explain some of them. The question will not be answered by local physics, which by construction cannot see it, but only by looking at the largest structures there are.

The habit worth carrying away is to ask whether a symmetry is a property of the laws or of the world. Relativity guarantees that no local experiment can single out a state of motion; it says nothing about experiments that reach round the whole of space, and a closed space provides one. The twin who ages most is the twin whose now closes up round the universe, and the turn in the ordinary paradox was only ever the most familiar way of making the two twins’ nows differ.