What if I fly by the Earth in a relativistic spaceship (no acceleration involved), sync the clocks with my immortal twin who is staying home, and then take a rather long trip around the closed universe? When I circle all the way to the Earth and sync the clocks again (still no acceleration), whose clock will be more vintage, mine or my twin's?

# [Physics] Twin paradox in closed universe

general-relativityspecial-relativitytime

#### Related Solutions

Let's have Bob travel out for 30 seconds and back for 30 seconds at 4/5 the speed of light while Alice stays at home. They each have clocks that (according to them) tick once per second.

I. When Bob leaves, his clock and Alice's both show time $0$.

II. As Bob travels and looks back through his telescope, he sees Alice's clock tick once every 3 seconds. But he figures that this is partly because the distance between him and Alice is increasing, so the light from each tick takes longer to reach him than the light from the previous tick. After correcting for this, he concludes that Alice's clock is ticking once every 1.67 seconds. That is, her clock is running slow, but not by as much as you'd think if you just measured the interval between arriving ticks and failed to correct for travel times.

III. When Bob's clock reads 30 seconds, he receives the light from Alice's tenth tick (that is, the image in his telescope shows Alice's clock reading 10), and calculates that Alice's clock now reads "18 seconds" (though the light from the most recent 8 ticks hasn't reached him yet).

IV. Now Bob turns around and decides that he and Alice are (and always have been) moving closer together, not farther apart. His telescope still shows Alice's clock reading "10", but now, unlike before, Bob believes that each tick has traveled **less** distance than the one before it. This causes him to recalculate that Alice's clock currently reads "82 seconds".

V. Bob now travels home, during which time his clock ticks 30 times, and he sees Alice's clock tick 90 times (advancing from 10 to 100). But (according to Bob) 72 of these ticks had already occurred before he turned around, though he had not yet received the light from those ticks. Therefore, according to Bob, the remaining 18 of those 90 ticks take place during his 30-second return journey. So he *sees* 3 ticks a second, but *calculates* that Alice's clock is ticking once every 1.67 seconds.

VI. Bob arrives home. His clock reads 60 seconds. Alice's clock reads 100 seconds.

Where did I get all this? I drew the spacetime diagram. So should you.

**Added in response to the followup comment by @Razor:**

1) * Say, for convenience, the waves/photons emitted by Alice are seperated by 1 second in Alice’s frame and Bob continuously receives them. So, at the moment Bob turns, does the time interval collapse? *

I can't answer this with confidence because I haven't the foggiest idea what it would mean for a time interval to "collapse". But my best guess is that whatever you mean by this, the answer is no.

2) * Say, for convenience, the waves/photons emitted by Alice are seperated by 1 second in Alice’s frame and Bob continuously receives them....Surely he cannot miss any photons (since photon absorption by Bob is something absolute). He absorbs all the photons or did he just miss Alice aging? *

Obviously, Bob's turning cannot affect the paths of the photons. So: Suppose that just as he turns, he receives a photon that Alice sent when she was exactly 20 years old.

Before the turn, Bob says: Here comes a photon that Alice sent when she was 20. Right now she's pretty far away, but when she was 20, she was a lot closer. So the photon has taken a pretty short time to get here, which means that right now she's a little older than 20.

After the turn, Bob says: Here comes a photon that Alice sent when she was 20. Right now she's pretty far away, but when she was 20, she was even farther. So the photon has taken a really long time to get here, which means that right now she's a lot older than 20.

Why does Bob say these things? Because before the turn, he's employing a frame in which Alice is moving **away** from him (and always has been), whereas after the turn, he's employing a frame in which Alice is moving **toward** him (and always has been. In the first frame, she used to be closer than she is now; in the second, she used to be farther than she is now.

3) As I said eighteen months ago when I first answered this question: If you want to discover these things for yourself, start by drawing the spacetime diagram.

## Best Answer

This is an excellent question! You are asking: if the universe were a large cylinder (it is sufficient that one direction is cyclic), how does the twin paradox resolve. Unlike the original twin paradox, now you can return to your twin without acceleration.

There is a pedagogical answer by Jeff Weeks in

The American Mathematical Monthly(Vol 108 p. 585, 2001), which is available here (pdf). I borrow images from that document, but encourage you to read it yourself.## Background: ordinary twin paradox with spacetime daigrams

The best way to see the solution is to first resolve the regular (infinite-space) twin paradox using spacetime diagrams. Here's a reminder:

The twins are Albert and Betty, where Albert stays put relative to the observer. What's drawn are the lines of constant time at intervals of 5 years. On the left are Albert's constant time slices, while on the right are Betty's constant time slices. These are now tilted because of time dilation. If you're not used to this, I very highly recommend

Very Special Relativityby Sander Bias for a readable introduction at any level.You can see that the twin paradox manifests when Betty changes direction (changes frames): she goes from appearing to by synchronous with an Albert who appears to be 16 when she is 25 to being synchronous with an Albert who is 34. (Betty is traveling at velocity $v=(3/5)c$, but the numbers aren't too important for us.)

## Cylinder universe twin paradox with spacetime diagrams

In the closed universe, Betty's constant time slices now form

helicesaround the cylinder. Thus she sees that she is contemporary withmany copies of Albert.Her helix of constant time intersects copies of Albert at different ages.Before going into "which one is older," we should stop here because this is the resolution to the paradox. What broke the symmetry between Albert and Betty? Albert's constant time curves are circles that are closed, his is a special frame where his world line has zero winding number. Betty's world line winds around the universe and her constant time curves also wind.

For the parameters in the example, Jeff Weeks used a cylinder with circumference 30 light years so that if Betty travels with $v=(3/5)c$, she intersects with Albert when she is 40 and he is 50.