# The twin paradox and positively curved space

I'm reading about the twin paradox in special relativity - if there are two identical twins, one of whom who sets off in a high speed rocket to a planet, and then heads back, will find the twin who remained on Earth to have aged comparatively more due to time dilation. However, from the perspective of the travelling twin, it was the Earth that was moving and so the Earth twin should be the younger one. The 'paradox' can be resolved by realising that the travelling twin does not remain in an inertial frame for all time, since he has to turn around at the planet and hence accelerate.

Now, there are three possible curvatures the universe could have - positive, negative, or flat. As far as I'm aware, we're quite sure it's flat. However, if it was positive, does this mean that the travelling twin, if he travelled long enough, would loop back to Earth eventually? And therefore he would be able to return to Earth without having changed inertial reference frame, and then we would actually have a paradox?

I'm thinking perhaps that travelling through curved space itself constitutes acceleration, since in a way you're changing direction? In which case the travelling twin wouldn't remain in an inertial reference frame and we could solve the 'paradox' as before?

You and your twin both start out on the equator at 0 degrees longitude, and you both set your car odometers to zero. You drive five degrees to the east. Your twin drives 355 degrees to the west. Even though you both started and ended at the same place, your odometers show different readings.

Would you count that an example of an "actual paradox"? If so, then yes, the scenario you described is an actual paradox. If not, not.

• This is different though - the Earth's surface is curved, and so in travelling around it you cannot remain in an inertial reference frame. My question is whether this also applies to curved space, rather than something 'physically curved' such as the Earth - does travelling through curved space automatically mean you cannot remain in an inertial reference frame? If that is the case then there is no paradox, if not then my question still applies.
– jl2
Commented Sep 19, 2015 at 19:55
• Yes, the earth's surface is curved---just like the universe in your scenario. Commented Sep 19, 2015 at 20:07
• @jl2 The word general in general relativity refers to the fact that there aren't global inertial frames, so you can't go around the universe and stay in a single global reference frame. Instead of frames you can use charts and compute (with the metric tensor) what a clock does along a path. And then the answer WillO gave really is that straight forward, different paths between the same points cab have different lengths and it isn't mysterious or paradoxical in the slightest. Commented Sep 19, 2015 at 20:07
• Incidentally --- although the earth's surface is curved, the motion I've described takes place entirely on the equator, which is a circle and therefore carries a flat metric. Commented Sep 20, 2015 at 0:43
• This is a poor analogy. Your odometers end up different, but your clocks don't.
– user854
Commented Mar 17, 2016 at 3:18

And therefore he would be able to return to Earth without having changed inertial reference frame, and then we would actually have a paradox?

No paradox. The equations of general relativity can be solved with a universe hat wraps around, even if it is flat.

What happens is that you see two earths, the one you are at, and an image of an older earth where light left long ago, wrapped all around the universe and just now got to you.

As you move towards it the image of the far away gets bluer an you see the images of the people on earth celebrating holidays arrive more often than you age.

As you move away from the nearby earth, you see the image of the nearby earth get redder and you see the images of the people on earth celebrating holidays arrive less often than you age.

Eventually the image of the earth that was initially farther away is an earth that is actually closer, and eventually you land on it. And then you could travel to the other side of the earth and sit and wait for the image of you landing to finally arrive from the long way around.

And you never see a paradox. If you focus on what actually happens there isn't a paradox.

• Sorry, but I don't see how this answer explains the twins and their clocks. Could you take a look at my similar question and see if you can clarify it for me?
– D.R
Commented Mar 18, 2019 at 2:03
• Your explanation though very thought-provoking, doesn't seem to clarify the posed problem. Commented Oct 22, 2023 at 12:40