Imagine a universe in which space has the shape of a circle (consider one dimension of space for simplicity, as we often do). Then notice that for Gracie to come back to George she does not need to stop, turn around and head back to Earth. Instead, she just keeps on going and because of the circular shape of space she subsequently passes by George (somewhat as traveling on the Earth’s surface in one direction ultimately takes you back to your starting point). That is, neither observer needs to relinquish constant velocity motion; each (apparently) can rightly claim to be at rest. So, in this example, how do you think the Twin Paradox works out?
Update: In this example, I’m imagining that space itself has a circular shape, not that Gracie is traveling along a circular track within space. If you’d like, think of space like a video game screen that “wraps around”: when you reach the right edge of the screen you reappear on the left edge, and so on. The point, then, is that Gracie does NOT accelerate/decelerate to return to George. She maintains constant velocity throughout the journey.
Both Gracie and George would claim the other sibling is younger. After all, each claims to be stationary while the other is moving. However, they cannot meet up and compare wrinkles. So each can claim to be correct. If one of them accelerates to meet the other, then that acceleration locks in the youth of that traveller. I picture the time-space diagram being like an arch. The stationary perspective (either can argue) shows the moving sibling as an arch, similar to tossing a baseball in the air, except much greater scale. The moving person leaves and magically returns to x=0.
If space is circular then time must, as well, be circular, and relativity won’t work any more.
We have said that from our perspectives, clocks that are forward in the direction of motion are lagging behind clocks that are backward. When space is circular, there won’t be “forward” and “backward” anymore, and so we will need to modify relativity and Lorentz Transformation to account for that.
For a mathematical example, assume Lambda=13/5, then if Gracie travels 5 years to get back to George, she will find his clock to be t’ = lambda(t – xv/c^2) = lambda(t – 0) = 13, but George will find the same result, and will say that Gracie has passed 13 years. Leading to an actual paradox this time