Question: Would relativity hold true if the universe had spherical geometry? For example, the famous twin paradox is resolved because the twin that leaves the Earth has to turn around (accelerate) to get back home. If the universe was spherical, then he could just travel in a straight line and come back to the Earth without ever having to turn around. — Christina
Answer: As you might expect, the twin paradox has been studied for the general global shape of space, whether it be cylindrical, toroidal, or spherical. The paradox enters when in a spherical dimensioned space when the travelling twin remains in an inertial frame, never accelerating. Since both twins are in inertial reference frames, they suffer the same time dilation, but both view the other as having suffered time dilation. Calculations show, though, that the travelling twin is actually younger due to the asymmetry in the inertial frames of the twins. The answer lies in the fact that these curved geometries possess multiply-connected topologies, which makes breaks the symmetry of the inertial reference frames for the two observers, which resolves the paradox. If you are interested in a very detailed physics-based explanation of the twin paradox, see Jean-Pierre Luminet’s nice overview.