I will not describe the twin paradox again. But let's suppose we have two twins, one stationary and the other moving with uniform velocity $c/2$ for instance. And let's suppose that they live in a hypertorus space, which is topological equivalent to an flat plane, so no curvature. But the hypertorus is a finite space, it means the moving twin will meet again at an instant $t$ in time in the future. What will be their age when they cross each other again?
Answer
The answer is in Time, Topology and the Twin Paradox by J.-P. Luminet
The twin paradox is the best known thought experiment associated with Einstein's theory of relativity. An astronaut who makes a journey into space in a high-speed rocket will return home to find he has aged less than a twin who stayed on Earth. This result appears puzzling, since the situation seems symmetrical, as the homebody twin can be considered to have done the travelling with respect to the traveller. Hence it is called a "paradox". In fact, there is no contradiction and the apparent paradox has a simple resolution in Special Relativity with infinite flat space. In General Relativity (dealing with gravitational fields and curved space-time), or in a compact space such as the hypersphere or a multiply connected finite space, the paradox is more complicated, but its resolution provides new insights about the structure of spacetime and the limitations of the equivalence between inertial reference frames.
The inertial frames for the twins are not symmetric.
In Special Relativity theory, two reference frames are equivalent if there is a Lorentz transformation from one to the other. The set of all Lorentz transformations is called the Poincaré group – a ten dimensional group which combines translations and homogeneous Lorentz transformations called “boosts”. The loss of equivalence between inertial frames is due to the fact that a multiply connected spatial topology globally breaks the Poincaré group.
In a multiply connected spatial topology, there are more than one straight path to join 2 points.
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