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Unformatted text preview: Physics 523, Introduction to Relativity Homework 1 Due Tuesday, 27 th September 2011 Hans Bantilan Twin Paradox in Compact Spaces Consider two inertial observers traveling on different geodesics in a flat spacetime, with observer A at rest in a coordinate system x = ( x , x i ) where we have the equivalence x i x i + L for i = 1 , 2 , 3, and observer B boosted relative to A with velocity v in the x 1-direction. The spacetime can thus be decomposed as the direct product M = IR T 3 , and since the spatial sections are compact, geodesics can intersect more than once. We are to evaluate the proper times experienced by each observer between subsequent intersection points of their worldlines. First, a bit of intuition. A healthy reaction might lead us to conclude that the proper times experienced by the two observers A and B ought to be the same, since two frames in special relativity are equivalent if there exists a Lorentz transformation between them. By construction, our case satisfies this requirement, but the subsequent conclusions do not hold. Seeing this amounts to reexamining the equivalence x i x i + L that was used to define our spacetime. Simply by defining this equivalence on the particular coordinate system ( x , x i ), we have picked out a preferred frame in our spacetime, since (i) only in this preferred frame will the equivalence x i x i + L relate points at the same time slice (for frames boosted relative to the preferred frame, the identifications relate events at different time slices), and (ii) by identifying points on spatial sections, space is now no longer invariant under rotations....
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