Eugene Shubert
the new William Miller
Joined: 06 Apr 2002
Posts: 1006
Location: Richardson Texas
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 Posted: Fri Apr 26, 2002 8:39 pm Post subject: How do you resolve the twin paradox in a toroidal universe? |
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| A student wrote: | In no popular physics book covering the special theory of relativity is the twin paradox discussed in the context of a closed, locally flat spacetime. By only assuming the constancy of the speed of light and the principle of relativity, what is shown is that “moving clocks run slow.” The mathematics of relativity theory describes how different observers moving at constant relative velocities will measure distances and times for the same event differently. An observer A who sees another person B moving with constant velocity will surmise that the moving person’s clock (or time) goes slower than the observer’s clock. If you look at the situation the other way around, person B thinks A's clock is going slower. This is the twin paradox.
The resolution of the paradox is that it is impossible to compare clocks using instantaneous communication. One must accelerate at least one of the observers and that destroys the perfect symmetry and ruins the paradox. However, what if the space is locally flat but is identified say along the x-direction: x <=> x + a. Then the two persons may compare their clocks again every time they meet and the paradox will remain? What exactly goes wrong in this example? |
The easiest way to explain this is to look at the simplest toroidal universe, SxR. In this case, where space S is just a circle and you’re only concerned with local issues, it’s easy to believe that all the postulates and conclusions of special relativity work perfectly fine. So let’s consider why the simplicity of special relativity doesn’t apply globally in any circular universe.
Let an inertial observer A overtake another inertial observer B with a small relative velocity v. At the instant A passes B, let B emit a photon in both the fore and aft directions. The photons will circumnavigate the universe and coincide at a unique event. The photons can’t arrive together at both A and B because these have moved some distance apart. It’s obvious that in this spatially compact (i.e., spatially closed and bounded) universe, there is a uniquely distinguished frame of reference that can only be seen on the global scale.
You asked about clocks. If you do actual calculations you’ll see that radical differences may come up when moving clocks in a spatially compact universe as opposed to the usual, spatially infinite spacetime that you read about in elementary relativity textbooks.
You should reach this conclusion:
In Minkowski space, it’s always the case that if you have two clocks at a point and take one of the clocks on a trip and then return it to its point of origin, then the elapsed time on the traveling clock will always be less than the stationary clock. This result is not always true in a universe that is spatially compact. If you pick the initial conditions correctly and the right path for travel, then the traveling clock could easily return reading more elapsed time than the clock at home.
I regard this result as rather beautiful. |
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