Visiting Princeton, the American home to Albert Einstein, I’m reminded of one of my favorite “paradoxes” of special relativity. And, even more so, one of my favorite versions of this paradox which, when I first heard it, it blew my mind. What paradox is this of which I speak? The twin paradox of course! Really just the plain old twin paradox? No. Much better than that: the twin paradox in donut space!
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\nThe twin “paradox” of special relativity is really one of the classics of undergraduate physics. Two twins, born minutes apart, over the years drift apart. One becomes an astronaut and the other a surf bum who enjoys hanging with his homies. Astronaut twin hops on a rocket ship which accelerates up to relativistic speeds, visits nearby star, says hi to the empty space around the star, and returns home to earth. Time dilation in special relativity tells us that the clock in the astronaut’s ship will tick slower than the one with the beach bum, and thus, when the astronaut returns home, he will be younger than his sibling. The “paradox” comes from thinking: but isn’t the whole experiment symmetric. Can’t I think about the whole experiment from the astronauts point of view, and won’t he see a similar symmetric path for his beach bum brother?
\nThe resolution of the twin paradox comes from realizing that the whole setup isn’t really symmetric: one of those twins undergoes acceleration in his path (the astronaut) while the other twin just sits on his lazy bum surfside. Cool. Standard stuff from undergraduate special relativity.
\nBut here is a fun twist to think about. Suppose that the universe we lived in was a donut (sit down Homer J, calm yourself.) A donut? Well suppose that if you traveled in one direction for a long time you would return to where you started. Just as if you travel on the surface of a donut (a two dimensional surface) you can walk “around” the donut (imagine yourself and ant on the donut) and end up where you started, you can imagine that the topology of the universe is such that if you walk in the right direction you will eventually end up back where you started. Such nontrivial topologies of the universe surprisingly aren’t ruled out by simple things like General Relativity, although experimental searches for such crazy topologies haven’t turned up any evidence that our universe has such a crazy topology.
\nBut back to the twins. Now suppose that you run the twin experiment, but with the twins entirely in inertial frames. In other words, the beach bum sits in an inertial frame (we probably need to put him in space to get a good approximation of this, but ignore this for now, this is a thought experiment, damnit!), and his astronaut buddy also is in an inertial frame, but one which is moving with a constant velocity with respect to his surfing brother. So now, since we are in a donut topology for our universe, it is possible to set up this experiment such that the astronaut brother eventually goes “all the way around the universe” and ends up beside his surfing bro. Now, the relationship between the twins really is symmetric: neither of them ever shifts reference frames or accelerates or such. And thus there really does seem to be a paradox. Which of the two twins is younger? And why?<\/p>\n","protected":false},"excerpt":{"rendered":"
Visiting Princeton, the American home to Albert Einstein, I’m reminded of one of my favorite “paradoxes” of special relativity. And, even more so, one of my favorite versions of this paradox which, when I first heard it, it blew my mind. What paradox is this of which I speak? The twin paradox of course! Really … <\/p>\n