However, it left this science fiction fan with a nagging idea for a setting: suppose we had 2 time dimensions? What sorts of things could happen?
Solving that puzzle had to wait until I got around to working out some of the dynamics, and when I finally did it turned out some other folks had already been playing with the idea. Including Sharshakavili (sp) and somebody named Bars.
To solve the dynamics you first have to decide whether you can violate causality on a macroscopic level--can somehow make your grandfather decide to become a monk before he met your grandmother? (Never mind quantum non-causality; I don't think I want to go there right now.)
I said no, and used a framework like the one Einstein used but with two time coordinates. Then you run into the question--how do you interact with a photon that's running along a slightly different timeline?
Hmm. I guessed that you would still be able to see it--but there'd be some probability that you'd miss seeing it entirely.
That was even cooler--you automatically would have trouble seeing some of the matter in the universe--it would be dark to you.
Presumably near things are mostly on the same timeline, and you'd only start to get major differences far away. OK, so what would a photon look like that came from a distant galaxy with a different timeline?
Oops. It would have the same number of "vibrations" in what would look like a shorter time--meaning it would be blue-shifted. Distant galaxies are red-shifted due to the expansion of the universe, and apparently the expansion is increasing (unless the supernova distribution is wrong somehow), which would make the redshift stronger, not bluer.
Plus, the blue shift complicates energy conservation. I think there might be a way to finesse that, but without either blue-shifted light or disappearing photons I think it safe to say we don't have macroscopic extra time dimensions. I gave up on the study. (If somebody wants to play around with a sci-fi setting in which multiple timelines interleave, feel free.)
On the other hand, if you've got 10 dimensions curled up in M-theory, Dr. Bars suggests that you can spare one for an extra time dimensions. He uses a formalism that uses position and momentum in symmetric ways, and argues from that symmetry that there should be another time dimension. (I think.) Unfortunately I don't share his optimizm about CERN discovering supersymmetry anytime soon--or anytime at all, for that matter.