I've seen a lot of the popular theories regarding how many dimensions there are, and it seems that they all have numerous spatial dimensions and one dimension of time. However, I haven't really found out much about why there is only ever a single dimension of time in any given theory. While I know that there is no consensus on what "multiple time dimensions" even means, I'm starting to get curious about what two or more time dimensions could possibly mean for physics as a whole. Are there any resources or papers I can read that might enlighten me on the different ways the extra time dimensions have been applied, and what changes/implications occur when they're applied?
We can look at this from a physical point of view or a mathematical point of view. From a physical point of view, this question doesn't seem to make much sense. Time is what is ticked off by clocks. How can this possible be more than 1 dimensional? From a mathematical point of view, a time-like dimension is one which carries an opposite sign in the metric to the spatial dimensions. In a mathematical point of view, then, a universe with 2 or more time-like dimensions would be one which is described by a metric of the form diag(-1,-1,...1,1,..) in an orthonormal basis. This would make the manifold non-Lorentzian in nature, which might provide many physical problems for us. However, in fact, this idea is not completely unheard of. Twistor theory, for example, uses a manifold with signature (2,2), or in our language, the metric has the form diag(-1,-1,1,1) in an orthonormal basis. But as far as I know, twistor theory does not try to say that there are 2 time dimensions, only that mapping objects from our normal 3+1 spacetime to a 2+2 manifold might yield meaningful physical results. I am not very familiar with twistor theory, so I can't comment much on it. But here's a wiki article: http://en.wikipedia.org/wiki/Twistor_theory