system
Given the following post, respond as if you're commenting on Reddit:
human
If there are 3 space dimensions and one time dimension, is it theoretically possible to have multiple time demensions and if so how would it work?
gpt
Yes, we can consider spacetimes with any number of temporal or spatial dimensions. The theory is set up essentially the same. Spacetime is modeled as a smooth *n*-dimensional manifold with a pseudo-Riemannian metric, and the metric satisfies the Einstein field equations (Einstein tensor = stress tensor).

A pseudo-Riemannian tensor is characterized by its signature, i.e., the number of negative quadratic forms in its metric and the number of positive quadratic forms. The coordinates with negative forms correspond to temporal dimensions. (This is a convention that is fixed from the start.) In general relativity, spacetime is 4-dimensional, and the signature is (1,3), so there is 1 temporal dimension and 3 spatial dimensions.

Okay, so that's a lot of math, but it all basically means that, yes, it makes sense to ask questions like "what does a universe with 2 time dimensions and 3 spatial dimensions look like?" It turns out that spacetimes with more than 1 temporal dimension are very pathological. For one, initial value problems do not generally have unique solutions. There is also generally no canonical way to pick out 1 of the infinitely many solutions to the equations of physics. This means that predictability is impossible (e.g., how do you know which solution is the correct one?). Essentially, there is no meaningful physics in a spacetime with more than 1 temporal dimension.

(Ultimately, the problems comes from the fact that allowing more than 1 temporal dimension allows the existence of closed timelike curves, i.e., time travel, even in perfectly regular spacetimes, like those with a flat metric. Regular metrics with CTC's do exist in GR, e.g., the Godel metric, but these metrics with CTC's generally do not correspond to anything reasonably physical, *or* the CTC's are hidden behind the event horizon of a black hole.)