system
Given the following post, respond as if you're commenting on Reddit:
human
Is it possible for a universe to have more than one dimension of time?

It is generally accepted that we live in a universe containing 4 dimensions (3 spatial, 1 time). Would it be possible for a universe to have two spatial dimensions and two dimensions of time? What about three spatial, two time?
gpt
Multiple time dimensions lead to CTCs (closed timelike curves) which allow for violation of causality (timetravel).

Assuming you're familiar with special relativity:

take a Universe like ours but with an additional time dimension (2+3). So

`[; x^\mu = (x^0, x^1, x^2, x^3, x^4) ;]`

with `[; x^0, x^1;]` both time dimensions. The spacetime interval is:

`[; ds^2 = (dx^0)^2 + (dx^1)^2 - (dx^2)^2 - (dx^3)^2 - (dx^4) ;]`

now, as in normal relativity, the sign of `[;ds^2;]` being +, 0, - tells you if that little segment can be followed by a massive object, massless object, or not possible to follow at all (timelike, lightlike, spacelike).

Any infinitesimal movement purely in the `[; (x^0,x^1) ;]` plane is obviously timelike, since `[; ds^2 = (dx^0)^2 + (dx^1)^2 > 0 ;]`. Therefore, taken a closed loop in the `[; (x^0,x^1) ;]` plane, (e.g. a circle), that loop is timelike.

So you've built a timelike loop, a time machine. An object following the loop would return to its past, without needing to travel faster than light (and in fact, without having any velocity at all, since it does not move in the space dimensions).