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There are some discussions on more than one time dimensions, e.g., Intuition for multiple temporal dimensions and More than one time dimension.

If we define that the parallel direction is time, of course, we can have only one time dimension but multiple space dimensions.

Then, the questions are:

Q1. Is the above definition reasonable?

Q2. If not, then why time is so different from space? And, how we define time?

Q3. Is multiple time dimensions possible? For example, are there any GUT support it?

Q4. If not, why only one time dimension? Or, which principle rules multiple time dimensions out?

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Re q1 and q2: in special relativity the key invarient is the line element $ds$ defined by:

$$ ds^2 = -c^2dt^2 + dx^2 + dy^2 + dz^2 $$

In this expression the timelike dimensions have a negative sign while the spacelike dimensions have a positive sign, so in the above expression there is only one timelike dimension, $dt$. This is how we define which dimensions are timelike. I'm not sure what you mean by saying "the parallel direction is time", but this is unlikely to be a helpful definition. The negative sign of the time dimension is responsible for all the "weird" effects in SR e.g. time dilation and length contraction, and it also implies a constant maximum speed (i.e. the speed of light).

Re q3: adding a second (macroscopic) timelike dimension is mathematically possible, though as far as I know no GUT uses this because it leads to physically unreasonable consequences, which takes us to ...

Re q4: a second macroscopic timelike dimension allows closed timelike curves and hence violations of causality, so any theory of this type wouldn't be a good description of the universe we observe. Various ways around this have been suggested. Itzhak Bars has suggested a theory with two time dimensions - I don't know the work well enough to know how he gets round causality problems. F Theory can be interpreted as having two time dimensions, but they are not macroscopic so there are no causality problems.

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Yes, I think I should give a rigid definition for parallel. Relativity is a closed theoretical system, where only 4-dimensions x,y,z and t are used. So, the relativity-version's answer for this problem is not enough for me. –  hsxie Feb 26 '13 at 13:06
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