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We know that space-time dimensions are 3+1 macroscopically, but what if 2+2? Obviously it is tough to imagine two time dimensions, but mathematically we can always imagine as either having two parameters $t_1$ and $t_2$ or else in Lorentz matrix $$\eta_{00} = \eta_{11} = -1$$ and, $$\eta_{22} = \eta_{33} = 1.$$

Is there any physical reason for not taking this, like the norms become negative or something else?

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Maybe you are interested the answer to this question too. –  Dilaton Nov 7 '12 at 14:41
    
The "norms" could become negative anyway with usual Lorentzian signature. –  c.p. Nov 8 '12 at 18:23
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You may want to look at the following paper arxiv.org/abs/hep-ph/9910207 –  Newman Nov 9 '12 at 2:20
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5 Answers

up vote 3 down vote accepted

As Cumrun Vafa explains in the video linked to below the picture of him in this article, F-theory works in a total of $10+2$ dimensions. The signature of the last two infinitesimal dimensions is ambiguous, so that they can indeed both be timelike. Since these are only infinitesimal dimensions, any causality issues etc are not a problem in this case.

And as Cumrun Vafa nicely explains in his talk, F-theory gives quite a nice phenomenology with an astonishingly realistic CKM-Matrix, coupling constants, etc; so it is NOT true that theories that operate in more than one time dimension are completely off base, as some people claim. There is no reason to dogmatically dismiss every theory that has more than one time dimension.

BTW, the talk is very accessible and enjoyable.

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The hyperbolicity of the associated classical field equations is lost in $2+2$ dimensions.

As a result, there is no way to implement causality (i.e., no way to enforce the limiting information transmission to a finite speed), and the resulting models have very little to do with the real world.

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Well, 2+2 dimensions is obviously not compatible with reality, since we observe 3 spacelike dimensions. The paper by Dvali (see the comment by Newman) discusses the possibility that a timelike dimension is curled up. It's far from obvious to me that our universe couldn't act like it had hyperbolicity if it was 3+2-dimensional, but with one of the timelike dimensions curled up. –  Ben Crowell Apr 15 '13 at 3:21
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There could actually be two time dimensions, though you have to implement those properly. Should you simply replace the domain of time from $\mathbb R$ to $\mathbb R^2$ you would completely mess up our notion of cause and effect, therefore giving rise to many logical and temporal paradoxes hard to explain. I don't even know if such a theory would be logically coherent.

However, these two time dimensions need not to belong to the same mathematical domain, nor to have an equivalent meaning. Maybe this link can be useful. See also this short review on wikipedia.

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Although space-like and time-like coordinates enter in the same foot on a relativistic theory, there are physical differences between them, for example

  • One can travel back and forward through space-like coordinated, while it is impossible in time-like coordinates.

  • If there exist more than one time-like dimensions, that would mean our time is a linear combination of those. Since one does not see other time-like coordinated, it implies that the other (transverse) time-like coordinates are compact.

The presence of closed time-like coordinates spoils causality... reason why no more than a single time coordinate is considered in physics.

Cheers.

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I do not agree. With a time cylinder model, with $T$ being the (constant) radius of the cylinder, the space-time interval would be (here $c = 1$) : $(\Delta s)² = (\Delta t)² + T (\Delta \theta)² - (\Delta r)²$. If $T$ is of the same order as the Planck time, the apparent violation of causality (in r and t) is only appreciable for a length separation of the same order as the Planck length. For standard lengths, the term $T (\Delta \theta)²$ is neglectable. –  Trimok Nov 14 '12 at 9:47
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The causality issues relevant to including a second dimension of time owe their incoherence to the controlling assumption that space is contiguous and time is continuous. A formulation that understands two dimensions of space and two of time works well when the spatial dimensions are considered separated by time and the temporal dimensions are separated by space. The resulting construct gives rise to a phenomenology of a continuous form of time and a contiguous form of space. Causality is not violated as causality is a phenomenological issue.

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