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There are a number of models (String theory, Cascading Gravity, Emergent Dimensions) that contain extra space like dimensions. Why do people tend to avoid considering extra timelike dimensions?

The only reason I have heard so far is that you want to avoid close timelike curves (which would violate causality). I would think that if causality is respected in our usual time dimension, we could (at least for the purpose of initial toy models) demand causality be respected in other time dimensions as well.

(The m+n dimensional question linked to was asking about what happens if m and n are be integers.)

Qmechanic
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Bob
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2 Answers2

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Causality cannot be respected in a spacetime with more than one timelike dimension. To show it just consider a convex normal neighbourhood around a point $p$, and take any closed curve lying in the plane defined by the tangent vectors $\partial_{t_1}$, $\partial_{t_2}$. Those curves will always be timelike and this will happen no matter the metric or global structure of the manifold.

There are many other problems involved, such as the lack of stable or unique solutions to the Cauchy problem, or the possibility of photon decay.

Slereah
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Itzhak Bars has a theory of this sort. It also should be noted the anti-de Sitter spacetime $AdS_4$ is a slice in a flat spacetime with metric $$ a^2~=~u^2~+~t^2~-~x^2~-~y^2~-~z^2 $$ where the $AdS_4$ is in a constant $u$ time slice. The $AdS_4$ spacetime is one time plus three space, but as $AdS_4~=~SO(4,2)/SO(4,1)$ is obeys an isometry group $SO(4,2)$ that has two time directions.

It could be said that an additional time lurks in the wings of physics. Bars claims it is more central to physics.

Lawrence B. Crowell
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