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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.)

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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.

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    $\begingroup$ That's interesting about photon decay, although photon decay isn't totally unthinkable even in 3+1 dimensions. See, e.g., arxiv.org/abs/1304.2821 . What seems more fundamentally problematic is that in m+n dimensions with both m and n $\ge2$, timelike vectors aren't topologically split by the light cone into past and future sets. That means you can't tell the difference between a timelike world-line and the spacetime diagram of the vacuum decaying into two real particles. $\endgroup$
    – user4552
    Commented Jul 8, 2017 at 15:50
  • $\begingroup$ What do you mean by "convex normal neighbourhood"? What comes to mind is something like the set of points immediately surrounding a point P in AdS spacetime, but I think this is not what you are talking about. $\endgroup$
    – Bob
    Commented Jul 8, 2017 at 15:51
  • $\begingroup$ @BenCrowell: If I first demand that both time dimensions respect causality, wouldn't the light cone still be respected (only now a timelike path is defined as $ds^2<-(dt_{(1)}^2+dt_{(2)}^2)+dx^i dx_i$ instead of $ds^2<-dt^2+dx^i dx_i$)? Also, when you say 'topologically split', do you mean that the light cone is always centered on the time axis? $\endgroup$
    – Bob
    Commented Jul 8, 2017 at 15:56
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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.

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