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Having two (or more) time-like dimensions doesn't create a problem with causality. The metric in ordinary 4-space is dx^2 + dy^2 + dz^2 -dt^2 if one is in a reference frame where the metric is diagonal (not moving too quickly works in most cases.) (dx means a difference in the x-coordinate, etc.) In moving frames (according to Einstein and Lorentz) one would have a quadratic form (dx,dy,dz,dt)^2(transpose)A(dx,dy,xz,dt) where A is a matrix with 3 non-negative eigenvalues and 1-non positive eigenvalue. If the metric is positive, the distance is space-like, if negative, time-like. Space-like separations are not causally connected. (In the sense of cause effeciens.)

A 5-dimensional metric would just be dx^2 + dy^2 + dz^2 - dt^2 - dw^2 (where w is the other time-like coordinate.) A positive distance is still space like and a negative distance timelike.

The real reason for (mostly) ignoring these models is that so far, they don't predict things actually observed.

Something I have thought about (but can't find in the literature) is the problem of how to do a Minkowskiation of the metric with two time coordinates. For example, normally one just uses, the coordinates (x,y,z,it) with i^2=-1. This works fine with one time coordinate. Using two corrdinates one may suggest (x,y,z,it,jw) where j^2=-1 too. Using iw would make the time coordinated indistinguishable. Now there is an algebraic problem; what is i*j? Hamilton (mid 1800s) discovered that there must be another term (k) where k^2=-1 and to be consistent, ij=k. I'm not sure this has physical meaning, but there seems to be a reason to think that having two time-like dimensions implies three such.

One may have to write out a complete Clifford algebra to make sense of this. Still, the predicted consequences aren't so far observed.

Thank you so much for the information on casuality!

A 5-dimensional metric would just be dx^2 + dy^2 + dz^2 - dt^2 - dw^2 (where w is the other time-like coordinate.) A positive distance is still space like and a negative distance timelike.

I don’t dispute what you have said, but wonder how it relates to brane theory, which is where I see the conventional time dimension rendered as a plane rather than a line and hence the cause-effect muddling.

In the linked article, the authors immediately proceed down the compactification path though I do not believe this is necessary and I really hadn’t considered how you would approach the eigen decomposition for 5D with 2 time dimensions or how you would figure a Minkowski metric with an extra dimension.

Still, the predicted consequences aren't so far observed.

Naturally, I’m not in the business of making predictions (and who would take me seriously anyway?) But of a truth, I would not be surprised to see an extra time dimension prediction involving a cosmological constant, especially with regard to dark energy.

Strangely, everytime I get into researching brane theory I keep coming back to the University of Pennsylvania ala Physicist, Tegmark, Ovrut and now Tianjun Li on a Time-Like Extra Dimension and Cosmological Constant in Brane Models

I haven’t completely waded through that article yet, but so far it sounds like it is heading in the direction I was pondering concerning dark energy. Hmmmm….

For example, normally one just uses, the coordinates (x,y,z,it) with i^2=-1. This works fine with one time coordinate. Using two corrdinates one may suggest (x,y,z,it,jw) where j^2=-1 too. Using iw would make the time coordinated indistinguishable. Now there is an algebraic problem; what is i*j? Hamilton (mid 1800s) discovered that there must be another term (k) where k^2=-1 and to be consistent, ij=k.

Why did I read this far? I don't know. My eyes had already glazed over.

If i and j are equal to the same thing (sqrt -1), why are they not equal to each other? Why / how can ij = k (which is also sqrt -1) if ij = i^2 = -1?

Am I misreading the * operator?