Just what would a universe be like if it had two time dimensions?
How would you generalize equations of motion? What would the action principle look like?
I've heard about something called "spooky at a distance".
I do not understand it at all, but from what I've heard, it seems to violate "my" notion of our one dimension of time.
Perhaps we already have two time dimensions.
hmmmm.... Perhaps this thread belongs in the sci-fi section.
The idea of two time directions is sometimes used in string theory/supergravity.
A paper by Itzhak Bars http://de.arxiv.org/abs/hep-th/9809034
And more recent results can be found at INSPIRE
I'm not sure how interesting it is but it does exist (in 12 dimensions or more)
For the sake of discussion, I would like to see how it works with two macroscopic time dimensions and flat space, with minkowsky product coming from the matrix diag(1,1,1,-1,-1).
I have absolutely no idea, I encountered a mention of it in a master thesis (and some articles I believe).
Maybe some day I'll find the time to check it out.
Where's the problem is a macroscopic diag(1,1,1,-1,-1) world?
Nothing from a mathematical point of view, I figure the physics would look quite different.
A guess would be that consistency of dimensional reductions can do some weird things.
I'm only a little aware of the consistency of those procedures so I have to look into it on Monday since the document that might give me a hint whether more time directions mess things up is in my dorm room.
For a classical non-relativistic particle, it seems that instead of conditions in two extremes [a,b] in the time-line we should impose them in a circle of the time-"plane".
Even if we get the geometry somehow explained, what about entropy?
From a purely mathematical point of view, having an additional time dimension would fundamentally change the form of the general wave equation, turning it into what I believe is called an "ultra-hyperbolic equation" ( correct me on this if I'm wrong ). The question then becomes how the solutions to this equation behave under a given set of initial values - for example, in a theory of electromagnetism in (3+2) dimensions, do we still get a nice, well behaved wave solution, or do we get something that evolves chaotically ? Or worse still, would we be faced with a situation where no general solution even exists ?
I am not qualified to answer any of this ( not having studied this type of PDE ), but I would also be interested to hear what the mathematicians here have to say on this. My intuition tells me that things won't work out nicely if you add an extra ( macroscopic ) time dimension, but I may well be wrong. And then of course there is the option of having the extra dimension compactified.
Two different universes.
Well, for one you'd have trivially easily occurring closed timelike curves.
The Cauchy problem is not well posed for such spacetime signatures. See e.g.
I suspected that, thank you ! So the (3+1) case really does seem to have a privileged character...
We could try some list of priviledged D = d + t combinations. To start with, let me note the classical signatures to have susy and other stuff related to division algebras:
R: d - t = 1
C: d - t = 2 This is our spacetime, 3-1=2
H: d - t = 4
O: d - t = 8. I think this is the 9-1 = 8 of string theory and perhaps the 10 - 2 mentioned by JorisL above
I am not sure if there is some mod8 periodicity here or the list just stops at 8.
The Cauchy problem is solved for linear functions. See here
For the more interested in 2 k, 4 n as 6 dimensional model you should look at Itzhak Bars work. He has interesting stringy ideas. Gravitational constant looks normal for observer but is a "shadow" of a dilaton field. He worked on many examples with many interpretations and seems to get a consistent framework for a working theory.
Here one impressing work as you see the mathematical beauty in it.
Isn't that related to the fact that AdS-space can be described as being embedded in Minkowski spacetime with two timelike directions? (the AdS group is SO(D-2,2) )
Seems like a good starting point to look at.
Maybe if I can find some time in the next couple of weeks I'll look into it.
This seems like the idea that underlies the arxiv-notes I linked.
I'm not well-versed in the terminology of a "target space" so I'm not sure this would lead to the AdS group?
Neither what ##d## means in this context, from the abstract it follows that ##D## is the number of spacetime dimensions.
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