B What would a universe with two time dimensions be like?

[Jupiter60]

Just what would a universe be like if it had two time dimensions?

 

[Paul Colby]

How would you generalize equations of motion? What would the action principle look like?

 

[OmCheeto]

Just what would a universe be like if it had two time dimensions?

hmmm...
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.

 

[JorisL]

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)

 

[arivero]

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).

 

[JorisL]

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.

 

[fresh_42]

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?

 

[JorisL]

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.

 

[arivero]

What would the action principle look like?

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".

 

[fresh_42]

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?

 

[Markus Hanke]

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.

 

[Fervent Freyja]

Two different universes.

 

[nikkkom]

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).

Well, for one you'd have trivially easily occurring closed timelike curves.

 

[Demystifier]

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.

The Cauchy problem is not well posed for such spacetime signatures. See e.g.
http://arxiv.org/abs/gr-qc/9702052
http://arxiv.org/abs/gr-qc/9901045

 

[Markus Hanke]

The Cauchy problem is not well posed for such spacetime signatures. See e.g.
http://arxiv.org/abs/gr-qc/9702052
http://arxiv.org/abs/gr-qc/9901045

I suspected that, thank you ! So the (3+1) case really does seem to have a privileged character...

 

[arivero]

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.

 

[Qcontinuum]

The Cauchy problem is not well posed for such spacetime signatures. See e.g.
http://arxiv.org/abs/gr-qc/9702052
http://arxiv.org/abs/gr-qc/9901045

The Cauchy problem is solved for linear functions. See here
http://rspa.royalsocietypublishing.org/content/royprsa/465/2110/3023.full.pdf

 

[Qcontinuum]

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.
http://arxiv.org/abs/0804.1585

 

[haushofer]

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)

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) )

 

[JorisL]

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.

The formalism is a simple $Sp(2, R)$ gauge theory for particles $X^M (\tau )$ (zero-branes) which arises from a basic idea as follows. $Sp(2, R)$ is the automorphism symmetry of the quantum relations $[x, p] = i$ and treats $(x, p)$ as a doublet. The idea is to turn this global symmetry of Quantum Mechanics
into a local symmetry of a theory. The 3-parameter local symmetry $Sp(2, R)$ includes $\tau$-reparametrizations as one of its local transformations, and therefore it can be regarded as a generalization of gravity on the worldline. The $Sp(2, R)$ gauge theory is non-trivial and physically consistent only if the zero
brane has two timelike coordinates $X^0(\tau)$, $X^{0\prime}(\tau)$ in target space, and has a global symmetry SO(d, 2), which is the Lorentz group with two times.

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.

 

[ohwilleke]

Just what would a universe be like if it had two time dimensions?

Flat (in the sense of lacking height, not necessarily topologically) and timeless, or one dimensional motion in space up and down a single line with time.

 

[rootone]

There could be no causality as we think of it, so no meaningful predictions of anything could be made.
Starting at event 'A', the following event 'B' could be any number of different things depending on the trajectory which an observer is traveling through the 2D time.
It would be possible to travel in a circular fashion through time, so that one could return to the exact same circumstances as 'last week' just by waiting.

 

[maline]

Starting at event 'A', the following event 'B' could be any number of different things depending on the trajectory which an observer is traveling through the 2D time.
It would be possible to travel in a circular fashion through time, so that one could return to the exact same circumstances as 'last week' just by waiting.

If a set of physical laws could be described at all, they would impose local or global constraints on variables over the manifold. We could, for instance, define two orthogonal time directions as "positive", and look for laws where events depend only on regions with lower values for both coordinates. An observer at any point would "remember" such a two dimensional region, rather than a trajectory. This would rule out circular causation.

 

[ohwilleke]

Flat (in the sense of lacking height, not necessarily topologically) and timeless, or one dimensional motion in space up and down a single line with time.

Apologies for not carefully reading "two <i>time</i> dimensions" as opposed to merely two dimensions. I wouldn't have been so flip if I had read more carefully.

 

[naima]

Haven't 3 timelike directions in a black hole?

 

[Ben Niehoff]

Haven't 3 timelike directions in a black hole?

No. Black hole spacetimes are globally Lorentzian; i.e., with one timelike direction.

In the Schwarzschild spacetime, for example, the timelike/spacelike nature of the coordinates t and r exchange roles as the horizon is crossed (note that the angular coordinates don't participate!); however, this is merely a coordinate artifact, and one can write the Schwarzschild metric in coordinates which are globally smooth across the horizon, with manifest globally Lorentzian signature (cf. Kruskal-Szekeres coordinates, or Eddington-Finkelstein coordinates).