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sevenofnine
2008-Aug-24, 07:02 PM
Hi everybody,
I was wondering about the cosmological implications in a Universe having two times (--+...+). In particular the causality principle of General Relativity.

m1omg
2008-Aug-24, 09:02 PM
Hi everybody,
I was wondering about the cosmological implications in a Universe having two times (--+...+). In particular the causality principle of General Relativity.

There are no "two times".You are misinterpretating the theory.It says that there is no "one" time, there is no absolute time, time is different for everyone, how you measure it is dependant on your perespective.

The differences are tiny (billionths of second) in normal life through.

Ken G
2008-Aug-24, 09:49 PM
Hi everybody,
I was wondering about the cosmological implications in a Universe having two times (--+...+).What do you mean by two times? How are they measured, and what phenomena require them?

sevenofnine
2008-Aug-24, 10:46 PM
A universe with two times is defined by an n-dimensional manifold together with a metric having signature (--+...+). I hope someone can understand this.

publius
2008-Aug-24, 11:07 PM
A universe with two times is defined by an n-dimensional manifold together with a metric having signature (--+...+). I hope someone can understand this.

That's what I thought you were getting at, a space-time with two (or more) time-like dimensions.

All I can say about that it is it entirely academic. Our own universe is modelled by a (curved) space-time with one time dimension and three spatial ones. The Einstein Field Equations(EFE) are all about manifolds with this 1T, 3S metric signature.

While one mathematically consider space-times with more than one time dimension, that's about all one can do (at least of as now).

One can consider the EFE in higher dimensions. That is write the equations, and have more dimensions in the various tensors. I'm pondering now just what the stress-energy tensor would "be" in a framework with two time dimensions, and if anything like that would make any physical sense at least as our universe works in a physical sense.

That's well beyond my knowledge. :)

An interesting sidebar about this is embedding space, embedding curved (1, 3) space-times in higher flat space-times. Many solutions of the EFE requried embedding spaces with more than one time dimension, actually. For example, simple Schwarzschild requires a (2, 4) flat space-time to embed.

But all of that has zero physical significance as embedding spaces are not required and intrinsic curvature, instrinisic geometry is all that matters. Space-time can be "curved" (which is just a mathematical metaphor, anyway) without requiring some "flat" higher dimensional embedding space into which to curve.

However, it's the curved subspaces that are solutions of the EFE in
(1, 3). The EFE itself in higher dimensions is something else entirely. Me, my mind is blown thinking about Minkowski with more than one time dimension. :)

-Richard

Drunk Vegan
2008-Aug-25, 01:57 AM
Philosophically speaking I'd consider the other time dimension to be nonlinear in nature. All points in time are happening simultaneously and causality is out the window.

Ken G
2008-Aug-26, 12:59 AM
A universe with two times is defined by an n-dimensional manifold together with a metric having signature (--+...+). I hope someone can understand this.
OK, I see what you mean. I cannot improve on publius' answer, and I agree with him that one cannot ask "what would physics be like in such a space", but rather, "what are the possible ways physics could work in such a space that would still work like the physics we see when restricted to the spacetime we observe". My guess is, the possibilities are too wide to be of much use in understanding the restriction we actually observe, but maybe some make "more sense" than others. That's kind of what string theorists do, it seems-- imagine a much wider array of possibilities and see if some seem "more aesthetic" than others. If some element of what we observe emerges "naturally" out of some aesthetically pleasing extention to what we cannot observe, they tend to feel good about that. Whether or not that is science can be debated.

astromark
2008-Aug-27, 06:18 AM
At the core of this question is the thought that time is something. To have two, you must except one... and you guessed it, I do not. Time is not some thing there could be more than one of. Our perception of its relentless passage might be altered, but time itself being that it does not actually exist can not be split or duplicated slowed or sped up. No. Only your reference frame can change.

publius
2008-Aug-27, 07:03 AM
And the same thing could be said about spatial dimensions. Flat landers might say there could be no more than two, and we 3-spacers might say there could be no more than three. Granted, it's much easier to think about more spatial dimensions than time dimensions, but it doesn't mean they can't exist. It just doesn't exist in our universe, as least as we perceive and have sucessfully modeled it.

As I said above, this is well beyond my knowlege, but thinking about it, I believe there is some "many worlds" type of framework that involves two timelike dimensions. The following is just vague ramblings based on possibly flawed and incomplete recollections of stuff I've read about: :)

Consider a Minkowski metric with 2Ts:

ds^2 = c(dt^2 + dT^2) -d|R|^2

Apply the same governing principle that things move along their world lines at c. We still have "one time" along the world line, just two degrees of time-like freedom in which those world lines can move.

Note we've got two Euclidean chunks. A 2D time-like side. Null paths, the speed of light, ds = 0 looks this, actually

dR/sqrt(dt^2 + dT^2) = c.

That's strange. Because of the extra time dimension, note something could move as fast as it wanted to against one time dimensions, so long as it moved in the other enough to keep ds^2 positive. IOW, things could move instantly relative to the proper time along one world line!

Does that violate causality? Well, I think you can see we've got some work to do to just figure out what causality *means* in this framework, much less worry about if we're violating it. :lol:

Heres's the many worlds utility of this. Note that any slice of the second time axis, some value of T, you've got a whole regular space-time with one time dimension and however many spatial dimensions that can play out from
-infinity < t < +infinty.

We've got a whole universe from beginning to end at each point in T time. There's our many worlds, every possibility can live at some different T coordinate.

It can also resolve time travel paradoxes. This depends on the rules you define about how you can move through the both time dimensions at once. To be completely general, you can move backwards in one time dimension so long as you move foward enough in the other one to keep the sum positive.

So, we could move backwards in the t coordinate, but we'd have to move foward in the T coordinate, and in doing so, "jump universes" and wind up in the past in some parallel universe.

I think you can keep some higher notion of causality -- no closed time-like curves doing it like this. The projection of that motion on one universe hyperplane would be a closed loop, but it is actually jumping universes, and you could never get back to the universe you left at an earlier t time than you left.

-Richard

Drunk Vegan
2008-Aug-27, 09:05 AM
We've got a whole universe from beginning to end at each point in T time. There's our many worlds, every possibility can live at some different T coordinate.

It can also resolve time travel paradoxes. This depends on the rules you define about how you can move through the both time dimensions at once. To be completely general, you can move backwards in one time dimension so long as you move foward enough in the other one to keep the sum positive.

So, we could move backwards in the t coordinate, but we'd have to move foward in the T coordinate, and in doing so, "jump universes" and wind up in the past in some parallel universe.

I think you can keep some higher notion of causality -- no closed time-like curves doing it like this. The projection of that motion on one universe hyperplane would be a closed loop, but it is actually jumping universes, and you could never get back to the universe you left at an earlier t time than you left.

You wouldn't happen to be working on an experimental project with John Titor (http://en.wikipedia.org/wiki/John_Titor) at the moment, are you?

If not, make sure you pay attention if you ever meet someone by that name - I expect that could be a dynamite partnership ;)

astromark
2008-Aug-27, 12:21 PM
It looks a little ATM for me... other universes ? other time dimensions ? Naa... I have only just grasped the accelerating expansion... I do not see room for other realities. Just the one. Perhaps next time...:)

sevenofnine
2008-Aug-27, 10:43 PM
Thank you publius,
I see you've been thinking about it! I agree with both of your posts! and there are nice observations too.

It's fun the strange thinks that can happen in such spacetime (such as the whole universe from the beginning to the end being at one point, escaping "normal causality"...)

The problem with two times is that you can not think linear! (this was my mistake in asking about causality)
Maybe here causality is nothing more that keeping both squared times positive, as you mentioned?

sevenofnine
2008-Aug-27, 10:46 PM
It looks a little ATM for me... other universes ? other time dimensions ? Naa... I have only just grasped the accelerating expansion... I do not see room for other realities. Just the one. Perhaps next time...:)

Well, in reply to this kind of post. My question was purely theoretical. Mathematical if you want.

The point is to consider a "kind of universe" which might not be ours, of course, and try to do physics there. Not to conjecture that we live in a two-timed universe.

Ken G
2008-Aug-28, 04:15 AM
Consider a Minkowski metric with 2Ts:

ds^2 = c(dt^2 + dT^2) -d|R|^2

Apply the same governing principle that things move along their world lines at c. We still have "one time" along the world line, just two degrees of time-like freedom in which those world lines can move.

Note we've got two Euclidean chunks. A 2D time-like side. Null paths, the speed of light, ds = 0 looks this, actually

dR/sqrt(dt^2 + dT^2) = c.
This does seem like a natural place to start, but I claim that this physics is actually quite identical to the one we already use, so long as we stipulate that clocks measure sqrt(dt^2 + dT^2). If they do that, and we have no other measuring devices, i.e., no way to distinguish dt from dT independently of each other, then two times aren't any different from one-- there are just ignorable degrees of freedom that we have no way to constrain any theory about so we get nothing we don't already have. Yes, you could have "loops", but we can already have those-- how do we know we have not been in this exact same situation, with the same configuration of memories in our brains, a thousand times already? How would science tell? It is completely ignorable to our science. In other words, science has no way to rule out one-dimensional "loops" in time where reality just retraces its steps any number of times-- we would never perceive anything the least bit unusual.

Thus to make "two times" a physically meaningful thing, we also have to have "two kinds of clock", one which measures each kind of time. We could then stipulate the invention of such clocks, such that the old-fashioned kind of clock actually measures intervals like sqrt(dt^2 + dT^2), so all the old physics still works. We would just need to conduct experiments with the new clocks to get a new theory about what t and T were doing separately from the previous combined version of time, and that theory is completely unconstrained by anything we have so far observed, including causality. Causality would apply to the "old" time coordinate, and still work fine-- the "new" coordinates could do anything you could care to imagine (the new clocks could follow any arbitrary rules you can name), and so long as the old clocks still measured sqrt(dt^2+dT^2), nothing would change in the old physics, including causality.

We've got a whole universe from beginning to end at each point in T time. There's our many worlds, every possibility can live at some different T coordinate. Note that's not quite the same as the "many worlds" interpretation of quantum mechanics, because the latter requires the many worlds to branch out from every current state of the universe. One could not simply attribute different t,T coordinates to each of that type of many world, because when those worlds subsequently branch out, the t,T values are in some sense "already in use" by some "other world". That's just what I find so dubious about the many worlds interpretation-- it is constantly spawning new worlds so copiously that the cardinality becomes completely unwieldy.

It can also resolve time travel paradoxes. This depends on the rules you define about how you can move through the both time dimensions at once. To be completely general, you can move backwards in one time dimension so long as you move foward enough in the other one to keep the sum positive. But does that resolve the paradox? If the sqrt(dt^2 + dT^2) coordinate, as it integrates, remains positive, then we do not have time travel in the conventional sense. It's true we could get the "new clocks" to go backward, but that wouldn't surprise us as we have not built our conceptions of time travel based on those kinds of clocks.

Indeed, the whole issue of what "two times" could possibly even mean gets right to the heart of what time means in the first place. What other thing would sound like a time other than the single coordinate we now call time, even in some imaginary universe? In a universe that did have two independent ways to make a clock, if they found the combination sqrt(dt^2 + dT^2) worked like our time, then that is what our concept of time would map into in that universe-- with no impact on our concept of causality.

Added: Perhaps the best way to summarize what I'm saying is that if we did discover a deeper complexity that required two coordinates to distinguish events happening at the same place (or even if one simply imagines such a universe that requires that complexity to understand, but obeyed Minkowski in the way you postulate), that would not actually constitute two times-- we'd still just have one time, the sqrt(dt^2 + dT^2) coordinate. It would be more like time itself had two internal degrees of freedom (which it already might, for all we know)-- but travel in time would still refer to what was happening to sqrt(dt^2 + dT^2), not t and T. So the causality of that universe could be exactly like ours, if the Minkowski metric applied-- there would just be some other set of rules to disentangle t and T and they could have any arbitrary additional causal-like relation you can imagine.