Can time be a periodic function in GRFE?

[Atlas3]

I am learning more about the Einstein Filed Equations. I do not know how to evaluate them. Can someone tell me if time can be a periodic function in the equation for spacetime? i may not have that spacetime correctly stated. Can time periodically vary like a SIN function and still work in the math? I am not just throwing math around. Continuous and counting just varying positive and negative? And what would this do if time parameter went from -1 to 0 to 1 and back again? I am trying to crush my thoughts here.

 

[PeterDonis]

Can someone tell me if time can be a periodic function in the equation for spacetime?

I'm not aware of any solution of the Einstein Field Equation where a timelike dimension is periodic in this way. Solutions of the EFE describe spacetime geometries; so time being a periodic function would mean that the geometry of spacetime was closed up along a timelike dimension--for example, if space were infinite, spacetime would be like a cylinder, "rolled up" along the time dimension. Again, I'm not aware of any solution that has this kind of geometry.

There are solutions, such as a closed FRW universe that expands and then recollapses, which have a finite length along a timelike dimension. But the timelike dimension is not periodic in these solutions; it doesn't go, in your words, from -1 to 0 to 1 and then back again. It just goes from (heuristically speaking) -1 to 0 to 1, then stops.

 

[Atlas3]

I'm not aware of any solution of the Einstein Field Equation where a timelike dimension is periodic in this way. Solutions of the EFE describe spacetime geometries; so time being a periodic function would mean that the geometry of spacetime was closed up along a timelike dimension--for example, if space were infinite, spacetime would be like a cylinder, "rolled up" along the time dimension. Again, I'm not aware of any solution that has this kind of geometry.

There are solutions, such as a closed FRW universe that expands and then recollapses, which have a finite length along a timelike dimension. But the timelike dimension is not periodic in these solutions; it doesn't go, in your words, from -1 to 0 to 1 and then back again. It just goes from (heuristically speaking) -1 to 0 to 1, then stops.

What is FRW universe?

 

[PeterDonis]

What is FRW universe?

The FRW spacetimes are a family of spacetimes used in cosmology; see here:

http://en.wikipedia.org/wiki/Friedmann–Lemaître–Robertson–Walker_metric

The particular solution I was describing is the one described in that article for $k > 0$.

 

[Atlas3]

I'm not aware of any solution of the Einstein Field Equation where a timelike dimension is periodic in this way. Solutions of the EFE describe spacetime geometries; so time being a periodic function would mean that the geometry of spacetime was closed up along a timelike dimension--for example, if space were infinite, spacetime would be like a cylinder, "rolled up" along the time dimension. Again, I'm not aware of any solution that has this kind of geometry.

There are solutions, such as a closed FRW universe that expands and then recollapses, which have a finite length along a timelike dimension. But the timelike dimension is not periodic in these solutions; it doesn't go, in your words, from -1 to 0 to 1 and then back again. It just goes from (heuristically speaking) -1 to 0 to 1, then stops.

That's the same thing as time being to 0 to 1 isn't it? if It was a cylinder.

 

[PeterDonis]

That's the same thing as time being to 0 to 1 isn't it?

Yes, you could assign a time coordinate either way, 0 to 1 or -1 to 1.

 

[Atlas3]

Thank you. It has been enlightening to read your replies.

 

[Atlas3]

I'm not aware of any solution of the Einstein Field Equation where a timelike dimension is periodic in this way. Solutions of the EFE describe spacetime geometries; so time being a periodic function would mean that the geometry of spacetime was closed up along a timelike dimension--for example, if space were infinite, spacetime would be like a cylinder, "rolled up" along the time dimension. Again, I'm not aware of any solution that has this kind of geometry.

There are solutions, such as a closed FRW universe that expands and then recollapses, which have a finite length along a timelike dimension. But the timelike dimension is not periodic in these solutions; it doesn't go, in your words, from -1 to 0 to 1 and then back again. It just goes from (heuristically speaking) -1 to 0 to 1, then stops.

I just wrapped my head around this to see the dimensions if time were "rolled up" and pictured that. But could it become a torus in a solution? time wrapped around the tube and space inside the ring? Is it something that can be reasonably solved for? I'm interested in understanding to work with this subject to find peace. That would have a problem too. Four of them I think. At each end of time and each end of the space dimensions?

 

[PeterDonis]

could it become a torus in a solution? time wrapped around the tube and space around the ring?

I don't know of a solution with this geometry either.

 

[Atlas3]

I don't know of a solution with this geometry either.

Can I find one? Are solutions proven to be exhausted at this point?

 

[PeterDonis]

Are solutions proven to be exhausted at this point?

No. There are probably many exact solutions out there that haven't been found. The EFE is a difficult equation to solve, and there are no general solution methods known; basically all of the known exact solutions have some kind of symmetry that simplifies the equations to something manageable.

Also, an exact solution is not the only way to extract useful information; numerical solutions are increasingly useful in that regard as computers continue to become more powerful.

 

[Atlas3]

No. There are probably many exact solutions out there that haven't been found. The EFE is a difficult equation to solve, and there are no general solution methods known; basically all of the known exact solutions have some kind of symmetry that simplifies the equations to something manageable.

Also, an exact solution is not the only way to extract useful information; numerical solutions are increasingly useful in that regard as computers continue to become more powerful.

Is there a starting point for this? I don't know where to begin. I could pursue numeric methods if pointed in the right direction also.

 

[Atlas3]

I don't know of a solution with this geometry either.

A torus could be quite flat. If the dimensions were large enough it would fill space much the same as a sphere and be quite flat as well, couldn't that be?

 

[Atlas3]

A torus could be quite flat. If the dimensions were large enough it would fill space much the same as a sphere and be quite flat as well, couldn't that be?

I found this information to pursue further. http://www.thephysicsforum.com/astrophysics-cosmology/6215-einsteins-equations.html

EDIT: that link was generous to op that needed much understanding however the description was informational.

 

[Orodruin]

I don't know of a solution with this geometry either.

You should just be able to put a flat metric on the torus. This would be locally equivalent to Minkowski space and the only differing thing would be the global geometry. Completely ruled out for our Universe of course, but an empty spacetime solution to the EFEs.

 

[Atlas3]

You should just be able to put a flat metric on the torus. This would be locally equivalent to Minkowski space and the only differing thing would be the global geometry. Completely ruled out for our Universe of course, but an empty spacetime solution to the EFEs.

To comprehend an EFEs it need not match our Universe to have significance. Not useful significance necessarily. I'm not bright enough to do the numbers. But I can comprehend the algebra in a simple situation. I really appreciate your reply. You aren't required to. Thank you it provided guidance.