What are closed time-like curves and how are they related to General Relativity?

[zepp0814]

I understand that GR allows for a method of time travel using closed time like curves (CTC)s. anyway i have a few question about this, first of is there some sort of (relativaly short) equation that discribes this. So my second question is based of a something i read in this thesis paper (http://digitalcommons.bucknell.edu/cgi/viewcontent.cgi?article=1083&context=honors_theses Part2.7) It stated mentioned something called "time-like 4-velocities" what is this and and how is it calculated.

 

[WannabeNewton]

There are different ways of getting closed time-like curves. The most simple way is to take a space-time and create some kind of a quotient space in a suitable way so as to create CTCs. For example, if you take Minkowski space-time $(\mathbb{R}^{4}, \eta_{ab})$ and identify the $t = 0$ and $t = 1$ slices, you will end up with the cylinder $S^{1}\times \mathbb{R}^{3}$ which you can picture intuitively as the two time slices "wrapping around each other". In this case, the integral curves of $\frac{\partial }{\partial t}$ will be closed time-like curves as you can probably visualize. See chapter 8 of Wald "General Relativity" for a further discussion of CTCs. To give a more physical example, CTCs also exist near the ring singularity of a kerr black hole and, if I recall correctly, are the integral curves of the axial killing vector field $\psi = \frac{\partial }{\partial \varphi}$ (which are closed by definition) since it turns time-like near the ring singularity.

 

[Bill_K]

A Closed Timelike Curve is something that works like this:

 

[stevebd1]

I understand that GR allows for a method of time travel using closed time like curves (CTC)s. anyway i have a few question about this, first of is there some sort of (relativaly short) equation that discribes this.

The following paper looks at closed timelike curves in Kerr-Newman metric where CTC's occur at $g_{\phi \phi}=0$ (which is at the ring singularity for Kerr metric and outside/near the ring for the Kerr-Newman metric)

'A twist in the geometry of rotating black holes: seeking the cause of acausality'
http://arxiv.org/abs/0708.2324v2

The following paper also has some info from page 33 onwards-

http://casa.colorado.edu/~ajsh/phys5770_08/bh.pdf

 

[BruceW]

the basic idea of a ctc is fairly simple. In relativity, objects move along timelike curves. and if this curve comes back on itself, then it is a closed curve. hence timetravel. the possibility for having a ctc depends on the geometry of the specific spacetime you are trying to model. (for example, the one WannabeNewton mentioned). I don't know much general relativity, but I don't think there is any general way to say what kind of spacetime geometries give rise to the possibility of ctc's. Of course, you can pick examples of geometries that do have ctc's. And you can also pick examples of geometries without ctc's. But I don't think there is a simple, general way to categorise any and all spacetime geometries into those with or without ctc's. I think for a general case, you just have to find out 'by inspection'.

 

[audioloop]

https://www.physicsforums.com/showthread.php?t=624880

 

[zepp0814]

Thanks everyone but I have one more question what part of EFE are the solutions (such as kerr ) for. Since most of them if not all don't require a mass or energy In there metric equation. I always just assume it was a solution to the metric tensor

 

[zepp0814]

Oh well I get swartzschild has his radius equation built into his metric wh8ch uses mass