Where do CTC's come from/ How do I interpret these tensors?

[space-time]

I recently derived the Einstein tensor and the stress energy momentum tensor for the Godel solution to the Einstein field equations. Now as usual I will give you the page where I got my line element from so you can have a reference: http://en.wikipedia.org/wiki/Gödel_metric

Here is what I got for my Einstein tensor G_μν:

G₀₀ , G₁₁ , and G₂₂ all equal 1/2
G₀₃ and G₃₀ = e^x / 2
G₃₃= (3/4) e^2x

Every other element was 0.

As a result of this being the Einstein tensor, the stress energy momentum tensor T_μν is as follows:

T₀₀ , T₁₁ , and T₂₂ all equal c⁴/(16πG)
T₀₃ and T₃₀ = ( c⁴e^x) / (16πG)
T₃₃= (3c⁴e^2x)/ (32πG)

Every other element was 0.

Now my research has told me that this metric contains closed time-like curves within it. Can someone please tell me how these tensors showcase the possibility of closed time-like curves? I suppose what I really need is a solid understanding of how to interpret the physical implications of these general relativistic tensors.

I notice that when you do dimensional analysis on my stress energy tensor, you'll find that it contains all force terms (assuming that the exponential terms are unit-less constants since the number e itself is a constant, correct me if I am wrong).

Does this mean that if I want to warp a region of space time into a Godel space time, then I would need to apply a force that is equivalent in magnitude to the elements in my stress energy tensor in the directions that said elements represent?

In other words, does this mean that I would have to apply the following forces in the following directions:

c⁴/(16πG) Newtons in the temporal direction, the xx direction, and the yy direction

( c⁴e^x) / (16πG) Newtons in the time-z direction and the z-time direction

(3c⁴e^2x)/ (32πG) Newtons in the zz direction

If my interpretation is correct, how exactly would one apply a force in the temporal direction considering that the temporal dimension is time itself?

Also, I notice that the angular velocity (ω) terms that I started out with just totally disappeared by the time I got to the Einstein tensor (though this may have to do with the fact that my tensors are in a coordinate basis). What becomes of these? Surely angular velocity has significance in interpreting the physical implications of these tensors (especially if this metric contains closed time-like curves).

Finally, what exactly do the Einstein tensor elements tell you about space-time curvature? This Einstein tensor in particular contains all constants.

Please help understand the physical meanings of these tensors, as well as where the CTC's come from. Thank you.

 

[PAllen]

I have no idea how you can look at a stress energy tensor or metric and see whether it contains CTCs, without finding them somehow. However, if your question is simply to exhibit them for this metric, the following writeup, on p.6, shows this in a quite simple way:

http://www.math.nyu.edu/~momin/stuff/grpaper.pdf

 

[aleazk]

I never studied the Gödel metric, to be honest, but I am familiar with the van Stockum dust solution. In this metric, it's quite easy to detect the CTC and also the solution itself (more precisely, something similar to it) seems to have some physical plausibility (at least when compared to the Gödel solution).

The general form of a metric that is both stationary and axisymmetric is (the Weyl-Papapetrou form):

$$
g=H(\mathrm{d}r\otimes\mathrm{d}r+\mathrm{d}z\otimes\mathrm{d}z)+L\mathrm{d}\varphi\otimes\mathrm{d}\varphi+M\mathrm{d}\varphi\otimes\mathrm{d}t-F\mathrm{d}t\otimes\mathrm{d}t
$$

where the coordinates used here have interpretations analogous to those of the usual cylindrical coordinates. In particular, $0\leq\varphi\leq2\pi$ and the curves with $r=constant,t=constant,z=constant$ are closed curves.

If we take the tanget vector field to these curves, the inner product of this vector field with itself is given by $g_{\varphi\varphi}=L$.

Intuitively, like in flat spacetime (where $g_{\varphi\varphi}=r^{2}>0$), one expects these closed curves to be spacelike. Nevertheless, there are solutions of the EFE in which $g_{\varphi\varphi}<0$ in some region, i.e., the tangent field is timelike and thus these closed curves become closed timelike curves.

An example is the van Stockum dust solution, where the SET is that of an infinitely long, rigid dust cylinder (of ordinary matter) rotating with angular velocity $\omega$. In the interior, $g_{\varphi\varphi}=L=r^{2}(1-\omega^{2}r^{2})$. So, if the boundary of the cylinder extends that far, the closed curves considered above become timelike when $r>\frac{1}{\omega}$.

It's conjectured that the case for the finite cylinder also contains CTC. So, why people is not building their own time machines at home? after all, all you need is a rotating cylinder.

The detail is that these solutions describe eternal cylinders, i.e., in the solution, the rotating cylinder always existed, there's nothing about its creation. If you include the creation of the cylinder, then the EFE require exotic matter for this.

Check this paper by Kip Thorne for a detailed and general study of the physical feasibility of CTC: http://www.its.caltech.edu/~kip/scripts/ClosedTimelikeCurves-II121.pdf

 

[space-time]

I have no idea how you can look at a stress energy tensor or metric and see whether it contains CTCs, without finding them somehow. However, if your question is simply to exhibit them for this metric, the following writeup, on p.6, shows this in a quite simple way:

http://www.math.nyu.edu/~momin/stuff/grpaper.pdf

Thanks for the link. I was reading it and I must ask: What is ε^μijk? The pdf does not define that term nor how to calculate it (if it is something that must be calculated).

Also, when I calculate a_ijk, wouldn't I have to take the cross product first and then do the dot product afterwards?

 

[DrGreg]

What is ε^μijk?

See Levi-Civita symbol. In particular, see the subsection on "Levi-Civita tensors".

 

[bcrowell]

What you're trying to do is infer global properties from the curvature. There are general techniques for doing this, e.g., Myers's theorem in Riemannian geometry, but I don't know much about them. Historically I think these techniques were developed relatively late (Myers's theorem is probably ca. 1940), so I don't think Godel would have used them. Ca. 1916-1960, I think the methods used for finding solutions to the Einstein field equations were that people simply played around with concrete coordinate systems.

 

[PAllen]

I thought of one general test: the Kip Thorne test. Google Kip Thorne + the name of the metric. If Kip Thorne wrote about it, it probably contains either CTCs or worm holes