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(adsbygoogle = window.adsbygoogle || []).push({});

Here is what I got for my Einstein tensor G_{μν}:

G_{00}, G_{11}, and G_{22}all equal 1/2

G_{03}and G_{30}= e^{x}/ 2

G_{33}= (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:

T_{00}, T_{11}, and T_{22}all equal c^{4}/(16πG)

T_{03}and T_{30}= ( c^{4}e^{x}) / (16πG)

T_{33}= (3c^{4}e^{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:

c^{4}/(16πG) Newtons in the temporal direction, the xx direction, and the yy direction

( c^{4}e^{x}) / (16πG) Newtons in the time-z direction and the z-time direction

(3c^{4}e^{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.

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

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