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I remember reading something about Stephen Hawking denying the fact you can't make CTC's (Closed Timelike Curves) without weak energy condition violation. If this is true, where do the light cones point to in the $t$ direction?

enter image description here

On the end of the right and left, the cone points up (future) but near the cylinder it tilts. In a region with negative energy density, do the cones tilt all the way to the $-t$ direction?

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Where do what light cones point to? Is there a specific scenario you are referring to? I assume T is timelike. Perhaps include more information? –  Jim Jun 6 at 19:45
    
Like in this image, inspirehep.net/record/1262683/files/Tipler2.png On the end of the right and left, the cone points up (future) but near the cylinder it tilts. In a region with negative energy density, do the cones tilt all the way to the -t direction? –  user122083 Jun 6 at 19:56
    
Weak energy condition is a property of the stress energy tensor, which you should get by computing the Einstein Tensor. All you gave in the question is a picture, from which is not possible to answer if it does violate or not WEC. On the other hand, you don't need to have the future directed light cones pointing in the $-t$ direction if you want closed causal curves, as the picture shows. –  cesaruliana Jun 7 at 18:09

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Stephen Hawking proved that closed timelike curves cannot be created in a finite system without using exotic matter. The Tippler cylinder doesn't use exotic matter, but it can create CTCs because it's infinite in length and therefore isn't a finite system.

I think the proof was in his paper on the Chronology Protection Conjecture.

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