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?
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?
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.
The theorem mentioned here, that in a finite range exotic matter, violating the energy conditions is necessary to form CTCs, i.e. that spacetime is suitable for time travel, is simply not true. Whoever supposedly proved it.
It has long been known that regular counterexamples, i.e. finite time machine spacetimes with normal matter, satisfying the energy conditions, are possible. For example this:
https://arxiv.org/pdf/gr-qc/0503077.pdf
