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?