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CTCs are found in the region where $r < 0$ . That should be just inside the ring singularity, since in Boyer Lindquist coordinate system $r = 0$ means ring singularity. Does that mean this image showing CTC outside the ringularity is wrong, or I am missing something?

https://www.researchgate.net/figure/This-is-how-light-cones-behave-in-Kerr-Newman-spacetimes-For-r-0-r-1-and-computations_fig2_1760606

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You are missing something. In a text where $r < 0$ is possible, the singular ring is a curve, not a surface: its equation in B-L coordinates is $(r = 0, \cos \theta = 0)$.

In contrast, $r = 0$ is the equation of a surface: it consists of two disks $(r = 0, \cos \theta > 0)$ and $(r = 0, \cos \theta < 0)$ together with the singular ring, their common boundary. The 2 disks are made of regular points: they are actually 2 manholes through which you can reach into negative space.

The picture is ambiguous: if the text from which you got it claims the CTC is in negative space (not sure, the pic is clipped), then it lies below the region $r = 0$ rather than just outside.

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  • $\begingroup$ Just to add that the planar disk inside the ring singularity is actually a discontinuity in the metric components. This is shown in H Ohanian’s book. $\endgroup$
    – Cham
    Dec 25, 2021 at 17:57
  • $\begingroup$ @Cham in such an interpretation, $r < 0$ is impossible. $\endgroup$ Dec 25, 2021 at 18:14

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