1. In correspondence to AdS black hole solutions, what does it mean by Misner string singularities? And when there are no Misner string singularities, what does this mean in terms of curvature singularities and event horizons and the black hole in general?

  2. A related question (as far as I'm aware), what do we do if our solution contains closed timeline curves (CTCs)? Should we really only consider cases when our solution doesn't contain closed timelike curves and strictly impose restrictions on our parameters that ensure this?

  3. (While on this topic, an elementary question, why does $g_{\phi \phi}<0$ imply CTCs, if we're in hyperbolic 4d space?)

  • $\begingroup$ Comment to the question (v3): Consider adding some references to make the question more accessible to a wider audience. $\endgroup$ – Qmechanic Aug 7 '14 at 14:36

If you construct your metric in such a way that $\phi$ is a killing vector generating an axisymmetry of the spacetime, and there is a value of the other three coordinates where $g_{\phi\phi} < 0$, then you have a timelike killing vector generating an axisymmetry. The curve traced out by this vector will then be a closed timelike curve by construction.

We tend to throw out solutions containing closed timelike curves as unphysical, because they break causality.

I haven't heard the term "misner string singularities" before, but I assume your authors are referring to a cosmic string spacetime, which will have naked singularities. You can derive a rudimentary version of this by going through the schwarzschild derivation, but assuming cylindrical symmetry rather than spherical symmetry.

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  • $\begingroup$ So we really want to enforce $g_{\phi \phi}>0$? And a paper I have here, $\text{Integrability in conformally coupled gravity: Taub-NUT spacetimes and rotating black holes}$ says the following: "the radial coordinate can be extended to range from $-\infty$ to $\infty$ starting and ending at a hyperbolic slicing of AdS. In this way the metric has up to four Killing horizons, but no space-time singularity whatsoever". Honestly, I really don't understand what this means. I mean, I'm assuming there is no value of $r$ (radius) s.t. the metric blows up. $\endgroup$ – Phibert Aug 6 '14 at 20:17
  • $\begingroup$ Sorry, forgot to tag you in my comment. I feel like you may be able to help me with the previous comment? $\endgroup$ – Phibert Aug 8 '14 at 11:44

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