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Googled "inner horizon" and found it used quite a bit in the context of Reissner-Nordstrom (electrically charged) black holes, which I have apparently neglected in my studies. The structure is somewhat more complicated than standard Schwarzschild black holes. E.G.
http://staff.science.uva.nl/~jpschaar/report/node7.html
"It turns out that the presence of the inner horizon introduces some very different features w.r.t. the Schwarzschild solution. One of the most important, from an infalling observer point of view, is the fact that the curvature singularity is timelike now, which means it can be avoided and even better; free falling observers cannot reach it. At first it was believed that this opens up the possibility to avoid ending up at the curvature singularity in the Schwarzschild space-time just by bringing a charge with you. The idea was that when you fall in the Schwarzschild black hole, it becomes charged and turns into a Reissner-Nordstrøm black hole. Sadly enough, this will probably not work, because from perturbation calculations it is suspected that slightly non-spherical collapse will turn the inner horizon into a singularity.
The outer horizon in this space-time behaves just as the event-horizon in the Schwarzschild case. When inside r+ you are forced to reach r-, just as an infalling observer in Schwarzschild is forced to reach the curvature singularity. When crossing the inner horizon the observer would be face to face with a curvature singularity, but as mentioned before an observer will not end up at the singularity. Instead of ending up at the singularity, the observer can continue into another asymptotically flat spacetime. (If it would be the same asymptotically flat space-time the causal structure would not be uniquely defined, it would depend on whether you came from the region inside r- or from outside r+). When all timelike curves can be extended to infinity (they never end up at a singularity), as is the case, the space-time is called (time-like) geodesically complete."
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Thanks, Metric. I don't think I can understand this anything more than superficially. But at least this confirms that when I heard the term "inner horizon" in the Nova documentary, I was right to think it was some (theoretically) real phenomenon that I had never heard of before. It's a pity the Nova documentary didn't clarify the physical scenario they were talking about, namely electrically charged black holes.