Coordinate projections sometimes have a habit of going wierd when you try to project them past a gravitational horizon. Sometimes you can do it, sometimes you can't, and sometimes the attempt turns various things inside out. A cool physical inversion that happens outside the horizon was used as the March 1993 cover story for Scientific American: Black Holes and the Centrifugal Force Paradox (by Marek Artur Abramowicz).
The effect isn't really paradoxical, but it's counter-intuitive until you think it through. Normally, if you orbit a body, you can break free of that body by firing up your spaceship's engines and going faster – too fast to be able to orbit at your current distance. What the BHCFP says is that if you're skimming too close to a black hole event horizon, and you fire up your engines, then the faster you try to circle, the more that your trajectory is deflected inwards, towards the hole. The centrifugal forces that would normally throw you away from the body, now seem to be inverted, pointing inwards rather than outwards.
The critical threshold beyond which this effect appears is the distance r=3M, exactly one-and-a-half times the radius of the horizon surface (which is at r=2M).
It turns out that the r=3M radius is the photon orbit. It's the critical distance at which light aimed at 90 degrees to the mass will be deflected enough by gravity to perform a complete orbit and end up at its starting-point. The SciAm article has some nice computer graphics showing what a circular self-supporting scaffolding tube constructed around the hole at r=3M would look like to an observer standing inside it ... it'd appear to be straight, and if the observer pulled out a telescope and looked far enough along the tube, they'd expect to see the back of their own head.
So r=3M is special. From the perspective of the observer at r=3M who's hovering with the aid of rocket engines, or standing in our circular tube up above the hole, the universe seems to be divided into two regions. On one side they see the black hole and its immediate surroundings, and on the other, they see the starfield that represents the outside universe. Topologically, both regions can be thought of as solid spheres, with their external parallel surfaces meeting at r=3M. Both regions are trying to impose their will on the observer's local geometry, but at r=3M, a stationary observer feels the geometrical competition between the effect of the two spheres as being in balance (although in order to maintain their position hovering above the hole, they're feeling rather a strong gravitational pull!). Spin either one of the two spheres, and the observer will be pulled towards it – spin both at exactly the same rotational rate – the effect that we'd see if we passed along the tube at high speed – and the radial gravitomagnetic effects of both spheres cancel.
So if you built an electric train to run around the interior of the tube, it'd feel the black hole's conventional gravitational attraction pulling it against one side of the tube ... but that pull would seem to be exactly the same no matter how quickly it circled the hole.
The author's moral is that if you're in a spaceship close to a black hole, and you want to escape, don't just throttle up your engines, actually point your ship away from the damned thing, or you're liable to get a nasty crashy surprise.
When we think about the optics of the situation, though, perhaps the hypothetical spaceship captain wouldn't be all that surprised:
See, if we imagine standing on a suspended non-orbiting platform at r=3m, we find ourselves looking along the r=3M surface in any (perpendicular) direction. The surface appears to us to be a flat plane cutting through our location. And because our view along r=3M circles around the hole indefinitely, our view along this apparent plane repeats indefinitely, too – the plane appears extend indefinitely far in all directions, showing us older and older views of the surface at greater distances, right back to the time that the black hole originally formed. So logically, anything that we see to one side of the plane corresponds to the interior of the r=3M sphere, and everything we see to the other corresponds to the contents of the "rest-of-the-universe" sphere. The outside universe only seems to exist on one side of this plane. On the other, gravitational lensing effects make the black hole's r=2M surface beneath us appear to be opened out into a second indefinitely-repeating surface, at some distance below the 3M plane.
Once we're at the 3M surface, there are two ways that we can go. If we slowly winch ourselves upwards away from the hole, then we see the flat 3M boundary of the outside universe curving itself back into a more normal-looking inward-facing enclosing sphere. But if we allow ourselves to be lowered further towards the black hole, to less than r=3M, then the 3M surface continues to distort past being a flat plane, to becoming a concave surface that curves above us, away from the hole. Instead of the universe surrounding the black hole, it now seems to us that the black hole (and the r=3M surface) is surrounding the universe! The region that we know ought to be just above the 2M surface appears visually to us to be part of a concave shell, apparently wrapped around a ball representing the remaining universe. The abstract, "topological" idea that our location can affect the choice of which sphere is "really" on the inside or outside now appears to us, visually, to be concrete reality!
The further we descend (slowly) towards 2M, the more pronounced the effect becomes, the more sharply the 2M surface appears to be curved around us, and the more that the outside starfield above appears to shrink to something that looks like a little bright ball suspended somewhere above, in the enveloping black-holey gloom directly above us, like a tiny planet or star.
So if we're hovering too close to r=2M, (or flying past in a spaceship) we shouldn't really be surprised if increasing our forward speed results in our colliding with part of the hole, because that's exactly what our forward view tells us is directly in front of us (and on every side, and directly behind us). If we want to escape from the hole's influence and get back to normal space, then we have to aim our spaceship at the little shrunken blob of compacted blueshifted starfield directly above us. All other directions point at the black hole.
So the rule-of-thumb for navigating within r=3M would seem to be: forget about your ship's fancy gyroscopic navigation systems, just look out of your window and make sure that the ship's nose appears to be pointed approximately at the part of the universe that you want to go to. But don't take your eyes off the forward view, because the harder your engines fire on your way out, the the stronger those distortion effects are going to become.