Spacetime geometry around two black holes For space-time geometry we all know images like he one below. But if I were to take a neutron star and put it right next to but not touching a black hole what effect would that have on the geometry of space time? Would the neutron star & black hole deform space-time more than the black hole did by itself? If the answer is yes, then if the neutron star is in a space-time well deeper than a black hole by itself would that now in turn make the neutron star a black hole as well?
I am curious about this because that would mean if you had a neutron star whiz past a black hole fast enough to not fall in, could we then see the neutron star turn into a black hole as it passes close to the black hole and then go back into a neutron star as it goes away from the gravity well of the black hole? 
Maybe a simpler question to the ones above, how do two large gravity wells behave when they get close to each other? 
and
At a certain "depth" of a gravity well does something turn into a black hole?   
image from sciencenews.org

 A: The pictures you have above are great ways to visualize the deformation of space-time by a mass, but they can be misleading.  A black hole, for example, is not just an extremely deep well, it's a point where the curvature grows infinite.  If you were to try to imagine what this would look like in the above pictures, it would be like poking a hole in the rubber sheet, or pulling it down so that it has infinite depth.  There is no finite "depth" that a gravitational well must have to suddenly become a black hole.
You also ask how two gravitational wells interact.  Since the Einstein field equations are nonlinear, the wells don't simply add.  Instead, they interact and interfere in some complex way.  However, to a first approximation, we can imagine just adding the "depths" of the wells to produce another picture similar to the ones in your post above.  This immediately shows that a neutron star whizzing past a black hole can't become a black hole and then turn back into a neutron star.  To become a black hole, the neutron star would have to have infinite "depth," and the only way to get to infinity from a finite depth is to add infinity.  Thus unless it is sucked into the black hole, no point in the neutron star will ever reach infinite curvature/"depth."  Whizzing past the black hole, on the other hand, could have other effects on the neutron star.  The black hole could strip a large part of the star's mass away for example.
Finally, the only known way for a black hole to dissipate is through Hawking radiation, so even if the neutron star somehow collapsed into a black hole, it certainly wouldn't spontaneously turn back into a neutron star.
A: Small point to add to this, but all matter has a Schwarzschild radius - the earth does (it's about 1/3rd of an inch in diameter), black holes do - it's their event horizon, and Neutron Stars do - but it's smaller than the star.   Any non-black hole object has a Schwarzschild radius that is smaller than it is.   Neutron Stars are dense enough that their Schwarzchild radius is a measurable percentage of their radius, not a teeny-tiny itty bitty percentage of the radius, like pretty much anything else, but the Schwarzchild radius is still smaller than the Neutron Star.
A quick calc, but the Schwarzchild radius of a small Neutron star - about 1.4 solar masses is about 4 KM, while the diameter is about 12-13 KM.   A larger Neutron Star, 2 solar masses, Schwarzchild is about 6 KM and diameter is about 10-11 KM.   Too much bigger than that and it probobly becomes a black hole.
A curious thing about the Schwarzchild Radius is that it's proportional to the mass, so a 3 stellar mass black hole has half the event horizon radius as a 6 stellar mass black hole.   The math is actually very neat.   For 2 black holes, to be close enough together to form 1 bigger black hole - at least in theory (in practice they'd spin around each other furiously fast), but in theory, for the mass to be big enough for them to merge, the Schwarzchild radii have to touch.   The same is true for a Neutron Star and a Black hole - for the combined mass of a Neutron Star and a black hole to create a bigger black hole, the Neutron Stars Schwarzchild Radius and the Black Holes Schwarzchild Radius have to touch and for that to happen, part of the Neutron star would have to be inside the black hole.
So flying past and creating a bigger black hole, it's not possible.   What is possible is that a Neutron Star flying past a black hole might get torn apart, which would be interesting to see what happens to the Neutron Star if it loses a chunk of it's mass and (perhaps) expands back into a white-dwarf's density - kind of thing, or perhaps, something else.
Now, if you were to, in theory, get several Neutron Stars to fly just past a black hole at the same time - that could be very different.   According to the "Kissing Problem", you could get 12 Spheres of equal size touching an inner sphere, so you have 13 Mass inside 3 radius.
Kissing Problem:  http://en.wikipedia.org/wiki/Kissing_number_problem
Taking the 1.4 stellar mass Neutron Star, 13 of them touching together would have a Schwarzchild radius of 52 miles and 13 of them could fit inside 39 miles - so if you have a few Neutron Stars fly close to a black hole at the same time, the combination of masses could be enough to create a bigger black hole - er, in theory, which once formed, nothing would escape.  It'd odd to think about but it's theoretically possible, if you shoot 1 Neutron Star right past a black hole, it could fly past - but if you shoot 2 near each other right past the black hole, it's possible that the combination of 3 Schwarzchild radius would expand around some of the 2 Neutron Stars before the stars touched the black hole's event horizon.
In a practical sense, such things are impossible, but theoretically, you'd need, bear minimum, 2 Neutron Stars by flying past a black hole, very close to each other and close to the black hole at the same time and they'd need to be dense - with Schwarzchild radii close to their actual radii.   In that impossible situation, the combined mass in theory could expand an event horizon that would envelope both the Neutron Stars.
In reality, it's probobly a lot more complicated, probobly for a few reasons, but one, for example, a Neutron star that flew inside the photon-sphere (1.5 times the Schwarzchild radius) could ever escape, though inside the photon-sphere it could get torn apart and some of it could explode it's way back outside and escape, but no object that flies inside the PhotonSphere can fly back out unless pushed.
The same principal would apply to 3 black holes that flew close together.  2 black holes and they wouldn't become 1 black hole till the event horizons touch, but 3 black holes all close together could become 1 black hole before the event horizons touch.   In reality, however, with black holes pulling space with them, the interaction of 2 black holes is probobly a bit more complicated - I'm using the simplest of possible estimations.
Another interesting sidebar to all of this is that if you take all the mass of the known universe and calculate the Schwarzchild radius it comes to about the size of the known universe, so, it's possible the universe is massive enough to create a black hole around it.   The trick is using 3 dimensions to combine mass - not just 2 object.  2 objects can't do what you're describing.   
(I hope that's not too confused and convoluted, it's kind of late, and that was longer than I meant it to be)
