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
Best Answer
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.