Question

As per Newton objects with mass attract each other, and per Einstein this is further explained by saying that mass warps space-time. So a massive object makes a "dent" into space-time, a gravity well. I have taken to visualizing this as placing a object on a rubber sheet and the resulting dent, being the gravity field. So obviously placing two objects on the sheet not to far from each other will make the dents overlap, and the object will roll towards each other. BUT for a BLACKHOLE, this is not a dent. It's a cut or rupture in the rubber sheet. Furthermore, space-time is constantly falling INTO the blackhole, and everything else that exists in space-time, including light. So a blackhole is not just a super-massive object, it's really a hole, and how can a hole move? How does it react to the gravity pull of a nearby object, when everything just falls thru it? Thanks!

Answer 1

Your mental picture is pretty correct except for the part "space-time is falling into the blackhole". This is completely wrong. Space-time is not falling anywhere. Consider Schwarzschild's solution. This one is static. So it's obvious that nothing happens with space-time at all, it just sits there.

In fact, you wouldn't be able to distinguish the black hole from an usual object of the same mass just by its gravitational effects on other distant objects alone. For objects that are far away from the BH the good old Newtonian limit will apply and it will attract them as described by Newton's gravitational law. In particular, it will attract another BH.

The picture you had in mind about the attraction of two objects as dents in rubber sheet that get closer is correct. This continues to hold even with black holes except that in this case it is not matter that bends the space-time but instead it's a self-sustaining space-time configuration. So a BH can in fact be thought as a special form of matter. Also, never mind that the sheet is ruptured at the singularity because on the outside of the horizon the space-time is perfectly regular and the inside of the horizon is causally disconnected from outside observers anyway (meaning that black hole is indeed black).

Einstein's equations will tell you how this curved space-time, which describes e.g. two (or more) black holes, will evolve into the future. Besides the usual Newtonian scattering you'll get relativistic corrections for their motion, you'll get gravitational waves flying away and other effects. E.g. the two black holes could join into one bigger hole.