But why assume spherical-symmetry? I last took physics 18 years ago, but my guess is that two spheres will deform each other, at least a little bit. As you say, forever is a long time. Over a long time, can you say for sure that that tiny deformation won’t make a difference?
Oh, now I see what you’re saying. Yes, in the two-body problem there will be some tidal deformations (which kill the exact spherical symmetry and probably make it analytically intractable).
My understanding is that dark energy is constant but the expanding universe will cause distance dependent forces (like gravity) between bodies to get weaker as the distance increases. But I could be wrong.
I’m still trying to wrap my head around the concept of once you’re inside the event horizon, time-space warps so that a collision with the singularity is inextricably in your future.
If you’re familiar with the concept of the light cone just think of it abruptly tipping at the event horizon, or if you’re using Finkelstein coordinates it slowly tips. But in either case it tips to the point that the time axis points toward the singularity. (sort of)
The gravitational field from a black hole is the same as any other gravitational field. The same orbital mechanics that govern the interaction of any two bodies applies.
A big glob of ordinary matter, lets call it the Glob, is traveling along a trajectory that is the result of all its previous encounters with objects. At some point along this trajectory it reaches a point in the gravity field of a black hole that makes the black hole’s gravity significantly greater in magnitude than any other matter in the universe, with respect to the Glob.
Many billions of years ago, the result of all the interactions between all the bits of matter in the universe caused both the black hole, and the Glob to have the velocity they have at that time. Doing the math is a tough one, but the fact is that the velocities are what they are, because of all the previous interactions. At this point in time, as in any other point of time, it is possible to predict the interactions with the Glob and the black hole, by solving the orbital characteristics of each of them with respect to their mutual center of mass. It’s not mysterious because one of them is a black hole.
A billion miles apart, one of the same three things are true: the Glob will orbit around the black hole, it will pass by the black hole in a hyperbolic orbit, or it will intersect the sphere represented by the Swartzchild radius. If both were non black holes, instead of the Swartzchild radius, we would be concerned with whether either object would come within Roche’s limit, and be fragmented by tidal forces. Below a certain limit, what would matter is whether the objects would pass close enough to actually impact. All of them still represent possible outcomes of the gravity of the pair, and the original velocity of the objects. It doesn’t change because of the Swartzchild radius. In fact, Roche’s limit might well be outside of the Swartzchild radius, and part of the Glob would be torn into sub atomic bits, part slung as whole matter to either an escape or orbital path, and part accrete to the black hole.
The only “mysterious” part of the thing is the fact that the Swartzchild radius is very important. It is the point at which the escape velocity is equal to the speed of light. That would not be important, except that nothing can go that fast and therefore everything that gets that close must remain within that radius. Another object that is moving toward the black hole at a lower velocity might well accrete as well, of course.
Tris
In the simplest models where the dark energy is constant, distant galaxies will become even more distant, and quickly become completely insignificant (since the force from the dark energy increases with distance, while normal gravity decreases). But nearby galaxies continue to orbit each other just like they always have, and never get far enough apart for the dark energy to take over. Likewise, the galaxies themselves, and all the objects within them, remain intact.
In the Big Rip models, however, the dark energy is steadily increasing in strength. Right now, it’s not strong enough to tear a galaxy cluster apart, but some time in the future, it will be. As it gets stronger, it in turn tears apart individual galaxies, and star clusters within those galaxies, and solar systems, and planets, all the way down to subatomic particles, in a spectacular singularity that ends the Universe.
Currently, the data actually slightly favor the Big Rip models (which is the main reason they get any attention at all), but it’s a very extraordinary claim, and the evidence for it is not yet extraordinary enough. So most cosmologists still acccept the simpler constant models.
“It’s a crazy theory.” “Unfortunately, not crazy enough.”