> Since we know that infinite densities cannot actually happen in the universe, we take this as a sign that Einstein's theory is incomplete. But after nearly a century of searching for extensions, we have not yet confirmed a better theory of gravity.
Is it really true that infinite densities are impossible? Consider a function that measures the volume of an object with respect to time, say V(t). If the force of gravity is strong enough to overcome every other force, wouldn't V(t) approach 0 as t goes to infinity? Isn't that essentially having infinite density?
It's what you would consider a mathematical limit. The limit is never actually reached. V(t) becomes vanishingly small, to the point that time effectively stands still, but mathematically it should still always remain a non-zero value, and thus density at the singularity isn't infinite. Anyway, at that point measuring things becomes exceedingly difficult and abstract.
I guess my point is that there isn't really an infinite quantity in mathematics, either. What we think about as "infinity" usually involves some type of unbounded limit.
The real question is: is the volume infinitely compressible or is there something that would eventually oppose enough counterforce to halt the collapse?
the infinite density is actually after a finite amount of time, since black hole collapse has already happened, which is why it matters that they are predicted to have infinitely dense points at their centers. The key is that this isn't a limit, GR seems to expect that there are infinitely dense objects just floating around breaking space-time.
Is it really true that infinite densities are impossible? Consider a function that measures the volume of an object with respect to time, say V(t). If the force of gravity is strong enough to overcome every other force, wouldn't V(t) approach 0 as t goes to infinity? Isn't that essentially having infinite density?