Pony said:
Maybe: take a big, finite ball, calculate the shrinking speed around the origin, and take the limit, and see if that tends to a finite, non zero number. I can calculate that
@PeterDonis has asked for clarification on the model. I am not
@Pony, but I want to guess at what he had in mind.
We take a finite spherical dust cloud with uniform density. The dust is initially at rest. We allow the system to evolve from this starting point and observe the result.
It is a well known consequence of the shell theorem that gravitational acceleration within a spherical body of uniform density scales directly with distance from the center. We would expect dust at the periphery of the cloud to be accelerating toward the center at a calculable rate: ##a = \frac{GM}{R^2}##. Here ##R## is the radius of the entire cloud.
Dust halfway out from the center would be accelerating inward at half rate. Dust 10% of the way out at 10% rate. ##a \propto r##
Similarly, the inward velocity of the dust will be proportional to radius: ##v_r \propto r##
So as the dust cloud evolves, it will remain spherical and will always have uniform density. Though that density will increase over time.
So the inward collapse will result in a point of infinite density in finite time. One could write a differential equation and solve for the collapse time as a function of ##G## and the initial density. The size of the spherical distribution does not enter in.
If we reverse time and "run the film backward" from a perfectly uniform starting point, we should see decellerating expansion and a moment of minimum density followed by an accelerating collapse and a "big crunch".