Big Crunch in an infinite, symmetrical, Newtonian universe?

[Pony]

Is there a Big Crunch in an infinity, Newtonian, symmetrical universe, e.g. an infinity grid, with stars in its nodes, due to the gravitational attraction?

One can argue that every net force is 0, so there isn't one.

But I think there is an another argument, that leads to a Big Crunch, that I fail to see rn.

 

[Ibix]

In a Newtonian universe filled with uniform density matter the amount of mass in a thin shell at radius ##r## scales as ##r^2##, and its contribution to the potential at your location scales as ##r^{-1}##. Thus the potential at your location (or indeed any other) looks like##\int_0^\infty rdr=\infty##. So I don't think such a Newtonian model is consistent to start with.

 

[Orodruin]

One can argue that every net force is 0, so there isn't one.

Not consistently.

In a Newtonian universe filled with uniform density matter the amount of mass in a thin shell at radius ##r## scales as ##r^2##, and its contribution to the potential at your location scales as ##r^{-1}##. Thus the potential at your location (or indeed any other) looks like##\int_0^\infty rdr=\infty##. So I don't think such a Newtonian model is consistent to start with.

The shell theorem essentially relies on being able to pick a potential that is zero at infinity. Unfortunately, this is incompatible with an infinite constant density (as you have shown here). Considering the Newtonian gravitational potential such that
$$
\nabla^2 \phi = 4\pi G \rho_0 = {\rm constant},
$$
the boundary condition at infinity necessarily breaks the homogeneity and isotropy assumption for any solution.

 

[Pony]

But I think there is an another argument, that leads to a Big Crunch, that I fail to see rn.

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]

Maybe: take a big, finite ball, calculate the shrinking speed around the origin

What shrinking speed around the origin are you talking about?

 

[berkeman]

Thread closed for Moderation...

 

[PeterDonis]

After some cleanup, thread is reopened.

 

[jbriggs444]

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".

 

[PeterDonis]

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.

We already know the result: this is the 1939 Oppenheimer-Snyder dust collapse model. The dust collapses to form a Schwarzschild black hole.

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.

Not in this scenario. You are thinking of a static spherical body. The spherical cloud of dust is not static.

as the dust cloud evolves, it will remain spherical and will always have uniform density. Though that density will increase over time.

This is true, but the computation that Oppenheimer and Snyder did to reach this conclusion is not the one you did. The correct computation makes use of the fact that the dust region at the initial instant is identical to a portion of a closed matter-dominated FRW universe at the instant of maximum expansion. The already known dynamics of that solution is then used to give the dynamics of the dust region, and Birkhoff's Theorem is sufficient to prove that the vacuum region exterior to the dust is the Schwarzschild vacuum. Those two things are all that is needed.

For more information, see this series of Insights articles (the link is to the first of three):

https://www.physicsforums.com/insig...-model-of-gravitational-collapse-an-overview/

 

[PeterDonis]

@PeterDonis has asked for clarification on the model.

The reason I asked is that I am not aware of any relevant model in which there is "shrinking speed around the origin".

 

[jbriggs444]

This is true, but the computation that Oppenheimer and Snyder did to reach this conclusion is not the one you did. The correct computation makes use of the fact that the dust region at the initial instant is identical to a portion of a closed matter-dominated FRW universe at the instant of maximum expansion.

Right. I understood @Pony to be working under the model of Newtonian mechanics rather than on the correct model of general relativity. My calculations are based on the Newtonian model.

And yes, I agree that an accelerating Newtonian collapse would not be well described as a fixed shrinking speed around the origin.