Spherical symmetric collapse of pressureless dust

[tom.stoer]

Is there an exact solution for the spherical symmetric collaps of pressureless dust? Can one see a Schwarzschild solution for r > R_dust with shrinking R_dust(t) ?

 

[Mentz114]

It is possible to embed a form of the Friedmann solution inside an exterior Schwarzschild spacetime. I can't remember if the interior bit can be made pressureless. My only reference right now is Stephani's "General Relativity".

[edit] The interior can be dust. So I guess this fits. This is the Tolman solution, it turns out.

 

[WannabeNewton]

Assuming you are looking for the collapse of a uniform ball of dust, I believe what you want is the Oppenheimer-Snyder collapse: http://grwiki.physics.ncsu.edu/wiki/Oppenheimer-Snyder_Collapse

See also if you can get access to Israel's paper on the collapse of a thin spherical shell of dust: http://adsabs.harvard.edu/abs/1967PhRv..153.1388I

Eric Poisson talks about both of these things (Oppenheimer-Snyder collapse and thin shell collapse) in sections 3.8 and 3.9 of his text "A Relativist's Toolkit: The Mathematics of Black Hole Mechanics".

Also, see the following passages from section 7.6 of Padmanabhan "Gravitation: Foundations and Frontiers":
http://postimg.org/image/48aevjpkp/
http://postimg.org/image/8swlah9a1/
http://postimg.org/image/upj4aumgp/

The rest of the section is about the nature of the collapse obtained from such solutions.

 

[tom.stoer]

Great - thx!

 

[sardar]

The spherical symmetric collapse of pressureless dust is a well-studied problem in general relativity. It describes the collapse of a cloud of dust particles under their own gravitational attraction, without any pressure or other external forces. This scenario has been extensively studied as it provides important insights into the behavior of matter in the extreme conditions of strong gravity.

There is an exact solution for the spherical symmetric collapse of pressureless dust, known as the Tolman-Bondi solution. This solution describes the evolution of a spherical shell of dust particles collapsing under their own gravity, and has been shown to accurately describe the behavior of collapsing stars and galaxies.

In this scenario, the collapsing dust particles continue to fall towards the center, causing the radius of the spherical shell to shrink with time. This process can be described by the Schwarzschild solution, which is a solution to Einstein's field equations for a spherically symmetric mass distribution. However, the Schwarzschild solution is only valid for r > Rdust, where Rdust is the radius of the collapsing dust shell. This means that the solution breaks down as the dust shell approaches the point of singularity, where the density and curvature become infinite.

Therefore, while the Schwarzschild solution can be used to describe the behavior of the collapsing dust shell for r > Rdust, it cannot be extended to the point of singularity. This is because the collapse of pressureless dust leads to the formation of a singularity, where the laws of physics as we know them break down. Therefore, it is not possible to see a Schwarzschild solution for r > Rdust with shrinking Rdust(t) in this scenario.

In summary, the spherical symmetric collapse of pressureless dust is a well-studied problem in general relativity with an exact solution known as the Tolman-Bondi solution. While the Schwarzschild solution can be used to describe the behavior of the collapsing dust shell for r > Rdust, it cannot be extended to the point of singularity where the solution breaks down. Further research is needed to fully understand the behavior of matter in the extreme conditions of strong gravity.