# Oppenheimer-Snyder Collapse

Oppenheimer-Snyder collapse is the gravitational collapse of a uniform ball of dust. The initial areal radius of the ball is $R_s \left(0\right)$ and the ADM mass is $M$. The Oppenheimer-Snyder solution is a piecing together of two other exact line element solutions at the surface boundary of a uniform spherical ball of uncharged dust. The external solution is the Schwarzschild solution. Internal to the matter the solution is the closed Fiedmann line element

$ds^2 = -dt'^2 + A^2 \left[ d\chi^2 + \sin^2\chi \, d\Omega^2 \right]$

"To model a pressureless ball" the $A$ and $t'$ are then parameterized in terms of a timelike coordinate $t''$ according to

$A=\frac{1}{2}A_0 \left( 1+cost'' \right)$
$t'=\frac{1}{2}A_0 \left( t''+sint'' \right)$

where matching boundary conditions we relate $A_0$ to the initial Schwarzschild radial coordinate radius of the matter by

$A_0 = \sqrt{\frac{R_s ^3 \left(0\right)}{2M}}$

and have

$\sin\chi _{s0} =\sqrt{\frac{2M}{R_s \left(0\right)}}$

The Schwarzschild radial coordinate radius of the matter as a function of our timelike coordinate parameter is

$R_s =\frac{1}{2}R_s \left(0\right)\left(1+cost''\right)$

And the Schwarzschild time evaluated at the surface can be related to our timelike coordinate parameter by

$t=2Mln\left(\frac{\sqrt{\frac{R_s \left(0\right)}{2M}-1}+tan\left(\frac{t''}{2}\right)}{\sqrt{\frac{R_s \left(0\right)}{2M}-1}-tan\left(\frac{t''}{2}\right)}\right)+2M\sqrt{\frac{R_s \left(0\right)}{2M}-1}\left(t''+\frac{R_s \left(0\right)}{4M}\left(t''+sint''\right)\right)$

and the Kerr-Schild coordinate velocity of the surface with respect to the remote observer is finally

$v=\sqrt{\frac{\frac{2M}{R_s }-\frac{2M}{R_s \left(0\right)}}{1-\frac{2M}{R_s \left(0\right)}}}$