Oppenheimer-Snyder Collapse

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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)}}}
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