Oppenheimer-Snyder model

The Oppenheimer-Snyder model describes a homogeneous ball of pressureless matter collapsing from rest, with a closed FLRW metric inside and a Schwarzschild metric outside. In geometric units its interior is $d{s}^2=-d{\tau }^2+a^2(d{\chi }^2+{\sin }^2\chi d{\Omega }^2)$, with $a=a_{\max }(1+\cos \eta )/2$ and $\tau =a_{\max }(\eta +\sin \eta )/2$. The boundary is a fixed comoving angle. Density conservation gives $\rho a^3$ constant, and the Friedmann equation at the initial turning point gives $8\pi {\rho }_0{a}_{\max }^2/3=1$.