= Homogeneous dust-ball matching
{title2=$2M=a_{\max}\sin^3\chi_0$}
At the boundary of an <Oppenheimer-Snyder cloud>, its <areal radius> is $R=a\sin\chi_0$. Matching to a <Schwarzschild metric> gives $M=4\pi\rho R^3/3=a_{\max}\sin^3\chi_0/2$. Thus $\sin^2\chi_0=2M/R_0$ and $a_{\max}=\sqrt{R_0^3/(2M)}$. The boundary is a radial <timelike geodesic> with specific <Killing energy> $\sqrt{1-2M/R_0}=\cos\chi_0$. Its radial equation is $\dot R^2=2M/R-2M/R_0$. The induced metric and the angular <extrinsic curvature> agree on both sides, so no surface layer is required.
Back to article page
