= Proper-time horizon crossing in homogeneous dust collapse
{title2=$\eta_H=2\arccos\sqrt{2M/R_0}$}
A boundary clock in <Oppenheimer-Snyder collapse> reads $\tau=a_{\max}(\eta+\sin\eta)/2$. Its <event horizon> crossing occurs at $R=2M$, giving the displayed angle, and the final <curvature singularity> occurs at $\eta=\pi$. The remaining <proper time> is $a_{\max}(\pi-\eta_H-\sin\eta_H)/2$. For $R_0\gg2M$ this approaches $4M/3$, but it is not exactly that value for finite release radius.
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