= Schwarzschild horizon-to-singularity proper time for fall from infinity
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{title2=$\Delta\tau=4M/3$}
A radial observer released from rest at infinity has $dr/d\tau=-\sqrt{2M/r}$. Integration from $r=2M$ to $r=0$ gives $\Delta\tau=\int_0^{2M}\sqrt{r/(2M)}dr=4M/3$ in geometric units. In ordinary units it is $4GM/(3c^3)$. The endpoint is the curvature singularity, not a regular spacetime centre; this value belongs to that specified energy, not every possible infalling worldline.
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