= Proper-time bound inside a Schwarzschild black hole
{title2=$\Delta s\leq2M\arcsin\sqrt{r_0/(2M)}-\sqrt{r_0(2M-r_0)}<\pi M$}
Inside the future black-hole region, the radius decreases along every future-directed timelike path. Its unit tangent obeys $|dr/ds|\geq\sqrt{2M/r-1}$ from the metric normalization, independently of radial acceleration or angular motion. Integrating from $r_0<2M$ to zero gives the displayed finite upper bound. The statement concerns future continuation within the black-hole region; reversing time in the white-hole region reverses the radial direction.
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