= Domain of the ingoing Painleve-Gullstrand chart
{title2=$V>0,\quad UV<1$}
Choose <Kruskal–Szekeres coordinates> with the right exterior at $U<0,V>0$ and $UV=(1-r/(2M))e^{r/(2M)}$. The ingoing <Painlevé–Gullstrand coordinates> cover that exterior and the future black-hole region $U>0,V>0$, together with their common future horizon. They exclude the other exterior, the white hole, the bifurcation sphere and the singularity. Indeed $\tau=v+F(r)-r_*(r)$, with $v=4M\log V$ and $F'=\sqrt{2M/r}/f$; the difference $F-r_*$ is finite at $r=2M$, whereas $V=0$ requires infinite coordinate time.
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