 Domain of the ingoing Painleve-Gullstrand chart

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 $\tau =v+F(r)-r_{\ast }(r)$, with $v=4M\log V$ and $F^{\prime }=\sqrt{2M/r}/f$; the difference $F-r_{\ast }$ is finite at $r=2M$, whereas $V=0$ requires infinite coordinate time.