Painlevé–Gullstrand coordinates

Painlevé–Gullstrand coordinates write the Schwarzschild metric asThe constant-$t$ spatial slices are flat, and the coordinates remain regular at the Schwarzschild radius.

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.

The inward radial geodesic congruence with unit Killing energy has $U=f^{-1}{\partial }_t-\sqrt{2M/r}{\partial }_r$, where $f=1-2M/r$. Its metric dual is $U^{♭}=-d\tau$, so $\tau$ is a synchronized proper time for the congruence. Locally $\tau =t+2\sqrt{2Mr}+2M\log |(\sqrt{r}-\sqrt{2M})/(\sqrt{r}+\sqrt{2M})|$. This yields Painlevé–Gullstrand coordinates with flat spatial slices and a regular future horizon.

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.

An orthonormal coframe is