Our diagonal chart had zero shift and curved radial rulers: \(ds^2=-A\,dt^2+A^{-1}dr^2+r^2 d\Omega^2\) with \(A=e^{2\Phi}\). Sometimes it’s nicer to make each constant-time slice Euclidean (flat rulers) and let the “flow” of space show up as a shift. Do a time change
\[ T \;=\; t + f(r),\qquad dT = dt + f'(r)\,dr, \]
and choose \(f'(r)\) so that the \(T=\text{const}\) spatial metric is \(dr^2+r^2 d\Omega^2\). This fixes
\[ \boxed{\,f'(r) \;=\; \pm \frac{\sqrt{\,1-A(r)\,}}{A(r)}\, }. \]
With this choice the metric becomes the (ingoing/outgoing) Painlevé–Gullstrand (PG) form:
\[ \boxed{\,ds^2 \;=\; -A\,dT^2 \;\pm\;2\sqrt{\,1-A\,}\,dT\,dr \;+\; dr^2 \;+\; r^2 d\Omega^2\, }. \]
Read as a \(3{+}1\) decomposition, PG slices have flat spatial metric \(\gamma_{rr}=1\), unit lapse \(N=1\), and a radial shift
\[ \boxed{\,N^r \;=\; \pm \sqrt{\,1-A(r)\,}\, }. \]
Interpretation: clocks are synchronized (\(N=1\)), rulers are flat, and gravity appears as a radial flow of space with speed \(|N^r|=\sqrt{1-A}\). For Schwarzschild \(A=1-2M/r\), this is \(|N^r|=\sqrt{2M/r}\) the Newtonian escape speed. At the horizon (\(A=0\)) the flow speed reaches light speed, which is why these coordinates remain regular there: the “river of space” simply carries everything inward across the horizon.
| Symbol | Name | Meaning | Value / Units | Metaphor | ||
|---|---|---|---|---|---|---|
| \(T\) | PG time | time coordinate with flat slices | \(T=t+f(r)\) | “Everyone’s wristwatches synced” | ||
| \(f'(r)\) | slice-tilt profile | how much we tilt time vs radius | \(\pm \sqrt{1-A}/A\) | “Dial that tips the slicing” | ||
| \(A(r)\) | redshift factor | from time-first potential \(A=e^{2\Phi}\) | dimensionless | “Light-cone tilt knob” | ||
| \(N\) | lapse (PG) | clock conversion on slices | \(N=1\) | “All clocks tick in unison” | ||
| \(N^r\) | radial shift | grid’s radial drift speed | \(\pm \sqrt{1-A}\) (speed) | “River of space flowing radially” | ||
| \(\gamma_{rr}\) | radial spatial metric | ruler on \(T=\)const slices | \(\gamma_{rr}=1\) | “Straight, uncurved ruler” | ||
| \(v(r)\) | flow speed (alias) | magnitude of the shift | (v= | N^r | =) | “Current speed of the river” |
| \(r\) | areal radius | sphere area \(4\pi r^2\) | length | “Radius painted on the shells” |