Plain English: the paper uses a lapse-first view where \(N=e^{\Phi}\) is the time gear, but it does not insist on a diagonal metric forever. Rotation and many time-dependent flows are cleaner if you allow a nonzero shift \(\omega\). The physics (including the flux→drift sign and the \(1/r\) falloff) is the same in any of the standard gauges—diagonal, Painlevé–Gullstrand (PG), or Eddington–Finkelstein (EF/Vaidya).
Diagonal spherical gauge (nice for static intuition). The paper introduces the diagonal form
\[ ds^2=-e^{2\Phi}\,dt^2+e^{-2\Phi}\,dr^2+r^2 d\Omega^2,\qquad N=e^{\Phi}. \]
It immediately switches to EF/Vaidya to state the observable far-field law, so you don’t confuse the coordinates with the physics.
EF/Vaidya (best for outward/ingoing flux). In EF coordinates the flux law reads
\[ \partial_v \Phi \;=\;-\frac{G}{c^4}\,\frac{L}{r}, \]
identical in content to the diagonal statement; here \(L=-c^2\,dm/dv\). This is the paper’s operational template and the origin of the sign-lock: outgoing \(L>0\Rightarrow\partial\Phi<0\) (redshift).
PG (flat spatial slices; shift carries the “river”). A simple time redefinition \(T=t+f(r)\) turns the diagonal metric into PG with unit lapse and a radial shift (“flow speed”) \(v(r)=\pm\sqrt{1-e^{2\Phi(r)}}\). It’s the same single function \(A=e^{2\Phi}=1-v^2\); you’ve only moved it from the lapse into the shift component.
Rotation & waves live in the shift (not in \(\Phi\)). The paper flags this explicitly: “Rotation/frame dragging lives in the shift \(\omega\); a spatially uniform \(\partial_t\Phi\) cannot generate a transverse \(B_g\)”. So “shaking \(\Phi\)” won’t fake Lense–Thirring; you need mass currents (nonzero \(\omega\)).
Why the sign-lock is gauge-independent. The EF/diagonal dictionary equates the same invariant mass-change \(dm\): \(T_{tr}\) in the diagonal chart maps to \(T_{vv}\) in EF, and both produce the same \(\partial\Phi\) statement once you account for the metric factor. This is why the minus sign and \(1/r\) stick no matter which chart you use.
Tiny caution the paper notes between the lines: In vacuum with the zero-shift diagonal choice, Birkhoff enforces \(\partial_t\Phi=0\). That’s a statement about that foliation, not a physical ban on dynamics—switch to PG or EF when modeling inflow/outflow or rotation.
| Symbol | Name | Meaning (units) | Typical value/example | Metaphor |
|---|---|---|---|---|
| \(N=e^{\Phi}\) | Lapse | Proper-time scale factor (\(d\tau=N\,dt\)) | \(N\approx 1\) in weak fields | “Time gear ratio” |
| \(\omega\) | Shift | Gravitomagnetic potential carrying rotation/flow | Nonzero for frame dragging | “River current of space” |
| EF/Vaidya | Ingoing/outgoing null gauge | Best for flux \(L\) and mass change \(dm/dv\) | \(\partial_v\Phi=-(G/c^4)L/r\) | “One-way streets (lightlike)” |
| PG | Painlevé–Gullstrand | Flat slices; unit lapse; radial shift \(v\) | \(A=1-v^2=e^{2\Phi}\) | “River model” |
| \(L(t)\) | Luminosity | Power crossing sphere at \(r\) (W) | Outgoing \(L>0\Rightarrow\partial\Phi<0\) | “Brightness pushing on time” |
Next up: Step 15 — The data-analysis bones: whitening, matched filters, and the exact things you compute for each lever.