Core point: the paper’s observables fall straight out of lapse-first general relativity and do not add a new propagating field.
Lapse-first bookkeeping. We choose the lapse \(N=e^{\Phi}\) as the “time gear,” allow nonzero shift \(\omega\) for rotation, and let spatial geometry follow from the ADM constraints. This makes many observables “clock-native.”
No extra DOF. In ADM, \(N=e^{\Phi}\) and \(N^i=\omega^i\) are Lagrange multipliers imposing the Hamiltonian/momentum constraints; only the two TT tensor modes propagate. The correlator \(C_\Phi\) that appears in visibility is just the constraint-projected (gauge-fixed) imprint of linearized metric/stress fluctuations on an observer’s clock—not a new scalar wave.
Worldline phase → visibility. Any system’s phase is \(\theta=\omega\!\int e^{\Phi}dt\). Small lapse fluctuations give \(\delta\theta\simeq \omega\!\int\delta\Phi\,dt\), and for two paths the Gaussian average yields the visibility kernel used for bounds.
Flux law (and why the sign is locked). From EF/Vaidya with \(m(v)\) one gets the far-field law
\[ \partial_t\Phi(t,r)\simeq-\frac{G}{c^4}\frac{L(t)}{r}, \]
with outgoing luminosity \(L>0\) implying \(\partial_t\Phi<0\) (redshift); ingoing flux flips the sign. This is the origin of the paper’s sign-locked, \(1/r\) template.
Scope reminder. The phrase “quantum temporal geometry” here means semiclassical temporal phase fluctuations of the lapse, constrained within GR; no beyond-GR mode is introduced.
| Symbol | Name | Meaning (units) | Typical value/example | Metaphor |
|---|---|---|---|---|
| \(N=e^{\Phi}\) | Lapse | Proper-time scale factor (\(d\tau=N\,dt\)) | \(N\approx1\) in weak fields | “Time gear ratio” |
| \(\Phi\) | Time potential | \(\ln N\); controls clock rate (–) | Set to 0 by convention today | “Altitude of time” |
| \(C_{\Phi}\) | Lapse correlator | \(\langle\delta\Phi(t)\delta\Phi(t')\rangle\) (–) | Constraint-projected GR fluctuations | “How time ripples echo” |
| TT modes | Propagating DOF | Two transverse–traceless tensors | Only waves that travel in vacuum | “The two violin strings of gravity” |
| \(L(t)\) | Luminosity | Power crossing sphere at \(r\) (W) | Drives \(\partial_t\Phi\propto-\,L/r\) | “Brightness pushing on time” |
Ready for Step 14 — Gauge notes (shift/rotation, diagonal vs EF/PG, and why “sign-lock” survives gauge moves).