Any physical system is a clock. Its quantum phase grows in proportion to the proper time it experiences. Since proper time is scaled by the lapse \(N=e^{\Phi}\), the phase you accumulate along a path picks up a factor \(e^{\Phi}\). Writing the rest-energy frequency as \(\omega \equiv E/\hbar\), the total phase is
\[ \theta=\frac{1}{\hbar}\!\int E\,d\tau =\omega\!\int e^{\Phi(t,\mathbf{x}(t))}\,dt. \]
For small fluctuations \(\Phi=\bar\Phi+\delta\Phi\) and a fixed path, the first-order phase change is simply
\[ \delta\theta \simeq \omega\!\int \delta\Phi(t)\,dt, \]
with \(\delta\Phi\) the constraint-projected \(h_{00}\) of linearized GR (i.e., not a new propagating scalar).
Why it matters: That last line is the bridge from geometry to experiments: any tiny fluctuation \(\delta\Phi\) directly writes phase onto any system at a strength set by its internal frequency \(\omega\). This is why optical transitions (huge \(\omega\)) are exquisitely sensitive phase probes, and why interferometer visibilities will give clean bounds on \(\langle\delta\Phi\,\delta\Phi\rangle\) in the next step.
| Symbol | Name | Meaning (units) | Typical value/example | Metaphor |
|---|---|---|---|---|
| \(\theta\) | Quantum phase | Action/\(\hbar\) accumulated along a path (rad) | Grows linearly with proper time | “Hand on a clock dial” |
| \(\omega\) | Clock frequency | \(E/\hbar\) or \(\Delta E/\hbar\) (rad·s\(^{-1}\)) | Optical line: \(\omega/2\pi\sim10^{15}\) Hz | “Metronome speed” |
| \(\Phi\) | Time potential | Sets lapse \(N=e^{\Phi}\) (dimensionless) | \(\Phi=0\) by convention today | “Altitude of time” |
| \(\delta\Phi\) | Lapse fluctuation | Constraint-projected \(h_{00}\) (dimensionless) | Bounded via visibility later | “Ripples in the time-field” |
| \(N=e^{\Phi}\) | Lapse | Proper-time scale factor \(d\tau=N\,dt\) | \(N\approx1\) in weak fields | “Time gear ratio” |
| \(d\tau\) | Proper time | Physical time along the worldline (s) | Integrates \(e^{\Phi}dt\) | “Time actually felt” |
Next up: Step 4 — From phase to visibility: the general visibility kernel \(V[f]=\exp\!\Big[-\tfrac12\omega^2\!\iint f\,C_\Phi\,f\Big]\) and how it turns \(\delta\Phi\) noise into measurable fringe loss.