If energy flows outward through a sphere of radius \(r\), think light, neutrinos, any luminosity \(L(t)\), then the local “clock-rate potential” \(\Phi\) at that radius must decrease at a rate proportional to that flux, with a strict \(1/r\) scaling. That decrease in \(\Phi\) shows up directly as a time-varying gravitational redshift between a distant observer and a clock at \(r\). Concretely,
\[ \partial_t \Phi(t,r)\;\simeq\;-\frac{G}{c^4}\,\frac{L(t)}{r}\quad\text{(outgoing; flip sign for ingoing flux).} \]
This is the operational law the paper templates against data.
Why the sign is locked: Outgoing energy \(L>0\) increases the exterior mass parameter in the Vaidya picture, which reduces \(A=e^{2\Phi}=1-\tfrac{2Gm}{rc^2}\). So \(\Phi\) goes down and \(\partial_t\Phi<0\): clocks redshift (run slower) as the burst passes. Ingoing flux flips the sign (blueshift). In Eddington–Finkelstein (Vaidya) coordinates this appears as \(L=-c^2\,dm/dv\) and \(\partial_v\Phi=-(G/c^4)L/r\), matching the law above.
Directly observable form: Redshift here is written as \(y\equiv \ln(\nu_\infty/\nu_r)\). The drift of that redshift equals \(\partial_t\Phi\):
\[ \frac{d}{dt}\ln\!\left(\frac{\nu_\infty}{\nu_r}\right)=\partial_t\Phi(t,r). \]
So the measured time-derivative of a clock-frequency ratio is the flux-driven \(\Phi\) drift.
Units/scale to keep in mind: The tiny coupling \(G/c^4 = 8.262\times10^{-45}\ \text{s}^2/(\text{kg}\,\text{m})\) sets the scale; that’s why everyday lamps are hopelessly small, while rare astrophysical bursts (e.g., a Galactic supernova) can scrape into the range of cutting-edge clock stability.
| Symbol | Name | Meaning (units) | Typical value/example | Metaphor |
|---|---|---|---|---|
| \(\Phi\) | Time potential | Controls proper-time: \(d\tau=e^{\Phi}dt\) (dimensionless) | Set to 0 "today" by convention | "Altitude" of time |
| \(\partial_t\Phi\) | Redshift drift | Time-rate of \(\Phi\) at fixed \(r\) \([\text{s}^{-1}]\) | Galactic SN: \(\partial_t\Phi \sim 8.3\times10^{-20}\ \text{s}^{-1}\) | "Slope" of the time-hill changing in time |
| \(L(t)\) | Luminosity | Power crossing sphere at \(r\) \([\text{W}]\) | Lab lamp (10 W \(\Rightarrow \partial_t\Phi \sim 8\times10^{-42}\ \text{s}^{-1}\)) (software-injected test) | "Brightness" pushing on time |
| \(r\) | Areal radius | Defined by \(4\pi r^2\) area \([\text{m}]\) | \(1/r\) weakening with distance | "Distance lever" (farther = weaker effect) |
| \(y=\ln(\nu_\infty/\nu_r)\) | Fractional redshift | Log frequency ratio (dimensionless) | \(\dot y=\partial_t\Phi\) is what we measure | "Pitch change of a note" |
| \(G/c^4\) | Coupling scale | \(8.262\times10^{-45}\ \text{s}^2/(\text{kg}\,\text{m})\) | Sets overall tiny amplitude | "Ultra-small gear ratio" |