In a homogeneous FRW universe we set \(a(\tau)=e^{-\Phi(\tau)}\) (Steps 3–4). Photons are perfect “flying clocks,” so their frequency simply scales with the local clock-rate field. The observed redshift of light emitted at \(\tau_{\mathrm{em}}\) and received “today” \(\tau_0\) is
\[ \boxed{\,1+z \;=\; \frac{1}{a(\tau_{\mathrm{em}})} \;=\; e^{\Phi(\tau_{\mathrm{em}})-\Phi(\tau_0)}\, }. \]
Choosing the usual normalization \(a(\tau_0)=1 \Leftrightarrow \Phi(\tau_0)=0\) gives the compact FRW rule
\[ \boxed{\,1+z \;=\; \frac{1}{a_{\mathrm{em}}} \;=\; e^{\Phi_{\mathrm{em}}}\, }. \]
This is achromatic (independent of wavelength) and ties directly to expansion: $ z/(1+z) = -H = $.
Unifying picture: in a static gravitational field (no cosmic expansion), the redshift between emitter and observer at different radii follows the spatial difference of the time potential:
\[ 1+z = \frac{\nu_{\mathrm{em}}}{\nu_{\mathrm{obs}}} = \frac{\sqrt{-g_{tt}(\mathrm{obs})}}{\sqrt{-g_{tt}(\mathrm{em})}} = e^{\Phi_{\mathrm{obs}}-\Phi_{\mathrm{em}}}. \]
Same idea, different specialization: FRW uses \(\Phi\) changing in time; static fields use \(\Phi\) changing in space.
| Symbol | Name | Meaning | Value / Units | Metaphor |
|---|---|---|---|---|
| \(z\) | redshift | fractional stretch of wavelength | \(1+z=\lambda_{\rm obs}/\lambda_{\rm em}\) | “How much the note drops in pitch” |
| \(a(\tau)\) | scale factor | spatial ruler on slices | \(a=e^{-\Phi}\) | “Rubber-sheet stretch” |
| \(\Phi(\tau)\) | time potential | log-lapse controlling clock rate | dimensionless | “Altitude of time” |
| \(\Phi_{\rm em},\,\Phi_0\) | emit/now values | time potential at emission/now | \(\Phi_0=0\) by convention | “Height then vs now” |
| \(H\) | Hubble rate | expansion rate | \(H=\dot a/a=-\dot\Phi\) | “Universe’s breathing rate” |
| \(\nu,\,\lambda\) | frequency, wavelength | photon measures | \(1+z=\nu_{\rm em}/\nu_{\rm obs}\) | “Tick rate of a flying clock” |
| \(g_{tt}\) | time metric entry | sets local clock speed | \(g_{tt}=-e^{2\Phi}\) | “Weight on the clock” |