To model a universe that looks the same everywhere and in every direction (FRW symmetry), we let the time potential depend on time only: \(\Phi=\Phi(t)\). With zero shift, the most general metric respecting these symmetries is
\[ \boxed{\,ds^2=-e^{2\Phi(t)}dt^2 \;+\; e^{-2\Phi(t)} \Big[\frac{dr^2}{1-k r^2}+r^2 d\Omega^2\Big]\, }. \]
This encodes “space follows time”: once \(\Phi(t)\) is chosen, spatial rulers scale by \(a(t)\equiv e^{-\Phi(t)}\) everywhere on each slice. The single constant \(k\in\{+1,0,-1\}\) selects closed/flat/open spatial curvature. Switching to proper (cosmic) time \(\tau\) via \(d\tau=e^{\Phi}dt\) makes the line element the usual FRW form,
\[ ds^2=-d\tau^2+a^2(\tau)\!\Big[\frac{dr^2}{1-k r^2}+r^2 d\Omega^2\Big],\quad a(\tau)=e^{-\Phi(\tau)}. \]
So in time-first language, all FRW dynamics reduce to how \(\Phi(t)\) (equivalently \(a\)) evolves; spatial geometry is then fixed uniformly across the slice.
| Symbol | Name | Meaning | Value / Units | Metaphor |
|---|---|---|---|---|
| \(\Phi(t)\) | time potential (homog.) | global clock-rate field on each slice | dimensionless | “Master tempo of the cosmos” |
| \(N(t)\) | lapse | converts \(dt\!\to\! d\tau\) | \(N=e^{\Phi(t)}\) | “Gear that sets cosmic beat” |
| \(a(t)\) | scale factor | spatial ruler on slices | \(a=e^{-\Phi}\) (dimensionless) | “Rubber-sheet stretch” |
| \(k\) | curvature index | spatial curvature sign | \(+1,0,-1\) | “Closed / flat / open setting” |
| \(r\) | comoving radius | coordinate fixed to the flow | dimensionless | “Painted grid on the sheet” |
| \(d\Omega^2\) | unit-sphere metric | \(d\theta^2+\sin^2\theta\,d\varphi^2\) | dimensionless | “Skin of a unit sphere” |
| \(\tau\) | cosmic proper time | time measured by comoving clocks | \(d\tau=e^{\Phi}dt\) | “Everyone’s wristwatch time” |
| \(H\) | Hubble rate | expansion rate | \(H=\dot a/a=-\dot\Phi\) | “Breathing rate of space” |