In a perfectly homogeneous and isotropic universe, the only background “gravity knob” is how fast time runs everywhere on a slice. We already encoded that with \(\Phi(\tau)\). To make the spatial geometry automatically track time, we define the scale factor as the inverse of the lapse’s log:
\[ a(\tau) \equiv e^{-\Phi(\tau)} . \]
Then all physical distances on a constant-\(\tau\) slice scale by \(a\). This choice is a convention (you could absorb constants into \(a\) or \(\Phi\)), but it makes bookkeeping clean: as \(\Phi\) increases (clocks run faster), \(a\) shrinks by the reciprocal factor, so the spatial “grid spacing” is tied directly to the state of time. Setting today’s normalization \(a(\tau_0)=1\) is equivalent to choosing \(\Phi(\tau_0)=0\). With comoving coordinates \(r\), small physical separations are \(\ell_{\text{phys}}(\tau)=a(\tau)\,r\).
| Symbol | Name | Meaning | Value / Units | Metaphor |
|---|---|---|---|---|
| \(a(\tau)\) | scale factor | global size of spatial slices | \(a=e^{-\Phi}\) (dimensionless) | “Rubber-sheet stretch factor” |
| \(\Phi(\tau)\) | time potential (homog.) | single function controlling clock rate | dimensionless; offset is a convention | “Altitude of time” dial |
| \(r\) | comoving coordinate | label that doesn’t expand/contract | dimensionless chart coordinate | “Grid paint on the rubber sheet” |
| \(\ell_{\text{phys}}(\tau)\) | physical distance | measured separation on a slice | \(\ell_{\text{phys}}=a\,r\) (length) | “Tape-measure distance between grid lines” |
| normalization | today’s choice | fixes the overall scale | \(a(\tau_0)=1 \Leftrightarrow \Phi(\tau_0)=0\) | “Set the ruler to read 1 today” |