To keep later equations compact, define a single factor
\[ A(t,r) \equiv e^{2\Phi(t,r)} = N^2 . \]
Then the spherical time-first metric from Step 5 reads
\[ ds^2 = -A\,dt^2 + A^{-1}dr^2 + r^2 d\Omega^2. \]
This bundles “how fast time runs” and the reciprocal radial stretching into one function. It also makes derivatives tidy:
\[ \partial_t A = 2A\,\partial_t\Phi,\qquad \partial_r A = 2A\,\partial_r\Phi, \]
which we’ll use when writing Einstein’s equations (the flux law and radial constraints) in a cleaner form. Intuitively, \(A\) is the light-cone tilt: larger \(A\) means time is “heavier” (clocks faster) and the radial ruler correspondingly lighter (since \(g_{rr}=A^{-1}\)).
| Symbol | Name | Meaning | Value / Units | Metaphor |
|---|---|---|---|---|
| \(A(t,r)\) | redshift factor | packs clock rate and radial stretch into one | \(A=e^{2\Phi}=N^2>0\) (dimensionless) | “Light-cone tilt knob” |
| \(\Phi(t,r)\) | time potential | log-lapse controlling \(A\) | \(A=e^{2\Phi}\) (dimensionless) | “Altitude of time” that sets \(A\) |
| \(N(t,r)\) | lapse | \(N=\sqrt{A}\) converts \(dt\!\to\! d\tau\) | \(d\tau=N\,dt\) | “Time gear ratio” |
| \(g_{tt}\) | time metric comp. | coefficient of \(dt^2\) | \(g_{tt}=-A\) | “Weight of time” |
| \(g_{rr}\) | radial metric comp. | coefficient of \(dr^2\) | \(g_{rr}=A^{-1}\) | “Elasticity of the radial ruler” |
| \(\partial_\mu A\) | slope of \(A\) | encodes time/space gradients via \(\Phi\) | \(\partial_\mu A=2A\,\partial_\mu\Phi\) | “How steep the tilt dial is turning” |