\[ \partial_t\Phi(t,r)\;=\;-\,4\pi\,r\,T_{tr}(t,r). \]
Einstein’s equation is \(G_{tr}=8\pi\,T_{tr}\). From Step 7 we already found the purely geometric identity \(G_{tr}=-\,\tfrac{2}{r}\,\partial_t\Phi\). Equating them gives a one-line evolution law for the time potential:
Sign intuition (conventions vary): if we take \(T_{tr}>0\) to mean outward energy flow, the minus sign says outward flow makes \(\Phi\) decrease in time at that radius (clocks locally slow relative to coordinate time); inward flow does the opposite. What matters operationally is the lockstep: wherever net flux crosses, \(\Phi\) must evolve; where flux vanishes (exactly or by boundary condition), \(\Phi\) freezes.
| Symbol | Name | Meaning | Value / Units | Metaphor |
|---|---|---|---|---|
| \(\partial_t\Phi\) | time-rate of time potential | how clock-rate field changes in coordinate time | 1/time | “How fast the time landscape is sinking/rising” |
| \(T_{tr}\) (or \(T^{t}{}_{r}\)) | radial energy-flux density | energy per area per time crossing a sphere (sign set by outward/inward) | energy/(area·time) | “Energy current through the spherical fence” |
| \(G_{tr}\) | mixed Einstein component | geometry’s response to flux | 1/length\(^2\) | “Shear hinge between time and radius” |
| \(r\) | areal radius | labels spheres of area \(4\pi r^2\) | length | “Radius painted on the shell” |
| \(4\pi\) | solid-angle factor | integrates flux over the sphere | dimensionless | “Whole-sky coverage” |
| \(\Phi\) | time potential | sets lapse \(N=e^{\Phi}\) and radial metric factor | dimensionless | “Altitude of time” |
| \(N\) | lapse | \(d\tau=N\,dt\) | dimensionless | “Clock gear ratio” |