\[ G_{tr} \;=\; -\,\frac{2}{r}\,\partial_t \Phi . \]
Take the spherical time-first metric from Steps 5–6,
\[ ds^2=-A\,dt^2+A^{-1}dr^2+r^2 d\Omega^2,\quad A=e^{2\Phi(t,r)}, \]
and compute the single off-diagonal Einstein component \(G_{tr}\). Because we chose areal radius \(r\) and reciprocal time–radial weights (\(g_{tt}g_{rr}=-1\)), all spatial-derivative clutter cancels. What’s left is purely kinematical: the only way to get a nonzero \(G_{tr}\) is if the time potential \(\Phi\) itself is changing in time. The size of that effect is fixed by geometry. \(-2\,\partial_t\Phi/r\).
Why this matters: this clean form sets up the next step. Einstein’s equation says \(G_{tr}=8\pi T_{tr}\). Plugging the identity above yields the flux law \(\partial_t\Phi = -4\pi r\,T_{tr}\): only radial energy flux can make \(\Phi\) evolve (and if no flux crosses a sphere, \(\partial_t\Phi=0\) there).
Quick checks:
| Symbol | Name | Meaning | Value / Units | Metaphor |
|---|---|---|---|---|
| \(G_{tr}\) | mixed Einstein component | geometry’s \(t\)-\(r\) coupling | curvature (1/length\(^2\)) | “Shear between time and radius” |
| \(\partial_t\Phi\) | time-slope of time potential | how the clock-rate field changes in time | \(d\Phi/dt\) (1/time) | “How quickly the time landscape tilts” |
| \(r\) | areal radius | spheres have area \(4\pi r^2\) | length | “Radius painted on spheres” |
| \(A\) | redshift factor | \(A=e^{2\Phi}=N^2\) | dimensionless | “Light-cone tilt knob” |
| (preview) \(T_{tr}\) | radial energy flux density | matter flow that sources \(G_{tr}\) next step | energy/(area·time) | “Energy current through the sphere” |