\[ \boxed{\;\frac{2}{r}\,\partial_r\Phi \;+\; \frac{1}{r^2} \;-\; \frac{1}{A\,r^2} \;=\; 8\pi\,T_{rr}\;} \qquad\text{with}\quad A=e^{2\Phi}. \]
This is the \(rr\)-Einstein equation for the spherical, time-first metric. It tells you how the radial gradient of the time potential (left) is balanced by matter’s radial normal stress \(T_{rr}\) (right). Geometrically, the combination \(\frac{1}{r^2}-\frac{1}{A r^2}\) measures how the circumference–radius relation is distorted by \(A\); the term \(\tfrac{2}{r}\partial_r\Phi\) is the slope of the time field felt by the radial ruler. Together they must equal the stress loading the spherical shell.
A handier, equivalent ODE for \(A(r)\) (using \(\partial_r A=2A\,\partial_r\Phi\)) is
\[ \boxed{\; \partial_r A \;=\; 8\pi\,A\,r\,T_{rr} \;-\; \frac{A-1}{r}\;} \]
This says: in vacuum (\(T_{rr}=0\)), the geometric “relaxation” term \(-\frac{A-1}{r}\) drives \(A\) toward \(1+\frac{C}{r}\), which is exactly the Schwarzschild profile (with \(C=-2M\)). Positive radial pressure (\(T_{rr}>0\)) pushes \(A\) down faster with \(r\) (tightening the radial ruler), while radial tension does the opposite.
Helpful intuition:
| Symbol | Name | Meaning | Value / Units | Metaphor |
|---|---|---|---|---|
| \(T_{rr}\) | radial normal stress | matter’s radial pressure/tension in the \(r\)-direction | energy density (pressure) | “Thumb pushing on the spherical shell” |
| \(\partial_r\Phi\) | radial time-slope | how the clock-rate field changes with radius | 1/length | “Tilt of the time landscape along \(r\)” |
| \(A(r)\) | redshift factor | \(A=e^{2\Phi}\), sets \(g_{tt}=-A,\ g_{rr}=A^{-1}\) | dimensionless | “Light-cone tilt dial” |
| \(\partial_r A\) | radial change of \(A\) | first-order ODE controlled by stress and geometry | 1/length | “How fast the dial turns as you move out” |
| \(r\) | areal radius | spheres have area \(4\pi r^2\) | length | “Radius painted on the shell” |
| \(A-1\) term | geometric relaxation | tendency toward flat space when stress=0 | dimensionless | “Spring pulling rulers back to flatness” |