Set vacuum outside the matter—so \(T_{\mu\nu}=0\).
\[ \partial_r A = -\,\frac{A-1}{r}. \]
Integrate: \(A(r)=1+\dfrac{C}{r}\). Asymptotic flatness demands \(A\!\to\!1\) at large \(r\), and matching the Newtonian limit fixes \(C=-2M\) (or \(C=-r_s\) with \(r_s\!=\!2GM/c^2\)). Therefore
\[ \boxed{\,A(r)=e^{2\Phi(r)} \;=\; 1-\frac{2M}{r} \;=\; 1-\frac{r_s}{r}\, } . \]
This is exactly the Schwarzschild profile. In words: no radial energy flux ⇒ \(\Phi\) cannot evolve in time, and the remaining radial equations force the unique static solution. That’s Birkhoff’s theorem in “time-first” clothing.
Quick checks:
| Symbol | Name | Meaning | Value / Units | Metaphor |
|---|---|---|---|---|
| \(T_{\mu\nu}=0\) | vacuum | no matter/flux outside the source | identically zero | “Silent universe outside the star” |
| \(\partial_t\Phi\) | time evolution of \(\Phi\) | vanishes in vacuum by flux law | \(0\) | “Time landscape stops moving” |
| \(A(r)\) | redshift factor | sets \(g_{tt}=-A,\ g_{rr}=A^{-1}\) | \(A=1-\dfrac{2M}{r}\) | “Light-cone tilt dial (fixed profile)” |
| \(M\) | mass parameter | total enclosed (ADM) mass seen at infinity | \(M= r_s/2 = GM/c^2\) (choose units) | “Gravitational charge” |
| \(r_s\) | Schwarzschild radius | horizon location in these coords | \(r_s=\dfrac{2GM}{c^2}\) | “Edge of no return (in this chart)” |
| \(\Phi(r)\) | time potential | log-lapse; \(e^{2\Phi}=A\) | \(\Phi=\tfrac12\ln(1-r_s/r)\) | “Altitude of time that carves the well” |
| \(g_{tt}, g_{rr}\) | metric entries | time & radial rulers | \(-A,\ A^{-1}\) | “Clock weight” & “ruler stretch” |