Outside a spherical source we just found \(A(r)=e^{2\Phi(r)}=1-\dfrac{r_s}{r}\) with \(r_s=\dfrac{2GM}{c^2}\). So
\[ \Phi(r)=\tfrac12\ln\!\Big(1-\frac{r_s}{r}\Big),\qquad g_{tt}=-(1-\tfrac{r_s}{r}),\qquad g_{rr}=(1-\tfrac{r_s}{r})^{-1}. \]
Immediate physical readouts:
Clock rate: \(d\tau=\sqrt{1-\tfrac{r_s}{r}}\;dt\) (clocks tick slower deeper in the well).
Newtonian limit: for \(r\gg r_s\),
\[ \Phi(r)=\tfrac12\ln(1-x)\approx -\tfrac12 x = -\,\frac{GM}{rc^2}, \]
i.e., \(\Phi\) reduces to the Newtonian potential divided by \(c^2\).
Gravitational redshift: \(1+z=\dfrac{1}{\sqrt{1-r_s/r}}\approx 1+\dfrac{GM}{rc^2}\).
Shapiro delay intuition: the stretched radial ruler \(g_{rr}=(1-r_s/r)^{-1}\) plus slowed clocks \(g_{tt}\) together lengthen light travel time past a mass.
These are the standard “sanity checks”: the exact Schwarzschild outside, Newtonian limit at large \(r\), and the textbook redshift/propagation effects. All read directly from \(\Phi\).
| Symbol | Name | Meaning | Value / Units | Metaphor |
|---|---|---|---|---|
| \(r_s\) | Schwarzschild radius | mass scale setting the horizon | \(r_s=2GM/c^2\) (length) | “Edge marker” of the well |
| \(A(r)\) | redshift factor | packs \(g_{tt}=-A,\ g_{rr}=A^{-1}\) | \(A=1-\tfrac{r_s}{r}\) | “Light-cone tilt dial” |
| \(\Phi(r)\) | time potential | log-lapse | \(\tfrac12\ln(1-\tfrac{r_s}{r})\) | “Altitude of time” shaping clocks |
| \(d\tau/dt\) | clock slowdown | proper vs coordinate time | \(\sqrt{1-\tfrac{r_s}{r}}\) | “Clock in thicker syrup” |
| \(z\) | grav. redshift | photon frequency shift to infinity | \(1+z=(1-\tfrac{r_s}{r})^{-1/2}\) | “Climbing out of a pit costs pitch” |