When gravity is weak and nothing changes rapidly in time (linearized, quasi-static limit), the \(tt\) Einstein equation reduces to a Poisson equation for the time potential:
\[ \boxed{\,\nabla^{2}\Phi(\mathbf{x}) = 4\pi\,\rho(\mathbf{x})\;}\qquad (G=c=1). \]
Restoring constants and comparing to Newton: \(g_{tt}\!\approx\!-(1+2\Phi)\) with \(\Phi=\phi_{\!N}/c^{2}\) and \(\nabla^{2}\phi_{\!N}=4\pi G\rho\). Equivalently,
\[ \boxed{\,\nabla^{2}\Phi = \frac{4\pi G}{c^{2}}\,\rho\, }. \]
So mass–energy density curves time: where \(\rho>0\), the “time landscape” \(\Phi\) bends downward (clocks slow). For a point mass \(M\),
\[ \nabla^{2}\Phi = \frac{4\pi G M}{c^{2}}\,\delta^{3}(\mathbf{x}) \;\Rightarrow\; \Phi(\mathbf{x}) = -\,\frac{GM}{c^{2}r}, \]
matching the large-\(r\) limit of Schwarzschild (Step 12). In this static regime the flux law (Step 8) gives \(\partial_{t}\Phi=0\): no net radial energy flow, so time doesn’t evolve; instead, \(\rho\) shapes the spatial profile of \(\Phi\).
| Symbol | Name | Meaning | Value / Units | Metaphor |
|---|---|---|---|---|
| \(\Phi(\mathbf{x})\) | time potential | log-lapse; sets local clock rate | dimensionless; \(\Phi=\phi_N/c^2\) | “Altitude of time” |
| \(\rho(\mathbf{x})\) | mass–energy density | rest-energy per volume (nonrelativistic limit) | energy/volume | “Stuff per room” |
| \(\nabla^{2}\) | Laplacian | divergence of gradient; measures curvature of a scalar | 1/length\(^2\) | “How bowl-shaped the landscape is” |
| \(\phi_{N}\) | Newtonian potential | classical gravitational potential | \(\nabla^{2}\phi_N=4\pi G\rho\) | “Height of a familiar potential well” |
| \(G,\,c\) | constants | gravity coupling, light speed | \(G\) (SI), \(c\) (m/s) | “Strength of glue; speed limit” |
| b.c. at \(\infty\) | boundary condition | fix zero of \(\Phi\) far away | \(\Phi\!\to\!0\) as \(r\!\to\!\infty\) | “Choose sea level for the landscape” |