We now choose coordinates adapted to spherical symmetry and set the shift to zero (no \(dt\,dr\) cross term). The metric is
\[ ds^2 = -\,e^{2\Phi(t,r)}\,dt^2 \;+\; e^{-2\Phi(t,r)}\,dr^2 \;+\; r^2 d\Omega^2, \]
with \(d\Omega^2 = d\theta^2+\sin^2\!\theta\,d\varphi^2\). Two key ideas are built in:
Setting the shift to zero (\(N^r=0\)) means our time coordinate runs orthogonal to the \(t=\text{const}\) slices (no radial drift of the slicing). That’s consistent with spherical problems without rotation and keeps the coming equations as simple as possible. If we later need infalling coordinates or rotation, we’ll reintroduce a nonzero shift.
| Symbol | Name | Meaning | Value / Units | Metaphor |
|---|---|---|---|---|
| \(ds^2\) | line element | squared spacetime interval between nearby events | length\(^2\) | “Infinitesimal spacetime ruler” |
| \(\Phi(t,r)\) | time potential | controls both clock rate and radial ruler via \(e^{\pm\Phi}\) | dimensionless | “Altitude of time” shaping space |
| \(g_{tt}\) | time–time metric comp. | coefficient of \(dt^2\) | \(g_{tt}=-e^{2\Phi}\) | “How fast local time runs” |
| \(g_{rr}\) | radial metric comp. | coefficient of \(dr^2\) | \(g_{rr}=e^{-2\Phi}\) | “How stretched the radial ruler is” |
| \(r\) | areal radius | spheres have area \(4\pi r^2\) | length | “Radius painted on spheres” |
| \(d\Omega^2\) | unit-sphere metric | angular piece \(d\theta^2+\sin^2\theta\,d\varphi^2\) | dimensionless | “Standard sphere skin” |
| \(N\) | lapse | \(d\tau/N = dt\) conversion | \(N=e^{\Phi}>0\) | “Clock gear ratio” |
| \(N^r\) (shift) | radial shift | mixes time and radius if nonzero | here \(N^r=0\) | “Conveyor belt of the grid” (turned off) |