$$\phi_{,ij} \approx \Phi_{,ij}/c^2$$

Equation 113 · Evolutionary Physics

The Bend an Elevator Cannot Fake

This equation gives an approximation: it relates the quantities while allowing an approximation.

Two known limits anchor it. First, the elevator: $\mathcal T_{ij}$ $\equiv$ 0 for any uniformly accelerated frame, proven directly above — the “uniform potential difference is not curvature” claim, derived rather than asserted. Second, weak-field gravity: expanding $\phi$ $\approx$ $\Phi$/$c^2$ for small $\Phi$/$c^2$ , the quadratic term $\phi_{,i}$$\phi_{,j}$ is second order in the already-small ratio $\Phi$/$c^2$ and negligible next to the first-order term $\phi_{,ij}$ $\approx$ $\Phi_{,ij}$/$c^2$ , so $\mathcal T_{ij}$ $\to$ $\Phi_{,ij}$ , exactly the ordinary Newtonian tidal tensor of tower and satellite geodesy [ 15 , 14 ] . The first derivative alone, meanwhile, recovers the textbook gravitational redshift these networks…

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