$$y_2 - y_1 = \phi_{,zz}\,h^2 + O(h^4)$$

Equation 123 · Evolutionary Physics

The Bend an Elevator Cannot Fake

This equation states an equality: the expressions on both sides have the same value under the article’s assumptions.

An exact analytic evaluation, not a simulation, fixes the scale this actually requires. Take three clocks stacked vertically at heights -h , 0 , and +h near Earth’s surface, giving two edges $y_1$ = $\phi(0)$-$\phi(-h)$ and $y_2$ = $\phi(h)$-$\phi(0)$ . Their difference is an exact finite-difference identity up to a Taylor remainder of order $h^4$ : $y_2 - y_1 = \phi_{,zz}\,h^2 + O(h^4)$. For Earth modelled as a point mass, $\Phi(r)$ = -GM/r gives a radial second derivative of magnitude 2GM/$R^3$ and, since $\Phi$ is harmonic in vacuum, transverse second derivatives of magnitude GM/$R^3$ with the opposite sign — a trace-free tensor in the exact ratio 2:{-1}:{-1} . With the standard gravitational parameter G$M_\oplus$ =…

Meanings in this article

• $y_1$: $\sqrt2$\,$\sigma_y$.

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