Equation 123 · Part 7 · The Bend an Elevator Cannot Fake

=

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

What this part means

The expressions on both sides represent the same quantity under the stated assumptions.

Its job in the formula

The equals sign connects the complete expression on the left with the complete expression on the right. Both sides must have compatible units.

Full expression=Article meaning

The passage around this formula

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 $M_\oplus$ =…

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Learn the underlying idea

An equals sign says that the expression on its left and the expression on its right have the same value under the stated definitions and assumptions.

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Sources cited in the surrounding passage

[15] Physical Geodesy ↗

These citations provide research context; check each source for the exact claim it supports.