Equation 123 · Part 6 · The Bend an Elevator Cannot Fake

Symbol h^4

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

h^4

What this part means

$h^4$ is one of the signed contributions combined to compute the quantity on the left.

Its job in the formula

$h^4$ is one of the signed contributions combined to compute the quantity on the left.

Full expressionSymbol h^4Article meaning

The passage around this formula

…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} .…

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Learn the underlying idea

An exponent tells how a base is used in multiplication. In x³, x is the base and 3 is the exponent: x³ = x × x × x.

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Sources cited in the surrounding passage

[15] Physical Geodesy ↗

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