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Equation 140 · The Bend an Elevator Cannot Fake

What does this equation mean?

h=2 σyϕ,zz=1.414×10−183.43×10−23 m−2≈203 m.h = \sqrt{\frac{\sqrt2\,\sigma_y}{\phi_{,zz}}} = \sqrt{\frac{1.414\times10^{-18}}{3.43\times10^{-23}\,\mathrm{m^{-2}}}} \approx 203\,\mathrm m.

Read the formula alongside the article passage below. Each part has a deeper page with its role in the equation, the supporting passage and nearby citations.

Start withsqrt2sigma_y
Divide byphi_,zz
This relates toh
How to read the two sides of this formula. Follow the article passage for the meaning of each quantity.

This equation gives an approximation: it relates the quantities while allowing an approximation. Read the equation part by part below; each part has a contextual explanation and a link to its mathematical background.

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hh

Symbol h

h occurs below the fraction bar. The quantity above the bar is divided by this expression; zero is excluded as a denominator.

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σy\sigma_y

Symbol sigma_y

sigmaya_y occurs above the fraction bar. The numerator is divided by the entire denominator below it.

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ϕ,zz\phi_{,zz}

Symbol phi_,zz

phi_,zz occurs below the fraction bar. The quantity above the bar is divided by this expression; zero is excluded as a denominator.

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mm

Symbol m

m is one of the signed contributions combined to compute the quantity on the left.

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=

=

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

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fraction

fraction

Divide the expression above the line by the one below it.

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√

√

Take a square root.

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≈

≈

Approximately equal to; the equality is not exact.

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subscript

subscript

The lower label selects a particular version, component, or indexed member of the quantity. For example, x₀ and xₜ can be values at different positions.

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superscript

superscript

A raised number can be a power. When it is a label or bound, it selects a case or the upper limit of a sum; the formula’s structure distinguishes these uses.

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2 σy\sqrt2\,\sigma_y

Numerator: sqrt2sigma_y

The complete quantity above the fraction bar.

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1.414×10−181.414\times10^{-18}

Numerator: 1.414 × 10^-18

The complete quantity above the fraction bar.

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3.43×10−23 m−23.43\times10^{-23}\,\mathrm{m^{-2}}

Denominator: 3.43 × 10^-23mathrmm^-2

The complete quantity below the fraction bar; it must be nonzero for this division.

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How to interpret it

With a fixed numerator, increasing a nonzero denominator reduces the fraction. Its accuracy depends on the assumptions and range of use described in the article. Read it with the definitions, units, and assumptions supplied by the article.

What the article says around this equation

​ σ y ​ , gives h=2 σyϕ,zz=1.414×10−183.43×10−23 m−2≈203 mh = \sqrt{\frac{\sqrt2\,\sigma_y}{\phi_{,zz}}} = \sqrt{\frac{1.414\times10^{-18}}{3.43\times10^{-23}\,\mathrm{m^{-2}}}} \approx 203\,\mathrm m. The fourth-derivative Taylor remainder at this height is smaller than the leading term by a factor of roughly 10^{-8} , confirming that the quadratic approximation used above is, for this purpose, exact. Two hundred metres of half-spacing — roughly four hundred metres end to end — is not a laboratory bench, but it is not exotic either: it is close to the height difference a transportable optical lattice clock actually used, linked to a second clock at the base of the Tokyo Skytree tower by fibre, to measure the ordinary first-derivative geopotential difference directly against independent levelling [ 9 ] . Comparable transportable-clock campaigns have…
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​ σ y ​ , gives h=2 σyϕ,zz=1.414×10−183.43×10−23 m−2≈203 mh = \sqrt{\frac{\sqrt2\,\sigma_y}{\phi_{,zz}}} = \sqrt{\frac{1.414\times10^{-18}}{3.43\times10^{-23}\,\mathrm{m^{-2}}}} \approx 203\,\mathrm m. The fourth-derivative Taylor remainder at this height is smaller than the leading term by a factor of roughly 10^{-8} , confirming that the quadratic approximation used above is, for this purpose, exact. Two hundred metres of half-spacing — roughly four hundred metres end to end — is not a laboratory bench, but it is not exotic either: it is close to the height difference a transportable optical lattice clock actually used, linked to a second clock at the base of the Tokyo Skytree tower by fibre, to measure the ordinary first-derivative geopotential difference directly against independent levelling [ 9 ] . Comparable transportable-clock campaigns have carried a lattice clock into an underground laboratory and compared it against a distant reference over fibre [ 6 ] , and connected two transportable clocks across a shorter urban baseline to test the redshift prediction itself against surveying [ 5 ] . None of those campaigns targeted the second-derivative signal computed here; the point of the calculation is that their baselines already sit at the right order of magnitude, not that any deployed network has yet reported this specific number.

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