Equation 129 · The Bend an Elevator Cannot Fake
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This equation states an equality: the expressions on both sides have the same value under the article’s assumptions. Read the equation part by part below; each part has a contextual explanation and a link to its mathematical background.
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Symbol G
G is part of the quantity the equation computes from the expression on the right.
Symbol M_oplus
plus is part of the quantity the equation computes from the expression on the right.
=
The expressions on both sides represent the same quantity under the stated assumptions.
See an illustrated explanation →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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What the article says around this equation
For Earth modelled as a point mass, = -GM/r gives a radial second derivative of magnitude 2GM/ and, since is harmonic in vacuum, transverse second derivatives of magnitude GM/ with the opposite sign — a trace-free tensor in the exact ratio 2:{-1}:{-1} . With the standard gravitational parameter G = 3.98610^{14}\, and mean radius = 6.37110^{6}\, , this gives G/ 1.54110^{-6}\, and a radial magnitude of 3.0810^{-6}\, , matching the standard geodetic free-air gravity gradient of about 3.08610^{-6}\, , or 0.3086 milligal per metre,…
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For Earth modelled as a point mass, = -GM/r gives a radial second derivative of magnitude 2GM/ and, since is harmonic in vacuum, transverse second derivatives of magnitude GM/ with the opposite sign — a trace-free tensor in the exact ratio 2:{-1}:{-1} . With the standard gravitational parameter G = 3.98610^{14}\, and mean radius = 6.37110^{6}\, , this gives G/ 1.54110^{-6}\, and a radial magnitude of 3.0810^{-6}\, , matching the standard geodetic free-air gravity gradient of about 3.08610^{-6}\, , or 0.3086 milligal per metre, to three figures [ 15 ] . So 3.0810^{-6}/ 3.4310^{-23}\, .
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