← Back to article

Equation 129 · The Bend an Elevator Cannot Fake

What does this equation mean?

GM⊕=3.986×1014 m3 s−2GM_\oplus = 3.986\times10^{14}\,\mathrm{m^3\,s^{-2}}

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.

Inputs and operations3.986 × 10^14mathrmm^3s^-2
Result or conditionGM_oplus
How to read the two sides of this formula. Follow the article passage for the meaning of each quantity.

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.

Read it piece by piece

GG

Symbol G

G is part of the quantity the equation computes from the expression on the right.

Understand this part →

M⊕M_\oplus

Symbol M_oplus

MoM_oplus is part of the quantity the equation computes from the expression on the right.

Understand this part →

=

=

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

Understand this part →

See an illustrated explanation →
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.

Understand this part →

See an illustrated explanation →

How to interpret it

Read it with the definitions, units, and assumptions supplied by the article.

What the article says around this equation

For Earth modelled as a point mass, Φ(r)\Phi(r) = -GM/r gives a radial second derivative of magnitude 2GM/R3R^3 and, since Φ\Phi is harmonic in vacuum, transverse second derivatives of magnitude GM/R3R^3 with the opposite sign — a trace-free tensor in the exact ratio 2:{-1}:{-1} . With the standard gravitational parameter GM⊕M_\oplus = 3.986×\times10^{14}\,m3 s−2\mathrm{m^3\,s^{-2}} and mean radius R⊕R_\oplus = 6.371×\times10^{6}\,m\mathrm m , this gives GM⊕M_\oplus/R⊕3R_\oplus^3 ≈\approx 1.541×\times10^{-6}\,s−2\mathrm{s^{-2}} and a radial magnitude of 3.08×\times10^{-6}\,s−2\mathrm{s^{-2}} , matching the standard geodetic free-air gravity gradient of about 3.086×\times10^{-6}\,s−2\mathrm{s^{-2}} , or 0.3086 milligal per metre,…
Read the full surrounding passage
For Earth modelled as a point mass, Φ(r)\Phi(r) = -GM/r gives a radial second derivative of magnitude 2GM/R3R^3 and, since Φ\Phi is harmonic in vacuum, transverse second derivatives of magnitude GM/R3R^3 with the opposite sign — a trace-free tensor in the exact ratio 2:{-1}:{-1} . With the standard gravitational parameter GM⊕M_\oplus = 3.986×\times10^{14}\,m3 s−2\mathrm{m^3\,s^{-2}} and mean radius R⊕R_\oplus = 6.371×\times10^{6}\,m\mathrm m , this gives GM⊕M_\oplus/R⊕3R_\oplus^3 ≈\approx 1.541×\times10^{-6}\,s−2\mathrm{s^{-2}} and a radial magnitude of 3.08×\times10^{-6}\,s−2\mathrm{s^{-2}} , matching the standard geodetic free-air gravity gradient of about 3.086×\times10^{-6}\,s−2\mathrm{s^{-2}} , or 0.3086 milligal per metre, to three figures [ 15 ] . So ϕ,zz\phi_{,zz} ≈\approx 3.08×\times10^{-6}/c2c^2 ≈\approx 3.43×\times10^{-23}\,m−2\mathrm{m^{-2}} .

Read the equation in its article →

Sources cited in the surrounding passage

These citations give research context. Read each source to check which claims it supports.

Return to The Bend an Elevator Cannot Fake

See this formula across 1 published context →

Browse the mathematical compendium →