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Published equation contexts

W(x):=Φgrav(x)−12∣Ω×r∣2W(\mathbf x) := \Phi_{\rm grav}(\mathbf x) - \tfrac12|\boldsymbol\Omega \times \mathbf r|^2

Why this formula appears here

Real terrestrial clocks are not simply static; they corotate with Earth, and their genuinely single-valued potential is the geopotential W(x\mathbf x) := Φgrav(x)\Phi_{\rm grav}(\mathbf x) - 12\tfrac12|Ω\boldsymbol\Omega ×\times r\mathbf r|^2 , gravitational potential combined with the centrifugal term for Earth’s rotation rate Ω\boldsymbol\Omega . Because W is a smooth, single-valued function of position in the corotating frame for any fixed Ω\Omega , folding it into the node potential ϕ\phi := W/c2c^2 preserves exact loop closure for any network of clocks that are themselves at rest relative to the rotating Earth — precisely the convention already standard in relativistic geodesy [ 4 , 3 ] .

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Φgrav\Phi_{\rm grav}

Symbol Phi_rm grav

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

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W(x):=Φgrav(x)−12∣Ω×r∣2W(\mathbf x) := \Phi_{\rm grav}(\mathbf x) - \tfrac12|\boldsymbol\Omega \times \mathbf r|^2

Equation 52 · Evolutionary Physics

The Bend an Elevator Cannot Fake

This equation states an equality: the expressions on both sides have the same value under the article’s assumptions.

Real terrestrial clocks are not simply static; they corotate with Earth, and their genuinely single-valued potential is the geopotential W(x\mathbf x) := Φgrav(x)\Phi_{\rm grav}(\mathbf x) - 12\tfrac12|Ω\boldsymbol\Omega ×\times r\mathbf r|^2 , gravitational potential combined with the centrifugal term for Earth’s rotation rate Ω\boldsymbol\Omega . Because W is a smooth, single-valued function of position in the corotating frame for any fixed Ω\Omega , folding it into the node potential ϕ\phi := W/c2c^2 preserves exact loop closure for any network of clocks that are themselves at rest relative to the rotating Earth — precisely the convention already standard in relativistic geodesy [ 4 , 3 ] .

Meanings in this article

  • WW: the because.
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