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Equation 52 · Part 7 · The Bend an Elevator Cannot Fake

multiplication

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

What this part means

Multiply the quantities on either side.

Its job in the formula

Multiply the quantities on either side.

The passage around this formula

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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Learn the underlying idea

Multiplication scales one quantity by another. A dot, a cross, or adjacent symbols can indicate a product.

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Sources cited in the article section

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