Equation 100 · Part 6 · The Bend an Elevator Cannot Fake

=

\mathcal T_{ij} := c^2\left(\phi_{,ij} + \phi_{,i}\,\phi_{,j}\right),

What this part means

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

Its job in the formula

The equals sign connects the complete expression on the left with the complete expression on the right. Both sides must have compatible units.

Full expression=Article meaning

The passage around this formula

Define the network’s curvature-diagnosing observable directly from this result, restoring physical units through the relation between coordinate acceleration and proper time for a static observer, $\mathcal T_{ij} := c^2\left(\phi_{,ij} + \phi_{,i}\,\phi_{,j}\right)$ . a symmetric rank-two tensor in the local spatial frame of the static observers, with units of inverse time squared — the Eötvös units already standard in gravity-gradient surveying, where 1\, $\mathrm E$ = 10^{-9}\, $\mathrm{s^{-2}}$ . Under a rotation of the local spatial axes it transforms as an ordinary rank-two tensor; under a change of which node is used as the additive reference for $\phi$ it is unchanged, because it depends only on derivatives of $\phi$ , never its value; and up…

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Learn the underlying idea

An equals sign says that the expression on its left and the expression on its right have the same value under the stated definitions and assumptions.

Open the illustrated equality: what the equals sign claims guide →

Sources cited in the article section

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