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Equation 100 · Part 9 · The Bend an Elevator Cannot Fake

superscript

Tij:=c2(ϕ,ij+ϕ,i ϕ,j),\mathcal T_{ij} := c^2\left(\phi_{,ij} + \phi_{,i}\,\phi_{,j}\right),
superscript

What this part means

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.

Its job in the formula

A raised mark can be a power or an index. Its position and the surrounding notation determine which.

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, Tij:=c2(ϕ,ij+ϕ,i ϕ,j)\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\,E\mathrm E = 10^{-9}\,s−2\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 ϕ\phi it is unchanged, because it depends only on derivatives of ϕ\phi , never its value; and up…

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Learn the underlying idea

An exponent tells how a base is used in multiplication. In x³, x is the base and 3 is the exponent: x³ = x × x × x.

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