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Equation 30 · Part 7 · The Bend an Elevator Cannot Fake

=

r:=[I−B(B⊤C−1B)+B⊤C−1]y,r := \left[I - B\left(B^\top C^{-1}B\right)^{+}B^\top C^{-1}\right] y,
=

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.

The passage around this formula

A tempting shortcut suggests itself immediately: pick any closed loop of clocks and sum the edge readings around it. If the sum fails to return to zero, something has broken the idea that a single scalar potential — curved or otherwise — assigns each clock one honest number. Formally, define the fitted residual r:=[I−B(B⊤C−1B)+B⊤C−1]yr := \left[I - B\left(B^\top C^{-1}B\right)^{+}B^\top C^{-1}\right] y. the part of y that weighted least squares could not explain with any node potential at all. It is tempting to read a nonzero r as curvature declaring itself. That reading is wrong, and the reason is worth deriving rather than waving at.

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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 →

The article lists its research sources here.