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Equation 71 · Part 3 · The Bend an Elevator Cannot Fake

Symbol U

ds2=−U(Z)2 dx02+dZ2+dx2+dy2,x0:=ct,U(Z):=1+gZc2.ds^2 = -U(Z)^2\,dx_0^2 + dZ^2 + dx^2 + dy^2, \qquad x_0 := ct, \qquad U(Z) := 1 + \frac{gZ}{c^2}.
UU

What this part means

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

Its job in the formula

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

The passage around this formula

…constant proper acceleration g : ds2=−U(Z)2 dx02+dZ2+dx2+dy2,x0:=ct,U(Z):=1+gZc2ds^2 = -U(Z)^2\,dx_0^2 + dZ^2 + dx^2 + dy^2, \qquad x_0 := ct, \qquad U(Z) := 1 + \frac{gZ}{c^2}. This is an exact solution describing flat Minkowski spacetime in non-inertial coordinates, not a weak-field approximation. A clock held static at height Z has dτ\tau = U(Z)\,dt exactly, so two such clocks compare frequencies as ν2\nu_2/ν1\nu_1 = U(Z1Z_1)/U(Z2Z_2) to all orders — the same exact redshift formula an accelerating rocket produces for light climbing or falling inside it. The node potential is ϕ(Z)\phi(Z) = ln⁡\ln U(Z) = ln⁡(1+gZ/c2)\ln(1 + gZ/c^2) , and its first derivative at…

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Learn the underlying idea

A function assigns an output to each allowed input. The expression f(x) means “apply f to x”.

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