subscript

The passage around this formula

Consider a uniformly accelerated laboratory in otherwise empty, flat spacetime — Einstein’s elevator, given by the exact Rindler metric in coordinates comoving with an observer under constant proper acceleration g : $ds^2 = -U(Z)^2\,dx_0^2 + dZ^2 + dx^2 + dy^2, \qquad x_0 := ct, \qquad U(Z) := 1 + \frac{gZ}{c^2}$. 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 d$\tau$ = U(Z)\,dt exactly, so two such clocks compare frequencies as $\nu_2$/$\nu_1$ = U($Z_1$)/U($Z_2$) to all orders — the same exact redshift formula an accelerating rocket produces for light climbing or falling inside it. The node potential is $\phi(Z)$ = $\ln$ U(Z) = $\ln(1 + gZ/c^2)$ , and its first derivative…

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