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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}

Why this formula appears here

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 : 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…

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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}.

Equation 71 · Evolutionary Physics

The Bend an Elevator Cannot Fake

This equation states an equality: the expressions on both sides have the same value under the article’s assumptions.

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 : 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…

Meanings in this article

  • ZZ: the height.
  • Z2Z^2: the square of Z; the height.
  • gg: the observer under constant proper acceleration.
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