← Mathematical compendium

Published equation contexts

ϕ(Z)=ln⁡U(Z)=ln⁡(1+gZ/c2)\phi(Z) = \ln U(Z) = \ln(1 + gZ/c^2)

Why this formula appears here

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 Z=0 is ϕ\phi' = g/c2c^2 : a nonzero, entirely real, measurable slope. A network of clocks bolted to this elevator’s walls would report exactly the nonzero edge data a network in a genuine gravitational field reports at the same order. If…

Read the full article-specific guide →

Read the representative guide

How to interpret it

Read it with the definitions, units, and assumptions supplied by the article.

Research cited beside this formula

Learn the mathematics behind it

Published contexts (1)

A symbol can carry a different meaning in another article. Each occurrence keeps its own guide and term definitions.

ϕ(Z)=ln⁡U(Z)=ln⁡(1+gZ/c2)\phi(Z) = \ln U(Z) = \ln(1 + gZ/c^2)

Equation 75 · 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.

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 Z=0 is ϕ\phi' = g/c2c^2 : a nonzero, entirely real, measurable slope. A network of clocks bolted to this elevator’s walls would report exactly the nonzero edge data a network in a genuine gravitational field reports at the same order. If…

Meanings in this article

  • ZZ: the height.
  • gg: the observer under constant proper acceleration.
Equation guide → · Article →