$$y_{ij} := \ln(\nu_j/\nu_i) = \phi(\mathbf x_i) - \phi(\mathbf x_j)$$

Equation 14 · 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.

For a static observer at position $\mathbf x$ in a stationary spacetime, define U($\mathbf x$) := d$\tau$/dt , the local rate of proper time relative to coordinate time. Two clocks compared through any phase-coherent link that depends only on their respective proper frequencies report the ratio $\nu_j$/$\nu_i$ = U($\mathbf x_j$)/U($\mathbf x_i$) — a fact about stationary spacetimes that follows from the timelike Killing vector generating t -translations, and one on which the standard treatments of relativity in a geodetic or navigational setting already rely [ 10 , 14 ] . The dependence on endpoints only, never on the path connecting them, is not an approximation: because $\partial$/$\partial$ t generates a…

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