$$\Delta z=1\,\mathrm m$$

Equation 111 · Evolutionary Physics

The Clock That Comes Back Wrong by Exactly Its Mass

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

To make this concrete without simulation, take the weak-field proper-time rate d$\tau$/dt$\approx$1+gz/$c^2$ for a static height z in Earth’s field, so that holding two branches at a height difference $\Delta$ z for coordinate time T gives $\Delta$$\tau$$\approx$ g\,$\Delta$ z\,T/$c^2$ . For $\Delta$ z=1\,$\mathrm m$ and T=1\,$\mathrm s$ , $\Delta$$\tau$$\approx$1.090$\times$10^{-16}\,$\mathrm s$ . Pair this with a single-photon optical clock transition near 698\,$\mathrm{nm}$ , comparable to the transition used in strontium-lattice-clock proposals for this kind of experiment [ 11 , 13 ] , giving $\Delta$$\nu$$\approx$4.295$\times$10^{14}\,$\mathrm{Hz}$ and $\Delta$ E=h$\Delta$$\nu$$\approx$2.846$\times$10^{-19}\,$\mathrm J$ , a mass excess $\Delta$…

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