Equation 117 · The Clock That Comes Back Wrong by Exactly Its Mass

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\Delta E/c^2\approx3.167\times10^{-36}\,\mathrm{kg}

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\Delta E

Symbol Δ E

the energy.

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c^2

Symbol c^2

The square of c: multiply c by itself.

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change

Capital delta attached to a quantity marks a difference between two values of that quantity; the article’s sign convention determines the order.

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superscript

A raised number can be a power. When it is a label or bound, it selects a case or the upper limit of a sum; the formula’s structure distinguishes these uses.

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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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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 $\Delta$ E/ $c^2$ $\approx$ 3.167 $\times$ 10^{-36}\, $\mathrm{kg}$ .

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