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T=1 sT=1\,\mathrm s

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To make this concrete without simulation, take the weak-field proper-time rate dτ\tau/dt≈\approx1+gz/c2c^2 for a static height z in Earth’s field, so that holding two branches at a height difference Δ\Delta z for coordinate time T gives Δ\Deltaτ\tau≈\approx g\,Δ\Delta z\,T/c2c^2 . For Δ\Delta z=1\,m\mathrm m and T=1\,s\mathrm s , Δ\Deltaτ\tau≈\approx1.090×\times10^{-16}\,s\mathrm s . Pair this with a single-photon optical clock transition near 698\,nm\mathrm{nm} , comparable to the transition used in strontium-lattice-clock proposals for this kind of experiment [ 11 , 13 ] , giving Δ\Deltaν\nu≈\approx4.295×\times10^{14}\,Hz\mathrm{Hz} and Δ\Delta E=hΔ\Deltaν\nu≈\approx2.846×\times10^{-19}\,J\mathrm J , a mass excess Δ\Delta…

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T=1 sT=1\,\mathrm s

Equation 112 · 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 dτ\tau/dt≈\approx1+gz/c2c^2 for a static height z in Earth’s field, so that holding two branches at a height difference Δ\Delta z for coordinate time T gives Δ\Deltaτ\tau≈\approx g\,Δ\Delta z\,T/c2c^2 . For Δ\Delta z=1\,m\mathrm m and T=1\,s\mathrm s , Δ\Deltaτ\tau≈\approx1.090×\times10^{-16}\,s\mathrm s . Pair this with a single-photon optical clock transition near 698\,nm\mathrm{nm} , comparable to the transition used in strontium-lattice-clock proposals for this kind of experiment [ 11 , 13 ] , giving Δ\Deltaν\nu≈\approx4.295×\times10^{14}\,Hz\mathrm{Hz} and Δ\Delta E=hΔ\Deltaν\nu≈\approx2.846×\times10^{-19}\,J\mathrm J , a mass excess Δ\Delta…

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T=1 sT=1\,\mathrm s

Equation 125 · 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.

The scale of what needs to be canceled is worth stating plainly, using the same illustrative parameters as the previous section. The laser/center-of-mass term for a single-photon transition near 698\,nm\mathrm{nm} ( keffk_{\rm eff}≈\approx9.00×\times10^6\,m−1\mathrm{m^{-1}} ) scales as keffk_{\rm eff}gT2T^2≈\approx8.8×\times10^{7}\,rad\mathrm{rad} for T=1\,s\mathrm s . The recoil term for a strontium-mass atom ( m≈\approx1.443×\times10^{-25}\,kg\mathrm{kg} ) scales as ℏ\hbar keff2k_{\rm eff}^2T/m≈\approx5.9×\times10^{4}\,rad\mathrm{rad} . The clock term computed above is ≈\approx0.29\,rad\mathrm{rad} . Both confound terms are, respectively, roughly eight and five orders of magnitude larger than the signal, which is exactly why…

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