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Δτ≈g Δz T/c2\Delta\tau\approx g\,\Delta z\,T/c^2

Why this formula appears here

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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Δτ\Delta\tau

Symbol Δτ

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gg

Symbol g

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Δz\Delta z

Symbol Δ z

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TT

Symbol T

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Published contexts (2)

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Δτ≈g Δz T/c2\Delta\tau\approx g\,\Delta z\,T/c^2

Equation 110 · Evolutionary Physics

The Clock That Comes Back Wrong by Exactly Its Mass

This equation gives an approximation: it relates the quantities while allowing an approximation.

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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Δτ≈g Δz T/c2\Delta\tau\approx g\,\Delta z\,T/c^2

Equation 133 · Evolutionary Physics

The Clock That Comes Back Wrong by Exactly Its Mass

This equation gives an approximation: it relates the quantities while allowing an approximation.

No experiment has run the differential two-height, two-internal-state protocol at the meter-and-second scale used illustratively above; it remains a proposal, not a result [ 10 , 11 ] . What existing data already constrain is the space this construction is allowed to occupy. Dual-species atom-interferometric tests of the weak equivalence principle have matched the free-fall trajectories of different atomic species to extraordinary precision by actively canceling the very laser/recoil terms this paper identifies as confounds, which is direct evidence that no large anomalous mass-dependence is hiding in the dominant phase term this construction must avoid [ 12 ] . Separately, ordinary…

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