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Equation 118 · The Clock That Comes Back Wrong by Exactly Its Mass

What does this equation mean?

Δϕclock=2π Δν Δτ≈0.294 rad.\Delta\phi_{\rm clock}=2\pi\,\Delta\nu\,\Delta\tau\approx0.294\ \text{rad}.

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Inputs and operations2piΔnuΔτ ≈ 0.294 rad
Result or conditionΔphi_rm clock
How to read the two sides of this formula. Follow the article passage for the meaning of each quantity.

This equation states an equality: the expressions on both sides have the same value under the article’s assumptions. Read the equation part by part below; each part has a contextual explanation and a link to its mathematical background.

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Δϕclock\Delta\phi_{\rm clock}

Symbol Δphi_rm clock

the resulting clock phase.

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π\pi

Symbol pi

pi is an input to the expression that computes the quantity on the left.

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Δν\Delta\nu

Symbol Δnu

Δnu is an input to the expression that computes the quantity on the left.

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Δτ\Delta\tau

Symbol Δτ

Δτ is an input to the expression that computes the quantity on the left.

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=

=

The expressions on both sides represent the same quantity under the stated assumptions.

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change

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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subscript

subscript

The lower label selects a particular version, component, or indexed member of the quantity. For example, x₀ and xₜ can be values at different positions.

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How to interpret it

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What the article says around this equation

The resulting clock phase is Δϕclock=2π Δν Δτ≈0.294 rad\Delta\phi_{\rm clock}=2\pi\,\Delta\nu\,\Delta\tau\approx0.294\ \text{rad}. This, again, is an exact evaluation of a stated formula under stated idealized assumptions — a single-photon clock transition, no recoil, no laser phase, a static height difference held exactly one second — offered as illustrative, not as a report of anything measured. No such experiment has been performed at this scale; it is precisely the regime the cited proposals argue is at the edge of feasibility with current optical-clock coherence and large-scale atom-fountain technology, not a regime already demonstrated.

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Sources cited in the article section

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