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Equation 70 · Part 3 · The Clock That Comes Back Wrong by Exactly Its Mass

Symbol Δphi

mop=ℏc2 ∂Δϕ∂Δτ,mop[LG]=ℏ ∂Φ∂(b⋅v).m_{\rm op}=\frac{\hbar}{c^2}\,\frac{\partial\Delta\phi}{\partial\Delta\tau},\qquad m_{\rm op}[\mathcal L_{\rm G}]=\hbar\,\frac{\partial\Phi}{\partial(\mathbf b\cdot\mathbf v)}.
Δϕ\Delta\phi

What this part means

Δphi occurs above the fraction bar. The numerator is divided by the entire denominator below it.

Its job in the formula

Δphi occurs above the fraction bar. The numerator is divided by the entire denominator below it.

The passage around this formula

…convention on the proper-time form follows the phase of a free relativistic particle, Δ\Deltaϕ\phi=-mc2c^2Δ\Deltaτ\tau/ℏ\hbar , so that mopm_{\rm op}=-(ℏ\hbar/c2c^2)\,∂\partialΔ\Deltaϕ\phi/∂\partialΔ\Deltaτ\tau recovers m identically when Δ\Deltaϕ\phi has no other dependence. Both forms require sweeping the loop’s own parameter — hold time, translation distance, boost velocity — across at least two settings and fitting the slope, exactly as a spectroscopist extracts an energy gap from a swept detuning rather than from one photon…

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Learn the underlying idea

A variable is a named place for a value. Its letter is a local label: x can mean position in one formula and a data point in another.

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Sources cited in the article section

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