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Equation 70 · Part 1 · The Clock That Comes Back Wrong by Exactly Its Mass

Symbol m_rm op

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)}.
mopm_{\rm op}

What this part means

mrm_rm op is part of the quantity the equation computes from the expression on the right.

Its job in the formula

mrm_rm op is part of the quantity the equation computes from the expression on the right.

The passage around this formula

Before pointing this construction at an interferometer, one methodological trap needs naming. A single measured phase Φ\Phi at one fixed loop area cannot, by itself, distinguish m from an unknown additive offset baked into the apparatus — any constant phase shift from an uncalibrated pulse timing, an unaccounted path-length difference, or a stray field looks identical to a rescaling of b\mathbf b⋅\cdotv\mathbf v . The estimator that survives this is a derivative, not a snapshot: 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)}. The sign 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 m_{\rm…

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