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Equation 70 · Part 5 · The Clock That Comes Back Wrong by Exactly Its Mass

Symbol L_rm G

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)}.
LGL_{\rm G}

What this part means

LrL_rm G is one factor in the product that computes the quantity on the left.

Its job in the formula

LrL_rm G is one factor in the product that computes the quantity on the left.

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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