$$\Delta\tau=0$$

Equation 79 · Evolutionary Physics

The Clock That Comes Back Wrong by Exactly Its Mass

This equation states an equality: the expressions on both sides have the same value under the article’s assumptions.

Two degenerate limits of the proper-time form confirm the derivative is doing the right job rather than hiding a division problem. As m$\to$0 , $\Delta$$\phi$$\to$0 at every fixed $\Delta$$\tau$ , so the slope — and therefore $m_{\rm op}$ — vanishes identically: a massless worldline is exactly the case with no proper time to speak of in the first place, and the formula correctly returns nothing to divide. And if the two branches are held at equal proper time, $\Delta$$\tau$=0 , then $\Delta$$\phi$=0 regardless of m ; a single measurement at that one setting is a 0/0 statement about mass, not a small number. Both limits are trivial once stated, and both are exactly why the estimator was built as a slope across a…

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