= Proper-time gauge for a massive worldline einbein
{title2=$ds=e\,d\lambda$}
For the <relativistic particle> action $S=(m/2)\int[g_{\mu\nu}\dot x^\mu\dot x^\nu/e-e]d\lambda$, varying the <worldline einbein> gives $g_{\mu\nu}\dot x^\mu\dot x^\nu=-e^2$. On the positive-einbein branch the <proper time> satisfies $ds=e\,d\lambda$. A <reparametrization> with $\lambda'=s$ sets the transformed einbein to one, making the <geodesic equation> affinely parametrized. This gauge choice is made after retaining the einbein <mass shell> constraint; fixing the action first would discard that equation. If the parameter interval is held fixed at both ends, its integrated <proper length> is a modulus and cannot be arbitrarily changed by endpoint-preserving transformations.
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