(50)–(51), p. 6 — Gordon's line element in the local rest frame

Let $c>0$, $n\geq1$, and let $u\in\mathbb R^3$ be a medium velocity with $|u|<c$. Write $\Lambda(u)$ for the Lorentz boost taking a displacement $v=(c\,dt',dx'_1,dx'_2,dx'_3)$ from the local rest frame to the laboratory frame. Gordon's covariant metric is $g_{\mu\nu}=\eta_{\mu\nu}+(n^{-2}-1)U_\mu U_\nu$, with $U=\gamma(1,u/c)$ and $\eta=\operatorname{diag}(1,-1,-1,-1)$. For every rest-frame displacement $v$,

This is the local-rest-frame line element (51), $ds^2=c^2dt'^2/n^2-|dx'|^2$. It exposes the light cone of the optical metric at one event.

Formalization Note The published boost is the identity at $u=0$. The subluminal condition makes its Lorentz factor physical; $n\geq1$ records the paper's regime but is not needed for the algebraic identity. This statement is also the pointwise second conjunct of the existing Open goal.