§III.D, p. 6 — the ray tangent has zero Gordon proper time

Let $c>0$ be the vacuum light speed, $n\geq1$ the refractive index, and $u\in\mathbb R^3$ a subluminal medium velocity. For a nonzero spatial wave vector $k$, let $H(k)$ be the positive comoving-frequency Hamiltonian (34), and let $v=\nabla_k H(k)$ be the laboratory ray velocity. The laboratory tangent in coordinates $x^0=ct$ is $(c,v)$. Then

where $g_{\mu\nu}=\eta_{\mu\nu}+(n^{-2}-1)U_\mu U_\nu$ is Gordon's covariant metric. This gives the pointwise null part of the ray-geodesic claim in §III.D.

Formalization Note The wave vector is nonzero because the gradient of the square-root Hamiltonian is not the ray direction at $k=0$. No condition $H(k)>0$ is imposed: the chosen branch has positive comoving frequency, while its laboratory frequency may be negative.