(30)–(33), pp. 4–5 — Gordon's null symbol is the ray dispersion polynomial

In a nondispersive dielectric, let $c>0$ be the vacuum light speed, $n\geq1$ the refractive index, $u\in\mathbb R^3$ the medium velocity with $|u|<c$, $k\in\mathbb R^3$ a spatial wave vector, and $\omega\in\mathbb R$ a laboratory frequency. Set $\gamma=(1-|u|^2/c^2)^{-1/2}$ and use the covariant wave vector $k_\nu=(\omega/c,-k)$. Gordon's contravariant metric is $g^{\mu\nu}=\eta^{\mu\nu}+(n^2-1)U^\mu U^\nu$, where $U=\gamma(1,u/c)$ and $\eta=\operatorname{diag}(1,-1,-1,-1)$. Then

Thus the eikonal null condition (30) is precisely the dispersion relation (33). This identifies the geometric and Hamiltonian descriptions of a wave covector.

Formalization Note The index range is $0,1,2,3$, with coordinate zero equal to $ct$. The $n\geq1$ and subluminal-flow assumptions record the paper's regime; the algebraic identity uses only $c\ne0$.