Optical metric

The optical metric was defined by German theoretical physicist Walter Gordon in 1923 to study the geometrical optics in curved space-time filled with moving dielectric materials.

Let u_a be the normalized (covariant) 4-velocity of the arbitrarily-moving dielectric medium filling the space-time, and assume that the fluid’s electromagnetic properties are linear, isotropic, transparent, nondispersive, and can be summarized by two scalar functions: a dielectric permittivity ε and a magnetic permeability μ.

Then the optical metric tensor is defined as

${\displaystyle {\hat {g}}_{ab}=g_{ab}\pm \left(1-{\frac {1}{\epsilon \mu }}\right)u_{a}u_{b},}$

where ${\displaystyle g_{ab}}$ is the physical metric tensor. The sign of ${\displaystyle \pm }$ is determined by the metric signature convention used: ${\displaystyle \pm }$ is replaced with a plus sign (+) for a metric signature (-,+,+,+), while a minus sign (-) is chosen for (+,-,-,-).

The inverse (contravariant) optical metric tensor is

${\displaystyle {\hat {g}}^{ab}=g^{ab}\pm (1-\epsilon \mu )u^{a}u^{b},}$

where u^a is the contravariant 4-velocity of the moving fluid. Note that the traditional refractive index is defined as n(x) ≡ √εμ .