The optical metric was defined by German theoretical physicist Walter Gordon in 1923 [1] to study the geometrical optics in curved space-time filled with moving dielectric materials.
Let ua 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.[2]
Then the optical metric tensor is defined as
where is the physical metric tensor. The sign of is determined by the metric signature convention used: is replaced with a plus sign (+) for a metric signature (-,+,+,+), while a minus sign (-) is chosen for (+,-,-,-).
The inverse (contravariant) optical metric tensor is
where ua is the contravariant 4-velocity of the moving fluid.
Note that the traditional refractive index is defined as n(x) ≡ √εμ.
^W. Gordon, 1923, Annals of Physics (New York), 22, 421
^J. D. Jackson, "Classical Electrodynamics", 1998, (John Wiley & Sons Inc, New York)
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