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 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. Then optical metric tensor is defined as
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where gab is the physical metric tensor. The sign of
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 .
Properties
An important fact about Gordon's optical metric is that in curved space-time filled with dielectric material, electromagnetic waves (under geometrical optics approximation) follows geodesics of the optical metric instead of the physical metric. Consequently, the study of geometric optics in curved space-time with dielectric material can sometimes be simplified by using optical metric (note that the dynamics of the physical system is still described by the physical metric). For example, optical metric can be used to study the radiative transfer in stellar atmospheres around compact astrophysical objects such as neutron stars and white dwarfs, and in accretion disks around black holes. In cosmology, optical metric can be used to study the distance-redshift relation in cosmological models in which the intergalactic or interstellar medium have a non-vanishing refraction index.
History
After the original introduction of the concept of optical metric by Gordon in 1923, the mathematical formalism of optical metric was further investigated by Jürgen Ehlers in 1967 including a detailed discussion of the geometrical optical approximation in curved space-time and the optical scalars transport equation. Gordon's optical metric was extended by Bin Chen and Ronald Kantowski to include light absorption. The original real optical metric was consequently extended into a complex one.
Applications
The first application of Gordon's optical metric theory to cosmology was also made by Bin Chen and Ronald Kantowski. The absorption corrected distance-redshift relation in the homogeneous and isotropic Friedman-Lemaitre-Robertson-Walker (FLRW) universe is called Gordon-Chen-Kantowski formalism and can be used to study the absorption of intergalactic medium (or cosmic opacity) in the Universe.
For example, the physical metric for Robertson-Walker metric can be written (using the metric signature (-,+,+,+))
where k = 1, 0, -1 for a closed, flat, or open universe. On the other hand, the optical metric for Robertson-Walker Universe filled with rest homogeneous refraction material is
where n(t) the cosmic-time dependent refraction index.
The luminosity distance-redshift relation in a Flat FLRW universe with dark absorption can be written
where z is the cosmological redshift, c is the light speed, H0 the Hubble Constant, τ is the optical depth caused by absorption (or the so-called cosmic opacity), and h(z) is the dimensionless Hubble curve. A non-zero cosmic opacity will render the standard candles such as Type Ia supernovae appear dimmer than expected from a transparent Universe. This can be used as an alternative explanation of the observed apparent acceleration of the cosmic expansion.