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.
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.
Gordon's optical metric was extended by Bin Chen and Ronald Kantowski[5] to include light absorption.
The optical metric was further generalized by Robert Thompson [6] from simple isotropic media described only by scalar-valued ε and μ to bianisotropic, magnetoelectrically coupled media residing in curved background space-times.
The first application of Gordon's optical metric theory to cosmology was also made by Bin Chen and Ronald Kantowski.
[7] The absorption corrected distance-redshift relation in the homogeneous and isotropic Friedman-Lemaitre-Robertson-Walker (FLRW) universe is called Gordon-Chen-Kantowski formalism [8] and can be used to study the absorption of intergalactic medium (or cosmic opacity) in the Universe.
On the other hand, the optical metric for Robertson-Walker Universe filled with rest homogeneous refraction material is where
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.
This is analogous to Kerr-Schild form, which uses a null vector field in place of timelike.
Schwarzschild spacetime, which describes a non-rotating black hole, can be expressed this way.
[9] There has been progress for Kerr spacetime which describes a rotating black hole, but this case remains elusive.
[10] The dielectric permittivity ε and magnetic permeability μ are usually understood within the 3-vector representation of electrodynamics via the relations
To obtain the tensorial optical metric, medium properties such as permittivity, permeability, and magnetoelectric couplings must first be promoted to 4-dimensional covariant tensors, and the electrodynamics of light propagation through such media residing within a background space-time must also be expressed in a compatible 4-dimensional way.
From the nilpotency of the exterior derivative one immediately has the homogeneous Maxwell equations
[11] Within dielectric media there exist charges bound up in otherwise neutral atoms.
This means that the distribution of dielectric material within the curved background space-time can be completely described functionally by giving
The corresponding 3-vectors are obtained in Minkowski space-time by taking the purely spatial (relative to the observer) parts of the contravariant versions of these 1-forms.
is an automorphism of a subspace of the cotangent space defined by orthogonality with respect to the observer.
A JWKB type approximation of plane wave solutions is assumed such that
Plugging this approximate solution into the wave equation, and retaining only the leading order terms in the limit
In fact, this determinant condition is satisfied identically because the antisymmetry in the second pair of indices on
This would appear to add a great deal of complexity to the problem, but it has been shown[6] that this adjugate has the form
What has been shown so far is that wave solutions of Maxwell's equations, in the ray limit, must satisfy one of these two polynomial conditions.
The fact that there are two of them implies a double light cone structure - one for each of the two polarization states, i.e. birefringence.
A key feature here is that the optical metric is not only a function of position, but also retains a dependency on
These pseudo-Finslerian optical metrics degenerate to a common, non-birefringent, pseudo-Riemannian optical metric for media that obey a curved space-time generalization of the Post conditions.