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 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 and a magnetic permeability . 
Then optical metric tensor is defined as
where formula_2 is the physical metric tensor. The sign of formula_3 is determined by the metric signature convention used: formula_3 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 is the contravariant 4-velocity of the moving fluid.
Note that the traditional refractive index is defined as .
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. The optical metric was further generalized by Robert Thompson from simple isotropic media described only by scalar-valued and to bianisotropic, magnetoelectrically coupled media residing in curved background space-times. 
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 a Robertson-Walker spacetime can be written (using the metric signature (-,+,+,+)) 
where formula_7 for a closed, flat, or open universe, and formula_8 is the scale factor.
On the other hand, the optical metric for Robertson-Walker Universe filled with rest homogeneous refraction material is
where formula_10 the cosmic-time dependent refraction index.
The luminosity distance-redshift relation in a Flat FLRW universe with dark absorption can be written
where is the cosmological redshift, is the light speed, the Hubble Constant, is the optical depth caused by absorption (or the so-called cosmic opacity), and 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.
Analogue gravity.
In analog models of gravity, the "Gordon form" expresses the metric for a curved spacetime as the sum of a flat (Minkowski) metric and a 4-velocity field u:
where n is the refractive index. This is analogous to Kerr-Schild form, which uses a null vector field in place of timelike. An open question is which spacetimes can be expressed in this way. The challenge is to pick coordinate systems for which the above relationship holds. Schwarzschild spacetime, which describes a non-rotating black hole, can be expressed this way. There has been progress for Kerr spacetime which describes a rotating black hole, but this case remains elusive.
Electrodynamics in media residing in curved space-times.
The dielectric permittivity and magnetic permeability are usually understood within the 3-vector representation of electrodynamics via the relations formula_13 and formula_14 where formula_15 and formula_16 are, respectively, the electric field, magnetic flux density, electric displacement, and magnetic field intensity, and where and could be matrices. On the other hand, general relativity is formulated in the language of 4-dimensional tensors. 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. Here, electrodynamic fields will be described in terms of differential forms, exterior algebra, and the exterior derivative. Similar to the way that 3-vectors are denoted with an arrow, as in formula_17 4-dimensional tensors will be denoted by bold symbols, for example formula_18 The musical isomorphisms will be used to indicate raising and lowering of indices with the metric, and a dot notation is used to denote contraction on adjacent indices, e.g. formula_19 The speed of light is set to formula_20 and the vacuum permeability and permittivity are likewise set to 1.
The fundamental quantity of electrodynamics is the potential 1-form formula_21 from which the field strength tensor is the 2-form formula_22 From the nilpotency of the exterior derivative one immediately has the homogeneous Maxwell equations formula_23 while a variation of the Yang-Mills action formula_24 with respect to formula_25 provides the inhomogeneous Maxwell equations formula_26 where formula_27 is the charge-current 3-form. Within dielectric media there exist charges bound up in otherwise neutral atoms. These charges are not free to move around very much, but distortions to the distribution of charge within the atom can allow dipole (or more generally multipole) moments to form, with which is associated a dipole field. Separating bound and free charges in the charge-current three form formula_28 the bound source is associated with a particular solution called the polarization field formula_29 satisfying formula_30 One may then write formula_31 with the constitutive equationformula_32 In linear media, the dipole moment is induced by the incident free field in such a way that the polarization field is linearly proportional to the free field, formula_33 (in indices this is formula_34). Then the constitutive equation can be written formula_35 The formula_36 tensor formula_37 is antisymmetric in each pair of indices, and the vacuum is seen to be a trivial dielectric such that formula_38 This means that the distribution of dielectric material within the curved background space-time can be completely described functionally by giving formula_39 and smooth transitions from vacuum into media can be described.
The electric and magnetic fields formula_15 and formula_41 as they are commonly understood in the 3-vector representation, have no independent existence. They are merely different parts of the 2-forms formula_42 and formula_43 as measured relative to a chosen observer. Let formula_44 be the contravariant velocity 4-vector of the observer. Then one may define the covariant 1-forms formula_45formula_46 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. These 1-form field definitions can be used to re-express the 2-form constitutive equation to a set of two 1-form equationsformula_47formula_48 where the formula_49 tensors formula_50 and formula_51 are formula_52formula_53formula_54formula_55 Note that each of these tensors is orthogonal, or transverse, to formula_56 meaning that formula_57 for each formula_58, which can be seen from the antisymmetry of formula_59 on each pair of indices. Since each of the 1-form fields defined above is also transverse to formula_56 we may conclude that each formula_61 is an automorphism of a subspace of the cotangent space defined by orthogonality with respect to the observer. In other words, everything operates in the observer's purely spatial 3-dimensional space. In terms of these parameters, formula_59 is found to beformula_63 Although the set of 1-form constitutive equations shown above are the ones that follow most naturally from the covariant 2-form constitutive equation formula_64, they are not the only possibility. Indeed, the traditional 3-vector formulation of the constitutive equations usually relates formula_65 and formula_66 by formula_67. Therefore, it could be desirable to rearrange the preceding set of relations into formula_68formula_69 where formula_70 are related to formula_71 by formula_72formula_73formula_74formula_75 The 4-dimensional inverse of these tensors does not exist, but the bar notation formula_76 denotes an inverse defined with respect to the subspace orthogonal to formula_56 which exists and is a valid operation since it was noted above that formula_78 is an automorphism of this subspace. In Minkowski space-time, the space-space part (relative to observer formula_44) of each of these tensors is equivalent to the traditional formula_80 constitutive matrices of 3-vector electrodynamics. In terms of this alternative set of constitutive tensors, formula_59 is found to be formula_82 Here, formula_83 is a projection operator that annihilates any tensor components parallel to formula_84 Since formula_85 then formula_86 also serves as the Kronecker delta on the subspace orthogonal to formula_84 In the vacuum, formula_88
Geometric optics and the optical metric.
For light propagating through linear dielectric media, Maxewell's inhomogeneous equation in the absence of free sources represents a wave equation for formula_89 in the Lorenz gauge, formula_90 (here formula_91 is the codifferential), given by formula_92 A JWKB type approximation of plane wave solutions is assumed such that formula_93 where the amplitude formula_94 is assumed to be slowly varying compared to the phase function formula_95 Plugging this approximate solution into the wave equation, and retaining only the leading order terms in the limit formula_96 leads to formula_97 where formula_98 The existence of a solution to this equation requires formula_99 In fact, this determinant condition is satisfied identically because the antisymmetry in the second pair of indices on formula_59 shows that formula_101 is already a trivial solution. Therefore, any non-trivial solutions must reside in the 3-dimensional subspace orthogonal to formula_102 so the tensor formula_103 is effectively only 3-dimensional. Thus, the determinant condition is insufficient to provide any information. However, the classical adjugate of a matrix formula_104 is related to its determinant by formula_105. Since in this case formula_106 but formula_104 is arbitrary, one obtains the secondary condition formula_108 Notice that the adjugate of a matrix is still a matrix, so the scalar determinant condition has now been replaced by a matrix condition. This would appear to add a great deal of complexity to the problem, but it has been shown that this adjugate has the form formula_109 where formula_110 is a fourth order polynomial in formula_111 The vanishing condition on the adjugate matrix is therefore equivalent to the scalar condition formula_112 The goal now is to demonstrate that the polynomial formula_110 takes the form formula_114 Then the condition formula_115 is satisfied by either of formula_116 (written with indices, formula_117). 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 tensors formula_118 therefore determine the lightcone structures. 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. In vacuum, it is readily found that formula_119 degenerates to the space-time metric. Since the formula_118 determine the lightcones in media in the way that formula_121 does for the vacuum, they are referred to as optical metrics. However, it is perhaps more appropriate to take the point of view that the space-time metric happens to also serve as the optical metric in vacuum, which is not so surprising considering that the space-time metric is the only available structure in vacuum.
So far, no assumptions have been imposed on the form of formula_122 or formula_123 so there are currently 36 freely specifiable parameters. To determine the optical metrics, Thompson imposes the conditions that formula_124 and formula_125 are antisymmetric with respect to formula_126 (i.e. antisymmetric when the indices on formula_124 and formula_125 are either both up or both down). The antisymmetry condition allows them to be written in the forms formula_129formula_130 With this restriction, it is found that formula_110 is biquadratic in formula_132 and can be factored to formula_133 where formula_134withformula_135 Finally, the optical metrics correspond to formula_136 The presence of the square root in formula_137 and consequently in formula_138 shows that the birefringent optical metrics are of the pseudo-Finslerian type. A key feature here is that the optical metric is not only a function of position, but also retains a dependency on formula_139. 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.