Searches for Lorentz violation involving photons are among the best tests of relativity. Examples range from modern versions of the classic Michelson-Morley experiment that utilize highly stable electromagnetic resonant cavities to searches for tiny deviations from c in the speed of light emitted by distant astrophysical sources. Due to the extreme distances involved, astrophysical studies have achieved sensitivities on the order of parts in 1038.
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Minimal Lorentz-violating electrodynamics
The most general framework for studies of relativity violations is an effective field theory called the Standard-Model Extension (SME). Lorentz-violating operators in the SME are classified by their mass dimension
The first term on the right-hand side is the conventional Maxwell lagrangian and gives rise to the usual source-free Maxwell equations. The next term violates both Lorentz and CPT invariance and is constructed from a dimension
The mathematics describing Lorentz violation in photons is similar to that of conventional electromagnetism in dielectrics. As a result, many of the effects of Lorentz violation are also seen in light passing through transparent materials. These include changes in the speed that can depend on frequency, polarization, and direction of propagation. Consequently, Lorentz violation can introduce dispersion in light propagating in empty space. It can also introduce birefringence, an effect seen in crystals such as calcite. The best constraints on Lorentz violation come from constraints on birefringence in light from astrophysical sources.
Nonminimal Lorentz-violating electrodynamics
The full SME incorporates general relativity and curved spacetimes. It also includes operators of arbitrary (nonrenormalizable) dimension
where the constant coefficients are promoted to operators
Vacuum birefringence
Birefringence of light occurs when the solutions to the modified Lorentz-violating Maxwell equations give rise to polarization-dependent speeds. Light propagates as the combination of two orthogonal polarizations that propagate at slightly different phase velocities. A gradual change in the relative phase results as one of the polarizations outpaces the other. The total polarization (the sum of the two) evolves as the light propagates, in contrast to the Lorentz-invariant case where the polarization of light remains fixed when propagating in a vacuum. In the CPT-odd case (d = odd), birefringence causes a simple rotation of the polarization. The CPT-even case (d = even) gives more complicated behavior as linearly polarized light evolves into elliptically polarizations.
The quantity determining the size of the effect is the change in relative phase,
Vacuum dispersion
Lorentz violation with
Dispersion may or may not be accompanied by birefringence. Polarization studies typically achieved sensitivities well beyond those achievable through dispersion. As a result, most searches for dispersion focus on Lorentz violation that leads to dispersion but not birefringence. The SME shows that dispersion without birefringence can only arise from operators of even dimension
Resonant cavities
While extreme sensitivity to Lorentz violation is achieved in astrophysical studies, most forms of Lorentz violation have little to no effect on light propagating in a vacuum. These types of violations cannot be tested using astrophysical tests, but can be sought in laboratory-based experiments involving electromagnetic fields. The primary examples are the modern [[Michelson-Morley experiments]] based on electromagnetic resonant cavities, which have achieved sensitivities on the order of parts in 1018 to Lorentz violation.
Resonant cavities support electromagnetic standing waves that oscillate at well-defined frequencies determined by the Maxwell equations and the geometry of the cavity. The Lorentz-violating modifications to the Maxwell equations lead to tiny shifts in the resonant frequencies. Experimenters search for these tiny shifts by comparing two or more cavities at different orientations. Since rotation-symmetry violation is a form of Lorentz violation, the resonant frequencies may depend on the orientation of the cavity. So, two cavities with different orientations may give different frequencies even if they are otherwise identical. A typical experiment compares the frequencies of two identical cavities oriented at right angles in the laboratory. To distinguish between frequency differences of more conventional origins, such as small defects in the cavities, and Lorentz violation, the cavities are typically placed on a turntable and rotated in the laboratory. The orientation dependence from Lorentz violation would cause the frequency difference to change as the cavities rotate.
Several classes of cavity experiment exist with different sensitivities to different types of Lorentz violation. Microwave and optical cavities have been used to constrain
Other experiments
A number of other searches for Lorentz violation in photons have been performed that do not fall under the above categories. These include accelerator based experiments, atomic clocks, and threshold analyses.
The results of experimental searches of Lorentz invariance violation in the photon sector of the SME are summarized in the Data Tables for Lorentz and CPT violation.