Searches for Lorentz violation involving photons provide one possible test of relativity. Examples range from modern versions of the classic
Michelson–Morley experiment
The Michelson–Morley experiment was an attempt to detect the existence of the luminiferous aether, a supposed medium permeating space that was thought to be the carrier of light waves. The experiment was performed between April and July 188 ...
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 10
38.
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
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are classified by their
mass dimension . To date, the most widely studied limit of the
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is the minimal SME, which limits attention to operators of renormalizable mass-dimension,
, in flat spacetime. Within the minimal
SME
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, photons are governed by the Lagrangian density
:
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
operator and a constant coefficient for Lorentz violation
. The second term introduces Lorentz violation, but preserves CPT invariance. It consists of a dimension
operator contracted with constant coefficients for Lorentz violation
. There are a total of four independent
coefficients and nineteen
coefficients. Both Lorentz-violating terms are invariant under observer Lorentz transformations, implying that the physics in independent of observer or coordinate choice. However, the coefficient tensors
and
are outside the control of experimenters and can be viewed as constant background fields that fill the entire Universe, introducing directionality to the otherwise isotropic spacetime. Photons interact with these background fields and experience frame-dependent effects, violating Lorentz invariance.
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
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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
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incorporates
general relativity
General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics ...
and curved spacetimes. It also includes operators of arbitrary (nonrenormalizable) dimension
. The general gauge-invariant
photon
A photon () is an elementary particle that is a quantum of the electromagnetic field, including electromagnetic radiation such as light and radio waves, and the force carrier for the electromagnetic force. Photons are massless, so they a ...
sector was constructed in 2009 by Kostelecky and Mewes.
It was shown that the more general theory could be written in a form similar to the minimal case,
:
where the constant coefficients are promoted to operators
and
, which take the form of power series in spacetime derivatives. The
operator contains all the CPT-odd
terms, while the CPT-even terms with
are in
. While the nonrenormalizable terms give many of the same types of signatures as the
case, the effects generally grow faster with frequency, due to the additional derivatives. More complex directional dependence typically also arises. Vacuum
dispersion
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of light without
birefringence is another feature that is found, which does not arise in the minimal
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.
Experiments
Vacuum birefringence
Birefringence of light occurs when the solutions to the modified Lorentz-violating Maxwell equations give rise to polarization-dependent speeds.
Light
Light or visible light is electromagnetic radiation that can be perceived by the human eye. Visible light is usually defined as having wavelengths in the range of 400–700 nanometres (nm), corresponding to frequencies of 750–420 te ...
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 (), birefringence causes a simple rotation of the polarization. The CPT-even case () 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,
, where
is the difference in phase speeds,
is the propagation time, and
is the wavelength. For
, the highest sensitivities are achieved by considering high-energy
photons from distant sources, giving large values to the ratio
that enhance the sensitivity to
. The best constraints on vacuum
birefringence from
Lorentz violation come from polarimetry studies of gamma-ray bursts (GRB).
For example, sensitivities of 10
−38 to the
coefficients for Lorentz violation have been achieved. For
, the velocity difference
is proportional to the wavelength, canceling the
dependence in the phase shift, implying there is no benefit to considering higher energies. As a result, maximum sensitivity is achieved by studying the most distant source available, the
cosmic microwave background
In Big Bang cosmology the cosmic microwave background (CMB, CMBR) is electromagnetic radiation that is a remnant from an early stage of the universe, also known as "relic radiation". The CMB is faint cosmic background radiation filling all spac ...
(CMB). Constraints on
coefficients for Lorentz violation from the CMB currently stand at around 10
−43 GeV.
Vacuum dispersion
Lorentz violation with
can lead to frequency-dependent light speeds.
To search for this effect, researchers compare the arrival times of photons from distant sources of pulsed radiation, such as GRB or pulsars. Assuming
photons of all energies are produced within a narrow window of time,
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would cause higher-energy photons to run ahead or behind lower-energy photons, leading to otherwise unexplained energy dependence in the arrival time. For two photons of two different energies, the difference in arrival times is approximately given by the ratio
, where
is the difference in the group velocity and
is the distance traveled. Sensitivity to Lorentz violation is then increased by considering very distant sources with rapidly changing time profiles. The speed difference
grows as
, so higher-energy sources provide better sensitivity to effects from
Lorentz violation, making GRB an ideal source.
Dispersion
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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
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focus on Lorentz violation that leads to
dispersion
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but not
birefringence. The
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shows that
dispersion
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without
birefringence can only arise from operators of even dimension
. Consequently, the energy dependence in the light speed from nonbirefringent Lorentz violation can be quadratic
or quartic
or any other even power of energy. Odd powers of energy, such as linear
and cubic
, do not arise in effective field theory.
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 10
18 to Lorentz violation.
Resonant cavities support electromagnetic standing waves that oscillate at well-defined frequencies determined by the
Maxwell equations
Maxwell's equations, or Maxwell–Heaviside equations, are a set of coupled partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, and electric circuits.
T ...
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
Microwave is a form of electromagnetic radiation with wavelengths ranging from about one meter to one millimeter corresponding to frequencies between 300 MHz and 300 GHz respectively. Different sources define different frequency ra ...
and
optical cavities have been used to constrain
violations. Microwave experiments have also placed some bounds on nonminimal
and
violations. However, for
, the effects of Lorentz violation grow with frequency, so optical cavities provide better sensitivity to nonrenormalizable violations, all else being equal. The geometrical symmetries of the cavity also affect the sensitivity since parity symmetric cavities are only directly sensitive to parity-even coefficients for Lorentz violation. Ring resonators provide a complementary class of cavity experiment that can test parity-odd violations. In a ring resonator, two modes propagating in opposites directions in the same ring are compared, rather than modes in two different cavities.
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
An atomic clock is a clock that measures time by monitoring the resonant frequency of atoms. It is based on atoms having different energy levels. Electron states in an atom are associated with different energy levels, and in transitions betwee ...
, and threshold analyses.
The results of experimental searches of Lorentz invariance violation in the photon sector of the
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are summarized in the Data Tables for Lorentz and CPT violation.
See also
*
Standard-Model Extension
*
Lorentz-violating neutrino oscillations
*
Antimatter tests of Lorentz violation
*
Bumblebee models
*
Tests of special relativity Special relativity is a physical theory that plays a fundamental role in the description of all physical phenomena, as long as gravitation is not significant. Many experiments played (and still play) an important role in its development and justifi ...
*
Test theories of special relativity Test theories of special relativity give a mathematical framework for analyzing results of experiments to verify special relativity.
An experiment to test the theory of relativity cannot assume the theory is true, and therefore needs some other fr ...
External links
Background information on Lorentz and CPT violationData Tables for Lorentz and CPT Violation
References
{{reflist, 25em
Electrodynamics
Special relativity
Hendrik Lorentz