Modern searches for Lorentz violation

Modern searches for Lorentz violation are scientific studies that look for deviations from

Both terrestrial and astronomical experiments have been carried out, and new experimental techniques have been introduced. No Lorentz violations have been measured thus far, and exceptions in which positive results were reported have been refuted or lack further confirmations. For discussions of many experiments, see Mattingly (2005).[1] For a detailed list of results of recent experimental searches, see Kostelecký and Russell (2008–2013).^[2] For a recent overview and history of Lorentz violating models, see Liberati (2013).^[3]

Early models assessing the possibility of slight deviations from Lorentz invariance have been published between the 1960s and the 1990s.^[3] In addition, a series of test theories of special relativity and effective field theories (EFT) for the evaluation and assessment of many experiments have been developed, including:

• The parameterized post-Newtonian formalism is widely used as a test theory for general relativity and alternatives to general relativity, and can also be used to describe Lorentz violating preferred frame effects.
• The Robertson-Mansouri-Sexl framework (RMS) contains three parameters, indicating deviations in the speed of light with respect to a preferred frame of reference.
• The c² framework (a special case of the more general THεμ framework) introduces a modified dispersion relation and describes Lorentz violations in terms of a discrepancy between the speed of light and the maximal attainable speed of matter, in presence of a preferred frame.^[4][5]
• Planck length
  as an invariant minimum length-scale, yet without having a preferred reference frame.
• Very special relativity describes space-time symmetries that are certain proper subgroups of the Poincaré group. It was shown that special relativity is only consistent with this scheme in the context of quantum field theory or CP conservation.
• Noncommutative geometry (in connection with Noncommutative quantum field theory or the Noncommutative standard model) might lead to Lorentz violations.
• Lorentz violations are also discussed in relation to
  Einstein aether theory, Hořava–Lifshitz gravity
  .

In addition, the Standard-Model Extension (SME) can be used to obtain a larger number of isotropy coefficients in the photon sector. It uses the even- and odd-parity coefficients (3×3 matrices) ${\displaystyle {\tilde {\kappa }}_{e-}}$, ${\displaystyle {\tilde {\kappa }}_{o+}}$ and ${\displaystyle {\tilde {\kappa }}_{tr}}$.^[8] They can be interpreted as follows: ${\displaystyle {\tilde {\kappa }}_{e-}}$ represent anisotropic shifts in the two-way (forward and backwards) speed of light, ${\displaystyle {\tilde {\kappa }}_{o+}}$ represent anisotropic differences in the one-way speed of counterpropagating beams along an axis,^[14][15] and ${\displaystyle {\tilde {\kappa }}_{tr}}$ represent isotropic (orientation-independent) shifts in the one-way phase velocity of light.^[16] It was shown that such variations in the speed of light can be removed by suitable coordinate transformations and field redefinitions, though the corresponding Lorentz violations cannot be removed, because such redefinitions only transfer those violations from the photon sector to the matter sector of SME.^[8] While ordinary symmetric optical resonators are suitable for testing even-parity effects and provide only tiny constraints on odd-parity effects, also asymmetric resonators have been built for the detection of odd-parity effects.^[16] For additional coefficients in the photon sector leading to birefringence of light in vacuum, which cannot be redefined as the other photon effects, see § Vacuum birefringence.

Another type of test of the ${\displaystyle {\tilde {\kappa }}_{o+}}$ related one-way light speed isotropy in combination with the electron sector of the SME was conducted by Bocquet et al. (2010).

).

Author	Year	RMS		SME
		Orientation	Velocity	${\displaystyle {\tilde {\kappa }}_{e-}}$	${\displaystyle {\tilde {\kappa }}_{o+}}$	${\displaystyle {\tilde {\kappa }}_{tr}}$
Michimura et al.[21]	2013				(0.7±1)×10−14	(−0.4±0.9)×10−10
Baynes et al.[22]	2012					(3±11)×10−10
Baynes et al.[23]	2011				(0.7±1.4)×10−12	(3.4±6.2)×10−9
Hohensee et al.[14]	2010			(0.8±0.6)×10−16	(−1.5±1.2)×10−12	(−1.50±0.74)×10−8
Bocquet et al.[17]	2010				≤1.6×10−14[24]
Herrmann et al.[25]	2009	(4±8)×10−12		(−0.31±0.73)×10−17	(−0.14±0.78)×10−13
Eisele et al.[26]	2009	(−1.6±6±1.2)×10−12		(0.0±1.0±0.3)×10−17	(1.5±1.5±0.2)×10−13
Tobar et al.[27]	2009		(−4.8±3.7)×10−8
Tobar et al.[28]	2009					(−0.3±3)×10−7
Müller et al.[29]	2007			(7.7±4.0)×10−16	(1.7±2.0)×10−12
Carone et al.[30]	2006					≲3×10−8[31]
Stanwix et al.[32]	2006	(9.4±8.1)×10−11		(−6.9±2.2)×10−16	(−0.9±2.6)×10−12
Herrmann et al.[33]	2005	(−2.1±1.9)×10−10		(−3.1±2.5)×10−16	(−2.5±5.1)×10−12
Stanwix et al.[34]	2005	(−0.9±2.0)×10−10		(−0.63±0.43)×10−15	(0.20±0.21)×10−11
Antonini et al.[35]	2005	(+0.5±3±0.7)×10−10		(−2.0±0.2)×10−14
Wolf et al.[36]	2004			(−5.7±2.3)×10−15	(−1.8±1.5)×10−11
Wolf et al.[37]	2004	(+1.2±2.2)×10−9	(3.7±3.0)×10−7
Müller et al.[38]	2003	(+2.2±1.5)×10−9		(1.7±2.6)×10−15	(14±14)×10−11
Lipa et al.[39]	2003			(1.4±1.4)×10−13	≤10−9
Wolf et al.[40]	2003	(+1.5±4.2)×10−9
Braxmaier et al.[41]	2002		(1.9±2.1)×10−5
Hils and Hall[42]	1990		6.6×10−5
Brillet and Hall[43]	1979	≲5×10−9		≲10−15

Solar System

Besides terrestrial tests also

In addition to the terrestrial Kennedy–Thorndike experiments mentioned above, Müller & Soffel (1995)[46] and Müller et al. (1999)^[47] tested the RMS velocity dependence parameter by searching for anomalous distance oscillations using LLR. Since time dilation is already confirmed to high precision, a positive result would prove that light speed depends on the observer's velocity and length contraction is direction dependent (like in the other Kennedy–Thorndike experiments). However, no anomalous distance oscillations have been observed, with a RMS velocity dependence limit of ${\displaystyle (-5\pm 12)\times 10^{-5}}$,^[47] comparable to that of Hils and Hall (1990, see table above on the right).

Another effect often discussed in connection with quantum gravity (QG) is the possibility of

${\displaystyle E_{\text{Pl}}\sim 1.22\times 10^{19}}$

coefficients ${\displaystyle k_{(V)00}^{(3)}}$ and ${\displaystyle k_{(V)00}^{(5)}}$ for Lorentz violation are given, 3 and 5 denote the mass dimensions employed. The latter corresponds to ${\displaystyle \xi }$ in the

${\textstyle k_{(V)00}^{(5)}={\frac {3{\sqrt {4\pi }}\xi }{5m_{\text{P}}}}}$

${\displaystyle m_{\text{P}}}$

Name	Year	SME bounds		EFT bound, ${\displaystyle \xi }$
		${\displaystyle k_{(V)00}^{(3)}}$ (GeV)	${\displaystyle k_{(V)00}^{(5)}}$ (GeV−1)
Götz et al.[64]	2013		≤5.9×10−35	≤3.4×10−16
Toma et al.[65]	2012		≤1.4×10−34	≤8×10−16
Laurent et al.[66]	2011		≤1.9×10−33	≤1.1×10−14
Stecker[63]	2011		≤4.2×10−34	≤2.4×10−15
Kostelecký et al.[12]	2009		≤1×10−32	≤9×10−14
QUaD[67]	2008	≤2×10−43
Kostelecký et al.[68]	2008	=(2.3±5.4)×10−43
Maccione et al.[69]	2008		≤1.5×10−28	≤9×10−10
Komatsu et al.[70]	2008	=(1.2±2.2)×10−43 [12]
Kahniashvili et al.[71]	2008	=(2.6±1.9)×10−43 [12]
Cabella et al.[72]	2007	=(2.5±3.0)×10−43 [12]
Fan et al.[73]	2007		≤3.4×10−26	≤2×10−7 [63]
Feng et al.[74]	2006	=(6.0±4.0)×10−43 [12]
Gleiser et al.[75]	2001		≤8.7×10−23	≤4×10−4 [63]
Carroll et al.[76]	1990	≤2×10−42

Lorentz violations could lead to differences between the speed of light and the limiting or maximal attainable speed (MAS) of any particle, whereas in special relativity the speeds should be the same. One possibility is to investigate otherwise forbidden effects at threshold energy in connection with particles having a charge structure (protons, electrons, neutrinos). This is because the dispersion relation is assumed to be modified in Lorentz violating EFT models such as SME. Depending on which of these particles travels faster or slower than the speed of light, effects such as the following can occur:^[77][78]

• Photon decay at superluminal speed. These (hypothetical) high-energy photons would quickly decay into other particles, which means that high energy light cannot propagate over long distances. So the mere existence of high energy light from astronomic sources constrains possible deviations from the limiting velocity.
• Vacuum Cherenkov radiation at superluminal speed of any particle (protons, electrons, neutrinos) having a charge structure. In this case, emission of Bremsstrahlung can occur, until the particle falls below threshold and subluminal speed is reached again. This is similar to the known Cherenkov radiation in media, in which particles are traveling faster than the phase velocity of light in that medium. Deviations from the limiting velocity can be constrained by observing high energy particles of distant astronomic sources that reach Earth.
• The rate of synchrotron radiation could be modified, if the limiting velocity between charged particles and photons is different.
• The Greisen–Zatsepin–Kuzmin limit could be evaded by Lorentz violating effects. However, recent measurements indicate that this limit really exists.

Since astronomic measurements also contain additional assumptions – like the unknown conditions at the emission or along the path traversed by the particles, or the nature of the particles –, terrestrial measurements provide results of greater clarity, even though the bounds are wider (the following bounds describe maximal deviations between the speed of light and the limiting velocity of matter):

By this kind of

These tests can be used to constrain deviations between the maximal attainable speed of matter and the speed of light,^[5] in particular with respect to the parameters of c_μν that are also used in the evaluations of the threshold effects mentioned above.^[81]

The classic

The current precision with which time dilation is measured (using the RMS test theory), is at the ~10⁻⁸ level. It was shown, that Ives-Stilwell type experiments are also sensitive to the ${\displaystyle {\tilde {\kappa }}_{tr}}$ isotropic light speed coefficient of the SME, as introduced above.[16] Chou et al. (2010) even managed to measure a frequency shift of ~10⁻¹⁶ due to time dilation, namely at everyday speeds such as 36 km/h.^[105]

Another fundamental symmetry of nature is CPT symmetry. It was shown that CPT violations lead to Lorentz violations in quantum field theory (even though there are nonlocal exceptions).^[110][111] CPT symmetry requires, for instance, the equality of mass, and equality of decay rates between matter and antimatter.

Modern tests by which CPT symmetry has been confirmed are mainly conducted in the neutral meson sector. In large particle accelerators, direct measurements of mass differences between top- and antitop-quarks have been conducted as well.

Using SME, also additional consequences of CPT violation in the neutral meson sector can be formulated.^[116] Other SME related CPT tests have been performed as well:

• Using
  cyclotron frequencies in proton-antiproton measurements, and couldn't find any deviation down to 9·10⁻¹¹.^[132]
• Hans Dehmelt et al. tested the anomaly frequency, which plays a fundamental role in the measurement of the electron's gyromagnetic ratio. They searched for sidereal variations, and differences between electrons and positrons as well. Eventually they found no deviations, thereby establishing bounds of 10⁻²⁴ GeV.^[133]
• Hughes et al. (2001) examined muons for sidereal signals in the spectrum of muons, and found no Lorentz violation down to 10⁻²³ GeV.^[134]
• The "Muon g-2" collaboration of the Brookhaven National Laboratory searched for deviations in the anomaly frequency of muons and anti-muons, and for sidereal variations under consideration of Earth's orientation. Also here, no Lorentz violations could be found, with a precision of 10⁻²⁴ GeV.^[135]

Lorentz violation bounds on Bhabha scattering have been given by Charneski et al. (2012).^[138] They showed that differential cross sections for the vector and axial couplings in QED become direction dependent in the presence of Lorentz violation. They found no indication of such an effect, placing upper limits on Lorentz violations of ${\displaystyle \scriptstyle \leq 10^{14}({\text{eV}})^{-1}}$.

The influence of Lorentz violation on gravitational fields and thus general relativity was analyzed as well. The standard framework for such investigations is the Parameterized post-Newtonian formalism (PPN), in which Lorentz violating preferred frame effects are described by the parameters ${\displaystyle \alpha _{1},\alpha _{2},\alpha _{3}}$ (see the

Also SME is suitable to analyze Lorentz violations in the gravitational sector. Bailey and Kostelecky (2006) constrained Lorentz violations down to ${\displaystyle 10^{-9}}$ by analyzing the perihelion shifts of Mercury and Earth, and down to ${\displaystyle 10^{-13}}$ in relation to solar spin precession.^[139] Battat et al. (2007) examined Lunar Laser Ranging data and found no oscillatory perturbations in the lunar orbit. Their strongest SME bound excluding Lorentz violation was ${\displaystyle (6.9\pm 4.5)\times 10^{-11}}$.^[140] Iorio (2012) obtained bounds at the ${\displaystyle 10^{-9}}$ level by examining Keplerian orbital elements of a test particle acted upon by Lorentz-violating gravitomagnetic accelerations.^[141] Xie (2012) analyzed the advance of periastron of binary pulsars, setting limits on Lorentz violation at the ${\displaystyle 10^{-10}}$ level.^[142]

Although neutrino oscillations have been experimentally confirmed, the theoretical foundations are still controversial, as it can be seen in the discussion related to sterile neutrinos. This makes predictions of possible Lorentz violations very complicated. It is generally assumed that neutrino oscillations require a certain finite mass. However, oscillations could also occur as a consequence of Lorentz violations, so there are speculations as to how much those violations contribute to the mass of the neutrinos.^[143]

Additionally, a series of investigations have been published in which a sidereal dependence of the occurrence of neutrino oscillations was tested, which could arise when there were a preferred background field. This, possible CPT violations, and other coefficients of Lorentz violations in the framework of SME, have been tested. Here, some of the achieved GeV bounds for the validity of Lorentz invariance are stated:

Since the discovery of neutrino oscillations, it is assumed that their speed is slightly below the speed of light. Direct velocity measurements indicated an upper limit for relative speed differences between light and neutrinos of ${\textstyle {|v-c|}/{c}<10^{-9}}$, see measurements of neutrino speed.

Also indirect constraints on neutrino velocity, on the basis of effective field theories such as SME, can be achieved by searching for threshold effects such as Vacuum Cherenkov radiation. For example, neutrinos should exhibit Bremsstrahlung in the form of electron-positron pair production.^[151] Another possibility in the same framework is the investigation of the decay of pions into muons and neutrinos. Superluminal neutrinos would considerably delay those decay processes. The absence of those effects indicate tight limits for velocity differences between light and neutrinos.^[152]

Velocity differences between neutrino flavors can be constrained as well. A comparison between muon- and electron-neutrinos by Coleman & Glashow (1998) gave a negative result, with bounds <6×10⁻²².^[9]

LSND, MiniBooNE

In 2001, the

In 2010, MINOS reported differences between the disappearance (and thus the masses) of neutrinos and antineutrinos at the 2.3 sigma level. This would violate CPT symmetry and Lorentz symmetry.[168]^[169][170] However, in 2011 MINOS updated their antineutrino results; after evaluating additional data, they reported that the difference is not as great as initially thought.^[171] In 2012, they published a paper in which they reported that the difference is now removed.^[172]

In 2007, the MAGIC Collaboration published a paper, in which they claimed a possible energy dependence of the speed of photons from the galaxy Markarian 501. They admitted, that also a possible energy-dependent emission effect could have cause this result as well.^[52][173] However, the MAGIC result was superseded by the substantially more precise measurements of the Fermi-LAT group, which couldn't find any effect even beyond the

In 1997, Nodland & Ralston claimed to have found a rotation of the polarization plane of light coming from distant radio galaxies. This would indicate an anisotropy of space.^{[174][175][176]} This attracted some interest in the media. However, some criticisms immediately appeared, which disputed the interpretation of the data, and who alluded to errors in the publication.^{[177][178][179][180][181][182][183]} More recent studies have not found any evidence for this effect (see section on Birefringence).

• Tests of special relativity
• Phenomenological quantum gravity