modern searches for Lorentz violation

Early models assessing the possibility of slight deviations from Lorentz invariance have been published between the 1960s and the 1990s. 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.{{Cite journal |author1=Haugan, Mark P. |author2=Will, Clifford M. |title=Modern tests of special relativity |journal=Physics Today |volume=40 |issue=5 |pages=69–86 |year=1987 |doi=10.1063/1.881074|bibcode = 1987PhT....40e..69H }}{{Cite journal |last=Will, C.M.|year=2006 |title=The Confrontation between General Relativity and Experiment |journal=Living Rev. Relativ. |volume =9 |issue=1 |page=12 |doi=10.12942/lrr-2006-3|doi-access=free |arxiv = gr-qc/0510072 |bibcode = 2006LRR.....9....3W |pmid=28179873 |pmc=5256066 }}
• Doubly special relativity (DSR) preserves the 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 Alternatives to general relativity such as Loop quantum gravity, Emergent gravity, Einstein aether theory, Hořava–Lifshitz gravity.

However, the Standard-Model Extension (SME) in which Lorentz violating effects are introduced by spontaneous symmetry breaking, is used for most modern analyses of experimental results. It was introduced by Kostelecký and colleagues in 1997 and the following years, containing all possible Lorentz and CPT violating coefficients not violating gauge symmetry.{{Cite journal |author1=Colladay, Don |author2=Kostelecký, V. Alan |title=CPT violation and the standard model |journal=Physical Review D |volume=55 |issue=11 |pages=6760–6774 |year=1997 |doi=10.1103/PhysRevD.55.6760 |arxiv=hep-ph/9703464 |bibcode = 1997PhRvD..55.6760C |s2cid=7651433 }}{{Cite journal |author1=Colladay, Don |author2=Kostelecký, V. Alan |title=Lorentz-violating extension of the standard model |journal=Physical Review D |volume=58 |issue=11 |pages=116002 |year=1998 |doi=10.1103/PhysRevD.58.116002|arxiv=hep-ph/9809521|bibcode = 1998PhRvD..58k6002C |s2cid=4013391 }} It includes not only special relativity, but the standard model and general relativity as well. Models whose parameters can be related to SME and thus can be seen as special cases of it, include the older RMS and c² models,{{cite journal|author1=Kostelecký, V. Alan |author2=Mewes, Matthew |title=Signals for Lorentz violation in electrodynamics|journal=Physical Review D|volume=66|issue=5|pages=056005|year=2002 |doi=10.1103/PhysRevD.66.056005 |arxiv=hep-ph/0205211|bibcode = 2002PhRvD..66e6005K |s2cid=21309077 }} the Coleman-Glashow model confining the SME coefficients to dimension 4 operators and rotation invariance,{{cite journal |author1=Coleman, Sidney |author2=Glashow, Sheldon L. |title=High-energy tests of Lorentz invariance|journal=Physical Review D|volume=59|issue=11|pages=116008|year=1999|doi=10.1103/PhysRevD.59.116008|arxiv=hep-ph/9812418|bibcode = 1999PhRvD..59k6008C |s2cid=1273409 }} and the Gambini-Pullin model{{Cite journal |author1=Gambini, Rodolfo |author2=Pullin, Jorge |title=Nonstandard optics from quantum space-time |journal=Physical Review D |volume=59 |issue=12 |pages=124021 |year=1999 |doi=10.1103/PhysRevD.59.124021|arxiv=gr-qc/9809038 |bibcode = 1999PhRvD..59l4021G |s2cid=32965963 }} or the Myers-Pospelov model{{Cite journal |author1=Myers, Robert C. |author2=Pospelov, Maxim |title=Ultraviolet Modifications of Dispersion Relations in Effective Field Theory |journal=Physical Review Letters |volume=90 |issue=21 |pages=211601 |year=2003 |arxiv=hep-ph/0301124|doi=10.1103/PhysRevLett.90.211601 |pmid=12786546 |bibcode=2003PhRvL..90u1601M|s2cid=37525861 }} corresponding to dimension 5 or higher operators of SME.{{cite journal |author1=Kostelecký, V. Alan |author2=Mewes, Matthew |title=Electrodynamics with Lorentz-violating operators of arbitrary dimension|journal=Physical Review D|volume=80|issue=1|year=2009|pages=015020 |doi=10.1103/PhysRevD.80.015020 |arxiv=0905.0031|bibcode=2009PhRvD..80a5020K|s2cid=119241509 }}

Many terrestrial experiments have been conducted, mostly with optical resonators or in particle accelerators, by which deviations from the isotropy of the speed of light are tested. Anisotropy parameters are given, for instance, by the Robertson-Mansouri-Sexl test theory (RMS). This allows for distinction between the relevant orientation and velocity dependent parameters. In modern variants of the Michelson–Morley experiment, the dependence of light speed on the orientation of the apparatus and the relation of longitudinal and transverse lengths of bodies in motion is analyzed. Also modern variants of the Kennedy–Thorndike experiment, by which the dependence of light speed on the velocity of the apparatus and the relation of time dilation and length contraction is analyzed, have been conducted; the recently reached limit for Kennedy-Thorndike test yields 7 10⁻¹².{{cite journal |author1=Gurzadyan, V.G. |author2=Margaryan, A.T. |title=The light speed versus the observer: the Kennedy–Thorndike test from GRAAL-ESRF|journal=Eur. Phys. J. C |volume=78|issue=8 |year=2018|pages=607|doi=10.1140/epjc/s10052-018-6080-x|arxiv=1807.08551|bibcode=2018EPJC...78..607G |s2cid=119374401 }} The current precision, by which an anisotropy of the speed of light can be excluded, is at the 10⁻¹⁷ level. This is related to the relative velocity between the Solar System and the rest frame of the cosmic microwave background radiation of ~368 km/s (see also Resonator Michelson–Morley experiments).

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) $\backslash tilde\{\backslash kappa\}\_\{e-\}$, $\backslash tilde\{\backslash kappa\}\_\{o+\}$ and $\backslash tilde\{\backslash kappa\}\_\{tr\}$. They can be interpreted as follows: $\backslash tilde\{\backslash kappa\}\_\{e-\}$ represent anisotropic shifts in the two-way (forward and backwards) speed of light, $\backslash tilde\{\backslash kappa\}\_\{o+\}$ represent anisotropic differences in the one-way speed of counterpropagating beams along an axis,{{cite journal|author=Hohensee|title=Improved constraints on isotropic shift and anisotropies of the speed of light using rotating cryogenic sapphire oscillators|journal=Physical Review D|volume=82|issue=7|pages=076001|year=2010 |doi=10.1103/PhysRevD.82.076001 |arxiv=1006.1376|bibcode = 2010PhRvD..82g6001H |s2cid=2612817|display-authors=etal}}{{cite arXiv|author=Hohensee|title=Covariant Quantization of Lorentz-Violating Electromagnetism|year=2010|class=quant-ph |eprint=1210.2683|display-authors=etal}}; Standalone version of work included in the Ph.D. Thesis of M.A. Hohensee. and $\backslash tilde\{\backslash kappa\}\_\{tr\}$ represent isotropic (orientation-independent) shifts in the one-way phase velocity of light.{{cite journal|author=Tobar|title=New methods of testing Lorentz violation in electrodynamics|journal=Physical Review D|volume=71|issue=2|pages=025004|year=2005 |doi=10.1103/PhysRevD.71.025004 |arxiv=hep-ph/0408006|bibcode = 2005PhRvD..71b5004T |display-authors=etal}} 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. 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. For additional coefficients in the photon sector leading to birefringence of light in vacuum, which cannot be redefined as the other photon effects, see {{section link|#Vacuum birefringence}}.

Another type of test of the $\backslash tilde\{\backslash kappa\}\_\{o+\}$ related one-way light speed isotropy in combination with the electron sector of the SME was conducted by Bocquet et al. (2010). They searched for fluctuations in the 3-momentum of photons during Earth's rotation, by measuring the Compton scattering of ultrarelativistic electrons on monochromatic laser photons in the frame of the cosmic microwave background radiation, as originally suggested by Vahe Gurzadyan and Amur Margarian {{cite journal|author1=Gurzadyan, V. G. |author2=Margarian, A. T. |title=Inverse Compton testing of fundamental physics and the cosmic background radiation |journal=Physica Scripta |volume=53 |issue=5 |pages=513–515 |year=1996|doi=10.1088/0031-8949/53/5/001|bibcode=1996PhyS...53..513G|s2cid=250775347 }} (for details on that 'Compton Edge' method and analysis see,{{cite journal |author=Gurzadyan, V. G. |title=A new limit on the light speed isotropy from the GRAAL experiment at the ESRF|journal=Proc. 12th M.Grossmann Meeting on General Relativity|volume=B|pages=1495–1499|isbn=978-981-4374-51-4|year=2012 |bibcode=2012mgm..conf.1495G |arxiv=1004.2867|display-authors=etal|doi=10.1142/9789814374552_0255 |s2cid=119219661}} discussion e.g.{{cite journal|author=Lingli Zhou|author2=Bo-Qiang Ma|title=A theoretical diagnosis on light speed anisotropy from GRAAL experiment |journal=Astroparticle Physics |volume=36|issue=1 |pages=37–41|year=2012 |doi=10.1016/j.astropartphys.2012.04.015|arxiv=1009.1675 |bibcode=2012APh....36...37Z |s2cid=118625197}}).

Besides terrestrial tests also astrometric tests using Lunar Laser Ranging (LLR), i.e. sending laser signals from Earth to Moon and back, have been conducted. They are ordinarily used to test general relativity and are evaluated using the Parameterized post-Newtonian formalism.{{cite journal|author1=Williams, James G. |author2=Turyshev, Slava G. |author3=Boggs, Dale H. |title=Lunar Laser Ranging Tests of the Equivalence Principle with the Earth and Moon|journal=International Journal of Modern Physics D|volume=18|issue=7|year=2009|pages=1129–1175|doi=10.1142/S021827180901500X|bibcode = 2009IJMPD..18.1129W|arxiv=gr-qc/0507083|s2cid=119086896 }} However, since these measurements are based on the assumption that the speed of light is constant, they can also be used as tests of special relativity by analyzing potential distance and orbit oscillations. For instance, Zoltán Lajos Bay and White (1981) demonstrated the empirical foundations of the Lorentz group and thus special relativity by analyzing the planetary radar and LLR data.{{cite journal|author1=Bay, Z. |author2=White, J. A. |title=Radar astronomy and the special theory of relativity|journal=Acta Physica Academiae Scientiarum Hungaricae|volume=51|issue=3|year=1981|pages=273–297|doi=10.1007/BF03155586|bibcode =1981AcPhy..51..273B|s2cid=119362077 }}

In addition to the terrestrial Kennedy–Thorndike experiments mentioned above, Müller & Soffel (1995){{cite journal|author1=Müller, J. |author2=Soffel, M. H. |title=A Kennedy-Thorndike experiment using LLR data|journal=Physics Letters A|volume=198|year=1995|pages=71–73|doi=10.1016/0375-9601(94)01001-B|issue=2|bibcode = 1995PhLA..198...71M }} and Müller et al. (1999){{cite journal|author=Müller, J. |author2=Nordtvedt, K. |author3=Schneider, M. |author4=Vokrouhlicky, D.|title=Improved Determination of Relativistic Quantities from LLR|journal=Proceedings of the 11th International Workshop on Laser Ranging Instrumentation|volume=10|year=1999 |pages=216–222|url=http://cddis.gsfc.nasa.gov/lw11/docs/lrw_llrpan.pdf}} 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 $(-5\backslash pm12)\backslash times10^\{-5\}$, 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 dispersion of light in vacuum (i.e. the dependence of light speed on photon energy), due to Lorentz-violating dispersion relations. This effect should be strong at energy levels comparable to, or beyond the Planck energy $E\_\backslash text\{Pl\}\; \backslash sim\; 1.22\; \backslash times\; 10^\{19\}$ GeV, while being extraordinarily weak at energies accessible in the laboratory or observed in astrophysical objects. In an attempt to observe a weak dependence of speed on energy, light from distant astrophysical sources such as gamma ray bursts and distant galaxies has been examined in many experiments. Especially the Fermi-LAT group was able show that no energy dependence and thus no observable Lorentz violation occurs in the photon sector even beyond the Planck energy, which excludes a large class of Lorentz-violating quantum gravity models.

Lorentz violating dispersion relations due to the presence of an anisotropic space might also lead to vacuum birefringence and parity violations. For instance, the polarization plane of photons might rotate due to velocity differences between left- and right-handed photons. In particular, gamma ray bursts, galactic radiation, and the cosmic microwave background radiation are examined. The SME coefficients $k\_\{(V)00\}^\{(3)\}$ and $k\_\{(V)00\}^\{(5)\}$ for Lorentz violation are given, 3 and 5 denote the mass dimensions employed. The latter corresponds to $\backslash xi$ in the EFT of Meyers and Pospelov by $k\_\{(V)00\}^\{(5)\}\; =\; \backslash frac\{3\backslash sqrt\{4\backslash pi\}\backslash xi\}\{5m\_\{\backslash text\{P\}\}\}$, $m\_\backslash text\{P\}$ being the Planck mass.

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:{{cite journal |author=Jacobson|title=Threshold effects and Planck scale Lorentz violation: Combined constraints from high energy astrophysics|journal=Physical Review D|volume=67|issue=12|year=2002|pages=124011 |doi=10.1103/PhysRevD.67.124011|arxiv=hep-ph/0209264|bibcode = 2003PhRvD..67l4011J |s2cid=119452240|display-authors=etal}}

• 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 spectroscopy experiments – sometimes called Hughes–Drever experiments as well – violations of Lorentz invariance in the interactions of protons and neutrons are tested by studying the energy levels of those nucleons in order to find anisotropies in their frequencies ("clocks"). Using spin-polarized torsion balances, also anisotropies with respect to electrons can be examined. Methods used mostly focus on vector spin interactions and tensor interactions, and are often described in CPT odd/even SME terms (in particular parameters of b_μ and c_μν).{{cite journal|author1=Kostelecký, V. Alan |author2=Lane, Charles D. |year=1999|title=Constraints on Lorentz violation from clock-comparison experiments|journal=Physical Review D|volume=60|issue=11|pages=116010|doi=10.1103/PhysRevD.60.116010|arxiv=hep-ph/9908504|bibcode = 1999PhRvD..60k6010K |s2cid=119039071 }} Such experiments are currently the most sensitive terrestrial ones, because the precision by which Lorentz violations can be excluded lies at the 10⁻³³ GeV level.

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

The classic time dilation experiments such as the Ives–Stilwell experiment, the Moessbauer rotor experiments, and the time dilation of moving particles, have been enhanced by modernized equipment. For example, the Doppler shift of lithium ions traveling at high speeds is evaluated by using saturated spectroscopy in heavy ion storage rings. For more information, see Modern Ives–Stilwell experiments.

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 $\backslash tilde\{\backslash kappa\}\_\{tr\}$ isotropic light speed coefficient of the SME, as introduced above. 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.{{cite journal|author=Chou| year= 2010| title=Optical Clocks and Relativity | journal=Science| volume = 329 | issue = 5999 | pages=1630–1633 | doi =10.1126/science.1192720 |bibcode = 2010Sci...329.1630C | pmid=20929843| s2cid= 206527813| url= https://zenodo.org/record/1230910|display-authors=etal}}

{{Main|Antimatter tests of Lorentz violation|Experimental testing of time dilation#Time dilation and CPT symmetry}}

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).{{Cite journal |author=Greenberg, O. W. |title=CPT Violation Implies Violation of Lorentz Invariance |journal=Physical Review Letters|volume=89|issue=23|pages=231602|year=2002|arxiv=hep-ph/0201258|pmid=12484997|s2cid=9409237 |doi=10.1103/PhysRevLett.89.231602 |bibcode=2002PhRvL..89w1602G}}{{Cite arXiv |author=Greenberg, O. W.|title=Remarks on a challenge to the relation between CPT and Lorentz violation |eprint=1105.0927 |year=2011|class=hep-ph }} 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.

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

• Using Penning traps in which individual charged particles and their counterparts are trapped, Gabrielse et al. (1999) examined cyclotron frequencies in proton-antiproton measurements, and couldn't find any deviation down to 9·10⁻¹¹.{{cite journal |author=Gabrielse|s2cid=55054189|title=Precision Mass Spectroscopy of the Antiproton and Proton Using Simultaneously Trapped Particles|journal=Physical Review Letters|volume=82|issue=16 |year=1999|doi=10.1103/PhysRevLett.82.3198|pages=3198–3201|bibcode=1999PhRvL..82.3198G|display-authors=etal}}
• 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.{{cite journal |author=Dehmelt|title=Past Electron-Positron g-2 Experiments Yielded Sharpest Bound on CPT Violation for Point Particles|journal=Physical Review Letters|volume=83|issue=23 |year=1999|doi=10.1103/PhysRevLett.83.4694|pages=4694–4696|arxiv=hep-ph/9906262|bibcode=1999PhRvL..83.4694D|s2cid=116195114|display-authors=etal}}
• Hughes et al. (2001) examined muons for sidereal signals in the spectrum of muons, and found no Lorentz violation down to 10⁻²³ GeV.{{cite journal |author=Hughes|title=Test of CPT and Lorentz Invariance from Muonium Spectroscopy|journal=Physical Review Letters|volume=87|issue=11 |year=2001|doi=10.1103/PhysRevLett.87.111804|pages=111804|arxiv=hep-ex/0106103|bibcode=2001PhRvL..87k1804H|display-authors=etal|pmid=11531514|s2cid=119501031}}
• 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.{{cite journal |author=Bennett|title=Search for Lorentz and CPT Violation Effects in Muon Spin Precession|journal=Physical Review Letters|volume=100|issue=9 |year=2008|doi=10.1103/PhysRevLett.100.091602|pages=091602|arxiv=0709.4670|bibcode=2008PhRvL.100i1602B|display-authors=etal|pmid=18352695|s2cid=26270066}}

Third generation particles have been examined for potential Lorentz violations using SME. For instance, Altschul (2007) placed upper limits on Lorentz violation of the tau of 10⁻⁸, by searching for anomalous absorption of high energy astrophysical radiation.{{Cite journal|author=Altschul, Brett|title=Astrophysical limits on Lorentz violation for all charged species|journal=Astroparticle Physics|volume=28|issue=3|pages=380–384|arxiv=hep-ph/0610324 |doi=10.1016/j.astropartphys.2007.08.003|year=2007|bibcode=2007APh....28..380A |s2cid=16235612}} In the BaBar experiment (2007), the D0 experiment (2015), and the LHCb experiment (2016), searches have been made for sidereal variations during Earth's rotation using B mesons (thus bottom quarks) and their antiparticles. No Lorentz and CPT violating signal were found with upper limits in the range 10⁻¹⁵ − 10⁻¹⁴ GeV.

Also top quark pairs have been examined in the D0 experiment (2012). They showed that the cross section production of these pairs doesn't depend on sidereal time during Earth's rotation.{{Cite journal|author=D0 Collaboration|title=Search for violation of Lorentz invariance in top quark pair production and decay|journal=Physical Review Letters|volume=108|issue=26|pages=261603|arxiv=1203.6106 |doi=10.1103/PhysRevLett.108.261603|year=2012|bibcode=2012PhRvL.108z1603A|pmid=23004960|s2cid=11077644}}

Lorentz violation bounds on Bhabha scattering have been given by Charneski et al. (2012).{{Cite journal|author=Charneski|title=Lorentz violation bounds on Bhabha scattering|journal=Physical Review D |volume=86|issue=4 |pages=045003|year=2012|arxiv=1204.0755|doi=10.1103/PhysRevD.86.045003 |bibcode=2012PhRvD..86d5003C|s2cid=119276343|display-authors=etal}} 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 $\backslash scriptstyle\backslash leq10^\{14\}(\backslash text\{eV\})^\{-1\}$.

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 $\backslash alpha\_1,\; \backslash alpha\_2,\; \backslash alpha\_3$ (see the PPN article on observational bounds on these parameters). Lorentz violations are also discussed in relation to Alternatives to general relativity such as Loop quantum gravity, Emergent gravity, Einstein aether theory or Hořava–Lifshitz gravity.

Also SME is suitable to analyze Lorentz violations in the gravitational sector. Bailey and Kostelecky (2006) constrained Lorentz violations down to $10^\{-9\}$ by analyzing the perihelion shifts of Mercury and Earth, and down to $10^\{-13\}$ in relation to solar spin precession.{{cite journal|author1=Bailey, Quentin G. |author2=Kostelecký, V. Alan |title=Signals for Lorentz violation in post-Newtonian gravity|journal=Physical Review D|volume=74|issue=4|pages=045001|year=2006|arxiv=gr-qc/0603030|doi=10.1103/PhysRevD.74.045001|bibcode = 2006PhRvD..74d5001B |s2cid=26268407 }} 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 $(6.9\backslash pm4.5)\; \backslash times\; 10^\{-11\}$.{{cite journal | author1=Battat, James B. R. |author2=Chandler, John F. |author3=Stubbs, Christopher W. |title=Testing for Lorentz Violation: Constraints on Standard-Model-Extension Parameters via Lunar Laser Ranging|journal=Physical Review Letters| volume=99| issue=24|year=2007|pages=241103|doi=10.1103/PhysRevLett.99.241103|bibcode =2007PhRvL..99x1103B|arxiv=0710.0702 | pmid=18233436| s2cid=19661431 }} Iorio (2012) obtained bounds at the $10^\{-9\}$ level by examining Keplerian orbital elements of a test particle acted upon by Lorentz-violating gravitomagnetic accelerations.{{cite journal|author=Iorio, L.|title=Orbital effects of Lorentz-violating standard model extension gravitomagnetism around a static body: a sensitivity analysis|journal=Classical and Quantum Gravity|volume=29|issue=17|pages=175007|year=2012|arxiv=1203.1859| doi=10.1088/0264-9381/29/17/175007|bibcode = 2012CQGra..29q5007I |s2cid=118516169}} Xie (2012) analyzed the advance of periastron of binary pulsars, setting limits on Lorentz violation at the $10^\{-10\}$ level.{{cite journal | author=Xie, Yi|title=Testing Lorentz violation with binary pulsars: constraints on standard model extension| journal = Research in Astronomy and Astrophysics|volume=13|issue=1|pages=1–4|year=2012|arxiv=1208.0736| doi=10.1088/1674-4527/13/1/001 | bibcode = 2013RAA....13....1X |s2cid=118469165}}

{{Main|Lorentz-violating neutrino oscillations}}

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.{{cite journal |author1=Díaz, Jorge S. |author2=Kostelecký, V. Alan |title=Lorentz- and CPT-violating models for neutrino oscillations|journal=Physical Review D|volume=85|issue=1 |year=2012|doi=10.1103/PhysRevD.85.016013|pages=016013|arxiv=1108.1799|bibcode = 2012PhRvD..85a6013D |s2cid=55890338 }}

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 $$

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.{{Cite journal|author=Mattingly|title=Possible cosmogenic neutrino constraints on Planck-scale Lorentz violation|journal=Journal of Cosmology and Astroparticle Physics|issue=2|pages=007| doi=10.1088/1475-7516/2010/02/007|arxiv=0911.0521|year=2010|bibcode = 2010JCAP...02..007M|volume=2010|s2cid=118457258|display-authors=etal}} 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.{{Cite journal|author1=Kostelecky, Alan |author2=Mewes, Matthew |title=Neutrinos with Lorentz-violating operators of arbitrary dimension|journal=Physical Review D|volume=85|issue=9|at=096005 |doi=10.1103/PhysRevD.85.096005|arxiv=1112.6395|date=May 25, 2012|bibcode = 2012PhRvD..85i6005K |s2cid=118474142 }}

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{{e|-22}}.

; LSND, MiniBooNE

In 2001, the LSND experiment observed a 3.8σ excess of antineutrino interactions in neutrino oscillations, which contradicts the standard model.{{cite journal |author=LSND Collaboration |title=Evidence for neutrino oscillations from the observation of ν¯e appearance in a ν¯μ beam |year=2001 |journal=Physical Review D |volume=64 |issue=11 |pages=112007 |arxiv=hep-ex/0104049 |doi=10.1103/PhysRevD.64.112007|bibcode =2001PhRvD..64k2007A|s2cid=118686517 }} First results of the more recent MiniBooNE experiment appeared to exclude this data above an energy scale of 450 MeV, but they had checked neutrino interactions, not antineutrino ones.{{cite journal |author=MiniBooNE Collaboration |title=Search for Electron Neutrino Appearance at the Δm2˜1eV2 Scale |year=2007|journal=Physical Review Letters |volume=98 |issue=23 |pages=231801 |arxiv=0704.1500 |doi=10.1103/PhysRevLett.98.231801|bibcode =2007PhRvL..98w1801A |pmid=17677898|s2cid=119315296 }} In 2008, however, they reported an excess of electron-like neutrino events between 200 and 475 MeV.{{cite journal |author=MiniBooNE Collaboration |title=Unexplained Excess of Electronlike Events from a 1-GeV Neutrino Beam |year=2009|journal=Physical Review Letters |volume=102 |issue=10 |pages=101802 |arxiv=0812.2243 |doi=10.1103/PhysRevLett.102.101802|bibcode =2009PhRvL.102j1802A |pmid=19392103|s2cid=3067551 }} And in 2010, when carried out with antineutrinos (as in LSND), the result was in agreement with the LSND result, that is, an excess at the energy scale from 450 to 1250 MeV was observed.{{cite web|title=MiniBooNE results suggest antineutrinos act differently|url =http://www.fnal.gov/pub/today/archive_2010/today10-06-18_readmore.html|publisher=Fermilab today |accessdate=14 December 2011 |date=June 18, 2010}}{{cite journal |author=MiniBooNE Collaboration |title=Event Excess in the MiniBooNE Search for ν¯μ→ν¯e Oscillations |year=2010 |journal=Physical Review Letters |volume=105 |issue=18 |pages=181801 |arxiv=1007.1150 |doi=10.1103/PhysRevLett.105.181801|pmid=21231096 |bibcode =2010PhRvL.105r1801A|s2cid=125243279 }} Whether those anomalies can be explained by sterile neutrinos, or whether they indicate Lorentz violations, is still discussed and subject to further theoretical and experimental researches.{{cite journal |author=Diaz, Jorge S. |title=Overview of Lorentz Violation in Neutrinos |year=2011 |journal=Proceedings of the DPF-2011 Conference |arxiv=1109.4620 |bibcode =2011arXiv1109.4620D}}

In 2011 the OPERA Collaboration published (in a non-peer reviewed arXiv preprint) the results of neutrino measurements, according to which neutrinos were traveling slightly faster than light.{{cite arXiv|author=OPERA collaboration |title=Measurement of the neutrino velocity with the OPERA detector in the CNGS beam|arxiv=1109.4897 |class=hep-ex |year=2011}} The neutrinos apparently arrived early by ~60 ns. The standard deviation was 6σ, clearly beyond the 5σ limit necessary for a significant result. However, in 2012 it was found that this result was due to measurement errors. The result was consistent with the speed of light;{{cite journal|author=OPERA collaboration |title=Measurement of the neutrino velocity with the OPERA detector in the CNGS beam|arxiv=1109.4897 |year=2012|bibcode=2012JHEP...10..093A |doi=10.1007/JHEP10(2012)093 |volume=2012|issue=10|journal=Journal of High Energy Physics|page=93|s2cid=17652398}} see Faster-than-light neutrino anomaly.

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.{{cite web|title=New measurements from Fermilab's MINOS experiment suggest a difference in a key property of neutrinos and antineutrinos|url =http://www.fnal.gov/pub/presspass/press_releases/minos-antineutrinos-20100614.html|publisher=Fermilab press release |accessdate=14 December 2011 |date=June 14, 2010}}{{cite journal |author=MINOS Collaboration |title=First Direct Observation of Muon Antineutrino Disappearance |year=2011 |journal=Physical Review Letters |volume=107 |issue=2 |pages=021801 |arxiv=1104.0344 |doi=10.1103/PhysRevLett.107.021801|bibcode=2011PhRvL.107b1801A |pmid=21797594 |s2cid=14782259}}{{cite journal |author=MINOS Collaboration |title=Search for the disappearance of muon antineutrinos in the NuMI neutrino beam |year=2011 |journal=Physical Review D |volume=84 |issue=7 |pages=071103 |arxiv=1108.1509 |doi=10.1103/PhysRevD.84.071103|bibcode =2011PhRvD..84g1103A|s2cid=6250231 }} However, in 2011 MINOS updated their antineutrino results; after evaluating additional data, they reported that the difference is not as great as initially thought.{{cite web|title=Surprise difference in neutrino and antineutrino mass lessening with new measurements from a Fermilab experiment|url =http://www.fnal.gov/pub/presspass/press_releases/2011/minos-antineutrinos-20110825.html|publisher=Fermilab press release |accessdate=14 December 2011 |date=August 25, 2011}} In 2012, they published a paper in which they reported that the difference is now removed.{{cite journal |author=MINOS Collaboration |title=An improved measurement of muon antineutrino disappearance in MINOS |journal=Physical Review Letters|volume=108|issue=19|pages=191801|year=2012 |arxiv=1202.2772|doi=10.1103/PhysRevLett.108.191801 |bibcode=2012PhRvL.108s1801A |pmid=23003026|s2cid=7735148 }}

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.

{{cite magazine

|title=Hints of a breakdown of relativity theory?

|url = http://www.scientificamerican.com/blog/post.cfm?id=hints-of-a-breakdown-of-relativity

|magazine=Scientific American

|author=George Musser

|accessdate=15 October 2011

|date=22 August 2007

}}

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 Planck energy. For details, see section Dispersion.

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.{{cite journal |author1=Nodland, Borge |author2=Ralston, John P. |title=Indication of Anisotropy in Electromagnetic Propagation over Cosmological Distances|journal=Physical Review Letters|volume=78|issue=16|year=1997|pages=3043–3046 |doi=10.1103/PhysRevLett.78.3043|arxiv=astro-ph/9704196|bibcode=1997PhRvL..78.3043N|s2cid=119410346 }}{{cite journal |author1=Nodland, Borge |author2=Ralston, John P. |title=Nodland and Ralston Reply|journal=Physical Review Letters|volume=79|issue=10|year=1997|pages=1958–1959|doi=10.1103/PhysRevLett.79.1958|arxiv=astro-ph/9705190|bibcode=1997PhRvL..79.1958N|s2cid=119418317 }}Borge Nodland, John P. Ralston (1997), Response to Leahy's Comment on the Data's Indication of Cosmological Birefringence, {{arxiv|astro-ph/9706126}}

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.{{cite web |author=J.P. Leahy |date=1997-09-16 |title=Is the Universe Screwy? |url=http://www.jb.man.ac.uk/~jpl/screwy.html}}{{cite web |author=Ted Bunn |url=https://facultystaff.richmond.edu/~ebunn/biref/ |title=Is the Universe Birefringent?}}{{cite journal |author1=Eisenstein, Daniel J. |author2=Bunn, Emory F. |title=Appropriate Null Hypothesis for Cosmological Birefringence|journal=Physical Review Letters|volume=79 |issue=10|year=1997|pages=1957–1958|s2cid=117874561 |doi=10.1103/PhysRevLett.79.1957|arxiv=astro-ph/9704247 |bibcode=1997PhRvL..79.1957E}}{{cite journal |author1=Carroll, Sean M. |author2=Field, George B. |title=Is There Evidence for Cosmic Anisotropy in the Polarization of Distant Radio Sources?|journal=Physical Review Letters|volume=79|issue=13|year=1997|pages=2394–2397|arxiv=astro-ph/9704263 |doi=10.1103/PhysRevLett.79.2394|bibcode=1997PhRvL..79.2394C|s2cid=13943605}}J. P. Leahy: (1997) Comment on the Measurement of Cosmological Birefringence, {{arxiv|astro-ph/9704285}}{{cite journal |author=Wardle|title=Observational Evidence against Birefringence over Cosmological Distances |journal=Physical Review Letters|volume=79|issue=10|year=1997|pages=1801–1804|arxiv=astro-ph/9705142 |doi=10.1103/PhysRevLett.79.1801|bibcode=1997PhRvL..79.1801W|s2cid=8589632|display-authors=etal}}{{cite journal |author=Loredo|title=Bayesian analysis of the polarization of distant radio sources: Limits on cosmological birefringence|journal=Physical Review D|volume=56|issue=12|year=1997|pages=7507–7512 |doi=10.1103/PhysRevD.56.7507 |arxiv=astro-ph/9706258|bibcode=1997PhRvD..56.7507L |s2cid=119372269|display-authors=etal}}

More recent studies have not found any evidence for this effect (see section on Birefringence).

• Tests of special relativity
• Phenomenological quantum gravity

{{reflist}}

• Kostelecký: [http://www.physics.indiana.edu/~kostelec/faq.html Background information on Lorentz and CPT violation]
• Roberts, Schleif (2006); Relativity FAQ: [http://math.ucr.edu/home/baez/physics/Relativity/SR/experiments.html What is the experimental basis of special relativity?]

{{Tests of special relativity}}

Category:Physics experiments

Category:Tests of special relativity