View source for Modern searches for Lorentz violation

{{Short description|Tests of special relativity}}
{{see also|Tests of special relativity}}
[[Image:GRB080319B illustration NASA.jpg|thumb|200px|Measurements on light from [[gamma-ray burst]]s show that the speed of light does not vary with energy]]
'''Modern searches for Lorentz violation''' are scientific studies that look for deviations from [[Lorentz covariance|Lorentz invariance or symmetry]], a set of fundamental frameworks that underpin modern [[science]] and fundamental [[physics]] in particular.  These studies try to determine whether violations or exceptions might exist for well-known [[physical law]]s such as [[special relativity]] and [[CPT symmetry]], as predicted by some variations of [[quantum gravity]], [[string theory]], and some [[alternatives to general relativity]].

Lorentz violations concern the fundamental predictions of special relativity, such as the [[principle of relativity]], the constancy of the [[speed of light]] in all [[inertial frames of reference]], and [[time dilation]], as well as the predictions of the [[standard model]] of [[particle physics]].  To assess and predict possible violations, [[test theories of special relativity]] and [[effective field theory|effective field theories]] (EFT) such as the [[Standard-Model Extension]] (SME) have been invented. These models introduce Lorentz and CPT violations through [[spontaneous symmetry breaking]] caused by hypothetical background fields, resulting in some sort of [[preferred frame]] effects. This could lead, for instance, to modifications of the [[dispersion relation]], causing differences between the maximal attainable speed of matter and the speed of light.

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).<ref name=mattingly>{{Cite journal| author=Mattingly, David| title=Modern Tests of Lorentz Invariance|journal=Living Rev. Relativ.| volume=8| year=2005| issue=5| pages=5| doi=10.12942/lrr-2005-5| doi-access=free| arxiv=gr-qc/0502097| bibcode=2005LRR.....8....5M| pmid=28163649| pmc=5253993}}</ref> For a detailed list of results of recent experimental searches, see Kostelecký and Russell (2008–2013).<ref name=kost0801>{{cite journal |first1=V.A. |last1=Kostelecky |first2=N. |last2=Russell |title=Data tables for Lorentz and ''CPT'' violation |year=2011 |journal=Reviews of Modern Physics |volume=83 |issue=1 |pages=11–31 |arxiv=0801.0287 |doi=10.1103/RevModPhys.83.11|bibcode=2011RvMP...83...11K |s2cid=3236027}}</ref> For a recent overview and history of Lorentz violating models, see Liberati (2013).<ref name=liberati>{{cite journal |author=Liberati, S. |title=Tests of Lorentz invariance: a 2013 update |year=2013 |journal=Classical and Quantum Gravity |volume=30 |issue=13 |pages=133001 |arxiv=1304.5795 |doi=10.1088/0264-9381/30/13/133001|bibcode=2013CQGra..30m3001L |s2cid=119261793}}</ref>

== Assessing Lorentz invariance violations ==
Early models assessing the possibility of slight deviations from Lorentz invariance have been published between the 1960s and the 1990s.<ref name=liberati /> In addition, a series of [[test theories of special relativity]] and [[effective field theory|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 [[Test theories of special relativity|Robertson-Mansouri-Sexl framework]] (RMS) contains three parameters, indicating deviations in the speed of light with respect to a preferred frame of reference.
* The c<sup>2</sup> 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.<ref>{{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 }}</ref><ref name=will06>{{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 }}</ref>
*[[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 violation|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]], [[Induced 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 [[Alan Kostelecký|Kostelecký]] and colleagues in 1997 and the following years, containing all possible Lorentz and CPT violating coefficients not violating [[Gauge theory|gauge symmetry]].<ref>{{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 }}</ref><ref>{{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 }}</ref> 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 c<sup>2</sup> models,<ref name=kost0205>{{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 }}</ref> the [[Sidney Coleman|Coleman]]-[[Sheldon Glashow|Glashow]] model confining the SME coefficients to dimension 4 operators and rotation invariance,<ref name=glas98>{{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 }}</ref> and the [[Rodolfo Gambini|Gambini]]-[[Jorge Pullin|Pullin]] model<ref>{{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 }}</ref> or the Myers-Pospelov model<ref name=meyers>{{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 }}</ref> corresponding to dimension 5 or higher operators of SME.<ref name=kost0905>{{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 }}</ref>

== Speed of light ==

=== Terrestrial ===
Many terrestrial experiments have been conducted, mostly with [[optical resonator]]s 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 [[Test theories of special relativity|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<sup>−12</sup>.<ref name=gm18>{{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 }}</ref> The current precision, by which an anisotropy of the speed of light can be excluded, is at the 10<sup>−17</sup> level. This is related to the relative velocity between the [[Solar System]] and the rest frame of the [[cosmic microwave background radiation]] of ~368&nbsp;km/s (see also [[Michelson–Morley experiment#Recent experiments|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) <math>\tilde{\kappa}_{e-}</math>, <math>\tilde{\kappa}_{o+}</math> and <math>\tilde{\kappa}_{tr}</math>.<ref name=kost0205 /> They can be interpreted as follows: <math>\tilde{\kappa}_{e-}</math> represent anisotropic shifts in the two-way (forward and backwards) speed of light, <math>\tilde{\kappa}_{o+}</math> represent anisotropic differences in the [[one-way speed of light|one-way speed]] of counterpropagating beams along an axis,<ref name=hohensee>{{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}}</ref><ref>{{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.</ref> and <math>\tilde{\kappa}_{tr}</math> represent isotropic (orientation-independent) shifts in the one-way phase velocity of light.<ref name=tobar>{{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}}</ref> 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.<ref name=kost0205 /> 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.<ref name="tobar"/> 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 <math>\tilde{\kappa}_{o+}</math> related one-way light speed isotropy in combination with the electron sector of the SME was conducted by Bocquet ''et al.'' (2010).<ref name=bocquet /> 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 <ref>{{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 }}</ref> (for details on that 'Compton Edge' method and analysis see,<ref>{{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}}</ref> discussion e.g.<ref>{{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}}</ref>).

{| class=wikitable
! rowspan=2 | Author
! rowspan=2 | Year
! colspan=2 | RMS
! colspan=3 | SME
|-
! Orientation
! Velocity
! <math>\tilde{\kappa}_{e-}</math>
! <math>\tilde{\kappa}_{o+}</math>
! <math>\tilde{\kappa}_{tr}</math>
|-
| Michimura ''et al.''<ref>{{cite journal|author=Michimura|title=New Limit on Lorentz Violation Using a Double-Pass Optical Ring Cavity|journal=Physical Review Letters|volume=110|issue=20|pages=200401|year=2013|arxiv=1303.6709|doi=10.1103/PhysRevLett.110.200401|bibcode=2013PhRvL.110t0401M|display-authors=etal|pmid=25167384|s2cid=34643297}}</ref> || 2013 || || || || (0.7±1){{e|−14}} || (−0.4±0.9){{e|−10}}
|-
| Baynes ''et al.''<ref>{{cite journal|author=Baynes|title=Oscillating Test of the Isotropic Shift of the Speed of Light|journal=Physical Review Letters|volume=108|issue=26|pages=260801|year=2012|doi=10.1103/PhysRevLett.108.260801|bibcode=2012PhRvL.108z0801B|display-authors=etal|pmid=23004951}}</ref> || 2012 || || || || || (3±11){{e|−10}}
|-
| Baynes ''et al.''<ref>{{cite journal|author=Baynes|title=Testing Lorentz invariance using an odd-parity asymmetric optical resonator|journal=Physical Review D|volume=84|issue=8|pages=081101|year=2011|doi=10.1103/PhysRevD.84.081101|arxiv=1108.5414|bibcode = 2011PhRvD..84h1101B |s2cid=119196989|display-authors=etal}}</ref> || 2011 || || || || (0.7±1.4){{e|−12}} || (3.4±6.2){{e|−9}}
|-
| Hohensee ''et al.''<ref name=hohensee /> || 2010 || || || (0.8±0.6){{e|−16}}  || (−1.5±1.2){{e|−12}} || (−1.50±0.74){{e|−8}}
|-
| Bocquet ''et al.''<ref name=bocquet>{{cite journal|author=Bocquet|title= Limits on Light-Speed Anisotropies from Compton Scattering of High-Energy Electrons|journal=Physical Review Letters|volume=104|issue=24|pages=24160|year=2010|doi= 10.1103/PhysRevLett.104.241601|arxiv=1005.5230|bibcode = 2010PhRvL.104x1601B|display-authors=etal|pmid=20867292|s2cid= 20890367}}</ref> || 2010 || || || || ≤1.6{{e|−14}}<ref><math>\tilde{\kappa}_{o+}</math> combined with electron coefficients</ref> ||
|-
| Herrmann ''et al.''<ref>{{cite journal|author=Herrmann|title=Rotating optical cavity experiment testing Lorentz invariance at the 10<sup>−17</sup> level|journal=Physical Review D|volume=80|issue=100|pages=105011|year=2009|doi=10.1103/PhysRevD.80.105011|arxiv=1002.1284|bibcode = 2009PhRvD..80j5011H |s2cid=118346408|display-authors=etal}}</ref> || 2009 || (4±8){{e|−12}} || || (−0.31±0.73){{e|−17}} || (−0.14±0.78){{e|−13}} ||
|-
| Eisele ''et al.''<ref>{{cite journal|author=Eisele|title=Laboratory Test of the Isotropy of Light Propagation at the 10<sup>−17</sup> level|journal=Physical Review Letters|volume=103|issue=9|pages=090401|year=2009|doi=10.1103/PhysRevLett.103.090401|bibcode=2009PhRvL.103i0401E|pmid=19792767|s2cid=33875626|url=http://www.exphy.uni-duesseldorf.de/Publikationen/2009/Eisele%20et%20al%20Laboratory%20Test%20of%20the%20Isotropy%20of%20Light%20Propagation%20at%20the%2010-17%20Level%202009.pdf|display-authors=etal|access-date=2012-07-21|archive-date=2022-10-09|archive-url=https://ghostarchive.org/archive/20221009/http://www.exphy.uni-duesseldorf.de/Publikationen/2009/Eisele%20et%20al%20Laboratory%20Test%20of%20the%20Isotropy%20of%20Light%20Propagation%20at%20the%2010-17%20Level%202009.pdf|url-status=dead}}</ref> || 2009 || (−1.6±6±1.2){{e|−12}} || || (0.0±1.0±0.3){{e|−17}} || (1.5±1.5±0.2){{e|−13}} ||
|-
| Tobar ''et al.''<ref>{{cite journal|author=Tobar|title=Testing local Lorentz and position invariance and variation of fundamental constants by searching the derivative of the comparison frequency between a cryogenic sapphire oscillator and hydrogen maser|journal=Physical Review D|volume=81|issue=2|year=2010|pages=022003|doi=10.1103/PhysRevD.81.022003|arxiv=0912.2803|bibcode = 2010PhRvD..81b2003T |s2cid=119262822|display-authors=etal}}</ref> || 2009 || || (−4.8±3.7){{e|−8}} || || ||
|-
| Tobar ''et al.''<ref>{{cite journal|author=Tobar|title=Rotating odd-parity Lorentz invariance test in electrodynamics|journal=Physical Review D|volume=80|issue=12|pages=125024|year=2009|doi=10.1103/PhysRevD.80.125024|arxiv=0909.2076|bibcode = 2009PhRvD..80l5024T |s2cid=119175604|display-authors=etal}}</ref> || 2009 || || || || || (−0.3±3){{e|−7}}
|-
| Müller ''et al.''<ref>{{cite journal|author=Müller|title=Relativity tests by complementary rotating Michelson-Morley experiments|journal=Phys. Rev. Lett.|volume=99|issue=5|pages=050401|year=2007|doi=10.1103/PhysRevLett.99.050401|arxiv=0706.2031|bibcode = 2007PhRvL..99e0401M|pmid=17930733 |s2cid=33003084|display-authors=etal}}</ref> || 2007 || || || (7.7±4.0){{e|−16}} || (1.7±2.0){{e|−12}} ||
|-
| Carone ''et al.''<ref>{{cite journal|author=Carone|title=New bounds on isotropic Lorentz violation|journal=Physical Review D|volume=74|issue=7|pages=077901|year=2006|arxiv=hep-ph/0609150|doi=10.1103/PhysRevD.74.077901|bibcode = 2006PhRvD..74g7901C |s2cid=119462975|display-authors=etal}}</ref> || 2006 || || || || || ≲3{{e|−8}}<ref>Measured by examining the [[Anomalous magnetic dipole moment|anomalous magnetic moment of the electron]].</ref>
|-
| Stanwix ''et al.''<ref>{{cite journal|author=Stanwix|title=Improved test of Lorentz invariance in electrodynamics using rotating cryogenic sapphire oscillators|journal=Physical Review D|volume=74|issue=8|year=2006|pages=081101|doi=10.1103/PhysRevD.74.081101|arxiv=gr-qc/0609072|bibcode = 2006PhRvD..74h1101S |s2cid=3222284|display-authors=etal}}</ref> || 2006 || (9.4±8.1){{e|−11}} || || (−6.9±2.2){{e|−16}} || (−0.9±2.6){{e|−12}} ||
|-
| Herrmann ''et al.''<ref>{{cite journal|author=Herrmann|title=Test of the Isotropy of the Speed of Light Using a Continuously Rotating Optical Resonator|journal=Phys. Rev. Lett.|volume=95|issue=15|year=2005|pages=150401|doi=10.1103/PhysRevLett.95.150401|arxiv=physics/0508097|bibcode = 2005PhRvL..95o0401H|pmid=16241700 |s2cid=15113821|display-authors=etal}}</ref> || 2005 || (−2.1±1.9){{e|−10}} || || (−3.1±2.5){{e|−16}} || (−2.5±5.1){{e|−12}} ||
|-
| Stanwix ''et al.''<ref>{{cite journal|author=Stanwix|title=Test of Lorentz Invariance in Electrodynamics Using Rotating Cryogenic Sapphire Microwave Oscillators|journal=Physical Review Letters|volume=95|issue=4|year=2005|pages=040404|doi=10.1103/PhysRevLett.95.040404|arxiv=hep-ph/0506074|bibcode = 2005PhRvL..95d0404S|pmid=16090785 |s2cid=14255475|display-authors=etal}}</ref> || 2005 || (−0.9±2.0){{e|−10}} || || (−0.63±0.43){{e|−15}} || (0.20±0.21){{e|−11}} ||
|-
| Antonini ''et al.''<ref>{{cite journal|author=Antonini|title=Test of constancy of speed of light with rotating cryogenic optical resonators|journal=Physical Review A|volume=71|issue=5|year=2005|pages=050101|doi=10.1103/PhysRevA.71.050101|arxiv=gr-qc/0504109|bibcode = 2005PhRvA..71e0101A |s2cid=119508308|display-authors=etal}}</ref> || 2005 || (+0.5±3±0.7){{e|−10}} || || (−2.0±0.2){{e|−14}} || ||
|-
| Wolf ''et al.''<ref>{{cite journal|author=Wolf|title=Improved test of Lorentz invariance in electrodynamics|journal=Physical Review D|volume=70|issue=5|year=2004|pages=051902|doi=10.1103/PhysRevD.70.051902|arxiv=hep-ph/0407232|bibcode = 2004PhRvD..70e1902W |s2cid=19178203|display-authors=etal}}</ref> || 2004 || || || (−5.7±2.3){{e|−15}} || (−1.8±1.5){{e|−11}} ||
|-
| Wolf ''et al.''<ref>{{cite journal|author=Wolf|title=Whispering Gallery Resonators and Tests of Lorentz Invariance|journal=General Relativity and Gravitation|volume=36|issue=10|year=2004|pages=2351–2372|doi=10.1023/B:GERG.0000046188.87741.51|arxiv=gr-qc/0401017|bibcode = 2004GReGr..36.2351W |s2cid=8799879|display-authors=etal}}</ref> || 2004 || (+1.2±2.2){{e|−9}} || (3.7±3.0){{e|−7}} || ||
|-
| Müller ''et al.''<ref>{{cite journal|author=Müller|title=Modern Michelson-Morley experiment using cryogenic optical resonators|journal=Physical Review Letters|volume=91|issue=2|pages=020401|year=2003|doi=10.1103/PhysRevLett.91.020401|arxiv=physics/0305117|bibcode = 2003PhRvL..91b0401M|pmid=12906465|s2cid=15770750|display-authors=etal}}</ref> || 2003 || (+2.2±1.5){{e|−9}} || || (1.7±2.6){{e|−15}} || (14±14){{e|−11}} ||
|-
| Lipa ''et al.''<ref>{{cite journal|author=Lipa|title=New Limit on Signals of Lorentz Violation in Electrodynamics|journal=Physical Review Letters|volume=90|issue=6|year=2003|pages=060403|doi=10.1103/PhysRevLett.90.060403|arxiv=physics/0302093|bibcode=2003PhRvL..90f0403L|display-authors=etal|pmid=12633280|s2cid=38353693}}</ref> || 2003 || || || (1.4±1.4){{e|−13}} || ≤{{10^|−9}} ||
|-
| Wolf ''et al.''<ref>{{cite journal|author=Wolf|title=Tests of Lorentz Invariance using a Microwave Resonator|journal=Physical Review Letters|volume=90|issue=6|year=2003|pages=060402|doi=10.1103/PhysRevLett.90.060402|arxiv=gr-qc/0210049|bibcode = 2003PhRvL..90f0402W|pmid=12633279 |s2cid=18267310|display-authors=etal}}</ref> || 2003 || (+1.5±4.2){{e|−9}} || || || ||
|-
| Braxmaier ''et al.''<ref>{{cite journal|author=Braxmaier|title=Tests of Relativity Using a Cryogenic Optical Resonator|journal=Phys. Rev. Lett.|volume=88|issue=1|pages=010401|year=2002|doi=10.1103/PhysRevLett.88.010401|pmid=11800924|bibcode=2001PhRvL..88a0401B|url=http://www.exphy.uni-duesseldorf.de/Publikationen/2002/Braxmaier-2002-PRL10401.pdf|display-authors=etal|access-date=2012-07-21|archive-date=2021-03-23|archive-url=https://web.archive.org/web/20210323200106/http://www.exphy.uni-duesseldorf.de/Publikationen/2002/Braxmaier-2002-PRL10401.pdf|url-status=dead}}</ref> || 2002 || || (1.9±2.1){{e|−5}} || || ||
|-
| Hils and Hall<ref>{{cite journal|author1=Hils, Dieter |author2=Hall, J. L. |title=Improved Kennedy-Thorndike experiment to test special relativity|journal=Phys. Rev. Lett.|volume=64|pages=1697–1700|year=1990|doi=10.1103/PhysRevLett.64.1697|bibcode = 1990PhRvL..64.1697H|issue=15|pmid=10041466 }}</ref> || 1990 || || 6.6{{e|−5}} || || ||
|-
| Brillet and Hall<ref>{{cite journal|author1=Brillet, A. |author2=Hall, J. L. |title=Improved laser test of the isotropy of space|journal=Phys. Rev. Lett.|volume=42|pages=549–552|year=1979|doi=10.1103/PhysRevLett.42.549|bibcode = 1979PhRvL..42..549B|issue=9 }}</ref> || 1979 || ≲5{{e|−9}} || || ≲{{10^|−15}} || ||
|}

=== Solar System ===
Besides terrestrial tests also [[astrometry|astrometric]] tests using [[Lunar Laser Ranging experiment|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]].<ref>{{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 }}</ref> 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.<ref>{{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 }}</ref>

In addition to the terrestrial Kennedy–Thorndike experiments mentioned above, Müller & Soffel (1995)<ref>{{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 }}</ref> and Müller et al. (1999)<ref name=muell99>{{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}}</ref> 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 <math>(-5\pm12)\times10^{-5}</math>,<ref name=muell99 /> comparable to that of Hils and Hall (1990, see table above on the right).

=== Vacuum dispersion ===
Another effect often discussed in connection with quantum gravity (QG) is the possibility of [[Dispersion (optics)|dispersion]] of light in vacuum (''i.e.'' the dependence of light speed on photon energy), due to Lorentz-violating [[dispersion relation]]s. This effect should be strong at energy levels comparable to, or beyond the [[Planck energy]] <math> E_\text{Pl} \sim 1.22 \times 10^{19}</math> 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 Gamma-ray Space Telescope|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,<ref name=lat /> which excludes a large class of Lorentz-violating quantum gravity models.

{| class=wikitable
! rowspan=2 | Name
! rowspan=2 | Year
! colspan=2 | QG Bounds (GeV)
|-
! 95% C.L.
! 99% C.L.
|-
| Vasileiou et al.<ref>{{cite journal |author=Vasileiou|title=Bounds on Spectral Dispersion from Fermi-Detected Gamma Ray Bursts |journal=Physical Review Letters|volume=87|issue=12 |year=2013|pages=122001|doi=10.1103/PhysRevD.87.122001|arxiv=1305.3463
| bibcode = 2013PhRvD..87l2001V |s2cid=119222087 |display-authors=etal}}</ref> || 2013 || >7.6 × {{math|''E''{{sub|Pl}}}} ||
|-
| Nemiroff et al.<ref>{{cite journal |author=Nemiroff|title=Constraints on Lorentz invariance violation from Fermi-Large Area Telescope observations of gamma-ray bursts |journal=Physical Review D|volume=108|issue=23 |year=2012|pages=231103|doi=10.1103/PhysRevLett.108.231103|arxiv=1109.5191
| bibcode = 2012PhRvL.108w1103N |display-authors=etal |pmid=23003941|s2cid=15592150 }}</ref> || 2012 ||   || >525 × {{math|''E''{{sub|Pl}}}} 
|-
| [[Fermi Gamma-ray Space Telescope|Fermi-LAT-GBM]]<ref name=lat>{{cite journal |author=Fermi LAT Collaboration|title=A limit on the variation of the speed of light arising from quantum gravity effects|journal=Nature|volume=462|issue=7271 |year=2009|pages=331–334|doi=10.1038/nature08574|arxiv=0908.1832|pmid=19865083
| bibcode = 2009Natur.462..331A |s2cid=205218977}}</ref> || 2009 || >3.42 × {{math|''E''{{sub|Pl}}}} || >1.19 × {{math|''E''{{sub|Pl}}}}
|-
| [[H.E.S.S.]]<ref>{{cite journal |author=HESS Collaboration|title=Limits on an Energy Dependence of the Speed of Light from a Flare of the Active Galaxy PKS 2155-304|journal=Physical Review Letters|volume=101|issue=17 |year=2008|pages=170402|doi=10.1103/PhysRevLett.101.170402|pmid=18999724|arxiv=0810.3475|bibcode=2008PhRvL.101q0402A
|s2cid=15789937}}</ref> || 2008 || ≥7.2{{e|17}} ||
|-
| [[MAGIC (telescope)|MAGIC]]<ref name=magic>{{cite journal |author=MAGIC Collaboration|title=Probing quantum gravity using photons from a flare of the active galactic nucleus Markarian 501 observed by the MAGIC telescope|journal=Physics Letters B|volume=668|issue=4 |year=2008|pages=253–257|doi=10.1016/j.physletb.2008.08.053|arxiv=0708.2889|bibcode=2008PhLB..668..253M
|s2cid=5103618}}</ref> || 2007 || ≥0.21{{e|18}} ||
|-
| Ellis ''et al.''<ref>{{cite journal |author=Ellis|title=Robust limits on Lorentz violation from gamma-ray bursts|journal=Astroparticle Physics|pages=402–411|volume=25|issue=6 |year=2006|doi=10.1016/j.astropartphys.2006.04.001|arxiv=astro-ph/0510172|bibcode = 2006APh....25..402E |display-authors=etal}}</ref><ref>{{cite journal |author=Ellis|title=Corrigendum to "Robust limits on Lorentz violation from gamma-ray bursts"|journal=Astroparticle Physics|pages=158–159|volume=29|issue=2 |year=2007|doi=10.1016/j.astropartphys.2007.12.003|arxiv=0712.2781|bibcode = 2008APh....29..158E |display-authors=etal}}</ref> || 2007 || ≥1.4{{e|16}} ||
|-
| Lamon ''et al.''<ref>{{cite journal |author=Lamon|title=Study of Lorentz violation in INTEGRAL gamma-ray bursts|journal=General Relativity and Gravitation|volume=40|issue=8 |year=2008|pages=1731–1743|doi=10.1007/s10714-007-0580-6|arxiv=0706.4039|bibcode = 2008GReGr..40.1731L |s2cid=1387664|display-authors=etal}}</ref> || 2007 || ≥3.2{{e|11}} ||
|-
| Martinez ''et al.''<ref>{{cite journal |author=Rodríguez Martínez|title=GRB 051221A and tests of Lorentz symmetry|journal=Journal of Cosmology and Astroparticle Physics|issue=5 |year=2006|doi=10.1088/1475-7516/2006/05/017|pages=017|arxiv=astro-ph/0601556|bibcode = 2006JCAP...05..017R|volume=2006|s2cid=18639701|display-authors=etal}}</ref> || 2006 || ≥0.66{{e|17}} ||
|-
| Boggs ''et al.''<ref>{{cite journal |author=Boggs|title=Testing Lorentz Invariance with GRB021206|journal=The Astrophysical Journal|pages=L77–L80|volume=611|issue=2 |year=2004|doi=10.1086/423933|arxiv=astro-ph/0310307|bibcode=2004ApJ...611L..77B|s2cid=15649601|display-authors=etal}}</ref> || 2004 || ≥1.8{{e|17}} ||
|-
| Ellis ''et al.''<ref>{{cite journal |author=Ellis|title=Quantum-gravity analysis of gamma-ray bursts using wavelets|journal=Astronomy and Astrophysics|volume=402|issue=2|year=2003|doi=10.1051/0004-6361:20030263|pages=409–424|arxiv=astro-ph/0210124|bibcode=2003A&A...402..409E|s2cid=15388873|display-authors=etal}}</ref> || 2003 || ≥6.9{{e|15}} ||
|-
| Ellis ''et al.''<ref>{{cite journal |author=Ellis|title=A Search in Gamma-Ray Burst Data for Nonconstancy of the Velocity of Light|journal=The Astrophysical Journal|volume=535|issue=1 |year=2000|pages=139–151|doi=10.1086/308825|arxiv=astro-ph/9907340|bibcode=2000ApJ...535..139E|s2cid=18998838|display-authors=etal}}</ref> || 2000 || ≥{{10^|15}} ||
|-
| Kaaret<ref>{{cite journal |author=Kaaret, Philip|title=Pulsar radiation and quantum gravity|journal=Astronomy and Astrophysics|volume=345|year=1999|pages=L32–L34|arxiv=astro-ph/9903464|bibcode = 1999A&A...345L..32K }}</ref> || 1999 || >1.8{{e|15}} ||
|-
| Schaefer<ref>{{cite journal |author=Schaefer, Bradley E.|title=Severe Limits on Variations of the Speed of Light with Frequency|journal=Physical Review Letters|volume=82|issue=25 |year=1999|doi=10.1103/PhysRevLett.82.4964|pages=4964–4966|arxiv=astro-ph/9810479|bibcode=1999PhRvL..82.4964S|s2cid=119339066}}</ref> || 1999 || ≥2.7{{e|16}} ||
|-
| Biller<ref>{{cite journal |author=Biller|title=Limits to Quantum Gravity Effects on Energy Dependence of the Speed of Light from Observations of TeV Flares in Active Galaxies|journal=Physical Review Letters|volume=83|issue=11|year=1999|doi=10.1103/PhysRevLett.83.2108|pages=2108–2111|arxiv=gr-qc/9810044|bibcode=1999PhRvL..83.2108B|s2cid=43423079 |display-authors=etal}}</ref> || 1999 || >4{{e|16}} ||
|}

=== Vacuum birefringence ===
Lorentz violating dispersion relations due to the presence of an anisotropic space might also lead to vacuum [[birefringence]] and [[parity violation]]s. For instance, the [[Polarization (waves)|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 [[Standard-Model Extension|SME]] coefficients <math> k_{(V)00}^{(3)}</math> and <math> k_{(V)00}^{(5)}</math> for Lorentz violation are given, 3 and 5 denote the mass dimensions employed. The latter corresponds to <math>\xi</math> in the [[effective field theory|EFT]] of Meyers and Pospelov<ref name=meyers /> by <math display="inline"> k_{(V)00}^{(5)} = \frac{3\sqrt{4\pi}\xi}{5m_{\text{P}}}</math>, <math>m_\text{P}</math> being the Planck mass.<ref name=steck />

{|class=wikitable 
! rowspan=2| Name
! rowspan=2| Year
! colspan=2| SME bounds
! rowspan=2| EFT bound, <math>\xi</math>
|-
! <math> k_{(V)00}^{(3)}</math> (GeV)
! <math> k_{(V)00}^{(5)}</math> (GeV<sup>−1</sup>)
|-
| Götz ''et al.''<ref>{{cite journal |author=Götz|title=The polarized gamma-ray burst GRB 061122|journal=[[Monthly Notices of the Royal Astronomical Society]]|volume=431|issue=4|year=2013|pages=3550–3556|doi=10.1093/mnras/stt439|doi-access=free |arxiv=1303.4186|bibcode = 2013MNRAS.431.3550G |s2cid=53499528|display-authors=etal}}</ref> || 2013 || || ≤5.9{{e|−35}} || ≤3.4{{e|−16}}
|-
| Toma ''et al.''<ref>{{cite journal |author=Toma|title=Strict Limit on CPT Violation from Polarization of γ-Ray Bursts| journal=Physical Review Letters|volume=109|issue=24|year=2012|pages=241104| doi=10.1103/PhysRevLett.109.241104|arxiv=1208.5288| bibcode=2012PhRvL.109x1104T|display-authors=etal|pmid=23368301|s2cid=42198517}}</ref> || 2012 || || ≤1.4{{e|−34}} || ≤8{{e|−16}} 
|-
| Laurent ''et al.''<ref>{{cite journal |author=Laurent|title=Constraints on Lorentz Invariance Violation using integral/IBIS observations of GRB041219A|journal=Physical Review D|volume=83|issue=12|year=2011|pages=121301|doi=10.1103/PhysRevD.83.121301| arxiv=1106.1068|bibcode = 2011PhRvD..83l1301L |s2cid=53603505|display-authors=etal}}</ref> || 2011 || || ≤1.9{{e|−33}} || ≤1.1{{e|−14}} 
|-
| Stecker<ref name=steck>{{cite journal |author=Stecker, Floyd W.|title=A new limit on Planck scale Lorentz violation from γ-ray burst polarization|journal=Astroparticle Physics|volume=35|issue=2|year=2011|pages=95–97| doi=10.1016/j.astropartphys.2011.06.007| arxiv=1102.2784|bibcode = 2011APh....35...95S |s2cid=119280055}}</ref> || 2011 || || ≤4.2{{e|−34}} || ≤2.4{{e|−15}} 
|-
| Kostelecký ''et al.''<ref name="kost0905"/> || 2009 || || ≤1{{e|−32}} || ≤9{{e|−14}} 
|-
| [[QUaD]]<ref>{{cite journal |author=QUaD Collaboration|title=Parity Violation Constraints Using Cosmic Microwave Background Polarization Spectra from 2006 and 2007 Observations by the QUaD Polarimeter|journal=Physical Review Letters|volume=102| issue=16| year=2009|pages=161302|doi=10.1103/PhysRevLett.102.161302| arxiv=0811.0618|bibcode=2009PhRvL.102p1302W| pmid=19518694| s2cid=84181915}}</ref> || 2008 || ≤2{{e|−43}} || ||
|-
| Kostelecký ''et al.''<ref>{{cite journal |author1=Kostelecký, V. Alan |author2=Mewes, Matthew |title=Astrophysical Tests of Lorentz and CPT Violation with Photons|journal=The Astrophysical Journal |volume=689|issue=1|year=2008|pages=L1–L4| doi=10.1086/595815|arxiv=0809.2846|bibcode=2008ApJ...689L...1K |s2cid=6465811}}</ref> || 2008 || =(2.3±5.4){{e|−43}} || ||
|-
| Maccione ''et al.''<ref>{{cite journal |author=Maccione|title=γ-ray polarization constraints on Planck scale violations of special relativity|journal=Physical Review D|volume=78|issue=10|year=2008|pages=103003| doi=10.1103/PhysRevD.78.103003| arxiv=0809.0220|bibcode = 2008PhRvD..78j3003M |s2cid=119277171|display-authors=etal}}</ref> || 2008 || || ≤1.5{{e|−28}} || ≤9{{e|−10}}
|-
| Komatsu ''et al.''<ref>{{cite journal |author=Komatsu|title=Five-Year Wilkinson Microwave Anisotropy Probe Observations: Cosmological Interpretation|journal=The Astrophysical Journal Supplement|volume=180 |issue=2|year=2009|pages=330–376|doi=10.1088/0067-0049/180/2/330|arxiv=0803.0547 |bibcode=2009ApJS..180..330K|s2cid=119290314|display-authors=etal}}</ref> || 2008 || =(1.2±2.2){{e|−43}} <ref name=kost0905 /> || ||
|-
| Kahniashvili ''et al.''<ref>{{cite journal |author=Kahniashvili|title=Testing Lorentz invariance violation with Wilkinson Microwave Anisotropy Probe five year data|journal=Physical Review D|volume=78 |issue=12|year=2008|pages=123009|doi=10.1086/589447 |arxiv=0803.2350 |bibcode=2008ApJ...679L..61X|s2cid=6069635|display-authors=etal}}</ref> || 2008 || =(2.6±1.9){{e|−43}} <ref name=kost0905 /> || ||
|-
| Cabella ''et al.''<ref>{{cite journal |author=Cabella|title=Constraints on CPT violation from Wilkinson Microwave Anisotropy Probe three year polarization data: A wavelet analysis|journal=Physical Review D|volume=76 |issue=12|year=2007| pages=123014| doi=10.1103/PhysRevD.76.123014|arxiv=0705.0810|bibcode = 2007PhRvD..76l3014C |s2cid=118717161|display-authors=etal}}</ref> || 2007 || =(2.5±3.0){{e|−43}} <ref name=kost0905 /> || ||
|-
| Fan ''et al.''<ref>{{cite journal |author=Fan|title=γ-ray burst ultraviolet/optical afterglow polarimetry as a probe of quantum gravity|journal=Monthly Notices of the Royal Astronomical Society|volume=376|issue=4 |year=2007|pages=1857–1860| doi=10.1111/j.1365-2966.2007.11576.x|doi-access=free |arxiv=astro-ph/0702006 |bibcode=2007MNRAS.376.1857F|s2cid=16384668|display-authors=etal}}</ref> || 2007 || || ≤3.4{{e|−26}} || ≤2{{e|−7}} <ref name=steck />
|-
| Feng ''et al.''<ref>{{cite journal |author=Feng|title=Searching for CPT Violation with Cosmic Microwave Background Data from WMAP and BOOMERANG|journal=Physical Review Letters|volume=96|issue=22|year=2006 |pages=221302|doi=10.1103/PhysRevLett.96.221302| arxiv=astro-ph/0601095|bibcode=2006PhRvL..96v1302F|display-authors=etal|pmid=16803298|s2cid=29494306}}</ref> || 2006 || =(6.0±4.0){{e|−43}} <ref name=kost0905 />  || ||
|-
| Gleiser ''et al.''<ref>{{cite journal |author1=Gleiser, Reinaldo J. |author2=Kozameh, Carlos N. |title=Astrophysical limits on quantum gravity motivated birefringence|journal=Physical Review D|volume=64|issue=8 |year=2001|doi=10.1103/PhysRevD.64.083007| pages=083007|arxiv=gr-qc/0102093|bibcode = 2001PhRvD..64h3007G |s2cid=9255863 }}</ref> || 2001 || || ≤8.7{{e|−23}} || ≤4{{e|−4}} <ref name=steck />
|-
| Carroll ''et al.''<ref name=carroll>{{cite journal |author=Carroll|title=Limits on a Lorentz- and parity-violating modification of electrodynamics|journal=Physical Review D|volume=41|issue=4|year=1990 |doi=10.1103/PhysRevD.41.1231| pmid=10012457| pages=1231–1240| bibcode = 1990PhRvD..41.1231C |display-authors=etal}}</ref> || 1990 || ≤2{{e|−42}} || ||
|}

== Maximal attainable speed ==

=== Threshold constraints ===
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 [[effective field theory|EFT]] models such as [[Standard-Model Extension|SME]]. Depending on which of these particles travels faster or slower than the speed of light, effects such as the following can occur:<ref name=JacNeu>{{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}}</ref><ref name=hohe />

*''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):

{|class=wikitable 
! rowspan=2 | Name
! rowspan=2 | Year
! colspan=4 | Bounds
! rowspan=2 | Particle
! rowspan=2 | Location
|-
! Photon decay
! Cherenkov
! Synchrotron
! GZK
|-
| Stecker<ref name=stecker14 /> || 2014 || || ≤5{{e|-21}} || || || Electron || Astronomical
|-
| Stecker & Scully<ref>{{cite journal |author1=Stecker, Floyd W. |author2=Scully, Sean T. |title=Searching for new physics with ultrahigh energy cosmic rays|journal=New Journal of Physics|volume=11|issue=8|year=2009|pages=085003|doi=10.1088/1367-2630/11/8/085003|arxiv=0906.1735|bibcode = 2009NJPh...11h5003S |s2cid=8009677 }}</ref> || 2009 || || || || ≤4.5{{e|-23}} || [[Ultra-high-energy cosmic ray|UHECR]] || Astronomical
|-
| Altschul<ref name=altschul0905>{{cite journal |author=Altschul, Brett|title=Bounding isotropic Lorentz violation using synchrotron losses at LEP|journal=Physical Review D|volume=80|issue=9|year=2009|pages=091901|doi=10.1103/PhysRevD.80.091901|arxiv=0905.4346|bibcode = 2009PhRvD..80i1901A |s2cid=18312444}}</ref> || 2009 || || || ≤5{{e|-15}} || || Electron || Terrestrial
|-
| Hohensee ''et al.''<ref name=hohe>{{cite journal |author=Hohensee|title=Particle-Accelerator Constraints on Isotropic Modifications of the Speed of Light|journal=Physical Review Letters|volume=102|issue=17|year=2009|pages=170402|doi=10.1103/PhysRevLett.102.170402|arxiv=0904.2031|bibcode=2009PhRvL.102q0402H|display-authors=etal|pmid=19518765|s2cid=13682668}}</ref> || 2009 || ≤−5.8{{e|-12}} || ≤1.2{{e|-11}} || || || Electron || Terrestrial
|-
| Bi ''et al.''<ref>{{cite journal |author1=Bi, Xiao-Jun |author2=Cao, Zhen |author3=Li, Ye |author4=Yuan, Qiang |title=Testing Lorentz invariance with the ultrahigh energy cosmic ray spectrum|journal=Physical Review D|volume=79|issue=8|year=2009|pages=083015|doi=10.1103/PhysRevD.79.083015|arxiv=0812.0121|bibcode = 2009PhRvD..79h3015B |s2cid=118587418 }}</ref> || 2008 || || || || ≤3{{e|-23}} || UHECR || Astronomical
|-
| Klinkhamer & Schreck<ref>{{cite journal |author1=Klinkhamer, F. R. |author2=Schreck, M. |title=New two-sided bound on the isotropic Lorentz-violating parameter of modified Maxwell theory|journal=Physical Review D|volume=78|issue=8|year=2008|pages=085026|doi=10.1103/PhysRevD.78.085026|arxiv=0809.3217|bibcode = 2008PhRvD..78h5026K |s2cid=119293488 }}</ref> || 2008 || ≤−9{{e|-16}} || ≤6{{e|-20}} || || || UHECR || Astronomical
|-
| Klinkhamer & Risse<ref>{{cite journal |author1=Klinkhamer, F. R. |author2=Risse, M. |title=Ultrahigh-energy cosmic-ray bounds on nonbirefringent modified Maxwell theory|journal=Physical Review D|volume=77|issue=1|year=2008|pages=016002|doi=10.1103/PhysRevD.77.016002|arxiv=0709.2502|bibcode = 2008PhRvD..77a6002K |s2cid=119109140 }}</ref> || 2007 || || ≤2{{e|-19}} || || || UHECR || Astronomical
|-
| Kaufhold ''et al.''<ref>{{cite journal |author1=Kaufhold, C. |author2=Klinkhamer, F. R. |title=Vacuum Cherenkov radiation in spacelike Maxwell-Chern-Simons theory|journal=Physical Review D|volume=76|issue=2|year=2007|pages=025024|doi=10.1103/PhysRevD.76.025024|arxiv=0704.3255|bibcode = 2007PhRvD..76b5024K |s2cid=119692639 }}</ref> || 2007 || || ≤{{10^|-17}} || || || UHECR || Astronomical
|-
| Altschul<ref>{{cite journal |author=Altschul, Brett|title=Lorentz violation and synchrotron radiation|journal=Physical Review D|volume=72|issue=8|year=2005|pages=085003|doi=10.1103/PhysRevD.72.085003|arxiv=hep-th/0507258|bibcode = 2005PhRvD..72h5003A |s2cid=2082044}}</ref> || 2005 || || || ≤6{{e|-20}} || || Electron || Astronomical
|-
| Gagnon ''et al.''<ref>{{cite journal |author1=Gagnon, Olivier |author2=Moore, Guy D. |title=Limits on Lorentz violation from the highest energy cosmic rays|journal=Physical Review D|volume=70|issue=6|year=2004|pages=065002|doi=10.1103/PhysRevD.70.065002|arxiv=hep-ph/0404196|bibcode = 2004PhRvD..70f5002G |s2cid=119104096 }}</ref> || 2004 || ≤−2{{e|-21}} || ≤5{{e|-24}} || || || UHECR || Astronomical
|-
| Jacobson ''et al.''<ref>{{cite journal |author=Jacobson|title=New Limits on Planck Scale Lorentz Violation in QED|journal=Physical Review Letters|volume=93|issue=2|year=2004|pages=021101|doi=10.1103/PhysRevLett.93.021101|arxiv=astro-ph/0309681|bibcode=2004PhRvL..93b1101J|display-authors=etal|pmid=15323893|s2cid=45952391}}</ref> || 2003 || ≤−2{{e|-16}} || ≤5{{e|-20}} || || || Electron || Astronomical
|-
| Coleman & Glashow<ref name="glas98"/> || 1997 || ≤−1.5{{e|-15}} || ≤5{{e|-23}} || || || UHECR || Astronomical
|}

=== Clock comparison and spin coupling ===

By this kind of [[spectroscopy]] experiments – sometimes called [[Hughes–Drever experiment]]s as well – violations of Lorentz invariance in the interactions of [[proton]]s and [[neutron]]s are tested by studying the [[energy level]]s of those [[nucleon]]s in order to find anisotropies in their frequencies ("clocks"). Using [[Spin polarization|spin-polarized]] torsion balances, also anisotropies with respect to [[electron]]s can be examined. Methods used mostly focus on vector spin interactions and tensor interactions,<ref name=smiciklas /> and are often described in [[CPT symmetry|CPT]] odd/even SME terms (in particular parameters of b<sub>μ</sub> and c<sub>μν</sub>).<ref>{{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 }}</ref> Such experiments are currently the most sensitive terrestrial ones, because the precision by which Lorentz violations can be excluded lies at the 10<sup>−33</sup> [[electron volt|GeV]] level.

These tests can be used to constrain deviations between the maximal attainable speed of matter and the speed of light,<ref name=will06 /> in particular with respect to the parameters of c<sub>μν</sub> that are also used in the evaluations of the threshold effects mentioned above.<ref name=altschul0905 />

{|class=wikitable
! rowspan=2 | Author
! rowspan=2 | Year
! colspan=3 | SME bounds
! rowspan=2 | Parameters
|-
!Proton!!Neutron!!Electron
|-
| Allmendinger ''et al.''<ref>{{cite journal |author=Allmendinger|title=New limit on Lorentz and CPT violating neutron spin interactions using a free precession 3He-129Xe co-magnetometer|journal=Physical Review Letters|volume=112|issue=11|year=2014|pages=110801|arxiv=1312.3225|doi=10.1103/PhysRevLett.112.110801|bibcode = 2014PhRvL.112k0801A|pmid=24702343|s2cid=8122573|display-authors=etal}}</ref> || 2013 || || <6.7{{e|−34}} || || b<sub>μ</sub>
|-
| Hohensee ''et al.''<ref name=hohensee1303>{{cite journal |author=Hohensee|title=Limits on violations of Lorentz symmetry and the Einstein equivalence principle using radio-frequency spectroscopy of atomic dysprosium|journal=Physical Review Letters|volume=111|issue=5|year=2013|pages=050401|arxiv=1303.2747|doi=10.1103/PhysRevLett.111.050401|bibcode=2013PhRvL.111e0401H|pmid=23952369|s2cid=27090952|display-authors=etal}}</ref> || 2013 || || || (−9.0±11){{e|−17}} || c<sub>μν</sub>
|-
| Peck ''et al.''<ref>{{cite journal|author=Peck|year=2012|title= New Limits on Local Lorentz Invariance in Mercury and Cesium|journal=Physical Review A|volume=86|issue=1|pages=012109|doi=10.1103/PhysRevA.86.012109|arxiv=1205.5022|bibcode=2012PhRvA..86a2109P|s2cid=118619087|display-authors=etal}}</ref> || 2012 || <4{{e|−30}} || <3.7{{e|−31}} || || b<sub>μ</sub>
|-
| Smiciklas ''et al.''<ref name=smiciklas>{{cite journal|author=M. Smiciklas|year=2011|title=New Test of Local Lorentz Invariance Using a 21Ne-Rb-K Comagnetometer|journal=Physical Review Letters|volume=107|issue=17|pages=171604|doi=10.1103/PhysRevLett.107.171604|arxiv=1106.0738|bibcode=2011PhRvL.107q1604S|pmid=22107506|s2cid=17459575|display-authors=etal}}</ref> || 2011 || || (4.8±4.4){{e|−32}} || || c<sub>μν</sub>
|-
| Gemmel ''et al.''<ref>{{cite journal|author=Gemmel|year=2010|title=Limit on Lorentz and CPT violation of the bound neutron using a free precession He3/Xe129 comagnetometer|journal=Physical Review D|volume=82|issue=11|pages=111901|doi=10.1103/PhysRevD.82.111901|arxiv=1011.2143|bibcode=2010PhRvD..82k1901G|s2cid=118438569|display-authors=etal}}</ref> || 2010 || || <3.7{{e|−32}} || || b<sub>μ</sub>
|-
| Brown ''et al.''<ref>{{cite journal|author=Brown|year=2010|title=New Limit on Lorentz- and CPT-Violating Neutron Spin Interactions|journal=Physical Review Letters|volume=105|issue=15|pages=151604|doi=10.1103/PhysRevLett.105.151604|arxiv=1006.5425|bibcode=2010PhRvL.105o1604B|pmid=21230893|s2cid=4187692|display-authors=etal}}</ref> || 2010 || <6{{e|−32}} || <3.7{{e|−33}} || || b<sub>μ</sub>
|-
| Altarev ''et al.''<ref>{{cite journal|author=Altarev, I.|year=2009|title=Test of Lorentz Invariance with Spin Precession of Ultracold Neutrons|journal=Physical Review Letters|volume=103|issue=8|pages=081602|doi=10.1103/PhysRevLett.103.081602|arxiv=0905.3221|bibcode=2009PhRvL.103h1602A|pmid=19792714|s2cid=5224718|display-authors=etal}}</ref> || 2009 || || <2{{e|−29}} || || b<sub>μ</sub>
|-
| Heckel ''et al.''<ref>{{cite journal|author=Heckel|year=2008|title=Preferred-frame and CP-violation tests with polarized electrons|journal=Physical Review D|volume=78|issue=9|pages=092006|doi=10.1103/PhysRevD.78.092006|arxiv=0808.2673|bibcode=2008PhRvD..78i2006H|s2cid=119259958|display-authors=etal}}</ref> || 2008 || || || (4.0±3.3){{e|−31}} || b<sub>μ</sub>
|-
| Wolf ''et al.''<ref>{{cite journal|author=Wolf|year=2006|title=Cold Atom Clock Test of Lorentz Invariance in the Matter Sector|journal=Physical Review Letters|volume=96|issue=6|pages=060801|doi=10.1103/PhysRevLett.96.060801|arxiv=hep-ph/0601024|bibcode=2006PhRvL..96f0801W|pmid=16605978|s2cid=141060|display-authors=etal}}</ref> || 2006 || (−1.8±2.8){{e|−25}} || || || c<sub>μν</sub>
|-
| Canè ''et al.''<ref>{{cite journal|author=Canè|year=2004|title=Bound on Lorentz and CPT Violating Boost Effects for the Neutron|journal=Physical Review Letters|volume=93|issue=23|pages=230801|doi=10.1103/PhysRevLett.93.230801|arxiv=physics/0309070|bibcode=2004PhRvL..93w0801C|pmid=15601138|s2cid=20974775|display-authors=etal}}</ref> || 2004 || || (8.0±9.5){{e|−32}} || || b<sub>μ</sub>
|-
| Heckel ''et al.''<ref>{{cite journal|author=Heckel|year=2006|title=New CP-Violation and Preferred-Frame Tests with Polarized Electrons|journal=Physical Review Letters|volume=97|issue=2|pages=021603|doi=10.1103/PhysRevLett.97.021603|arxiv=hep-ph/0606218|bibcode=2006PhRvL..97b1603H|pmid=16907432|s2cid=27027816|display-authors=etal}}</ref> || 2006 || || || <5{{e|−30}} || b<sub>μ</sub>
|-
| Humphrey ''et al.''<ref name=hum>{{cite journal|author=Humphrey|year=2003|title=Testing CPT and Lorentz symmetry with hydrogen masers|journal=Physical Review A|volume=68|issue=6|pages=063807|doi=10.1103/PhysRevA.68.063807|arxiv=physics/0103068|bibcode=2003PhRvA..68f3807H|s2cid=13659676|display-authors=etal}}</ref> || 2003 || || || <2{{e|−27}} || b<sub>μ</sub>
|-
| Hou ''et al.''<ref>{{cite journal|author=Hou|year=2003|title=Test of Cosmic Spatial Isotropy for Polarized Electrons Using a Rotatable Torsion Balance|journal=Physical Review Letters|volume=90|issue=20|pages=201101|doi=10.1103/PhysRevLett.90.201101|arxiv=physics/0009012|bibcode=2003PhRvL..90t1101H|pmid=12785879|s2cid=28211115|display-authors=etal}}</ref> || 2003 || || || (1.8±5.3){{e|−30}} || b<sub>μ</sub>
|-
| Phillips ''et al.''<ref>{{cite journal|author=Phillips|year=2001|title=Limit on Lorentz and CPT violation of the proton using a hydrogen maser|journal=Physical Review D|volume=63|issue=11|pages=111101|doi=10.1103/PhysRevD.63.111101|arxiv=physics/0008230|bibcode=2001PhRvD..63k1101P|s2cid=10665017|display-authors=etal}}</ref> || 2001 || <2{{e|−27}} || || || b<sub>μ</sub>
|-
| Bear ''et al.''<ref>{{cite journal|author=Bear|year=2000|title=Limit on Lorentz and CPT Violation of the Neutron Using a Two-Species Noble-Gas Maser|journal=Physical Review Letters|volume=85|issue=24|pages=5038–5041|doi=10.1103/PhysRevLett.85.5038|arxiv=physics/0007049|bibcode=2000PhRvL..85.5038B|pmid=11102181|s2cid=41363493|display-authors=etal}}</ref> || 2000 || || (4.0±3.3){{e|−31}} || || b<sub>μ</sub>
|}

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

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

{| class=wikitable
! Author !! Year !! Velocity !! Maximum deviation <br />from time dilation !! Fourth order <br />RMS bounds
|-
|Novotny ''et al.''<ref>{{Cite journal|author =Novotny, C.| title = Sub-Doppler laser spectroscopy on relativistic beams and tests of Lorentz invariance| journal=Physical Review A|volume = 80| issue = 2| pages =022107| year = 2009|doi =10.1103/PhysRevA.80.022107|bibcode = 2009PhRvA..80b2107N |display-authors=etal}}</ref>||2009||0.34c||≤1.3{{e|-6}}||≤1.2{{e|-5}}
|-
|Reinhardt ''et al.''<ref>{{Cite journal|author =Reinhardt| title = Test of relativistic time dilation with fast optical atomic clocks at different velocities| journal=Nature Physics|volume = 3| issue = 12| pages =861–864| year = 2007|doi =10.1038/nphys778| bibcode=2007NatPh...3..861R|display-authors=etal}}</ref>||2007||0.064c||≤8.4{{e|-8}} ||
|-
|Saathoff ''et al.''<ref>{{Cite journal|author =Saathoff| title = Improved Test of Time Dilation in Special Relativity| journal = Phys. Rev. Lett.| volume = 91| issue = 19| pages =190403| year = 2003|doi = 10.1103/PhysRevLett.91.190403| bibcode=2003PhRvL..91s0403S|display-authors=etal| pmid=14611572}}</ref>||2003||0.064c||≤2.2{{e|-7}} ||
|-
|Grieser ''et al.''<ref>{{Cite journal|author =Grieser| title =A test of special relativity with stored lithium ions| journal =Applied Physics B: Lasers and Optics| volume = 59| issue = 2| pages =127–133| year = 1994|doi = 10.1007/BF01081163|bibcode = 1994ApPhB..59..127G | s2cid =120291203|display-authors=etal| url =https://cds.cern.ch/record/258041/files/P00021223.pdf}}</ref>||1994||0.064c||≤1{{e|-6}}||≤2.7{{e|-4}}
|}

==CPT and antimatter tests==
{{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).<ref>{{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}}</ref><ref>{{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 }} </ref> 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 quark|top- and antitop-quarks]] have been conducted as well.

{|
| valign=top|
{| class=wikitable
! colspan=2 |Neutral [[B meson]]s
|-
!Author!!Year
|-
|[[LHCb experiment|LHCb]]<ref name=lhcb16>{{Cite journal|author=LHCb Collaboration|title=Search for violations of Lorentz invariance and CPT symmetry in B(s) mixing |journal=Physical Review Letters|volume=116|issue=24|pages=241601 |year=2016|arxiv=1603.04804|doi=10.1103/PhysRevLett.116.241601|pmid=27367382 |bibcode = 2016PhRvL.116x1601A |s2cid=206276472 }}</ref>||2016
|-
|[[BaBar experiment|BaBar]]<ref>{{Cite journal|author=BaBar Collaboration|title=Tests of CPT symmetry in B0-B0bar mixing and in B0 -> c cbar K0 decays|journal=Physical Review D|volume=94|issue=3|pages=011101 |year=2016|arxiv=1605.04545|doi=10.1103/PhysRevD.94.011101|bibcode=|s2cid=104928733 }}</ref>||2016
|-
|[[D0 experiment|D0]]<ref name=dzero15>{{Cite journal|author=D0 Collaboration|title=Search for Violation of CPT and Lorentz invariance in Bs meson oscillations |journal=Physical Review Letters|volume=115|issue=16|pages=161601 |year=2015|arxiv=1506.04123|doi=10.1103/PhysRevLett.115.161601|pmid=26550864 |bibcode = 2015PhRvL.115p1601A |s2cid=5422866 }}</ref>||2015
|-
|[[Belle experiment|Belle]]<ref>{{Cite journal|author=Belle Collaboration|title=Search for time-dependent CPT violation in hadronic and semileptonic B decays |journal=Physical Review D|volume=85|issue=7|pages=071105|year=2012|arxiv=1203.0930|doi=10.1103/PhysRevD.85.071105|bibcode = 2012PhRvD..85g1105H |s2cid=118453351 }}</ref>||2012
|-
|Kostelecký et al.<ref name=koste10>{{Cite journal|author1=Kostelecký, V. Alan |author2=van Kooten, Richard J. |title=CPT violation and B-meson oscillations|journal=Physical Review D|volume=82|issue=10|pages=101702|year=2010|arxiv=1007.5312|doi=10.1103/PhysRevD.82.101702|bibcode = 2010PhRvD..82j1702K |s2cid=55598299 }}</ref>||2010
|-
|[[BaBar experiment|BaBar]]<ref name=babar07>{{Cite journal|author=BaBar Collaboration|title=Search for CPT and Lorentz Violation in B0-Bmacr0 Oscillations with Dilepton Events|journal=Physical Review Letters|volume=100|issue=3|pages=131802|year=2008|arxiv=0711.2713|doi=10.1103/PhysRevLett.100.131802|bibcode=2008PhRvL.100m1802A|pmid=18517935|s2cid=118371724}}</ref>||2008
|-
|[[BaBar experiment|BaBar]]<ref>{{Cite journal|author=BaBar Collaboration|title=Search for T, CP and CPT violation in B0-B0 mixing with inclusive dilepton events|journal=Physical Review Letters|volume=96|issue=25|pages= 251802|year=2006|arxiv=hep-ex/0603053|doi=10.1103/PhysRevLett.96.251802|pmid=16907295|bibcode=2006PhRvL..96y1802A|s2cid=21907946}}</ref>||2006
|-
|[[BaBar experiment|BaBar]]<ref>{{Cite journal|author=BaBar Collaboration|title=Limits on the decay-rate difference of neutral-B Mesons and on CP, T, and CPT Violation in B0-antiB0 oscillations|journal=Physical Review D|volume=70|issue=25|pages= 012007|year=2004|arxiv=hep-ex/0403002|doi=10.1103/PhysRevD.70.012007|bibcode=2004PhRvD..70a2007A|s2cid=119469038}}</ref>||2004
|-
|Belle<ref>{{Cite journal|author=Belle Collaboration|title=Studies of B0-B0 mixing properties with inclusive dilepton events |journal=Physical Review D|volume=67|issue=5|pages=052004|year=2003|arxiv=hep-ex/0212033|doi=10.1103/PhysRevD.67.052004|bibcode = 2003PhRvD..67e2004H |s2cid=119529021 }}</ref>||2003
|}
| valign=top|
{| class=wikitable
! colspan=2 |Neutral [[D meson]]s
|-
!Author!!Year
|-
|FOCUS<ref>{{Cite journal|author=FOCUS Collaboration|title=Charm system tests of CPT and Lorentz invariance with FOCUS |journal=Physics Letters B|volume=556|issue=1–2|pages=7–13|year=2003|arxiv=hep-ex/0208034|doi=10.1016/S0370-2693(03)00103-5|bibcode = 2003PhLB..556....7F |s2cid=119339001 }}</ref>||2003
|}
| valign=top|
{| class=wikitable
! colspan=2 |Neutral [[kaon]]s
|-
!Author!!Year
|-
|KTeV<ref>{{Cite journal|author=KTeV Collaboration|title=Precise measurements of direct CP violation, CPT symmetry, and other parameters in the neutral kaon system|journal=Physical Review D|volume=83|issue=9|pages=092001|year=2011|arxiv=1011.0127|doi=10.1103/PhysRevD.83.092001|bibcode = 2011PhRvD..83i2001A |s2cid=415448}}</ref>||2011
|-
|KLOE<ref>{{Cite journal|author=KLOE Collaboration|title=First observation of quantum interference in the process ϕ→KK→ππππ: A test of quantum mechanics and CPT symmetry|journal=Physics Letters B|volume=642|issue=4|pages=315–321|year=2006|arxiv=hep-ex/0607027|doi=10.1016/j.physletb.2006.09.046|bibcode = 2006PhLB..642..315K |s2cid=119508337 }}</ref>||2006
|-
|CPLEAR<ref>{{Cite journal|author=CPLEAR Collaboration|title=Physics at CPLEAR|journal=Physics Reports|volume=374|issue=3|pages=165–270|year=2003|doi=10.1016/S0370-1573(02)00367-8|bibcode = 2003PhR...374..165A }}</ref>||2003
|-
|KTeV<ref>{{Cite journal|author=KTeV Collaboration|title=Measurements of direct CP violation, CPT symmetry, and other parameters in the neutral kaon system|journal=Physical Review D|volume=67|issue=1|pages=012005|year=2003|arxiv=hep-ex/0208007|doi=10.1103/PhysRevD.67.012005|bibcode = 2003PhRvD..67a2005A }}</ref>||2003
|-
|NA31<ref>{{Cite journal|author=NA31 Collaboration|title=A measurement of the phases of the CP-violating amplitudes in K0-->2π decays and a test of CPT invariance|journal=Physics Letters B|volume=237|issue=2|pages=303–312|year=1990|doi=10.1016/0370-2693(90)91448-K|bibcode = 1990PhLB..237..303C |url=https://cds.cern.ch/record/204342/files/cer-000116017.pdf}}</ref>||1990
|}
| valign=top|
{| class=wikitable
! colspan=2 |[[Top quark|Top- and antitop quarks]]
|-
!Author!!Year
|-
|[[Collider Detector at Fermilab|CDF]]<ref>{{Cite journal|author=CDF Collaboration|title=Measurement of the Mass Difference Between Top and Anti-top Quarks|journal=Physical Review D|volume=87|issue=5|pages=052013|year=2013|arxiv=1210.6131|doi=10.1103/PhysRevD.87.052013|bibcode = 2013PhRvD..87e2013A |s2cid=119239216}}</ref>||2012
|-
|[[Compact Muon Solenoid|CMS]]<ref>{{Cite journal|author=CMS Collaboration|title=Measurement of the Mass Difference between Top and Antitop Quarks|journal=Journal of High Energy Physics|pages=109|year=2012|arxiv=1204.2807|doi=10.1007/JHEP06(2012)109|bibcode = 2012JHEP...06..109C|volume=2012|issue=6|s2cid=115913220}}</ref>||2012
|-
|[[D0 experiment|D0]]<ref>{{Cite journal|author=D0 Collaboration|title=Direct Measurement of the Mass Difference between Top and Antitop Quarks|journal=Physical Review D|volume=84|issue=5|pages=052005|year=2011|arxiv=1106.2063|doi=10.1103/PhysRevD.84.052005|bibcode = 2011PhRvD..84e2005A |s2cid=3911219}}</ref>||2011
|-
|CDF<ref>{{Cite journal|author=CDF Collaboration|title=Measurement of the Mass Difference between t and t¯ Quarks|journal=Physical Review Letters|volume=106|issue=15|pages=152001|year=2011|arxiv=1103.2782|doi=10.1103/PhysRevLett.106.152001|bibcode=2011PhRvL.106o2001A|pmid=21568546|s2cid=9823674}}</ref>||2011
|-
|D0<ref>{{Cite journal|author=D0 Collaboration|title=Direct Measurement of the Mass Difference between Top and Antitop Quarks|journal=Physical Review Letters|volume=103|issue=13|pages=132001|year=2009|arxiv=0906.1172|doi=10.1103/PhysRevLett.103.132001|bibcode=2009PhRvL.103m2001A|pmid=19905503|s2cid=3911219}}</ref>||2009
|}
|}

Using SME, also additional consequences of CPT violation in the neutral meson sector can be formulated.<ref name=koste10 /> Other SME related CPT tests have been performed as well:
* Using [[Penning trap]]s in which individual charged particles and their counterparts are trapped, Gabrielse ''et al.'' (1999) examined [[Gyroradius|cyclotron frequencies]] in proton-[[antiproton]] measurements, and couldn't find any deviation down to 9·10<sup>−11</sup>.<ref>{{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}}</ref>
*[[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 time|sidereal]] variations, and differences between electrons and positrons as well. Eventually they found no deviations, thereby establishing bounds of 10<sup>−24</sup> GeV.<ref>{{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}}</ref>
*Hughes ''et al.'' (2001) examined [[muon]]s for sidereal signals in the spectrum of muons, and found no Lorentz violation down to 10<sup>−23</sup> GeV.<ref>{{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}}</ref>
*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<sup>−24</sup> GeV.<ref>{{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}}</ref>

== Other particles and interactions ==

[[Generation (particle physics)|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 (particle)|tau]] of 10<sup>−8</sup>, by searching for anomalous absorption of high energy astrophysical radiation.<ref>{{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}}</ref> In the [[BaBar experiment]] (2007),<ref name=babar07/> the [[D0 experiment]] (2015),<ref name=dzero15/> and the [[LHCb experiment]] (2016),<ref name=lhcb16 /> searches have been made for sidereal variations during Earth's rotation using [[B meson]]s (thus [[bottom quark]]s) and their antiparticles. No Lorentz and CPT violating signal were found with upper limits in the range 10<sup>−15</sup> − 10<sup>−14</sup> 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.<ref>{{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}}</ref>

Lorentz violation bounds on [[Bhabha scattering]] have been given by Charneski ''et al''. (2012).<ref>{{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}}</ref> 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 <math>\scriptstyle\leq10^{14}(\text{eV})^{-1}</math>.

== Gravitation ==
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 <math>\alpha_1, \alpha_2, \alpha_3</math> (see the [[Parameterized post-Newtonian formalism|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]], [[Induced 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 <math> 10^{-9}</math> by analyzing the [[Two-body problem in general relativity|perihelion shifts of Mercury]] and Earth, and down to <math> 10^{-13}</math> in relation to solar spin precession.<ref>{{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 }}</ref> 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 <math>(6.9\pm4.5) \times 10^{-11}</math>.<ref>{{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 }}</ref> Iorio (2012) obtained bounds at the <math> 10^{-9}</math> level by examining Keplerian orbital elements of a test particle acted upon by Lorentz-violating [[Gravitoelectromagnetism|gravitomagnetic]] accelerations.<ref>{{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}}</ref> Xie (2012) analyzed the advance of [[Apsis|periastron]] of [[binary pulsar]]s, setting limits on Lorentz violation at the <math> 10^{-10}</math> level.<ref>{{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}}</ref>

==Neutrino tests==

===Neutrino oscillations===
{{Main|Lorentz-violating neutrino oscillations}}

Although [[neutrino oscillation]]s have been experimentally confirmed, the theoretical foundations are still controversial, as it can be seen in the discussion related to [[sterile neutrino]]s. 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.<ref>{{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 }}</ref>

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:

{| class=wikitable
! Name
! Year
! SME bounds (GeV)
|-
| [[Double Chooz]]<ref>{{cite journal |author=Double Chooz collaboration|title=First test of Lorentz violation with a reactor-based antineutrino experiment|journal=Physical Review D|volume=86|issue=11 |year=2012|pages=112009|doi=10.1103/PhysRevD.86.112009|arxiv=1209.5810|bibcode = 2012PhRvD..86k2009A |s2cid=3282231}}</ref> || 2012 || ≤10{{sup|−20}}
|-
| [[MINOS]]<ref>{{cite journal |author=MINOS collaboration|title=Search for Lorentz invariance and CPT violation with muon antineutrinos in the MINOS Near Detector|journal=Physical Review D|volume=85|issue=3 |year=2012|doi=10.1103/PhysRevD.85.031101|pages=031101|arxiv=1201.2631|bibcode = 2012PhRvD..85c1101A |s2cid=13726208}}</ref> || 2012 || ≤10{{sup|−23}}
|-
| [[MiniBooNE]]<ref>{{cite journal |author=MiniBooNE Collaboration|title=Test of Lorentz and CPT violation with Short Baseline Neutrino Oscillation Excesses|journal=Physics Letters B|volume=718|issue=4 |year=2013|doi=10.1016/j.physletb.2012.12.020|pages=1303–1308|arxiv=1109.3480|bibcode = 2013PhLB..718.1303A |s2cid=56363527}}</ref> || 2012 || ≤10{{sup|−20}}
|-
| [[IceCube]]<ref>{{cite journal |author=IceCube Collaboration|title=Search for a Lorentz-violating sidereal signal with atmospheric neutrinos in IceCube|journal=Physical Review D|volume=82|issue=11|year=2010|doi=10.1103/PhysRevD.82.112003|pages=112003|arxiv=1010.4096|bibcode = 2010PhRvD..82k2003A |s2cid=41803841}}</ref> || 2010 || ≤10{{sup|−23}}
|-
| [[MINOS]]<ref>{{cite journal |author=MINOS collaboration|title=Search for Lorentz Invariance and CPT Violation with the MINOS Far Detector|journal=Physical Review Letters|volume=105|issue=15 |year=2010|doi=10.1103/PhysRevLett.105.151601|pages=151601|arxiv=1007.2791|bibcode=2010PhRvL.105o1601A|pmid=21230890|s2cid=728955}}</ref> || 2010 || ≤10{{sup|−23}}
|-
| [[MINOS]]<ref>{{cite journal |author=MINOS collaboration|title=Testing Lorentz Invariance and CPT Conservation with NuMI Neutrinos in the MINOS Near Detector|journal=Physical Review Letters|volume=101|issue=15 |year=2008|doi=10.1103/PhysRevLett.101.151601|pages=151601|arxiv=0806.4945|bibcode=2008PhRvL.101o1601A|pmid=18999585|s2cid=5924748}}</ref> || 2008 || ≤10{{sup|−20}}
|-
| [[LSND]]<ref>{{cite journal |author=LSND collaboration|title=Tests of Lorentz violation in ν¯μ→ν¯e oscillations|journal=Physical Review D|volume=72|issue=7 |year=2005|doi=10.1103/PhysRevD.72.076004|pages=076004|arxiv=hep-ex/0506067|bibcode = 2005PhRvD..72g6004A |s2cid=117963760}}</ref> || 2005 || ≤10{{sup|−19}}
|}

===Neutrino speed===
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 <math display="inline"> {|v - c|}/{c} < 10^{-9}</math>, 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]].<ref>{{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}}</ref> Another possibility in the same framework is the investigation of the decay of [[pion]]s 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.<ref>{{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 }}</ref>

Velocity differences between neutrino [[Flavour (particle physics)|flavor]]s 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}}.<ref name=glas98 />

{| class=wikitable
|-
! rowspan=2| Name || rowspan=2| Year || rowspan=2| Energy || colspan=2| SME bounds for (v − c)/c
|-
! width=100|Vacuum Cherenkov || width=100|Pion decay
|-
| Stecker ''et al.''<ref name=stecker14>{{cite journal |author=Stecker, Floyd W.|title=Constraining Superluminal Electron and Neutrino Velocities using the 2010 Crab Nebula Flare and the IceCube PeV Neutrino Events|journal=Astroparticle Physics| volume=56|year=2014|pages=16–18|arxiv=1306.6095|doi=10.1016/j.astropartphys.2014.02.007|bibcode = 2014APh....56...16S |s2cid=35659438}}</ref> || 2014 || 1&nbsp;PeV || <5.6{{e|−19}} ||
|-
| Borriello ''et al.''<ref>{{Cite journal|author=Borriello|title=Stringent constraint on neutrino Lorentz invariance violation from the two IceCube PeV neutrinos|journal=Physical Review D|volume=87|issue=11|pages=116009|doi=10.1103/PhysRevD.87.116009| arxiv=1303.5843|year=2013|bibcode = 2013PhRvD..87k6009B |s2cid=118521330|display-authors=etal}}</ref> || 2013 || 1&nbsp;PeV || {{10^|−18}} ||
|-
| Cowsik ''et al.''<ref>{{Cite journal|author=Cowsik|title=Testing violations of Lorentz invariance with cosmic rays| journal=Physical Review D|volume=86|issue=4|pages=045024|doi=10.1103/PhysRevD.86.045024|arxiv=1206.0713|year=2012|bibcode = 2012PhRvD..86d5024C |s2cid=118567883|display-authors=etal}}</ref> || 2012 || 100&nbsp;TeV || colspan=2 style="text-align:center"|{{10^|−13}}
|-
| Huo ''et al.''<ref>{{Cite journal|author1=Huo, Yunjie |author2=Li, Tianjun |author3=Liao, Yi |author4=Nanopoulos, Dimitri V. |author5=Qi, Yonghui |title=Constraints on neutrino velocities revisited|journal=Physical Review D|volume=85|issue=3|pages=034022 | doi=10.1103/PhysRevD.85.034022|arxiv=1112.0264|year=2012|bibcode = 2012PhRvD..85c4022H |s2cid=118501796 }}</ref> || 2012 || 400&nbsp;TeV || <7.8{{e|−12}} ||
|-
| [[ICARUS (experiment)|ICARUS]]<ref>{{Cite journal|author=ICARUS Collaboration|title=A search for the analogue to Cherenkov radiation by high energy neutrinos at superluminal speeds in ICARUS |journal=Physics Letters B|volume=711|issue=3–4| pages=270–275 | doi=10.1016/j.physletb.2012.04.014|arxiv=1110.3763|year=2012|bibcode = 2012PhLB..711..270I |s2cid=118357662 }}</ref> || 2011 || 17&nbsp;GeV || <2.5{{e|−8}} ||
|-
| Cowsik ''et al.''<ref>{{Cite journal|author1=Cowsik, R. |author2=Nussinov, S. |author3=Sarkar, U. |title=Superluminal neutrinos at OPERA confront pion decay kinematics |year=2011|journal=Physical Review Letters|volume=107|issue=25|pages=251801 |doi=10.1103/PhysRevLett.107.251801|arxiv=1110.0241|bibcode=2011PhRvL.107y1801C |pmid=22243066|s2cid=6226647 }}</ref> || 2011 || 400&nbsp;TeV || || {{10^|−12}}
|-
| Bi ''et al.''<ref>{{Cite journal|author1=Bi, Xiao-Jun |author2=Yin, Peng-Fei |author3=Yu, Zhao-Huan |author4=Yuan, Qiang |title=Constraints and tests of the OPERA superluminal neutrinos|year=2011|journal=Physical Review Letters|volume=107| issue=24|pages=241802|doi=10.1103/PhysRevLett.107.241802|arxiv=1109.6667|bibcode=2011PhRvL.107x1802B |pmid=22242991|s2cid=679836 }}</ref> || 2011 || 400&nbsp;TeV || || {{10^|−12}}
|-
| Cohen/Glashow<ref name=coh>{{cite journal |author1=Cohen, Andrew G. |author2=Glashow, Sheldon L. |title=Pair Creation Constrains Superluminal Neutrino Propagation|journal=Physical Review Letters|volume=107|issue=18|year=2011 |pages=181803 |doi=10.1103/PhysRevLett.107.181803|arxiv=1109.6562|bibcode=2011PhRvL.107r1803C |pmid=22107624|s2cid=56198539 }}</ref> || 2011 || 100&nbsp;TeV || <1.7{{e|−11}} ||
|}

==Reports of alleged Lorentz violations==

=== Open reports ===

; LSND, MiniBooNE 
In 2001, the [[LSND]] experiment observed a 3.8σ excess of antineutrino interactions in neutrino oscillations, which contradicts the standard model.<ref>{{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 }}</ref> 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.<ref>{{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 }}</ref> In 2008, however, they reported an excess of electron-like neutrino events between 200 and 475 MeV.<ref>{{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 }}</ref> 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.<ref>{{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}}</ref><ref>{{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 }}</ref> Whether those anomalies can be explained by [[sterile neutrino]]s, or whether they indicate Lorentz violations, is still discussed and subject to further theoretical and experimental researches.<ref>{{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}}</ref>

=== Solved reports ===
In 2011 the [[OPERA experiment|OPERA Collaboration]] published (in a [[peer review|non-peer reviewed]] [[arXiv]] preprint) the results of neutrino measurements, according to which neutrinos were traveling slightly [[faster than light]].<ref name="neutrino velocity">{{cite arXiv<!--Intentionally citing the preprint -->|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}}</ref> The neutrinos apparently arrived early by ~60&nbsp;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;<ref>{{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}}</ref> 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.<ref>{{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}}</ref><ref>{{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}}</ref><ref>{{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 }}</ref> However, in 2011 MINOS updated their antineutrino results; after evaluating additional data, they reported that the difference is not as great as initially thought.<ref>{{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}}</ref> In 2012, they published a paper in which they reported that the difference is now removed.<ref>{{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 }}</ref>

In 2007, the [[MAGIC (telescope)|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.<ref name=magic /><ref>
{{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
}}</ref>
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]].<ref name=lat/> For details, see section [[#Dispersion|Dispersion]].

In 1997, Nodland & Ralston claimed to have found a rotation of the polarization plane of light coming from distant [[Radio galaxy|radio galaxies]]. This would indicate an anisotropy of space.<ref>{{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 }}</ref><ref>{{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 }}</ref><ref>Borge Nodland, John P. Ralston (1997), ''Response to Leahy's Comment on the Data's Indication of Cosmological Birefringence'', {{arxiv|astro-ph/9706126}}</ref>
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.<ref>{{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}}</ref><ref>{{cite web |author=Ted Bunn |url=https://facultystaff.richmond.edu/~ebunn/biref/ |title=Is the Universe Birefringent?}}</ref><ref>{{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}}</ref><ref>{{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}}</ref><ref>J. P. Leahy: (1997) ''Comment on the Measurement of Cosmological Birefringence'', {{arxiv|astro-ph/9704285}}</ref><ref>{{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}}</ref><ref>{{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}}</ref>
More recent studies have not found any evidence for this effect (see section on [[#Vacuum birefringence|Birefringence]]).

==See also==
*[[Tests of special relativity]]
*[[Phenomenological quantum gravity]]

==References==
{{reflist}}

==External links==
*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]]