Tests of Lorentz invariance: a 2013 update

We present an updated review of Lorentz invariance tests in effective field theories (EFTs) in the matter as well as in the gravity sector. After a general discussion of the role of Lorentz invariance and a derivation of its transformations along the so-called von Ignatovski theorem, we present the dynamical frameworks developed within local EFT and the available constraints on the parameters governing the Lorentz breaking effects. In the end, we discuss two specific examples: the OPERA ‘affaire’ and the case of Hořava–Lifshitz gravity. The first case will serve as an example, and a caveat, of the practical application of the general techniques developed for constraining Lorentz invariance violation to a direct observation potentially showing these effects. The second case will show how the application of the same techniques to a specific quantum gravity scenario has far-reaching implications not foreseeable in a purely phenomenological EFT approach.

[1]  Lee Smolin,et al.  Principle of relative locality , 2011, 1101.0931.

[2]  L. Sindoni.  Emergent gravitational dynamics from multi-BEC hydrodynamics? , 2010, 1011.4411.

[3]  P. Sokolsky.  Observation of the GZK cutoff by the HiRes Experiment , 2009 .

[4]  B. Ward.  Estimates of radiation by superluminal neutrinos , 2012, 1201.1322.

[5]  G. W. Gibbons,et al.  General very special relativity is Finsler geometry , 2007, 0707.2174.

[6]  J. Bjorken.  A Dynamical origin for the electromagnetic field , 1963 .

[7]  J. Pullin,et al.  Small Lorentz violations in quantum gravity: do they lead to unacceptably large effects? , 2011, 1106.1417.

[8]  J. Ellis,et al.  Erratum (astro-ph/0510172): Robust Limits on Lorentz Violation from Gamma-Ray Bursts , 2007, 0712.2781.

[9]  G. Amelino-Camelia.  Relativity in space-times with short distance structure governed by an observer independent (Planckian) length scale , 2000, gr-qc/0012051.

[10]  J. Ellis,et al.  Uncertainties in the proton lifetime , 1980 .

[11]  Matt Visser,et al.  Faster-than-c Signals, Special Relativity, and Causality , 2002 .

[12]  T. Jacobson,et al.  New limits on Planck scale Lorentz violation in QED. , 2003, Physical review letters.

[13]  A. Perez.  Minimal Length Scale Scenarios for Quantum Gravity , 2013 .

[14]  S. Hossenfelder.  Comment on "Relative locality and the soccer ball problem" , 2013 .

[15]  P. Lipari,et al.  OPERA-REASSESSING DATA ON THE ENERGY DEPENDENCE OF THE SPEED OF NEUTRINOS , 2011, 1109.5172.

[16]  Lee Smolin,et al.  Relative locality: a deepening of the relativity principle , 2011, 1106.0313.

[17]  S. Covino,et al.  Constraints on Lorentz Invariance Violation using integral/IBIS observations of GRB041219A , 2011, 1106.1068.

[18]  J. Collins,et al.  Lorentz invariance: an additional fine-tuning problem , 2004 .

[19]  Q. Yuan,et al.  Constraints and tests of the OPERA superluminal neutrinos. , 2011, Physical review letters.

[20]  Distance Measurement and Wave Dispersion in a Liouville-String Approach to Quantum Gravity , 1996, hep-th/9605211.

[21]  Sidney Coleman,et al.  High-energy tests of Lorentz invariance , 1999 .

[22]  L. Maccione,et al.  Scale hierarchy in Hořava-Lifshitz gravity: strong constraint from synchrotron radiation in the Crab Nebula. , 2012, Physical review letters.

[23]  U. Sarkar,et al.  Testing Violations of Lorentz Invariance with Cosmic-Rays , 2012, 1206.0713.

[24]  T. Jacobson.  Erratum: Extended Hořava gravity and Einstein-aether theory [Phys. Rev. D 81, 101502 (2010)] , 2010 .

[25]  C. Rovelli,et al.  Reconcile Planck scale discreteness and the Lorentz-Fitzgerald contraction , 2002, gr-qc/0205108.

[26]  J. Tasson,et al.  Prospects for large relativity violations in matter-gravity couplings. , 2008, Physical review letters.

[27]  J. Kowalski-Glikman.  Curved Momentum Space and Relative Locality , 2012, 1205.1304.

[28]  H. Kitamura,et al.  Extension of the Cosmic-Ray Energy Spectrum Beyond the Predicted Greisen-Zatsepin-Kuz'min Cutoff , 1998, astro-ph/9807193.

[29]  L. Sindoni,et al.  Emergence of Lorentzian signature and scalar gravity , 2008, 0806.4239.

[30]  T. Jacobson,et al.  A strong astrophysical constraint on the violation of special relativity by quantum gravity , 2003, Nature.

[31]  R. Sundrum.  From Fixed Points to the Fifth Dimension , 2011, 1106.4501.

[32]  K. Greisen.  End to the cosmic ray spectrum , 1966 .

[33]  Physical and Cosmological Implications of a Possible Class of Particles Able to Travel Faster than Light , 1996, astro-ph/9601090.

[34]  A. Kostelecký,et al.  Electrodynamics with Lorentz-violating operators of arbitrary dimension , 2009, 0905.0031.

[35]  Probing The Structure of Space-Time with Cosmic Rays , 2000, astro-ph/0001258.

[36]  H. Nielsen,et al.  Lorentz non-invariance , 1983 .

[37]  J. Lévy-Leblond.  One more derivation of the Lorentz transformation , 1976 .

[38]  P. Dirac,et al.  Is there an Æther? , 1951, Nature.

[39]  KINEMATICAL SOLUTION OF THE UHE-COSMIC-RAY PUZZLE WITHOUT A PREFERRED CLASS OF INERTIAL OBSERVERS , 2002, astro-ph/0209232.

[40]  M. Romalis,et al.  New limit on Lorentz- and CPT-violating neutron spin interactions. , 2010, Physical review letters.

[41]  G. Sigl,et al.  Lorentz Violation in the Photon Sector and Ultra-High Energy Cosmic Rays , 2007 .

[42]  G. Bogoslovsky.  Lorentz symmetry violation without violation of relativistic symmetry , 2005, hep-th/0511151.

[43]  Generalized Lorentz invariance with an invariant energy scale , 2002, gr-qc/0207085.

[44]  An interferometric gravitational wave detector as a quantum-gravity apparatus , 1998, gr-qc/9808029.

[45]  V. Tioukov,et al.  Measurement of the neutrino velocity with the OPERA detector in the CNGS beam , 2011, 1109.4897.

[46]  Dynamical formation of horizons in recoiling D-branes , 2000, gr-qc/0006004.

[47]  O W Greenberg.  CPT violation implies violation of Lorentz invariance. , 2002, Physical review letters.

[48]  J. Ellis,et al.  Probing Quantum Gravity using Photons from a Mkn 501 Flare Observed by MAGIC , 2007 .

[49]  Tatehiro Mihara,et al.  Strict limit on CPT violation from polarization of γ-ray bursts. , 2012, Physical review letters.

[50]  J. Ellis,et al.  Space-Time Foam may Violate the Principle of Equivalence , 2003, gr-qc/0312044.

[51]  Alan D. Martin,et al.  Review of Particle Physics , 2010 .

[52]  W. H. McMaster,et al.  MATRIX REPRESENTATION OF POLARIZATION , 1961 .

[53]  T. Jacobson.  Extended Horava gravity and Einstein-aether theory , 2010, 1001.4823.

[54]  E. Barausse,et al.  No-go theorem for slowly rotating black holes in Hořava-Lifshitz gravity. , 2012, Physical review letters.

[55]  L. Chappell.  George F. Will , 1997 .

[56]  John Ellis,et al.  String theory modifies quantum mechanics , 1992 .

[57]  R. Montemayor,et al.  Synchrotron radiation in Myers-Pospelov effective electrodynamics , 2004, hep-ph/0410143.

[58]  M. Pospelov,et al.  On Lorentz violation in Horava-Lifshitz type theories , 2010, 1010.5249.

[59]  R. Bluhm,et al.  Constraints and stability in vector theories with spontaneous Lorentz violation , 2008, 0802.4071.

[60]  Gonzalo Palma,et al.  Loop quantum gravity and ultrahigh energy cosmic rays , 2003 .

[61]  S. Carroll,et al.  Noncommutative field theory and Lorentz violation. , 2001, Physical review letters.

[62]  J. Ellis,et al.  Exploration of possible quantum gravity effects with neutrinos II: Lorentz violation in neutrino propagation , 2009, 0903.5048.

[63]  T. M. Roganova,et al.  Upper limit on the ultrahigh-energy photon flux from AGASA and yakutsk data , 2006 .

[64]  T. Jacobson,et al.  Gravity with a dynamical preferred frame , 2000, gr-qc/0007031.

[65]  L. Maccione,et al.  Violations of Lorentz invariance in the neutrino sector after OPERA , 2011 .

[66]  D. Hooper,et al.  Cosmogenic photons as a test of ultra-high energy cosmic ray composition , 2010, 1007.1306.

[67]  V. Tioukov,et al.  Measurement of the neutrino velocity with the OPERA detector in the CNGS beam using the 2012 dedicated data , 2012, 1212.1276.

[68]  Igor G. Mitrofanov,et al.  Condensed-matter physics (communication arising): Spurious magnetism in high-Tc superconductor , 2003, Nature.

[69]  A. Peters,et al.  Measurement of gravitational acceleration by dropping atoms , 1999, Nature.

[70]  Stefano Liberati,et al.  Lorentz Violation: Motivation and New Constraints , 2009, 0906.0681.

[71]  S. Covino,et al.  The polarized gamma-ray burst GRB 061122 , 2013, 1303.4186.

[72]  Longo,et al.  New precision tests of the Einstein equivalence principle from SN1987A. , 1988, Physical review letters.

[73]  Einstein-Aether theory , 2004, gr-qc/0410001.

[74]  A. Kostelecký.  Gravity, Lorentz violation, and the standard model , 2003, hep-th/0312310.

[75]  T. Jacobson.  Einstein-aether gravity: a status report , 2008, 0801.1547.

[76]  Reinaldo J. Gleiser,et al.  Astrophysical limits on quantum gravity motivated birefringence , 2001 .

[77]  M. Chaichian,et al.  CPT violation does not lead to violation of Lorentz invariance and vice versa , 2011, 1103.0168.

[78]  W. Unruh.  Experimental black hole evaporation , 1981 .

[79]  Relativistic Gravity With a Dynamical Preferred Frame , 2001, gr-qc/0112012.

[80]  Loop-generated bounds on changes to the graviton dispersion relation , 2002, hep-ph/0201082.

[81]  Tasi lectures on D-branes , 1996, hep-th/9611050.

[82]  M. Antonelli,et al.  A search for directional violations of the Lorentz invariance through the study of a possible asymmetry of particle lifetimes , 2010, 1011.3720.

[83]  S. Troitsky,et al.  Muon content of ultrahigh-energy air showers: Yakutsk data versus simulations , 2007, 0710.5508.

[84]  J. Polchinski.  Comment on ‘Small Lorentz violations in quantum gravity: do they lead to unacceptably large effects?’ , 2011, 1106.6346.

[85]  Rodolfo Gambini,et al.  Nonstandard optics from quantum space-time , 1999 .

[86]  F. Stecker,et al.  Searching for new physics with ultrahigh energy cosmic rays , 2009, 0906.1735.

[87]  E. Barausse,et al.  No-go theorem for slowly rotating black holes in Hořava-Lifshitz gravity. , 2012, Physical review letters.

[88]  Stefano Liberati,et al.  Gamma-ray polarization constraints on Planck scale violations of special relativity , 2008, 0809.0220.

[89]  S. Barwick.  ARIANNA: A New Concept for UHE Neutrino Detection , 2006, astro-ph/0610631.

[90]  L. Stodolsky.  The speed of light and the speed of neutrinos , 1988 .

[91]  Daniel Sudarsky,et al.  Lorentz invariance and quantum gravity: an additional fine-tuning problem? , 2004, Physical review letters.

[92]  D. Anselmi.  Weighted scale invariant quantum field theories , 2008, 0801.1216.

[93]  F. T. Collaboration,et al.  Measurement of the UHECR energy spectrum using data from the surface detector of the Pierre Auger Observatory , 2007, 0706.2096.

[94]  Assaf Shomer,et al.  A Pedagogical explanation for the non-renormalizability of gravity , 2007, 0709.3555.

[95]  A. Dabrowska,et al.  Measurement of the neutrino velocity with the ICARUS detector at the CNGS beam , 2012, Physics Letters B.

[96]  D. Mattingly.  Have we tested Lorentz invariance enough , 2008, 0802.1561.

[97]  L. Maccione,et al.  Planck-scale Lorentz violation constrained by Ultra-High-Energy Cosmic Rays , 2009, 0902.1756.

[98]  G. Rubtsov,et al.  Calculation of cross sections in Lorentz-violating theories , 2012, 1204.5782.

[99]  Minimal modifications of the primordial power spectrum from an adiabatic short distance cutoff , 2002, hep-th/0206149.

[100]  C. Lämmerzahl,et al.  Constraining the energy-momentum dispersion relation with Planck-scale sensitivity using cold atoms. , 2009, Physical review letters.

[101]  Roberto Torretti,et al.  Relativity and geometry , 1984 .

[102]  N. Afshordi.  Cuscuton and low energy limit of Horava-Lifshitz gravity , 2009, 0907.5201.

[103]  V. Gorini,et al.  Reciprocity Principle and the Lorentz Transformations , 1969 .

[104]  CPT Violation and decoherence in quantum gravity , 2009, 0904.0606.

[105]  D. Ikeda.  Recent Results from Telescope Array , 2012 .

[106]  Synchrotron radiation from the Crab Nebula discriminates between models of space–time foam , 2003, astro-ph/0308403.

[107]  THE ROBUSTNESS OF INFLATION TO CHANGES IN SUPER-PLANCK-SCALE PHYSICS , 2000, astro-ph/0005432.

[108]  T. Sotiriou.  Hořava-Lifshitz gravity: a status report , 2010, 1010.3218.

[109]  M. Romalis,et al.  New test of local Lorentz invariance using a 21Ne-Rb-K comagnetometer. , 2011, Physical review letters.

[110]  J. Stephen,et al.  Polarized Gamma-Ray Emission from the Crab , 2008, Science.

[111]  J. Niemeyer,et al.  Transplanckian dispersion and scale invariance of inflationary perturbations , 2001, astro-ph/0101451.

[112]  Michele Maltoni,et al.  Atmospheric Neutrino Oscillations and New Physics , 2004 .

[113]  WAVES ON NONCOMMUTATIVE SPACE–TIME AND GAMMA-RAY BURSTS , 1999, hep-th/9907110.

[114]  Einstein-aether waves , 2004, gr-qc/0402005.

[115]  G. Sigl,et al.  Lorentz violation and ultrahigh-energy photons , 2008, 0807.1210.

[116]  A. Kostelecký,et al.  Data Tables for Lorentz and CPT Violation , 2008, 0801.0287.

[117]  Post-Newtonian parameters and constraints on Einstein-aether theory , 2005, gr-qc/0509083.

[118]  I. Area,et al.  Some light on "Measurement of the neutrino velocity with the OPERA detector in the CNGS beam" , 2011 .

[119]  About Lorentz invariance in a discrete quantum setting , 2004, gr-qc/0405085.

[120]  Chu,et al.  Atomic interferometry using stimulated Raman transitions. , 1991, Physical review letters.

[121]  D. Blokhint͡sev.  BASIS FOR SPECIAL RELATIVITY THEORY PROVIDED BY EXPERIMENTS IN HIGH ENERGY PHYSICS , 1966 .

[122]  Luis Javier Garay Elizondo,et al.  Quantum-gravity and minimum length , 1995 .

[123]  M. Visser,et al.  Analogue Gravity , 2005, Living reviews in relativity.

[124]  D. Mazon,et al.  Locality and the relativity principle beyond special relativity , 2011, 1107.0939.

[125]  The Pierre Auger Collaboration.  Upper limit on the cosmic-ray photon flux above 10^19 eV using the surface detector of the Pierre Auger Observatory , 2007, 0712.1147.

[126]  Stefano Liberati,et al.  Relativistic Bose–Einstein condensates: a new system for analogue models of gravity , 2010, 1001.1044.

[127]  J. Provost,et al.  Additivity, rapidity, relativity , 1979 .

[128]  S. Glashow,et al.  Pair creation constrains superluminal neutrino propagation. , 2011, Physical review letters.

[129]  G. Gelmini,et al.  On photon splitting in theories with Lorentz invariance violation , 2005, hep-ph/0503130.

[130]  G. Gelmini,et al.  GZK photons as ultra-high-energy cosmic rays , 2005, astro-ph/0506128.

[131]  G. Gelmini,et al.  GZK photons in the minimal ultra-high energy cosmic rays model , 2007, astro-ph/0702464.

[132]  J. Ellis,et al.  Comments on ultra-high-energy photons and D-foam models , 2010, 1004.4167.

[133]  M. Phillips,et al.  Observational Evidence from Supernovae for an Accelerating Universe and a Cosmological Constant , 1998, astro-ph/9805201.

[134]  V. Baccetti,et al.  Lorentz violating kinematics: threshold theorems , 2011, 1111.6340.

[135]  W. Kluźniak.  Transparency of the universe to TeV photons in some models of quantum gravity , 1999 .

[136]  Felix Aharonian,et al.  The Crab Nebula and Pulsar between 500 GeV and 80 TeV: Observations with the HEGRA Stereoscopic Air Cerenkov Telescopes , 2004 .

[137]  L. Smolin,et al.  Relative locality and the soccer ball problem , 2011, 1104.2019.

[138]  I. H. Park,et al.  SEARCH FOR ANISOTROPY OF ULTRAHIGH ENERGY COSMIC RAYS WITH THE TELESCOPE ARRAY EXPERIMENT , 2012, 1205.5984.

[139]  Lorentz invariance as a low energy phenomenon , 1983 .

[140]  Steven Weinberg.  Quantum contributions to cosmological correlations , 2005 .

[141]  M. Serone,et al.  Renormalization group in Lifshitz-type theories , 2009, 0906.3477.

[142]  M. Visser.  Power-counting renormalizability of generalized Horava gravity , 2009, 0912.4757.

[143]  R. Parentani.  Constructing QFT's wherein Lorentz Invariance is broken by dissipative effects in the UV , 2007, 0709.3943.

[144]  Alan D. Martin,et al.  Review of Particle Physics , 2014 .

[145]  W. C. Brown,et al.  Measurement of the Depth of Maximum of Extensive Air Showers above 1018 eV ( The Pierre Auger Collaboration ) , 2022 .

[146]  F. Aharonian et al..  The Crab Nebula and Pulsar between 500 GeV and 80 TeV: Observations with the HEGRA stereoscopic air Cherenkov telescopes , 2004 .

[147]  Eric Perlmutter,et al.  Black holes in three dimensional higher spin gravity: a review , 2012, 1208.5182.

[148]  L. Maccione,et al.  Ultrahigh-energy photons as probes of Lorentz symmetry violations in stringy space-time foam models. , 2010, Physical review letters.

[149]  J. Ellis,et al.  Probes of Lorentz violation in neutrino propagation , 2008, 0805.0253.

[150]  S. Sonego,et al.  Foundations of anisotropic relativistic mechanics , 2008, 0812.1294.

[151]  A. Banday,et al.  Dipole Anisotropy in the COBE Differential Microwave Radiometers First-Year Sky Maps , 1993 .

[152]  Petr Hořava.  Quantum Gravity at a Lifshitz Point , 2009, 0901.3775.

[153]  E. Adelberger,et al.  Preferred-Frame and CP-Violation Tests with Polarized Electrons , 2007, 0808.2673.

[154]  W. Huelsnitz.  Search for Quantum Gravity with IceCube and High Energy Atmospheric Neutrinos , 2010 .

[155]  C. Kennel,et al.  Confinement of the Crab pulsar's wind by its supernova remnant , 1984 .

[156]  Varying G and Other Constants , 1997, gr-qc/9711084.

[157]  Robert C Myers,et al.  Ultraviolet modifications of dispersion relations in effective field theory. , 2003, Physical review letters.

[158]  Harvey R. Brown,et al.  Physical Relativity: Space-Time Structure from a Dynamical Perspective , 2005 .

[159]  David Mattingly,et al.  Modern Tests of Lorentz Invariance , 2005, Living reviews in relativity.

[160]  M. Gasperini.  Broken Lorentz symmetry and the dimension of space-time , 1986 .

[161]  G. Moore,et al.  Constraining the New Aether: gravitational Cherenkov radiation , 2005, hep-ph/0505211.

[162]  A. Polyakov,et al.  The string dilation and a least coupling principle , 1994, hep-th/9401069.

[163]  β-Function in a non-covariant Yang-Mills theory , 1978 .

[164]  S. Hossenfelder.  Comment on arXiv:1104.2019,"Relative locality and the soccer ball problem,"by Amelino-Camelia et al , 2012, 1202.4066.

[165]  P. Cochat,et al.  Et al , 2008, Archives de pediatrie : organe officiel de la Societe francaise de pediatrie.

[166]  G. Amelino-Camelia.  Gravity-wave interferometers as quantum-gravity detectors , 1999, Nature.

[167]  Joe Henson,et al.  QUANTUM GRAVITY PHENOMENOLOGY, LORENTZ INVARIANCE AND DISCRETENESS , 2003, gr-qc/0311055.

[168]  J. L. Kelley,et al.  Searching for Quantum Gravity with High Energy Atmospheric Neutrinos and AMANDA-II , 2008 .

[169]  D. Mazon,et al.  Uncertainties in Constraints from Pair Production on Superluminal Neutrinos , 2012, 1203.2585.

[170]  W C Brown,et al.  Correlation of the highest-energy cosmic rays with nearby extragalactic objects. , 2007, Science.

[171]  M. Pospelov,et al.  EXPERIMENTAL CHALLENGES FOR QUANTUM GRAVITY , 2004, gr-qc/0402028.

[172]  Clifford M. Will,et al.  The Confrontation between General Relativity and Experiment , 2001, Living reviews in relativity.

[173]  G. Sigl,et al.  Lorentz violation for photons and ultrahigh-energy cosmic rays. , 2007, Physical review letters.

[174]  M. Visser.  Lorentz symmetry breaking as a quantum field theory regulator , 2009, 0902.0590.

[175]  F. Stecker,et al.  Lorentz Invariance Violation and the Observed Spectrum of Ultrahigh Energy Cosmic Rays , 2008, 0811.2230.

[176]  John Ellis,et al.  Tests of quantum gravity from observations of γ-ray bursts , 1998, Nature.

[177]  Floyd W. Stecker,et al.  A new limit on Planck scale Lorentz violation from γ-ray burst polarization , 2011, 1102.2784.

[178]  J. P. Rodrigues,et al.  Search for a Lorentz-violating sidereal signal with atmospheric neutrinos in IceCube , 2010, Physical Review D.

[179]  J Schovancova,et al.  Measurement of the depth of maximum of extensive air showers above 10{18} eV. , 2010, Physical review letters.

[180]  M. Gasperini.  Inflation and broken Lorentz symmetry in the very early universe , 1985 .

[181]  F. W. Stecker,et al.  New tests of Lorentz invariance following from observations of the highest energy cosmic γ-rays , 2001 .

[182]  F. Klinkhamer,et al.  New two-sided bound on the isotropic Lorentz-violating parameter of modified Maxwell theory , 2008, 0809.3217.

[183]  Lorentz Symmetry Violation and High-Energy Cosmic Rays , 1997, physics/9712005.

[184]  D. Blas,et al.  Models of non-relativistic quantum gravity: the good, the bad and the healthy , 2010, 1007.3503.

[185]  A. Celotti,et al.  New constraints on Planck-scale Lorentz violation in QED from the Crab Nebula , 2007, 0707.2673.

[186]  E. Barausse,et al.  Erratum: No-Go Theorem for Slowly Rotating Black Holes in Hor̆ava-Lifshitz Gravity [Phys. Rev. Lett.109, 181101 (2012)] , 2013 .

[187]  Ghost condensation and a consistent infrared modification of gravity , 2003, hep-th/0312099.

[188]  J. Ellis,et al.  Potential Sensitivity of Gamma-Ray Burster Observations to Wave Dispersion in Vacuo , 1997, astro-ph/9712103.

[189]  Alan Kostelecky,et al.  Neutrinos with Lorentz-violating operators of arbitrary dimension , 2011, 1112.6395.

[190]  A. Zee.  Perhaps proton decay violates Lorentz invariance , 1982 .

[191]  K. Sfetsos,et al.  Relativistic Quantum Gravity at a Lifshitz Point , 2009, 0906.1201.

[192]  Alan Kostelecky,et al.  Lorentz-Violating Extension of the Standard Model , 1998 .

[193]  CPT violation and decoherence in quantum gravity , 2004, gr-qc/0407005.

[194]  H. Nielsen,et al.  The Rédei-like model and testing Lorentz invariance , 1982 .

[195]  D. Blas,et al.  Consistent extension of Horava gravity. , 2009, Physical review letters.

[196]  Samuel,et al.  Spontaneous breaking of Lorentz symmetry in string theory. , 1989, Physical review. D, Particles and fields.

[197]  J. Lukierski,et al.  Classical and Quantum Mechanics of Free κ-Relativistic Systems , 1993, hep-th/9312153.

[198]  M. C. Maccarone,et al.  Correlation of the highest energy cosmic rays with nearby extragalactic objects , 2007 .

[199]  Threshold effects and Planck scale Lorentz violation: Combined constraints from high-energy astrophysics , 2002, hep-ph/0209264.

[200]  Fate of Lorentz symmetry in relative-locality momentum spaces , 2011, 1110.5081.

[201]  L. Smolin.  Could deformed special relativity naturally arise from the semiclassical limit of quantum gravity , 2008, 0808.3765.

[202]  M. Gasperini.  Singularity prevention and broken Lorentz symmetry , 1987 .

[203]  W. Yao,et al.  Review of Particle Physics (2006) , 2006 .

[204]  Spacetime discreteness, Lorentz invariance and locality , 2011 .

[205]  G. Tino,et al.  Probing the quantum-gravity realm with slow atoms , 2010, 1004.0847.

[206]  L. Maccione,et al.  Possible cosmogenic neutrino constraints on Planck-scale Lorentz violation , 2009, 0911.0521.

[207]  L. Maccione,et al.  GZK photon constraints on Planck-scale Lorentz violation in QED , 2008, 0805.2548.

[208]  E. al.,et al.  Measurement of neutrino velocity with the MINOS detectors and NuMI neutrino beam , 2007, 0706.0437.

[209]  C. Barceló,et al.  A Real Lorentz-FitzGerald Contraction , 2007, 0705.4652.

[210]  M. Pospelov,et al.  Lorentz violation in supersymmetric field theories. , 2004, Physical review letters.

[211]  F. T. Collaboration,et al.  Highlights from the Pierre Auger Observatory , 2011, 1310.4620.

[212]  R. Ellis,et al.  Measurements of $\Omega$ and $\Lambda$ from 42 high redshift supernovae , 1998, astro-ph/9812133.

[213]  A. Dabrowska,et al.  Precision measurement of the neutrino velocity with the ICARUS detector in the CNGS beam , 2012, 1208.2629.

[214]  V. Mitsou,et al.  A Search in Gamma-Ray Burst Data for Nonconstancy of the Velocity of Light , 2000 .

[215]  I. Grenier,et al.  Polarization of the Crab Pulsar and Nebula as Observed by the INTEGRAL/IBIS Telescope , 2008, 0809.1292.

[216]  L. Sindoni,et al.  Gravitational dynamics in Bose Einstein condensates , 2008, 0807.4910.

[217]  Stefano Liberati,et al.  Lorentz violation at high energy: Concepts, phenomena, and astrophysical constraints , 2006 .

[218]  S. Sibiryakov,et al.  Emergent Lorentz invariance from strong dynamics: holographic examples , 2013, 1305.0011.

[219]  G. Prodi,et al.  Gravitational bar detectors set limits to Planck-scale physics on macroscopic variables , 2012, Nature Physics.

[220]  M. Visser,et al.  Quantum gravity without Lorentz invariance , 2009, 0905.2798.