# Modern searches for Lorentz violation

Modern searches for Lorentz violation

Important motivations for modern searches for Lorentz violation are deviations from Lorentz invariance (and thus special relativity) predicted by some variations of quantum gravity, string theory, and some alternatives to general relativity. Besides 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, also the predictions of the standard model of particle physics are considered to be modified by possible Lorentz violations.

To assess and predict possible violations, test theories of special relativity and effective field theories (EFT) such as the Standard-Model Extension (SME), have been invented. These models operate by introducing preferred frame effects. For example, modifications of the dispersion relation should lead to differences between the limiting velocity of matter and the speed of light. Also the parameterized post-Newtonian formalism as a test theory for general relativity can be used to describe preferred frame effects. Another model including different Lorentz violations is doubly special relativity (DSR), which preserves the Planck energy as an invariant maximum energy-scale, yet without having a preferred reference frame.

Both terrestrial as well as astronomical experiments have been carried out, and new experimental techniques have been introduced. No Lorentz violations could be measured thus far, and exceptions in which positive results were reported, have been refuted or lack further confirmations. For discussions of many experiments, see Mattingly (2005).[1] For a detailed list of results of recent experimental searches, see Alan Kostelecký and Russell (2011).[2] See also the main article Tests of special relativity.

## Isotropy of the speed of light

Experiments with optical resonators, by which deviations from the isotropy of the speed of light are tested, can be seen as modern variants of the Michelson–Morley experiment to test orientation dependence of the speed of light, and the Kennedy–Thorndike experiment to test velocity dependence. The anisotropy parameters are given using the Robertson-Mansouri-Sexl test theory (RMS), which allows to distinguish between the relevant orientation- and velocity dependent values. The current precision, by which an anisotropy of the speed of light can be excluded, is at the 10 − 17 level. This is related to the relative velocity between the solar system and the rest frame of the cosmic microwave background radiation of ∼368 km/s (see also Resonator Michelson–Morley experiments)

Orientation dependence
Author Year RMS bounds Anisotropy of c
Herrmann et al.[3] 2009 $\scriptstyle(4\pm8)\times10^{-12}$
$\scriptstyle\lesssim10^{-17}$
Eisele et al.[4] 2009 $\scriptstyle(-1.6\pm6\pm1.2)\times10^{-12}$
Müller et al.[5] 2007
$\scriptstyle\lesssim10^{-16}$
Stanwix et al.[6] 2006 $\scriptstyle9.4(8.1)\times10^{-11}$
Herrmann et al.[7] 2005 $\scriptstyle(-2.1\pm1.9)\times10^{-10}$
Stanwix et al.[8] 2005 $\scriptstyle-0.9(2.0)\times10^{-10}$
Antonini et al.[9] 2005 $\scriptstyle(+0.5\pm3\pm0.7)\times10^{-10}$
Wolf et al.[10] 2004
$\scriptstyle\lesssim10^{-15}$
Wolf et al.[11] 2004 $\scriptstyle(+1.2\pm2.2)\times10^{-9}$
Müller et al.[12] 2003 $\scriptstyle(+2.2\pm1.5)\times10^{-9}$
Wolf et al.[13] 2003 $\scriptstyle(+1.5\pm4.2)\times10^{-9}$
Brillet & Hall[14] 1979 $\scriptstyle\leq3.2\times10^{-9}$
Velocity dependence
Author Year RMS bounds
Tobar et al. [15] 2009 $\scriptstyle\leq-4,8(3,7)\times10^{-8}$
Wolf et al.[16] 2004 $\scriptstyle\leq(3.7\pm3.0)\times10^{-7}$
Braxmeier et al.[17] 2002 $\scriptstyle\leq(1.9\pm2.1)\times10^{-5}$
Hils and Hall[18] 1990 $\scriptstyle\leq6.6\times10^{-5}$

## Time dilation

The classic time dilation experiments such as the Ives–Stilwell experiment, the Moessbauer rotor experiments, and the Time dilation of moving particles, have been enhanced by modernized equipment. For example, the Doppler shift of lithium ions traveling at high speeds is evaluated by using saturated spectroscopy in heavy ion storage rings. The current precision, with which time dilation is measured, is at the 10 − 8 level. Chou et al. (2010) even managed to measure a frequency shift of ∼10 − 16 due to time dilation, namely at every day's speeds such as 36 km/h.[19] For more information, see Modern Ives–Stilwell experiments.

Author Year Velocity Maximum deviation
from time dilation
Novotny et al.[20] 2009 0,34c $\scriptstyle \leq1.3\times10^{-6}$
Reinhardt et al.[21] 2007 0,064c $\scriptstyle \leq8.4\times10^{-8}$
Saathoff et al.[22] 2003 0,064c $\scriptstyle \leq2.2\times10^{-7}$
Grieser et al.[23] 1994 0,064c $\scriptstyle \leq1\times10^{-6}$

## Clock anisotropy

By this kind of spectroscopy experiments – called Hughes–Drever experiments as well – violations of Lorentz invariance in the interactions of protons and neutrons are investigated, caused by a possible existence of a preferred frame. The energy levels of those nucleons are studied in order to find anisotropies in their frequencies ("clocks"). Clock anisotropy experiments are currently the most sensitive terrestrial ones, because the current precision by which Lorentz violations can be excluded, lies at the 10 − 33 GeV level. Using spin-polarized torsion balances, also anisotropies with respect to electrons can be examined.

Author Year SME-Anisotropy in GeV
Proton Neutron
Smiciklas et al.[24] 2011 10−29
Gemmel et al.[25] 2010 10−32
Brown et al.[26] 2010 10−32 10−33
Altarev et al.[27] 2009 10−20
Wolf et al.[28] 2006 10−25
Canè et al.[29] 2004 10−32
Humphrey et al.[30] 2003 10−27
Walsworth et al.[31] 2000 10−27 10−31
Bear et al.[32] 2000 10−31
Berglund et al.[33] 1995 10−27 10−30
Chupp et al.[34] 1989 10−27
Lamoreaux et al.[35] 1989 10−29
Prestage et al.[36] 1985 10−27
Electron
Author Year SME-Anisotropy
in GeV
Heckel et al.[37] 2008 10−31
Heckel et al.[38] 2006 10−30
Hou et al.[39] 2003 10−29
Humphrey et al.[30] 2003 10−27
Phillips et al.[40] 2000 10−27
Berglund et al.[33] 1995 10−27
Wang et al.[41] 1993 10−27
Wineland et al.[42] 1991 10−25
Phillips[43] 1987 10−27

## Antimatter tests

Lorentz symmetry is closely connected to CPT symmetry, from which a symmetry between matter and antimatter follows. This can be tested by using Penning traps, by which individual charged particles and there counterparts are trapped. Gabrielse et al. (1999) examined cyclotron frequencies in proton-antiproton measurements, and couldn't find any deviation down to $9\cdot10^{-11}$.[44]

Hans Dehmelt et al. tested the anomaly frequency, which plays a fundamental role in the measurement of the electron's gyromagnetic ratio. They searched for sidereal variations, and differences between electrons and positrons as well. Eventually they found no deviations, thereby establishing bounds of 10 − 24 GeV.[45]

Hughes et al. (2001) examined muons for sidereal signals in the spectrum of muons, and found no Lorentz violation down to 10 − 23 GeV.[46] The "Muon g-2" collaboration of the Brookhaven National Laboratory searched for deviations in the anomaly frequency of muons and anti-muons, and for sidereal variations under consideration of Earth's orientation. Also here, no Lorentz violations could be found, with a precision of 10 − 24 GeV.[47]

## Astronomic photon tests

### Dispersion

Dispersion of light (i.e. the dependence of light speed on its energy) from distant astronomic sources as a possible consequence of Lorentz violation has been tested in many experiments. In the context of some models of quantum gravity (QG), it was assumed that such violations should occur at energy levels similar to, or beyond the Planck energy of $\sim1.22\times10^{19}$ GeV. In the following papers, light from gamma ray bursts and from distant galaxies etc. are used to measure such relations. Especially the Fermi-LAT group was able show, that no energy dependence and thus no Lorentz violation occurs in the photon sector even beyond the Planck energy limit, since they observed photon energies up to 31 GeV.[48] So a large class of Lorentz violating quantum gravity models is excluded by that.

Name Year QG-Bounds in GeV
Fermi-LAT-GBM-Collaboration[48] 2009 $\scriptstyle >1.46\times10^{19}$
H.E.S.S.-Collaboration[49] 2008 $\scriptstyle \geq7.2\times10^{17}$
MAGIC-Collaboration[50] 2007 $\scriptstyle \geq0.21\times10^{18}$
Lamon et al.[51] 2008 $\scriptstyle \geq3.2\times10^{11}$
Martinez et al.[52] 2006 $\scriptstyle \geq0.66\times10^{17}$
Ellis et al.[53][54] 2006/8 $\scriptstyle \geq1.4\times10^{16}$
Boggs et al.[55] 2004 $\scriptstyle \geq1.8\times10^{17}$
Ellis et al.[56] 2003 $\scriptstyle \geq6.9\times10^{15}$
Ellis et al.[57] 2000 $\scriptstyle \geq10^{15}$
Schaefer[58] 1999 $\scriptstyle \geq2.7\times10^{16}$
Biller[59] 1999 $\scriptstyle >4\times10^{16}$
Kaaret[60] 1999 $\scriptstyle >1.8\times10^{15}$

### Birefringence

Due to Lorentz violations, such as the presence of an anisotropic space, also vacuum birefringence and parity violations could occur. Researches have been undertaken to detect deviations in the polarization of photons, for instance the rotation of the polarization plane due to velocity differences between left- and right photons. In the following publications, gamma ray bursts, galactic radiation, and the Cosmic microwave background radiation are examined. The SME coefficients $\scriptstyle k_{(V)00}^{(3)}$ and $\scriptstyle k_{(V)00}^{(5)}$ (the latter corresponds to ξ in another EFT[61]) for Lorentz violation are given, where 3 and 5 denote the mass dimensions used. No significant Lorentz violations could be measured up to now.[2]

Name Year SME bounds EFT bounds
Laurent et al.[62] 2011 $\scriptstyle k_{(V)00}^{(5)}\leq1.9\times10^{-33}$ GeV-1 $\scriptstyle \xi\leq1.1\times10^{-14}$
Stecker[61] 2011 $\scriptstyle k_{(V)00}^{(5)}\leq4.2\times10^{-34}$ GeV-1 $\scriptstyle \xi\leq2.4\times10^{-15}$
Kostelecký et al.[63] 2009 $\scriptstyle k_{(V)00}^{(5)}\leq1\times10^{-32}$ GeV-1 $\scriptstyle \xi\leq9\times10^{-14}$
QUaD Collaboration[64] 2008 $\scriptstyle k_{(V)00}^{(3)}\leq2\times10^{-43}$ GeV
Kostelecký et al.[65] 2008 $\scriptstyle k_{(V)00}^{(3)}=(2.3\pm5.4)\times10^{-43}$ GeV
Maccione et al.[66] 2008 $\scriptstyle k_{(V)00}^{(5)}\leq1.5\times10^{-28}$ GeV-1 $\scriptstyle \xi\leq9\times10^{-10}$
Komatsu et al.[67] 2008 $\scriptstyle k_{(V)00}^{(3)}=(1.2\pm2.2)\times10^{-43}$ GeV
Kahniashvili et al.[68] 2008 $\scriptstyle k_{(V)00}^{(3)}\leq2.5\times10^{-43}$ GeV
Xia et al.[69] 2008 $\scriptstyle k_{(V)00}^{(3)}=(2.6\pm1.9)\times10^{-43}$ GeV
Cabella et al.[70] 2007 $\scriptstyle k_{(V)00}^{(3)}=(2.5\pm3.0)\times10^{-43}$ GeV
Fan et al.[71] 2007 $\scriptstyle k_{(V)00}^{(5)}\leq3.4\times10^{-26}$ GeV-1 $\scriptstyle \xi\leq2\times10^{-7}$
Feng et al.[72] 2006 $\scriptstyle k_{(V)00}^{(3)}=(6.0\pm4.0)\times10^{-43}$ GeV
Gleiser et al.[73] 2001 $\scriptstyle k_{(V)00}^{(5)}\leq8.7\times10^{-23}$ GeV-1 $\scriptstyle \xi\leq2\times10^{-4}$
Carroll et al.[74] 1990 $\scriptstyle k_{(V)00}^{(3)}\leq2\times10^{-42}$ GeV

Exceedance of the threshold energy could lead to a difference between the speed of photons, and the limiting velocity of any particle having a charge structure (protons, electrons, neutrinos), since the dispersion relation is assumed to be modified in Lorentz violating EFT models such as SME. Depending on which of these particles travels faster or slower than the speed of light, otherwise forbidden processes can occur:[75][76]

• Decay at superluminal speed of photons. These photons 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.

However, 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 higher significance, even though the bounds are lower (the following bounds describe maximal deviations between the speed of light and the limiting velocity of matter):

Name Year SME bounds Particle Astr./Terr.
Photon decay Cherenkov Synchrotron
Altschul[77] 2009 $\scriptstyle \leq5\times10^{-15}$ Electron Terr.
Hohensee et al.[76] 2009 $\scriptstyle \leq-5.8\times10^{-12}$ $\scriptstyle \leq1.2\times10^{-11}$ Electron Terr.
Klinkhamer & Schreck[78] 2008 $\scriptstyle \leq-9\times10^{-16}$ $\scriptstyle \leq6\times10^{-20}$ UHECR Astr.
Klinkhamer & Risse[79] 2007 $\scriptstyle \leq2\times10^{-19}$ UHECR Astr.
Kaufhold et al.[80] 2007 $\scriptstyle \leq10^{-17}$ UHECR Astr.
Altschul[81] 2005 $\scriptstyle \leq6\times10^{-20}$ Electron Astr.
Gagnon et al.[82] 2004 $\scriptstyle \leq-2\times10^{-21}$ $\scriptstyle \leq5\times10^{-24}$ UHECR Astr.
Jacobson et al.[83] 2003 $\scriptstyle \leq-2\times10^{-16}$ $\scriptstyle \leq5\times10^{-20}$ Electron Astr.
Coleman & Glashow[84] 1997 $\scriptstyle \leq-1.5\times10^{-15}$ $\scriptstyle \leq5\times10^{-23}$ UHECR Astr.

## Neutrino tests

### Neutrino oscillations

Although neutrino oscillations have been experimentally confirmed, the theoretical foundations are still controversial, as it can be seen in the discussion related to sterile neutrinos. This makes predictions of possible Lorentz violations very complicated. However, some succeeded in deriving a possible scenario for neutrino Lorentz violations, such as the EFT of Sidney Coleman & Sheldon Lee Glashow,[84] or SME[85]. 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.

Additionally, a series of investigations was 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:

Name Year SME bounds
in GeV
MiniBooNE[86] 2011 $\scriptstyle \leq10^{-20}$
IceCube[87] 2010 $\scriptstyle \leq10^{-23}$
MINOS[88] 2010 $\scriptstyle \leq10^{-23}$
MINOS[89] 2008 $\scriptstyle \leq10^{-20}$
LSND[90] 2005 $\scriptstyle \leq10^{-19}$

### Neutrino speed

It was thought for a long time in accordance with the standard model, that neutrinos are massless and thus traveling at the speed of light. However, since the discovery of neutrino oscillations, neutrinos are considered as having a certain mass, and thus their speed should be less than the speed of light. Some non-standard models also predict faster than light speeds, for instance, in models where the neutrinos are assumed to be tachyons, or in Lorentz violating models with modified dispersion relations, or models in which neutrinos take "shortcuts" through extra dimensions.

First time of flight measurements were made in the 1970s, and the observed interactions of 25-GeV muon neutrino gave $(v-c)/c < 4\times10^{-5}$ as maximal deviation from the speed of light.[91][92] A much higher agreement with the speed of light was achieved in the course of observations of 10-MeV neutrinos coming from SN 1987A, giving bounds of $(v-c)/c < 2\times10^{-9}$.[93][94] A measurement of the absolute transit time over a distance of 734 km was carried out by MINOS (2007). They determined the speed of 3-GeV neutrinos as being 1,000051(29) c, so they apparently traveled with superluminal speeds, with a deviation from the speed of light of $(v-c)/c = (5.1\pm2.9)2\times10^{-5}$. However, the standard deviation was only 1.8σ, clearly below the 5σ-limit necessary for a significant result. Thus, as argued by the MINOS-group,[95] the deviation is not significant and is also consistent with the speed of light. For the alleged measurement of superluminal neutrinos, see section OPERA below.

Another possible consequence of Lorentz violation could be the occurrence of velocity differences between neutrino flavors. A comparison between muon- and electron-neutrinos by Coleman & Glashow (1998) gave a negative result, with bounds $<6\times10^{-22}$.[84]

## Reports of alleged Lorentz violations

### OPERA

In 2011, the OPERA Collaboration published (in a non-peer reviewed arXiv preprint) the results of neutrino measurements, according to which neutrinos are traveling faster than light.[96] Traversing a distance of 730 km, the neutrinos arrived early by $60.7\pm6.9\pm7.4$ ns. The relative difference to the speed of light was $(v-c)/c=(2.48\pm0.28\pm0.3)\times10^{-5}$, corresponding to ∼7.44 km/s. The standard deviation was 6σ, so unlike the MINOS experiment the result is clearly beyond the 5σ limit necessary for a significant result. They also reported that no significant energy dependence (between 13.9 and 42.9 GeV) of neutrino speeds was measured. They argued that earlier experiments were executed with much lower neutrino energies (see Neutrino speed above). Eventually, they stress that they don't want to draw far reaching conclusions from their experiment, and await further research by the scientific community. Fermilab has announced to repeat the MINOS experiment in the next months to check the OPERA result.[95] Also a series of arXiv preprints[97] concerning this subject have been published.[98]

### MAGIC

In 2007, the MAGIC Collaboration published a paper, in which they claimed a possible energy dependence of the speed of photons from the galaxy Markarian 501. They admitted, that also a possible energy dependent emission effect could have cause this result as well.[50][99] 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.[48] For details, see section Dispersion.

### Nodland & Ralston

In 1997, Nodland & Ralston claimed to have found a rotation of the polarization plane of light coming from distant radio galaxies. This would indicate an anisotropy of space.[100][101][102] 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.[103][104][105][106][107][108][109] More recent researches also haven't found any evidence for this effect, see section Birefringence.

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