Modern tests of Lorentz invariance

David Mattingly

REVIEW

Modern tests of Lorentz invariance

David Mattingly

General Relativity and Quantum Cosmology · Submitted 22 Feb 2005 · 281 references

Pith's verdict: The canonical 2005 survey of Lorentz-violation phenomenology; its real payload is a hedged naturalness argument in §4.3/§8, and the one soft spot is a 10σ table entry waved off on the authority of a preprint.

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arXiv IDgr-qc/0502097 (version 2)

Pith ID4NQWEA73, Pith's permanent record of this paper and its review

CitationDavid Mattingly. "Modern tests of Lorentz invariance." arXiv:gr-qc/0502097 (2005). https://pith.science/paper/4NQWEA73

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Editor's note

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This is the survey you hand a new student who wants to work on Lorentz violation. It is still the cleanest map of the field: frameworks first (SME, RMS, c², DSR, non-systematic dispersion), then terrestrial bounds, then astrophysics, then gravity. The service it performs is not the catalogue but the bookkeeping — every bound is tied to the framework it actually tests, and Mattingly is explicit that a bound in one does not transfer to another. That distinction is where most of the confusion in this literature comes from, and he gets it right. The threshold theorems in §6.5.3, the asymmetric/hard-Čerenkov/upper-threshold taxonomy in §6.5.4, and the order-of-magnitude Čerenkov table are the parts people actually cite, and they hold up on checking. Eqs. 101–102 and 118 are correct as far as I can verify them by hand. The intellectual payload is §4.3 and §8: absent a custodial symmetry, radiative corrections should have dragged Planck-scale violation down to dimension ≤4 at levels we would already have seen. That is a naturalness heuristic, not a derivation, and the text says so — 'it therefore seems hard to believe,' plus the explicit SUSY window (M at Planck, m_s ≤ 1 TeV) and the braneworld escape in [70]. Worth keeping in mind that the reader's framing ('should have generated... at levels experiment would already have seen') is a notch stronger than what Mattingly wrote. The soft spot I would actually flag is small but real: Table 1's κ̃^e−_ZZ = −20 ± 2 ×10⁻¹⁵ is a 10σ nonzero entry dismissed as systematics on the authority of [32], which is cited as an arXiv preprint with no journal reference. That is the one place where the 'no evidence for Lorentz violation' premise leans on something unrefereed. The GZK numbers in §6.5.6 are labelled order-of-magnitude and the author says a proper rate recomputation over the CMBR spectrum has not been done; the §6.4 rate argument is likewise an acknowledged scope limit, with neutrino Čerenkov, photon splitting, and GZK scattering carved out by name. None of that is hidden. For a review this is about as honest as it gets. Send it to referees — the framework-to-bound mapping and the naturalness synthesis deserve the scrutiny, and the Table 1 citation wants a journal reference before it hardens into folklore.

Referee report

Significance

As a review the paper's contribution is not a new derivation but a synthesis, and the synthesis is genuinely useful: it states explicitly which experiment constrains which coefficient of which framework, and it is candid about where the frameworks are incomplete (DSR has no dynamics, §3.4; non-systematic dispersion has no rate calculation, §6.6.2). Two features deserve credit. First, §6.5.2 lists the assumptions (rotation invariance, monotonicity, one energetic incoming particle) before any threshold result is used, rather than burying them. Second, §4.3 confronts the naturalness problem head-on: without a custodial symmetry, dimension <=4 Lorentz-violating operators should have been radiatively generated at levels already excluded, and the paper names the two escape hatches (SUSY+CPT, and the braneworld hiding of [70]) instead of asserting the problem away. The §8 conclusion is properly hedged ('it therefore seems hard to believe'), and the §6.4 discussion of when threshold kinematics alone suffices is quantified (eleven orders of magnitude of tolerance in the matrix element) and then explicitly carved out for neutrino Cerenkov, photon splitting and GZK scattering. The paper also,

Major objections (3)

[§6.2, Eq. (80)] Eq. (80) is dimensionally inconsistent with Eq. (79). Eq. (79) gives v = 1 + (n-1) f^(n) E^(n-2)/(2 E_Pl^(n-2)), so the velocity difference between two energies scales as E^(n-2), but Eq. (80) is printed with (E_1^(n-1) - E_2^(n-1))/(2 E_Pl^(n-2)), which for n = 3 has dimensions of energy rather than being dimensionless. With the corrected exponent the Markarian 421 numbers (1 and 2 TeV, 280 s variability, a light-travel time of order 1.4 x 10^17 s for z = 0.03) give |f^(3)| of order 20-30 rather than the quoted 128. Please correct the exponent and state explicitly which energy pair and which distance/travel time were used to obtain |f^(3)| < 128, since the quoted number is the one most often cited from this section.
[§6.5.5, Eq. (102)] Eq. (102) inverts the dependence on the Lorentz-violating coefficient. Minimizing Eq. (101) over z gives max|z^3 - z| = 2/(3 sqrt 3) at |z| = 1/sqrt 3, hence k_thresh = (6 sqrt 3 m^2 E_Pl/|f^(3)_eR|)^(1/3), with f in the denominator; the printed expression has f in the numerator. The quoted numerical bound |f^(3)_eR| < 0.25 (and |eta_R| < 0.125) is the one that follows from the corrected form for m_e and k_thresh = 50 TeV, so the number is right and only the displayed formula is wrong, but as printed Eq. (102) would give |f^(3)_eR| of order 400 for the same threshold. This should be corrected since the equation is presented as the derivation of the bound.
[§5.3, Table 1] The entry kappa-tilde^e-_ZZ = -20 +/- 2 (in units of 10^-15) is a ten-sigma nonzero value, and the caption sets it aside as systematics on the authority of Ref. [32], which is cited as arXiv:gr-qc/0504109 with no journal reference. For a table whose purpose is to record the state of the constraint, a ten-sigma entry dismissed on the strength of an unpublished preprint needs more than a parenthetical: please add a footnote saying what the reported systematic is and how it was identified, give a published reference if one now exists, and state what the bound on the relevant combination of c_mu-nu and kappa-tilde^e-_ZZ would become if the effect were real. The same request applies to the other headline numbers that currently rest on preprints, in particular the proton bound of §5.2 attributed to [241] and the dimension-five bounds of §5.4 and §5.8 attributed to [52] and [65].

Minor objections (7)

[§3.2, Eqs. (18)-(20)] The third expansion is printed as 'a = 1 + delta_RMS v^2', but the three parameters being expanded are a, b and d; the third line should read d = 1 + delta_RMS v^2.
[§6.5.5, Table 2] The n = 2 entry for electrons is listed as 10^-15, but Eq. (87) with m_e = 0.511 MeV and p_obs = 50 TeV gives m^2/p^2 = 1.0 x 10^-16. The other entries in the table reproduce Eq. (87) to within the stated order of magnitude (for example the proton n = 3 entry gives 5 x 10^-15 against the tabulated 10^-14), so this one entry appears to be a factor of ten high; please check it.
[§5.2] The bound is printed as '|b-tilde_perp| = 6.4 +/- 5.4 x 10^-32 GeV'; the error bar should be inside the parentheses, i.e. (6.4 +/- 5.4) x 10^-32 GeV, to avoid reading as 6.4 plus a quantity of 5.4 x 10^-32 GeV.
[§6.5.5] The step from the birefringence bound |xi| <= 10^-4 to setting xi = 0 in the photon-decay analysis deserves one sentence of justification. The threshold bound being derived is of order 10^-2, so the birefringence-allowed range of xi is not negligible compared with it, and the reader is entitled to see why discarding xi is conservative rather than merely convenient.
[§6.3] The bound |k_a| <= 10^-32 is quoted without units or normalization. Please state that k_a is dimensionless and specify the frame (the sun-centered equatorial frame used for Table 1) in which the ten components are defined, so the number can be compared with the cavity entries of §5.3.
[§6.8.1] The remark that 'It is also seems likely that the LSND result is simply incorrect' states as settled a question that was, at the time of writing, the motivation for MiniBooNE. A review should present this as contested and note that the forthcoming test will decide it; there is also a typographical slip ('It is also seems').
[§2.7.1] The causality discussion would be easier to follow if the distinction between the breakdown of microcausality and the preservation of cluster decomposition in a concordant frame were stated once in a single sentence before the aether example, since the two are easy to conflate on a first reading.

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The paper

The paper claims

The evidence now leaves no easy room for Planck-scale Lorentz violation at or below dimension six.

The reading

This review asks whether the last decade of Lorentz-invariance tests has closed the door on Planck-scale violations of relativity. It catalogues the theoretical frameworks—the Standard Model Extension, Robertson–Mansouri–Sexl, doubly special relativity, non-systematic dispersion, and dynamical aether—and maps each class of experiment onto the coefficients those frameworks define. Its central synthesis is that terrestrial clock, cavity, torsion-balance, and Penning-trap measurements now bound the lowest-dimension coefficients so tightly, and astrophysical birefringence and threshold observations reach so far into Planck-suppressed operators, that there is no easy way to embed dimension-six-or-lower Lorentz violation coming solely from Planck-scale physics. The sharpest consequence is that any such violation would need a custodial symmetry, such as supersymmetry combined with CPT, or an extra-dimensional hiding place, to explain why low-energy experiments have not already seen it.

What carries the argument

The load-bearing machinery is the Standard Model Extension (SME), especially its minimal version, which writes every Lorentz-violating coupling as a fixed tensor coefficient multiplying a Standard Model operator. The review uses the SME to translate disparate experiments into common parameters: a clock comparison constrains neutron $b_J$, a cavity resonance constrains photon $κ_{e-}$ and $κ_{o+}$, and a cosmic-ray threshold constrains a dispersion coefficient $f^{(n)}$. For astrophysics it adds the modified dispersion relation $E^2=m^2+p^2+f^{(n)}p^n/E_{Pl}^{n-2}$ and threshold theorems stating that at a reaction threshold the outgoing momenta are parallel and the incoming momenta anti-parallel. The concordant frame—the frame in which all Lorentz-violating coefficients are small—lets the review compare limits without committing to a preferred observer. Alternative frameworks, doubly special relativity and the dynamical aether, are carried along because they evade some of these constraints and predict different signatures.

Extended reading notes

Core claim

The paper's central claim is a synthesis rather than a single measurement: the existing web of constraints leaves no natural, unsuppressed route for Planck-scale Lorentz violation to appear as operators of dimension six or below. The review supports this by collecting terrestrial limits on the minimal Standard Model Extension—components of $c_{μν}$, $b_J$, $κ_{e-}$, and $κ_{o+}$ at levels from roughly $10^{-15}$ to $10^{-27}$ GeV—and astrophysical limits from photon time of flight, birefringence, photon decay, vacuum Cerenkov emission, the GZK cutoff, synchrotron emission, and neutrino oscillations. The astrophysical bounds reach coefficients of order $10^{-14}$ or better for dimension-five and dimension-six dispersion, even though the Planck scale is far above any observed particle energy. The conclusion is that a viable Lorentz-violating extension of known physics must either be protected by supersymmetry and CPT, hidden in an extra dimension, or fine-tuned; otherwise the same radiative corrections that generate low-dimension operators should already have produced a signal.

Load-bearing premise

The synthesis rests on the premise that, without a protective symmetry, radiative corrections will drag Planck-scale Lorentz violation down to dimension-four operators large enough for existing terrestrial experiments to see, so the absence of a low-energy signal is evidence against the whole programme.

Editorial extensions

If this is right

If the synthesis is right, quantum-gravity models that break Lorentz invariance at the Planck scale must come with a custodial symmetry or a screening mechanism, or they are already excluded by existing data.
The same bounds turn every improved clock, cavity, torsion-balance, and Penning-trap measurement into a sharper probe of Planck-scale physics, even though those experiments operate at laboratory energies.
Astrophysical thresholds and birefringence will remain the only practical route to dimension-five and dimension-six operators, but their interpretation depends on assuming reaction dynamics close to the Lorentz-invariant case.
Doubly special relativity stays viable precisely because it has no preferred frame and evades threshold constraints; an energy-dependent time of flight without birefringence would be its clean signature.
In the gravitational sector, dynamical Lorentz violation predicts extra gravitational-wave polarizations and preferred-frame post-Newtonian parameters, giving a separate channel for tests.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The review's naturalness argument cuts both ways: if no Lorentz violation is found, it says more about which quantum-gravity models are natural than about whether relativity is exact; if a signal appears only astrophysically, the burden shifts to finding a low-energy screening mechanism.
The catalogue of bounds is not statistically combined; a joint likelihood over the SME coefficients could close parameter space more sharply than the individual limits quoted here.
The concordant-frame assumption may be weakest in the early universe or near compact objects, where a dynamical Lorentz-violating field could vary in space and time and evade terrestrial bounds.
A confirmed DSR time-of-flight signal would not just constrain a model; it would show that the effective-field-theory logic underlying most of the review's strongest bounds is the wrong description, making the absence of birefringence a decisive cross-check.

Original abstract

Motivated by ideas about quantum gravity, a tremendous amount of effort over the past decade has gone into testing Lorentz invariance in various regimes. This review summarizes both the theoretical frameworks for tests of Lorentz invariance and experimental advances that have made new high precision tests possible. The current constraints on Lorentz violating effects from both terrestrial experiments and astrophysical observations are presented.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 2,888 input tokens, $0.0001 reported.

What the abstract itself states, as probabilities: claims first 0.02 · states limitation 0.06 · human participants 0.04 · controlled comparison 0.03 · randomized assignment 0.01 · states quantitative result 0.05 · compares with prior results 0.13 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v2. Results and the main claim are underlined; show every role.

Motivated by ideas about quantum gravity, a tremendous amount of effort over the past decade has gone into testing Lorentz invariance in various regimes. This review summarizes both the theoretical frameworks for tests of Lorentz invariance and experimental advances that have made new high precision tests possible. The current constraints on Lorentz violating effects from both terrestrial experiments and astrophysical observations are presented.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	Motivated by ideas about quantum gravity, a tremendous amount of effort over the past decade has gone into testing Lorentz invariance in various regimes.	background 1.00	0.04	0.04	asserted 0.99
s1	This review summarizes both the theoretical frameworks for tests of Lorentz invariance and experimental advances that have made new high precision tests possible.	contribution 1.00	0.38	0.13	asserted 0.97
s2	The current constraints on Lorentz violating effects from both terrestrial experiments and astrophysical observations are presented.	result 0.58	0.28	0.24	asserted 0.92

What kind of paper the classifier reports this is: review 1.00 · evidence from literature 0.97.

What was examined, and what was not

sources read

arXiv listing abstract v2

not examined

full text
figures and tables
references and citing sentences
relations to other papers

levels run

P abstract

not run

S sections
T body sentences
C citations
R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures

**Figure 1.** Elements involved in threshold constraints.

**Figure 2.** Total outgoing particle energy in symmetric and as

**Figure 3.** An example of an upper and lower threshold.

**Figure 4.** Constraints on LV in QED at n = 3 on a log-log plot. For negative parameters minus the logarithm of the absolute value is plotted, and region of width 10−10 is excised around each axis. The constraints in solid lines apply to ξ and both η±, and are symmetric about both the ξ and the η axis. At least one of the two pairs (η±, ξ) must lie within the union of the dashed bell-shaped region and its reflection about the ξ axis. Intersecting lines are truncated where they cross. 50

Figures reproduced from arXiv:gr-qc/0502097 by the authors.

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Forward citations

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How later papers used it

261 later papers cite it

Evidence: some. 257 of the citing papers share no author with it, and 4 of those carry a confident label from Pith's citation classifier.

Pith read 39 of the 362 citing sentences, choosing only those whose citation markers point at this paper alone, and wrote the paraphrases from them. The summary, the counts, the evidence level, the links and the quotes come from code; the summary counts the papers listed below. A later paper is listed under a label when Pith's citation classifier gives one of its sentences that label at probability 0.8 or more (0.95 for support), Pith's reading finds that the citing paper itself did it, and a second check confirms the paraphrase against the sentence. This section lists uses only. It does not list criticism or disagreement, so their absence here says nothing about how the paper has been received. Papers that share an author with this one are listed apart, under "By the same authors". A quote is the citing authors' own sentence, shown when their paper's licence is CC BY, CC BY-SA or CC0. JSON

1 later paper by other authors is listed below: 1 cited it as support.

Cites it as support 1 of 3 papers with this label

Used this paper's strong experimental constraints on the explicit breaking of Lorentz symmetry as evidence that statistical realization of the symmetries is preferable, given a quantum-gravity motivation.

A sprinkling of hybrid-signature discrete spacetimes in real-world networks (2021) · 0.98 · paraphrased: the licence does not allow quoting

How the literature uses it

219 arXiv papers cite this in 315 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 172 sentences without a confident label

Compares with it 1 paper

This is in fact what had been obtained in previous studies [ref] .

Modified energy-momentum conservation laws and vacuum Cherenkov radiation · Excluded regions for particular cases of the modified kinematics · 0.86

Cites it as support 2 papers

The linear energy dependence in EFT is largely unfavored [ref] .

Lorentz violation effects on astrophysical propagation of very high energy photons · Some Selected Theoretical Models · 0.99
With a quantum-gravity motivation in mind, the statistical realization of the symmetries is preferable, because of strong experimental constraints on the explicit breaking of Lorentz symmetry [ref] .

A sprinkling of hybrid-signature discrete spacetimes in real-world networks · Networks encoding Euclidean and Minkowski spacetime · 0.98

Cites it as background 125 papers

The Lorentz invariance is one of the most experimentally well established and tested feature of Nature [ref] .

Breaking the light speed barrier · Introduction · 0.83
One particularly intriguing class of modified gravity theories involves the explicit or spontaneous violation of Lorentz symmetry [ref] .

Primordial black hole in Lorentz-violating theories: Insights from Bumblebee gravity · Introduction · 0.98
Aside from these theoretical restrictions, there are phenomenological constraints stemming from experimental bounds on Lorentz violation (LV), see e.g. [ref] .

Towards a unitary, renormalizable and ultraviolet-complete quantum theory of gravity · Particle spectrum, dispersion relations and phenomenological constraints · 0.92

Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.

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Phillips, D.F. et al., “Limit on Lorentz and CPT violat ion of the proton using a hydrogen maser”, Phys. Rev. D , 63, 111101, (2001). Related online version (cited on 22 Februa ry 2005): http://arXiv.org/abs/physics/0008230. 75

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