Potential Sensitivity of Gamma-Ray Burster Observations to Wave Dispersion in Vacuo

D.V. Nanopoulos; G. Amelino-Camelia; John Ellis; N.E. Mavromatos; Subir Sarkar

REVIEW

Potential Sensitivity of Gamma-Ray Burster Observations to Wave Dispersion in Vacuo

G. Amelino-Camelia, John Ellis, N.E. Mavromatos, D.V. Nanopoulos, Subir Sarkar

Astrophysics · Submitted 7 Dec 1997 · 30 references

Pith's verdict: The founding paper of the GRB time-of-flight programme: the scaling is right, and the Planck-scale reach is an aspiration the paper mostly, but not entirely, labels as one.

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arXiv IDastro-ph/9712103 (version 2)

Pith IDBX33CEZ3, Pith's permanent record of this paper and its review

CitationD.V. Nanopoulos, G. Amelino-Camelia, John Ellis, N.E. Mavromatos, Subir Sarkar. "Potential Sensitivity of Gamma-Ray Burster Observations to Wave Dispersion in Vacuo." arXiv:astro-ph/9712103 (1997). https://pith.science/paper/BX33CEZ3

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Editor's note

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This is the paper that started the GRB time-of-flight business, and it should be read as such. The core is two lines: a linear-in-energy deformed dispersion relation gives δv ~ ξE/E_QG, and integrating over a distance L gives Δt ~ ξ(E/E_QG)(L/c). That is a standard expansion, correctly done, and the order-of-magnitude reach — millisecond structure at ~20 MeV from ~10^10 light years probes E_QG ~ E_P — has held up. The genuinely new move relative to the authors' own ref [17] is the GRB application, and the paper says so explicitly: ref [17] did not consider GRBs because their cosmological distances were not then established. That is a real, dated contribution, not a repackaging of the Liouville-string framework. The discussion of how to separate a quantum-gravity shift from source and plasma effects — energy dependence, linearity — is the right instinct and is what the later literature actually used. What it does well: it is clear about what it is. The abstract says 'might be sensitive,' not 'is sensitive,' and the 10^16 GeV number for GRB 920229 is explicitly conditional on a redshift that has not been measured ('If a burst such as this were to be demonstrated in future to lie at a redshift z ~ 1'). The stress-test is right that there are two tiers here and only the lower one touches data, but the paper is not hiding that; the gap is between the abstract's framing and the body, not between the body and the data. The one thing I would actually press on is the Δt <~ 10^-2 s limit: 'we think it should be possible to extract' is an assertion, not a derivation, and the 10^16 GeV figure scales directly with it. That should be shown or sourced. Second, the 100 μs microburst comes from ref [23], a conference proceedings in press, not reproduced here, and the sensitivity argument leans on it. Minor: the sentence about the 'medium effect' being linear in photon energy is muddled — conventional plasma dispersion goes the other way — and could confuse a careful reader. The heavy self-citation of the Liouville-string framework does not make the estimate circular: the reach calculation uses only the generic δv ~ E/E_QG form and external inputs. Who it is for: anyone entering quantum-gravity phenomenology, and anyone who wants the origin of the GRB timing constraint. Yes to referees — this is a Nature letter, short by design, and the physics is sound.

Referee report

Significance

If the dispersion relation of Eq. (1) is realized in nature, this paper identifies a genuinely new observational window on Planck-scale physics, and it does so with an estimate that is parameter-free apart from E_QG itself. The scaling argument is transparent and the arithmetic is reproducible: I checked the η values quoted for GRBs, pulsars, supernovae, and the CMB against Eq. (2) with the stated inputs, and they follow. The paper makes a falsifiable prediction and, importantly, states the sign distinction from conventional plasma dispersion (the quantum-gravity delay increases with energy, the plasma delay decreases), which is what makes the test more than a null search. The proposal to use lensed GRBs as an emission-mechanism-independent probe is a concrete and useful addition. These are real strengths and should be weighed in the assessment.

Major objections (2)

[Paragraph beginning 'Significant sensitivities may already be attainable with the present GRB data' (the GRB 920229 / 10] The only sensitivity number in the paper that is tied to actual data is E_QG ~ 10^16 GeV, and it rests on two inputs that are not established here. The first is the assumed redshift z ~ 1 for GRB 920229; the authors flag this explicitly ('If a burst such as this were to be demonstrated in future to lie at a redshift z ~ 1'), and that hedge is appropriate. The second is the arrival-time limit Δt <~ 10^-2 s, which is asserted rather than derived ('we think it should be possible to extract an upper limit of Δt <~ 10^-2 sec'). Since the reach scales linearly in 1/Δt, the quoted 10^16 GeV is a statement about what a future analysis of ref. [23]'s light curve could yield, not a limit obtained in this paper. The gap to the abstract's headline (abstract:s2, E_QG comparable to E_P) is then three orders of magnitude, and a reader could take the Planck-scale number as the paper's finding rather than its aspiration. I would ask the authors to present the reach as an explicit function of Δt and z, ideally in a small table, and either to derive the Δt <~ 10^-2 s from the published light curve or to label it unambiguously as an estimate of what a future analysis could achieve.
[Final paragraph, 'To distinguish any quantum-gravity shift from effects due to the source'] The discrimination strategy rests on the statement that source-intrinsic time shifts 'would not in general' be linear in photon energy, and this is the weakest link in the proposed observational programme. Energy-dependent lags in GRB pulses are well documented and are often approximately linear in E, with hard photons leading, which is the same sign as the ξ = +1 branch of Eq. (1); the degeneracy between intrinsic lag and dispersion is therefore not removed by the energy dependence alone. The paper would be strengthened by a concrete test that exploits what does differ: the dispersion term is a deterministic function of z and E common to all bursts, whereas intrinsic lags vary burst to burst and correlate with luminosity. A population regression of measured lag on (1+z)-weighted distance at fixed observed energy band would separate the two, and stating this explicitly would make the observational proposal considerably more persuasive.

Minor objections (9)

[Paragraph on κ-Poincaré deformations] The stated dispersion relation c^2 p^2 = E_QG^2 [1 - e^{E/E_QG}]^2 expands to E^2 (1 + 2E/E_QG + ...), i.e. ξ = 2, not ξ = 1 as the text concludes. The sign is right and the order of magnitude is unaffected, but the coefficient should be corrected or the normalization of E_QG stated explicitly.
[Final paragraph] The word 'medium' is used in two different senses in adjacent sentences: first for conventional plasma effects, which 'decrease with energy', and then for the quantum-gravity effect, which 'would be linear in the photon energy'. Rewording one of the two would remove an ambiguity that sits exactly where the reader needs clarity.
[Eq. (2)] Equation (2) uses L/c with L the light-travel distance and does not integrate over redshift. At z ~ 1 the cosmological correction is of order unity and does not change the conclusions, but the text should say so, since the quoted numbers are order-of-magnitude estimates.
[Paragraph on photon mass] The remark that the data impose an upper limit of order 10^-6 eV on a possible photon mass is not derived. Using the stated ΔE ~ 200 keV and Δt ~ 10^-2 s I obtain a bound closer to 10^-5 eV; since the paper's conclusion (that this is much weaker than existing limits) is unaffected, a one-line derivation would suffice.
[Abstract] The abstract refers to 'the fine-scale time structure and hard spectra of GRB emissions', but the argument uses high photon energy and short time structure, not spectral hardness as such. Consider rewording to match what is actually computed.
[Reference [23] and the microburst paragraph] The 100 μs microburst in GRB 920229 is the key data input for the only data-anchored number, and it comes from a conference proceedings listed as in press. The text should state the statistical significance of the feature as reported in that source, so that the reader can judge how firm the timescale is.
[Reference [25]] The claim that string theory favours effective quantum-gravity scales 'typically in the range 10^16 to 10^18 GeV' is attributed to ref. [25], a paper on strong-coupling Calabi-Yau compactification. The derivation of that range from that source is not transparent and should be spelled out or the citation adjusted.
[Comparisons with other probes] The η values for GRBs, pulsars, supernovae, and the CMB are scattered through the prose. A single small table collecting the assumed E, L, δt, and resulting η for each probe would make the comparison much easier to check.
[References and text] Several typographical corrections are needed: 'van Paradis' should be 'van Paradijs' (ref. [1]), 'Metzeger' should be 'Metzger' (refs. [3], [4]), 'dipersion' should be 'dispersion', 'Robertson-Walker-Friedman' should be 'Friedmann', and 'It has been recently suggested [8] the vacuum' is missing 'that'.

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The paper

The paper claims

The paper argues that the fine time structure of gamma-ray bursts, combined with their cosmological distances, makes them sensitive probes of a possible energy-dependent speed of light, with reach extending to the Planck scale of about $10^

The reading

Some approaches to quantum gravity suggest that empty space is not perfectly transparent to light: photons of different energies would travel at slightly different speeds, with a fractional velocity difference of order $E/E_{QG}$ where $E_{QG}$ is an effective quantum-gravity energy scale. The paper's claim is that gamma-ray bursts are unusually good places to look for this, because they combine high photon energies, cosmological distances, and time structure down to the millisecond. A signal at tens of MeV, varying on millisecond timescales, arriving from about $10^{10}$ light years would acquire an energy-dependent arrival delay comparable to its own variability if $E_{QG}$ is near the Planck energy $10^{19}$ GeV. That is a reach no laboratory experiment can approach, and the authors sketch an observational programme to exploit it.

What carries the argument

The machinery is the pair of equations relating a deformed photon dispersion relation to an observable arrival-time lag. The dispersion relation $c^2p^2 = E^2[1 + ξE/E_{QG} + O(E^2/E_{QG}^2)]$ yields an energy-dependent velocity $v ∼ c(1 - ξE/E_{QG})$, and integrating along the line of sight gives the delay $Δt ∼ ξ(E/E_{QG})(L/c)$. The argument then turns on the ratio $η = |Δt|/δt$: a phenomenon is a useful probe to the extent that its intrinsic variability $δt$ is short, its photon energies $E$ are high, and its distance $L$ is large. Gamma-ray bursts are claimed to be the best known combination of all three, and the paper tabulates the competition — pulsars reach $η ∼ 10^{-10}$, supernova neutrinos $η ∼ 10^{-4}$, and the cosmic microwave background gives a spectral distortion of order $ν/E_P ∼ 10^{-32}$, negligible.

Extended reading notes

Core claim

The central claim is a sensitivity estimate, not a detection. If photons obey a deformed dispersion relation $c^2p^2 = E^2[1 + f(E/E_{QG})]$, then to leading order in $E/E_{QG}$ their speed becomes $v = c(1 - ξ E/E_{QG})$, and a signal of energy $E$ travelling a distance $L$ acquires a time delay $Δt ∼ ξ (E/E_{QG})(L/c)$ relative to the energy-independent case. The useful figure of merit is $η ≡ |Δt|/δt$, the delay divided by the signal's own variability timescale. The authors estimate that a burst with millisecond structure at about 20 MeV from $10^{10}$ light years has $η ∼ 1$ when $E_{QG} ∼ 10^{19}$ GeV, i.e. of order the Planck energy. Applying this to existing data — the $100$ $μ$s microburst seen in three BATSE channels in GRB 920229, and an arrival-time limit $Δt ≲ 10^{-2}$ s across an energy separation of about 200 keV — they estimate a reach of $E_{QG} ∼ 10^{16}$ GeV if such a burst were confirmed at redshift $z ∼ 1$, improvable to about $10^{18}$ GeV if the time difference could be resolved down to the reported rise time. They also note that a gravitationally lensed burst would give an emission-mechanism-independent probe, with sensitivity $η ∼ 10^{-6}$.

Load-bearing premise

The whole sensitivity estimate divides by the burst's variability timescale, and it assumes that the observed hundred-microsecond to millisecond structure is intrinsic to the source and measurable at that precision, rather than a product of detector resolution, scattering, or propagation — an assumption the authors concede is not backed by a complete theory of burst emission.

Editorial extensions

If this is right

A confirmed energy-dependent lag in a burst of known redshift, scaling linearly with photon energy, would set a lower bound on $E_{QG}$ and rule out any quantum-gravity scheme predicting dispersion well below that scale.
Because the effect grows with energy while conventional plasma dispersion falls with energy, the two can in principle be separated by the sign and shape of the lag-versus-energy relation.
A gravitationally lensed gamma-ray burst would provide a delay measurement independent of the emission mechanism, since conventional lensing delays are achromatic.
The programme sets a target for instrument design: fine time resolution at the highest attainable photon energies, extending burst spectroscopy toward TeV energies.
The same logic applied to other transient sources gives a ranked list of astrophysical probes, with gamma-ray bursts far ahead of pulsars and supernova neutrinos.

Reading between the lines

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

The estimate scales linearly with the assumed intrinsic variability timescale, so the reach quoted is really a statement about detector time resolution and about how much of the observed microstructure is emission rather than propagation or scattering.
If the effect is real and linear in energy, archival burst data with simultaneous multi-channel coverage could be reanalysed for a lag without waiting for new instruments, since the relevant quantity is a cross-channel time shift rather than absolute timing.
The same dispersion relation would shift the arrival times of different-energy photons from any transient at cosmological distance, so a null result in bursts would constrain the underlying quantum-gravity proposals as much as a positive one.
Because the predicted lag is tiny but cumulative over cosmological distances, the natural next step is a population-level search: stacking many bursts with known redshifts to beat down the per-burst timing uncertainty.

Original abstract

The recent confirmation that at least some gamma-ray bursters (GRBs) are indeed at cosmological distances raises the possibility that observations of these could provide interesting constraints on the fundamental laws of physics. Here we demonstrate that the fine-scale time structure and hard spectra of GRB emissions are very sensitive to the possible dispersion of electromagnetic waves in vacuo with velocity differences $\delta v \sim E/E_{\QG}$, as suggested in some approaches to quantum gravity. A simple estimate shows that GRB measurements might be sensitive to a dispersion scale $E_{QG}$ comparable to the Planck energy scale $E_{P} \sim 10^{19}$ GeV, sufficient to test some of these theories, and we outline aspects of an observational programme that could address this goal.

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.

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What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.23 · human participants 0.02 · controlled comparison 0.03 · randomized assignment 0.01 · states quantitative result 0.55 · compares with prior results 0.11 · 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.

The recent confirmation that at least some gamma-ray bursters (GRBs) are indeed at cosmological distances raises the possibility that observations of these could provide interesting constraints on the fundamental laws of physics. Here we demonstrate that the fine-scale time structure and hard spectra of GRB emissions are very sensitive to the possible dispersion of electromagnetic waves in vacuo with velocity differences $\delta v \sim E/E_{\QG}$, as suggested in some approaches to quantum gravity. A simple estimate shows that GRB measurements might be sensitive to a dispersion scale $E_{QG}$ comparable to the Planck energy scale $E_{P} \sim 10^{19}$ GeV, sufficient to test some of these theories, and we outline aspects of an observational programme that could address this goal.

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	The recent confirmation that at least some gamma-ray bursters (GRBs) are indeed at cosmological distances raises the possibility that observations of these could provide interesting constraints on the fundamental laws of physics.	background 1.00	0.05	0.04	hedged possibility 0.95
s1	Here we demonstrate that the fine-scale time structure and hard spectra of GRB emissions are very sensitive to the possible dispersion of electromagnetic waves in vacuo with velocity differences $\delta v \sim E/E_{\QG}$, as suggested in some approaches to quantum gravity.	contribution 0.60	0.85	0.52	asserted 0.99
s2	A simple estimate shows that GRB measurements might be sensitive to a dispersion scale $E_{QG}$ comparable to the Planck energy scale $E_{P} \sim 10^{19}$ GeV, sufficient to test some of these theories, and we outline aspects of an observational programme that could address this goal.	result 0.49	0.75	0.68	speculative 0.66

What kind of paper the classifier reports this is: position or commentary 0.55 · evidence from argument 0.77.

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not examined

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

levels run

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S sections
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C citations
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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

References

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astro-ph.HE 2024-11 conditional novelty 5.0 of 10

Using profile likelihood on GRB 160625B spectral lags, the authors report no global chi-square minimum below the Planck scale and set 95% lower limits of 2.55e16 GeV (linear) and 1.85e7 GeV (quadratic) on the LIV ener...
Generalized framework for testing gravity with gravitational-wave propagation. I. Formulation
gr-qc 2017-10 conditional novelty 5.0 of 10

A WKB solution to the generalized gravitational-wave propagation equation packages modified gravity into an amplitude damping factor and a phase delay, and Fisher forecasts show a single loud binary can measure the da...
Hamilton geometry: Phase space geometry from modified dispersion relations
gr-qc 2015-07 accept novelty 5.0 of 10

Hamilton geometry provides a consistent framework for deriving phase space geometry from modified dispersion relations, with explicit computations showing intertwined spacetime and momentum space curvature for q-de Si...
OPERA-reassessing data on the energy dependence of the speed of neutrinos
hep-ph 2011-09 conditional novelty 5.0 of 10

Combining OPERA, MINOS and 1979 Fermilab muon-neutrino speed data, a momentum-independent superluminal shift fits best, while tachyon and quadratic energy-dependence models are disfavored.
Constructing local bulk observables in interacting AdS/CFT
hep-th 2011-02 conditional novelty 5.0 of 10

Local bulk observables in interacting AdS/CFT can be built by adding higher-dimension multi-trace boundary operators that cancel spacelike commutators order by order in 1/N.
Strong gravitational lensing by black hole in F(R) Euler Heisenberg Gravity's Rainbow
astro-ph.GA 2025-07 conditional novelty 4.0 of 10

Applying the strong deflection limit to the F(R)-Euler-Heisenberg-Rainbow black hole gives lensing observables that increase with the Euler-Heisenberg parameter and decrease with charge.
Examining Lorentz invariance violation with three remarkable GRB photons
astro-ph.HE 2025-04 conditional novelty 4.0 of 10

Adding three extreme GRB photons preserves the fitted Lorentz-violation scale E_LV ~ 3e17 GeV, but only when an energy-dependent intrinsic delay term, itself a fitting construct, absorbs the new TeV photons.
Spectra for Reactions in Astrophysical Electromagnetic Cascades with Lorentz Invariance Violation: The Vacuum Cherenkov Effect
astro-ph.HE 2024-12 conditional novelty 4.0 of 10

The authors provide closed-form total rates and simulated photon and electron spectra for vacuum Cherenkov radiation under second-order Lorentz invariance violation.
Below the Schwinger critical magnetic field value, quantum vacuum and gamma-ray bursts delay
hep-ph 2025-01 reject novelty 3.0 of 10

The authors claim a 2.4-hour GRB delay from a 10^6 Tesla intergalactic magnetic field, but the computation actually yields about 100 days, and the assumed field is many orders of magnitude stronger than observed.
White Paper and Roadmap for Quantum Gravity Phenomenology in the Multi-Messenger Era
gr-qc 2023-12 accept novelty 3.0 of 10

The paper lays out a community roadmap for testing quantum gravity with gamma rays, neutrinos, cosmic rays, and gravitational waves, and argues that dedicated experiments, better astrophysical modelling, and open data...
Teleparallel Gravity: From Theory to Cosmology
gr-qc 2021-06 conditional novelty 3.0 of 10

A comprehensive review that organizes teleparallel gravity from metric-affine geometry through modified theories to cosmological observations, and lists corrections to several published equations along the way.
Fundamental Physics with High-Energy Cosmic Neutrinos
astro-ph.HE 2019-03 accept novelty 3.0 of 10

High-energy cosmic neutrinos can test neutrino cross sections, flavor mixing, Lorentz and CPT symmetry, neutrino stability, and dark matter, provided detectors grow large enough to collect EeV events.
A Critical Review of Classical Bouncing Cosmologies
astro-ph.CO 2014-06 accept novelty 3.0 of 10

Most bouncing cosmologies do not yet match current data or avoid known instabilities, but a few, notably the super-bounce and S-brane bounce, remain viable research directions.
Tests of Lorentz invariance: a 2013 update
gr-qc 2013-04 accept novelty 3.0 of 10

Lorentz-violation constraints from particle physics and astrophysics are collected and tabulated, and applying them to Horava-Lifshitz gravity is argued to force that theory's Lorentz-breaking scale well below the Pla...
Doubly-Special Relativity: Facts, Myths and Some Key Open Issues
gr-qc 2010-03 conditional novelty 3.0 of 10

Doubly special relativity is a coherent but still unformalized program in which an observer-independent Planck-scale length coexists with the speed of light, and the review separates valid DSR scenarios from look-alikes.
Revisiting Varying Speed of Light in Cosmology: Insights from the Friedmann-Lema\^itre-Robertson-Walker Metric
physics.gen-ph 2025-05 reject novelty 2.0 of 10

A varying speed of light in FLRW is presented as gauge freedom via the lapse function, but the key variational derivation omits the √-g measure and the claimed Hubble-tension resolution contradicts Eq. (30).
High-Energy and Ultra-High-Energy Neutrinos
hep-ph 2022-03 accept novelty 2.0 of 10

High-energy and ultra-high-energy astrophysical neutrinos offer a unique window on beyond-Standard-Model physics and on the sources of ultra-high-energy cosmic rays, and the paper lays out the detector program needed ...
Quantum gravity phenomenology at the dawn of the multi-messenger era -- A review
hep-ph 2021-11 accept novelty 2.0 of 10

The review argues that multi-messenger astronomy now supplies workable empirical constraints on Planck-scale quantum gravity, chiefly through energy-dependent propagation effects accumulated over cosmological distances.
Prospects for Fundamental Physics with LISA
gr-qc 2020-01 unverdicted novelty 2.0 of 10

The LISA fundamental-physics working group maps six gravitational-wave source classes onto the beyond-standard-model physics each could probe, without quantitative forecasts.
Black Holes in Loop Quantum Gravity
gr-qc 2017-03 conditional novelty 2.0 of 10

Loop quantum gravity's discrete quantum geometry can reproduce the Bekenstein-Hawking area law and frame black hole evaporation as information degradation rather than loss, though the program remains incomplete.
Modern tests of Lorentz invariance
gr-qc 2005-02 accept novelty 2.0 of 10

No experiment has found Lorentz violation, and the combined terrestrial and astrophysical bounds are tight enough to exclude a generic Planck-scale origin for the effect at mass dimension six or below.
Condensates: A Bridge to Quantum Gravity
gr-qc 2026-09 unverdicted novelty 1.0 of 10

A review essay arguing that spacetime is a Bose-Einstein-condensate-like emergent phase and that quantum gravity will be probed in condensed matter laboratories, not colliders.

How later papers used it

213 later papers cite it

Evidence: some. 187 of the citing papers share no author with it, and 4 of those carry a confident label from Pith's citation classifier.

Pith read 17 of the 350 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 used its method or tool. 1 more paper by the same authors is listed: 1 built on it.

Builds on it 1 paper · 1 by the same authors

By the same authors

Followed this paper's suggestion for photons while avoiding subscribing to a specific theoretical approach.

Probing Lorentz invariance with a high-energy neutrino flare (2024) (same authors) · 0.87 · paraphrased: the licence does not allow quoting

Uses its method or tool 1 paper

Here, we obtain the eigenvalue solution of the above discussed quantum system using the following pair of rainbow function given by [ref] [equation] Thereby, substituting this rainbow function into the relation ([ref]), we obtain [equation]

Study of scalar particles through the Klein-Gordon equation under rainbow gravity effects in Bonnor-Melvin-Lambda space-time (2023) · Quantum Motions of Scalar particles: The Klein-Gordon Equation · 0.93

How the literature uses it

162 arXiv papers cite this in 265 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 · 174 sentences without a confident label

Builds on it 1 paper

In so doing, we follow the suggestion for photons made in [ref] , but avoid subscribing to a specific theoretical approach.

Probing Lorentz invariance with a high-energy neutrino flare · Why high-energy astrophysical neutrino flares? · 0.87

Compares with it 3 papers

However, if $f(E)≈1+αE/E_P, g(E)=1$, as in [ref] , the temperature is modified but there is no remnant.

Remnants of Black Rings from Gravity's Rainbow · Spinning Black Rings in Gravity's Rainbow · 0.91
This distinguishing feature sets meVSL apart from traditional VSL models, which often assume or necessitate an explicit mechanism driving the evolution of the speed of light [ref] .

Revisiting Varying Speed of Light in Cosmology: Insights from the Friedmann-Lema\^itre-Robertson-Walker Metric · Introduction · 0.83
While our optical time delay constraints are not competitive with observations of gamma-ray bursts [ref] or TeV flares from blazars [ref] , our approach, which may be unique in the literature, does constrain both time delays and maximum observed polarization with simultaneously obtained photometry and polarimetry using the same pair of broadband optical filters.

Constraints on Lorentz Invariance and CPT Violation using Optical Photometry and Polarimetry of Active Galaxies BL Lacertae and S5 B0716+714 · Introduction · 0.93

Cites it as background 77 papers

Over these two years past since a consensus [ref] was reached on this reformulation for expanding spacetimes of the original LSB proposal first given in Ref. [ref] (for non-expanding spacetimes), the prediction ([ref]) was the unchallenged benchmark used in numerous data analyses [ref] testing the hypothesis of momentum dependence of the speed of photons.

Testing quantum-spacetime relativity with gamma-ray telescopes · Introduction · 0.84
This can be tested by comparing the arrival times of different energy photons that have been emitted simultaneously [ref] .

Lorentz Invariance Violation Limits from GRB 221009A · Introduction · 0.86
Various choices of Rainbow functions motivated by physical considerations have been studied in [ref] .

Maximal acceleration in Rainbow gravity · Rainbow Gravity · 0.93

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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