Generalized Lorentz invariance with an invariant energy scale

Joao Magueijo; Lee Smolin

REVIEW

Generalized Lorentz invariance with an invariant energy scale

Joao Magueijo, Lee Smolin

General Relativity and Quantum Cosmology · Submitted 22 Jul 2002 · 37 references

Pith's verdict: The U-map framework is real and useful, but eq. (63), the example offered as meeting all five requirements, does not actually deliver a maximal momentum or a diverging speed of light.

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CitationJoao Magueijo, Lee Smolin. "Generalized Lorentz invariance with an invariant energy scale." arXiv:gr-qc/0207085 (2002). https://pith.science/paper/N3U7YGQM

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Editor's note

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The formal core is worth your time. Any isotropic dispersion relation E²f1² − p²f2² = m² can be written as a non-linear realization of the ordinary Lorentz group, K_i = U⁻¹ L_i0 U with U(E,p) = (E f1, p f2). That is a clean generalization of their PRL, it swallows κ-Poincaré as a special case, and it gives you a machine for turning a measured dispersion relation into a transformation law. The conditions for an invariant energy scale (U singular at E_P) and for a maximal momentum (E/f3 has a maximum) are derived correctly. The composite rule in §4.1 — an N-particle system transforms with λ/N — is genuinely non-obvious and fixes a real paradox: without it, a pile of sub-Planckian particles can never carry more than a Planck energy. And §5.3's negative result is the best part of the paper: no single-scale f1 with a Planck-scale parameter can move the threshold in a relativistic theory, because the required shift is frame-independent. That undercuts the Amelino-Camelia–Piran mechanism as stated, and the authors say so plainly rather than burying it. Credit for that; it is the honest half of the section and the more valuable one. Now the problem. Eq. (63) is offered as the explicit example meeting all five requirements, and it does not. f3 = f2/f1 is not free: with f1 = 1/((1+λ1E)(1−λE)) and f2 = e^{E/E_P}, you get f3 = e^{E/E_P}(1+λ1E)(1−λE). Then p = E/f3 diverges at E = 1/λ = E_P, so there is no maximal momentum, and c → 0 there rather than diverging. §5.3 instead prescribes f3 = e^{E/E_P}, which does give p_max and a diverging c, but that requires f2 = e^{E/E_P} f1, which is singular at E = 1/λ1 ≈ 10¹¹ GeV/N_p. So the claimed decoupling of the VSL sector from the threshold sector does not hold. This is a bookkeeping error rather than a conceptual one, but it sits in the sentence the abstract leans on. Your report checked the boosts and the threshold formulas but not (63); I would drop soundness a notch. Second, the UHECR explanation is a fit — λ1 tuned so that N_p λ1⁻¹ ~ 10¹¹ GeV — and N_p is left undefined. The paper admits both, so this is not a hidden flaw, but the abstract's 'explains the anomaly' is stronger than §5.3 supports. Who should read it: anyone working on doubly special relativity, Planck-scale phenomenology, or deformed dispersion relations. The framework is citable; the worked example needs fixing before the headline claim stands. Send it out.

Referee report

Major objections (4)

[§7, Eq. (63)] The explicit example does not satisfy requirements 4 and 5. With f1 = 1/((1+λ1E)(1−λE)) and f2 = e^{E/E_P}, the definition f3 = f2/f1 (§2, eq. 11) gives f3 = e^{E/E_P}(1+λ1E)(1−λE). Then p = E/f3 = E e^{−E/E_P}/[(1+λ1E)(1−λE)] diverges as E → 1/λ = E_P, so there is no maximal momentum, contrary to the criterion stated in §2.2. Likewise c = f3/(1 − E f3'/f3) → 0 at E_P, not the diverging speed of light required by requirement 4. §5.3 instead prescribes f3 = e^{E/E_P}, which does give p_max ≈ E_P/e and a diverging c; but that requires f2 = f3 f1, not f2 = e^{E/E_P}. The example as printed therefore does not establish the paper's headline claim, and the correct version should be given explicitly.
[§5.3] The statement that 'the function f2 does not enter the discussion of threshold anomalies, and so the issues of VSL and of the existence of a maximal momentum are decoupled from threshold anomalies' holds in only one direction. Thresholds depend on f1 alone, but f3 = f2/f1, so f1 necessarily enters the VSL and maximal-momentum sector. Choosing f2 = f3 f1 to cancel f1's effect gives f2 = e^{E/E_P}/((1+λ1E)(1−λE)), which is singular at E = 1/λ1 ≈ 10^11 GeV/N_p, inside the physical domain P_phys; there U maps every nonzero momentum to infinity and is not invertible. A different f1 (for example an exponential rather than a rational function of λ1E) would avoid the pole, but the paper should either exhibit such an f1 or state that the decoupling is only partial.
[Abstract (s4) and §5.3] The claim that the theory 'explains' the UHECR anomaly overstates what is shown. The paper's own §5.1 argument — that with a single dimensional parameter λ and no small dimensionless numbers, λ^{-1} must be of the same order as the threshold — implies that the resolution requires a second scale λ1 with N_p/λ1 ≈ 10^11 GeV, that is, a parameter fitted to the anomaly being explained. The paper is commendably explicit about this in §5.3, but the abstract and the introduction should say 'parameterizes' rather than 'explains,' and should foreground the no-go result, which is the substantive finding.
[§4.1, Eq. (31), and §5.2, Eq. (50)] The composition rule p(1...n) = U^{-1}[λ/n](Σ U[p_i;λ](p_i)) is postulated, not derived; the paper states that f(n,λ) 'can be any function leading to suitable limiting properties.' The UHECR threshold formula (50) depends on this rule through λ → λ/N_p, and N_p is left undefined ('may require the application of this theory to the full quantum field theory of QCD,' §5.2). The authors acknowledge this limitation, but the abstract's claim that the theory 'explains the anomaly' rests on an unquantified N_p, and the abstract should be tempered accordingly.

Minor objections (8)

[§3.1, Eq. (14)] As printed, p' = γ(1−v)p1 + λp(γ(1−v)−1) appears to be missing a division; the expression following from U = exp(λp₀D) is p' = γ(1−v)p/[1+λp(γ(1−v)−1)]. Please check the typesetting.
[§3.1, Eq. (15)] Please show the derivation of p = mv/(1+λm) and of E = mc² with m = m₀γ from U = exp(λp₀D); inverting U at rest gives E = m/(1+λm), which does not obviously reproduce the stated relations.
[§6, Eq. (62)] From eq. (56) I obtain [x⁰,p₀] = (1−λp₀)(1−2λp₀), not (1−λp₀)². Eqs. (60) and (61) are consistent with the stated t, so please re-check (62).
[§5.3] The text says f3 = e^{E/E_P} gives 'a maximum momentum, equal to E_P'; in fact p = E e^{−E/E_P} is maximized at E = E_P with p_max = E_P/e.
[§5.2] The statement that 'N_p for the proton is likely of order 3' is not supported; please give the basis for this estimate or remove it.
[References] References [20] and [22] are the same preprint (gr-qc/0205067), and [35] 'Snyder, Phys. Rev. 1947' is incomplete (volume and page missing).
[§2.1] The requirement that 'the image of U must include [0,∞] for both energy and momentum' should state whether this is required for both signs of energy, since §3.1 relies on E ∈ [−∞,∞] to preserve the group property.
[Throughout] There are numerous typographical errors, presumably typesetting artifacts: 'moﬁﬁed' (p.16), 'analagous' (p.4), 'defntion' (p.15), 'threfore' (p.14), 'transforamtions' (p.6), 'Phyiscs' (author affiliation), and 'sc ale' (title). These should be corrected before publication.

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The paper

The paper claims

Deformed Lorentz symmetry can keep all inertial frames equal

The reading

The paper asks whether one can modify special relativity at the Planck scale — giving particles a maximum energy or momentum and an energy-dependent speed of light — without reintroducing a preferred rest frame. Its answer is yes, provided the ordinary Lorentz group acts non-linearly on momentum space through an invertible map $U$; the familiar quadratic invariant is replaced by a non-quadratic one, and the group algebra itself is untouched. The authors turn this into a working recipe: read $U$ off from a proposed dispersion relation, check two conditions (invertibility on a physical sector, and an image covering all energies and momenta), and the theory follows. They then show how to keep energy-momentum conservation and how to stop composite bodies from inheriting the elementary-particle cutoff, and they use the machinery to ask whether such theories can explain the observed ultra-high-energy cosmic ray and gamma ray threshold anomalies. The answer there is more guarded: with a single dimensional parameter the modified thresholds cannot be moved far from their ordinary values, so the authors construct a two-scale example that raises the cosmic ray threshold while keeping an invariant Planck energy, a maximal momentum, and a speed of light that grows with energy.

What carries the argument

The map $U$ on momentum space, and the conjugated generators $K^i = U^{-1} L^{i0} U$ that it induces. $U$ is the object that carries the whole argument: it converts a proposed dispersion relation into a set of transformation laws, it decides whether an invariant energy or a maximal momentum exists (through the behaviour of $E f_1$ and $E/f_3$), and it fixes how momenta compose. Two conditions on $U$ do the load-bearing work — it must be invertible on a physical sector $P_{phys}$ (typically $E < E_{Planck}$ and/or $|p| < E_{Planck}$), and its image must cover $[0,∞]$ in both energy and momentum, since otherwise some boost parameter is bounded, the group property fails, and a preferred frame appears. The second piece of machinery is the composition rule $p_{(1...n)} = U^{-1}[λ/n] (U[p_1;λ](p_1) + … + U[p_n;λ](p_n))$, which replaces ordinary addition of momenta and rescales the deformation parameter by the number of constituents so that macroscopic bodies recover ordinary Lorentz behaviour.

Extended reading notes

Core claim

The central result is that all five properties the authors list — relativity of inertial frames, an invariant energy scale of order the Planck energy, a raised ultra-high-energy cosmic ray threshold, a varying and preferably diverging speed of light, and a maximal momentum — can coexist in a single theory. The construction rests on writing the modified Lorentz generators as $K^i = U^{-1} L^{i0} U$, where $L^{ab}$ are the standard generators and $U$ is a generally non-unitary, non-linear map of momentum space. Any isotropic dispersion relation $E^2 f_1^2(E;λ) - p^2 f_2^2(E;λ) = m^2$ defines such a map via $U(E,p) = (E f_1, p f_2)$, so the dispersion relation becomes the experimental input and the transformation laws the output. An invariant Planck energy appears when $E f_1$ diverges at some finite $E_P$; a maximal momentum appears when $E/f_3$ has a maximum, with $f_3 = f_2/f_1$, and the authors note this forces the speed of light to diverge somewhere. The paper then shows that a single-scale $f_1$ cannot move a threshold far from its special-relativistic value without putting the scale at the threshold itself, and proposes $f_1 = 1/[(1+λ_1 E)(1-λ E)]$ with $λ_1$ much larger than $λ$, which raises the cosmic ray threshold at $N_p λ_1^{-1} ≈ 10^{11}$ GeV while leaving an invariant positive energy $E_P = λ^{-1}$ of order $10^{19}$ GeV.

Load-bearing premise

The multi-particle composition rule, which rescales the deformation parameter to $λ/n$ and is generalized to an arbitrary function $f(n,λ)$ that the authors say 'can be any function leading to suitable limiting properties' — the cosmic ray threshold formula depends entirely on this postulated rule, and the number of quanta in a proton, $N_p$, is left undetermined pending a QCD treatment.

Editorial extensions

If this is right

Any proposed deformed dispersion relation can be converted into a full set of transformation laws, so the transformation law no longer has to be guessed independently of the dispersion relation.
A theory with an invariant Planck energy automatically contains three distinguished high-energy states: photons at the invariant energy, particles of infinite rest mass, and 'infinitions' — finite-mass particles that cannot be brought to rest.
Because a maximal momentum forces the speed of light to diverge at some energy, a theory with a granular short-distance cutoff and a theory with a rising speed of light are not independent options.
Energy and momentum conservation survives, but additivity does not: conservation is written $p^a \oplus q^a = p'^a \oplus q'^a$, so a particle near the Planck energy becomes progressively harder to accelerate in a collision.
Defining spatial coordinates as generators of translations in momentum space yields a commutative spacetime, $[x^a, x^b] = 0$, at the price of an energy-dependent effective Planck constant $\hbar = 1 - λ p_0$ in $[x^i, p^j] = δ^{ij}(1 - λ p_0)$.

Reading between the lines

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

The two-scale construction $f_1 = 1/[(1+λ_1 E)(1-λ E)]$ is a template rather than a prediction: any function that is approximately one below the Planck energy and diverges there would do the same job, so the framework predicts a family of threshold behaviours rather than a number.
The energy-dependent $\hbar$ in the real-space formulation suggests a route to laboratory tests: if $\hbar$ falls with momentum, the uncertainty relations themselves become momentum-dependent, which is in principle a different signature from a modified dispersion relation alone.
The requirement that $U$'s image cover all energies is a sharp selection rule that any candidate quantum-gravity dispersion relation must pass; it could be applied as a filter to dispersion relations derived from other approaches before asking about their phenomenology.
The proposal that the deformation parameter scales as $λ/n$ for an $n$-constituent system implies that the same physical body would need a different kinematical description depending on how its internal structure is counted, which is a strong and unusual claim about where the quantum-classical boundary sits.

Original abstract

The hypothesis that the Lorentz transformations may be modified at Planck scale energies is further explored. We present a general formalism for theories which preserve the relativity of inertial frames with a non-linear action of the Lorentz transformations on momentum space. Several examples are discussed in which the speed of light varies with energy and elementary particles have a maximum momenta and/or energy. Energy and momentum conservation are suitably generalized and a proposal is made for how the new transformation laws apply to composite systems. We then use these results to explain the ultra high energy cosmic ray anomaly and we find a form of the theory that explains the anomaly, and leads also to a maximum momentum and a speed of light that diverges with energy. We finally propose that the spatial coordinates be identified as the generators of translation in Minkowski spacetime. In some examples this leads to a commutative geometry, but with an energy dependent Planck constant.

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, 5,518 input tokens, $0.0002 reported.

What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.14 · human participants 0.01 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.06 · compares with prior results 0.15 · mentions code or data availability 0.01.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; show every role.

The hypothesis that the Lorentz transformations may be modified at Planck scale energies is further explored. We present a general formalism for theories which preserve the relativity of inertial frames with a non-linear action of the Lorentz transformations on momentum space. Several examples are discussed in which the speed of light varies with energy and elementary particles have a maximum momenta and/or energy. Energy and momentum conservation are suitably generalized and a proposal is made for how the new transformation laws apply to composite systems. We then use these results to explain the ultra high energy cosmic ray anomaly and we find a form of the theory that explains the anomaly, and leads also to a maximum momentum and a speed of light that diverges with energy. We finally propose that the spatial coordinates be identified as the generators of translation in Minkowski spacetime. In some examples this leads to a commutative geometry, but with an energy dependent Planck constant.

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 hypothesis that the Lorentz transformations may be modified at Planck scale energies is further explored.	background 0.96	0.22	0.04	hedged possibility 0.94
s1	We present a general formalism for theories which preserve the relativity of inertial frames with a non-linear action of the Lorentz transformations on momentum space.	contribution 0.96	0.92	0.17	asserted 0.99
s2	Several examples are discussed in which the speed of light varies with energy and elementary particles have a maximum momenta and/or energy.	result 0.68	0.64	0.10	asserted 0.95
s3	Energy and momentum conservation are suitably generalized and a proposal is made for how the new transformation laws apply to composite systems.	contribution 0.89	0.90	0.09	asserted 0.79
s4	We then use these results to explain the ultra high energy cosmic ray anomaly and we find a form of the theory that explains the anomaly, and leads also to a maximum momentum and a speed of light that diverges with energy.	result 0.87	0.95	0.40	asserted 1.00
s5	We finally propose that the spatial coordinates be identified as the generators of translation in Minkowski spacetime.	contribution 0.58	0.78	0.12	speculative 0.48
s6	In some examples this leads to a commutative geometry, but with an energy dependent Planck constant.	result 0.61	0.76	0.10	hedged possibility 0.60

What kind of paper the classifier reports this is: theoretical result 0.99 · evidence from proof or derivation 0.93.

What was examined, and what was not

sources read

arXiv listing abstract v1

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

References

Works cited (37)

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Forward citations

Cited by 12 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

The principle of relative locality
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A boundary-term prescription makes relative-locality theories translation-invariant for causally connected interactions, and in kappa-Poincare momentum space the torsion makes photon arrival-time delays depend on the ...
A winding number analysis of Schwarzschild black hole stability in light of Planck-scale modified kinematics
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For the cubic entropy correction S=πr_h²−αr_h³ arising from a Planck-scale modified dispersion relation, all physically allowed Schwarzschild-like branches have winding number w=−1, so no stable phase appears.

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From finite deterministic updates, deformed bosonic and fermionic canonical algebras emerge after coarse-graining, both normalized by the same effective Planck constant.
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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.
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How later papers used it

97 later papers cite it

Evidence: some. 95 of the citing papers share no author with it, and 8 of those carry a confident label from Pith's citation classifier.

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1 later paper by other authors is listed below: 1 compared its own results with it.

Compares with it 1 of 4 papers with this label

Compared its own energy-momentum undeformed dispersion relations result, derived from a general noncommutative differential geometry construction, with the Deformed Special Relativity proposed in this paper, also known as DSR2.

Observables and Dispersion Relations in k-Minkowski Spacetime (2017) · 1.00 · paraphrased: the licence does not allow quoting

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78 arXiv papers cite this in 120 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 · 56 sentences without a confident label

Uses its method or tool 3 papers

In this paper, the model we adopt is DSR2, which was proposed and generalized by Magueijo and Smolin [ref] .

Photon Gas Thermodynamics in Doubly Special Relativity · Modified Dispersion Relation For Photon Gas · 0.80
Let us start with the all important map [ref] [equation] which in explicit form reads [equation] where $n_μ=(1, 0, 0, 0)$ is introduced to express the map in a covariant way and we use the notation $a_μb^μ=(ab)$, $(n p)=p^0$, $(n P)=P^0$.

Effects of a Maximal Energy Scale in Thermodynamics for Photon Gas and Construction of Path Integral · Deformations in Lorentz symmetry: noncommutative spacetime · 0.96
By using the non-linear Lorentz transformation in the momentum space, the MDR can be compactly written as [ref]

Black hole complementarity with the generalized uncertainty principle in gravity's rainbow · black hole complementarity with GUP in Gravity's Rainbow · 0.80

Compares with it 3 papers

Thus the DSR model considered in [ref] has a more theoretical motivation and it can be developed starting from the NC phase space variables [ref] whereas the models considered in [ref] are phenomenological in nature and as far as we know, there is no fundamental phase space structures to describe these models.

Effects of a Maximal Energy Scale in Thermodynamics for Photon Gas and Construction of Path Integral · Thermodynamics of photon gas in $κ$-Minkowski spacetime · 0.82
Finally, we compare our results with the pioneering approach in [ref] .

Observables and Dispersion Relations in k-Minkowski Spacetime · Introduction · 1.00
In particular it turns out that the invariant dispersion relation for the four-momentum vector is the same as in SR ($E^2 = p^2 + m^2$), which is not the case in most DSR studies [ref] .

Minimum Length - Maximum Velocity · Reference frame invariance of the GUP formalism · 0.91

Cites it as support 1 paper

Also the speed of light $c$ is an invariant quantity in the DSR model [ref] .

Effects of a Maximal Energy Scale in Thermodynamics for Photon Gas and Construction of Path Integral · Thermodynamics of photon gas in $κ$-Minkowski spacetime · 0.97

Cites it as background 52 papers

For this purpose we return here to the simple perspective and formalization of the first studies [ref] which noticed that some known deformations of relativistic symmetries, some of the ones studied within the “doubly-special relativity" research programme (see, e.g., Refs. [ref] ), could result in the property that coincidences of events established by nearby observers might not appear as coincidences of events in the coordinatization of spacetime by distant observers.

Modeling transverse relative locality · Introduction · 0.80
This modification of Special Relativity, which does not introduce a preferred frame but instead postulates the Planck mass as an observer-independent invariant, has become known as “Deformed Special Relativity” (DSR) [ref] .

The Box-Problem in Deformed Special Relativity · Introduction · 0.90
On the other hand, a relativistic description for the time delay predictions [ref] , i.e., observer independent, is possible within the Double Special Relativity (DSR) framework introduced in [ref] (see also [ref] ).

Rainbows without unicorns: Metric structures in theories with Modified Dispersion Relations · Introduction · 0.86

Discussion

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