Lorentz invariance with an invariant energy scale

Joao Magueijo; Lee Smolin

REVIEW

Lorentz invariance with an invariant energy scale

Joao Magueijo, Lee Smolin

High Energy Physics - Theory · Submitted 11 Dec 2001 · 12 references

Pith's verdict: A clean, honest construction that genuinely makes the Planck energy an invariant — the weak point is the physics of composite systems, which the paper asserts rather than derives.

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arXiv IDhep-th/0112090 (version 2)

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

CitationJoao Magueijo, Lee Smolin. "Lorentz invariance with an invariant energy scale." arXiv:hep-th/0112090 (2001). https://pith.science/paper/ELHFNVRC

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Editor's note

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Worth your time, and worth citing. The core move is simple and correct: add a dilatation to each boost, K^i = L^{i0} + l_P p^i D, so the algebra closes on the ordinary Lorentz structure constants while the action on momentum space becomes non-linear. The Planck energy is then a fixed point by construction, the invariant in Eq. 11 follows, and the Doppler formula (15)-(16) has E = 1/l_P as a fixed point. I checked 12, 15, 16 against 11 and they hold. The Fock analogy is apt and the paper is careful to say why the x-space and p-space versions do different things. It also states plainly that the choice of U(p0) is not unique and that other choices give other dispersion relations. That is the right kind of honesty for a proposal paper, and it is why this one launched a program rather than dying in a footnote. [abstract:s0, abstract:s1, abstract:s2] are all accurate as stated; nothing in the abstract overreaches the body. The soft spot is the one the stress-test flags, and it lands. After Eq. 14 the paper says macroscopic bodies made of low-energy particles 'do not feel the transformations (7) because these, being non-linear, are not additive.' That is an assertion standing in for a composition law. Without one, the transformation of a total momentum under a boost is undetermined, and the claim that E_P is invariant for all observers and all systems is established only for single particles. This is the soccer-ball problem, and it is not a small thing — it is exactly where the program later got stuck. The paper does not list it among the open questions at the end, which it should have. Two smaller things. Eq. 18 does not look like the field-theory image of Eq. 11: the invariant has (1 - l_P p0)^{-2}, the action has (1 - l_P^2 ∂0φ)^{+2}. That is an inversion and a stray power of l_P; either a typo or a real inconsistency, and a referee should make them fix it. And the relation to Amelino-Camelia's independent 2001 proposal is not engaged with, though the timing makes that forgivable. Who should read it: anyone working on quantum-gravity phenomenology or deformed symmetries, and anyone who wants a clean worked example of a non-linear group action. Send it out. The construction is sound, the open problems are real but honestly framed, and the paper is short enough that a referee can check the algebra in an afternoon.

Referee report

Significance

If the construction holds, it is a significant conceptual result: it shows that an invariant energy scale need not be purchased by breaking Lorentz invariance or by introducing a preferred frame. The explicit algebraic construction, the unchanged structure constants, and the identification of the transformation with Fock's (applied to momentum space rather than position space) are genuine strengths. The paper also makes falsifiable statements — the fixed point at E = E_P in Eq. (16), the impossibility of Doppler-crossing E_P, and the modified gravitational redshift in Eq. (17) — which is more than many proposals in this area offer. The manuscript is a letter, so the sketchiness of the field-theory and GR sections is partly acceptable, but the physical extension to composite systems is not merely a loose end.

Major objections (4)

[After Eq. (14)] The assertion that macroscopic bodies 'may have masses larger than E_P, but if they are made of particles with E << E_P they do not feel the transformations (7) because these, being non-linear, are not additive' is asserted rather than derived. Non-additivity of the transformation does not imply that composite systems are unaffected; it implies that the transformation of the total momentum is not the sum of the transformations of the constituents. Without a composition law for total momentum, the transformation of a composite system under a boost is undetermined by Eqs. (7)-(10) alone, and the claim that E_P is an invariant for all physical systems, not just single particles, is not established. A concrete test would help: take two particles each with E << E_P moving in opposite directions, state what the total momentum is and how it transforms, and say whether the total energy is bounded by E_P via Eq. (13) applied to the total mass. If it is, then the total energy is not the sum of constituent energies, which needs to be stated and justified.
[Eq. (18)] The displayed scalar-field action does not follow from the invariant (11). The invariant has the factor (1 - l_P p_0)^{-2} multiplying eta_ab p^a p^b, whereas the action as printed has [1 - l_P^2 (partial_0 phi)]^2 multiplying eta^{ab}(partial_a phi)(partial_b phi), i.e. a numerator factor rather than the inverse denominator. The factor is also dimensionally inconsistent: partial_0 phi has mass dimension 2 while l_P has dimension -1, so l_P partial_0 phi is not dimensionless. The natural identification is with the wave vector k_a (for a plane wave partial_a phi = -i k_a phi), not with partial_a phi itself. The action should be corrected and the identification stated explicitly.
[Eq. (17) and the modified equivalence principle] Eq. (17) is derived by applying the standard equivalence principle (v = Delta phi) to the Doppler formula, but later the paper proposes a modified equivalence principle in which the non-linear realization acts on partial_a phi in orthonormal frames. These two statements should be reconciled: does Eq. (17) follow from the modified equivalence principle, or from the standard one? If the latter, the two are inconsistent, and the GR embedding is not determined by the principles stated. A short derivation of Eq. (17) from the modified equivalence principle, or an explicit statement that it is only a heuristic first-order estimate, would resolve the tension.
[End of the paper, discussion of U(p0)] The paper acknowledges that many non-linear realizations exist, obtained by other choices of U(p0) in Eq. (4), leading to different invariants and dispersion relations. What remains unresolved is whether any principle in the paper selects this U: the correspondence principle, the requirement that the algebra be unmodified, and the invariance of c are all satisfied by other choices as well. If the choice is arbitrary, then the specific predictions (Eqs. 15-17) are predictions of a particular model rather than of the framework, and the paper should say so explicitly and identify what additional principle would fix U. This matters for assessing how much of the paper's physical content is forced by the stated postulates.

Minor objections (7)

[Title page and throughout] Several typographical errors should be corrected: 'Maguejo' should be 'Magueijo', 'Phyiscs' should be 'Physics', 'Pound Rebbka' should be 'Pound-Rebka', 'surpising' should be 'surprising', 'Riemanian' should be 'Riemannian', and 'Kilowski-Gilkman' is presumably 'Kowalski-Glikman'.
[Eq. (2)] The notation 'lP piD' should be written with explicit indices and spacing, e.g. l_P p^i D, to avoid ambiguity about which index is contracted.
[Eq. (5)] The action of U(p0) is easy to misread. As written, exp(l_P p_0 D) with D = p^a partial_a would give e^{l_P p_0} p^a, not p^a/(1 - l_P p_0). The result in Eq. (5) is correct only under the field-dependent reading stated in the text ('p0 always means the time component of the vector immediately to the right'), and a clarifying sentence or an explicit flow derivation would help the reader.
[Eq. (3)] The sign of the commutator [K^i, K^j] should be checked against the stated conventions (antihermitian generators, metric signature (+,-,-,-)); the standard result in these conventions is [K^i, K^j] = -epsilon^{ijk} J^k. If the paper's convention differs, it should be stated.
[Paragraph before Eq. (1)] The claim that 'the only six parameter group that has these characteristics is the Lorentz group itself' is used to argue that the symmetry group must be the Lorentz group acting non-linearly. This should be qualified: there are other six-dimensional Lie groups containing SO(3) (for example SO(4) and ISO(3)), and the argument needs a precise statement of which characteristics are meant and why they exclude those alternatives.
[Discussion of super-Planckian photons, after Eq. (16)] The statement that for a receding source there is a speed for which E' = infinity and beyond which E' < 0 is a striking consequence and deserves a sentence of physical interpretation. Negative energies are a strong claim, and the reader should be told whether they are regarded as unphysical (and hence as excluding that regime) or as a genuine feature of the model.
[Introduction and references] The paper does not cite the literature on non-linear realizations of the Lorentz and Poincare groups (for example Wigner's 1939 work and the standard discussion in Weinberg's textbook), which is the mathematical framework being used. A citation would help place the construction.

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The paper

The paper claims

The paper proposes that the Planck energy can be an invariant of motion, sitting alongside the speed of light, without giving up the relativity of inertial frames.

The reading

Special relativity says no preferred frame exists, and that the speed of light is the same for everyone. Quantum gravity suggests there is also a special energy, the Planck energy, at which spacetime stops behaving classically. The puzzle the paper confronts is whose frame that threshold lives in: if one observer calls it the Planck energy, an ordinary Lorentz boost seems to hand a different observer a different number. The authors argue the answer is to keep the Lorentz group but let it act non-linearly on momentum space, adding a stretch (a dilatation) to every boost so that the Planck energy is carried back to itself. If that works, every free-falling observer agrees on where the classical description breaks down, while ordinary Lorentz behaviour survives at everyday energies.

What carries the argument

The load-bearing device is the energy-dependent transformation $U(p_0) = exp(l_P p_0 D)$, which conjugates the ordinary Lorentz generators into modified ones, $K^i = U^{-1}(p_0) L^{i0} U(p_0)$. Because $U$ acts as a rescaling on momentum components, $U(p_0) circ p^a = p^a/(1 - l_P p_0)$, and the group action $W[omega_{ab}] = U^{-1} e^{omega_{ab} L^{ab}} U$ inherits the ordinary algebra while acquiring a non-linear action on the momentum vector. The singularity of $U$ at $p_0 = 1/l_P$ is precisely the new invariant scale: the transformation blows up there, which is how the Planck energy is pinned. The same conjugation pattern is then applied to derivatives of fields in curved spacetime, $partial_a phi = e^mu_a partial_mu phi$, to state a modified equivalence principle in which the non-linear realization acts on orthonormal-frame field derivatives rather than on the frame components themselves.

Extended reading notes

Core claim

The paper constructs a six-parameter symmetry group with exactly the ordinary Lorentz algebra, but acting non-linearly on momentum space. Each boost generator is modified by adding a dilatation term, $K^i = L^{i0} + l_P p^i D$, where $l_P$ is the Planck length and $D = p^a partial_{p^a}$; the commutators $[J^i, K^j] = epsilon^{ijk} K^k$ and $[K^i, K^j] = epsilon^{ijk} J^k$ are unchanged, so this is a non-linear realization rather than a deformation of the algebra. Exponentiating gives modified boost equations (the paper's equations 7 to 10) that reduce to the usual ones for small momenta and that map a particle of energy $E_P$ to itself. The quadratic invariant is replaced by $|p|^2 = eta_{ab} p^a p^b / (1 - l_P p_0)^2$, which diverges at $E = 1/l_P$, so the Planck energy becomes an invariant and the notion of a quadratic metric collapses near it. The resulting kinematics gives $E = m/(1 + m/E_P)$ and $p = m v/(1 + m/E_P)$, so no particle's energy can reach $E_P$; the photon Doppler shift becomes $1/E' - 1/E_P = (1/(gamma(1-v)))(1/E - 1/E_P)$, so sub- and super-Planckian photons never mix. The authors note the boost transformation is identical to one Fock wrote down, but applied to momentum space instead of position space, which is what makes the invariant a large energy rather than a large length.

Load-bearing premise

The specific choice of the dilatation factor, $U(p_0) = exp(l_P p_0 D)$, is not derived from any deeper principle: the paper itself says many other choices of $U(p_0)$ give other invariants and other dispersion relations, and nothing in the argument selects this one over the alternatives.

Editorial extensions

If this is right

No particle can be accelerated to or past the Planck energy; a body's energy approaches $E_P$ asymptotically as its rest mass grows without bound, and at that limit energy and momentum become frame-independent.
Photons still satisfy $E = |p|$ and travel at $c$ in this particular realization, so the theory does not predict an energy-dependent speed of light, unlike other Planck-scale proposals the paper mentions.
The Doppler shift is modified to $1/E' - 1/E_P = (1/(gamma(1-v)))(1/E - 1/E_P)$, and via the equivalence principle the gravitational redshift becomes $Delta E/E = Delta phi (1 - l_P E)$, a suppression that grows with photon energy.
The scalar field action acquires the factor $[1 - l_P^2 (partial_0 phi)]^2$ multiplying the kinetic term, so the field equation is non-linear and the superposition principle fails, though a single plane wave remains a solution for a massless field.
In general relativity, the modified equivalence principle is stated as a rule for how the non-linear Lorentz action applies to orthonormal-frame derivatives of fields, leaving the spin connection and curvature formalism intact.
The sign of $l_P$ is not fixed by the theory; the two signs give physically distinct predictions, with $E_P > 0$ making particle energies lower than $m c^2$ and $E_P < 0$ making them higher and divergent at the Planck mass.

Reading between the lines

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

Because the algebra's structure constants are untouched, the modification is a choice of coordinates on momentum space rather than a new symmetry; one could ask whether the non-linear realization is a field redefinition of ordinary special relativity with no observable content, which is the sharpest version of the question the paper leaves open.
The paper's own remark that many choices of $U(p_0)$ give different dispersion relations turns the proposal into a family of theories, so the natural next step is to classify which choices preserve $E = |p|$ for photons and which introduce a varying speed of light, and then use that classification to organise gamma-ray-burst and ultra-high-energy-cosmic-ray searches.
The prediction that super-Planckian photons redshift when the source moves toward the observer, and cannot be redshifted below $E_P$, is an unusual signature that would be visible in the spectra of extreme accelerators if such photons exist at all.
Because macroscopic bodies made of low-energy particles do not feel the non-linear transformation (it is not additive), the proposal predicts no anomalous behaviour for bulk matter, which suggests the observable window is narrow: single particles near the Planck energy, or cumulative propagation effects over cosmological distances.

Original abstract

We propose a modification of special relativity in which a physical energy, which may be the Planck energy, joins the speed of light as an invariant, in spite of a complete relativity of inertial frames and agreement with Einstein's theory at low energies. This is accomplished by a non-linear modification of the action of the Lorentz group on momentum space, generated by adding a dilatation to each boost in such a way that the Planck energy remains invariant. The associated algebra has unmodified structure constants, and we highlight the similarities between the group action found and a transformation previously proposed by Fock. We also discuss the resulting modifications of field theory and suggest a modification of the equivalence principle which determines how the new theory is embedded in general relativity.

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, 3,700 input tokens, $0.0002 reported.

What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.23 · human participants 0.01 · controlled comparison 0.04 · randomized assignment 0.01 · states quantitative result 0.05 · compares with prior results 0.90 · mentions code or data availability 0.01.

The abstract, sentence by sentence

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

We propose a modification of special relativity in which a physical energy, which may be the Planck energy, joins the speed of light as an invariant, in spite of a complete relativity of inertial frames and agreement with Einstein's theory at low energies. This is accomplished by a non-linear modification of the action of the Lorentz group on momentum space, generated by adding a dilatation to each boost in such a way that the Planck energy remains invariant. The associated algebra has unmodified structure constants, and we highlight the similarities between the group action found and a transformation previously proposed by Fock. We also discuss the resulting modifications of field theory and suggest a modification of the equivalence principle which determines how the new theory is embedded in general relativity.

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	We propose a modification of special relativity in which a physical energy, which may be the Planck energy, joins the speed of light as an invariant, in spite of a complete relativity of inertial frames and agreement with Einstein's theory at low energies.	contribution 1.00	0.84	0.79	hedged possibility 0.63
s1	This is accomplished by a non-linear modification of the action of the Lorentz group on momentum space, generated by adding a dilatation to each boost in such a way that the Planck energy remains invariant.	method 0.93	0.89	0.18	asserted 1.00
s2	The associated algebra has unmodified structure constants, and we highlight the similarities between the group action found and a transformation previously proposed by Fock.	result 0.76	0.80	0.07	asserted 0.99
s3	We also discuss the resulting modifications of field theory and suggest a modification of the equivalence principle which determines how the new theory is embedded in general relativity.	contribution 0.67	0.80	0.09	hedged possibility 0.80

What kind of paper the classifier reports this is: theoretical result 0.96 · evidence from proof or derivation 0.87.

What was examined, and what was not

sources read

arXiv listing abstract v2

not examined

full text
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references and citing sentences
relations to other papers

levels run

P abstract

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S sections
T body sentences
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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

Works cited (12)

[1]

C. Rovelli and L. Smolin Discreteness of area and volume in quantum gravity Nuclear Physics B 442 (1995) 593. Erratum: Nucl. Phys. B 456 (1995) 734

View on Pith 1995
[2]

Giovanni Amelino-Camelia and Tsvi Piran, Planck-scale deformation of Lorentz symmetry as a solution to the UHECR and the TeV- γ paradoxes, astro-ph/0008107 Phys.Rev. D64 (2001) 036005

View on Pith Pith review arXiv 2001
[3]

G. Amelino-Camelia et al, Int.J.Mod.Phys.A12:607- 624,1997; G. Amelino-Camelia et al Nature 393:763- 765,1998

View on Pith 1997
[4]

J. Ellis et al, Astrophys.J.535:139-151,2000

View on Pith 2000
[5]

G. Amelino-Camelia, S. Majid, Int.J. Mod. Phys. A15:4301-4324, 2000

View on Pith 2000
[6]

J. Ellis, N.E. Mavromatos and D. Nanopoulos, Phys.Rev.D63:124025,2001; ibidem astro-ph/0108295

View on Pith Pith review arXiv 2001
[7]

V. Fock, The theory of space-time and gravitation, Perg- amon Press, 1964

View on Pith 1964
[8]

S. N. Manida, gr-qc/9905046

View on Pith Pith review arXiv
[9]

S. S. Stepanov, physics/9909009, astro-ph/9909311

View on Pith Pith review arXiv
[10]

J. Moﬀat, Int. J. of Physics D 2, 351 (1993); J. Moﬀat, Foundations of Physics, 23, 411 (1993)

View on Pith 1993
[11]

A. Albrecht and J. Magueijo, Phys. Rev. D 59, 043516 4 (1999)

View on Pith 1999
[12]

J. Magueijo, Phys.Rev.D63, 043502 (2001). 5

View on Pith 2001

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How later papers used it

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Evidence: substantial. 187 of the citing papers share no author with it, and 16 of those carry a confident label from Pith's citation classifier.

Pith read 17 of the 281 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

2 later papers by other authors are listed below: 1 built on it and 1 used its method or tool.

Builds on it 1 of 2 papers with this label

Extended this paper's momentum-space construction to arbitrary representations, including position space, since its quantization map does not depend on the representation considered.

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

Uses its method or tool 1 of 5 papers with this label

Followed this paper's MS formalism in its own work.

Invariant ultraviolet scale corrections to the thermodynamics of degenerate Fermi gas and its implications (2018) · 0.99 · paraphrased: the licence does not allow quoting

How the literature uses it

151 arXiv papers cite this in 219 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 · 105 sentences without a confident label

Contests it 1 paper

Consequently, the refutation of the paradox raised by Amelino-Camelia [ref] and by Magueijo and Smolin [ref] , presented in [ref] , appears ineffective in this context, since the paradox itself is formulated precisely in Minkowski spacetime, where the Poisson bracket is assumed to vanish.

On the non-commutativity of geometric observables in different Lorentz frames · Summary and discussion · contest 0.85

Builds on it 1 paper

Since the quantization map $D$ does not depend on the representation considered, this allows to extend the construction in [ref] , based on momentum space, to arbitrary representations; in particular to position space.

Observables and Dispersion Relations in k-Minkowski Spacetime · Introduction · 1.00

Uses its method or tool 2 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
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 4 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
This implies that massless fields in $κ$-Minkowski spacetime have no modified dispersion relations, in agreement with [ref] and disagreement with [ref] .

Observables and Dispersion Relations in k-Minkowski Spacetime · Introduction · 0.97
Such a curved momentum space is reminiscent of the structures appearing in doubly special relativity [ref] and in approaches based on the Born reciprocity principle [ref] .

Geometric structure of the relativistic quantum phase space · Geometric equation of the quantum phase space · 0.83

Cites it as support 2 papers

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
Indeed, there exist models which include a minimum length directly in what we would call the effective spacetime without violating Lorentz invariance [ref] .

On the condensed matter scheme for emergent gravity and interferometry · Discreteness of spacetime · 0.97

Cites it as background 90 papers

In RG, the modified dispersion relation of the test particle takes the form [ref] E^2 f^2(E/E_p) - P^2 g^2(E/E_p) = m^2, where m is the mass and P is the 3-momentum of the test particle.

Maximal acceleration in Rainbow gravity · Rainbow Gravity · 0.80
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
The alternative possibility is a “deformation" of relativistic symmetries, in the sense of the “Doubly-Special Relativity" [ref] (DSR) scenario, which insists on a relativistic formulation of laws, as required by the first postulate, and confines all revisions of relativity to revisions of the second postulate.

Testing quantum-spacetime relativity with gamma-ray telescopes · Introduction · 0.91

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.

[1] [1]

C. Rovelli and L. Smolin Discreteness of area and volume in quantum gravity Nuclear Physics B 442 (1995) 593. Erratum: Nucl. Phys. B 456 (1995) 734

View on Pith 1995

[2] [2]

Giovanni Amelino-Camelia and Tsvi Piran, Planck-scale deformation of Lorentz symmetry as a solution to the UHECR and the TeV- γ paradoxes, astro-ph/0008107 Phys.Rev. D64 (2001) 036005

View on Pith Pith review arXiv 2001

[3] [3]

G. Amelino-Camelia et al, Int.J.Mod.Phys.A12:607- 624,1997; G. Amelino-Camelia et al Nature 393:763- 765,1998

View on Pith 1997

[4] [4]

J. Ellis et al, Astrophys.J.535:139-151,2000

View on Pith 2000

[5] [5]

G. Amelino-Camelia, S. Majid, Int.J. Mod. Phys. A15:4301-4324, 2000

View on Pith 2000

[6] [6]

J. Ellis, N.E. Mavromatos and D. Nanopoulos, Phys.Rev.D63:124025,2001; ibidem astro-ph/0108295

View on Pith Pith review arXiv 2001

[7] [7]

V. Fock, The theory of space-time and gravitation, Perg- amon Press, 1964

View on Pith 1964

[8] [8]

S. N. Manida, gr-qc/9905046

View on Pith Pith review arXiv

[9] [9]

S. S. Stepanov, physics/9909009, astro-ph/9909311

View on Pith Pith review arXiv

[10] [10]

J. Moﬀat, Int. J. of Physics D 2, 351 (1993); J. Moﬀat, Foundations of Physics, 23, 411 (1993)

View on Pith 1993

[11] [11]

A. Albrecht and J. Magueijo, Phys. Rev. D 59, 043516 4 (1999)

View on Pith 1999

[12] [12]

J. Magueijo, Phys.Rev.D63, 043502 (2001). 5

View on Pith 2001