Phonons mimicking doubly special relativity kinematics

Francesco Marino

REVIEW 4 major objections 4 minor 43 references

Phonons mimicking doubly special relativity kinematics

T0 review · 4 major / 4 minor · reviewed 2026-08-16 · deepseek-v4-flash

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Pith's one-line read Phonons in tailored fluids can reproduce doubly special relativity

desk verdict Current version fails on its own equations, but the DSR-to-hydrodynamics mapping is a genuinely new idea worth a corrective rewrite. read the letter →

arxiv 2505.02026 v2 pith:32W34YUD submitted 2025-05-04 gr-qc hep-th

Francesco Marino This is my paper · ORCID

classification gr-qchep-th

keywords doublyspecialrelativityphononsanaloguegravitydispersionrelationviscoelasticfluidsstorageviscosityPlanckscalehydrodynamic

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper constructs two hydrodynamic models in which the collective sound waves (phonons) obey the deformed energy-momentum relations of doubly special relativity (DSR), with the fluid's viscoelastic parameters playing the role of the Planck scale. The first model adds an elastic storage term to the Euler equation and yields the DSR1 dispersion relation to order $E^3/E_p^3$; the second adds an elastic restoring force and yields the DSR2 relation after a rescaling of sound speed and mass. The demonstrations use the standard analogue-gravity setup of a barotropic, irrotational, inviscid fluid, extended with repulsive dipolar interactions that give the phonons a rest mass. If the constructions hold, quantum-gravity-inspired kinematics could be simulated in ordinary fluids rather than requiring Planck-scale physics.

What carries the argument

The argument is carried by the two linearized interaction potentials added to the Euler equation. The first, $\Phi_1=i\gamma\nabla^2\psi_1$, is called the storage-viscosity term: it is meant to represent momentum storage without dissipation, and its convective derivative supplies the $-i\gamma\partial_T\nabla^2\psi_1$ term in the modified Klein-Gordon equation. The second, $\Phi_1=i\gamma'\psi_1$, is an elastic restoring force analogous to a spring, and it leads to the DSR2 dispersion after redefining the sound speed and the phonon mass. The dipolar potential $\varphi_1$ satisfying the Poisson equation $\nabla^2\varphi_1=-\beta\rho_1$ provides the rest frequency $\Omega_0=\sqrt{\beta\rho_0}$, and the analogue Planck frequency for the DSR2 model is $\Omega_p=(4\beta\rho_0+\gamma')/(2\gamma')$.

What would settle it

Measure the complex shear modulus of a candidate viscoelastic fluid: if its real (dissipative) part is non-negligible at the phonon frequencies, or if its response violates the causality relations linking real and imaginary parts, the purely imaginary storage force needed for the DSR1 model cannot be physical. Alternatively, a direct search for the predicted dispersion $E^2=c_s^2p^2(1+E/E_p)+m^2c_s^4$ in a real fluid's phonon spectrum would fail if the linear-in-$E/E_p$ correction is absent.

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Extended reading notes

Core claim

The central discovery is that two simple additions to the fluid momentum equation turn the usual acoustic Lorentz-invariant phonon dispersion into the two canonical DSR laws. With a reactive storage term $\Phi_1=i\gamma\nabla^2\psi_1$ in the linearized Euler equation, the phonon wave equation becomes $\partial_T^2\psi_1=c_s^2\nabla^2\psi_1-\Omega_0^2\psi_1-i\gamma\partial_T\nabla^2\psi_1$, whose dispersion is the DSR1 relation $E^2\simeq c^2p^2(1+E/E_p)+m^2c^4$ at order $E^3/E_p^3$. With a spring-like restoring term $\Phi_1=i\gamma'\psi_1$ instead, the dispersion becomes the DSR2 relation after the rescalings $c_s^2\to c_s^2(1-m^2c_s^4/E_p)$ and $m^2\to m^2(1-m^2c_s^4/E_p)$, with $E_p=2m^2c_s^4/\gamma'$. In both models the rest mass comes from a repulsive dipolar interaction $\nabla^2\varphi_1=-\beta\rho_1$, and the analogue Planck scale is fixed by the viscoelastic coefficients $\gamma$ and $\gamma'$ together with $\beta$ and the background density.

Load-bearing premise

The construction assumes that a fluid can exert a purely reactive imaginary viscous force $i\gamma\nabla^2 v$ with no dissipative counterpart and no constitutive model, so that the linearized Euler equation can contain an imaginary term while the velocity potential remains real.

Editorial extensions

If this is right

A fluid with engineered dipolar and viscoelastic interactions would display phonon dispersions identical to DSR1 and DSR2 near its sonic Planck scale, making Planck-scale kinematics accessible in a tabletop system.
The analogue Planck scale is not fixed by atomic discreteness but by rheological parameters, so it can in principle be tuned across many orders of magnitude.
The two models show complementary deformations: the DSR1 relation can be read as an energy-dependent inertial mass, while the DSR2 relation corresponds to an energy-dependent rest mass; a fluid analogue could switch between the two by changing one interaction term.
The full nonlinear equations (continuity, Euler, and Poisson equations with either $\Phi=i\gamma\nabla^2\psi$ or $\Phi=i\gamma'\psi$) provide a concrete starting point for numerical or experimental study of analogue DSR kinematics.

Reading between the lines

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

The same design strategy should extend to other deformed dispersion relations: replacing the storage and restoring potentials with other interaction kernels would generate families of phonon laws with different high-energy corrections.
If a photon-fluid or atomic-condensate analogue is built, one could probe DSR predictions such as deformed boosts or an energy-dependent speed of sound; the fluid would be the first experimental system where a Planck-like scale is a tunable knob rather than a fixed constant.
A rheological consistency test follows from the model itself: a purely imaginary, frequency-independent viscosity would violate causality because the real and imaginary parts of any passive linear response are linked by integral relations, so the construction can be made fully physical only if a constitutive model produces a reactive force that is imaginary in the linearized equation but still cau
Because the DSR1 realization is stated only to order $E^3/E_p^3$, measuring the next-order term in the phonon dispersion would distinguish the analogue model from exact DSR1 and reveal the underlying microscopic cutoff.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Quick take: the construction as written doesn't do what it claims. The sign errors are real, and the iγ∇²v term isn't a legitimate time-domain force for a real fluid. But the underlying idea is not silly, and the paper is clearly written and well placed in the analogue-gravity literature.

What's new: the specific mapping of the DSR1 and DSR2 dispersions to a storage-viscosity term and an elastic restoring force is, as far as I know, not in the cited literature. That is a real conceptual suggestion. The paper also does a clean job of setting up the linearized hydrodynamics and connecting the effective Planck scale to the viscoelastic parameters.

The soft spots are load-bearing. First, Eq. (3) gives a subluminal correction of the wrong sign; for p² = E²/c_s² it produces E² = c_s²p² + ... − ℏ E³/E_p, whereas DSR1 has a plus. Second, Eq. (16) with Φ₁ = iγ'ψ₁ yields E² = c_s²p² + Ω₀² + γ'E; the printed Eq. (17) has the opposite sign. You can't fix that with the rescaling in (18); that rescaling just renames c_s and m and doesn't remove the sign mismatch. Third, and more fundamentally, the force iγ∇²v (or iγ∇²ψ₁) in a real-fluid Euler equation has no real solution for real ψ₁; a complex velocity potential forces a complex density, so the model is not a fluid in any ordinary sense. Rheological 'storage viscosity' is a frequency-domain object; a causal time-domain constitutive law is a convolution, not a bare imaginary term. The Conclusion's note about an 'oversimplified description' doesn't address this structural problem.

What should happen: I'd reject this version. The idea might be salvageable if the author can find a real constitutive model (e.g., a relaxational viscoelastic response) whose low-frequency/linearized limit gives the desired dispersion with the correct sign, but that requires substantial rewriting. As is, the derivation fails on its own terms.

Audience: someone working on analogue gravity and DSR phenomenology might get a useful idea out of the mapping, but they should treat the equations with suspicion.

Recommendation: don't desk-reject out of hand—send to a referee who can check the signs and the constitutive question. But the referee should understand the paper is currently not correct.

Referee Report

4 major / 4 minor

Summary. The manuscript claims to construct two hydrodynamic models in which phonons obey doubly special relativistic (DSR) kinematics. The first model adds a repulsive dipolar interaction and a purely imaginary 'storage viscosity' term iγ∇²v to the Euler equation, purportedly yielding the Amelino-Camelia (DSR1) dispersion relation up to O(E³/E_p³). The second model replaces this term with an elastic restoring force iγ'ψ, purportedly yielding the Magueijo-Smolin (DSR2) relation. The analogue Planck scale is identified with viscoelastic parameters. Section II contains the algebraic construction, and Section III summarizes the two models and discusses idealizations. The paper is written as a Letter and includes references to the analogue gravity and DSR literature.

Significance. If the reported construction were sound, it would be a noteworthy extension of the analogue gravity program, showing that DSR-type modified dispersion relations can emerge from non-Newtonian fluid mechanics and potentially offering a condensed-matter testbed for DSR phenomenology. The paper has strengths: it states clear target relations, builds on established analogue gravity concepts, and explicitly identifies the parameters that control the putative Planck scale. However, the central hydrodynamic derivation relies on an imaginary force term with no real-fluid time-domain realization, and the dispersion algebra does not reproduce the claimed DSR relations. As a result, the promised significance is not achieved by the present manuscript.

major comments (4)

[Section II, Eq. (3)] The dispersion relation obtained from Eq. (3) does not match DSR1. Fourier transforming Eq. (3) with ∂_T → -iω and ∇² → -k² gives ω² = c_s²k² + Ω₀² - γωk², i.e., in energy-momentum variables E² = c_s²p² + m²c_s⁴ - (c_s²p²E)/E_p (using γ = ℏc_s²/E_p). The correction is negative, whereas the DSR1 relation (1) requires E² ≈ c_s²p²(1 + E/E_p) + m²c_s⁴, i.e., a positive correction. Thus Eq. (3) yields a subluminal modification, not the Amelino-Camelia relation claimed in the abstract and Section III.
[Section II, Eqs. (16)-(18)] The claimed reduction to DSR2 is not supported by the equations. Fourier transforming Eq. (16) gives ω² = c_s²k² + Ω₀² + γ'ω, or E² = c_s²p² + m²c_s⁴ + ℏγ'E, with a positive correction proportional to ℏE. The paper's Eq. (17) instead states E² = c_s²p² + m²c_s⁴(1 - γ'E/(m²c_s⁴)), which has a negative correction with a different coefficient. The rescaling (18) only redefines c_s and m; it cannot flip the sign of the γ' term or change its coefficient from ℏγ'E to the form in Eq. (17). The claimed equivalence at linear order therefore fails.
[Section II, Eq. (14) and the storage-viscosity paragraph] The term iγ∇²v is not a legitimate time-domain force in a real fluid. In the linearized Euler equation (8), setting Φ₁ = iγ∇²ψ₁ makes the right-hand side imaginary for a real velocity potential ψ₁; the only real solution is ∇²ψ₁ = 0, which suppresses density dynamics via Eq. (7). If ψ₁ is instead taken complex, the continuity equation (7) forces ρ₁ to be complex, contradicting the interpretation of ρ₁ as a real density fluctuation. A causal viscoelastic constitutive relation is necessarily a memory convolution in the time domain; the frequency-domain storage modulus cannot be transplanted into the real-time Euler equation as a bare imaginary coefficient. This is a structural, not quantitative, problem: without this term, Eq. (3) is not derived from the hydrodynamics.
[Section III, Discussion and Conclusions] The matching of the dispersion relations is achieved by reverse engineering: the parameters γ and γ' are fixed a posteriori to reproduce the target DSR Planck scale (γ = ℏc_s²/E_p and E_p = 2m²c_s⁴/γ'), and the dipolar strength β sets the mass term. The statement that DSR 'naturally emerges' is therefore true by construction rather than by prediction. To support the claim of a physical analogy, the paper would need an independent identification of the viscoelastic parameters—e.g., from a rheological measurement—that did not presuppose the DSR scale.

minor comments (4)

[Section II, after Eq. (4)] The text says 'Substituting these expressions into Eq. (2)' but the substitution is into the continuity equation (4), not the DSR2 dispersion relation (2).
[Section II, Eq. (5)] Equation (5) is called the linearized continuity equation but contains the second-order term ρ₁v₁; the linearized equation should read ∂_tρ₁ + ∇·(ρ₀v₁) = 0.
[Section II, Eq. (14)] For γ = 0, Eqs. (14)-(15) reduce to the Euler equations for an inviscid fluid, not to the Navier-Stokes equations; Navier-Stokes would require a real viscous term η∇²v.
[Section II, text near Eq. (18)] The equivalence of the rescaling (18) to the new barotropic law P' = (1 + γ'/4βρ₀)P is stated without derivation; this step should be shown explicitly or removed.

Circularity Check

2 steps flagged · score 7.0 of 10

The DSR phonon models are reverse-engineered: the target dispersion is written as a modified Klein-Gordon equation and the force potentials are set to match it, so the claimed derivation is an identity.

self definitional [Section II, Eqs. (3), (12)-(13)]
"Substituting Eqs. (10)-(12) in (3) and defining the rest frequency Ω0 = √βρ0 and Φ1 = iγ∇2ψ1 we obtain ∂2TTψ1 = −1/ρ0 ∂TP1 −∂Tφ1 −∂TΦ1, which reduces to (8) after integration with respect to T."

Equation (3) is obtained by taking the DSR1 relation (1) and replacing E and p by quantum operators, so it already contains the claimed DSR1 kinematics. The free interaction potential Φ1 is then defined to be exactly the leftover term iγ∇2ψ1, with γ already fixed as ℏcs²/Ep. This makes the linearized Euler equation reduce to Eq. (3) by construction. The subsequent statement that phonons obey DSR1 is therefore not an emergent result but a restatement of the input dispersion relation (1).
self definitional [Section II, Eqs. (16)-(18)]
"By replacing the storage viscosity term iγ∇2ψ1 with the restoring force term Φ1 = iγ′ψ1 in Eq. (13) ... Defining Ep = 2m2cs4/γ′ and rescaling the sound speed and phonon mass as ... Eq. (17) reduces to the DSR2 energy-momentum relation (2)."

The DSR2 construction is explicitly a post hoc fit. The force term Φ1 = iγ′ψ1 is inserted to produce Eq. (16), and then Ep and the sound-speed/mass rescalings in Eq. (18) are chosen after the fact so that Eq. (17) matches the target relation (2). No independent hydrodynamic input fixes γ′, Ep, or the rescaling; they are all determined by the requirement that the final dispersion be DSR2. Thus the 'natural emergence' of Magueijo-Smolin kinematics is equivalent to imposing those kinematics through the free parameters.

full rationale

The paper is transparent that it is constructing models by 'carefully selecting the interaction potentials,' but the central derivations are not independent of the claimed results. For DSR1, the starting point Eq. (3) is just the DSR1 dispersion relation rewritten as a Klein-Gordon equation, and the potential Φ1 is defined to be the exact term needed to make the hydrodynamics reproduce Eq. (3). For DSR2, the parameters Ep and the rescalings in Eq. (18) are chosen after the target relation, so Eq. (17) reduces to Eq. (2) by definition. In both cases the phonon kinematics are reverse-engineered: the claimed 'demonstration' that phonons obey DSR is equivalent to inserting DSR into the model through the free force terms. This is a legitimate design exercise but not a prediction from first principles. No self-citation chain is load-bearing here. Separately, the imaginary force term iγ∇2v in the real time-domain Euler equation raises a physical-consistency concern about whether any real fluid realizes the proposed potential, but that concern is about correctness, not circularity.

Assumptions & free parameters 3 free parameters · 4 assumptions · 2 invented entities

The central claim rests on three interaction parameters (β, γ, γ') that are chosen by hand to reproduce DSR dispersions, and on two invented force terms that lack a real constitutive derivation. In addition, the explicit imaginary coefficients make the time-domain equations inconsistent with a real velocity potential. The DSR relations are the target input, not an emergent output, which explains the high circularity burden.

free parameters (3)

β (dipolar interaction strength) = chosen to set Ω0 = sqrt(βρ0)
Sets the phonon rest mass to the DSR mass m; not derived from independent data.
γ (storage viscosity coefficient) = γ = ℏc_s²/E_p
Chosen so that the modified dispersion has a Planck-scale correction equal to the DSR1 scale; the DSR dispersion is the target fitted here.
γ' (elastic restoring frequency) = γ' = 2m²c_s⁴/E_p
Chosen to match the linear correction of the Magueijo-Smolin relation; the DSR2 dispersion is the target.

assumptions (4)

domain assumption Barotropic, irrotational, inviscid base fluid with constant density and constant sound speed
Needed for a global Lorentz invariant phonon dispersion; stated in Section II around Eq. (9).
ad hoc to paper The fluid can support a purely imaginary (storage) viscosity with no dissipation, realized as a real hydrodynamic force
Introduced at Eq. (14) and the storage-viscosity paragraph; no constitutive relation or real-field derivation is provided, and an explicit i in the time-domain equation is inconsistent with a real velocity potential.
ad hoc to paper The DSR dispersion relations (1) and (2) are the target relations to be mimicked
The author chooses these as the goal; they are not derived within the paper and serve as the input for engineering the potentials.
standard math Linear perturbation theory applies with ρ1 ≪ ρ0 and v1 ≪ v0
Standard in analogue gravity; stated at the start of Section II.

invented entities (2)

Storage viscosity term iγ∇²v (reactive momentum-storage force)
purpose: Produces the DSR1-type correction to the phonon dispersion.
The paper postulates this force to reproduce Amelino-Camelia's DSR1 dispersion; no experimental observation or microscopic derivation is given, and the imaginary coefficient cannot act on a real velocity field.
Elastic restoring force Φ1 = iγ'ψ1
purpose: Produces the DSR2-type dispersion.
A spring-like term added by hand to the Euler equation to realize Magueijo-Smolin kinematics; it is not derived from a known fluid, and the sign in the resulting dispersion is opposite to the claimed one.

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Cite this review

Pith. "Pith review of Phonons mimicking doubly special relativity kinematics." pith.science (2026). https://pith.science/paper/32W34YUD

@misc{pith2026250502026,
  author       = {Pith},
  title        = {Pith review of: Phonons mimicking doubly special relativity kinematics},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/32W34YUD}},
  note         = {Machine review of arXiv:2505.02026}
}

read the original abstract

Collective excitations (phonons) in barotropic, irrotational, inviscid fluids exhibit an effective Lorentz invariance, where the sound speed plays the role of the invariant speed of light in special relativity. By carefully selecting the interaction potentials, we explicitly construct two hydrodynamic models in which phonons obey doubly special relativistic kinematics, with the analogue Planck scale emerging from non-Newtonian behaviour at high energies. Specifically, we demonstrate that elastic storage leads to an approximate realization of Amelino-Camelia's scenario, while the Magueijo-Smolin model naturally emerges in the presence of elastic restoring forces.

Discussion (0). Continue with ORCID to comment.

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Pith tools

Reviewed August 16, 2026 · model on record in the stance chip above.

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and Magueijo-Smolin [8], commonly referred to as DSR1 and DSR2. The DSR1 energy-momentum relation at O(E3/E 3 p) is E2 ≃c2p2 ( 1 + E Ep ) +m2c4 (1) where E is the total energy of a particle with momentum p and rest mass m. At this order of approximation, Eq. (1) can be directly compared with the DSR2 dis persion relation E2 =c2p2 +m2c4 ( 1 − E Ep ) 2 (2) ...

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