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The Lorentzian geometry of relaxation

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Pith's one-line read Causality forces dispersion curves of relaxational media to travel spacelike on a Lorentzian plane of imaginary frequency and wavenumber, yielding sharp universal bounds on diffusivity, viscosity, and the reach of hydrodynamics.

desk verdict Clean geometric reformulation that turns causality constraints on relaxational spectra into sharp, saturated bounds; the math holds under the stated assumptions. read the letter →

arxiv 2607.10148 v1 pith:GHZBQCZW submitted 2026-07-11 nucl-th gr-qchep-thmath-phmath.MP

Lorenzo Gavassino This is my paper · ORCID
classification nucl-thgr-qchep-thmath-phmath.MP
keywords relativistichydrodynamicscausalitydispersionrelationskinetictheorytransportcoefficientsrelaxationplaneLorentziangeometry
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

The paper shows that any relativistic medium whose linear modes are purely relaxational (real iω whenever ik is real) naturally turns the plane of imaginary frequency and wavenumber into a Lorentzian geometry just like ordinary spacetime. Modes fall into three classes—relaxation-like, unstable-like, and evanescent-like—exactly as vectors fall into future, past, and spacelike. Causality then forces every isolated dispersion curve to stay spacelike on that plane. Once that single geometric rule is in hand, longstanding questions about boosted spectra, the breakdown of hydrodynamics, maximal transport coefficients, and the presence of ballistic cuts become elementary constructions that can often be read off a diagram. The resulting bounds are sharp: diffusivity cannot exceed w^{2} times the non-hydrodynamic gap, acoustic diffusivity cannot exceed (w^{2}−c_s^{2}) times the gap, and the hydrodynamic radius of validity is at least half that gap divided by the maximal signal speed.

What carries the argument

The Lorentzian structure of the (iω, ik) plane together with Theorems 1 and 2: the spectrum expands at most at speed w (Hausdorff distance), and every isolated branch therefore has spacelike tangent. Self-adjointness of σ and E guarantees that iω remains real for real ik, so the entire geometry lives on a real plane.

What would settle it

Find a causal, finite-order linear PDE whose spectrum is purely real for real imaginary wavenumber yet whose isolated dispersion curve has slope steeper than the light cone somewhere on the (iω, ik) plane; or construct a kinetic theory that saturates the diffusivity bound D = w^{2} τ_g while remaining causal and stable.

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Extended reading notes

Core claim

Under the structural assumptions that linearized dynamics take the Boltzmann-like form ∂_t Ψ = −σΨ − E ∂_x Ψ with self-adjoint operators σ and E and operator norm ||E|| ≤ 1, causality forces every isolated dispersion relation on the relaxation plane to be spacelike: |d(iω)/d(ik)| ≤ w ≤ 1. This single constraint is the source of all subsequent universal bounds on transport and spectral geometry.

Load-bearing premise

The linearized dynamics must be writable with two self-adjoint operators—one for collisions, one for free streaming—so that imaginary frequency stays real whenever imaginary wavenumber is real; if that self-adjointness fails, the Lorentzian plane itself disappears.

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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

0 major / 5 minor

Summary. The paper shows that relativistic theories with purely relaxational spectra (kinetic theory, transient hydrodynamics, causal viscoelasticity) equip the real plane {iω, ik} with a Lorentzian structure. Timelike future-directed, past-directed, and spacelike vectors are identified with relaxation-like, unstable-like, and evanescent-like modes. Under the structural assumption that the linearized dynamics take the Boltzmann-like form ∂_t Ψ = −σ Ψ − E ∂_x Ψ with self-adjoint σ and E and ||E|| ≤ 1, causality forces the spectrum to expand at most at speed w and forces every isolated dispersion relation to be spacelike: |d(iω)/d(ik)| ≤ w (Theorems 1–2). From this geometry the paper derives universal bounds on diffusivity and acoustic diffusivity, maximal deviations from time dilation, observer dependence of spectral hierarchies, the radius of validity of hydrodynamics in boosted frames, and the necessity of ballistic cuts in RTA-type and Boltzmann kinetic theory. Theorem 8 extends the representation to any finite-order linear PDE that is causal and purely relaxational.

Significance. If the structural hypotheses hold, the work supplies a single geometric language that converts several longstanding questions in relativistic matter physics into elementary statements about causal trajectories on the relaxation plane. The bounds D ≤ w^{2} τ_g and D ≤ (w^{2} − c_s^{2}) τ_g are sharp (saturated by the explicit Cattaneo and four-dimensional models (23) and (27)), improve on existing hydrohedron estimates, and apply uniformly to kinetic theory and transient hydrodynamics. The lower bound R ≥ 1/(2w τ_g) on the hydrodynamic radius, the time-dilation window (12), and the no-go theorems for cuts (Theorems 5–7) are likewise parameter-free and falsifiable. The derivations rest on standard Kato perturbation theory and the proven Lax conjecture, with the spectral-correlator and causality equivalences supplied in the appendices. This is a genuine conceptual advance for the linear-response theory of relativistic media.

minor comments (5)
  1. The abstract and introduction list applications to viscosity, yet the shear-viscosity bound (32) appears only in Sec. V.D; a one-sentence forward pointer would help the reader locate it.
  2. Figure 4 (lower right) and Figure 9 would benefit from a short caption note stating the precise values of w and c_s used, so that the saturation of the bounds can be verified by eye.
  3. In Sec. IV.A the phrase “mild structural assumptions” is used; it would be clearer to list the three ingredients (self-adjointness of σ and E, boundedness of E, and reality of iω for real ik) explicitly at first occurrence.
  4. Appendix C assumes well-posedness in every inertial frame when proving that causality implies ||E|| ≤ 1. A brief remark that this is the standard relativistic requirement (rather than an extra hypothesis) would forestall possible confusion.
  5. A few typographical inconsistencies appear (e.g., “Milne and Rindler coordinates” versus later “Milne coordinates”; occasional missing spaces around ±). These are purely cosmetic.

Circularity Check

1 steps flagged · score 1.0 of 10

No significant circularity: geometric bounds follow from self-adjoint Boltzmann form plus Kato perturbation theory; self-citations supply independent background lemmas.

  1. self citation load bearing [Sec. II (covariant stability bound) and Appendix C]
    "This bound is particularly useful in relativity, because it is Lorentz invariant. Indeed, it lies at the heart of the modern stability-causality theorem [10] and of the first universal rigorous bounds on relativistic transport coefficients [11–13]. … Here, we prove that equation (6) is causal if and only if w ≡ ||E|| does not exceed 1."

    The paper invokes the author’s prior stability-causality theorem [10] and related transport bounds [11–13] as background for the geometric picture, and Appendix C re-derives the operator-norm characterization of causality. These citations are load-bearing for the claim that ||E|| ≤ 1 is equivalent to causality, yet they are independently published, parameter-free results that do not assume the new geometric bounds on D, au'/ au or R. The circularity is therefore only the mild self-citation of background lemmas, not a definitional loop.

full rationale

The central claims (Theorems 1–2, the D ≤ w^{2} au_g and D ≤ (w^{2} − c_s^{2}) au_g bounds, the time-dilation window, and the hydrodynamic radius R ≥ 1/(2w au_g)) are derived inside the paper from the structural assumptions of Sec. IV.A (self-adjoint au and E with ||E|| ≤ 1) by applying standard Kato perturbation theory to the family au_ik = au + ik E. The Lorentzian structure itself is read off from the Lorentz transformation law of the pair (iω, ik) and is not postulated. Explicit models (Cattaneo (23) and the four-dimensional sound model (27)) saturate the bounds, confirming sharpness without fitting. Self-citations to the author’s earlier causality papers supply the covariant stability bound and the operator-norm characterization of causality (Appendix C); those results are independently published, parameter-free, and used only as background lemmas, not as definitions of the quantities being bounded. Theorem 8 shows that any finite-order causal PDE with real coefficients and purely relaxational spectrum can be recast in the same form, so the geometric constraints are not an artifact of a particular kinetic representation. No step reduces a claimed prediction to a fitted input or to a self-citation that itself assumes the target result. The single minor self-citation load is therefore non-circular and scores 1.

Assumptions & free parameters 0 free parameters · 5 assumptions · 1 invented entities

The paper introduces no free parameters and no new dynamical entities. Its load-bearing content consists of three domain assumptions (Boltzmann form, self-adjointness, purely relaxational spectrum) plus standard results from operator theory and hyperbolic PDE theory. Once those are granted, every subsequent bound follows by elementary geometry and Kato perturbation theory.

assumptions (5)
  • domain assumption Linearized dynamics admit the first-order form ∂_t Ψ = −σ Ψ − E ∂_x Ψ with σ, E self-adjoint on a Hilbert space H and E bounded.
    Sec. IV.A; justified by Onsager reciprocity + PT symmetry and by the characteristic analysis of hyperbolic systems. Theorem 8 shows this form is equivalent (for spectral purposes) to any finite-order linear PDE with real coefficients and real spectrum.
  • domain assumption Causality is equivalent to the operator-norm bound ||E|| = w ≤ 1.
    Appendix C; proved by energy estimates for the finite-dimensional case and by a contradiction argument involving boosted initial-value problems for the infinite-dimensional case.
  • domain assumption The spectrum is purely relaxational: iω ∈ ℝ whenever ik ∈ ℝ.
    Follows at once from self-adjointness of σ + ik E; stated as the defining property of the class of theories under study.
  • standard math Standard results of Kato perturbation theory (Hausdorff continuity of spectra under bounded perturbations, analyticity of isolated eigenvalues of finite multiplicity, preservation of compact resolvent).
    Invoked throughout Sec. IV and Theorems 1–2, 5–7; citations to Kato’s monograph are given.
  • standard math The Lax conjecture (now theorem) on hyperbolic polynomials.
    Used in the proof of Theorem 8 to guarantee the existence of Hermitian matrices realizing a given hyperbolic polynomial.
invented entities (1)
  • Relaxation plane equipped with Lorentzian causal structure independent evidence
    purpose: To classify modes as relaxation-like / unstable-like / evanescent-like and to convert spectral questions into geometric cone arguments.
    The plane itself is just the set of real pairs (iω, ik); the Lorentzian metric is induced by the boost transformation law of the plane-wave factor. It is therefore a derived geometric object rather than an independent postulate, but it is the central conceptual invention of the paper.

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Cite this review

Pith. "Pith review of The Lorentzian geometry of relaxation." pith.science (2026). https://pith.science/paper/GHZBQCZW

@misc{pith2026260710148,
  author       = {Pith},
  title        = {Pith review of: The Lorentzian geometry of relaxation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GHZBQCZW}},
  note         = {Machine review of arXiv:2607.10148}
}
abstract

We show that relativistic theories with purely relaxational excitation spectra, such as kinetic theory and transient hydrodynamics, naturally endow the dispersion plane $\{i\omega,ik\}$ with a Lorentzian geometric structure analogous to that of the Minkowski plane $\{t,x\}$. In this picture, timelike future-directed, timelike past-directed, and spacelike directions correspond respectively to relaxation-like, unstable-like, and evanescent-like modes. Under mild structural assumptions on the underlying theory, causality constrains dispersion relations to follow spacelike trajectories on the plane. This geometric viewpoint recasts longstanding problems in relativistic matter physics as elementary geometric ones that can often be solved graphically. As applications, we derive universal constraints on dispersion relations, deviations from time dilation, the observer dependence of spectral hierarchies, the regime of validity of hydrodynamics in boosted frame, the maximal allowed diffusivity and viscosity of relativistic media, and the presence of non-hydrodynamic branch cuts in kinetic theory.

Figures

Figures reproduced from arXiv: 2607.10148 by the authors.

Figure 1
Figure 1. FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Dispersion relations of two causal and covariantly stable theories: Cattaneo diffusion, [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Representation of the relaxation-like sector in Milne coordinates (upper panels) and of the evanescent-like sectors in [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figures from the paper (9 more)
Figure 4
Figure 4. Figure 4: FIG. 4. Some simple illustrations of how Theorem 1 constrains the spectrum to propagate causally on the relaxation plane. [PITH_FULL_IMAGE:figures/full_fig_p007_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5. Using causality to constrain dispersion relations. [PITH_FULL_IMAGE:figures/full_fig_p008_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6. Deviations from exact time dilation induced by a finite propagation speed [PITH_FULL_IMAGE:figures/full_fig_p009_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7. An example of a model in which a rest-frame spectral hierarchy breaks down at large boosts. At [PITH_FULL_IMAGE:figures/full_fig_p010_7.png]
Figure 8
Figure 8. Figure 8: FIG. 8. The regime of validity of hydrodynamics. [PITH_FULL_IMAGE:figures/full_fig_p011_8.png]
Figure 9
Figure 9. Figure 9: FIG. 9. Comparison between the sound sector of the maximally viscous model (27) (blue) and Israel–Stewart theory with bulk [PITH_FULL_IMAGE:figures/full_fig_p014_9.png]
Figure 10
Figure 10. Figure 10: FIG. 10. Dispersion relations of the model (36), describing photons propagating on a plane and undergoing elastic scattering. [PITH_FULL_IMAGE:figures/full_fig_p017_10.png]
Figure 11
Figure 11. Figure 11: FIG. 11. Wake generated behind a pulse that perturbs a photon distribution governed by (36). [PITH_FULL_IMAGE:figures/full_fig_p018_11.png]
Figure 12
Figure 12. Figure 12: FIG. 12. Minkowski diagram illustrating that assuming causality, covariant well-posedness, and [PITH_FULL_IMAGE:figures/full_fig_p023_12.png]

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Forward citations

Cited by 4 Pith papers

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

  1. Relativistic transport near moving interfaces

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    Interface-localized linear solutions in relativistic media are superpositions of imaginary-frequency modes selected by intersecting the spectrum with the line iω = v ik.

  2. Near-Light-Cone Nonhydrodynamic Structure from Boosted Hydrodynamics

    hep-th 2026-07 accept novelty 7.0 of 10

    Large boosts select near-light-cone nonhydrodynamic singularities that set the hydrodynamic radius of convergence, with RTA giving a finite offset and holography a vanishing one after Lorentz rescaling.

  3. Lorentz-Covariant Spectral Bounds from Thermal Quantum Field Theory

    nucl-th 2026-07 reject novelty 6.0 of 10

    Boosting a hot quantum fluid spreads a rest-frame relaxation mode into a band of modes; the boosted gap is bounded by Γ_gap/[γ(1+v v_max)] and the gradient-expansion radius is squeezed by the sound speed.

  4. Lorentz-Covariant Spectral Bounds from Thermal Quantum Field Theory: Retarded Green's Functions, Kubo Relations, and Holographic Constraints

    nucl-th 2026-07 conditional novelty 5.0 of 10

    The paper derives Lorentz-covariant bounds on boosted-frame non-hydrodynamic relaxation rates from thermal QFT axioms, and shows in a holographic N=4 SYM plasma that relaxation speeds up under boost rather than slowing down.

Reference graph

Works this paper leans on

92 extracted references · 1 canonical work pages · cited by 4 Pith papers

  1. [1]

    Minkowski

    (numerically truncated atn max = 250) at the sample points (x−t)/τ=−0.0005 (blue),−0.001 (magenta),−0.01 (red), and−∞(dashed). Near the front, the photons propagate almost entirely toward positivex, and progressively isotropize farther behind it. At infinity, the distribution approaches a constant isotropic state with perturbed photon density (1, δI) = 1,...

  2. [2]

    Heinz and R

    U. Heinz and R. Snellings, Ann. Rev. Nucl. Part. Sci.63, 123 (2013), arXiv:1301.2826 [nucl-th]

    arXiv 2013

  3. [3]

    Romatschke and U

    P. Romatschke and U. Romatschke,Relativistic Fluid Dynamics In and Out of Equilibrium, Cambridge Monographs on Mathematical Physics (Cambridge University Press, 2019) arXiv:1712.05815 [nucl-th]. 23 FIG. 12. Minkowski diagram illustrating that assuming causality, covariant well-posedness, and||E||>1 leads to a contradiction. Indeed, well-posedness in boost...

    arXiv 2019

  4. [4]

    Florkowski, M

    W. Florkowski, M. P. Heller, and M. Spali´ nski, Reports on Progress in Physics81, 046001 (2018), arXiv:1707.02282 [hep-ph]

    arXiv 2018

  5. [5]

    Rezzolla and O

    L. Rezzolla and O. Zanotti,Relativistic Hydrodynamics(Oxford University Press, Oxford, 2013)

    2013

  6. [6]

    Shibata and K

    M. Shibata and K. Hotokezaka, Ann. Rev. Nucl. Part. Sci.69, 41 (2019), arXiv:1908.02350 [astro-ph.HE]

    arXiv 2019

  7. [7]

    P. B. Arnold, G. D. Moore, and L. G. Yaffe, JHEP11, 001 (2000), arXiv:hep-ph/0010177

    arXiv 2000

  8. [8]

    Hannestad, Ann

    S. Hannestad, Ann. Rev. Nucl. Part. Sci.56, 137 (2006), arXiv:hep-ph/0602058

    arXiv 2006

Show all 92 references
  1. [9]

    S. Pu, T. Koide, and D. H. Rischke, Phys. Rev. D81, 114039 (2010), arXiv:0907.3906 [hep-ph]

    2010 arXiv

  2. [10]

    Gavassino, M

    L. Gavassino, M. Antonelli, and B. Haskell, Phys. Rev. Lett.128, 010606 (2022), arXiv:2105.14621 [gr-qc]

    2022 arXiv

  3. [11]

    Gavassino, Phys

    L. Gavassino, Phys. Rev. X12, 041001 (2022), arXiv:2111.05254 [gr-qc]

    2022 arXiv

  4. [12]

    M. P. Heller, A. Serantes, M. Spali´ nski, and B. Withers, Phys. Rev. Lett.130, 261601 (2023), arXiv:2212.07434 [hep-th]

    2023 arXiv

  5. [13]

    Gavassino, Phys

    L. Gavassino, Phys. Lett. B840, 137854 (2023), arXiv:2301.06651 [hep-th]

    2023 arXiv

  6. [14]

    M. P. Heller, A. Serantes, M. Spali´ nski, and B. Withers, Nature Phys.20, 1948 (2024), arXiv:2305.07703 [hep-th]

    1948 arXiv

  7. [15]

    Brants, Phys

    R. Brants, Phys. Rev. D110, 116027 (2024), arXiv:2409.09022 [hep-th]

    2024 arXiv

  8. [16]

    L. Hui, A. Nicolis, A. Podo, and S. Zhou, JHEP07, 188 (2025), arXiv:2502.04215 [hep-th]

    2025 arXiv

  9. [17]

    Gavassino, Phys

    L. Gavassino, Phys. Rev. D111, 103011 (2025), arXiv:2502.08740 [nucl-th]

    2025 arXiv

  10. [18]

    R. E. Hoult, (2025), arXiv:2512.24819 [hep-th]

    2025 arXiv

  11. [19]

    Bhattacharyya, S

    S. Bhattacharyya, S. Mitra, S. Roy, and R. Singh, (2025), arXiv:2512.12450 [hep-th]

    2025 arXiv

  12. [20]

    Bajec and A

    M. Bajec and A. Soloviev, Phys. Rev. D112, 065002 (2025), arXiv:2506.15531 [hep-th]

    2025 arXiv

  13. [21]

    Gavassino, Phys

    L. Gavassino, Phys. Rev. Lett.137, 022301 (2026), arXiv:2601.03081 [gr-qc]

    2026 arXiv

  14. [22]

    Brants, (2026), arXiv:2605.21377 [hep-th]

    R. Brants, (2026), arXiv:2605.21377 [hep-th]

    2026 arXiv

  15. [23]

    Sommerfeld, Annalen der Physik349, 177 (1914)

    A. Sommerfeld, Annalen der Physik349, 177 (1914)

    1914

  16. [24]

    Brillouin, Annalen der Physik349, 203 (1914)

    L. Brillouin, Annalen der Physik349, 203 (1914)

    1914

  17. [25]

    Brillouin,Wave Propagation and Group Velocity(Academic Press, New York, 1960)

    L. Brillouin,Wave Propagation and Group Velocity(Academic Press, New York, 1960)

    1960

  18. [26]

    S. A. Bludman and M. A. Ruderman, Phys. Rev.170, 1176 (1968)

    1968

  19. [27]

    R. Fox, C. G. Kuper, and S. G. Lipson, Nature223, 597 (1969)

    1969

  20. [28]

    Aharonov, A

    Y. Aharonov, A. Komar, and L. Susskind, Phys. Rev.182, 1400 (1969)

    1969

  21. [29]

    R. Fox, C. G. Kuper, and S. G. Lipson, Proceedings of the Royal Society of London A316, 515 (1970)

    1970

  22. [30]

    Krotscheck and W

    E. Krotscheck and W. Kundt, Communications in Mathematical Physics60, 171 (1978)

    1978

  23. [31]

    Hiscock and L

    W. Hiscock and L. Lindblom, Physical review D: Particles and fields31, 725 (1985)

    1985

  24. [32]

    Kost¨adt and M

    P. Kost¨adt and M. Liu, Phys. Rev. D62, 023003 (2000), arXiv:cond-mat/0010276 [cond-mat.stat-mech]

    2000 arXiv

  25. [33]

    Adams, N

    A. Adams, N. Arkani-Hamed, S. Dubovsky, A. Nicolis, and R. Rattazzi, JHEP10, 014 (2006), arXiv:hep-th/0602178

    2006 arXiv

  26. [34]

    Gavassino, M

    L. Gavassino, M. M. Disconzi, and J. Noronha, Phys. Rev. Lett.132, 162301 (2024), arXiv:2307.05987 [hep-th]

    2024 arXiv

  27. [35]

    Gavassino, Phys

    L. Gavassino, Phys. Rev. Lett.137, 022302 (2026), arXiv:2601.19464 [gr-qc]

    2026 arXiv

  28. [36]

    Gavassino, (2026), arXiv:2604.07031 [nucl-th]

    L. Gavassino, (2026), arXiv:2604.07031 [nucl-th]

    2026 arXiv

  29. [37]

    N. N. Bogolyubov, A. A. Logunov, A. I. Oksak, and I. T. Todorov,General principles of quantum field theory(1990)

    1990

  30. [38]

    Lowdon, Phys

    P. Lowdon, Phys. Rev. D96, 065013 (2017)

    2017

  31. [39]

    Lowdon, Nucl

    P. Lowdon, Nucl. Phys. B935, 242 (2018), arXiv:1711.07569 [hep-th]

    2018 arXiv

  32. [40]

    Buchel, R

    A. Buchel, R. E. Hoult, and P. Kovtun, (2026), arXiv:2606.19049 [hep-th]

    2026 arXiv

  33. [41]

    Israel and J

    W. Israel and J. Stewart, Annals of Physics118, 341 (1979). 24

    1979

  34. [42]

    D. Jou, J. Casas-V´ azquez, and G. Lebon, Reports on Progress in Physics51, 1105 (1999)

    1999

  35. [43]

    Muller and T

    I. Muller and T. Ruggeri,Rational Extended Thermodynamics, 2nd ed. (Springer-Verlag New York, 1998)

    1998

  36. [44]

    Rauch,Partial Differential Equations, Graduate Texts in Mathematics (Springer New York, 2012)

    J. Rauch,Partial Differential Equations, Graduate Texts in Mathematics (Springer New York, 2012)

    2012

  37. [45]

    Christodoulou,The Formation of Shocks in 3-Dimensional Fluids, EMS Monographs in Mathematics, Vol

    D. Christodoulou,The Formation of Shocks in 3-Dimensional Fluids, EMS Monographs in Mathematics, Vol. 2 (European Mathematical Society Publishing House, Z ¨urich, Switzerland, 2007)

    2007

  38. [46]

    M. M. Disconzi, C. Luo, G. Mazzone, and J. Speck, Selecta Mathematica28(2022), 10.1007/s00029-021-00733-3

    2022 doi

  39. [47]

    S. W. Hawking and G. F. R. Ellis,The Large Scale Structure of Space-Time, Cambridge Monographs on Mathematical Physics (Cambridge University Press, 2011)

    2011

  40. [48]

    Babichev, V

    E. Babichev, V. Mukhanov, and A. Vikman, JHEP02, 101 (2008), arXiv:0708.0561 [hep-th]

    2008 arXiv

  41. [49]

    M. M. Disconzi, Living Rev. Rel.27, 6 (2024), arXiv:2308.09844 [math.AP]

    2024 arXiv

  42. [50]

    Dudynski and M

    M. Dudynski and M. L. Ekiel-Jez`ewska, Phys. Rev. Lett.55, 2831 (1985)

    1985

  43. [51]

    F. S. Bemfica, M. M. Disconzi, and J. Noronha, Phys. Rev. D100, 104020 (2019)

    2019

  44. [52]

    F. S. Bemfica, M. M. Disconzi, and J. Noronha, Phys. Rev. X12, 021044 (2022)

    2022

  45. [53]

    Kovtun, Journal of High Energy Physics2019, 34 (2019), arXiv:1907.08191 [hep-th]

    P. Kovtun, Journal of High Energy Physics2019, 34 (2019), arXiv:1907.08191 [hep-th]

    2019 arXiv

  46. [54]

    Gavassino, A

    L. Gavassino, A. D. Kov´ acs, and H. S. Reall, Phys. Rev. D113, 124022 (2026)

    2026

  47. [55]

    C. Cattaneo,Sur une forme de l’´ equation de la chaleur ´ eliminant le paradoxe d’une propagation instantan´ ee, Comptes rendus hebdomadaires des s´ eances de l’Acad´ emie des sciences (Gauthier-Villars, 1958)

    1958

  48. [56]

    Novak, J

    I. Novak, J. Sonner, and B. Withers, Phys. Rev. D98, 086023 (2018), arXiv:1806.08655 [hep-th]

    2018 arXiv

  49. [57]

    D. J. Korteweg and G. de Vries, The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science 5, 39, 422 (1895)

  50. [58]

    772 (Springer, New York, 2010)

    Øyvind Grøn,Lecture Notes on the General Theory of Relativity: From Newton’s Attractive Gravity to the Repulsive Gravity of Vacuum Energy, Lecture Notes in Physics, Vol. 772 (Springer, New York, 2010)

    2010

  51. [59]

    Gavassino, M

    L. Gavassino, M. Antonelli, and B. Haskell, Phys. Rev. D106, 056010 (2022), arXiv:2207.14778 [gr-qc]

    2022 arXiv

  52. [60]

    Gavassino, Phys

    L. Gavassino, Phys. Rev. D107, 065013 (2023), arXiv:2210.05067 [nucl-th]

    2023 arXiv

  53. [61]

    Gavassino, M

    L. Gavassino, M. M. Disconzi, and J. Noronha, Phys. Rev. Lett.132, 222302 (2024), arXiv:2302.03478 [nucl-th]

    2024 arXiv

  54. [62]

    Soares Rocha, L

    G. Soares Rocha, L. Gavassino, and N. Mullins, Phys. Rev. D110, 016020 (2024), arXiv:2405.10878 [nucl-th]

    2024 arXiv

  55. [63]

    Gavassino, Phys

    L. Gavassino, Phys. Rev. D110, 094012 (2024), arXiv:2408.14316 [nucl-th]

    2024 arXiv

  56. [64]

    Dudy´ nski and M

    M. Dudy´ nski and M. L. Ekiel-Jezewska, Communications in Mathematical Physics102, 17 (1985)

    1985

  57. [65]

    Kato,Perturbation Theory for Linear Operators, 2nd ed., Grundlehren der mathematischen Wissenschaften, Vol

    T. Kato,Perturbation Theory for Linear Operators, 2nd ed., Grundlehren der mathematischen Wissenschaften, Vol. 132 (Springer-Verlag, Berlin, Heidelberg, New York, 1980) corrected printing of the second edition

    1980

  58. [66]

    Kato, Progress of Theoretical Physics4, 514 (1949)

    T. Kato, Progress of Theoretical Physics4, 514 (1949)

    1949

  59. [67]

    Gavassino, Phys

    L. Gavassino, Phys. Rev. D114, 014018 (2026), arXiv:2601.19474 [nucl-th]

    2026 arXiv

  60. [68]

    Bajec, S

    M. Bajec, S. Grozdanov, and A. Soloviev, JHEP08, 065 (2024), arXiv:2403.17769 [hep-th]

    2024 arXiv

  61. [69]

    E. A. Spiegel, ApJ126, 202 (1957)

    1957

  62. [70]

    Dudy´ nski, Journal of Statistical Physics57, 199 (1989)

    M. Dudy´ nski, Journal of Statistical Physics57, 199 (1989)

    1989

  63. [71]

    Hartman, S

    T. Hartman, S. A. Hartnoll, and R. Mahajan, Phys. Rev. Lett.119, 141601 (2017), arXiv:1706.00019 [hep-th]

    2017 arXiv

  64. [72]

    Kato, Progress of Theoretical Physics5, 207 (1950)

    T. Kato, Progress of Theoretical Physics5, 207 (1950)

    1950

  65. [73]

    Hippert, J

    M. Hippert, J. Noronha, and P. Romatschke, Phys. Lett. B860, 139184 (2025), arXiv:2402.14085 [nucl-th]

    2025 arXiv

  66. [74]

    Ghiglieri, G

    J. Ghiglieri, G. D. Moore, and D. Teaney, Phys. Rev. Lett.121, 052302 (2018), arXiv:1805.02663 [hep-ph]

    2018 arXiv

  67. [75]

    Kovtun, D

    P. Kovtun, D. T. Son, and A. O. Starinets, Phys. Rev. Lett.94, 111601 (2005), arXiv:hep-th/0405231

    2005 arXiv

  68. [76]

    P. K. Kovtun and A. O. Starinets, Phys. Rev. D72, 086009 (2005)

    2005

  69. [77]

    M. P. Heller, R. A. Janik, M. Spali´ nski, and P. Witaszczyk, Phys. Rev. Lett.113, 261601 (2014)

    2014

  70. [78]

    G. D. Moore, JHEP05, 084 (2018), arXiv:1803.00736 [hep-ph]

    2018 arXiv

  71. [79]

    Kurkela and U

    A. Kurkela and U. A. Wiedemann, Eur. Phys. J. C79, 776 (2019), arXiv:1712.04376 [hep-ph]

    2019 arXiv

  72. [80]

    G. S. Rocha, I. Danhoni, K. Ingles, G. S. Denicol, and J. Noronha, Phys. Rev. D110, 076003 (2024), arXiv:2404.04679 [nucl-th]

    2024 arXiv

  73. [81]

    Gavassino, Phys

    L. Gavassino, Phys. Rev. Res.6, L042043 (2024), arXiv:2404.12327 [nucl-th]

    2024 arXiv

  74. [82]

    Romatschke, Eur

    P. Romatschke, Eur. Phys. J. C76, 352 (2016), arXiv:1512.02641 [hep-th]

    2016 arXiv

  75. [83]

    J. B. Conway,A Course in Functional Analysis, 2nd ed., Graduate Texts in Mathematics, Vol. 96 (Springer-Verlag, New York, 1990)

    1990

  76. [84]

    Struchtrup, Continuum Mechanics and Thermodynamics17, 43 (2005)

    H. Struchtrup, Continuum Mechanics and Thermodynamics17, 43 (2005)

    2005

  77. [85]

    Struchtrup and P

    H. Struchtrup and P. Taheri, IMA Journal of Applied Mathematics76, 672 (2011)

    2011

  78. [86]

    A. S. Lewis, P. A. Parrilo, and M. V. Ramana, arXiv Mathematics e-prints , math/0304104 (2003), arXiv:math/0304104 [math.OC]

    2003 arXiv

  79. [87]

    Courant and D

    R. Courant and D. Hilbert,Methods of Mathematical Physics, Vol 2: Partial Differential Equations(John Wiley and Sons, New York, NY, 1989)

    1989

  80. [88]

    Gavassino, M

    L. Gavassino, M. M. Disconzi, and J. Noronha, Phys. Rev. D109, 096041 (2024), arXiv:2302.05332 [nucl-th]

    2024 arXiv

  81. [89]

    Mullins, M

    N. Mullins, M. Hippert, and J. Noronha, Phys. Rev. D108, 076013 (2023), arXiv:2306.08635 [nucl-th]

    2023

  82. [90]

    Gavassino, N

    L. Gavassino, N. Mullins, and M. Hippert, Phys. Rev. D109, 125002 (2024), arXiv:2402.06776 [nucl-th]

    2024 arXiv

  83. [91]

    W. A. Hiscock and L. Lindblom, Annals of Physics151, 466 (1983)

    1983

  84. [92]

    Teschl,Mathematical Methods in Quantum Mechanics With Applications to Schr ¨odinger Operators”, Graduate Studies in Mathematics, Vol

    G. Teschl,Mathematical Methods in Quantum Mechanics With Applications to Schr ¨odinger Operators”, Graduate Studies in Mathematics, Vol. 99 (American Mathematical Society, Providence, RI, 2009)

    2009

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