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Maximally symmetric boost-invariant solutions of the Boltzmann equation in foliated geometries

Mauricio Martinez

Christopher Plumberg

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Phys. Rev. D 113, 105001 – Published 4 May, 2026

DOI: https://doi.org/10.1103/rdz6-2hbx

Abstract

In this work we study the relativistic kinetic theory of a boost-invariant conformal gas on a static, maximally symmetric background dS3×R, considering all constant-curvature slicings of dS3—flat, spherical, or hyperbolic—and their associated symmetry groups. Using a symmetry-driven cotangent bundle approach, we show that the isometry group of each slicing acts on phase space in such a way that only its Casimir invariants and the timelike coordinate are unconstrained, so the distribution function depends solely on these quantities. This yields a unified boost-invariant exact solution of the Boltzmann equation valid for each constant–curvature foliation of dS3×R. Specializing this general solution to the flat and spherical foliations reproduces the Bjorken and Gubser flows, respectively, while its restriction to the hyperbolic foliation produces a genuinely new analytic solution (“Grozdanov flow”). Hydrodynamics and free streaming emerge naturally as limiting regimes of this novel exact solution. We further comment on several relevant aspects of the new boost-invariant solution on the hyperbolic slicing and on their interpretation once mapped back to Minkowski space.

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References (63)

  1. M. Krook and T. T. Wu, Formation of Maxwellian tails, Phys. Rev. Lett. 36, 1107 (1976).
  2. M. Krook and T. T. Wu, Exact solutions of the Boltzmann equation, Phys. Fluids 20, 1589 (1977).
  3. A. V. Bobylev, Fourier transform method in the theory of the Boltzmann equation for Maxwellian molecules, Sov. Phys. Dokl. 20, 820 (1976).
  4. G. S. Denicol, U. W. Heinz, M. Martinez, J. Noronha, and M. Strickland, New exact solution of the relativistic Boltzmann equation and its hydrodynamic limit, Phys. Rev. Lett. 113, 202301 (2014).
  5. G. S. Denicol, U. W. Heinz, M. Martinez, J. Noronha, and M. Strickland, Studying the validity of relativistic hydrodynamics with a new exact solution of the Boltzmann equation, Phys. Rev. D 90, 125026 (2014).
  6. D. Bazow, G. S. Denicol, U. Heinz, M. Martinez, and J. Noronha, Analytic solution of the Boltzmann equation in an expanding system, Phys. Rev. Lett. 116, 022301 (2016).
  7. D. Bazow, G. S. Denicol, U. Heinz, M. Martinez, and J. Noronha, Nonlinear dynamics from the relativistic Boltzmann equation in the Friedmann-Lemaître-Robertson-Walker spacetime, Phys. Rev. D 94, 125006 (2016).
  8. Y. Wang, X. Zhao, Z. Xu, and J. Hu, Analytical solution and lie algebra of the relativistic Boltzmann equation, arXiv:2511.08652.
  9. Y. Hatta, M. Martinez, and B.-W. Xiao, Analytic solutions of the relativistic Boltzmann equation, Phys. Rev. D 91, 085024 (2015).
  10. J. Noronha and G. S. Denicol, Perfect fluidity of a dissipative system: Analytical solution for the Boltzmann equation in AdS2⊗S2, Phys. Rev. D 92, 114032 (2015).
  11. J. Hu, Analytical solution of the nonlinear relativistic Boltzmann equation, J. High Energy Phys. 07 (2025) 066.
  12. S. Chen and S. Shi, Exact solution of Boltzmann equation in a longitudinal expanding system, Phys. Rev. C 109, L051901 (2024).
  13. G. Baym, Thermal equilibration in ultrarelativistic heavy ion collisions, Phys. Lett. 138B, 18 (1984).
  14. J. Tindall, J. M. Torres-Rincon, J. B. Rose, and H. Petersen (SMASH Collaboration), Equilibration and freeze-out of an expanding gas in a transport approach in a Friedmann–Robertson–Walker metric, Phys. Lett. B 770, 532 (2017).
  15. S. S. Gubser and A. Yarom, Conformal hydrodynamics in Minkowski and de Sitter spacetimes, Nucl. Phys. B846, 469 (2011).
  16. J. D. Bjorken, Highly relativistic nucleus-nucleus collisions: The central rapidity region, Phys. Rev. D 27, 140 (1983).
  17. S. S. Gubser, Symmetry constraints on generalizations of Bjorken flow, Phys. Rev. D 82, 085027 (2010).
  18. S. Grozdanov, A family of three maximally symmetric boost-invariant flows in relativistic hydrodynamics, arXiv:2510.10769.
  19. P. Rioseco and O. Sarbach, Accretion of a relativistic, collisionless kinetic gas into a Schwarzschild black hole, Classical Quantum Gravity 34, 095007 (2017).
  20. P. Rioseco, Relativistic Kinetic Theory with Applications in Astrophysics, Ph.D. thesis, Universidad Michoacana de San Nicolás de Hidalgo, 2019.
  21. R. O. Acuña-Cárdenas, C. Gabarrete, and O. Sarbach, An introduction to the relativistic kinetic theory on curved spacetimes, Gen. Relativ. Gravit. 54, 23 (2022).
  22. O. Sarbach and T. Zannias, Relativistic kinetic theory: An introduction, AIP Conf. Proc. 1548, 134 (2013).
  23. O. Sarbach and T. Zannias, Tangent bundle formulation of a charged gas, AIP Conf. Proc. 1577, 192 (2015).
  24. O. Sarbach and T. Zannias, The geometry of the tangent bundle and the relativistic kinetic theory of gases, Classical Quantum Gravity 31, 085013 (2014).
  25. S. R. de Groot, W. A. van Leewen, and C. G. van Weert, Relativistic Kinetic Theory: Principles and Applications (Elsevier, North-Holland, 1980).
  26. C. Cercignani and G. Medeiros Kremer, The Relativistic Boltzmann Equation: Theory and Applications (Birkhäuser Verlag, Basel, Switzerland, 2002).
  27. F. Debbasch and W. van Leeuwen, General relativistic Boltzmann equation, I: Covariant treatment, Physica (Amsterdam) 388A, 1079 (2009).
  28. F. Debbasch and W. van Leeuwen, General relativistic Boltzmann equation, II: Manifestly covariant treatment, Physica (Amsterdam) 388A, 1818 (2009).
  29. P. L. Bhatnagar, E. P. Gross, and M. Krook, A model for collision processes in gases. I. Small amplitude processes in charged and neutral one-component systems, Phys. Rev. 94, 511 (1954).
  30. J. Anderson and H. Witting, A relativistic relaxation-time model for the Boltzmann equation, Physica (Utrecht) 74, 466 (1974).
  31. G. S. Denicol, J. Noronha, H. Niemi, and D. H. Rischke, Origin of the relaxation time in dissipative fluid dynamics, Phys. Rev. D 83, 074019 (2011).
  32. W. Florkowski, R. Ryblewski, and M. Strickland, Testing viscous and anisotropic hydrodynamics in an exactly solvable case, Phys. Rev. C 88, 024903 (2013).
  33. W. Florkowski, R. Ryblewski, and M. Strickland, Anisotropic hydrodynamics for rapidly expanding systems, Nucl. Phys. A916, 249 (2013).
  34. J. Berndt, J. C. Díaz-Ramos, and M. J. Vanaei, Cohomogeneity one actions on minkowski spaces, Monatshefte fur Mathematik 184, 185 (2017).
  35. J. Diaz-Ramos, S. Kashani, and M. Vanaei, Cohomogeneity one actions on anti de sitter spacetimes, Results Math. 72, 515 (2017).
  36. J. E. Marsden and A. Weinstein, Reduction of symplectic manifolds with symmetry, Rep. Math. Phys. 5, 121 (1974).
  37. J. E. Marsden and T. S. Ratiu, Introduction to Mechanics and Symmetry (Springer, New York, 1999).
  38. R. Abraham and J. E. Marsden, Foundations of Mechanics (Benjamin/Cummings, New York, 1978).
  39. J.-P. Ortega and T. S. Ratiu, Momentum Maps and Hamiltonian Reduction (Birkhäuser, Boston, MA, 2004).
  40. K. R. Meyer, Symmetries and integrals in mechanics, Dyn. Stab. Syst. 5, 259 (1973).
  41. R. Cushman and L. Bates, Global Aspects of Classical Integrable Systems (Springer, Basel, 2015).
  42. E. Beltrametti and A. Blasi, On the number of Casimir operators associated with any Lie group, Phys. Lett. 20, 62 (1966).
  43. J. E. Humphreys, Introduction to Lie algebras and representation theory, Vol. 9 (Springer Science & Business Media, New York, 2012).
  44. M. Martinez, M. McNelis, and U. Heinz, Anisotropic fluid dynamics for Gubser flow, Phys. Rev. C 95, 054907 (2017).
  45. M. Martinez and C. Plumberg (to be published).
  46. B. Banerjee, R. S. Bhalerao, and V. Ravishankar, Equilibration of the quark-gluon plasma produced in relativistic heavy ion collisions, Phys. Lett. B 224, 16 (1989).
  47. I. Muller, Zum Paradoxon der Warmeleitungstheorie, Z. Phys. 198, 329 (1967).
  48. W. Israel and J. M. Stewart, Transient relativistic thermodynamics and kinetic theory, Ann. Phys. (N.Y.) 118, 341 (1979).
  49. H. Marrochio, J. Noronha, G. S. Denicol, M. Luzum, S. Jeon, and C. Gale, Solutions of conformal Israel-Stewart relativistic viscous fluid dynamics, Phys. Rev. C 91, 014903 (2015).
  50. A. Soloviev, Close encounters with attractors of the third kind, Phys. Rev. D 113, 074006 (2026).
  51. J. F. Carinena and A. Ramos, Integrability of Riccati equation from a group theoretical viewpoint, Int. J. Mod. Phys. A 14, 1935 (1999).
  52. V. V. Kisil, Geometry of Möbius Transformations: Elliptic, Parabolic and Hyperbolic Actions of SL2(R) (World Scientific, Singapore, 2012).
  53. F. S. Bemfica, M. M. Disconzi, and J. Noronha, Causality and existence of solutions of relativistic viscous fluid dynamics with gravity, Phys. Rev. D 98, 104064 (2018).
  54. R. Baier, P. Romatschke, D. T. Son, A. O. Starinets, and M. A. Stephanov, Relativistic viscous hydrodynamics, conformal invariance, and holography, J. High Energy Phys. 04 (2008) 100.
  55. U. W. Heinz and M. Martinez, Investigating the domain of validity of the Gubser solution to the Boltzmann equation, Nucl. Phys. A943, 26 (2015).
  56. A. Behtash, C. N. Cruz-Camacho, and M. Martinez, Far-from-equilibrium attractors and nonlinear dynamical systems approach to the Gubser flow, Phys. Rev. D 97, 044041 (2018).
  57. A. Behtash, S. Kamata, M. Martinez, and H. Shi, Global flow structure and exact formal transseries of the Gubser flow in kinetic theory, J. High Energy Phys. 07 (2019) 226.
  58. A. Behtash, S. Kamata, M. Martinez, and H. Shi, Dynamical systems and nonlinear transient rheology of the far-from-equilibrium Bjorken flow, Phys. Rev. D 99, 116012 (2019).
  59. A. Behtash, C. N. Cruz-Camacho, S. Kamata, and M. Martinez, Non-perturbative rheological behavior of a far-from-equilibrium expanding plasma, Phys. Lett. B 797, 134914 (2019).
  60. A. Behtash, S. Kamata, M. Martinez, T. Schäfer, and V. Skokov, Transasymptotics and hydrodynamization of the Fokker-Planck equation for gluons, Phys. Rev. D 103, 056010 (2021).
  61. S. Kamata, J. Jankowski, and M. Martinez, Novel features of attractors and transseries in nonconformal Bjorken flows, Phys. Rev. D 107, 116004 (2023).
  62. M. P. Do Carmo, Riemannian Geometry (Birkhäuser, Boston, 1992).
  63. J. Lee, Introduction to Riemannian Manifolds (Springer, New York, 2018).

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Phys. Rev. D 113, 105001– Published 4 May, 2026
  • Received 12 March 2026
  • Accepted 7 April 2026
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Funded by SCOAP3.

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