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  • Open Access

Carroll hydrodynamics with spin

Ashish Shukla1,2,*, Rajeev Singh3,†, and Pushkar Soni4,‡

  • *Contact author: ashish.shukla@saha.ac.in
  • †Contact author: rajeev.singh@e-uvt.ro
  • ‡Contact author: pushkars21@iitk.ac.in
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Phys. Rev. D 114, 065001 – Published 2 September, 2026

DOI: https://doi.org/10.1103/mp5n-98rc

Abstract

We formulate Carroll hydrodynamics with the inclusion of a spin current. Our strategy relies on the fact that the c→0 limit of relativistic hydrodynamics yields the equations of Carroll hydrodynamics. Starting with the preultralocal parametrization of the background geometry and the hydrodynamic degrees of freedom for a relativistic fluid endowed with a spin current, the c→0 limit produces Carroll hydrodynamics with spin. It is known that boost-invariant hydrodynamic models for ultrarelativistic fluids relevant for the physics of quark-gluon plasma, such as Bjorken and Gubser flow, are manifestations of Carroll hydrodynamics under appropriate geometric choices for the underlying Carrollian structure. In this work, we further this mapping between such boost-invariant models and Carroll hydrodynamics, now with the inclusion of a spin current.

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References (88)

  1. L. Adamczyk et al. (STAR Collaboration), Global Λ hyperon polarization in nuclear collisions: evidence for the most vortical fluid, Nature (London) 548, 62 (2017).
  2. J. Adam et al. (STAR Collaboration), Global polarization of Λ hyperons in Au+Au collisions at sNN=200  GeV, Phys. Rev. C 98, 014910 (2018).
  3. C. Gale, S. Jeon, and B. Schenke, Hydrodynamic modeling of heavy-ion collisions, Int. J. Mod. Phys. A 28, 1340011 (2013).
  4. J. E. Parkkila, A. Onnerstad, and D. J. Kim, Bayesian estimation of the specific shear and bulk viscosity of the quark-gluon plasma with additional flow harmonic observables, Phys. Rev. C 104, 054904 (2021).
  5. P. Kovtun, D. T. Son, and A. O. Starinets, Viscosity in strongly interacting quantum field theories from black hole physics, Phys. Rev. Lett. 94, 111601 (2005).
  6. F. Becattini, W. Florkowski, and E. Speranza, Spin tensor and its role in nonequilibrium thermodynamics, Phys. Lett. B 789, 419 (2019).
  7. F. Becattini, Does the spin tensor play any role in non-gravitational physics?, Nucl. Phys. A1005, 121833 (2021).
  8. N. Weickgenannt, E. Speranza, X.-l. Sheng, Q. Wang, and D. H. Rischke, Derivation of the nonlocal collision term in the relativistic Boltzmann equation for massive spin-1/2 particles from quantum field theory, Phys. Rev. D 104, 016022 (2021).
  9. B. Betz, M. Gyulassy, and G. Torrieri, Polarization probes of vorticity in heavy ion collisions, Phys. Rev. C 76, 044901 (2007).
  10. F. Becattini, F. Piccinini, and J. Rizzo, Angular momentum conservation in heavy ion collisions at very high energy, Phys. Rev. C 77, 024906 (2008).
  11. F. Becattini, V. Chandra, L. Del Zanna, and E. Grossi, Relativistic distribution function for particles with spin at local thermodynamical equilibrium, Ann. Phys. (Amsterdam) 338, 32 (2013).
  12. W. Florkowski, B. Friman, A. Jaiswal, and E. Speranza, Relativistic fluid dynamics with spin, Phys. Rev. C 97, 041901 (2018).
  13. W. Florkowski, B. Friman, A. Jaiswal, R. Ryblewski, and E. Speranza, Spin-dependent distribution functions for relativistic hydrodynamics of spin-1/2 particles, Phys. Rev. D 97, 116017 (2018).
  14. W. Florkowski, A. Kumar, and R. Ryblewski, Thermodynamic versus kinetic approach to polarization-vorticity coupling, Phys. Rev. C 98, 044906 (2018).
  15. S. Bhadury, W. Florkowski, A. Jaiswal, A. Kumar, and R. Ryblewski, Relativistic dissipative spin dynamics in the relaxation time approximation, Phys. Lett. B 814, 136096 (2021).
  16. S. Bhadury, W. Florkowski, A. Jaiswal, A. Kumar, and R. Ryblewski, Dissipative spin dynamics in relativistic matter, Phys. Rev. D 103, 014030 (2021).
  17. A. Das, W. Florkowski, A. Kumar, R. Ryblewski, and R. Singh, Semi-classical kinetic theory for massive spin-half fermions with leading-order spin effects, Acta Phys. Pol. B 54, 8 (2023).
  18. N. Weickgenannt, D. Wagner, E. Speranza, and D. H. Rischke, Relativistic dissipative spin hydrodynamics from kinetic theory with a nonlocal collision term, Phys. Rev. D 106, L091901 (2022).
  19. W. Florkowski and M. Hontarenko, Generalized thermodynamic relations for perfect spin hydrodynamics, Phys. Rev. Lett. 134, 082302 (2025).
  20. Z. Drogosz, W. Florkowski, and M. Hontarenko, Hybrid approach to perfect and dissipative spin hydrodynamics, Phys. Rev. D 110, 096018 (2024).
  21. S. Bhadury, Z. Drogosz, W. Florkowski, S. K. Kar, and V. Mykhaylova, Local equilibrium Wigner function for spin-1/2 particles, arXiv:2505.02657.
  22. K. Hattori, M. Hongo, X.-G. Huang, M. Matsuo, and H. Taya, Fate of spin polarization in a relativistic fluid: An entropy-current analysis, Phys. Lett. B 795, 100 (2019).
  23. R. Biswas, A. Daher, A. Das, W. Florkowski, and R. Ryblewski, Relativistic second-order spin hydrodynamics: An entropy-current analysis, Phys. Rev. D 108, 014024 (2023).
  24. F. Becattini, A. Daher, and X.-L. Sheng, Entropy current and entropy production in relativistic spin hydrodynamics, Phys. Lett. B 850, 138533 (2024).
  25. A. Daher, X.-L. Sheng, D. Wagner, and F. Becattini, Dissipative currents and transport coefficients in relativistic spin hydrodynamics, Phys. Rev. D 112, 094020 (2025).
  26. F. Becattini and R. Singh, On the local thermodynamic relations in relativistic spin hydrodynamics, Eur. Phys. J. C 85, 1338 (2025).
  27. S. Fang, K. Fukushima, S. Pu, and D.-L. Wang, Relativistic spin hydrodynamics with antisymmetric spin tensors and an extension of the Bargmann-Michel-Telegdi equation, Phys. Rev. Lett. 136, 182301 (2026).
  28. A. D. Gallegos and U. Gürsoy, Holographic spin liquids and Lovelock Chern-Simons gravity, J. High Energy Phys. 11 (2020) 151.
  29. J. M. Maldacena, The large N limit of superconformal field theories and supergravity, Adv. Theor. Math. Phys. 2, 231 (1998).
  30. N. Abboud, L. Gavassino, R. Singh, and E. Speranza, Perfect spinfluid: A divergence-type approach, Phys. Rev. D 112, 094043 (2025).
  31. A. Chiarini, J. Sammet, and M. Shokri, Semiclassical spin hydrodynamics in flat and curved spacetime: Covariance, linear waves, and Bjorken background, Phys. Rev. D 113, 036009 (2026).
  32. A. D. Gallegos, U. Gürsoy, and A. Yarom, Hydrodynamics of spin currents, SciPost Phys. 11, 041 (2021).
  33. M. Hongo, X.-G. Huang, M. Kaminski, M. Stephanov, and H.-U. Yee, Relativistic spin hydrodynamics with torsion and linear response theory for spin relaxation, J. High Energy Phys. 11 (2021) 150.
  34. A. D. Gallegos, U. Gursoy, and A. Yarom, Hydrodynamics, spin currents and torsion, J. High Energy Phys. 05 (2022) 139.
  35. J. D. Bjorken, Highly relativistic nucleus-nucleus collisions: The central rapidity region, Phys. Rev. D 27, 140 (1983).
  36. S. S. Gubser, Symmetry constraints on generalizations of Bjorken flow, Phys. Rev. D 82, 085027 (2010).
  37. S. S. Gubser and A. Yarom, Conformal hydrodynamics in Minkowski and de Sitter spacetimes, Nucl. Phys. B846, 469 (2011).
  38. W. Florkowski, A. Kumar, R. Ryblewski, and R. Singh, Spin polarization evolution in a boost invariant hydrodynamical background, Phys. Rev. C 99, 044910 (2019).
  39. R. Singh, G. Sophys, and R. Ryblewski, Spin polarization dynamics in the Gubser-expanding background, Phys. Rev. D 103, 074024 (2021).
  40. D.-L. Wang, X.-Q. Xie, S. Fang, and S. Pu, Analytic solutions of relativistic dissipative spin hydrodynamics with radial expansion in Gubser flow, Phys. Rev. D 105, 114050 (2022).
  41. A. Bagchi, K. S. Kolekar, and A. Shukla, Carrollian origins of Bjorken flow, Phys. Rev. Lett. 130, 241601 (2023).
  42. A. Bagchi, K. S. Kolekar, T. Mandal, and A. Shukla, Heavy-ion collisions, Gubser flow, and Carroll hydrodynamics, Phys. Rev. D 109, 056004 (2024).
  43. J.-M. Lévy-Leblond, Une nouvelle limite non-relativiste du groupe de poincaré, Ann. l’Inst. Henri Poincaré A Phys. Théor. 3, 1 (1965).
  44. N. D. Sen Gupta, On an analogue of the Galilei group, Nuovo Cimento A 44, 512 (1966).
  45. A. Bagchi, A. Banerjee, P. Dhivakar, S. Mondal, and A. Shukla, The Carrollian kaleidoscope, Eur. Phys. J. C 86, 429 (2026).
  46. K. Nguyen, Lectures on Carrollian holography, Eur. Phys. J. C 86, 733 (2026).
  47. L. Ciambelli and P. Jai-akson, Foundations of Carrollian geometry, Phys. Rep. 1188, 1 (2026).
  48. L. Ciambelli, C. Marteau, A. C. Petkou, P. M. Petropoulos, and K. Siampos, Covariant Galilean versus Carrollian hydrodynamics from relativistic fluids, Classical Quantum Gravity 35, 165001 (2018).
  49. A. C. Petkou, P. M. Petropoulos, D. R. Betancour, and K. Siampos, Relativistic fluids, hydrodynamic frames and their Galilean versus Carrollian avatars, J. High Energy Phys. 09 (2022) 162.
  50. J. Armas and E. Have, Carrollian fluids and spontaneous breaking of boost symmetry, Phys. Rev. Lett. 132, 161606 (2024).
  51. L. Bidussi, J. Hartong, E. Have, J. Musaeus, and S. Prohazka, Fractons, dipole symmetries and curved spacetime, SciPost Phys. 12, 205 (2022).
  52. A. Bagchi, A. Banerjee, R. Basu, M. Islam, and S. Mondal, Magic fermions: Carroll and flat bands, J. High Energy Phys. 03 (2022) 227.
  53. N. Ara, A. Banerjee, R. Basu, and B. Krishnan, Flat bands and compact localised states: A Carrollian roadmap, SciPost Phys. 19, 046 (2025).
  54. A. Bagchi, A. Banerjee, S. Mondal, and S. Sarkar, Carroll in shallow water, arXiv:2411.04190.
  55. S. Biswas, A. Dubey, S. Mondal, A. Banerjee, A. Kundu, and A. Bagchi, Carroll at phase separation, Phys. Rev. B 113, L241115 (2026).
  56. J. de Boer, J. Hartong, N. A. Obers, W. Sybesma, and S. Vandoren, Carroll symmetry, dark energy and inflation, Front. Phys. 10, 810405 (2022).
  57. K. S. Kolekar, T. Mandal, A. Shukla, and P. Soni, Hydrodynamics in the Carrollian regime, Int. J. Mod. Phys. A 41, 2650016 (2026).
  58. D. Hansen, N. A. Obers, G. Oling, and B. T. Sogaard, Carroll expansion of general relativity, SciPost Phys. 13, 055 (2022).
  59. A. Jaiswal, Spin-hydrodynamics of electrons in graphene and magnetization due to thermal vorticity, arXiv:2409.07764.
  60. N. Banerjee, J. Bhattacharya, S. Bhattacharyya, S. Jain, S. Minwalla, and T. Sharma, Constraints on fluid dynamics from equilibrium partition functions, J. High Energy Phys. 09 (2012) 046.
  61. K. Jensen, M. Kaminski, P. Kovtun, R. Meyer, A. Ritz, and A. Yarom, Towards hydrodynamics without an entropy current, Phys. Rev. Lett. 109, 101601 (2012).
  62. P. Kovtun, Thermodynamics of polarized relativistic matter, J. High Energy Phys. 07 (2016) 028.
  63. P. Kovtun and A. Shukla, Kubo formulas for thermodynamic transport coefficients, J. High Energy Phys. 10 (2018) 007.
  64. A. Shukla, Equilibrium thermodynamic susceptibilities for a dense degenerate Dirac field, Phys. Rev. D 100, 096010 (2019).
  65. P. Kovtun and A. Shukla, Einstein’s equations in matter, Phys. Rev. D 101, 104051 (2020).
  66. S. R. De Groot, W. A. Van Leeuwen, and C. G. Van Weert, Relativistic Kinetic Theory. Principles and Applications (North-Holland Publishing Company, Amsterdam, 1980).
  67. M. Henneaux, Geometry of zero signature space-times, Bull. Soc. Math. Belg. 31, 47 (1979).
  68. C. Duval, G. W. Gibbons, P. A. Horvathy, and P. M. Zhang, Carroll versus Newton and Galilei: Two dual non-Einsteinian concepts of time, Classical Quantum Gravity 31, 085016 (2014).
  69. X. Bekaert and K. Morand, Connections and dynamical trajectories in generalised Newton-Cartan gravity II. An ambient perspective, J. Math. Phys. (N.Y.) 59, 072503 (2018).
  70. J. Hartong, Gauging the Carroll algebra and ultra-relativistic gravity, J. High Energy Phys. 08 (2015) 069.
  71. I. L. Shapiro, Physical aspects of the space-time torsion, Phys. Rep. 357, 113 (2002).
  72. F. Becattini and L. Tinti, Nonequilibrium thermodynamical inequivalence of quantum stress-energy and spin tensors, Phys. Rev. D 87, 025029 (2013).
  73. J. Hu, Kubo formulae for first-order spin hydrodynamics, Phys. Rev. D 103, 116015 (2021).
  74. P.-X. Hao, W. Song, Z. Xiao, and X. Xie, BMS-invariant free fermion models, Phys. Rev. D 109, 025002 (2024).
  75. A. Banerjee, S. Dutta, and S. Mondal, Carroll fermions in two dimensions, Phys. Rev. D 107, 125020 (2023).
  76. E. A. Bergshoeff, A. Campoleoni, A. Fontanella, L. Mele, and J. Rosseel, Carroll fermions, SciPost Phys. 16, 153 (2024).
  77. E. Ekiz, E. O. Kahya, and U. Zorba, Quantization of Carrollian fermions, Phys. Rev. D 111, 105019 (2025).
  78. D. Grumiller, L. Mele, and L. Montecchio, Carroll spinors, Mod. Phys. Lett. A 41, 2540003 (2026).
  79. J. de Boer, J. Hartong, N. A. Obers, W. Sybesma, and S. Vandoren, Carroll stories, J. High Energy Phys. 09 (2023) 148.
  80. R. Arnowitt, S. Deser, and C. W. Misner, Dynamical structure and definition of energy in general relativity, Phys. Rev. 116, 1322 (1959).
  81. R. L. Arnowitt, S. Deser, and C. W. Misner, The dynamics of general relativity, Gen. Relativ. Gravit. 40, 1997 (2008).
  82. A. Nicolis, R. Penco, F. Piazza, and R. Rattazzi, Zoology of condensed matter: Framids, ordinary stuff, extra-ordinary stuff, J. High Energy Phys. 06 (2015) 155.
  83. F. W. Hehl, On the energy tensor of spinning massive matter in classical field theory and general relativity, Rep. Math. Phys. 9, 55 (1976).
  84. F. Becattini and L. Tinti, Thermodynamical inequivalence of quantum stress-energy and spin tensors, Phys. Rev. D 84, 025013 (2011).
  85. E. Speranza and N. Weickgenannt, Spin tensor and pseudo-gauges: From nuclear collisions to gravitational physics, Eur. Phys. J. A 57, 155 (2021).
  86. R. Singh, On the nonuniqueness of the energy–momentum and spin currents, Int. J. Mod. Phys. A 41, 2650021 (2026).
  87. J. Armas and A. Jain, Thermodynamics of ideal spin fluids and pseudo-gauge ambiguity, arXiv:2601.14421.
  88. C. Duval, G. W. Gibbons, and P. A. Horvathy, Conformal Carroll groups, J. Phys. A 47, 335204 (2014).

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Phys. Rev. D 114, 065001– Published 2 September, 2026
  • Received 7 May 2026
  • Accepted 7 August 2026
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