{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,7]],"date-time":"2026-04-07T19:53:44Z","timestamp":1775591624297,"version":"3.50.1"},"reference-count":62,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2020,9,18]],"date-time":"2020-09-18T00:00:00Z","timestamp":1600387200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>The formulation of a universal theory for bulk viscosity and heat conduction represents a theoretical challenge for our understanding of relativistic fluid dynamics. Recently, it was shown that the multifluid variational approach championed by Carter and collaborators has the potential to be a general and natural framework to derive (hyperbolic) hydrodynamic equations for relativistic dissipative systems. Furthermore, it also allows keeping direct contact with non-equilibrium thermodynamics, providing a clear microscopic interpretation of the elements of the theory. To provide an example of its universal applicability, in this paper we derive the fundamental equations of the radiation hydrodynamics directly in the context of Carter\u2019s multifluid theory. This operation unveils a novel set of thermodynamic constraints that must be respected by any microscopic model. Then, we prove that the radiation hydrodynamics becomes a multifluid model for bulk viscosity or heat conduction in some appropriate physical limits.<\/jats:p>","DOI":"10.3390\/sym12091543","type":"journal-article","created":{"date-parts":[[2020,9,18]],"date-time":"2020-09-18T10:22:23Z","timestamp":1600424543000},"page":"1543","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":23,"title":["Multifluid Modelling of Relativistic Radiation Hydrodynamics"],"prefix":"10.3390","volume":"12","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-6603-9253","authenticated-orcid":false,"given":"Lorenzo","family":"Gavassino","sequence":"first","affiliation":[{"name":"Nicolaus Copernicus Astronomical Center of the Polish Academy of Sciences, Bartycka 18, 00-716 Warszawa, Poland"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5470-4308","authenticated-orcid":false,"given":"Marco","family":"Antonelli","sequence":"additional","affiliation":[{"name":"Nicolaus Copernicus Astronomical Center of the Polish Academy of Sciences, Bartycka 18, 00-716 Warszawa, Poland"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Brynmor","family":"Haskell","sequence":"additional","affiliation":[{"name":"Nicolaus Copernicus Astronomical Center of the Polish Academy of Sciences, Bartycka 18, 00-716 Warszawa, Poland"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2020,9,18]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"1","DOI":"10.12942\/lrr-2007-1","article-title":"Relativistic Fluid Dynamics: Physics for Many Different Scales","volume":"10","author":"Andersson","year":"2007","journal-title":"Living Rev. Relativ."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"175","DOI":"10.1086\/151073","article-title":"Entropy Generation and the Survival of Protogalaxies in an Expanding Universe","volume":"168","author":"Weinberg","year":"1971","journal-title":"Astrophys. J."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"725","DOI":"10.1103\/PhysRevD.31.725","article-title":"Generic instabilities in first-order dissipative relativistic fluid theories","volume":"31","author":"Hiscock","year":"1985","journal-title":"Phys. Rev. D"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"043018","DOI":"10.1103\/PhysRevD.102.043018","article-title":"When the entropy has no maximum: A new perspective on the instability of the first-order theories of dissipation","volume":"102","author":"Gavassino","year":"2020","journal-title":"Phys. Rev. D"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"023003","DOI":"10.1103\/PhysRevD.62.023003","article-title":"Causality and stability of the relativistic diffusion equation","volume":"62","author":"Liu","year":"2000","journal-title":"Phys. Rev. D"},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Stewart, W.I.J. (1979). Transient relativistic thermodynamics and kinetic theory. Ann. Phys., 341\u2013372.","DOI":"10.1016\/0003-4916(79)90130-1"},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Gavassino, L., Antonelli, M., and Haskell, B. (2020). Bulk viscosity in relativistic fluids: From thermodynamics to hydrodynamics. arXiv.","DOI":"10.1088\/1361-6382\/abe588"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"4536","DOI":"10.1103\/PhysRevD.45.4536","article-title":"Equivalence of convective and potential variational derivations of covariant superfluid dynamics","volume":"45","author":"Carter","year":"1992","journal-title":"Phys. Rev. D"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"3687","DOI":"10.1103\/PhysRevD.41.3687","article-title":"Stability, causality, and hyperbolicity in Carter\u2019s \u201cregular\u201d theory of relativistic heat-conducting fluids","volume":"41","author":"Olson","year":"1990","journal-title":"Phys. Rev. D"},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Carter, B. (1989). Covariant Theory of Conductivity in Ideal Fluid or Solid Media, Springer.","DOI":"10.1007\/BFb0084028"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"5855","DOI":"10.1103\/PhysRevD.51.5855","article-title":"Equation of state for cool relativistic two-constituent superfluid dynamics","volume":"51","author":"Carter","year":"1995","journal-title":"Phys. Rev. D"},{"key":"ref_12","first-page":"267","article-title":"Relativistic dynamics of vortex defects in superfluids","volume":"549","author":"Bunkov","year":"2000","journal-title":"NATO Advanced Science Institutes (ASI) Series C"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"064033","DOI":"10.1103\/PhysRevD.93.064033","article-title":"Relativistic formulation of the Hall-Vinen-Bekarevich-Khalatnikov superfluid hydrodynamics","volume":"93","author":"Gusakov","year":"2016","journal-title":"Phys. Rev. D"},{"key":"ref_14","first-page":"1601","article-title":"The relativistic hydrodynamics of a superfluid","volume":"83","author":"Lebedev","year":"1982","journal-title":"Zhurnal Eksperimental\u2019noi i Teoreticheskoi Fiziki"},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Khalatnikov, B.C.I. (1992). Momentum, vorticity, and helicity in covariant superfluid dynamics. Ann. Phys., 243\u2013265.","DOI":"10.1016\/0003-4916(92)90348-P"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"917","DOI":"10.1007\/s10714-006-0272-7","article-title":"Relativistic kinetics of phonon gas in superfluids","volume":"38","author":"Popov","year":"2006","journal-title":"Gen. Relativ. Gravit."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"025014","DOI":"10.1088\/1361-6382\/ab5f23","article-title":"Thermodynamics of uncharged relativistic multifluids","volume":"37","author":"Gavassino","year":"2020","journal-title":"Class. Quantum Gravity"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"969","DOI":"10.1088\/0264-9381\/18\/6\/302","article-title":"Slowly rotating general relativistic superfluid neutron stars","volume":"18","author":"Andersson","year":"2001","journal-title":"Class. Quantum Gravity"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"043005","DOI":"10.1103\/PhysRevD.71.043005","article-title":"Relativistic numerical models for stationary superfluid neutron stars","volume":"71","author":"Prix","year":"2005","journal-title":"Phys. Rev. D"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"083004","DOI":"10.1103\/PhysRevD.93.083004","article-title":"Numerical models for stationary superfluid neutron stars in general relativity with realistic equations of state","volume":"93","author":"Sourie","year":"2016","journal-title":"Phys. Rev. D"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"1189","DOI":"10.1046\/j.1365-8711.1998.01575.x","article-title":"Differential rotation of relativistic superfluid in neutron stars","volume":"297","author":"Langlois","year":"1998","journal-title":"Mon. Not. R. Astron. Soc."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"103005","DOI":"10.1103\/PhysRevD.62.103005","article-title":"Covariant vortex in superconducting-superfluid-normal fluid mixtures with a stiff equation of state","volume":"62","author":"Prix","year":"2000","journal-title":"Phys. Rev. D"},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Carter, B., Langlois, D., and Prix, R. (2001). Relativistic solution of Iordanskii problem in multi-constituent superfluid mechanics. arXiv.","DOI":"10.1007\/978-3-662-04665-4_10"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"4641","DOI":"10.1093\/mnras\/stw2613","article-title":"Global numerical simulations of the rise of vortex-mediated pulsar glitches in full general relativity","volume":"464","author":"Sourie","year":"2017","journal-title":"Mon. Not. R. Astron. Soc."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"5403","DOI":"10.1093\/mnras\/sty130","article-title":"Effects of general relativity on glitch amplitudes and pulsar mass upper bounds","volume":"475","author":"Antonelli","year":"2018","journal-title":"Mon. Not. R. Astron. Soc."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"3562","DOI":"10.1093\/mnras\/staa886","article-title":"A universal formula for the relativistic correction to the mutual friction coupling time-scale in neutron stars","volume":"494","author":"Gavassino","year":"2020","journal-title":"Mon. Not. R. Astron. Soc."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"1950078","DOI":"10.1142\/S0218271819500780","article-title":"Multi-fluid theory and cosmology: A convective variational approach to interacting dark-sector","volume":"28","author":"Osano","year":"2019","journal-title":"Int. J. Mod. Phys. D"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"42","DOI":"10.1007\/s10714-020-02694-y","article-title":"A transient phase in cosmological evolution: A multi-fluid approximation for a quasi-thermodynamical equilibrium","volume":"52","author":"Osano","year":"2020","journal-title":"Gen. Relativ. Gravit."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"45","DOI":"10.1098\/rspa.1991.0034","article-title":"Convective variational approach to relativistic thermodynamics of dissipative fluids","volume":"433","author":"Carter","year":"1991","journal-title":"Proc. R. Soc. Lond. Ser. A"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"1223","DOI":"10.1103\/PhysRevD.43.1223","article-title":"Comparison between variational and traditional approaches to relativistic thermodynamics of dissipative fluids","volume":"43","author":"Priou","year":"1991","journal-title":"Phys. Rev. D"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"195023","DOI":"10.1088\/0264-9381\/28\/19\/195023","article-title":"A consistent first-order model for relativistic heat flow","volume":"28","author":"Andersson","year":"2011","journal-title":"Class. Quantum Gravity"},{"key":"ref_32","first-page":"738","article-title":"Thermal dynamics in general relativity","volume":"467","author":"Andersson","year":"2011","journal-title":"Proc. R. Soc. Lond. Ser. A"},{"key":"ref_33","unstructured":"Carter, B. (2012). The regular conducting fluid model for relativistic thermodynamics. arXiv."},{"key":"ref_34","unstructured":"Mihalas, D., and Weibel Mihalas, B. (1984). Foundations of Radiation Hydrodynamics, Dover Publications INC."},{"key":"ref_35","unstructured":"Huang, K. (1987). Statistical Mechanics, John Wiley & Sons. [2nd ed.]."},{"key":"ref_36","unstructured":"Clayton, D.D. (1983). Principles of Stellar Evolution and Nucleosynthesis, The University of Chicago Press."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"792","DOI":"10.1119\/1.1479743","article-title":"Teaching the photon gas in introductory physics","volume":"70","author":"Leff","year":"2002","journal-title":"Am. J. Phys."},{"key":"ref_38","doi-asserted-by":"crossref","unstructured":"Rezzolla, L., and Zanotti, O. (2013). Relativistic Hydrodynamics, Oxford University Press.","DOI":"10.1093\/acprof:oso\/9780198528906.001.0001"},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"1858","DOI":"10.1103\/PhysRevD.40.1858","article-title":"Thermal radiation from stellar collapse to a black hole","volume":"40","author":"Shapiro","year":"1989","journal-title":"Phys. Rev. D"},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"024023","DOI":"10.1103\/PhysRevD.78.024023","article-title":"Relativistic radiation magnetohydrodynamics in dynamical spacetimes: Numerical methods and tests","volume":"78","author":"Farris","year":"2008","journal-title":"Phys. Rev. D"},{"key":"ref_41","doi-asserted-by":"crossref","unstructured":"Andersson, N., Haskell, B., Comer, G.L., and Samuelsson, L. (2018). The dynamics of neutron star crusts: Lagrangian perturbation theory for a relativistic superfluid-elastic system. arXiv.","DOI":"10.1088\/1361-6382\/ab12a1"},{"key":"ref_42","first-page":"542","article-title":"Hydrodynamics of Solutions of Two Superfluid Liquids","volume":"5","author":"Khalatnikov","year":"1957","journal-title":"Sov. J. Exp. Theor. Phys."},{"key":"ref_43","first-page":"164","article-title":"Three-velocity hydrodynamics of superfluid solutions","volume":"42","author":"Andreev","year":"1976","journal-title":"Sov. J. Exp. Theor. Phys."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"1468","DOI":"10.1103\/PhysRev.94.1468","article-title":"General Relativistic Variational Principle for Perfect Fluids","volume":"94","author":"Taub","year":"1954","journal-title":"Phys. Rev."},{"key":"ref_45","unstructured":"Cattaneo, C. (1958). Sur Une Forme De L\u2019\u00e9quation De La Chaleur \u00e9liminant Le Paradoxe D\u2019une Propagation Instantan\u00e9e, Gauthier-Villars. Comptes Rendus Hebdomadaires Des S\u00e9ances De l\u2019Acad\u00e9mie Des Sciences."},{"key":"ref_46","unstructured":"Callen, H.B. (1985). Thermodynamics and an Introduction to Thermostatistics, Wiley. [2nd ed.]."},{"key":"ref_47","unstructured":"Khinchin, A. (1949). Mathematical Foundations of Statistical Mechanics, Dover Publications. Dover Books on Mathematics."},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"149","DOI":"10.1016\/0022-4073(84)90112-2","article-title":"Relating Eddington factors to flux limiters","volume":"31","author":"Levermore","year":"1984","journal-title":"J. Quant. Spectrosc. Radiat. Transf."},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"3533","DOI":"10.1093\/mnras\/sts632","article-title":"Semi-implicit scheme for treating radiation under M1 closure in general relativistic conservative fluid dynamics codes","volume":"429","author":"Narayan","year":"2013","journal-title":"Mon. Not. R. Astron. Soc."},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"22","DOI":"10.1088\/0004-637X\/796\/1\/22","article-title":"Numerical Simulations of Optically Thick Accretion onto a Black Hole. II. Rotating Flow","volume":"796","author":"Fragile","year":"2014","journal-title":"Astrophys. J."},{"key":"ref_51","doi-asserted-by":"crossref","unstructured":"Gavassino, L. (2020). The zeroth law of thermodynamics in special relativity. arXiv.","DOI":"10.1007\/s10701-020-00393-x"},{"key":"ref_52","unstructured":"De Groot, S.R., van Leeuwen, W.A., and van Weert, C.G. (1980). Relativistic Kinetic Theory: Principles and Applications, North-Holland Publishing Company."},{"key":"ref_53","doi-asserted-by":"crossref","first-page":"117","DOI":"10.1007\/BF00644827","article-title":"Chemical Potential Effects on Neutrino Diffusion in Supernovae","volume":"35","author":"Mazurek","year":"1975","journal-title":"Astrophys. Space Sci."},{"key":"ref_54","unstructured":"Landau, L., and Lifshitz, E. (2013). Fluid Mechanics, Elsevier Science. Number v. 6."},{"key":"ref_55","doi-asserted-by":"crossref","unstructured":"Feugeas, B.D.J.L. (1999). Etude th\u00e9orique et num\u00e9rique d\u2019une hi\u00e9rarchie de mod\u00e8les aux moments pour le transfert radiatif. Comptes Rendus L\u2019Acad\u00e9mie Des Sci. Ser. I Math., 915\u2013920.","DOI":"10.1016\/S0764-4442(00)87499-6"},{"key":"ref_56","doi-asserted-by":"crossref","first-page":"308","DOI":"10.1086\/178065","article-title":"Radiation from Stellar Collapse to a Black Hole","volume":"472","author":"Shapiro","year":"1996","journal-title":"Astrophys. J."},{"key":"ref_57","doi-asserted-by":"crossref","first-page":"2372","DOI":"10.1093\/mnras\/stv2022","article-title":"Photon-conserving Comptonization in simulations of accretion discs around black holes","volume":"454","author":"Narayan","year":"2015","journal-title":"Mon. Not. R. Astron. Soc."},{"key":"ref_58","doi-asserted-by":"crossref","first-page":"075008","DOI":"10.1088\/0264-9381\/32\/7\/075008","article-title":"A covariant action principle for dissipative fluid dynamics: From formalism to fundamental physics","volume":"32","author":"Andersson","year":"2015","journal-title":"Class. Quantum Gravity"},{"key":"ref_59","doi-asserted-by":"crossref","first-page":"620","DOI":"10.1103\/PhysRev.106.620","article-title":"Information Theory and Statistical Mechanics","volume":"106","author":"Jaynes","year":"1957","journal-title":"Phys. Rev."},{"key":"ref_60","doi-asserted-by":"crossref","first-page":"541","DOI":"10.1016\/0022-4073(78)90024-9","article-title":"Maximum entropy Eddington factors","volume":"20","author":"Minerbo","year":"1978","journal-title":"J. Quant. Spectrosc. Radiat. Transf."},{"key":"ref_61","doi-asserted-by":"crossref","first-page":"466","DOI":"10.1016\/0031-8914(74)90355-3","article-title":"A relativistic relaxation-time model for the Boltzmann equation","volume":"74","author":"Anderson","year":"1974","journal-title":"Physica"},{"key":"ref_62","doi-asserted-by":"crossref","unstructured":"Cercignani, C., and Kremer, G.M. (2002). The Relativistic Boltzmann Equation: Theory and Applications, Birkhauser Verlag.","DOI":"10.1007\/978-3-0348-8165-4"}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/12\/9\/1543\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T10:11:15Z","timestamp":1760177475000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/12\/9\/1543"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,9,18]]},"references-count":62,"journal-issue":{"issue":"9","published-online":{"date-parts":[[2020,9]]}},"alternative-id":["sym12091543"],"URL":"https:\/\/doi.org\/10.3390\/sym12091543","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2020,9,18]]}}}