Six-Field Rational Extended Thermodynamics of Polyatomic Gases in Curved Spacetime

Abstract: We formulate a generally covariant six-field Rational Extended Thermodynamics model (RET$_6$) for relativistic polyatomic gases, with the dynamical pressure as the only non-equilibrium variable. The model is based on a polyatomic extension of the Boltzmann-Chernikov kinetic equation, where the one-particle distribution depends also on an internal-energy variable, and on the Maximum Entropy closure of the associated relativistic moment hierarchy. The resulting field equations, closure relations, and production term are therefore fixed by the underlying kinetic structure rather than postulated phenomenologically. We extend the RET$_6$ model from Minkowski spacetime to a general curved spacetime by the minimal coupling prescription and couple it to the Einstein equations. As a first structural result, we prove a kinetic-theory no-go theorem in this polyatomic RET setting stating that any stress-energy tensor induced by a non-negative relativistic one-particle distribution function satisfies the strong energy condition. We then specialize the theory to a homogeneous and isotropic Friedmann-Lemaître-Robertson-Walker (FLRW) spacetime. In this setting the dynamical pressure modifies the expansion dynamics with respect to the perfect-fluid Euler case, but the no-go theorem excludes acceleration driven by the RET$_6$ gas alone. Finally, we reintroduce a cosmological constant and study the combined $Λ$RET$_6$ model, proving the existence and local stability of a de Sitter attractor at late times. Numerical integrations show that, for physically motivated post-recombination initial data and relaxation times, the expansion history rapidly approaches that of $Λ$CDM, with small non-equilibrium corrections controlled by the relaxation time and by the initial value of the dynamical pressure.

• The paper presents a kinetic-theory based RET6 framework that systematically derives dynamical pressure and non-equilibrium relations for polyatomic gases in curved spacetime.
• By applying the Maximum Entropy Principle and minimal coupling in FLRW cosmology, the model confirms a robust convergence to a de Sitter attractor, aligning with ΛCDM evolution.
• Numerical analyses show that non-equilibrium corrections rapidly damp out, validating the strong energy condition and limiting bulk viscous effects from driving cosmic acceleration.

Six-Field Rational Extended Thermodynamics of Polyatomic Gases in Curved Spacetime: An Expert Summary

Overview and Motivation

This paper introduces a generally covariant six-field Rational Extended Thermodynamics (RET$_6$) framework for relativistic polyatomic gases in curved spacetime (2607.09463). The approach incorporates the dynamical pressure as the canonical non-equilibrium variable, systematically derived from a polyatomic extension of the Boltzmann–Chernikov kinetic equation and closed via the Maximum Entropy Principle (MEP). Unlike phenomenological Müller–Israel-type theories, RET$_6$ enforces that all constitutive and production relations arise from kinetic foundations, not ad hoc hypotheses. The authors further extend RET$_6$ to curved backgrounds by minimal coupling and analyze its cosmological implications within the homogeneous and isotropic FLRW geometry.

The principal results are: (i) a kinetic-theory no-go theorem demonstrating that any stress-energy tensor from a non-negative relativistic one-particle distribution satisfies the strong energy condition (SEC), excluding self-driven acceleration by kinetic matter alone; (ii) local and asymptotic properties of the $\Lambda$RET$_6$ model (RET$_6$ with a cosmological constant), including a rigorous demonstration of a late-time de Sitter attractor; and (iii) detailed numerical investigations confirming that, with physically motivated parameters, non-equilibrium corrections rapidly decay, yielding an expansion history essentially indistinguishable from $\Lambda$CDM.

RET$_6$ Construction from Kinetic Theory

RET$_6$ is derived as a principal subsystem of the fifteen-field polyatomic RET$_{15}$ hierarchy, distinguished by setting the trace-free components of the Lagrange multipliers to zero while retaining the scalar dynamical pressure $_6$0. The system is governed by particle, energy-momentum, and projected third-order moment conservation laws, with independent fields $_6$1. The underlying polyatomic Boltzmann–Chernikov equation includes an internal-energy variable $_6$2, whose state density $_6$3 generalizes the phase-space structure for non-monatomic gases.

Explicitly, the energy-momentum tensor takes the form

$_6$4

with $_6$5 the equilibrium pressure, $_6$6 the total energy density, and $_6$7 the spatial projector in the local rest frame. The closure and production terms for higher moments, including the dynamical pressure relaxation, are determined unambiguously by MEP closure, differentiating this approach from phenomenological models.

Hyperbolicity and causality are explicitly verified in the admissible state space. The region ensuring characteristic velocities $_6$8 is delineated in the $_6$9-plane, where $_6$0 is the dimensionless inverse temperature and $_6$1.

Figure 1: Phase portrait of the de Sitter attractor for the $_6$2RET$_6$3 dynamical system, showing the system's trajectory approaching $_6$4.

Extension to Curved Spacetimes and FLRW Cosmology

Minimal coupling promotes the RET$_6$5 system to general relativity, replacing $_6$6 and all derivatives with covariant ones. In FLRW geometry, dissipative variables other than $_6$7 decay due to isotropy and homogeneity, so RET$_6$8 provides the unique hyperbolic non-equilibrium extension of perfect-fluid Euler cosmology.

The system reduces to evolution equations for the scale factor $_6$9, temperature-like variable $\Lambda$0, and dynamical pressure $\Lambda$1, supplemented by the generalized Friedmann equations. In dimensionless units, the dynamical pressure acts as a source of bulk viscous corrections, with a relaxation equation gauged by the kinetic relaxation time $\Lambda$2.

Strong numerical evidence is given that for cosmological initial data, non-equilibrium corrections from $\Lambda$3 are rapidly damped, leaving only transient departures from equilibrium.

Figure 2: Evolution of $\Lambda$4, $\Lambda$5, and $\Lambda$6 for $\Lambda$7, $\Lambda$8, and initial negative bulk pressure, showing rapid relaxation.

Figure 3: Evolution with $\Lambda$9, $_6$0; the larger relativistic effects yield higher magnitude but still rapidly damped non-equilibrium transients.

Figure 4: Non-monotonic relaxation of $_6$1 for initial positive values; the dynamical pressure briefly overshoots before decaying.

Kinetic-Theory No-Go Theorem and Implications

A key theoretical result is the kinetic-theory no-go theorem: any macroscopic stress-energy constructed from a non-negative, admissible polyatomic one-particle distribution satisfies the SEC—

$_6$2

(pointwise, in any local inertial frame). The theorem is independent of closure details, applying to all finite-moment RET models with a kinetic foundation. In FLRW applications, this implies that a RET$_6$3 polyatomic gas cannot drive accelerated cosmic expansion; regardless of the magnitude of allowed bulk pressure, the deceleration parameter remains $_6$4 in the absence of a cosmological constant or truly exotic matter.

$_6$5RET$_6$6 Analysis and de Sitter Attractor

Upon restoring a positive cosmological constant to the Einstein equations, the $_6$7RET$_6$8 system is shown to possess a de Sitter attractor at late times. A dynamical-systems analysis, conducted in rescaled variables suitable for late-time asymptotics, yields that all eigenvalues of the linearized Jacobian at the attractor are negative definite provided $_6$9. Thus, the system is locally asymptotically stable and cosmic expansion generically converges to de Sitter behavior at late times, even with non-zero initial $_6$0.

Numerical Results: Convergence to $_6$1CDM

Numerical integrations, both for arbitrary initial non-equilibrium states and for realistic post-recombination parameters, confirm the analytic predictions.

Figure 5: Direct comparison of $_6$2 obtained with $_6$3RET$_6$4 and standard $_6$5CDM evolution from recombination, showing indistinguishable expansion histories.

Figure 6: Temperature and dynamical pressure evolution: $_6$6 returns to equilibrium ($_6$7) rapidly, and $_6$8 decreases monotonically.

Figure 7: Density parameters $_6$9, $\Lambda$0, $\Lambda$1 and deceleration parameter $\Lambda$2 under $\Lambda$3RET$\Lambda$4 evolution; dynamical corrections are negligible after early times.

Over a range of realistic relaxation times and initial conditions, the non-equilibrium contributions from the dynamical pressure are consistently found to be subdominant and quickly erased by relaxation. The convergence to $\Lambda$5CDM is robust, and deviations are at most transient.

Conclusions and Outlook

This work provides a rigorous, kinetic-theory-consistent construction of a six-field relativistic dissipative gas in general relativity that incorporates polyatomic internal structure. All field equations and closure relations are derived from first principles, without phenomenological input. The kinetic-theory no-go result sets a sharp boundary for what types of cosmic acceleration can be realized by kinetic matter: in standard RET closures, bulk viscosity is insufficient to drive accelerated expansion.

Practical implications include the validation of $\Lambda$6CDM as a late-time attractor even in the presence of kinetic-theory-motivated non-equilibrium corrections, with RET$\Lambda$7 offering a theoretically robust mechanism for bulk viscosity and transient deviations from equilibrium. Theoretically, this framework demonstrates that RET-based cosmologies, once coupled to gravity, naturally inherit foundational energy conditions from their kinetic microstructure, reinforcing the hierarchy between dark energy and standard matter models.

Future developments may explore: alternative closure schemes to circumvent the SEC; inclusion of more complex non-equilibrium variables (e.g., heat flux, shear stress in less symmetric cosmologies); and the impact on early-universe processes such as baryogenesis, structure formation, or nonequilibrium phase transitions. The RET$\Lambda$8 paradigm offers a promising base for mathematically rigorous, physically motivated cosmological fluid modeling beyond ideal hydrodynamics.