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A general relativistic kinetic theory approach to linear transport in generic hydrodynamic frame

  • Long Cui , Xin Hao EMAIL logo and Liu Zhao
Published/Copyright: April 9, 2025
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Journal of Non-Equilibrium Thermodynamics
From the journal Journal of Non-Equilibrium Thermodynamics Volume 50 Issue 3

Abstract

In this study, we investigate the linear transport of neutral system within the framework of relativistic kinetic theory. Under the relaxation time approximation, we obtain an iterative solution to the relativistic Boltzmann equation in generic stationary spacetime. This solution provides a scheme to study non-equilibrium system order by order. Our calculations are performed in generic hydrodynamic frame, and the results can be reduced to a specific hydrodynamic frame by imposing constraints. As a specific example, we analytically calculated the covariant expressions of the particle flow and the energy momentum tensor up to the first order in relaxation time. Finally and most importantly, we present all 14 kinetic coefficients for a neutral system, which are verified to satisfy the Onsager reciprocal relation in a generic hydrodynamic frame and guarantee a non-negative entropy production in the frame where the first order conservation laws are restored.

Keywords: relativistic kinetic theory; linear transport in curved spacetime; Onsager reciprocal relations
Corresponding author: Xin Hao, School of Physics, Hebei Normal University, Shijiazhuang 050024, China, E-mail: 

  1. Research ethics: Not applicable.

  2. Author contributions: The authors have accepted responsibility for the entire content of this manuscript and approved its submission.

  3. Use of Large Language Models, AI and Machine Learning Tools: None declared.

  4. Conflict of interest: The authors state no conflict of interest.

  5. Research funding: This work is supported by the National Natural Science Foundation of China under the grant No. 12275138 and by the Hebei NSF under the Grant No. A2021205037.

  6. Data availability: Not applicable.

Appendix A: Calculation of particle flow and energy momentum tensor

We work in orthonormal basis ( e a ̂ ) μ obeying η a ̂ b ̂ = g μ ν ( e a ̂ ) μ ( e b ̂ ) ν . Without loss of generality, we require that U μ = c ( e 0 ̂ ) μ , which implies that the induced metric can be expressed as Δ μ ν = δ i ̂ j ̂ ( e i ̂ ) μ ( e j ̂ ) ν .

For massive particles, the momentum p a ̂ = p μ ( e a ̂ ) μ can be parameterized by mass shell conditions

(65) p a ̂ = m c ( cosh ⁡ ϑ , n i ̂ ⁡ sinh ⁡ ϑ ) ,

where n i ̂ ∈ S d − 1 is a spacelike unit vector. Then the momentum space volume element be represented as

(66) ϖ = ( d p ) d | p 0 ̂ | = | p | d − 1 d | p | d Ω d − 1 p 0 ̂ = ( m c ⁡ sinh ⁡ ϑ ) d − 1 d ϑ d Ω d − 1 ,

where dΩ d−1 is the volume element of the (d − 1)-dimensional unit sphere S d−1. In this way, the integration in the momentum space can be decomposed into integration over ϑ ∈ (0, ∞) and integration over the unit sphere S d−1.

Here we list some useful integration formulae,

(67) ∫ d Ω d − 1 = A d − 1 ,

(68) ∫ n i ̂ n j ̂ d Ω d − 1 = 1 d A d − 1 δ i ̂ j ̂ ,

(69) ∫ n i ̂ n j ̂ n k ̂ n l ̂ d Ω d − 1 = 3 ( d + 2 ) d A d − 1 δ ( i ̂ j ̂ δ k ̂ l ̂ ) ,

(70) ∫ n i ̂ d Ω d − 1 = ∫ n i ̂ n j ̂ n k ̂ d Ω d − 1 = 0 .

Using eqs. (66)–(71), we can further calculate

(71) ∫ ϖ p μ f ( 0 ) = g h d A d − 1 ( m c ) d J d − 1,1 1 c U μ ,

(72) ∫ ϖ p μ p ν f ( 0 ) = g h d A d − 1 ( m c ) d + 1 J d − 1,2 1 c 2 U μ U ν + 1 d J d + 1,0 Δ μ ν ,

(73) ∫ ϖ 1 p 0 ̂ p μ p ν f ( 0 ) = g h d A d − 1 ( m c ) d J d − 1,1 1 c 2 U μ U ν + 1 d J d + 1 , − 1 Δ μ ν ,

(74) ∫ ϖ 1 p 0 ̂ p μ p ν p σ f ( 0 ) = g h d A d − 1 ( m c ) d + 1 J d − 1,2 1 c 3 U μ U ν U σ + 3 d J d + 1,0 1 c U ( μ Δ ν σ ) ,

(75) ∫ ϖ 1 p 0 ̂ p μ p ν p σ p ρ f ( 0 ) = g h d A d − 1 ( m c ) d + 2 J d − 1,3 1 c 4 U μ U ν U σ U ρ   + 6 d J d + 1,1 1 c 2 U ( μ U ν Δ σ ρ ) + 3 ( d + 2 ) d J d + 3 , − 1 Δ ( μ ν Δ σ ρ ) .

The following calculations are then straightforward,

(76) N ( 1 ) μ = c ∫ ϖ p μ f ( 1 ) = − c 3 τ ∫ ϖ 1 ε p μ − p ν p σ ∇ ν B σ + p ν ∇ ν α ∂ f ( 0 ) ∂ α = − c 2 τ − ∇ ν B σ ∂ ∂ α ∫ ϖ 1 p 0 ̂ p μ p ν p σ f ( 0 ) + ∇ ν α ∂ ∂ α ∫ ϖ 1 p 0 ̂ p μ p ν f ( 0 ) = − c 2 τ g h d ( m c ) d A d − 1 ∂ J d − 1,1 ∂ α 1 c 2 U ν ∇ ν α − m ∂ J d − 1,2 ∂ α 1 c 2 U ν U σ ∇ ν B σ − m 1 d ∂ J d + 1,0 ∂ α Δ ν σ ∇ ν B σ U μ   + 1 d − 2 m ∂ J d + 1,0 ∂ α U σ ∇ ( σ B ν ) + ∂ J d + 1 , − 1 ∂ α ∇ ν α Δ μ ν = − τ g h d ( m c ) d A d − 1 ∂ J d − 1,1 ∂ α U ν ∇ ν α + m c 2 ∂ J d − 1,2 ∂ α U ν ∇ ν β − m c 2 1 d ∂ J d + 1,0 ∂ α β ∇ ν U ν U μ   + c 2 d ∂ J d + 1 , − 1 ∂ α ∇ ν α + m c 2 ∂ J d + 1,0 ∂ α ∇ ν β − 1 c 2 β U ρ ∇ ρ U ν Δ μ ν ,

(77) T ( 1 ) μ ν = c ∫ ϖ p μ p ν f ( 1 ) = − c 3 τ ∫ ϖ 1 ε p μ p ν − p ρ p σ ∇ ρ B σ + p σ ∇ σ α ∂ f ( 0 ) ∂ α = − c 2 τ − ∇ ρ B σ ∂ ∂ α ∫ ϖ 1 p 0 ̂ p μ p ν p ρ p σ f ( 0 ) + ∇ σ α ∂ ∂ α ∫ ϖ 1 p 0 ̂ p μ p ν p σ f ( 0 ) = − c τ g h d A d − 1 ( m c ) d + 1 1 c 2 − m c 2 ∂ J d − 1,3 ∂ α 1 c 2 U σ U ρ ∇ ρ B σ − m c 2 1 d ∂ J d + 1,1 ∂ α ∇ ρ B σ Δ σ ρ + ∂ J d − 1,2 ∂ α U σ ∇ σ α U μ U ν + 1 d − 2 m c 2 ∂ J d + 1,1 ∂ α 1 c 2 U σ ∇ ( ρ B σ ) + ∂ J d + 1,0 ∂ α ∇ ρ α 2 U ( μ Δ ν ) ρ + 1 d − m c 2 ∂ J d + 1,1 ∂ α 1 c 2 ∇ ρ B σ U σ U ρ − m c 2 1 d ∂ J d + 3 , − 1 ∂ α ∇ ρ B σ Δ σ ρ + ∂ J d + 1,0 ∂ α U σ ∇ σ α Δ μ ν   − m c 2 2 ( d + 2 ) d ∂ J d + 3 , − 1 ∂ α ∇ ( ρ B σ ) Δ μ σ Δ ν ρ − 1 d ∇ ρ B σ Δ σ ρ Δ μ ν = − c τ g h d A d − 1 ( m c ) d + 1 1 c 2 ∂ J d − 1,2 ∂ α U σ ∇ σ α + m c 2 ∂ J d − 1,3 ∂ α U σ ∇ σ β − m c 2 1 d ∂ J d + 1,1 ∂ α β ∇ ρ U ρ U μ U ν + 1 d ∂ J d + 1,0 ∂ α ∇ ρ α + m c 2 ∂ J d + 1,1 ∂ α ∇ ρ β − 1 c 2 β U σ ∇ σ U ρ ( Δ ρ ν U μ + Δ ρ μ U ν ) + 1 d ∂ J d + 1,0 ∂ α U ρ ∇ ρ α + m c 2 ∂ J d + 1,1 ∂ α U ρ ∇ ρ β − m c 2 1 d ∂ J d + 3 , − 1 ∂ α β ∇ ρ U ρ Δ μ ν   − m c 2 2 ( d + 2 ) d ∂ J d + 3 , − 1 ∂ α β Δ ρ μ Δ σ ν ∇ ( ρ U σ ) − 1 d ∇ ρ U ρ Δ μ ν .

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Received: 2024-04-10
Accepted: 2025-03-11
Published Online: 2025-04-09
Published in Print: 2025-07-28

© 2025 Walter de Gruyter GmbH, Berlin/Boston

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https://doi.org/10.1515/jnet-2024-0024
Keywords for this article
relativistic kinetic theory; linear transport in curved spacetime; Onsager reciprocal relations

Articles in the same Issue

  1. Frontmatter
  2. Original Research Articles
  3. Heat transfer at nano-scale and boundary conditions: a comparison between the Guyer-Krumhansl model and the Thermomass theory
  4. Exergy-based efficient ecological-function optimization for endoreversible Carnot refrigerators
  5. Effect of depositional nanoparticles on heat transfer at the solid–liquid interface using molecular dynamics simulations
  6. Optimization of injection parameters, and ethanol shares for cottonseed biodiesel fuel in diesel engine utilizing artificial neural network (ANN) and taguchi grey relation analysis (GRA)
  7. A general relativistic kinetic theory approach to linear transport in generic hydrodynamic frame
  8. Asymmetric quantum harmonic Otto engine under hot squeezed thermal reservoir
  9. Approaches of finite-time thermodynamics in conceptual design of heat exchange systems
  10. Thermal transport in a silicon/diamond micro-flake with quantum dots inserts
  11. Finite element analysis on generalized piezothermoelastic interactions in an unbounded piezoelectric medium containing a spherical cavity
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