Relativistic kinetic theory: An introduction
Abstract
We present a brief introduction to the relativistic kinetic theory of gases with emphasis on the underlying geometric and Hamiltonian structure of the theory. Our formalism starts with a discussion on the tangent bundle of a Lorentzian manifold of arbitrary dimension. Next, we introduce the Poincaré one-form on this bundle, from which the symplectic form and a volume form are constructed. Then, we define an appropriate Hamiltonian on the bundle which, together with the symplectic form yields the Liouville vector field. Whenever the corresponding flow is restricted to special submanifolds of the tangent bundle, referred to as future mass shells, its projection onto the base manifold describes families of future-directed timelike geodesics. A collisionless simple gas is described by a distribution function on a particular mass shell, satisfying the Liouville equation. Fibre integrals of the distribution function determine the particle current density and the stress-energy tensor. We show that the stress-energy tensor satisfies the familiar energy conditions and that both the current density and stress-energy tensor are divergence-free. Our discussion also includes the generalization to charged gases, a summary of the Einstein-Maxwell-Vlasov system in any dimensions, as well as a brief introduction to the general relativistic Boltzmann equation for a simple gas.
- Publication:
-
IX Mexican School on Gravitation and Mathematical Physics: Cosmology for the XXIst Century: Gravitation and Mathematical Physics Division of the Mexican Physical Society DGFM-SMF
- Pub Date:
- July 2013
- DOI:
- 10.1063/1.4817035
- arXiv:
- arXiv:1303.2899
- Bibcode:
- 2013AIPC.1548..134S
- Keywords:
-
- Boltzmann equation;
- Einstein field equations;
- kinetic theory;
- Liouville equation;
- Vlasov equation;
- 04.20.-q;
- 04.25.-g;
- 04.40.-b;
- 05.20.Dd;
- Classical general relativity;
- Approximation methods;
- equations of motion;
- Self-gravitating systems;
- continuous media and classical fields in curved spacetime;
- Kinetic theory;
- General Relativity and Quantum Cosmology;
- Condensed Matter - Statistical Mechanics;
- High Energy Physics - Theory
- E-Print:
- Prepared for the proceedings of the IX Mexican School of the Gravitation and Mathematical Physics Division of the Mexican Physical Society, 30 pages, 1 figure