Tangent bundle formulation of a charged gas

Abstract

We discuss the relativistic kinetic theory for a simple, collisionless, charged gas propagating on an arbitrary curved spacetime geometry. Our general relativistic treatment is formulated on the tangent bundle of the spacetime manifold and takes advantage of its rich geometric structure. In particular, we point out the existence of a natural metric on the tangent bundle and illustrate its role for the development of the relativistic kinetic theory. This metric, combined with the electromagnetic field of the spacetime, yields an appropriate symplectic form on the tangent bundle. The Liouville vector field arises as the Hamiltonian vector field of a natural Hamiltonian. The latter also defines natural energy surfaces, called mass shells, which turn out to be smooth Lorentzian submanifolds. A simple, collisionless, charged gas is described by a distribution function which is defined on the mass shell and satisfies the Liouville equation. Suitable fibre integrals of the distribution function define observable fields on the spacetime manifold, such as the current density and stress-energy tensor. Finally, the geometric setting of this work allows us to discuss the relationship between the symmetries of the electromagnetic field, those of the spacetime metric, and the symmetries of the distribution function. Taking advantage of these symmetries, we construct the most general solution of the Liouville equation an a Kerr-Newman black hole background.

16 pages, 2 figures, prepared for the proceedings of the Fifth Leopoldo García-Colín Mexican Meeting on Mathematical and Experimental Physics, Mexico, September 2013

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