Geometric formulation for the relativistic kinetic theory of photons

Abstract

We provide a geometric foundation for the kinetic theory of photons. Although the induced metric $\hat{h}$ on the light cone bundle ${\Gamma }_0^{+}$ is degenerate, which causes the corresponding volume element to vanish, we can still use a method similar to the Hodge dual to construct a volume element ${\eta }_{{\Gamma }_0^{+}}$ for the light cone bundle. Based on this geometric structure $({\Gamma }_0^{+},{\eta }_{{\Gamma }_0^{+}},\hat{h})$, we establish the fully covariant Boltzmann equation for photons. More importantly, the volume element and the induced metric are linked in a nontrivial way, which allows the physical distributions to be defined consistently. This yields the corresponding hydrodynamic quantities and their divergences, which take the same form as in the case of massive particles.

23 pages