Relativistic kinetic theory

Relativistic kinetic theory describes particles by a scalar phase-space distribution function on the future mass shell. Its Boltzmann equation uses geodesic streaming and a collision operator. Integrating its momentum moments supplies a kinetic stress-energy tensor and particle-number currents. An orthonormal tetrad makes the local momentum integrals identical in form to those of special relativity.

A scalar function on the future mass shell describing the expected particle count in a phase-space distribution cell. In a local orthonormal tetrad, the number crossing a spacelike surface element is $f{p}^{\mu }d{\Sigma }_{\mu }{d}^{3}p/E$. On a constant-time slice it becomes $f{d}^{3}x{d}^{3}p$. Including degeneracy and phase-space normalization in $f$ gives the stress-energy tensor $T^{\mu \nu }=\int (d^{3}p/E)f{p}^{\mu }{p}^{\nu }$. Other conventions may put the degeneracy or $(2\pi {)}^{-3}$ explicitly in the measure.

The temporal-temporal component of the photon stress-energy tensor in a local orthonormal tetrad. For an isotropic one-particle distribution function expressed using comoving energy $ϵ=aE$, it is ${\bar{\rho }}_{\gamma }=4\pi a^{-4}\int dϵ{ϵ}^3\bar{f}(ϵ)$. A thermal photon temperature perturbation gives a first-order directional energy perturbation four times its fractional temperature change. Boundary conditions on the distribution justify this factor through integration by parts.

The invariant future mass-shell measure is $d^{3}p/E$, up to fixed normalization factors absorbed in $f$. Its relation to the Lorentz-invariant phase-space measure follows by integrating $2\theta (p^0)\delta (p_{\mu }{p}^{\mu }+m^2)d^{4}p$ over $p^0$ in a local orthonormal tetrad. A particle carries four-momentum $p^{\mu }$ and crosses a local spatial surface with velocity $p^i/p^0$. Integrating the associated momentum flux gives the displayed stress-energy tensor. For photons, $p^{\hat{a}}=E(1,\mathbf{e})$ and $|\mathbf{e}|=1$.