Past exam of the mathematics course of the University of Cambridge / 2014 / iii / Paper 53 / 2 / i / Solution

The one-particle distribution function is a scalar function on the future mass shell. Choose its normalization so that, in a local orthonormal tetrad, $f{d}^{3}x{d}^{3}p$ is the expected photon number in a phase-space cell on a constant-time slice. Spin degeneracy and any phase-space normalization factors are included in $f$; with an occupation-number convention they would instead appear explicitly in the measure.

More covariantly, the photon number crossing a spacelike element is $f{p}^{\mu }d{\Sigma }_{\mu }{d}^{3}p/E$. The factor $p^{\mu }d{\Sigma }_{\mu }$ reduces to $E{d}^{3}x$ on a local constant-time slice, recovering the cell interpretation. The measure $d^{3}p/E$ is Lorentz invariant on the mass shell.

A photon carries energy $E$ and momentum $p^i$, while its velocity is $p^i/E$. Energy density, momentum density and momentum flux are thereforeThey assemble into the stress-energy tensorThe tetrad momenta are related to coordinate components by the tetrad basis. For an isotropic distribution, angular averaging gives $T^{\hat{i}\hat{j}}={\rho }_{\gamma }{\delta }^{ij}/3$ and $T^{\hat{0}\hat{i}}=0$. Thus the photon gas has pressure ${\rho }_{\gamma }/3$, consistent with its massless equation of state.