Dispersion relations of relativistic radiation hydrodynamics

We compute the linearised dispersion relations of shear waves, heat waves, and sound waves in relativistic ''matter+radiation'' fluids with grey absorption opacities. This is done by solving radiation hydrodynamics perturbatively in the ratio ''radiation stress-energy''/''matter stress-energy''. The resulting expressions $\omega \, {=} \, \omega(k)$ accurately describe the hydrodynamic evolution for any $k\, {\in}\, \mathbb{R}$. General features of the dynamics (e.g., covariant stability, propagation speeds, and damping of discontinuities) are argued directly from the analytic form of these dispersion relations.