By employing the invariant four-dimensional representation of the photon-electron interaction, obtained from lowest order quantum electrodynamics, the Compton scattering kernel is easily found in any coordinate frame. This procedure provides a simple alternative to the usual Lorentz transformation of the scattering kernel (from electron rest frame to frame of interest) used in radiation-hydrodynamics computations and associated moving-media problems in transport theory. Furthermore, arbitrary distributions of electrons can be conveniently handled in this representation, and standard predictions for electrons initially at rest can be recovered easily.