Fully relativistic derivation of the thermal Sunyaev-Zel'dovich effect

We present the first fully and inherently relativistic derivation of the thermal Sunyaev-Zel'dovich effect. This work uses the formalism historically applied to compute radiation spectra emerging from inverse Thomson/Compton sources of x-ray radiation. Comparing our results to the traditional approach based on relativistically-corrected classical Kompaneets equation, we find small, but systematic differences. Most notable are the modest ( ⩽ 10 % ) differences in the crossover frequency where the spectral distortion due to the Sunyaev-Zel'dovich effect vanishes, and the quadratic scaling of the energy shift with the electron cloud temperatures.