  We present the first fully and inherently relativistic derivation of the
thermal Sunyaev-Zel'dovich effect. This work uses the formalism historically
used 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 energy increase of the distribution at high electron
cloud temperatures.
