Gauge invariance and the cosmic microwave background radiation temperature

In recent years the Sachs-Wolfe anisotropy of the cosmic microwave radiation has been carefully calculated using gauge-invariant formalisms, like that of Bardeen (1980). We carefully reexamine the most detailed of these, that by Panek (1986), particularly the underlying gauge-invariance properties of the calculation, the gauge choice of the observed background temperature, the definition of the last-scattering surface, and the key steps of the analysis. We also treat other somewhat neglected aspects of the usual calculations -- the directional differencing which is an important feature of the anisotropy observations, the limitations of scalar-vector-tensor decomposition, and the compensation effect. Our work essentially confirms the result Panek gives in the adiabatic case, but suggests certain improvements and corrections of subtle details to the way the calculation is actually performed and interpreted. We also extend the calculation to the nonadiabatic case in an explicitly gauge-invariant way.