Perturbation Classification

  As is well known (see e.g. [7,15]), a general symmetric tensor such as the metric and stress-energy perturbations can be separated into scalar, vector and tensor pieces through their coordinate transformation properties. We now review the properties of their normal modes so that they may be related to those of the radiation. We find that the $m=0,\pm 1,\pm 2$ modes of the radiation couple to the scalar, vector and tensor modes of the metric. Although we consider flat geometries here, we preserve a covariant notation that ensures straightforward generalization to open geometries through the replacement of $\delta_{ij}$ with the curved three metric and ordinary derivatives with covariant derivatives [6,7].