Dynamical versus auxiliary fields in gravitational waves around a black hole

The auxiliary/dynamic decoupling method of Kol (2006 Preprint hep-th/0609001) applies to perturbations of any co-homogeneity 1 background (such as a spherically symmetric spacetime or a homogeneous cosmology). Here it is applied to compute the perturbations around a Schwarzschild black hole in an arbitrary dimension. The method provides a clear insight for the existence of master equations. The computation is straightforward, coincides with previous results of Regge-Wheeler, Zerilli and Kodama-Ishibashi but does not require any ingenuity in either the definition of variables or in fixing the gauge. We note that the method's emergent master fields are canonically conjugate to the standard ones. In addition, our action approach yields the auxiliary sectors.