could choose the most generic case where the energy-momentum tensor of the scalar field shares the symmetries of the geometry and not the scalar field but, as a matter of simplicity, we will consider that the scalar field inherits the spacetime symmetries, which implies $\phi=\phi(r)$. We will not substitute in our calculations $C=r^2$ because various solutions in the literature can not be written in coordinate system where $C=r^2$, like the Fisher solution for a massless quintessence scalar field [@Fisher:1948yn].

Replacing the metric (\[eq:metricB\]) in the action (\[eq:action\]) and varying the action with respect to the functions A, B and C, we get three equations of motion which we will denote as ${\cal E}_A=0$, ${\cal E}_B=0$ and ${\cal E}_C=0$ respectively. Their complete expressions are given in the Appendix \[appenB\].

Perturbations
=============

Given the background equations of motion, we can now derive the equations of perturbations. These perturbations decompose into two types which separates at linear order, and therefore can be studied separately. One type of perturbations induce a rotation of the black hole while the second type impart no such rotation. They are called axial (or odd) and polar (or even) perturbations respectively. The metric can be decomposed as $g_{\mu \nu}=\bar 