  Quasi-Normal Modes (QNM) or ringdown phase of gravitational waves provide
critical information about the structure of compact objects like Black Holes.
Thus, QNMs can be a tool to test General Relativity (GR) and possible
deviations from it. In the case of GR, it is known for a long time that a
relation between two types of Black Hole perturbations: scalar (Zerilli) and
vector (Regge-Wheeler), leads to an equal share of emitted gravitational
energy. With the direct detection of Gravitational waves, it is now natural to
ask: whether the same relation (between scalar and vector perturbations) holds
for modified gravity theories? If not, whether one can use this as a way to
probe deviations from General Relativity. As a first step, we show explicitly
that the above relation between Regge-Wheeler and Zerilli breaks down for a
general f (R) model, and hence the two perturbations do not share equal amounts
of emitted gravitational energy. We discuss the implication of this imbalance
on observations and the no-hair conjecture.
