F_1 \over dt} \nonumber \\
H_0 & = &  {1 \over 3(r-2M)^2} F_1 +{r^3 \over 3M(r-2M)^2} {d^2 F_1
\over dt^2}  \nonumber \\
H_2 & = & {1 \over (r-2M)^2} F_1(t), \end{aligned}$$ where $F_1(t)$ is an arbitrary function. Remarkably, one can show that this solution is pure gauge. Choosing $f(t)=-F_1(t)/(6M)$ in (\[eqxi1\]) leads to vanishing gauge transformed quantities.

We therefore see that the $\ell=1$ perturbations are pure gauge. This result was first noticed by Zerilli [@zerilli]. This was to be expected in physical grounds since one could always imagine setting coordinates boosted in such a way that the black hole would not move. It has the further implication that we can compute the radiated energy by computing the $\ell=2$ perturbations in a gauge in which the $\ell=1$ perturbations vanish, and studying their evolution.

Multipolar decomposition of the initial data : the $\ell=2$ contributions
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The relevant $\ell=2$ perturbations are of second order in our perturbation parameter, $P$. In principle, when one works out second order perturbations of a given metric, the evolution equations one gets have the general form of a linear operator (similar to the one that acts in first order giving rise to the Zerilli equation) acting on second 