“Schwarzschild” radial coordinates $r$, with $R=(\sqrt{r}+\sqrt{r-2M})^2/4$, where $M = 2 a$, and choose $N=\sqrt{1-2M/r}$. It can be checked that with these choices, for $P=0$ the extrinsic curvature vanishes and we recover the Schwarzschild metric in the usual Schwarzschild coordinates. This last expression for the metric in $(r,\theta,\phi)$ coordinates (or, rather, the initial data) has, therefore, the appropriate form for a perturbative treatment of the Regge - Wheeler type, as extended to second order in [@physrep].

The final form for the initial data that results from this construction has thefollowing multipolar components: to zeroth order in the momentum, we only have $\ell=0$ components; to first order in the linear momentum,only $\ell=1$ contributions; to order ${\cal O}(P^2)$, we have $\ell=0,2$ contributions. All contributions are even-parity and are analyzed in more detail in the following Sections.

Multipolar decomposition of the initial data : the $\ell=1$ contributions
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The contribution to the three-metric of zeroth order in $P$ is just the Schwarzschild solution. The first apparently non-trivial contribution is given at ${\cal O}(P^1)$ and corresponds as we discussed in the previous subsection to an $\ell=1$ multipole. Let us analyze the $\ell=1$ perturbations of a spherically symmetric spacetime. In order to do this we use 