Einstein equations produced by the initial data as a Taylor expansion in $t$, up to the appropriate order. One gets, using the usual Regge-Wheeler notation [@ReWh] the following expansions for the $\ell=1$ components of the metric,

$$\begin{aligned}
H_2 &=& -{3\over 2} {(\sqrt{r}+\sqrt{r-2M})^2 \sqrt{r-2M}\over r^{7/2}} P t
+O(t^3)\\
K&=&{3\over 4} {(\sqrt{r}+\sqrt{r-2M})^2 \sqrt{r-2M}\over r^{7/2}} P t
+O(t^3)\\
h_1 &=& 
-{3\over 4} {(\sqrt{r}+\sqrt{r-2M})^2 \over r^{2}} P t
+O(t^3)\end{aligned}$$

all the other components being $O(t^3)$.

The components of the gauge vector generating the first order gauge transformation that makes the initial $\ell=1$ data purely $O(t^3)$ are, $$\begin{aligned}
\xi_t &=& {\cal{M}}_0 \cos\theta\\
\xi_r &=& {\cal{M}}_1 \cos\theta\\
\xi_\theta &=& -{\cal{M}}_2 \sin\theta\\
\xi_t &=& 0,\end{aligned}$$ where, $$\begin{aligned}
{\cal{M}}_0 &=&
-{\frac {P\left (-3\,\sqrt {r-2\,M}rM-5\,{r}^{3/2}M+2\,{M}^{2}\sqrt {r}
+2\,{r}^{5/2}+2\,\sqrt {r-2\,M}{r}^{2}\right )}{4\,\sqrt {r}M\left (r-2\,
M\right )}}\\
{\cal{M}}_1 &=& 
-{\frac {P\left (-6\,\sqrt {r-2\,M}rM+2\,{r}^{5/2}-8\,{r}^{3/2}M+8\,{M}^{2}
\sqrt {r}+3\,\sqrt {r-2\,M}{M}^{2}+2\,\sqrt {r-2\,M}{r}^{2}\right )t}{4\,
{r}^{5/2}M}}\\
{\cal{M}}_2 &=&
-{\frac {P\left (-3\,\sqrt {r-2\,M}rM-5\,{r}^{3/2}M+2\,{M}^{2}\sqrt {r}
+2\,{r}^{5/2}+2\,\sqrt {r-2\,M}{r}^{2}\right )t}{4\,{r}^{7/2}M}}.\end{aligned}$$

Performing a first order gauge transformation with this generator, one eliminates the first order $\ell=1$ component of the metric. Therefore the leading terms in the initial data become second order. The latter have two contributions, both of $\ell=2$ multipolar order. One contribution simply comes from the expansion to second order of the initial data 