to that gauge. In other words, we need to carry out a first order gauge transformation on the initial data that provides a new initial data corresponding to a gauge where the $\ell=1$ perturbations vanish. Since all perturbations satisfy equations that are second order in time, this requires that the $\ell=1$ terms of the metric, and their first time derivative vanish on the initial slice. If we consider (\[gauge1\]), this requires that the gauge vector components ${\cal{M}}_i$ are such that both the left hand sides in (\[gauge1\]), and their first time derivatives, vanish when evaluated at $t=0$. This, in principle requires only the knowledge of the metric functions on the right hand side of (\[gauge1\]), and their first time derivative, at the fiducial time $t=0$. It turns out, however, that we also need second order time derivatives,

to implement the second order gauge transformation required to obtain the $\ell=2$ initial data. These second order time derivatives may be straighforwardly evaluated from the corresponding Einstein equations for the $\ell=1$ perturbations.

For the particular case in question, the $\ell=1$ initial data (and first time derivative) is determined by (\[Ksphe\]), and one can use this information to construct the space-time solution of the 