$^{\varphi}\eta^{m+1}=0$, $^{\varphi}\zeta^m=0$, $^{\varphi}\zeta^{m+1}=1$ but, except for these conditions, $^{\varphi}\eta,\,^{\varphi}\zeta$ remain arbitrary. For convenience, we shall also refer to such a pair of vector fields as a gauge. It is always possible to take the chart above defined such that $\,^{\varphi}\eta^{\mu}=\,^{\varphi}
\zeta^{\mu}=0$. So, in this chart, point of different submanifolds $\cM_{\lambda,\Omega}$ connected by the diffeomorphism $\varphi$ have the same $\cM$–coordinates $x^0,\dots,x^{m-1}$, and differ only by the value of the coordinates $\lambda,\,\Omega$. We call such a chart “adapted to the gauge $\varphi$”: this is what is always used in practice.

The perturbation in $T$ can now be defined simply as \^\_0 T\^\_[,]{} := .\^\*\_[,]{}T|\_[\_0]{} -T\_0. \[defperturbation\] The first term on the right–hand side of (\[defperturbation\]) can be Taylor–expanded using (\[Taylorgroup\]) to get \^\_0 T\^\_[,]{}=\_[k,k’=0]{}\^ \^[(k,k’)]{}\_T-T\_0, \[perturbationexpansion\] where \^[(k,k’)]{}\_T:=\_[=0,=0,\_0]{} = .\^k\_[\^]{} \^[k’]{}\_[\^]{}T|\_[\_0]{}, \[defperturbationk\] which defines the perturbation of order $(k,k')$ of $T$ (notice that $\delta^{(0,0)}_\varphi T=T_0$). It is worth noticing $\Delta_0 T^\varphi_{\lambda,\Omega}$ and $\delta^{(k,k')}_\varphi T$ are defined on $\cM_0$; this formalizes the statement one commonly finds in the literature that “perturbations are fields living in the background”. It is important to appreciate that the parameters $\lambda,\,\Omega$ labelling the various spacetime models also serve to perform the expansion (\[perturbationexpansion\]), and therefore determine what one 