# Can the perturbation of the metric be treated as an tensor to validate MakeRule?

I am doing variation of the Lagrangian, while it seems like the perturbation of the metric can not be identified as a tensor, such that MakeRule is invalid to apply towards the perturbation. the specific code is like

<<xActxPert
dim=4
DefManifold[M4,
dim, {\[Alpha], \[Beta], \[Gamma], \[Mu], \[Nu], \[Rho], \
\[Sigma], \[Tau], \[Kappa], \[Xi], \[Zeta], \[Delta]}]
DefMetric[-1, Met[-\[Mu], -\[Nu]],
CD, {";", "\[Del]"},
PrintAs -> "g"];
DefMetricPerturbation[Met, \[Delta]Met, \[Epsilon], PrintAs -> "h"];


I want to change the object that taking derivative with respect to,

LL = CD[-\[Alpha]][Perturbation[Met[-\[Gamma], -\[Mu]]]] CD[-\[Beta]][
Perturbation[Met[-\[Rho], -\[Delta]]]]


expect to becomes

rule1 = MakeRule[{CD[-\[Alpha]][
Perturbation[Met[-\[Beta], -\[Gamma]]]] CD[-\[Rho]][
Perturbation[Met[-\[Mu], -\[Sigma]]]], -Perturbation[
Met[-\[Beta], -\[Gamma]]] CD[-\[Alpha]][
CD[-\[Rho]][Perturbation[Met[-\[Mu], -\[Sigma]]]]]},
MetricOn -> All, ContractMetrics -> None]
LL/.rule1


but the rule1 can not be identified since the perturbation of the metric is not same to the tensor that yield by DefTensor

Is there any method to do such replacement? Thanks

I find a way to do it is by DefTensor[h1[-a,-b],M4,Symmetric[{-a,-b}]], define the perturbation as a tensor enable applying MakeRule towards the expression.