When doing perturbation using xPert, how to neglect higher order term of metric perturbation

Question

I am doing the perturbation of the metric tensor and its determinant, my code is like this, using the xPert in the xAct suite,

<< xAct`xPert`
DefManifold[M4,4,{α,β,γ,μ,ν,ρ,τ,κ}]
DefMetric[{3,1,0},g[-α,-β],{";","CD"}]
DefMetricPerturbation[g, δg, ε, PrintAs -> "h"];
Perturbed[Detg[], 2]
ExpandPerturbation[%]
Simplify[%]
SeparateMetric[][%]

then I get the result like however I want to keep the product term like but neglect the terms like 2nd $h^{2}_{\;κτ}$ of metric perturbation quantities in the expression, how to conveniently cancel them by some commands?

Answer 1

Perturbation Expansion Control: In xPert, when you expand the perturbations of a quantity like the determinant of the metric, it includes all perturbation orders up to the specified limit (in this case, 2nd order). However, if you only want the background field and first-order perturbations, the higher-order terms need to be removed.

Setting up a Rule to Nullify Higher-Order Terms: The rule

bgfield = h[LI[order_], __] :> 0 /; order > 1;

effectively removes perturbations of any order higher than one by setting them to zero.

Application of the Rule: Once the rule is defined, you can apply it to the perturbation result, which ensures that only the first-order terms remain. This is done by using the replacement rule /. bgfield.

Implementation:

(* Define the perturbation vanishing rule *) bgfield = h[LI[order_], __] :> 0 /; order > 1;

(* Apply it to the perturbed metric *) Perturbed[Detg[], 4] /. bgfield

This solution was extracted from https://josmar493.dreamhosters.com/xPert/xPert.nb