Exit

<<xAct`xPert`
<<xAct`xCoba`

$PrePrint=ScreenDollarIndices

Abstract part

DefManifold[M,4,IndexRange[a,n]]

DefMetric[-1,background[-a,-b],cd,PrintAs->"g"]

DefParameter[ϵ]

DefMetricPerturbation[background,pert,ϵ,PrintAs->"h"]

DefTensor[G1[-a,-b],M]

G1[-a,-b]=ToCanonical[Perturbed[Einsteincd[-a,-b],1]//ExpandPerturbation//SeparateMetric[background],UseMetricOnVBundle->None]

Component part

DefChart[ch,M,{0,1,2,3},{t[],r[],θ[],ϕ[]},ChartColor->Cyan]

DefConstantSymbol[eps,PrintAs->"ϵ"];
DefConstantSymbol[ome,PrintAs->"ω"];
DefScalarFunction[h0];
DefScalarFunction[h1];
DefScalarFunction[Y];
DefConstantSymbol[mass,PrintAs->"M"]

backgroundmetric=CTensor[{{-(1-2mass/r[]),0,0,0},{0,(1-2mass/r[])^-1,0,0},{0,0,r[]^2,0},{0,0,0,r[]^2 Sin[θ[]]^2}},{-ch,-ch}]

backgroundmetric[-a,-b]

MetricCompute[backgroundmetric,ch,All]

bgcd=CovDOfMetric[backgroundmetric]

(axperturbation=CTensor[{{0,0,(h0[t[],r[]]/Sin[θ[]])Derivative[0,0,0,1][Y][t[],r[],θ[],ϕ[]],-h0[t[],r[]]Sin[θ[]]Derivative[0,0,1,0][Y][t[],r[],θ[],ϕ[]]},{0,0,(h1[t[],r[]]/Sin[θ[]])Derivative[0,0,0,1][Y][t[],r[],θ[],ϕ[]],-h1[t[],r[]]Sin[θ[]]Derivative[0,0,1,0][Y][t[],r[],θ[],ϕ[]]},{(h0[t[],r[]]/Sin[θ[]])Derivative[0,0,0,1][Y][t[],r[],θ[],ϕ[]],(h1[t[],r[]]/Sin[θ[]])Derivative[0,0,0,1][Y][t[],r[],θ[],ϕ[]],0,0},{-h0[t[],r[]]Sin[θ[]]Derivative[0,0,1,0][Y][t[],r[],θ[],ϕ[]],-h1[t[],r[]]Sin[θ[]]Derivative[0,0,1,0][Y][t[],r[],θ[],ϕ[]],0,0}},{-ch,-ch}])[-a,-b]

We need to replace abstract indices with component values. This metric g is still different from the backgroundmetric used in the perturbation.

rules={
background[-a_Symbol,-b_Symbol]->backgroundmetric[-a,-b],
background[a_Symbol,b_Symbol]->Inv[backgroundmetric][a,b],
cd->bgcd,
Riccicd[inds__]->Ricci[bgcd][inds],
RicciScalarcd[]->RicciScalar[bgcd][],
Einsteincd[inds__]->Einstein[bgcd][inds],
pert[LI[1],inds__]->axperturbation[inds]
};

$CVSimplify=Expand;

res=G1[-a,-b]/.rules;

Collect[Head[res][[1]],ϵ,Simplify]