Quit[]

Block[{Print},<<xAct`xTensor`;]
Block[{Print},<<xAct`xTras`]

Block[{Print},DefManifold[M,6,{a,b,c,d,e,g,h,i,j,k,l,m,n,o,p}]];
Block[{Print},DefMetric[-1,metric[-c,-d],CD,PrintAs->"g",CurvatureRelations->True]];
PrintAs[RiemannCD]^="R";
PrintAs[RicciCD]^="R";
PrintAs[RicciScalarCD]^="R";
PrintAs[ChristoffelCD]^="Γ";
PrintAs[EinsteinCD]^="G";
Block[{Print},DefScalarFunction [F];]
Block[{Print},DefTensor[fp[],M,PrintAs->"f'[R]"];]
F/:Perturbation[F[RicciScalarCD[]]]:=fp[]Perturbation[RicciScalarCD[]]

abseom=FullSimplification[]@VarL[metric[a,b],CD][F[RicciScalarCD[]]](*abstract filed equation*)

Block[{Print},DefChart[sp,M,{0,1,2,3,4,5},{t[],x1[],x2[],x3[],y[],θ[]}]]
Block[{Print},DefScalarFunction[{A,B}]]
Block[{Print},DefConstantSymbol[L]]
bg=CTensor[DiagonalMatrix[{-A[y[]]^2,A[y[]]^2,A[y[]]^2,A[y[]]^2,1,L^2 B[y[]]^2}],{-sp,-sp}];
invbg=Inv[bg];
cd=LC[bg];
ricciscalar=RicciScalar[cd];
riccicd=Ricci[cd];(*anstaz*)

rule1={
metric[-a_,-b_]:>bg[-a,-b],
metric[a_,b_]:>invbg[a,b],
RicciScalarCD[]:>ricciscalar[],
RicciCD[-a_,-b_]:>riccicd[-a,-b],
CD[-a_]:>cd[-a],
CD[a_]:>cd[a]
(*Christoffel[CD][a_,-b_,-c_]:>Christoffel[cd][a,-b,-c]*)
};
rule2={metric[{a_, sp}, {b_, sp}]:>ContractBasis[bg[-c,-d]Basis[{a,-sp},c]Basis[{b,-sp},d]]};(*convert abstract metric components into concret brackground metric*)
notation={F[RicciScalarCD[]]:>ff,fp[]:>ffr};

eom=abseom/.notation/.rule1/.rule2

equ1=FullSimplify[ContractBasis[eom Basis[{1,-sp},a]Basis[{1,-sp},b]]]
equ4=FullSimplify[ContractBasis[eom Basis[{4,-sp},a]Basis[{4,-sp},b]]]
equ5=FullSimplify[ContractBasis[eom Basis[{5,-sp},a]Basis[{5,-sp},b]]]