Perturbation Theory from Curvature Wave Equations in Spherical Symmetry

Quit[]

Fundamental Equations and Examples

<<xAct`xTensor`
<<xAct`xPert`
<<xAct`xPerm`
<<xAct`TexAct`

0. Definitions, main functions and background relations [RUN]

0.0. Manifolds, metrics and some convenient functions

We introduce the total manifold as a warped product of a pair of 2-dimensional manifolds:

DefManifold[M1,2,{a,b,c,d,e,f,i,j,k,v}];
DefManifold[M2,2,{A,B,F,G,H,J,L,M,P,Q,S,T}];
DefManifold[M3,{M1,M2},{μ,ν,ρ,σ,η,α,β,γ,χ}];
M1Indices={a,b,c,d,e,f,i,j,k,v};
M2Indices={A,B,F,G,H,J,L,M,P,Q,S,T};
M3Indices={μ,ν,ρ,σ,η,α,β,γ,χ};

We introduce the background data for both the metric and the energy-momentum tensor:

DefMetric[-1,g1[-a,-b],CD1,{":","▽"},PrintAs ->"g"];
DefMetric[1,g2[-A,-B],CD2,{"|","D"},PrintAs ->"Ω" ];
DefTensor[r[],M1];
DefTensor[𝓉[-a],M1];
DefTensor[Τ[-a,-b],M1,Symmetric[{-a,-b}]];
DefTensor[𝒫[],M1];
DefProductMetric[g3[-μ,-ν],{{TangentM1,1},{TangentM2,r[]}},CD3,{";","𝒟"},PrintAs -> "(g^((4)))"];

Some convenient redefinitions of functions:

SkewSymetrise[expr_,Indices_]:=xAct`xTensor`Antisymmetrize[expr,Indices]
Symetrise[expr_,Indices_]:=xAct`xTensor`Symmetrize[expr,Indices]

0.1. Special Relations of Background 2D Curvature

The Riemann tensor of a 2D manifold is entirely determined by its symmetries, and the only degree of freedom is its Ricci scalar:

RieN$2D=MakeRule[{RiemannCD1[-a,-b,-c,-d],RicciScalarCD1[](g1[-a,-c]g1[-d,-b]-g1[-a,-d]g1[-c,-b])/2}];
RieS$2D=MakeRule[{RiemannCD2[-A,-B,-F,-G],g2[-A,-F]g2[-B,-G]-g2[-A,-G]g2[-B,-F]}];
RicN$2D=MakeRule[{RicciCD1[-a,-b],((RicciScalarCD1[])/2)g1[-a,-b]}];
RicS$2D=MakeRule[{RicciCD2[-A,-B],g2[-A,-B]}];
RicScalS$2D=MakeRule[{RicciScalarCD2[],2}];
CurvS$2D=Flatten[{RieS$2D,RicS$2D,RicScalS$2D}];
CurvN$2D=Flatten[{RieN$2D,RicN$2D}];

0.2. Volume-Form Expansion Rules

xAct built-in functions do not seem to include the total volume form in terms of the 2D volume forms. Write this replacements here, always with indices down:

Vol1=MakeRule[{epsilong3[-a,-b,-c,-d],0}];
Vol21=MakeRule[{epsilong3[-a,-b,-c,-F],0}];
Vol22=MakeRule[{epsilong3[-a,-b,-F,-c],0}];
Vol23=MakeRule[{epsilong3[-a,-F,-b,-c],0}];
Vol24=MakeRule[{epsilong3[-F,-a,-b,-c],0}];
Vol31=MakeRule[{epsilong3[-a,-A,-B,-F],0}];
Vol32=MakeRule[{epsilong3[-A,-a,-B,-F],0}];
Vol33=MakeRule[{epsilong3[-A,-B,-a,-F],0}];
Vol34=MakeRule[{epsilong3[-A,-B,-F,-a],0}];
Vol4=MakeRule[{epsilong3[-A,-B,-F,-G],0}];
Vol51=MakeRule[{epsilong3[-a,-b,-A,-B],((r[])^2)epsilong1[-a,-b]epsilong2[-A,-B]}];
Vol52=MakeRule[{epsilong3[-A,-B,-a,-b],((r[])^2)epsilong1[-a,-b]epsilong2[-A,-B]}];
Vol53=MakeRule[{epsilong3[-a,-A,-b,-B],-((r[])^2)epsilong1[-a,-b]epsilong2[-A,-B]}];
Vol54=MakeRule[{epsilong3[-A,-a,-B,-b],-((r[])^2)epsilong1[-a,-b]epsilong2[-A,-B]}];
Vol55=MakeRule[{epsilong3[-a,-A,-B,-b],((r[])^2)epsilong1[-a,-b]epsilong2[-A,-B]}];
Vol56=MakeRule[{epsilong3[-A,-a,-b,-B],((r[])^2)epsilong1[-a,-b]epsilong2[-A,-B]}];
Vol$Rep=Flatten[{Vol1,Vol21,Vol22,Vol23,Vol24,Vol31,Vol32,Vol33,Vol34,Vol4,Vol51,Vol52,Vol53,Vol54,Vol55,Vol56}];

0.3. Spherical Harmonics and their equations

Definitions

We first define the harmonic tensors abstractly as tensors on M2:

DefTensor[Y[],M2,PrintAs->"𝕐",Dagger->Complex];
DefTensor[Z[-A],M2,PrintAs->"ℤ",Dagger->Complex];
DefTensor[X[-A],M2,PrintAs->"𝕏",Dagger->Complex];
DefTensor[U[-A,-B],M2,Symmetric[{-A,-B}],PrintAs->"𝕌",Dagger->Complex];
DefTensor[V[-A,-B],M2,Symmetric[{-A,-B}],PrintAs->"𝕍",Dagger->Complex];
DefTensor[W[-A,-B],M2,Symmetric[{-A,-B}],PrintAs->"𝕎",Dagger->Complex];
DefConstantSymbol[Λ];
Harms$Tot={Y[],Z[-A],X[-A],U[-A,-B],V[-A,-B],W[-A,-B]};

We introduce their definition in terms of the usual, complex spherical harmonics:

Dir$Harm$Def$Z=MakeRule[{Z[-B],CD2[-B][Y[]]},MetricOn->All];
Dir$Harm$Def$X=MakeRule[{X[-B],epsilong2[-B,A]CD2[-A][Y[]]},MetricOn->All];
Dir$Harm$Def$U=MakeRule[{U[-A,-B],g2[-A,-B] Y[]},MetricOn->All];
Dir$Harm$Def$V=MakeRule[{V[-A,-B],CD2[-A][CD2[-B][Y[]]]+((Λ^2)Y[]g2[-A,-B])/2},MetricOn->All];
Dir$Harm$Def$W=MakeRule[{W[-A,-B],(CD2[-A][epsilong2[-B,F]CD2[-F][Y[]]]+CD2[-B][epsilong2[-A,F]CD2[-F][Y[]]])/2},MetricOn->All];
Dir$Harm$Def$Total=Flatten[{Dir$Harm$Def$Z,Dir$Harm$Def$X,Dir$Harm$Def$U,Dir$Harm$Def$V,Dir$Harm$Def$W}];

Harmonic derivative reductions

The fundamental equations in this approach to perturbation theory contain at most 4th order derivatives of Y[], and have at most 2 free spherical indices (other than volume form ones).
Once an equation is entirely expressed in terms of Y[] and its (up to 4) derivatives, we use the harmonic equations of motion to obtain the components. We implement such equations here up to 4th order.

(*second derivative contractions*)
Harm$1$1=MakeRule[{CD2[-A][CD2[A][Y[]]],-(Λ^2)Y[]},MetricOn->All];
Harm$1$2=MakeRule[{epsilong2[A,B]CD2[-A][CD2[-B][Y[]]],0},MetricOn->All];
(*The following function is useful in deriving the higher-order reductions given below*)
Harm$Derivative$Canonical$Reduction[expr_]:=(SortCovDs[expr,CD2]//ContractMetric//ToCanonical)//.Flatten[{CurvS$2D,Harm$1$1}]//ContractMetric//ToCanonical

(*third derivative contractions*)
Harm$2$1=MakeRule[{CD2[A][CD2[-A][CD2[-F][Y[]]]],CD2[-F][Y[]]-Λ^2*CD2[-F][Y[]]},MetricOn->All];
Harm$2$2=MakeRule[{CD2[A][CD2[-F][CD2[-A][Y[]]]],CD2[-F][Y[]]-Λ^2*CD2[-F][Y[]]},MetricOn->All];
Harm$2$3=MakeRule[{CD2[-F][CD2[A][CD2[-A][Y[]]]],-(Λ^2*CD2[-F][Y[]])},MetricOn->All];
Harm$2$4=MakeRule[{epsilong2[A,B]CD2[-A][CD2[-B][CD2[-F][Y[]]]],epsilong2[-F,-A]*CD2[A][Y[]]},MetricOn->All];
Harm$2$5=MakeRule[{epsilong2[A,B]CD2[-A][CD2[-F][CD2[-B][Y[]]]],epsilong2[-F,-A]*CD2[A][Y[]]},MetricOn->All];
Harm$2$6=MakeRule[{epsilong2[A,B]CD2[-F][CD2[-A][CD2[-B][Y[]]]],0},MetricOn->All];
(*fourth derivative contractions*)
(*We only consider the case with two indices free, the full contractions follow by repeated use these and lower-order ones*)
(*contraction with an epsilon*)
Harm$3$1=MakeRule[{epsilong2[A,B]CD2[-A][CD2[-B][CD2[-P][CD2[-Q][Y[]]]]],epsilong2[-Q,A]*CD2[-P][CD2[-A][Y[]]]+epsilong2[-P,A]*CD2[-Q][CD2[-A][Y[]]]},MetricOn->All];
Harm$3$2=MakeRule[{epsilong2[A,B]CD2[-A][CD2[-P][CD2[-B][CD2[-Q][Y[]]]]],2*epsilong2[-Q,A]*CD2[-P][CD2[-A][Y[]]]+epsilong2[-P,A]*CD2[-Q][CD2[-A][Y[]]]},MetricOn->All];
Harm$3$3=MakeRule[{epsilong2[A,B]CD2[-A][CD2[-P][CD2[-Q][CD2[-B][Y[]]]]],2*epsilong2[-Q,A]*CD2[-P][CD2[-A][Y[]]]+epsilong2[-P,A]*CD2[-Q][CD2[-A][Y[]]]},MetricOn->All];
Harm$3$4=MakeRule[{epsilong2[P,Q]CD2[-A][CD2[-B][CD2[-P][CD2[-Q][Y[]]]]],0},MetricOn->All];
Harm$3$5=MakeRule[{epsilong2[A,B]CD2[-P][CD2[-A][CD2[-Q][CD2[-B][Y[]]]]],epsilong2[-Q,B]*CD2[-P][CD2[-B][Y[]]]},MetricOn->All];
Harm$3$6=MakeRule[{epsilong2[A,B]CD2[-P][CD2[-A][CD2[-B][CD2[-Q][Y[]]]]],epsilong2[-Q,B]*CD2[-P][CD2[-B][Y[]]]},MetricOn->All];
(*contraction with a metric*)
Harm$3$7=MakeRule[{CD2[-A][CD2[A][CD2[-P][CD2[-Q][Y[]]]]],2*Λ^2*g2[-P,-Q]*Y[]+4*CD2[-Q][CD2[-P][Y[]]]-Λ^2*CD2[-Q][CD2[-P][Y[]]]},MetricOn->All];
Harm$3$8=MakeRule[{CD2[-A][CD2[-P][CD2[A][CD2[-Q][Y[]]]]],Λ^2*g2[-P,-Q]*Y[]+3*CD2[-Q][CD2[-P][Y[]]]-Λ^2*CD2[-Q][CD2[-P][Y[]]]},MetricOn->All];
Harm$3$9=MakeRule[{CD2[-A][CD2[-P][CD2[-Q][CD2[A][Y[]]]]],Λ^2*g2[-P,-Q]*Y[]+3*CD2[-Q][CD2[-P][Y[]]]-Λ^2*CD2[-Q][CD2[-P][Y[]]]},MetricOn->All];
Harm$3$10=MakeRule[{CD2[-A][CD2[-B][CD2[P][CD2[-P][Y[]]]]],-2*Λ^2*g2[-A,-B]*Y[]-4*CD2[-B][CD2[-A][Y[]]]+CD2[-P][CD2[P][CD2[-B][CD2[-A][Y[]]]]]},MetricOn->All];
Harm$3$11=MakeRule[{CD2[-P][CD2[-A][CD2[-Q][CD2[A][Y[]]]]],CD2[-Q][CD2[-P][Y[]]]-Λ^2*CD2[-Q][CD2[-P][Y[]]]},MetricOn->All];
Harm$3$12=MakeRule[{CD2[-P][CD2[-A][CD2[A][CD2[-Q][Y[]]]]],CD2[-Q][CD2[-P][Y[]]]-Λ^2*CD2[-Q][CD2[-P][Y[]]]},MetricOn->All];
(*Full contraction with metrics*)
Harm$3$13=MakeRule[{CD2[-A][CD2[A][CD2[-P][CD2[P][Y[]]]]],Λ^4*Y[]},MetricOn->All];
Harm$3$14=MakeRule[{CD2[-A][CD2[-P][CD2[A][CD2[P][Y[]]]]],-(Λ^2*Y[])+Λ^4*Y[]},MetricOn->All];
Harm$3$15=MakeRule[{CD2[-A][CD2[P][CD2[-P][CD2[A][Y[]]]]],-(Λ^2*Y[])+Λ^4*Y[]},MetricOn->All];
(*Full contraction one metric and one epsilon epsilons*)
Harm$3$16=MakeRule[{epsilong2[P,Q]CD2[-A][CD2[A][CD2[-P][CD2[-Q][Y[]]]]],0},MetricOn->All];
Harm$3$17=MakeRule[{epsilong2[P,Q]CD2[-A][CD2[-P][CD2[A][CD2[-Q][Y[]]]]],0},MetricOn->All];
Harm$3$18=MakeRule[{epsilong2[P,Q]CD2[-A][CD2[-P][CD2[-Q][CD2[A][Y[]]]]],0},MetricOn->All];
Harm$3$19=MakeRule[{epsilong2[P,Q]CD2[-P][CD2[A][CD2[-A][CD2[-Q][Y[]]]]],0},MetricOn->All];
Harm$3$20=MakeRule[{epsilong2[P,Q]CD2[-P][CD2[A][CD2[-Q][CD2[-A][Y[]]]]],0},MetricOn->All];
Harm$3$21=MakeRule[{epsilong2[P,Q]CD2[-P][CD2[-Q][CD2[-A][CD2[A][Y[]]]]],0},MetricOn->All];

Harm$EQ$Y$Total=Flatten[{Harm$1$1,Harm$1$2,Harm$2$1,Harm$2$2,Harm$2$3,Harm$2$4,Harm$2$5,Harm$2$6,Harm$3$1,Harm$3$2,Harm$3$3,Harm$3$4,Harm$3$5,Harm$3$6,Harm$3$7,Harm$3$8,Harm$3$9,Harm$3$10,Harm$3$11,Harm$3$12,Harm$3$13,Harm$3$14,Harm$3$15,Harm$3$16,Harm$3$17,Harm$3$18,Harm$3$19,Harm$3$20,Harm$3$21}];

0.4. Null frames

4-d null frames

We first introduce the 4d null frames, working in the sign conventions of the Stephani’s book:

DefTensor[ℓ[-μ],M3,PrintAs->"l"]
DefTensor[𝓃[-μ],M3,PrintAs->"n"]
DefTensor[𝓂[-μ],M3,PrintAs->"m"]
DefTensor[𝓂C[-μ],M3,PrintAs->"(OverBar[m])"]

DefTensor[dℓ[-μ],M3,PrintAs->"(OverDot[l])"]
DefTensor[d𝓃[-μ],M3,PrintAs->"(OverDot[n])"]
DefTensor[d𝓂[-μ],M3,PrintAs->"(OverDot[m])"]
DefTensor[d𝓂C[-μ],M3,PrintAs->"(OverDot[(OverBar[m])])"]

Pert$Frame=Flatten[{MakeRule[{Perturbation[ℓ[-μ]],dℓ[-μ]}],MakeRule[{Perturbation[𝓃[-μ]],d𝓃[-μ]}],MakeRule[{Perturbation[𝓂[-μ]],d𝓂[-μ]}],MakeRule[{Perturbation[𝓂C[-μ]],d𝓂C[-μ]}]}];

(**)

No$Pert$Frame=Flatten[{MakeRule[{Perturbation[ℓ[-μ]],0}],MakeRule[{Perturbation[ℓ[μ]],0}],MakeRule[{Perturbation[𝓃[-μ]],0}],MakeRule[{Perturbation[𝓃[μ]],0}],MakeRule[{Perturbation[𝓂[-μ]],0}],MakeRule[{Perturbation[𝓂[μ]],0}],MakeRule[{Perturbation[𝓂C[-μ]],0}],MakeRule[{Perturbation[𝓂C[μ]],0}],MakeRule[{Perturbation[ℓ[-a]],0}],MakeRule[{Perturbation[ℓ[a]],0}],MakeRule[{Perturbation[𝓃[-a]],0}],MakeRule[{Perturbation[𝓃[a]],0}],MakeRule[{Perturbation[𝓂[-a]],0}],MakeRule[{Perturbation[𝓂[a]],0}],MakeRule[{Perturbation[𝓂C[-a]],0}],MakeRule[{Perturbation[𝓂C[a]],0}],MakeRule[{Perturbation[ℓ[-A]],0}],MakeRule[{Perturbation[ℓ[A]],0}],MakeRule[{Perturbation[𝓃[-A]],0}],MakeRule[{Perturbation[𝓃[A]],0}],MakeRule[{Perturbation[𝓂[-A]],0}],MakeRule[{Perturbation[𝓂[A]],0}],MakeRule[{Perturbation[𝓂C[-A]],0}],MakeRule[{Perturbation[𝓂C[A]],0}]}];

ℓℓ$Rule=MakeRule[{ℓ[-μ]ℓ[μ],0}];
ℓ𝓃$Rule=MakeRule[{ℓ[-μ]𝓃[μ],-1}];
ℓ𝓂$Rule=MakeRule[{ℓ[-μ]𝓂[μ],0}];
ℓ𝓂C$Rule=MakeRule[{ℓ[-μ]𝓂C[μ],0}];
𝓃𝓃$Rule=MakeRule[{𝓃[-μ]𝓃[μ],0}];
𝓃𝓂$Rule=MakeRule[{𝓃[-μ]𝓂[μ],0}];
𝓃𝓂C$Rule=MakeRule[{𝓃[-μ]𝓂C[μ],0}];
𝓂𝓂$Rule=MakeRule[{𝓂[-μ]𝓂[μ],0}];
𝓂𝓂C$Rule=MakeRule[{𝓂[-μ]𝓂C[μ],1}];
𝓂C𝓂C$Rule=MakeRule[{𝓂C[-μ]𝓂C[μ],0}];
AutomaticRules[ℓ,Flatten[{ℓℓ$Rule,ℓ𝓃$Rule,ℓ𝓂$Rule,ℓ𝓂C$Rule}]];
AutomaticRules[𝓃,Flatten[{𝓃𝓃$Rule,ℓ𝓃$Rule,𝓃𝓂$Rule,𝓃𝓂C$Rule}]];
AutomaticRules[𝓂,Flatten[{𝓂𝓂$Rule,ℓ𝓂$Rule,𝓃𝓂$Rule,𝓂𝓂C$Rule}]];
AutomaticRules[𝓂C,Flatten[{𝓂C𝓂C$Rule,ℓ𝓂C$Rule,𝓃𝓂C$Rule,𝓂𝓂C$Rule}]];

Vol$To$Frame=MakeRule[{epsilong3[-μ,-ν,-ρ,-σ],I*(ℓ[-σ]*𝓂[-ρ]*𝓂C[-ν]*𝓃[-μ]-ℓ[-ρ]*𝓂[-σ]*𝓂C[-ν]*𝓃[-μ]-ℓ[-σ]*𝓂[-ν]*𝓂C[-ρ]*𝓃[-μ]+ℓ[-ν]*𝓂[-σ]*𝓂C[-ρ]*𝓃[-μ]+ℓ[-ρ]*𝓂[-ν]*𝓂C[-σ]*𝓃[-μ]-ℓ[-ν]*𝓂[-ρ]*𝓂C[-σ]*𝓃[-μ]-ℓ[-σ]*𝓂[-ρ]*𝓂C[-μ]*𝓃[-ν]+ℓ[-ρ]*𝓂[-σ]*𝓂C[-μ]*𝓃[-ν]+ℓ[-σ]*𝓂[-μ]*𝓂C[-ρ]*𝓃[-ν]-ℓ[-μ]*𝓂[-σ]*𝓂C[-ρ]*𝓃[-ν]-ℓ[-ρ]*𝓂[-μ]*𝓂C[-σ]*𝓃[-ν]+ℓ[-μ]*𝓂[-ρ]*𝓂C[-σ]*𝓃[-ν]+ℓ[-σ]*𝓂[-ν]*𝓂C[-μ]*𝓃[-ρ]-ℓ[-ν]*𝓂[-σ]*𝓂C[-μ]*𝓃[-ρ]-ℓ[-σ]*𝓂[-μ]*𝓂C[-ν]*𝓃[-ρ]+ℓ[-μ]*𝓂[-σ]*𝓂C[-ν]*𝓃[-ρ]+ℓ[-ν]*𝓂[-μ]*𝓂C[-σ]*𝓃[-ρ]-ℓ[-μ]*𝓂[-ν]*𝓂C[-σ]*𝓃[-ρ]-ℓ[-ρ]*𝓂[-ν]*𝓂C[-μ]*𝓃[-σ]+ℓ[-ν]*𝓂[-ρ]*𝓂C[-μ]*𝓃[-σ]+ℓ[-ρ]*𝓂[-μ]*𝓂C[-ν]*𝓃[-σ]-ℓ[-μ]*𝓂[-ρ]*𝓂C[-ν]*𝓃[-σ]-ℓ[-ν]*𝓂[-μ]*𝓂C[-ρ]*𝓃[-σ]+ℓ[-μ]*𝓂[-ν]*𝓂C[-ρ]*𝓃[-σ])}];
(*Vol$To$Frame=I Factorial[4]SkewSymetrise[ℓ[-μ]𝓃[-ν]𝓂[-ρ]𝓂C[-σ],{-μ,-ν,-ρ,-σ}]*)

2-d null frames

DefTensor[l[-a],M1,PrintAs->"l"]
DefTensor[n[-a],M1,PrintAs->"n"]

ll$Rule=MakeRule[{l[-a]l[a],0},MetricOn->All];
ln$Rule=MakeRule[{l[-a]n[a],-1},MetricOn->All];
nn$Rule=MakeRule[{n[-a]n[a],0},MetricOn->All];
laϵab$Rule=MakeRule[{l[a]epsilong1[-a,-b],l[-b]},MetricOn->All];
naϵab$Rule=MakeRule[{n[a]epsilong1[-a,-b],-n[-b]},MetricOn->All];
AutomaticRules[l,Flatten[{ll$Rule,ln$Rule,laϵab$Rule}]];
AutomaticRules[n,Flatten[{nn$Rule,ln$Rule,naϵab$Rule}]];

DefTensor[m[-A],M2,PrintAs->"m"]
DefTensor[mC[-A],M2,PrintAs->"(OverBar[m])"]

mm$Rule=MakeRule[{m[-A]m[A],0},MetricOn->All];
mmC$Rule=MakeRule[{m[-A]mC[A],2},MetricOn->All];(*Notice unconventional normalisation, to follow the convetions of [2306.17847]*)
mCmC$Rule=MakeRule[{mC[-A]mC[A],0},MetricOn->All];
mAϵAB$Rule=MakeRule[{m[A]epsilong2[-A,-B],-I m[-B]},MetricOn->All];
mCAϵAB$Rule=MakeRule[{mC[A]epsilong2[-A,-B],I mC[-B]},MetricOn->All];
AutomaticRules[m,Flatten[{mm$Rule,mmC$Rule,mAϵAB$Rule}]];
AutomaticRules[mC,Flatten[{mCmC$Rule,mmC$Rule,mCAϵAB$Rule}]];

FourD$To$TwoD$l=Flatten[{MakeRule[{ℓ[-a],l[-a]}],MakeRule[{ℓ[-A],0}]}];
FourD$To$TwoD$n=Flatten[{MakeRule[{𝓃[-a],n[-a]}],MakeRule[{𝓃[-A],0}]}];
(*Notice unconventional normalisation, to follow the convetions of [2306.17847]*)
FourD$To$TwoD$m=Flatten[{MakeRule[{𝓂[-a],0}],MakeRule[{𝓂[-A],((r[])/(Sqrt[2]))m[-A]}]}];
FourD$To$TwoD$mC=Flatten[{MakeRule[{𝓂C[-a],0}],MakeRule[{𝓂C[-A],((r[])/(Sqrt[2]))mC[-A]}]}];
FourD$To$TwoD=Flatten[{FourD$To$TwoD$l,FourD$To$TwoD$n,FourD$To$TwoD$m,FourD$To$TwoD$mC}];

2-d GHP expansion of frame and warp-factor derivatives

DefTensor[ϵ$sc[],M1,PrintAs->"ϵ"]
DefTensor[ϵp$sc[],M1,PrintAs->"ϵ'"]
DefTensor[ρ$sc[],M1,PrintAs->"ρ"]
DefTensor[ρp$sc[],M1,PrintAs->"ρ'"]

Dl$ToFrame=MakeRule[{CD1[-a][l[-b]],2ϵp$sc[]l[-a]l[-b]-2ϵ$sc[]n[-a]l[-b]},MetricOn->All]
Dn$ToFrame=MakeRule[{CD1[-a][n[-b]],2ϵ$sc[]n[-a]n[-b]-2ϵp$sc[]l[-a]n[-b]},MetricOn->All]
Dr$ToFrame=MakeRule[{CD1[-a][r[]],r[](ρ$sc[] n[-a]+ρp$sc[]l[-a])},MetricOn->All]

ρ$sc$To$Dr=MakeRule[{ρ$sc[],-((1/(r[])))l[a]CD1[-a][r[]]}]
ρp$sc$To$Dr=MakeRule[{ρp$sc[],-((1/(r[])))n[a]CD1[-a][r[]]}]

CD1[-a][n[a]]/.Dn$ToFrame

DefTensor[ω$a[-a],M1,PrintAs->"ω"]

ω$a$ToFrame=MakeRule[{ω$a[-a],2(-ϵ$sc[]n[-a]+ϵp$sc[]l[-a])},MetricOn->All];

Relation to spin-weighted spherical harmonics

Following conventions of [2306.17847]:

DefTensor[m1Y[],M2,PrintAs->"(Subscript[Y, -1])"]
DefTensor[p1Y[],M2,PrintAs->"(Subscript[Y, 1])"]
DefTensor[m2Y[],M2,PrintAs->"(Subscript[Y, -2])"]
DefTensor[p2Y[],M2,PrintAs->"(Subscript[Y, 2])"]

DefConstantSymbol[LL,PrintAs->"L"]

Λ$To$L=MakeRule[{Λ,(Sqrt[LL(LL+1)])}]

DY$To$sY=MakeRule[{CD2[-A][Y[]],(Λ/2)(m1Y[]m[-A]-p1Y[]mC[-A])},MetricOn->All];
DDY$To$sY=MakeRule[{CD2[-A][CD2[-B][Y[]]],((Λ(Sqrt[(LL+2)(LL-1)]))/4)(m2Y[]m[-A]m[-B]+p2Y[]mC[-A]mC[-B])-((Λ^2)/2)Y[]g2[-A,-B]},MetricOn->All];
Y$To$sY=Flatten[{DY$To$sY,DDY$To$sY}];

0.5. Minimal Background Substitution Rules and Background Harmonic Projectors

First it might be useful to define the dual of (Subscript[r, a]) a

Dr$To$t=MakeRule[{CD1[-a][r[]],-epsilong1[-a,-b]𝓉[b]}]
t$To$Dr=MakeRule[{𝓉[-a],-epsilong1[-a,-b]CD1[b][r[]]}]

We first introduce the energy-momentum tensor (Subscript[𝒯, μν]):

DefTensor[𝒯[-μ,-ν],M3,Symmetric[{-μ,-ν}]];

And define its background harmonic components:

BG$𝒯=Flatten[{MakeRule[{𝒯[-a,-b],Τ[-a,-b]}],MakeRule[{𝒯[-a,-A],0}],MakeRule[{𝒯[-A,-a],0}],MakeRule[{𝒯[-A,-B],((r[])^2)𝒫[]g2[-A,-B]}]}];

Our convention for imposing the on-shell condition consists of replacing second derivatives of r by first derivatives only + the background energy-momentum tensor:

BG$1=MakeRule[{CD1[-a][CD1[-b][r[]]],((1-CD1[-c][r[]]CD1[c][r[]])/(2 r[]))g1[-a,-b]-((r[])/2)(Τ[-a,-b]-Τ[c,-c] g1[-a,-b])},MetricOn->All];
BG$2=MakeRule[{RicciScalarCD1[],Τ[a,-a]-2 𝒫[]+((2 )/((r[])^2))(1-CD1[-a][r[]]CD1[a][r[]])},MetricOn->All];
BG$T$Cons=MakeRule[{CD1[-b][Τ[-a,b]],(2*𝒫[]*CD1[-a][r[]])/r[]-(2*Τ[-a,-b]*CD1[b][r[]])/r[]},MetricOn->All];
(*BG$T$Cons is not regarded as minimal*)
BG$Minimal=Flatten[{BG$1,BG$2,BG$𝒯,CurvN$2D,CurvS$2D}];

The following function computes the product-metric expansion of a background quantity that generically depends on the metric and the energy-momentum tensor only. By convention, in this code all tensor fields introduced are assumed to carry their defining indices downstairs. This is important, in order to avoid conflict with SeparateMetric[] and ExpandPerturbation[].

Exp$Sep$BG$minimal[Expr_,SpacetimeIndices_,SubMfldIndices_]:=Module[{IndicesDown,Expanded,Separated},IndicesDown=((Expr//SeparateMetric[])/.Thread[SpacetimeIndices->SubMfldIndices])//TraceProductDummy;
Expanded=(ExpandProductMetric[IndicesDown/.Vol$Rep,g3])//.BG$Minimal;
Separated=(Expanded)//CheckZeroDerivative;
Separated//ContractMetric//ToCanonical
]

0.6. Background Harmonic Components of (Subscript[𝕁, μ ν ρ]), (Subscript[ℂ, μ ν ρ σ]), and total background substitution rules

(Subscript[𝒬, μ α β])

DefTensor[𝒬[-μ,-α,-β],M3,Antisymmetric[{-α,-β}]];

𝒬$To$𝒯=MakeRule[{𝒬[-μ,-α,-β],((1/2))(CD3[-α][𝒯[-β,-μ]-((𝒯[-ν,ν])/3)g3[-β,-μ]]-CD3[-β][𝒯[-α,-μ]-((𝒯[-ν,ν])/3)g3[-α,-μ]])}];

(Subscript[𝒬, a b c])

BG$𝒬abc=MakeRule[{𝒬[-a,-b,-c],-1/3*(g1[-a,-c]*CD1[-b][𝒫[]])+CD1[-b][Τ[-a,-c]]/2-(g1[-a,-c]*CD1[-b][Τ[d,-d]])/6+(g1[-a,-b]*CD1[-c][𝒫[]])/3-CD1[-c][Τ[-a,-b]]/2+(g1[-a,-b]*CD1[-c][Τ[d,-d]])/6}]
(*Exp$Sep$BG$minimal[𝒬[-μ,-α,-β]/.𝒬$To$𝒯,{μ,α,β},{a,b,c}]
ToCanonical[%-(𝒬[-a,-b,-c]/.BG$𝒬abc)]==0//Simplify*)

(Subscript[𝒬, a b A])

BG$𝒬abA=MakeRule[{𝒬[-a,-b,-A],0}];
(*Exp$Sep$BG$minimal[𝒬[-μ,-α,-β]/.𝒬$To$𝒯,{μ,α,β},{a,b,A}]
ToCanonical[%-(𝒬[-a,-b,-A]/.BG$𝒬abA)]==0//Simplify*)

(Subscript[𝒬, a A b])

BG$𝒬aAb=MakeRule[{𝒬[-a,-A,-b],0}];
(*Exp$Sep$BG$minimal[𝒬[-μ,-α,-β]/.𝒬$To$𝒯,{μ,α,β},{a,A,b}]
ToCanonical[%-(𝒬[-a,-A,-b]/.BG$𝒬aAb)]==0//Simplify*)

(Subscript[𝒬, a A B])

BG$𝒬aAB=MakeRule[{𝒬[-a,-A,-B],0}];
(*Exp$Sep$BG$minimal[𝒬[-μ,-α,-β]/.𝒬$To$𝒯,{μ,α,β},{a,A,B}]
ToCanonical[%-(𝒬[-a,-A,-B]/.BG$𝒬aAB)]==0//Simplify*)

(Subscript[𝒬, A P Q])

BG$𝒬APQ=MakeRule[{𝒬[-A,-P,-Q],0}];
(*Exp$Sep$BG$minimal[𝒬[-μ,-α,-β]/.𝒬$To$𝒯,{μ,α,β},{A,P,Q}]
ToCanonical[%-(𝒬[-A,-P,-Q]/.BG$𝒬APQ)]==0//Simplify*)

(Subscript[𝒬, A P a])

BG$𝒬APa=MakeRule[{𝒬[-A,-P,-a],-1/2*(g2[-A,-P]*r[]*𝒫[]*CD1[-a][r[]])-(g2[-A,-P]*r[]^2*CD1[-a][𝒫[]])/6+(g2[-A,-P]*r[]^2*CD1[-a][Τ[b,-b]])/6+(g2[-A,-P]*r[]*Τ[-a,-b]*CD1[b][r[]])/2}];
(*Exp$Sep$BG$minimal[𝒬[-μ,-α,-β]/.𝒬$To$𝒯,{μ,α,β},{A,P,a}]
ToCanonical[%-(𝒬[-A,-P,-a]/.BG$𝒬APa)]==0//Simplify*)

(Subscript[𝒬, A a P])

BG$𝒬AaP=MakeRule[{𝒬[-A,-a,-P],-(-1/2*(g2[-A,-P]*r[]*𝒫[]*CD1[-a][r[]])-(g2[-A,-P]*r[]^2*CD1[-a][𝒫[]])/6+(g2[-A,-P]*r[]^2*CD1[-a][Τ[b,-b]])/6+(g2[-A,-P]*r[]*Τ[-a,-b]*CD1[b][r[]])/2)}];
(*Exp$Sep$BG$minimal[𝒬[-μ,-α,-β]/.𝒬$To$𝒯,{μ,α,β},{A,a,P}]
ToCanonical[%-(𝒬[-A,-a,-P]/.BG$𝒬AaP)]==0//Simplify*)

(Subscript[𝒬, A a b])

BG$𝒬Aab=MakeRule[{𝒬[-A,-a,-b],0}];
(*Exp$Sep$BG$minimal[𝒬[-μ,-α,-β]/.𝒬$To$𝒯,{μ,α,β},{A,a,b}]
ToCanonical[%-(𝒬[-A,-a,-b]/.BG$𝒬Aab)]==0//Simplify*)

BG$𝒬=Flatten[{BG$𝒬abc,BG$𝒬abA,BG$𝒬aAb,BG$𝒬aAB,BG$𝒬APQ,BG$𝒬APa,BG$𝒬AaP,BG$𝒬Aab}];

(Subscript[ℳ, μ α β ])

DefTensor[ℳ[-μ,-α,-β],M3,Antisymmetric[{-α,-β}]];

ℳ$To$𝒯=MakeRule[{ℳ[-μ,-α,-β],-1/12*(epsilong3[-μ,-ρ,-α,-β]*CD3[ρ][𝒯[-γ,γ]])+(epsilong3[-σ,-ρ,-α,-β]*CD3[ρ][𝒯[σ,-μ]])/4+(epsilong3[-σ,-μ,-α,-β]*CD3[σ][𝒯[-γ,γ]])/12-(epsilong3[-σ,-ρ,-α,-β]*CD3[σ][𝒯[ρ,-μ]])/4}];

(Subscript[ℳ, a b c])

BG$ℳabc=MakeRule[{ℳ[-a,-b,-c],0}];
(*Exp$Sep$BG$minimal[ℳ[-μ,-α,-β]/.ℳ$To$𝒯,{μ,α,β},{a,b,c}]
ToCanonical[%-(ℳ[-a,-b,-c]/.BG$ℳabc)]==0//Simplify*)

(Subscript[ℳ, a b  A])

BG$ℳabA=MakeRule[{ℳ[-a,-b,-A],0}];
(*Exp$Sep$BG$minimal[ℳ[-μ,-α,-β]/.ℳ$To$𝒯,{μ,α,β},{a,b,A}]
ToCanonical[%-(ℳ[-a,-b,-A]/.BG$ℳabA)]==0//Simplify*)

(Subscript[ℳ, a A b])

BG$ℳaAb=MakeRule[{ℳ[-a,-A,-b],0}];
(*Exp$Sep$BG$minimal[ℳ[-μ,-α,-β]/.ℳ$To$𝒯,{μ,α,β},{a,A,b}]
ToCanonical[%-(ℳ[-a,-A,-b]/.BG$ℳaAb)]==0//Simplify*)

(Subscript[ℳ, a A b])

BG$ℳaAb=MakeRule[{ℳ[-a,-A,-b],0}];
(*Exp$Sep$BG$minimal[ℳ[-μ,-α,-β]/.ℳ$To$𝒯,{μ,α,β},{a,A,b}]
ToCanonical[%-(ℳ[-a,-A,-b]/.BG$ℳaAb)]==0//Simplify*)

(Subscript[ℳ, a A B])

BG$ℳaAB=MakeRule[{ℳ[-a,-A,-B],-1/6*(epsilong1[-a,b]*epsilong2[-A,-B]*r[]^2*CD1[-b][Τ[c,-c]])-(epsilong1[-a,-b]*epsilong2[-A,-B]*r[]^2*CD1[b][𝒫[]])/3+(epsilong1[b,c]*epsilong2[-A,-B]*r[]^2*CD1[-c][Τ[-a,-b]])/2}];
(*Exp$Sep$BG$minimal[ℳ[-μ,-α,-β]/.ℳ$To$𝒯,{μ,α,β},{a,A,B}]
ToCanonical[%-(ℳ[-a,-A,-B]/.BG$ℳaAB)]==0//Simplify*)

(Subscript[ℳ, A P Q])

BG$ℳAPQ=MakeRule[{ℳ[-A,-P,-Q],0}];
(*Exp$Sep$BG$minimal[ℳ[-μ,-α,-β]/.ℳ$To$𝒯,{μ,α,β},{A,P,Q}]
ToCanonical[%-(ℳ[-A,-P,-Q]/.BG$ℳAPQ)]==0//Simplify*)

(Subscript[ℳ, A P a])

BG$ℳAPa=MakeRule[{ℳ[-A,-P,-a],-1/6*(epsilong1[-a,b]*epsilong2[-A,-P]*r[]^2*CD1[-b][Τ[c,-c]])+(epsilong1[-a,-b]*epsilong2[-A,-P]*r[]*𝒫[]*CD1[b][r[]])/2-(epsilong1[-a,c]*epsilong2[-A,-P]*r[]*Τ[-b,-c]*CD1[b][r[]])/2+(epsilong1[-a,-b]*epsilong2[-A,-P]*r[]^2*CD1[b][𝒫[]])/6}];
(*Exp$Sep$BG$minimal[ℳ[-μ,-α,-β]/.ℳ$To$𝒯,{μ,α,β},{A,P,a}]
ToCanonical[%-(ℳ[-A,-P,-a]/.BG$ℳAPa)]==0//Simplify*)

(Subscript[ℳ, A a P])

BG$ℳAaP=MakeRule[{ℳ[-A,-a,-P],-(-1/6*(epsilong1[-a,b]*epsilong2[-A,-P]*r[]^2*CD1[-b][Τ[c,-c]])+(epsilong1[-a,-b]*epsilong2[-A,-P]*r[]*𝒫[]*CD1[b][r[]])/2-(epsilong1[-a,c]*epsilong2[-A,-P]*r[]*Τ[-b,-c]*CD1[b][r[]])/2+(epsilong1[-a,-b]*epsilong2[-A,-P]*r[]^2*CD1[b][𝒫[]])/6)}];
(*Exp$Sep$BG$minimal[ℳ[-μ,-α,-β]/.ℳ$To$𝒯,{μ,α,β},{A,a,P}]
ToCanonical[%-(ℳ[-A,-a,-P]/.BG$ℳAaP)]==0//Simplify*)

(Subscript[ℳ, A a b])

BG$ℳAab=MakeRule[{ℳ[-A,-a,-b],0}];
(*Exp$Sep$BG$minimal[ℳ[-μ,-α,-β]/.ℳ$To$𝒯,{μ,α,β},{A,a,b}]
ToCanonical[%-(ℳ[-A,-a,-b]/.BG$ℳAab)]==0//Simplify*)

BG$ℳ=Flatten[{BG$ℳabc,BG$ℳabA,BG$ℳaAb,BG$ℳaAB,BG$ℳAPQ,BG$ℳAPa,BG$ℳAaP,BG$ℳAab}];

(Subscript[𝕁, μ α β])

DefTensor[𝕁[-μ,-α,-β],M3,Antisymmetric[{-α,-β}]];(*stands for S=Q+i M*)
DefTensor[𝓆[-a],M1];
DefTensor[𝓅[-a],M1];

𝕁$To$𝒯=MakeRule[{𝕁[-μ,-α,-β],CD3[-α][𝒯[-β,-μ]]/2-(g3[-μ,-β]*CD3[-α][𝒯[-γ,γ]])/6-CD3[-β][𝒯[-α,-μ]]/2+(g3[-μ,-α]*CD3[-β][𝒯[-γ,γ]])/6+I*(-1/12*(epsilong3[-μ,-η,-α,-β]*CD3[η][𝒯[-γ,γ]])+(epsilong3[-ν,-η,-α,-β]*CD3[η][𝒯[ν,-μ]])/4+(epsilong3[-ν,-μ,-α,-β]*CD3[ν][𝒯[-γ,γ]])/12-(epsilong3[-ν,-η,-α,-β]*CD3[ν][𝒯[η,-μ]])/4)}];

BG$𝓆=MakeRule[{𝓆[-a],(1/2)(epsilong1[d,e]CD1[-e][Τ[-a,-d]]-(1/3)(CD1[d][Τ[e,-e]+2𝒫[]])epsilong1[-a,-d])}];
BG$𝓅=MakeRule[{𝓅[-a],-1/2*(r[]*𝒫[]*CD1[-a][r[]])-(r[]^2*CD1[-a][𝒫[]])/6+(r[]^2*CD1[-a][Τ[b,-b]])/6+(r[]*Τ[-a,-b]*CD1[b][r[]])/2}];
BG$𝕁abc=MakeRule[{𝕁[-a,-b,-c],((1/2)(epsilong1[d,e]CD1[-e][Τ[-a,-d]]-(1/3)(CD1[d][Τ[e,-e]+2𝒫[]])epsilong1[-a,-d]))epsilong1[-b,-c]}];
BG$𝕁aAb=MakeRule[{𝕁[-a,-A,-b],0}];
BG$𝕁abA=MakeRule[{𝕁[-a,-b,-A],0}];
BG$𝕁aAB=MakeRule[{𝕁[-a,-A,-B],I ((r[])^2)((1/2)(epsilong1[d,e]CD1[-e][Τ[-a,-d]]-(1/3)(CD1[d][Τ[e,-e]+2𝒫[]])epsilong1[-a,-d]))epsilong2[-A,-B]}];
BG$𝕁Aab=MakeRule[{𝕁[-A,-a,-b],0}];
BG$𝕁PQa=MakeRule[{𝕁[-P,-Q,-a],(-1/2*(r[]*𝒫[]*CD1[-a][r[]])-(r[]^2*CD1[-a][𝒫[]])/6+(r[]^2*CD1[-a][Τ[b,-b]])/6+(r[]*Τ[-a,-b]*CD1[b][r[]])/2)g2[-P,-Q]-I epsilong1[-a,c](-1/2*(r[]*𝒫[]*CD1[-c][r[]])-(r[]^2*CD1[-c][𝒫[]])/6+(r[]^2*CD1[-c][Τ[b,-b]])/6+(r[]*Τ[-c,-b]*CD1[b][r[]])/2)epsilong2[-P,-Q]}];
BG$𝕁PaQ=MakeRule[{𝕁[-P,-a,-Q],-((-1/2*(r[]*𝒫[]*CD1[-a][r[]])-(r[]^2*CD1[-a][𝒫[]])/6+(r[]^2*CD1[-a][Τ[b,-b]])/6+(r[]*Τ[-a,-b]*CD1[b][r[]])/2)g2[-P,-Q]-I epsilong1[-a,c](-1/2*(r[]*𝒫[]*CD1[-c][r[]])-(r[]^2*CD1[-c][𝒫[]])/6+(r[]^2*CD1[-c][Τ[b,-b]])/6+(r[]*Τ[-c,-b]*CD1[b][r[]])/2)epsilong2[-P,-Q])}];
BG$𝕁APQ=MakeRule[{𝕁[-A,-P,-Q],0}];
BG$𝕁=Flatten[{BG$𝓆,BG$𝓅,BG$𝕁abc,BG$𝕁aAb,BG$𝕁abA,BG$𝕁aAB,BG$𝕁Aab,BG$𝕁PQa,BG$𝕁PaQ,BG$𝕁APQ}];

(*ToCanonical[(Exp$Sep$BG$minimal[𝕁[-μ,-α,-β]/.𝕁$To$𝒯,{μ,α,β},{a,b,c}])-(𝕁[-a,-b,-c]/.BG$𝕁)]==0//Simplify
ToCanonical[(Exp$Sep$BG$minimal[𝕁[-μ,-α,-β]/.𝕁$To$𝒯,{μ,α,β},{a,A,b}])-(𝕁[-a,-A,-b]/.BG$𝕁)]==0//Simplify
ToCanonical[(Exp$Sep$BG$minimal[𝕁[-μ,-α,-β]/.𝕁$To$𝒯,{μ,α,β},{a,b,A}])-(𝕁[-a,-b,-A]/.BG$𝕁)]==0//Simplify
ToCanonical[(Exp$Sep$BG$minimal[𝕁[-μ,-α,-β]/.𝕁$To$𝒯,{μ,α,β},{a,A,B}])-(𝕁[-a,-A,-B]/.BG$𝕁)]==0//Simplify
ToCanonical[(Exp$Sep$BG$minimal[𝕁[-μ,-α,-β]/.𝕁$To$𝒯,{μ,α,β},{A,a,b}])-(𝕁[-A,-a,-b]/.BG$𝕁)]==0//Simplify
ToCanonical[(Exp$Sep$BG$minimal[𝕁[-μ,-α,-β]/.𝕁$To$𝒯,{μ,α,β},{P,Q,a}])-(𝕁[-P,-Q,-a]/.BG$𝕁)]==0//Simplify
ToCanonical[(Exp$Sep$BG$minimal[𝕁[-μ,-α,-β]/.𝕁$To$𝒯,{μ,α,β},{P,a,Q}])-(𝕁[-P,-a,-Q]/.BG$𝕁)]==0//Simplify
ToCanonical[(Exp$Sep$BG$minimal[𝕁[-μ,-α,-β]/.𝕁$To$𝒯,{μ,α,β},{A,P,Q}])-(𝕁[-A,-P,-Q]/.BG$𝕁)]==0//Simplify*)

(Subscript[ℂ, μ ν α β])

DefTensor[CC[-μ,-ν,-ρ,-σ],M3, RiemannSymmetric[{1,2,3,4}],PrintAs->"ℂ"];

Definition in terms of the Weyl

ℂ$To$Weyl=MakeRule[{CC[-μ,-ν,-ρ,-σ],WeylCD3[-μ,-ν,-ρ,-σ]-(I/2) epsilong3[-μ,-ν,α,β]WeylCD3[-α,-β,-ρ,-σ]}]

DefTensor[𝒲[],M1];

W$To$BG=MakeRule[{𝒲[],3(1-CD1[a][r[]]CD1[-a][r[]])+((r[])^2)(Τ[a,-a]-𝒫[])}];
DW$To$BG=MakeRule[{CD1[-a][𝒲[]],(-3*CD1[-a][r[]])/r[]-2*r[]*𝒫[]*CD1[-a][r[]]-r[]*Τ[b,-b]*CD1[-a][r[]]-r[]^2*CD1[-a][𝒫[]]+r[]^2*CD1[-a][Τ[b,-b]]+3*r[]*Τ[-a,-b]*CD1[b][r[]]+(3*CD1[-a][r[]]*CD1[-b][r[]]*CD1[b][r[]])/r[]}];
DDW$To$BG=MakeRule[{CD1[-a][CD1[-b][𝒲[]]],(-3*g1[-a,-b])/(2*r[]^2)-g1[-a,-b]*𝒫[]+3*Τ[-a,-b]+r[]^2*𝒫[]*Τ[-a,-b]-(3*r[]^2*Τ[-a,c]*Τ[-b,-c])/2-2*g1[-a,-b]*Τ[c,-c]-g1[-a,-b]*r[]^2*𝒫[]*Τ[c,-c]+2*r[]^2*Τ[-a,-b]*Τ[c,-c]-(g1[-a,-b]*r[]^2*Τ[c,-c]*Τ[d,-d])/2+(6*CD1[-a][r[]]*CD1[-b][r[]])/r[]^2-2*𝒫[]*CD1[-a][r[]]*CD1[-b][r[]]+2*Τ[c,-c]*CD1[-a][r[]]*CD1[-b][r[]]-2*r[]*CD1[-a][𝒫[]]*CD1[-b][r[]]-r[]*CD1[-a][Τ[c,-c]]*CD1[-b][r[]]-2*r[]*CD1[-a][r[]]*CD1[-b][𝒫[]]+2*r[]*CD1[-a][r[]]*CD1[-b][Τ[c,-c]]-r[]^2*CD1[-b][CD1[-a][𝒫[]]]+r[]^2*CD1[-b][CD1[-a][Τ[c,-c]]]+3*Τ[-b,-c]*CD1[-a][r[]]*CD1[c][r[]]+3*r[]*CD1[-a][Τ[-b,-c]]*CD1[c][r[]]-3*Τ[-a,-c]*CD1[-b][r[]]*CD1[c][r[]]+(3*g1[-a,-b]*CD1[-c][r[]]*CD1[c][r[]])/r[]^2+g1[-a,-b]*𝒫[]*CD1[-c][r[]]*CD1[c][r[]]-3*Τ[-a,-b]*CD1[-c][r[]]*CD1[c][r[]]+2*g1[-a,-b]*Τ[d,-d]*CD1[-c][r[]]*CD1[c][r[]]-(6*CD1[-a][r[]]*CD1[-b][r[]]*CD1[-c][r[]]*CD1[c][r[]])/r[]^2-(3*g1[-a,-b]*CD1[-c][r[]]*CD1[c][r[]]*CD1[-d][r[]]*CD1[d][r[]])/(2*r[]^2)}];

Its background substitution rules (the derivation has been carried out below):

BG$Cabcd=MakeRule[{CC[-a,-b,-c,-d],-(r[]^(-2)-𝒫[]/3+Τ[e,-e]/3-(CD1[-e][r[]]*CD1[e][r[]])/r[]^2)epsilong1[-a,-b]epsilong1[-c,-d]}];
(**)
BG$CabAB$1=MakeRule[{CC[-a,-b,-A,-B],-I ((r[])^2)(r[]^(-2)-𝒫[]/3+Τ[e,-e]/3-(CD1[-e][r[]]*CD1[e][r[]])/r[]^2)epsilong1[-a,-b]epsilong2[-A,-B]}];
BG$CabAB$2=MakeRule[{CC[-A,-B,-a,-b],-I ((r[])^2)(r[]^(-2)-𝒫[]/3+Τ[e,-e]/3-(CD1[-e][r[]]*CD1[e][r[]])/r[]^2)epsilong1[-a,-b]epsilong2[-A,-B]}];
(**)
BG$CaAbB$1=MakeRule[{CC[-a,-A,-b,-B],((-((r[])^2))/2)(r[]^(-2)-𝒫[]/3+Τ[e,-e]/3-(CD1[-e][r[]]*CD1[e][r[]])/r[]^2)(g1[-a,-b]g2[-A,-B]+I epsilong1[-a,-b]epsilong2[-A,-B])}];
BG$CaAbB$2=MakeRule[{CC[-A,-a,-b,-B],(((r[])^2)/2)(r[]^(-2)-𝒫[]/3+Τ[e,-e]/3-(CD1[-e][r[]]*CD1[e][r[]])/r[]^2)(g1[-a,-b]g2[-A,-B]+I epsilong1[-a,-b]epsilong2[-A,-B])}];
BG$CaAbB$3=MakeRule[{CC[-a,-A,-B,-b],(((r[])^2)/2)(r[]^(-2)-𝒫[]/3+Τ[e,-e]/3-(CD1[-e][r[]]*CD1[e][r[]])/r[]^2)(g1[-a,-b]g2[-A,-B]+I epsilong1[-a,-b]epsilong2[-A,-B])}];
BG$CaAbB$4=MakeRule[{CC[-A,-a,-B,-b],((-((r[])^2))/2)(r[]^(-2)-𝒫[]/3+Τ[e,-e]/3-(CD1[-e][r[]]*CD1[e][r[]])/r[]^2)(g1[-a,-b]g2[-A,-B]+I epsilong1[-a,-b]epsilong2[-A,-B])}];
(**)
BG$CaABF$1=MakeRule[{CC[-a,-A,-B,-F],0}];
BG$CaABF$2=MakeRule[{CC[-A,-a,-B,-F],0}];
BG$CaABF$3=MakeRule[{CC[-A,-B,-a,-F],0}];
BG$CaABF$4=MakeRule[{CC[-A,-B,-F,-a],0}];
(**)
(**)
BG$CAabc$1=MakeRule[{CC[-A,-a,-b,-c],0}];
BG$CAabc$2=MakeRule[{CC[-a,-A,-b,-c],0}];
BG$CAabc$3=MakeRule[{CC[-a,-b,-A,-c],0}];
BG$CAabc$4=MakeRule[{CC[-a,-b,-c,-A],0}];
(**)
BG$CABFG=MakeRule[{CC[-A,-B,-F,-G],((r[])^4)(r[]^(-2)-𝒫[]/3+Τ[e,-e]/3-(CD1[-e][r[]]*CD1[e][r[]])/r[]^2)( epsilong2[-A,-B]epsilong2[-F,-G])}];
BG$ℂ=Flatten[{BG$Cabcd,BG$CabAB$1,BG$CabAB$2,BG$CaAbB$1,BG$CaAbB$2,BG$CaAbB$3,BG$CaAbB$4,BG$CaABF$1,BG$CaABF$2,BG$CaABF$3,BG$CaABF$4,BG$CAabc$1,BG$CAabc$2,BG$CAabc$3,BG$CAabc$4,BG$CABFG}];

(*(**)
ToCanonical[epsilong1[a,b]epsilong1[c,d]((Exp$Sep$BG$minimal[WeylToRiemann[CC[-μ,-ν,-ρ,-σ]/.ℂ$To$Weyl],{μ,ν,ρ,σ},{a,b,c,d}])-(CC[-a,-b,-c,-d]/.BG$ℂ))//ContractMetric]==0//Simplify
(**)
ToCanonical[((Exp$Sep$BG$minimal[WeylToRiemann[CC[-μ,-ν,-ρ,-σ]/.ℂ$To$Weyl],{μ,ν,ρ,σ},{A,b,c,d}])-(CC[-A,-b,-c,-d]/.BG$ℂ))//ContractMetric]==0//Simplify
ToCanonical[((Exp$Sep$BG$minimal[WeylToRiemann[CC[-μ,-ν,-ρ,-σ]/.ℂ$To$Weyl],{μ,ν,ρ,σ},{b,A,c,d}])-(CC[-b,-A,-c,-d]/.BG$ℂ))//ContractMetric]==0//Simplify
ToCanonical[((Exp$Sep$BG$minimal[WeylToRiemann[CC[-μ,-ν,-ρ,-σ]/.ℂ$To$Weyl],{μ,ν,ρ,σ},{b,c,A,d}])-(CC[-b,-c,-A,-d]/.BG$ℂ))//ContractMetric]==0//Simplify
ToCanonical[((Exp$Sep$BG$minimal[WeylToRiemann[CC[-μ,-ν,-ρ,-σ]/.ℂ$To$Weyl],{μ,ν,ρ,σ},{b,c,d,A}])-(CC[-b,-c,-d,-A]/.BG$ℂ))//ContractMetric]==0//Simplify
(**)
(Symetrise[Exp$Sep$BG$minimal[WeylToRiemann[CC[-μ,-ν,-ρ,-σ]/.ℂ$To$Weyl],{μ,ν,ρ,σ},{A,B,a,b}],{-a,-b}]//ToCanonical)==0//Simplify
ToCanonical[epsilong1[a,b]((Exp$Sep$BG$minimal[WeylToRiemann[CC[-μ,-ν,-ρ,-σ]/.ℂ$To$Weyl],{μ,ν,ρ,σ},{A,B,a,b}])-(CC[-A,-B,-a,-b]/.BG$ℂ))//ContractMetric]==0//Simplify
ToCanonical[((Exp$Sep$BG$minimal[WeylToRiemann[CC[-μ,-ν,-ρ,-σ]/.ℂ$To$Weyl],{μ,ν,ρ,σ},{a,b,A,B}])-(CC[-a,-b,-A,-B]/.BG$ℂ))//ContractMetric]==0//Simplify
ToCanonical[((Exp$Sep$BG$minimal[WeylToRiemann[CC[-μ,-ν,-ρ,-σ]/.ℂ$To$Weyl],{μ,ν,ρ,σ},{a,A,b,B}])-(CC[-a,-A,-b,-B]/.BG$ℂ))//ContractMetric]==0//Simplify
ToCanonical[((Exp$Sep$BG$minimal[WeylToRiemann[CC[-μ,-ν,-ρ,-σ]/.ℂ$To$Weyl],{μ,ν,ρ,σ},{a,A,B,b}])-(CC[-a,-A,-B,-b]/.BG$ℂ))//ContractMetric]==0//Simplify
ToCanonical[((Exp$Sep$BG$minimal[WeylToRiemann[CC[-μ,-ν,-ρ,-σ]/.ℂ$To$Weyl],{μ,ν,ρ,σ},{A,a,b,B}])-(CC[-A,-a,-b,-B]/.BG$ℂ))//ContractMetric]==0//Simplify
ToCanonical[((Exp$Sep$BG$minimal[WeylToRiemann[CC[-μ,-ν,-ρ,-σ]/.ℂ$To$Weyl],{μ,ν,ρ,σ},{A,a,B,b}])-(CC[-A,-a,-B,-b]/.BG$ℂ))//ContractMetric]==0//Simplify
(**)
ToCanonical[((Exp$Sep$BG$minimal[WeylToRiemann[CC[-μ,-ν,-ρ,-σ]/.ℂ$To$Weyl],{μ,ν,ρ,σ},{a,A,P,Q}])-(CC[-a,-A,-P,-Q]/.BG$ℂ))//ContractMetric]==0//Simplify
ToCanonical[((Exp$Sep$BG$minimal[WeylToRiemann[CC[-μ,-ν,-ρ,-σ]/.ℂ$To$Weyl],{μ,ν,ρ,σ},{A,a,P,Q}])-(CC[-A,-a,-P,-Q]/.BG$ℂ))//ContractMetric]==0//Simplify
ToCanonical[((Exp$Sep$BG$minimal[WeylToRiemann[CC[-μ,-ν,-ρ,-σ]/.ℂ$To$Weyl],{μ,ν,ρ,σ},{A,P,a,Q}])-(CC[-A,-P,-a,-Q]/.BG$ℂ))//ContractMetric]==0//Simplify
ToCanonical[((Exp$Sep$BG$minimal[WeylToRiemann[CC[-μ,-ν,-ρ,-σ]/.ℂ$To$Weyl],{μ,ν,ρ,σ},{A,P,Q,a}])-(CC[-A,-P,-Q,-a]/.BG$ℂ))//ContractMetric]==0//Simplify
(**)
ToCanonical[((Exp$Sep$BG$minimal[WeylToRiemann[CC[-μ,-ν,-ρ,-σ]/.ℂ$To$Weyl],{μ,ν,ρ,σ},{A,B,P,Q}])-(CC[-A,-B,-P,-Q]/.BG$ℂ))//ContractMetric]==0//Simplify*)

Total Background Substitutions

BG$Total=Flatten[{BG$1,BG$2,BG$𝒯,BG$𝕁,BG$ℂ,FourD$To$TwoD}];

Tests

(CD3[μ][CC[-μ,-ν,-ρ,-σ]]-𝕁[-ν,-ρ,-σ])//SeparateMetric[];
%/.{ν->a,ρ->b,σ->c}//TraceProductDummy;
ExpandProductMetric[%,g3]/.Flatten[{BG$ℂ,BG$𝕁}]//CheckZeroDerivative;
%/.Flatten[{BG$1,BG$2}]//ContractMetric//ToCanonical;
epsilong1[b,c]%//ContractMetric//ToCanonical;
epsilong1[a,-f]%//ContractMetric//ToCanonical;
%/.BG$T$Cons//ContractMetric//ToCanonical

(CD3[μ][CC[-μ,-ν,-ρ,-σ]]-𝕁[-ν,-ρ,-σ])//SeparateMetric[];
%/.{ν->A,ρ->B,σ->P}//TraceProductDummy;
ExpandProductMetric[%,g3]/.Flatten[{BG$ℂ,BG$𝕁}]//CheckZeroDerivative;
%/.Flatten[{BG$1,BG$2}]//ContractMetric//ToCanonical

(CD3[μ][CC[-μ,-ν,-ρ,-σ]]-𝕁[-ν,-ρ,-σ])//SeparateMetric[];
%/.{ν->A,ρ->a,σ->b}//TraceProductDummy;
ExpandProductMetric[%,g3]/.Flatten[{BG$ℂ,BG$𝕁}]//CheckZeroDerivative;
%/.Flatten[{BG$1,BG$2}]//ContractMetric//ToCanonical

(CD3[μ][CC[-μ,-ν,-ρ,-σ]]-𝕁[-ν,-ρ,-σ])//SeparateMetric[];
%/.{ν->a,ρ->A,σ->B}//TraceProductDummy;
ExpandProductMetric[%,g3]/.Flatten[{BG$ℂ,BG$𝕁}]//CheckZeroDerivative;
epsilong1[a,-f]epsilong2[A,B]%/.Flatten[{BG$1,BG$2}]//ContractMetric//ToCanonical;
%/.BG$T$Cons//ContractMetric//ToCanonical

0.7. Definition of First-order Fluctuations

The main idea of perturbation theory in spherical symmetry is to decompose tensors in their harmonic components on the sphere. All fundamental tensors are defined with indices down.

Metric

Unprotect[IndexForm];
IndexForm[LI[x_]]:=ColorString[ToString[x],RGBColor[0,0,1]];
Protect[IndexForm];
DefMetricPerturbation[g3,h,ϵ,PrintAs->"h"]

We introduce the harmonic components of the metric:

DefTensor[𝒽$ab$p[-a,-b],M1,Symmetric[{-a,-b}],PrintAs->"(SuperPlus[𝒽])"]
DefTensor[𝒽$a$m[-a],M1,PrintAs->"(SuperMinus[𝒽])"]
DefTensor[𝒽$p[],M1,PrintAs->"(SuperPlus[𝒽])"]
𝒽$Components={𝒽$ab$p[-a,-b],𝒽$a$m[-a],𝒽$p[]};

We define the replacement rules exchanging the general metric perturbation by its decomposition into harmonics:

𝒽$Harm$Rep=Flatten[{MakeRule[{h[LI[1],-a,-b],𝒽$ab$p[-a,-b]Y[]}],MakeRule[{h[LI[1],-a,-A],𝒽$a$m[-a]X[-A]}],MakeRule[{h[LI[1],-A,-a],𝒽$a$m[-a]X[-A]}],MakeRule[{h[LI[1],-A,-B],𝒽$p[]U[-A,-B]}]}];

Finally we introduce a splitting of the metric in traceless and trace parts:

DefTensor[Η[-a,-b],M1,Symmetric[{-a,-b}],PrintAs->"(SuperPlus[Η])"];

Η$Rule$Trace=MakeRule[{Η[-a,a],0}];

AutomaticRules[Η,Η$Rule$Trace];

𝒽$To$Η=MakeRule[{𝒽$ab$p[a,b],Η[a,b]+(1/2)𝒽$ab$p[d,-d]g1[a,b]}];

Plus$Part=Flatten[{MakeRule[{𝒽$a$m[-a],0},MetricOn->All],MakeRule[{𝒽$ab$p[-a,-b],2𝒽$ab$p[-a,-b]},MetricOn->All],MakeRule[{𝒽$p[],2𝒽$p[]},MetricOn->All]}];
Minus$Part=Flatten[{MakeRule[{𝒽$a$m[-a],2𝒽$a$m[-a]},MetricOn->All],MakeRule[{𝒽$ab$p[-a,-b],0},MetricOn->All],MakeRule[{𝒽$p[],0},MetricOn->All]}];

We also introduce the ``pure gauge’’ components of the metric fluctuation, since these will be useful later in the discussion on gauge symmetries

DefTensor[𝒽$a$p[-a],M1,PrintAs->"(SuperPlus[𝒽])"]
DefTensor[𝒽$m[],M1,PrintAs->"(SuperMinus[𝒽])"]
DefTensor[𝒽$o[],M1,PrintAs->"(𝒽^⊕)"]

Energy-momentum Tensor

We proceed as for the metric to define the fluctuation of the energy-momentum tensor:

DefTensor[Φ$ab$p[-a,-b],M1,Symmetric[{-a,-b}],PrintAs->"(SuperPlus[Φ])"]
DefTensor[Φ$a$p[-a],M1,PrintAs->"(SuperPlus[Φ])"]
DefTensor[Φ$a$m[-a],M1,PrintAs->"(SuperMinus[Φ])"]
DefTensor[Φ$p[],M1,PrintAs->"(SuperPlus[Φ])"]
DefTensor[Φ$m[],M1,PrintAs->"(SuperMinus[Φ])"]
DefTensor[Φ$o[],M1,PrintAs->"(Φ^⊕)"]
𝒯$Components={Φ$ab$p[-a,-b],Φ$a$p[-a],Φ$a$m[-a],Φ$p[],Φ$m[],Φ$o[]};

𝒯$Harm$Rep=Flatten[{MakeRule[{Perturbation[𝒯[-a,-b]],Φ$ab$p[-a,-b]Y[]}],MakeRule[{Perturbation[𝒯[-a,-A]],Φ$a$p[-a]Z[-A]+Φ$a$m[-a]X[-A]}],MakeRule[{Perturbation[𝒯[-A,-a]],Φ$a$p[-a]Z[-A]+Φ$a$m[-a]X[-A]}],MakeRule[{Perturbation[𝒯[-A,-B]],Φ$p[]U[-A,-B]+Φ$o[]V[-A,-B]+Φ$m[]W[-A,-B]}]}];

Plus$Part=Flatten[{Plus$Part,MakeRule[{Φ$a$m[-a],0},MetricOn->All],MakeRule[{Φ$m[],0},MetricOn->All],MakeRule[{Φ$ab$p[-a,-b],2Φ$ab$p[-a,-b]},MetricOn->All],MakeRule[{Φ$a$p[-a],2Φ$a$p[-a]},MetricOn->All],MakeRule[{Φ$p[],2Φ$p[]},MetricOn->All],MakeRule[{Φ$o[],2Φ$o[]},MetricOn->All]}];
Minus$Part=Flatten[{Minus$Part,MakeRule[{Φ$a$m[-a],2Φ$a$m[-a]},MetricOn->All],MakeRule[{Φ$m[],2Φ$m[]},MetricOn->All],MakeRule[{Φ$ab$p[-a,-b],0},MetricOn->All],MakeRule[{Φ$a$p[-a],0},MetricOn->All],MakeRule[{Φ$p[],0},MetricOn->All],MakeRule[{Φ$o[],0},MetricOn->All]}];

Self-dual Weyl currents

DefTensor[Π$ab$p[-a,-b],M1,Symmetric[{-a,-b}],PrintAs->"(SuperPlus[Π])"]
DefTensor[Π$ab$m[-a,-b],M1,Symmetric[{-a,-b}],PrintAs->"(SuperMinus[Π])"]
DefTensor[Π$a$p[-a],M1,PrintAs->"(SuperPlus[Π])"]
DefTensor[Π$a$m[-a],M1,PrintAs->"(SuperMinus[Π])"]
DefTensor[Π$p[],M1,PrintAs->"(SuperPlus[Π])"]
DefTensor[Π$m[],M1,PrintAs->"(SuperMinus[Π])"]
DefTensor[Π$a$o[-a],M1,PrintAs->"(Π^⊕)"]
DefTensor[Γ$a$p[-a],M1,PrintAs->"(SuperPlus[Γ])"]
DefTensor[Γ$a$m[-a],M1,PrintAs->"(SuperMinus[Γ])"]
DefTensor[Γ$p[],M1,PrintAs->"(SuperPlus[Γ])"]
DefTensor[Γ$m[],M1,PrintAs->"(SuperMinus[Γ])"]

𝕁$Components={Π$ab$p[-a,-b],Π$ab$m[-a,-b],Π$a$p[-a],Π$a$m[-a],Π$p[],Π$m[],Π$a$o[-a],Γ$a$p[-a],Γ$a$m[-a],Γ$p[],Γ$m[]};

𝕁$Rep$abc=MakeRule[{Perturbation[𝕁[-a,-b,-c]],Π$a$p[-a]Y[]epsilong1[-b,-c]}];
𝕁$Rep$abP=MakeRule[{Perturbation[𝕁[-a,-b,-P]],(Π$p[]epsilong1[-a,-b]+Π$ab$p[-a,-b])Z[-P]+(Π$m[]epsilong1[-a,-b]+Π$ab$m[-a,-b])X[-P]}];
𝕁$Rep$aPb=MakeRule[{Perturbation[𝕁[-a,-P,-b]],-((Π$p[]epsilong1[-a,-b]+Π$ab$p[-a,-b])Z[-P]+(Π$m[]epsilong1[-a,-b]+Π$ab$m[-a,-b])X[-P])}];
𝕁$Rep$aPQ=MakeRule[{Perturbation[𝕁[-a,-P,-Q]],-2 Π$a$m[-a]Y[]epsilong2[-P,-Q]}];

𝕁$Rep$Pab=MakeRule[{Perturbation[𝕁[-P,-a,-b]],-2(Π$p[]Z[-P]+Π$m[]X[-P])epsilong1[-a,-b]}];

𝕁$Rep$PQa=MakeRule[{Perturbation[𝕁[-P,-Q,-a]],Π$a$m[-a]Y[]epsilong2[-P,-Q]+Γ$a$p[-a]U[-P,-Q]+Γ$a$m[-a]W[-P,-Q]+Π$a$o[-a]V[-P,-Q]}];
𝕁$Rep$PaQ=MakeRule[{Perturbation[𝕁[-P,-a,-Q]],-(Π$a$m[-a]Y[]epsilong2[-P,-Q]+Γ$a$p[-a]U[-P,-Q]+Γ$a$m[-a]W[-P,-Q]+Π$a$o[-a]V[-P,-Q])}];
𝕁$Rep$APQ=MakeRule[{Perturbation[𝕁[-A,-P,-Q]],(Γ$m[]Z[-A]+Γ$p[]X[-A])epsilong2[-P,-Q]}];

𝕁$Harm$Rep=Flatten[{𝕁$Rep$abc,𝕁$Rep$abP,𝕁$Rep$aPb,𝕁$Rep$aPQ,𝕁$Rep$Pab,𝕁$Rep$PQa,𝕁$Rep$PaQ,𝕁$Rep$APQ}];

Self-dual Weyl

We proceed as before to introduce the fluctuation of the self-dual Weyl, already implementing some of its symmetries

DefTensor[Ψ$ab$p[-a,-b],M1,Symmetric[{-a,-b}],PrintAs->"(SuperPlus[Ψ])"]
DefTensor[Ψ$ab$m[-a,-b],M1,Symmetric[{-a,-b}],PrintAs->"(SuperMinus[Ψ])"]
DefTensor[Ψ$ab$o[-a,-b],M1,Symmetric[{-a,-b}],PrintAs->"(Ψ^⊕)"]
DefTensor[Ψ$a$p[-a],M1,PrintAs->"(SuperPlus[Ψ])"]
DefTensor[Ψ$a$m[-a],M1,PrintAs->"(SuperMinus[Ψ])"]
DefTensor[Ψ$p[],M1,PrintAs->"(SuperPlus[Ψ])"]
DefTensor[Ψ$m[],M1,PrintAs->"(SuperMinus[Ψ])"]
DefTensor[Ξ$a$p[-a],M1,PrintAs->"(SuperPlus[Ξ])"]
DefTensor[Ξ$a$m[-a],M1,PrintAs->"(SuperMinus[Ξ])"]
DefTensor[Ξ$p[],M1,PrintAs->"(SuperPlus[Ξ])"]

ℂ$Components={Ψ$ab$p[-a,-b],Ψ$ab$m[-a,-b],Ψ$ab$o[-a,-b],Ψ$a$p[-a],Ψ$a$m[-a],Ψ$p[],Ψ$m[],Ξ$a$p[-a],Ξ$a$m[-a],Ξ$p[]}

We introduce the replacement rules for the fluctuations of the sd-Weyl:

C$abcd$1=MakeRule[{Perturbation[CC[-a,-b,-c,-d]],Ψ$p[]Y[]epsilong1[-a,-b]epsilong1[-c,-d]}];
(**)
C$Abcd$1=MakeRule[{Perturbation[CC[-A,-b,-c,-d]],(Ψ$a$p[-b]Z[-A]+Ψ$a$m[-b]X[-A])epsilong1[-c,-d]}];
C$Abcd$2=MakeRule[{Perturbation[CC[-b,-A,-c,-d]],-(Ψ$a$p[-b]Z[-A]+Ψ$a$m[-b]X[-A])epsilong1[-c,-d]}];
C$Abcd$3=MakeRule[{Perturbation[CC[-c,-d,-A,-b]],(Ψ$a$p[-b]Z[-A]+Ψ$a$m[-b]X[-A])epsilong1[-c,-d]}];
C$Abcd$4=MakeRule[{Perturbation[CC[-c,-d,-b,-A]],-(Ψ$a$p[-b]Z[-A]+Ψ$a$m[-b]X[-A])epsilong1[-c,-d]}];
(**)
C$AaBb$1=MakeRule[{Perturbation[CC[-A,-a,-B,-b]],((Ψ$m[])/2)epsilong1[-a,-b]epsilong2[-A,-B]Y[]+Ψ$ab$p[-a,-b]U[-A,-B]+Ψ$ab$o[-a,-b]V[-A,-B]+Ψ$ab$m[-a,-b]W[-A,-B]}];
C$AaBb$2=MakeRule[{Perturbation[CC[-a,-A,-b,-B]],(((Ψ$m[])/2)epsilong1[-a,-b]epsilong2[-A,-B]Y[]+Ψ$ab$p[-a,-b]U[-A,-B]+Ψ$ab$o[-a,-b]V[-A,-B]+Ψ$ab$m[-a,-b]W[-A,-B])}];
C$AaBb$3=MakeRule[{Perturbation[CC[-a,-A,-B,-b]],-(((Ψ$m[])/2)epsilong1[-a,-b]epsilong2[-A,-B]Y[]+Ψ$ab$p[-a,-b]U[-A,-B]+Ψ$ab$o[-a,-b]V[-A,-B]+Ψ$ab$m[-a,-b]W[-A,-B])}];
C$AaBb$4=MakeRule[{Perturbation[CC[-A,-a,-b,-B]],-(((Ψ$m[])/2)epsilong1[-a,-b]epsilong2[-A,-B]Y[]+Ψ$ab$p[-a,-b]U[-A,-B]+Ψ$ab$o[-a,-b]V[-A,-B]+Ψ$ab$m[-a,-b]W[-A,-B])}];
(**)
C$abAB$1=MakeRule[{Perturbation[CC[-a,-b,-A,-B]],Ψ$m[]Y[]epsilong1[-a,-b]epsilong2[-A,-B]}];
C$abAB$2=MakeRule[{Perturbation[CC[-A,-B,-a,-b]],Ψ$m[]Y[]epsilong1[-a,-b]epsilong2[-A,-B]}];
(**)
C$ABGb$1=MakeRule[{Perturbation[CC[-A,-B,-G,-b]],(Ξ$a$m[-b]Z[-G]+Ξ$a$p[-b]X[-G])epsilong2[-A,-B]}];
C$ABGb$2=MakeRule[{Perturbation[CC[-A,-B,-b,-G]],-((Ξ$a$m[-b]Z[-G]+Ξ$a$p[-b]X[-G])epsilong2[-A,-B])}];
C$ABGb$3=MakeRule[{Perturbation[CC[-G,-b,-A,-B]],((Ξ$a$m[-b]Z[-G]+Ξ$a$p[-b]X[-G])epsilong2[-A,-B])}];
C$ABGb$4=MakeRule[{Perturbation[CC[-b,-G,-A,-B]],-((Ξ$a$m[-b]Z[-G]+Ξ$a$p[-b]X[-G])epsilong2[-A,-B])}];
(**)
C$ABGF$1=MakeRule[{Perturbation[CC[-A,-B,-G,-F]],Ξ$p[]Y[]epsilong2[-A,-B]epsilong2[-G,-F]}];
ℂ$Harm$Rep=Flatten[{C$abcd$1,C$Abcd$1,C$Abcd$2,C$Abcd$3,C$Abcd$4,C$AaBb$1,C$AaBb$2,C$AaBb$3,C$AaBb$4,C$abAB$1,C$abAB$2,C$ABGb$1,C$ABGb$2,C$ABGb$3,C$ABGb$4,C$ABGF$1}];

Zero-rules and variable-content functions for self-dual components

This is a useful function to identify the content of self-dual variables in long equations

selfdual$ZeroRules=Flatten[{MakeRule[{#,0}]}&/@Flatten[{ℂ$Components,𝕁$Components}]];
selfdual$variable$content[x_]:=Module[{eqs={},diff},Do[diff=ToCanonical[x-(x/. {selfdual$ZeroRules[[i]]})];
If[diff=!=0,AppendTo[eqs,diff]],{i,Length[selfdual$ZeroRules]}];
eqs]

0.8. Harmonic Projectors

The following functions compute the harmonic components of an expression which has 1 or 2 free spherical indices and the expression has been written in terms of Y[] and derivatives only (with a maximum of 4 derivatives, as used in this work). The names of the free indices should be necessarily P, and (P,Q). More practical functions could have been implemented using the Harmonics packages of xAct, not employed here for other reasons.

ZA$Component[exprP_]:=Module[{A$int},A$int=DummyIn[TangentM2];(((((-1)/(Λ^2)))((g2[A$int,P]CD2[-A$int][exprP])//CheckZeroDerivative//ContractMetric)//.Harm$EQ$Y$Total//ContractMetric//ToCanonical)/Y[])//ToCanonical]
XA$Component[exprP_]:=Module[{A$int},A$int=DummyIn[TangentM2];((((1/(Λ^2)))((epsilong2[A$int,P]CD2[-A$int][exprP])//CheckZeroDerivative//ContractMetric)//.Harm$EQ$Y$Total//ContractMetric//ToCanonical)/Y[])//ToCanonical]
UAB$Component[exprPQ_]:=((((1/2))(((g2[P,Q]exprPQ)//CheckZeroDerivative//ContractMetric)//.Harm$EQ$Y$Total//ContractMetric//ToCanonical))/Y[])//ToCanonical
ϵAB$Component[exprPQ_]:=((((1/2))(((epsilong2[P,Q]exprPQ)//CheckZeroDerivative//ContractMetric)//.Harm$EQ$Y$Total//ContractMetric//ToCanonical))/Y[])//ToCanonical
VAB$Component[exprPQ_]:=(((((-1)/(1-((Λ^2)/2))))((((1/(Λ^2))CD2[Q][(CD2[P][exprPQ]//CheckZeroDerivative//ContractMetric)//.Harm$EQ$Y$Total//ContractMetric]+((g2[P,Q]exprPQ)/2))//CheckZeroDerivative//ContractMetric)//.Harm$EQ$Y$Total//ContractMetric//ToCanonical))/Y[])//ToCanonical
WAB$Component[exprPQ_]:=Module[{A$int},A$int=DummyIn[TangentM2];((((1/(1-((Λ^2)/2))))((((1/(Λ^2))epsilong2[A$int,Q]CD2[-A$int][(CD2[P][exprPQ]//CheckZeroDerivative//ContractMetric)//.Harm$EQ$Y$Total//ContractMetric]+((epsilong2[P,Q]exprPQ)/2))//CheckZeroDerivative//ContractMetric)//.Harm$EQ$Y$Total//ContractMetric//ToCanonical))/Y[])//ToCanonical]

0.9. Perturb, expand in manifold products, and separate harmonic dependence

The following function perturbs, expands and separates any functional tensor that depends on the fundamental tensors introduced above. It is important to recall how SeparateMetrics[] and ExpandPerturbation[] work here, to understand each step and why in that precise order. The output is an expression that depends on the tensor harmonics 𝕐, (Subscript[ℤ, A]), (Subscript[𝕏, A]),(Subscript[𝕌, A B]),(Subscript[𝕍, A B]),(Subscript[𝕎, A B])and their derivatives. In order to obtain the harmonic components, on the resulting expression one just acts with /.Dir$Harm$Def$Total to replace  (Subscript[ℤ, A]), (Subscript[𝕏, A]),(Subscript[𝕌, A B]),(Subscript[𝕍, A B]),(Subscript[𝕎, A B]) by their definitions in terms of 𝕐 and derivatives only, then //.Harm$EQ$Y$Total to reduce them to fundamental harmonic form, and finally with the functions #$Component[] to extract the components. Examples of use are given below.

Pert$Exp$Sep[Expr_,SpacetimeIndices_,SubMfldIndices_,XX$Harm$Rep_]:=Module[{IndicesDown,Expanded,Separated,Reduced},IndicesDown=((Perturbation[Expr]//ExpandPerturbation//SeparateMetric[])/.Thread[SpacetimeIndices->SubMfldIndices])//TraceProductDummy;
Expanded=ExpandProductMetric[IndicesDown/.Vol$Rep,g3];
Separated=(Expanded/.Flatten[{XX$Harm$Rep}])//.Flatten[{BG$Total,CurvN$2D,CurvS$2D}];
Reduced=(Separated)//CheckZeroDerivative;
Reduced//ContractMetric//ToCanonical]

0.10. Lorentzian 2-d GHP formalism

Defining GHP components of fundamental tensors

(Subscript[(SuperPlus[𝒽]), a b]) and (Subscript[(SuperPlus[𝒽]), a]):

DefTensor[𝒽$ll$p[],M1,PrintAs->"(Subscript[(SuperPlus[𝒽]), ℓℓ])"]
DefTensor[𝒽$ln$p[],M1,PrintAs->"(Subscript[(SuperPlus[𝒽]), ℓ𝓃])"]
DefTensor[𝒽$nn$p[],M1,PrintAs->"(Subscript[(SuperPlus[𝒽]), 𝓃𝓃])"]

𝒽$ab$p$ToFrame=MakeRule[{𝒽$ab$p[-a,-b],𝒽$nn$p[]l[-a]l[-b]+𝒽$ll$p[]n[-a]n[-b]+𝒽$ln$p[](l[-a]n[-b]+n[-a]l[-b])},MetricOn->All]

𝒽$ln$p$ToTrace=MakeRule[{𝒽$ln$p[],-(1/2)𝒽$ab$p[a,-a]},MetricOn->All]

DefTensor[𝒽$l$m[],M1,PrintAs->"(Subscript[(SuperMinus[𝒽]), ℓ])"]
DefTensor[𝒽$n$m[],M1,PrintAs->"(Subscript[(SuperMinus[𝒽]), 𝓃])"]

𝒽$a$m$ToFrame=MakeRule[{𝒽$a$m[-a],-𝒽$n$m[]l[-a]-𝒽$l$m[]n[-a]},MetricOn->All]

(Subscript[(SuperPlus[Φ]), a b]), (Subscript[(SuperPlus[Φ]), a]), and (Subscript[(SuperMinus[Φ]), a]):

DefTensor[Φ$ll$p[],M1,PrintAs->"(Subscript[(SuperPlus[Φ]), ℓℓ])"]
DefTensor[Φ$ln$p[],M1,PrintAs->"(Subscript[(SuperPlus[Φ]), ℓ𝓃])"]
DefTensor[Φ$nn$p[],M1,PrintAs->"(Subscript[(SuperPlus[Φ]), 𝓃𝓃])"]

Φ$ab$p$ToFrame=MakeRule[{Φ$ab$p[-a,-b],Φ$nn$p[]l[-a]l[-b]+Φ$ll$p[]n[-a]n[-b]+Φ$ln$p[](l[-a]n[-b]+n[-a]l[-b])},MetricOn->All]

Φ$ln$p$ToTrace=MakeRule[{Φ$ln$p[],-(1/2)Φ$ab$p[a,-a]},MetricOn->All];

DefTensor[Φ$l$p[],M1,PrintAs->"(Subscript[(SuperPlus[Φ]), ℓ])"]
DefTensor[Φ$n$p[],M1,PrintAs->"(Subscript[(SuperPlus[Φ]), 𝓃])"]

Φ$a$p$ToFrame=MakeRule[{Φ$a$p[-a],-Φ$n$p[]l[-a]-Φ$l$p[]n[-a]},MetricOn->All]

DefTensor[Φ$l$m[],M1,PrintAs->"(Subscript[(SuperMinus[Φ]), ℓ])"]
DefTensor[Φ$n$m[],M1,PrintAs->"(Subscript[(SuperMinus[Φ]), 𝓃])"]

Φ$a$m$ToFrame=MakeRule[{Φ$a$m[-a],-Φ$n$m[]l[-a]-Φ$l$m[]n[-a]},MetricOn->All]

(Subscript[(SuperPlus[Ψ]), a]), (Subscript[(SuperMinus[Ψ]), a]), (Subscript[(SuperPlus[Ξ]), a]), (Subscript[(SuperMinus[Ξ]), a]):

DefTensor[Ψ$l$p[],M1,PrintAs->"(Subscript[(SuperPlus[Ψ]), ℓ])"]
DefTensor[Ψ$n$p[],M1,PrintAs->"(Subscript[(SuperPlus[Ψ]), 𝓃])"]

Ψ$a$p$ToFrame=MakeRule[{Ψ$a$p[-a],-Ψ$n$p[]l[-a]-Ψ$l$p[]n[-a]},MetricOn->All]

DefTensor[Ψ$l$m[],M1,PrintAs->"(Subscript[(SuperMinus[Ψ]), ℓ])"]
DefTensor[Ψ$n$m[],M1,PrintAs->"(Subscript[(SuperMinus[Ψ]), 𝓃])"]

Ψ$a$m$ToFrame=MakeRule[{Ψ$a$m[-a],-Ψ$n$m[]l[-a]-Ψ$l$m[]n[-a]},MetricOn->All]

DefTensor[Ξ$l$p[],M1,PrintAs->"(Subscript[(SuperPlus[Ξ]), ℓ])"]
DefTensor[Ξ$n$p[],M1,PrintAs->"(Subscript[(SuperPlus[Ξ]), 𝓃])"]

Ξ$a$p$ToFrame=MakeRule[{Ξ$a$p[-a],-Ξ$n$p[]l[-a]-Ξ$l$p[]n[-a]},MetricOn->All]

DefTensor[Ξ$l$m[],M1,PrintAs->"(Subscript[(SuperMinus[Ξ]), ℓ])"]
DefTensor[Ξ$n$m[],M1,PrintAs->"(Subscript[(SuperMinus[Ξ]), 𝓃])"]

Ξ$a$m$ToFrame=MakeRule[{Ξ$a$m[-a],-Ξ$n$m[]l[-a]-Ξ$l$m[]n[-a]},MetricOn->All]

(Subscript[(SuperPlus[Ψ]), a b]), (Subscript[(SuperMinus[Ψ]), a b]), (Subscript[(Ψ^⊕), a b]):

DefTensor[Ψ$ll$p[],M1,PrintAs->"(Subscript[(SuperPlus[Ψ]), ℓℓ])"]
DefTensor[Ψ$ln$p[],M1,PrintAs->"(Subscript[(SuperPlus[Ψ]), ℓ𝓃])"]
DefTensor[Ψ$nn$p[],M1,PrintAs->"(Subscript[(SuperPlus[Ψ]), 𝓃𝓃])"]

Ψ$ab$p$ToFrame=MakeRule[{Ψ$ab$p[-a,-b],Ψ$nn$p[]l[-a]l[-b]+Ψ$ll$p[]n[-a]n[-b]+Ψ$ln$p[](l[-a]n[-b]+n[-a]l[-b])},MetricOn->All]

Ψ$ln$p$ToTrace=MakeRule[{Ψ$ln$p[],-(1/2)Ψ$ab$p[a,-a]},MetricOn->All]

DefTensor[Ψ$ll$m[],M1,PrintAs->"(Subscript[(SuperMinus[Ψ]), ℓℓ])"]
DefTensor[Ψ$nn$m[],M1,PrintAs->"(Subscript[(SuperMinus[Ψ]), 𝓃𝓃])"]

Ψ$ab$m$ToFrame=MakeRule[{Ψ$ab$m[-a,-b],Ψ$nn$m[]l[-a]l[-b]+Ψ$ll$m[]n[-a]n[-b]},MetricOn->All]

DefTensor[Ψ$ll$o[],M1,PrintAs->"(Subscript[(Ψ^⊕), ℓℓ])"]
DefTensor[Ψ$nn$o[],M1,PrintAs->"(Subscript[(Ψ^⊕), 𝓃𝓃])"]

Ψ$ab$o$ToFrame=MakeRule[{Ψ$ab$o[-a,-b],Ψ$nn$o[]l[-a]l[-b]+Ψ$ll$o[]n[-a]n[-b]},MetricOn->All]

(Subscript[(SuperPlus[Π]), ab]),(Subscript[(SuperMinus[Π]), ab]):

DefTensor[Π$ll$p[],M1,PrintAs->"(Subscript[(SuperPlus[Π]), ℓℓ])"]
DefTensor[Π$ln$p[],M1,PrintAs->"(Subscript[(SuperPlus[Π]), ℓ𝓃])"]
DefTensor[Π$nn$p[],M1,PrintAs->"(Subscript[(SuperPlus[Π]), 𝓃𝓃])"]

Π$ab$p$ToFrame=MakeRule[{Π$ab$p[-a,-b],Π$nn$p[]l[-a]l[-b]+Π$ll$p[]n[-a]n[-b]+Π$ln$p[](l[-a]n[-b]+n[-a]l[-b])},MetricOn->All]

Π$ln$p$ToTrace=MakeRule[{Π$ln$p[],-(1/2)Π$ab$p[a,-a]},MetricOn->All]

DefTensor[Π$ll$m[],M1,PrintAs->"(Subscript[(SuperMinus[Π]), ℓℓ])"]
DefTensor[Π$ln$m[],M1,PrintAs->"(Subscript[(SuperMinus[Π]), ℓ𝓃])"]
DefTensor[Π$nn$m[],M1,PrintAs->"(Subscript[(SuperMinus[Π]), 𝓃𝓃])"]

Π$ab$m$ToFrame=MakeRule[{Π$ab$m[-a,-b],Π$nn$m[]l[-a]l[-b]+Π$ll$m[]n[-a]n[-b]+Π$ln$m[](l[-a]n[-b]+n[-a]l[-b])},MetricOn->All]

Π$ln$m$ToTrace=MakeRule[{Π$ln$m[],-(1/2)Π$ab$m[a,-a]},MetricOn->All]

(Subscript[(SuperPlus[Π]), a]),(Subscript[(SuperMinus[Π]), a]),(Subscript[(Π^⊕), a]),(Subscript[(SuperPlus[Γ]), a]),(Subscript[(SuperMinus[Γ]), a]):

DefTensor[Π$l$p[],M1,PrintAs->"(Subscript[(SuperPlus[Π]), ℓ])"]
DefTensor[Π$n$p[],M1,PrintAs->"(Subscript[(SuperPlus[Π]), 𝓃])"]

Π$a$p$ToFrame=MakeRule[{Π$a$p[-a],-Π$n$p[]l[-a]-Π$l$p[]n[-a]},MetricOn->All]

DefTensor[Π$l$m[],M1,PrintAs->"(Subscript[(SuperMinus[Π]), ℓ])"]
DefTensor[Π$n$m[],M1,PrintAs->"(Subscript[(SuperMinus[Π]), 𝓃])"]

Π$a$m$ToFrame=MakeRule[{Π$a$m[-a],-Π$n$m[]l[-a]-Π$l$m[]n[-a]},MetricOn->All]

DefTensor[Π$l$o[],M1,PrintAs->"(Subscript[(Π^⊕), ℓ])"]
DefTensor[Π$n$o[],M1,PrintAs->"(Subscript[(Π^⊕), 𝓃])"]

Π$a$o$ToFrame=MakeRule[{Π$a$o[-a],-Π$n$o[]l[-a]-Π$l$o[]n[-a]},MetricOn->All]

DefTensor[Γ$l$p[],M1,PrintAs->"(Subscript[(SuperPlus[Γ]), ℓ])"]
DefTensor[Γ$n$p[],M1,PrintAs->"(Subscript[(SuperPlus[Γ]), 𝓃])"]

Γ$a$p$ToFrame=MakeRule[{Γ$a$p[-a],-Γ$n$p[]l[-a]-Γ$l$p[]n[-a]},MetricOn->All]

DefTensor[Γ$l$m[],M1,PrintAs->"(Subscript[(SuperMinus[Γ]), ℓ])"]
DefTensor[Γ$n$m[],M1,PrintAs->"(Subscript[(SuperMinus[Γ]), 𝓃])"]

Γ$a$m$ToFrame=MakeRule[{Γ$a$m[-a],-Γ$n$m[]l[-a]-Γ$l$m[]n[-a]},MetricOn->All]

Total:

GHP$𝒽$ab$p={𝒽$ll$p[],𝒽$nn$p[],𝒽$ln$p[]}
GHP$𝒽$a$m={𝒽$l$m[],𝒽$n$m[]}
GHP$Φ$ab$p={Φ$ll$p[],Φ$nn$p[],Φ$ln$p[]}
GHP$Φ$a$p={Φ$l$p[],Φ$n$p[]}
GHP$Φ$a$m={Φ$l$m[],Φ$n$m[]}
GHP$ℂ={Ψ$l$p[],Ψ$n$p[],Ψ$l$m[],Ψ$n$m[],Ψ$ll$p[],Ψ$ln$p[],Ψ$nn$p[],Ψ$ll$m[],Ψ$nn$m[],Ψ$ll$o[],Ψ$nn$o[],Ξ$l$p[],Ξ$n$p[],Ξ$l$m[],Ξ$n$m[]}
GHP$Π$ab$p={Π$ll$p[],Π$ln$p[],Π$nn$p[]}
GHP$Π$ab$m={Π$ll$m[],Π$ln$m[],Π$nn$m[]}
GHP$Π$a$p={Π$l$p[],Π$n$p[]}
GHP$Π$a$m={Π$l$m[],Π$n$m[]}
GHP$Π$a$o={Π$l$o[],Π$n$o[]}
GHP$Γ$a$p={Γ$l$p[],Γ$n$p[]}
GHP$Γ$a$m={Γ$l$m[],Γ$n$m[]}

0.11. Diffeomorphism generator

DefTensor[ξ[-μ],M3]

DefTensor[ξ$a$p[-a],M1,PrintAs->"(SuperPlus[ξ])"]
DefTensor[ξ$p[],M1,PrintAs->"(SuperPlus[ξ])"]
DefTensor[ξ$m[],M1,PrintAs->"(SuperMinus[ξ])"]

ξ$Harm$Rep=Flatten[{MakeRule[{ξ[-a],ξ$a$p[-a]Y[]}],MakeRule[{ξ[-A],ξ$p[]Z[-A]+ξ$m[]X[-A]}]}]

The following is the gauge-compensating diffeo generator:

ξ$To$Met=Flatten[{MakeRule[{ξ$a$p[-a],-𝒽$a$p[-a]+(((r[])^2)/2)CD1[-a][((𝒽$o[])/((r[])^2))]},MetricOn->All],MakeRule[{ξ$p[],-((𝒽$o[])/2)},MetricOn->All],MakeRule[{ξ$m[],-((𝒽$m[])/2)},MetricOn->All]}]

1. Fundamental Equations [RUN]

1.0. Perturbed special identities of 2D curvature

Commutators

CommCovs$𝒽$ab$p$FreeSecond=MakeRule[{CD1[-a][CD1[c][𝒽$ab$p[-b,-c]]],(-2*𝒽$ab$p[-a,-b])/r[]^2+(g1[-a,-b]*𝒽$ab$p[c,-c])/r[]^2+2*𝒽$ab$p[-a,-b]*𝒫[]-g1[-a,-b]*𝒽$ab$p[c,-c]*𝒫[]-𝒽$ab$p[-a,-b]*Τ[c,-c]+(g1[-a,-b]*𝒽$ab$p[c,-c]*Τ[d,-d])/2+CD1[-c][CD1[-a][𝒽$ab$p[-b,c]]]+(2*𝒽$ab$p[-a,-b]*CD1[-c][r[]]*CD1[c][r[]])/r[]^2-(g1[-a,-b]*𝒽$ab$p[d,-d]*CD1[-c][r[]]*CD1[c][r[]])/r[]^2},MetricOn->All];
CommCovs$𝒽$ab$p$FreeFirst=MakeRule[{CD1[c][CD1[-a][𝒽$ab$p[-b,-c]]],(2*𝒽$ab$p[-a,-b])/r[]^2-(g1[-a,-b]*𝒽$ab$p[c,-c])/r[]^2-2*𝒽$ab$p[-a,-b]*𝒫[]+g1[-a,-b]*𝒽$ab$p[c,-c]*𝒫[]+𝒽$ab$p[-a,-b]*Τ[c,-c]-(g1[-a,-b]*𝒽$ab$p[c,-c]*Τ[d,-d])/2+CD1[-a][CD1[-c][𝒽$ab$p[-b,c]]]-(2*𝒽$ab$p[-a,-b]*CD1[-c][r[]]*CD1[c][r[]])/r[]^2+(g1[-a,-b]*𝒽$ab$p[d,-d]*CD1[-c][r[]]*CD1[c][r[]])/r[]^2},MetricOn->All];
CommCovs$𝒽$a$m$FreeFirst=MakeRule[{CD1[-b][CD1[-a][𝒽$a$m[b]]],𝒽$a$m[-a]/r[]^2-𝒽$a$m[-a]*𝒫[]+(𝒽$a$m[-a]*Τ[b,-b])/2+CD1[-a][CD1[-b][𝒽$a$m[b]]]-(𝒽$a$m[-a]*CD1[-b][r[]]*CD1[b][r[]])/r[]^2},MetricOn->All];
CommCovs$𝒽$a$m$FreeSecond=MakeRule[{CD1[-a][CD1[-b][𝒽$a$m[b]]],-(𝒽$a$m[-a]/r[]^2)+𝒽$a$m[-a]*𝒫[]-(𝒽$a$m[-a]*Τ[b,-b])/2+CD1[-b][CD1[-a][𝒽$a$m[b]]]+(𝒽$a$m[-a]*CD1[-b][r[]]*CD1[b][r[]])/r[]^2},MetricOn->All];
CommCovs$𝒽$p$FreeFirst=Flatten[{MakeRule[{CD1[b][CD1[-b][CD1[-a][𝒽$p[]]]],CD1[-a][𝒽$p[]]/r[]^2-𝒫[]*CD1[-a][𝒽$p[]]+(Τ[b,-b]*CD1[-a][𝒽$p[]])/2+CD1[-a][CD1[-b][CD1[b][𝒽$p[]]]]-(CD1[-a][𝒽$p[]]*CD1[-b][r[]]*CD1[b][r[]])/r[]^2,MetricOn->All}],MakeRule[{CD1[b][CD1[-b][CD1[-a][𝒽$p[]]]],CD1[-a][𝒽$p[]]/r[]^2-𝒫[]*CD1[-a][𝒽$p[]]+(Τ[b,-b]*CD1[-a][𝒽$p[]])/2+CD1[-a][CD1[-b][CD1[b][𝒽$p[]]]]-(CD1[-a][𝒽$p[]]*CD1[-b][r[]]*CD1[b][r[]])/r[]^2},MetricOn->All]}];
CommCovs$𝒽$p$FreeSecond=MakeRule[{CD1[-a][CD1[b][CD1[-b][𝒽$p[]]]],-(CD1[-a][𝒽$p[]]/r[]^2)+𝒫[]*CD1[-a][𝒽$p[]]-(Τ[b,-b]*CD1[-a][𝒽$p[]])/2+CD1[-b][CD1[-a][CD1[b][𝒽$p[]]]]+(CD1[-a][𝒽$p[]]*CD1[-b][r[]]*CD1[b][r[]])/r[]^2},MetricOn->All];

Gauss-Bonnet identity

FirstOrder$Gauss$Bonnet$Rule=MakeRule[{CD1[c][CD1[-c][𝒽$ab$p[-a,-b]]],2*(𝒽$ab$p[-a,-b]/r[]^2-(g1[-a,-b]*𝒽$ab$p[c,-c])/(2*r[]^2)-𝒽$ab$p[-a,-b]*𝒫[]+(g1[-a,-b]*𝒽$ab$p[c,-c]*𝒫[])/2+(𝒽$ab$p[-a,-b]*Τ[c,-c])/2-(g1[-a,-b]*𝒽$ab$p[c,-c]*Τ[d,-d])/4+CD1[-a][CD1[-c][𝒽$ab$p[-b,c]]]/2-CD1[-b][CD1[-a][𝒽$ab$p[c,-c]]]/2+CD1[-b][CD1[-c][𝒽$ab$p[-a,c]]]/2-(𝒽$ab$p[-a,-b]*CD1[-c][r[]]*CD1[c][r[]])/r[]^2+(g1[-a,-b]*𝒽$ab$p[d,-d]*CD1[-c][r[]]*CD1[c][r[]])/(2*r[]^2)-(g1[-a,-b]*CD1[-d][CD1[-c][𝒽$ab$p[c,d]]])/2+(g1[-a,-b]*CD1[-d][CD1[d][𝒽$ab$p[c,-c]]])/2)},MetricOn->All]

1.1. δ (Subscript[ℂ, μ ν ρ σ]) in terms of (Subscript[𝒽, μ ν ])

(Subscript[(SuperPlus[Ψ]), a b])

Ψ$ab$p$To$Met=MakeRule[{Ψ$ab$p[-a,-b],(-1/6*(g1[-a,-b]*𝒽$ab$p[c,-c])+(Λ^2*g1[-a,-b]*𝒽$ab$p[c,-c])/12-(g1[-a,-b]*𝒽$p[])/(3*r[]^2)-(Λ^2*g1[-a,-b]*𝒽$p[])/(6*r[]^2)-(2*r[]^2*𝒽$ab$p[-a,-b]*𝒫[])/3+(g1[-a,-b]*r[]^2*𝒽$ab$p[c,-c]*𝒫[])/3+(g1[-a,-b]*𝒽$p[]*𝒫[])/3-(g1[-a,-b]*r[]^2*𝒽$ab$p[c,d]*Τ[-c,-d])/6+(r[]^2*𝒽$ab$p[-a,-b]*Τ[c,-c])/6-(g1[-a,-b]*𝒽$p[]*Τ[c,-c])/6+(I*Λ^2*epsilong1[-a,-c]*𝒽$a$m[c]*CD1[-b][r[]])/r[]-(I/2)*Λ^2*epsilong1[-a,-c]*CD1[-b][𝒽$a$m[c]]+(r[]^2*CD1[-b][CD1[-a][𝒽$ab$p[c,-c]]])/2-(r[]^2*CD1[-c][CD1[-a][𝒽$ab$p[-b,c]]])/2-(r[]^2*CD1[-c][CD1[-b][𝒽$ab$p[-a,c]]])/2+(r[]^2*CD1[-c][CD1[c][𝒽$ab$p[-a,-b]]])/2-(g1[-a,-b]*CD1[-c][CD1[c][𝒽$p[]]])/6-(I*Λ^2*epsilong1[-a,-c]*𝒽$a$m[-b]*CD1[c][r[]])/r[]+(g1[-a,-b]*𝒽$ab$p[d,-d]*CD1[-c][r[]]*CD1[c][r[]])/6-(g1[-a,-b]*𝒽$p[]*CD1[-c][r[]]*CD1[c][r[]])/(3*r[]^2)-(g1[-a,-b]*r[]*CD1[-c][𝒽$ab$p[d,-d]]*CD1[c][r[]])/6+(2*g1[-a,-b]*CD1[-c][𝒽$p[]]*CD1[c][r[]])/(3*r[])+(I/2)*Λ^2*epsilong1[-a,-c]*CD1[c][𝒽$a$m[-b]]+(g1[-a,-b]*r[]*CD1[c][r[]]*CD1[-d][𝒽$ab$p[-c,d]])/3+(g1[-a,-b]*r[]^2*CD1[-d][CD1[-c][𝒽$ab$p[c,d]]])/3-(g1[-a,-b]*r[]^2*CD1[-d][CD1[d][𝒽$ab$p[c,-c]]])/3-(g1[-a,-b]*𝒽$ab$p[-c,-d]*CD1[c][r[]]*CD1[d][r[]])/3)/2},MetricOn->All];

(*Pert$Exp$Sep[WeylToRiemann[CC[-μ,-ν,-ρ,-σ]/.ℂ$To$Weyl],{μ,ν,ρ,σ},{P,a,Q,b},{𝒽$Harm$Rep}];
%/.Dir$Harm$Def$Total//ContractMetric//ToCanonical;
%//.Harm$EQ$Y$Total//ContractMetric//ToCanonical;
UAB$Component[%];
ToCanonical[%-(Ψ$ab$p[-a,-b]/.Ψ$ab$p$To$Met)]*)

(Subscript[(SuperMinus[Ψ]), a b])

Ψ$ab$m$To$Met=MakeRule[{Ψ$ab$m[-a,-b],((-2*I)*epsilong1[-a,c]*𝒽$ab$p[-b,-c]+I*epsilong1[-a,-b]*𝒽$ab$p[c,-c]+2*(CD1[-a][𝒽$a$m[-b]]+CD1[-b][𝒽$a$m[-a]]-g1[-a,-b]*CD1[-c][𝒽$a$m[c]]))/4},MetricOn->All];

(*Pert$Exp$Sep[WeylToRiemann[CC[-μ,-ν,-ρ,-σ]/.ℂ$To$Weyl],{μ,ν,ρ,σ},{P,a,Q,b},{𝒽$Harm$Rep}];
%/.Dir$Harm$Def$Total//ContractMetric//ToCanonical;
%//.Harm$EQ$Y$Total//ContractMetric//ToCanonical;
WAB$Component[%]//Simplify;
ToCanonical[%-(Ψ$ab$m[-a,-b]/.Ψ$ab$m$To$Met)]*)

(Subscript[(Ψ^⊕), a b])

Ψ$ab$o$To$Met=MakeRule[{Ψ$ab$o[-a,-b],-((𝒽$ab$p[-a,-b]/2-(Λ^2*𝒽$ab$p[-a,-b])/4-(g1[-a,-b]*𝒽$ab$p[c,-c])/4+(Λ^2*g1[-a,-b]*𝒽$ab$p[c,-c])/8+(I/2)*epsilong1[-a,-c]*CD1[-b][𝒽$a$m[c]]-(I/4)*Λ^2*epsilong1[-a,-c]*CD1[-b][𝒽$a$m[c]]-(I/2)*epsilong1[-a,-b]*CD1[-c][𝒽$a$m[c]]+(I/4)*Λ^2*epsilong1[-a,-b]*CD1[-c][𝒽$a$m[c]]+(I/2)*epsilong1[-a,-c]*CD1[c][𝒽$a$m[-b]]-(I/4)*Λ^2*epsilong1[-a,-c]*CD1[c][𝒽$a$m[-b]])/(1-Λ^2/2))},MetricOn->All];

(*Pert$Exp$Sep[WeylToRiemann[CC[-μ,-ν,-ρ,-σ]/.ℂ$To$Weyl],{μ,ν,ρ,σ},{P,a,Q,b},{𝒽$Harm$Rep}];
%/.Dir$Harm$Def$Total//ContractMetric//ToCanonical;
%//.Harm$EQ$Y$Total//ContractMetric//ToCanonical;
VAB$Component[%];
ToCanonical[%-(Ψ$ab$o[-a,-b]/.Ψ$ab$o$To$Met)]*)

(Subscript[(SuperPlus[Ψ]), a])

Ψ$a$p$To$Met=MakeRule[{Ψ$a$p[-b],(((3*I)/4)*𝒽$a$m[-b])/r[]^2+((I/4)*Λ^2*𝒽$a$m[-b])/r[]^2-(I/2)*𝒽$a$m[-b]*𝒫[]-(I/4)*𝒽$a$m[c]*Τ[-b,-c]+(I/2)*𝒽$a$m[-b]*Τ[c,-c]-((I/2)*𝒽$a$m[c]*CD1[-b][r[]]*CD1[-c][r[]])/r[]^2+(epsilong1[-b,c]*CD1[-c][𝒽$ab$p[d,-d]])/4+((I/2)*CD1[-b][r[]]*CD1[-c][𝒽$a$m[c]])/r[]-(I/4)*CD1[-c][CD1[-b][𝒽$a$m[c]]]+(I/4)*CD1[-c][CD1[c][𝒽$a$m[-b]]]-(epsilong1[-c,d]*𝒽$ab$p[-b,-d]*CD1[c][r[]])/(2*r[])-(epsilong1[-b,-c]*𝒽$ab$p[d,-d]*CD1[c][r[]])/(4*r[])-(epsilong1[-b,-c]*𝒽$p[]*CD1[c][r[]])/(2*r[]^3)-(((3*I)/4)*𝒽$a$m[-b]*CD1[-c][r[]]*CD1[c][r[]])/r[]^2-((I/2)*CD1[-c][𝒽$a$m[-b]]*CD1[c][r[]])/r[]+(epsilong1[-b,-c]*CD1[c][𝒽$p[]])/(4*r[]^2)-(epsilong1[c,d]*CD1[-d][𝒽$ab$p[-b,-c]])/2-(epsilong1[-b,c]*CD1[-d][𝒽$ab$p[-c,d]])/4},MetricOn->All];

(*Pert$Exp$Sep[WeylToRiemann[CC[-μ,-ν,-ρ,-σ]/.ℂ$To$Weyl],{μ,ν,ρ,σ},{P,b,c,d},{𝒽$Harm$Rep}];
%/.Dir$Harm$Def$Total//ContractMetric//ToCanonical;
((-1/2)epsilong1[c,d]%//ContractMetric//ToCanonical)//.Harm$EQ$Y$Total//ContractMetric//ToCanonical;
ZA$Component[%];
ToCanonical[%-(Ψ$a$p[-b]/.Ψ$a$p$To$Met)]*)

(Subscript[(SuperMinus[Ψ]), a])

Ψ$a$m$To$Met=MakeRule[{Ψ$a$m[-b],(epsilong1[-b,-c]*𝒽$a$m[c])/(4*r[]^2)-(Λ^2*epsilong1[-b,-c]*𝒽$a$m[c])/(4*r[]^2)+(epsilong1[-b,-c]*𝒽$a$m[c]*𝒫[])/6+(epsilong1[-c,d]*𝒽$a$m[c]*Τ[-b,-d])/2-(epsilong1[-b,d]*𝒽$a$m[c]*Τ[-c,-d])/4+(epsilong1[-b,-c]*𝒽$a$m[c]*Τ[d,-d])/3+((I/4)*𝒽$ab$p[c,-c]*CD1[-b][r[]])/r[]-((I/2)*𝒽$p[]*CD1[-b][r[]])/r[]^3-(I/4)*CD1[-b][𝒽$ab$p[c,-c]]+((I/4)*CD1[-b][𝒽$p[]])/r[]^2+(epsilong1[-c,-d]*CD1[-b][CD1[d][𝒽$a$m[c]]])/2+(I/4)*CD1[-c][𝒽$ab$p[-b,c]]-(epsilong1[-b,-d]*CD1[-c][CD1[d][𝒽$a$m[c]]])/4-((I/2)*𝒽$ab$p[-b,-c]*CD1[c][r[]])/r[]+(epsilong1[-c,-d]*CD1[-b][𝒽$a$m[d]]*CD1[c][r[]])/(2*r[])+(epsilong1[-b,-c]*CD1[c][r[]]*CD1[-d][𝒽$a$m[d]])/(2*r[])+(epsilong1[-b,-c]*CD1[-d][CD1[d][𝒽$a$m[c]]])/4+(epsilong1[-b,-d]*𝒽$a$m[c]*CD1[-c][r[]]*CD1[d][r[]])/(2*r[]^2)-(epsilong1[-b,-c]*𝒽$a$m[c]*CD1[-d][r[]]*CD1[d][r[]])/(4*r[]^2)+(epsilong1[-c,-d]*CD1[c][r[]]*CD1[d][𝒽$a$m[-b]])/(2*r[])-(epsilong1[-b,-d]*CD1[c][r[]]*CD1[d][𝒽$a$m[-c]])/(2*r[])},MetricOn->All];

(*Pert$Exp$Sep[WeylToRiemann[CC[-μ,-ν,-ρ,-σ]/.ℂ$To$Weyl],{μ,ν,ρ,σ},{P,b,c,d},{𝒽$Harm$Rep}];
%/.Dir$Harm$Def$Total//ContractMetric//ToCanonical;
((-1/2)epsilong1[c,d]%//ContractMetric//ToCanonical)//.Harm$EQ$Y$Total//ContractMetric//ToCanonical;
XA$Component[%];
ToCanonical[%-(Ψ$a$m[-b]/.Ψ$a$m$To$Met)]*)

(SuperPlus[Ψ])

Ψ$p$To$Met=MakeRule[{Ψ$p[],(-2*𝒽$ab$p[a,-a])/(3*r[]^2)+(Λ^2*𝒽$ab$p[a,-a])/(12*r[]^2)+(2*𝒽$p[])/(3*r[]^4)-(Λ^2*𝒽$p[])/(6*r[]^4)+(𝒽$ab$p[a,-a]*𝒫[])/6-(𝒽$ab$p[a,b]*Τ[-a,-b])/6+(𝒽$p[]*Τ[a,-a])/(6*r[]^2)-(𝒽$ab$p[a,-a]*Τ[b,-b])/12-CD1[-a][CD1[a][𝒽$p[]]]/(6*r[]^2)+(2*𝒽$ab$p[b,-b]*CD1[-a][r[]]*CD1[a][r[]])/(3*r[]^2)-(4*𝒽$p[]*CD1[-a][r[]]*CD1[a][r[]])/(3*r[]^4)-(CD1[-a][𝒽$ab$p[b,-b]]*CD1[a][r[]])/(6*r[])+(2*CD1[-a][𝒽$p[]]*CD1[a][r[]])/(3*r[]^3)+(CD1[a][r[]]*CD1[-b][𝒽$ab$p[-a,b]])/(3*r[])-CD1[-b][CD1[-a][𝒽$ab$p[a,b]]]/6+CD1[-b][CD1[b][𝒽$ab$p[a,-a]]]/6-(I*Λ^2*epsilong1[-a,-b]*𝒽$a$m[a]*CD1[b][r[]])/r[]^3-(𝒽$ab$p[-a,-b]*CD1[a][r[]]*CD1[b][r[]])/(3*r[]^2)+((I/2)*Λ^2*epsilong1[-a,-b]*CD1[b][𝒽$a$m[a]])/r[]^2},MetricOn->All];

(*Pert$Exp$Sep[WeylToRiemann[CC[-μ,-ν,-ρ,-σ]/.ℂ$To$Weyl],{μ,ν,ρ,σ},{a,b,c,d},{𝒽$Harm$Rep}];
%/.Dir$Harm$Def$Total//ContractMetric//ToCanonical;
%//.Harm$EQ$Y$Total//ContractMetric//ToCanonical;
(1/4)epsilong1[a,b]epsilong1[c,d]%//ContractMetric//ToCanonical;
ToCanonical[%/Y[]];
ToCanonical[%-(Ψ$p[]/.Ψ$p$To$Met)]*)

(SuperMinus[Ψ])

Ψ$m$To$Met=MakeRule[{Ψ$m[],(-1/6*I)*𝒽$ab$p[a,-a]+(I/12)*Λ^2*𝒽$ab$p[a,-a]-((I/3)*𝒽$p[])/r[]^2-((I/6)*Λ^2*𝒽$p[])/r[]^2+(I/3)*𝒽$p[]*𝒫[]-(I/6)*r[]^2*𝒽$ab$p[a,b]*Τ[-a,-b]-(I/6)*𝒽$p[]*Τ[a,-a]+(I/12)*r[]^2*𝒽$ab$p[a,-a]*Τ[b,-b]-(I/6)*CD1[-a][CD1[a][𝒽$p[]]]+(I/6)*𝒽$ab$p[b,-b]*CD1[-a][r[]]*CD1[a][r[]]-((I/3)*𝒽$p[]*CD1[-a][r[]]*CD1[a][r[]])/r[]^2-(I/6)*r[]*CD1[-a][𝒽$ab$p[b,-b]]*CD1[a][r[]]+(((2*I)/3)*CD1[-a][𝒽$p[]]*CD1[a][r[]])/r[]+(I/3)*r[]*CD1[a][r[]]*CD1[-b][𝒽$ab$p[-a,b]]-(I/6)*r[]^2*CD1[-b][CD1[-a][𝒽$ab$p[a,b]]]+(I/6)*r[]^2*CD1[-b][CD1[b][𝒽$ab$p[a,-a]]]+(Λ^2*epsilong1[-a,-b]*𝒽$a$m[a]*CD1[b][r[]])/r[]-(I/3)*𝒽$ab$p[-a,-b]*CD1[a][r[]]*CD1[b][r[]]-(Λ^2*epsilong1[-a,-b]*CD1[b][𝒽$a$m[a]])/2},MetricOn->All];

(*Pert$Exp$Sep[WeylToRiemann[CC[-μ,-ν,-ρ,-σ]/.ℂ$To$Weyl],{μ,ν,ρ,σ},{a,b,A,B},{𝒽$Harm$Rep}];
%/.Dir$Harm$Def$Total//ContractMetric//ToCanonical;
((-1/4)epsilong2[A,B]epsilong1[a,b]%//ContractMetric//ToCanonical)//.Harm$EQ$Y$Total//ContractMetric//ToCanonical;
ToCanonical[%/Y[]];
ToCanonical[%-(Ψ$m[]/.Ψ$m$To$Met)]*)

(Subscript[(SuperPlus[Ξ]), a])

Ξ$a$p$To$Met=MakeRule[{Ξ$a$p[-a],((3*I)/4)*epsilong1[-a,-b]*𝒽$a$m[b]-(I/4)*Λ^2*epsilong1[-a,-b]*𝒽$a$m[b]+(I/2)*epsilong1[-b,c]*r[]^2*𝒽$a$m[b]*Τ[-a,-c]-(I/4)*epsilong1[-a,c]*r[]^2*𝒽$a$m[b]*Τ[-b,-c]+(I/2)*epsilong1[-a,-b]*r[]^2*𝒽$a$m[b]*Τ[c,-c]-(r[]*𝒽$ab$p[b,-b]*CD1[-a][r[]])/4+(𝒽$p[]*CD1[-a][r[]])/(2*r[])+(r[]^2*CD1[-a][𝒽$ab$p[b,-b]])/4-CD1[-a][𝒽$p[]]/4-(r[]^2*CD1[-b][𝒽$ab$p[-a,b]])/4-(I/4)*epsilong1[-a,-c]*r[]^2*CD1[-b][CD1[c][𝒽$a$m[b]]]+(r[]*𝒽$ab$p[-a,-b]*CD1[b][r[]])/2+(I/2)*epsilong1[-b,-c]*r[]*CD1[-a][𝒽$a$m[c]]*CD1[b][r[]]+(I/2)*epsilong1[-a,-b]*r[]*CD1[b][r[]]*CD1[-c][𝒽$a$m[c]]+(I/4)*epsilong1[-a,-b]*r[]^2*CD1[-c][CD1[c][𝒽$a$m[b]]]+(I/2)*epsilong1[-a,-c]*𝒽$a$m[b]*CD1[-b][r[]]*CD1[c][r[]]-((3*I)/4)*epsilong1[-a,-b]*𝒽$a$m[b]*CD1[-c][r[]]*CD1[c][r[]]+(I/2)*epsilong1[-b,-c]*r[]*CD1[b][r[]]*CD1[c][𝒽$a$m[-a]]-(I/2)*epsilong1[-a,-c]*r[]*CD1[b][r[]]*CD1[c][𝒽$a$m[-b]]+(I/2)*epsilong1[-b,-c]*r[]^2*CD1[c][CD1[-a][𝒽$a$m[b]]]-(I/2)*epsilong1[-b,-c]*r[]^2*CD1[c][CD1[b][𝒽$a$m[-a]]]},MetricOn->All];

(*Pert$Exp$Sep[WeylToRiemann[CC[-μ,-ν,-ρ,-σ]/.ℂ$To$Weyl],{μ,ν,ρ,σ},{A,B,P,a},{𝒽$Harm$Rep}];
%/.Dir$Harm$Def$Total//ContractMetric//ToCanonical;
((1/2)epsilong2[A,B]%//ContractMetric//ToCanonical)//.Harm$EQ$Y$Total//ContractMetric//ToCanonical;
XA$Component[%];
ToCanonical[%-(Ξ$a$p[-a]/.Ξ$a$p$To$Met)]*)

(Subscript[(SuperMinus[Ξ]), a])

Ξ$a$m$To$Met=MakeRule[{Ξ$a$m[-a],-1/4*𝒽$a$m[-a]-(Λ^2*𝒽$a$m[-a])/4+(r[]^2*𝒽$a$m[-a]*𝒫[])/3+(r[]^2*𝒽$a$m[b]*Τ[-a,-b])/4-(r[]^2*𝒽$a$m[-a]*Τ[b,-b])/3+(𝒽$a$m[b]*CD1[-a][r[]]*CD1[-b][r[]])/2+(I/4)*epsilong1[-a,b]*r[]^2*CD1[-b][𝒽$ab$p[c,-c]]-(r[]*CD1[-a][r[]]*CD1[-b][𝒽$a$m[b]])/2+(r[]^2*CD1[-b][CD1[-a][𝒽$a$m[b]]])/4-(r[]^2*CD1[-b][CD1[b][𝒽$a$m[-a]]])/4-(I/2)*epsilong1[-b,c]*r[]*𝒽$ab$p[-a,-c]*CD1[b][r[]]-(I/4)*epsilong1[-a,-b]*r[]*𝒽$ab$p[c,-c]*CD1[b][r[]]-((I/2)*epsilong1[-a,-b]*𝒽$p[]*CD1[b][r[]])/r[]+(𝒽$a$m[-a]*CD1[-b][r[]]*CD1[b][r[]])/4+(r[]*CD1[-b][𝒽$a$m[-a]]*CD1[b][r[]])/2+(I/4)*epsilong1[-a,-b]*CD1[b][𝒽$p[]]-(I/2)*epsilong1[b,c]*r[]^2*CD1[-c][𝒽$ab$p[-a,-b]]-(I/4)*epsilong1[-a,b]*r[]^2*CD1[-c][𝒽$ab$p[-b,c]]},MetricOn->All];

(*Pert$Exp$Sep[WeylToRiemann[CC[-μ,-ν,-ρ,-σ]/.ℂ$To$Weyl],{μ,ν,ρ,σ},{A,B,P,a},{𝒽$Harm$Rep}];
%/.Dir$Harm$Def$Total//ContractMetric//ToCanonical;
((1/2)epsilong2[A,B]%//ContractMetric//ToCanonical)//.Harm$EQ$Y$Total//ContractMetric//ToCanonical;
ZA$Component[%];
ToCanonical[%-(Ξ$a$m[-a]/.Ξ$a$m$To$Met)]*)

(SuperPlus[Ξ])

Ξ$p$To$Met=MakeRule[{Ξ$p[],-1/3*(r[]^2*𝒽$ab$p[a,-a])-(Λ^2*r[]^2*𝒽$ab$p[a,-a])/12+(4*𝒽$p[])/3+(Λ^2*𝒽$p[])/6+(r[]^4*𝒽$ab$p[a,-a]*𝒫[])/6-(2*r[]^2*𝒽$p[]*𝒫[])/3+(r[]^4*𝒽$ab$p[a,b]*Τ[-a,-b])/6+(r[]^2*𝒽$p[]*Τ[a,-a])/2-(r[]^4*𝒽$ab$p[a,-a]*Τ[b,-b])/4+(r[]^2*CD1[-a][CD1[a][𝒽$p[]]])/6+(r[]^2*𝒽$ab$p[b,-b]*CD1[-a][r[]]*CD1[a][r[]])/3-(2*𝒽$p[]*CD1[-a][r[]]*CD1[a][r[]])/3+(r[]^3*CD1[-a][𝒽$ab$p[b,-b]]*CD1[a][r[]])/6-(2*r[]*CD1[-a][𝒽$p[]]*CD1[a][r[]])/3-(r[]^3*CD1[a][r[]]*CD1[-b][𝒽$ab$p[-a,b]])/3+(r[]^4*CD1[-b][CD1[-a][𝒽$ab$p[a,b]]])/6-(r[]^4*CD1[-b][CD1[b][𝒽$ab$p[a,-a]]])/6+I*Λ^2*epsilong1[-a,-b]*r[]*𝒽$a$m[a]*CD1[b][r[]]+(r[]^2*𝒽$ab$p[-a,-b]*CD1[a][r[]]*CD1[b][r[]])/3-(I/2)*Λ^2*epsilong1[-a,-b]*r[]^2*CD1[b][𝒽$a$m[a]]},MetricOn->All];

(*Pert$Exp$Sep[WeylToRiemann[CC[-μ,-ν,-ρ,-σ]/.ℂ$To$Weyl],{μ,ν,ρ,σ},{A,B,P,Q},{𝒽$Harm$Rep}];
%/.Dir$Harm$Def$Total//ContractMetric//ToCanonical;
((1/4)epsilong2[A,B]epsilong2[P,Q]%//ContractMetric//ToCanonical)//.Harm$EQ$Y$Total//ContractMetric//ToCanonical;
ToCanonical[%/Y[]];
ToCanonical[%-(Ξ$p[]/.Ξ$p$To$Met)]*)

Total

Pert$ℂ$To$Met=Flatten[{Ψ$ab$p$To$Met,Ψ$ab$m$To$Met,Ψ$ab$o$To$Met,Ψ$a$p$To$Met,Ψ$a$m$To$Met,Ψ$p$To$Met,Ψ$m$To$Met,Ξ$a$p$To$Met,Ξ$a$m$To$Met,Ξ$p$To$Met}];

1.2. δ (Subscript[𝕁 , μ ν ρ]) in terms of (Subscript[𝒽, μ ν]) and δ (Subscript[(Subscript[𝒯 , μ ν ]),  ])

(Subscript[(SuperPlus[Π]), a b])

Π$ab$p$To$Met$T=MakeRule[{Π$ab$p[-a,-b],(𝒽$ab$p[-a,-b]*𝒫[])/4-(g1[-a,-b]*𝒽$p[]*𝒫[])/(3*r[]^2)+(𝒽$ab$p[-b,c]*Τ[-a,-c])/8+(𝒽$ab$p[-a,c]*Τ[-b,-c])/8-(g1[-a,-b]*𝒽$ab$p[c,d]*Τ[-c,-d])/6-Φ$ab$p[-a,-b]/2+(g1[-a,-b]*Φ$ab$p[c,-c])/6+(g1[-a,-b]*Φ$p[])/(3*r[]^2)+((I/4)*epsilong1[-b,d]*𝒽$a$m[c]*Τ[-c,-d]*CD1[-a][r[]])/r[]-((I/4)*epsilong1[-b,-c]*Φ$a$m[c]*CD1[-a][r[]])/r[]+(Φ$a$p[-b]*CD1[-a][r[]])/(4*r[])+(I/8)*epsilong1[-b,-c]*𝒫[]*CD1[-a][𝒽$a$m[c]]-(I/8)*epsilong1[-b,d]*Τ[-c,-d]*CD1[-a][𝒽$a$m[c]]+CD1[-a][Φ$a$p[-b]]/4+((I/4)*epsilong1[-a,d]*𝒽$a$m[c]*Τ[-c,-d]*CD1[-b][r[]])/r[]-((I/4)*epsilong1[-a,-c]*Φ$a$m[c]*CD1[-b][r[]])/r[]+(Φ$a$p[-a]*CD1[-b][r[]])/(4*r[])+(I/8)*epsilong1[-a,-c]*𝒫[]*CD1[-b][𝒽$a$m[c]]-(I/8)*epsilong1[-a,d]*Τ[-c,-d]*CD1[-b][𝒽$a$m[c]]+CD1[-b][Φ$a$p[-a]]/4-(I/4)*epsilong1[-b,d]*𝒽$a$m[c]*CD1[-c][Τ[-a,-d]]-(I/4)*epsilong1[-a,d]*𝒽$a$m[c]*CD1[-c][Τ[-b,-d]]+(I/8)*epsilong1[-b,-c]*𝒫[]*CD1[c][𝒽$a$m[-a]]+(I/8)*epsilong1[-b,d]*Τ[-c,-d]*CD1[c][𝒽$a$m[-a]]+(I/8)*epsilong1[-a,-c]*𝒫[]*CD1[c][𝒽$a$m[-b]]+(I/8)*epsilong1[-a,d]*Τ[-c,-d]*CD1[c][𝒽$a$m[-b]]-(I/4)*epsilong1[-b,-c]*CD1[c][Φ$a$m[-a]]-(I/4)*epsilong1[-a,-c]*CD1[c][Φ$a$m[-b]]+(I/4)*epsilong1[-b,d]*𝒽$a$m[c]*CD1[-d][Τ[-a,-c]]+(I/4)*epsilong1[-a,d]*𝒽$a$m[c]*CD1[-d][Τ[-b,-c]]},MetricOn->All];

(*Pert$Exp$Sep[𝕁[-μ,-ν,-ρ]/.𝕁$To$𝒯,{μ,ν,ρ},{a,b,P},{𝒽$Harm$Rep,𝒯$Harm$Rep}];
%/.Dir$Harm$Def$Total//ContractMetric//ToCanonical;
%//.Harm$EQ$Y$Total//ContractMetric//ToCanonical;
Symetrise[ZA$Component[%],{-a,-b}]//ToCanonical;
ToCanonical[%-(Π$ab$p[-a,-b]/.Π$ab$p$To$Met$T)]*)

(Subscript[(SuperMinus[Π]), a b])

Π$ab$m$To$Met$T=MakeRule[{Π$ab$m[-a,-b],(I/8)*epsilong1[-b,c]*𝒽$ab$p[-a,-c]*𝒫[]+(I/8)*epsilong1[-a,c]*𝒽$ab$p[-b,-c]*𝒫[]+(I/8)*epsilong1[-b,c]*𝒽$ab$p[-a,d]*Τ[-c,-d]+(I/8)*epsilong1[-a,c]*𝒽$ab$p[-b,d]*Τ[-c,-d]-(I/4)*epsilong1[-b,c]*Φ$ab$p[-a,-c]-(I/4)*epsilong1[-a,c]*Φ$ab$p[-b,-c]-(𝒽$a$m[c]*Τ[-b,-c]*CD1[-a][r[]])/(4*r[])+(Φ$a$m[-b]*CD1[-a][r[]])/(4*r[])+((I/4)*epsilong1[-b,-c]*Φ$a$p[c]*CD1[-a][r[]])/r[]-(𝒫[]*CD1[-a][𝒽$a$m[-b]])/4+(Τ[-b,-c]*CD1[-a][𝒽$a$m[c]])/8-(𝒽$a$m[-b]*CD1[-a][𝒫[]])/6-(𝒽$a$m[-b]*CD1[-a][Τ[c,-c]])/12+CD1[-a][Φ$a$m[-b]]/4-(𝒽$a$m[c]*Τ[-a,-c]*CD1[-b][r[]])/(4*r[])+(Φ$a$m[-a]*CD1[-b][r[]])/(4*r[])+((I/4)*epsilong1[-a,-c]*Φ$a$p[c]*CD1[-b][r[]])/r[]-(𝒫[]*CD1[-b][𝒽$a$m[-a]])/4+(Τ[-a,-c]*CD1[-b][𝒽$a$m[c]])/8-(𝒽$a$m[-a]*CD1[-b][𝒫[]])/6-(𝒽$a$m[-a]*CD1[-b][Τ[c,-c]])/12+CD1[-b][Φ$a$m[-a]]/4-(Τ[-b,-c]*CD1[c][𝒽$a$m[-a]])/8-(Τ[-a,-c]*CD1[c][𝒽$a$m[-b]])/8+(I/4)*epsilong1[-b,-c]*CD1[c][Φ$a$p[-a]]+(I/4)*epsilong1[-a,-c]*CD1[c][Φ$a$p[-b]]},MetricOn->All];

(*Pert$Exp$Sep[𝕁[-μ,-ν,-ρ]/.𝕁$To$𝒯,{μ,ν,ρ},{a,b,P},{𝒽$Harm$Rep,𝒯$Harm$Rep}];
%/.Dir$Harm$Def$Total//ContractMetric//ToCanonical;
%//.Harm$EQ$Y$Total//ContractMetric//ToCanonical;
Symetrise[XA$Component[%],{-a,-b}]//ToCanonical;
ToCanonical[%-(Π$ab$m[-a,-b]/.Π$ab$m$To$Met$T)]*)

(Subscript[(SuperPlus[Π]), a])

Π$a$p$To$Met$T=MakeRule[{Π$a$p[-a],((I/2)*Λ^2*𝒽$a$m[-a]*𝒫[])/r[]^2-((I/2)*Λ^2*Φ$a$m[-a])/r[]^2-(epsilong1[b,c]*Τ[-b,d]*CD1[-a][𝒽$ab$p[-c,-d]])/4+(epsilong1[-a,b]*Τ[c,d]*CD1[-b][𝒽$ab$p[-c,-d]])/6+(epsilong1[-a,b]*𝒽$ab$p[c,d]*CD1[-b][Τ[-c,-d]])/6-(epsilong1[-a,b]*CD1[-b][Φ$ab$p[c,-c]])/6-(2*epsilong1[-a,-b]*𝒽$p[]*𝒫[]*CD1[b][r[]])/(3*r[]^3)+(2*epsilong1[-a,-b]*Φ$p[]*CD1[b][r[]])/(3*r[]^3)+(epsilong1[-a,-b]*𝒫[]*CD1[b][𝒽$p[]])/(3*r[]^2)+(epsilong1[-b,c]*𝒽$ab$p[-a,-c]*CD1[b][𝒫[]])/3+(epsilong1[-a,-b]*𝒽$p[]*CD1[b][𝒫[]])/(3*r[]^2)-(epsilong1[-a,-b]*CD1[b][Φ$p[]])/(3*r[]^2)-(epsilong1[b,c]*Τ[-b,d]*CD1[-c][𝒽$ab$p[-a,-d]])/4-(epsilong1[b,c]*𝒽$ab$p[-a,-b]*CD1[-c][Τ[d,-d]])/6+(epsilong1[b,c]*CD1[-c][Φ$ab$p[-a,-b]])/2+(epsilong1[b,c]*Τ[-b,d]*CD1[-d][𝒽$ab$p[-a,-c]])/4},MetricOn->All];

(*Pert$Exp$Sep[𝕁[-μ,-ν,-ρ]/.𝕁$To$𝒯,{μ,ν,ρ},{a,b,c},{𝒽$Harm$Rep,𝒯$Harm$Rep}];
%/.Dir$Harm$Def$Total//ContractMetric//ToCanonical;
((-1/2)epsilong1[b,c]%//ContractMetric//ToCanonical)//.Harm$EQ$Y$Total//ContractMetric//ToCanonical;
ToCanonical[%/Y[]];
ToCanonical[%-(Π$a$p[-a]/.Π$a$p$To$Met$T)]*)

(Subscript[(SuperMinus[Π]), a])

Π$a$m$To$Met$T=MakeRule[{Π$a$m[-a],(Λ^2*𝒽$a$m[-a]*𝒫[])/4-(Λ^2*Φ$a$m[-a])/4+(I/4)*Λ^2*epsilong1[-a,-b]*Φ$a$p[b]+(I/6)*epsilong1[-a,b]*r[]^2*Τ[c,d]*CD1[-b][𝒽$ab$p[-c,-d]]+(I/6)*epsilong1[-a,b]*r[]^2*𝒽$ab$p[c,d]*CD1[-b][Τ[-c,-d]]-(I/6)*epsilong1[-a,b]*𝒽$p[]*CD1[-b][Τ[c,-c]]-(I/12)*epsilong1[-a,b]*r[]^2*𝒽$ab$p[c,-c]*CD1[-b][Τ[d,-d]]-(I/6)*epsilong1[-a,b]*r[]^2*CD1[-b][Φ$ab$p[c,-c]]-(I/2)*epsilong1[-a,c]*r[]*𝒽$ab$p[-b,-c]*𝒫[]*CD1[b][r[]]+(I/4)*epsilong1[-a,-b]*r[]*𝒽$ab$p[c,-c]*𝒫[]*CD1[b][r[]]-((I/6)*epsilong1[-a,-b]*𝒽$p[]*𝒫[]*CD1[b][r[]])/r[]-(I/4)*epsilong1[-a,c]*r[]*𝒽$ab$p[d,-d]*Τ[-b,-c]*CD1[b][r[]]+(I/2)*epsilong1[-a,c]*r[]*𝒽$ab$p[-c,d]*Τ[-b,-d]*CD1[b][r[]]+(I/2)*epsilong1[-a,c]*r[]*𝒽$ab$p[-b,d]*Τ[-c,-d]*CD1[b][r[]]-(I/2)*epsilong1[-a,c]*r[]*Φ$ab$p[-b,-c]*CD1[b][r[]]+((I/6)*epsilong1[-a,-b]*Φ$p[]*CD1[b][r[]])/r[]+(I/12)*epsilong1[-a,-b]*𝒫[]*CD1[b][𝒽$p[]]-(I/4)*epsilong1[-a,c]*Τ[-b,-c]*CD1[b][𝒽$p[]]-(I/6)*epsilong1[-a,c]*r[]^2*𝒽$ab$p[-b,-c]*CD1[b][𝒫[]]+(I/12)*epsilong1[-a,-b]*r[]^2*𝒽$ab$p[c,-c]*CD1[b][𝒫[]]+(I/6)*epsilong1[-a,-b]*CD1[b][Φ$p[]]+(I/6)*epsilong1[-a,b]*r[]^2*𝒽$ab$p[-b,c]*CD1[-c][Τ[d,-d]]},MetricOn->All];

(*Pert$Exp$Sep[𝕁[-μ,-ν,-ρ]/.𝕁$To$𝒯,{μ,ν,ρ},{P,Q,a},{𝒽$Harm$Rep,𝒯$Harm$Rep}];
%/.Dir$Harm$Def$Total//ContractMetric//ToCanonical;
%//.Harm$EQ$Y$Total//ContractMetric//ToCanonical;
ϵAB$Component[%];
ToCanonical[%-(Π$a$m[-a]/.Π$a$m$To$Met$T)]*)

(Subscript[(Π^⊕), a])

Π$a$o$To$Met$T=MakeRule[{Π$a$o[-a],(I*epsilong1[-a,c]*r[]*𝒽$a$m[b]*Τ[-b,-c]+r[]*Φ$a$p[-a]+Φ$o[]*CD1[-a][r[]]-r[]*CD1[-a][Φ$o[]]-I*epsilong1[-a,-b]*(r[]*Φ$a$m[b]+Φ$m[]*CD1[b][r[]]-r[]*CD1[b][Φ$m[]]))/(2*r[])},MetricOn->All];

(*Pert$Exp$Sep[𝕁[-μ,-ν,-ρ]/.𝕁$To$𝒯,{μ,ν,ρ},{P,Q,a},{𝒽$Harm$Rep,𝒯$Harm$Rep}];
%/.Dir$Harm$Def$Total//ContractMetric//ToCanonical;
%//.Harm$EQ$Y$Total//ContractMetric//ToCanonical;
VAB$Component[%];
ToCanonical[%-(Π$a$o[-a]/.Π$a$o$To$Met$T)]//Simplify*)

(SuperPlus[Π])

Π$p$To$Met$T=MakeRule[{Π$p[],-1/8*(epsilong1[a,b]*𝒽$ab$p[-a,c]*Τ[-b,-c])+((I/4)*Φ$a$m[a]*CD1[-a][r[]])/r[]-(I/4)*𝒫[]*CD1[-a][𝒽$a$m[a]]-(I/3)*𝒽$a$m[a]*CD1[-a][𝒫[]]+(I/12)*𝒽$a$m[a]*CD1[-a][Τ[b,-b]]+(I/4)*CD1[-a][Φ$a$m[a]]-(I/4)*𝒽$a$m[a]*CD1[-b][Τ[-a,b]]-((I/4)*𝒽$a$m[a]*Τ[-a,-b]*CD1[b][r[]])/r[]+(epsilong1[-a,-b]*Φ$a$p[a]*CD1[b][r[]])/(4*r[])-(epsilong1[-a,-b]*CD1[b][Φ$a$p[a]])/4},MetricOn->All];

(*Pert$Exp$Sep[𝕁[-μ,-ν,-ρ]/.𝕁$To$𝒯,{μ,ν,ρ},{a,b,P},{𝒽$Harm$Rep,𝒯$Harm$Rep}];
%/.Dir$Harm$Def$Total//ContractMetric//ToCanonical;
((-1/2)epsilong1[a,b]%//ContractMetric//ToCanonical)//.Harm$EQ$Y$Total//ContractMetric//ToCanonical;
ZA$Component[%];
ToCanonical[%-(Π$p[]/.Π$p$To$Met$T)]*)

(SuperMinus[Π])

Π$m$To$Met$T=MakeRule[{Π$m[],(-1/8*I)*𝒽$ab$p[a,-a]*𝒫[]+((I/3)*𝒽$p[]*𝒫[])/r[]^2+(I/24)*𝒽$ab$p[a,b]*Τ[-a,-b]+(I/12)*Φ$ab$p[a,-a]-((I/3)*Φ$p[])/r[]^2-((I/4)*Φ$a$p[a]*CD1[-a][r[]])/r[]-(I/4)*CD1[-a][Φ$a$p[a]]+(epsilong1[-a,b]*𝒽$a$m[a]*CD1[-b][Τ[c,-c]])/12+(epsilong1[-b,c]*𝒽$a$m[a]*Τ[-a,-c]*CD1[b][r[]])/(4*r[])+(epsilong1[-a,-b]*Φ$a$m[a]*CD1[b][r[]])/(4*r[])-(epsilong1[-b,c]*Τ[-a,-c]*CD1[b][𝒽$a$m[a]])/8+(epsilong1[-a,c]*Τ[-b,-c]*CD1[b][𝒽$a$m[a]])/8+(epsilong1[-a,-b]*𝒽$a$m[a]*CD1[b][𝒫[]])/6-(epsilong1[-a,-b]*CD1[b][Φ$a$m[a]])/4},MetricOn->All];

(*Pert$Exp$Sep[𝕁[-μ,-ν,-ρ]/.𝕁$To$𝒯,{μ,ν,ρ},{a,b,P},{𝒽$Harm$Rep,𝒯$Harm$Rep}];
%/.Dir$Harm$Def$Total//ContractMetric//ToCanonical;
((-1/2)epsilong1[a,b]%//ContractMetric//ToCanonical)//.Harm$EQ$Y$Total//ContractMetric//ToCanonical;
XA$Component[%];
ToCanonical[%-(Π$m[]/.Π$m$To$Met$T)]*)

(Subscript[(SuperPlus[Γ]), a])

Γ$a$p$To$Met$T=MakeRule[{Γ$a$p[-a],(I/4)*Λ^2*epsilong1[-a,-b]*𝒽$a$m[b]*𝒫[]-(I/4)*Λ^2*epsilong1[-a,-b]*Φ$a$m[b]-(Λ^2*Φ$a$p[-a])/4+(𝒽$p[]*𝒫[]*CD1[-a][r[]])/(6*r[])-(Φ$p[]*CD1[-a][r[]])/(6*r[])-(r[]^2*Τ[b,c]*CD1[-a][𝒽$ab$p[-b,-c]])/6-(𝒫[]*CD1[-a][𝒽$p[]])/12-(r[]^2*𝒽$ab$p[b,c]*CD1[-a][Τ[-b,-c]])/6+(𝒽$p[]*CD1[-a][Τ[b,-b]])/6+(r[]^2*CD1[-a][Φ$ab$p[b,-b]])/6-CD1[-a][Φ$p[]]/6-(r[]*𝒽$ab$p[-b,c]*Τ[-a,-c]*CD1[b][r[]])/2+(r[]*Φ$ab$p[-a,-b]*CD1[b][r[]])/2+(Τ[-a,-b]*CD1[b][𝒽$p[]])/4},MetricOn->All];

(*Pert$Exp$Sep[𝕁[-μ,-ν,-ρ]/.𝕁$To$𝒯,{μ,ν,ρ},{P,Q,a},{𝒽$Harm$Rep,𝒯$Harm$Rep}];
%/.Dir$Harm$Def$Total//ContractMetric//ToCanonical;
%//.Harm$EQ$Y$Total//ContractMetric//ToCanonical;
UAB$Component[%];
ToCanonical[%-(Γ$a$p[-a]/.Γ$a$p$To$Met$T)]*)

(Subscript[(SuperMinus[Γ]), a])

Γ$a$m$To$Met$T=MakeRule[{Γ$a$m[-a],(-(r[]*𝒽$a$m[b]*Τ[-a,-b])+r[]*Φ$a$m[-a]+Φ$m[]*CD1[-a][r[]]-r[]*CD1[-a][Φ$m[]]+I*epsilong1[-a,-b]*(r[]*Φ$a$p[b]+Φ$o[]*CD1[b][r[]]-r[]*CD1[b][Φ$o[]]))/(2*r[])},MetricOn->All];

(*Pert$Exp$Sep[𝕁[-μ,-ν,-ρ]/.𝕁$To$𝒯,{μ,ν,ρ},{P,Q,a},{𝒽$Harm$Rep,𝒯$Harm$Rep}];
%/.Dir$Harm$Def$Total//ContractMetric//ToCanonical;
%//.Harm$EQ$Y$Total//ContractMetric//ToCanonical;
WAB$Component[%];
ToCanonical[%-(Γ$a$m[-a]/.Γ$a$m$To$Met$T)]//Simplify*)

(SuperPlus[Γ])

Γ$p$To$Met$T=MakeRule[{Γ$p[],(𝒽$p[]*𝒫[])/6-(r[]^2*𝒽$ab$p[a,b]*Τ[-a,-b])/6+(r[]^2*Φ$ab$p[a,-a])/6+Φ$o[]/2-(Λ^2*Φ$o[])/4-Φ$p[]/6+(r[]*Φ$a$p[a]*CD1[-a][r[]])/2-(I/2)*epsilong1[-a,-b]*r[]*𝒽$a$m[a]*𝒫[]*CD1[b][r[]]-(I/2)*epsilong1[-b,c]*r[]*𝒽$a$m[a]*Τ[-a,-c]*CD1[b][r[]]+(I/2)*epsilong1[-a,c]*r[]*𝒽$a$m[a]*Τ[-b,-c]*CD1[b][r[]]-(I/2)*epsilong1[-a,-b]*r[]*Φ$a$m[a]*CD1[b][r[]]+(I/4)*epsilong1[-b,c]*r[]^2*Τ[-a,-c]*CD1[b][𝒽$a$m[a]]-(I/4)*epsilong1[-a,c]*r[]^2*Τ[-b,-c]*CD1[b][𝒽$a$m[a]]-(I/2)*epsilong1[-a,-b]*r[]^2*𝒽$a$m[a]*CD1[b][𝒫[]]+(I/2)*epsilong1[-a,-b]*r[]^2*CD1[b][Φ$a$m[a]]},MetricOn->All];

(*Pert$Exp$Sep[𝕁[-μ,-ν,-ρ]/.𝕁$To$𝒯,{μ,ν,ρ},{P,A,B},{𝒽$Harm$Rep,𝒯$Harm$Rep}];
%/.Dir$Harm$Def$Total//ContractMetric//ToCanonical;
((1/2)epsilong2[A,B]%//ContractMetric//ToCanonical)//.Harm$EQ$Y$Total//ContractMetric//ToCanonical;
XA$Component[%];
ToCanonical[%-(Γ$p[]/.Γ$p$To$Met$T)]//Simplify*)

(SuperMinus[Γ])

Γ$m$To$Met$T=MakeRule[{Γ$m[],(I/4)*epsilong1[a,b]*r[]^2*𝒽$ab$p[-a,c]*Τ[-b,-c]-Φ$m[]/2+(Λ^2*Φ$m[])/4+(r[]*𝒽$a$m[a]*𝒫[]*CD1[-a][r[]])/2-(r[]*Φ$a$m[a]*CD1[-a][r[]])/2-(I/2)*epsilong1[-a,-b]*r[]*Φ$a$p[a]*CD1[b][r[]]+(I/2)*epsilong1[-a,-b]*r[]^2*CD1[b][Φ$a$p[a]]},MetricOn->All];

(*Pert$Exp$Sep[𝕁[-μ,-ν,-ρ]/.𝕁$To$𝒯,{μ,ν,ρ},{P,A,B},{𝒽$Harm$Rep,𝒯$Harm$Rep}];
%/.Dir$Harm$Def$Total//ContractMetric//ToCanonical;
((1/2)epsilong2[A,B]%//ContractMetric//ToCanonical)//.Harm$EQ$Y$Total//ContractMetric//ToCanonical;
ZA$Component[%];
ToCanonical[%-(Γ$m[]/.Γ$m$To$Met$T)]//Simplify*)

Total

Pert𝕁$To$metTmn=Flatten[{Π$ab$p$To$Met$T,Π$ab$m$To$Met$T,Π$a$p$To$Met$T,Π$a$m$To$Met$T,Π$a$o$To$Met$T,Π$p$To$Met$T,Π$m$To$Met$T,Γ$a$p$To$Met$T,Γ$a$m$To$Met$T,Γ$p$To$Met$T,Γ$m$To$Met$T}];

1.3. Linearised Einstein Equations

DefTensor[Pert$Einstein$ab[-a,-b],M1]
DefTensor[Pert$Einstein$aZA[-a],M1]
DefTensor[Pert$Einstein$aXA[-a],M1]
DefTensor[Pert$Einstein$UAB[],M1]
DefTensor[Pert$Einstein$VAB[],M1]
DefTensor[Pert$Einstein$WAB[],M1]

ab

Eval$Pert$Einstein$ab=MakeRule[{Pert$Einstein$ab[-a,-b],𝒽$ab$p[-a,-b]/r[]^2+(Λ^2*𝒽$ab$p[-a,-b])/(2*r[]^2)-(g1[-a,-b]*𝒽$ab$p[c,-c])/r[]^2-(Λ^2*g1[-a,-b]*𝒽$ab$p[c,-c])/(2*r[]^2)-(Λ^2*g1[-a,-b]*𝒽$p[])/(2*r[]^4)-(𝒽$p[]*Τ[-a,-b])/r[]^2+g1[-a,-b]*𝒽$ab$p[c,d]*Τ[-c,-d]+𝒽$ab$p[-a,-b]*Τ[c,-c]-g1[-a,-b]*𝒽$ab$p[c,-c]*Τ[d,-d]-Φ$ab$p[-a,-b]-(2*𝒽$p[]*CD1[-a][r[]]*CD1[-b][r[]])/r[]^4+(CD1[-a][𝒽$p[]]*CD1[-b][r[]])/r[]^3+(CD1[-a][r[]]*CD1[-b][𝒽$p[]])/r[]^3-CD1[-b][CD1[-a][𝒽$p[]]]/r[]^2+(g1[-a,-b]*CD1[-c][CD1[c][𝒽$p[]]])/r[]^2+(CD1[-a][𝒽$ab$p[-b,-c]]*CD1[c][r[]])/r[]+(CD1[-b][𝒽$ab$p[-a,-c]]*CD1[c][r[]])/r[]-(𝒽$ab$p[-a,-b]*CD1[-c][r[]]*CD1[c][r[]])/r[]^2+(g1[-a,-b]*𝒽$ab$p[d,-d]*CD1[-c][r[]]*CD1[c][r[]])/r[]^2+(g1[-a,-b]*𝒽$p[]*CD1[-c][r[]]*CD1[c][r[]])/r[]^4-(CD1[-c][𝒽$ab$p[-a,-b]]*CD1[c][r[]])/r[]+(g1[-a,-b]*CD1[-c][𝒽$ab$p[d,-d]]*CD1[c][r[]])/r[]-(g1[-a,-b]*CD1[-c][𝒽$p[]]*CD1[c][r[]])/r[]^3-(2*g1[-a,-b]*CD1[c][r[]]*CD1[-d][𝒽$ab$p[-c,d]])/r[]-(g1[-a,-b]*𝒽$ab$p[-c,-d]*CD1[c][r[]]*CD1[d][r[]])/r[]^2},MetricOn->All];

(*Pert$Exp$Sep[RicciCD3[-μ,-ν]-(1/2)RicciScalarCD3[]g3[-μ,-ν]-𝒯[-μ,-ν],{μ,ν},{a,b},{𝒽$Harm$Rep,𝒯$Harm$Rep}];
%/.Dir$Harm$Def$Total//ContractMetric//ToCanonical ;
%//.Harm$EQ$Y$Total//ContractMetric//ToCanonical ;
ToCanonical[%/Y[]];
%/.CommCovs$𝒽$ab$p$FreeFirst//ContractMetric//ToCanonical;
%/.FirstOrder$Gauss$Bonnet$Rule //ContractMetric//ToCanonical;
ToCanonical[%-(Pert$Einstein$ab[-a,-b]/.Eval$Pert$Einstein$ab)]*)

ZA and XA

Eval$Pert$Einstein$aZA=MakeRule[{Pert$Einstein$aZA[-a],-Φ$a$p[-a]+(𝒽$ab$p[b,-b]*CD1[-a][r[]])/(2*r[])+(𝒽$p[]*CD1[-a][r[]])/r[]^3-CD1[-a][𝒽$ab$p[b,-b]]/2-CD1[-a][𝒽$p[]]/(2*r[]^2)+CD1[-b][𝒽$ab$p[-a,b]]/2},MetricOn->All];
Eval$Pert$Einstein$aXA=MakeRule[{Pert$Einstein$aXA[-a],-1/2*𝒽$a$m[-a]/r[]^2+(Λ^2*𝒽$a$m[-a])/(2*r[]^2)+𝒽$a$m[-a]*𝒫[]+(𝒽$a$m[b]*Τ[-a,-b])/2-Φ$a$m[-a]-(𝒽$a$m[b]*CD1[-a][r[]]*CD1[-b][r[]])/r[]^2-(CD1[-a][r[]]*CD1[-b][𝒽$a$m[b]])/r[]+CD1[-b][CD1[-a][𝒽$a$m[b]]]/2-CD1[-b][CD1[b][𝒽$a$m[-a]]]/2+(CD1[-a][𝒽$a$m[-b]]*CD1[b][r[]])/r[]+(𝒽$a$m[-a]*CD1[-b][r[]]*CD1[b][r[]])/(2*r[]^2)},MetricOn->All];

(*Pert$Exp$Sep[RicciCD3[-μ,-ν]-(1/2)RicciScalarCD3[]g3[-μ,-ν]-𝒯[-μ,-ν],{μ,ν},{a,P},{𝒽$Harm$Rep,𝒯$Harm$Rep}] ;
%/.Dir$Harm$Def$Total//ContractMetric//ToCanonical ;
Dum=%//.Harm$EQ$Y$Total//ContractMetric//ToCanonical;
ZA$Component[Dum];
ToCanonical[%-(Pert$Einstein$aZA[-a]/.Eval$Pert$Einstein$aZA)]
XA$Component[Dum];
ToCanonical[%-(Pert$Einstein$aXA[-a]/.Eval$Pert$Einstein$aXA)]*)

UAB VAB WAB ϵAB

Eval$Pert$Einstein$UAB=MakeRule[{Pert$Einstein$UAB[],-1/4*(Λ^2*𝒽$ab$p[a,-a])-𝒽$p[]/r[]^2-(r[]^2*𝒽$ab$p[a,-a]*𝒫[])/2+𝒽$p[]*𝒫[]+(r[]^2*𝒽$ab$p[a,b]*Τ[-a,-b])/2-(𝒽$p[]*Τ[a,-a])/2-(r[]^2*𝒽$ab$p[a,-a]*Τ[b,-b])/4-Φ$p[]+CD1[-a][CD1[a][𝒽$p[]]]/2+(2*𝒽$p[]*CD1[-a][r[]]*CD1[a][r[]])/r[]^2+(r[]*CD1[-a][𝒽$ab$p[b,-b]]*CD1[a][r[]])/2-(CD1[-a][𝒽$p[]]*CD1[a][r[]])/r[]-r[]*CD1[a][r[]]*CD1[-b][𝒽$ab$p[-a,b]]-(r[]^2*CD1[-b][CD1[-a][𝒽$ab$p[a,b]]])/2+(r[]^2*CD1[-b][CD1[b][𝒽$ab$p[a,-a]]])/2},MetricOn->All];
Eval$Pert$Einstein$VAB=MakeRule[{Pert$Einstein$VAB[],-1/2*𝒽$ab$p[a,-a]-Φ$o[]},MetricOn->All];
Eval$Pert$Einstein$WAB=MakeRule[{Pert$Einstein$WAB[],-Φ$m[]+CD1[-a][𝒽$a$m[a]]},MetricOn->All];

(*Pert$Exp$Sep[RicciCD3[-μ,-ν]-(1/2)RicciScalarCD3[]g3[-μ,-ν]-𝒯[-μ,-ν],{μ,ν},{P,Q},{𝒽$Harm$Rep,𝒯$Harm$Rep}] ;
%/.Dir$Harm$Def$Total//ContractMetric//ToCanonical ;
Dum=%//.Harm$EQ$Y$Total//ContractMetric//ToCanonical ;
UAB$Component[Dum]//ContractMetric//ToCanonical//Simplify;
ToCanonical[%-(Pert$Einstein$UAB[]/.Eval$Pert$Einstein$UAB)]
VAB$Component[Dum]//ContractMetric//ToCanonical//Simplify;
ToCanonical[%-(Pert$Einstein$VAB[]/.Eval$Pert$Einstein$VAB)]
WAB$Component[Dum]//ContractMetric//ToCanonical//Simplify;
ToCanonical[%-(Pert$Einstein$WAB[]/.Eval$Pert$Einstein$WAB)]
ϵAB$Component[Dum]//ContractMetric//ToCanonical//Simplify*)

Total

Eval$Pert$Einstein$Total=Flatten[{Eval$Pert$Einstein$ab,Eval$Pert$Einstein$aZA,Eval$Pert$Einstein$aXA,Eval$Pert$Einstein$UAB,Eval$Pert$Einstein$VAB,Eval$Pert$Einstein$WAB}];

Substitution rules version 1

EinsteinLin$Φ$ab$p=MakeRule[{Φ$ab$p[-a,-b],𝒽$ab$p[-a,-b]/r[]^2+(Λ^2*𝒽$ab$p[-a,-b])/(2*r[]^2)-(g1[-a,-b]*𝒽$ab$p[c,-c])/r[]^2-(Λ^2*g1[-a,-b]*𝒽$ab$p[c,-c])/(2*r[]^2)-(Λ^2*g1[-a,-b]*𝒽$p[])/(2*r[]^4)-(𝒽$p[]*Τ[-a,-b])/r[]^2+g1[-a,-b]*𝒽$ab$p[c,d]*Τ[-c,-d]+𝒽$ab$p[-a,-b]*Τ[c,-c]-g1[-a,-b]*𝒽$ab$p[c,-c]*Τ[d,-d]-(2*𝒽$p[]*CD1[-a][r[]]*CD1[-b][r[]])/r[]^4+(CD1[-a][𝒽$p[]]*CD1[-b][r[]])/r[]^3+(CD1[-a][r[]]*CD1[-b][𝒽$p[]])/r[]^3-CD1[-b][CD1[-a][𝒽$p[]]]/r[]^2+(g1[-a,-b]*CD1[-c][CD1[c][𝒽$p[]]])/r[]^2+(CD1[-a][𝒽$ab$p[-b,-c]]*CD1[c][r[]])/r[]+(CD1[-b][𝒽$ab$p[-a,-c]]*CD1[c][r[]])/r[]-(𝒽$ab$p[-a,-b]*CD1[-c][r[]]*CD1[c][r[]])/r[]^2+(g1[-a,-b]*𝒽$ab$p[d,-d]*CD1[-c][r[]]*CD1[c][r[]])/r[]^2+(g1[-a,-b]*𝒽$p[]*CD1[-c][r[]]*CD1[c][r[]])/r[]^4-(CD1[-c][𝒽$ab$p[-a,-b]]*CD1[c][r[]])/r[]+(g1[-a,-b]*CD1[-c][𝒽$ab$p[d,-d]]*CD1[c][r[]])/r[]-(g1[-a,-b]*CD1[-c][𝒽$p[]]*CD1[c][r[]])/r[]^3-(2*g1[-a,-b]*CD1[c][r[]]*CD1[-d][𝒽$ab$p[-c,d]])/r[]-(g1[-a,-b]*𝒽$ab$p[-c,-d]*CD1[c][r[]]*CD1[d][r[]])/r[]^2},MetricOn->All];
EinsteinLin$Φ$a$p=MakeRule[{Φ$a$p[-a],((𝒽$ab$p[b,-b]*CD1[-a][r[]])/r[]+(2*𝒽$p[]*CD1[-a][r[]])/r[]^3-CD1[-a][𝒽$ab$p[b,-b]]-CD1[-a][𝒽$p[]]/r[]^2+CD1[-b][𝒽$ab$p[-a,b]])/2},MetricOn->All];
EinsteinLin$Φ$a$m=MakeRule[{Φ$a$m[-a],(r[]^2*𝒽$a$m[b]*Τ[-a,-b]-2*𝒽$a$m[b]*CD1[-a][r[]]*CD1[-b][r[]]-2*r[]*CD1[-a][r[]]*CD1[-b][𝒽$a$m[b]]+r[]^2*CD1[-b][CD1[-a][𝒽$a$m[b]]]-r[]^2*CD1[-b][CD1[b][𝒽$a$m[-a]]]+2*r[]*CD1[-a][𝒽$a$m[-b]]*CD1[b][r[]]+𝒽$a$m[-a]*(-1+Λ^2+2*r[]^2*𝒫[]+CD1[-b][r[]]*CD1[b][r[]]))/(2*r[]^2)},MetricOn->All];
EinsteinLin$Φ$p=MakeRule[{Φ$p[],-(𝒽$p[]/r[]^2)+𝒽$p[]*𝒫[]+(r[]^2*𝒽$ab$p[a,b]*Τ[-a,-b])/2-(𝒽$p[]*Τ[a,-a])/2-(𝒽$ab$p[a,-a]*(Λ^2+2*r[]^2*𝒫[]+r[]^2*Τ[b,-b]))/4+CD1[-a][CD1[a][𝒽$p[]]]/2+(2*𝒽$p[]*CD1[-a][r[]]*CD1[a][r[]])/r[]^2+(r[]*CD1[-a][𝒽$ab$p[b,-b]]*CD1[a][r[]])/2-(CD1[-a][𝒽$p[]]*CD1[a][r[]])/r[]-r[]*CD1[a][r[]]*CD1[-b][𝒽$ab$p[-a,b]]-(r[]^2*CD1[-b][CD1[-a][𝒽$ab$p[a,b]]])/2+(r[]^2*CD1[-b][CD1[b][𝒽$ab$p[a,-a]]])/2},MetricOn->All];
EinsteinLin$Φ$m=MakeRule[{Φ$m[],CD1[-a][𝒽$a$m[a]]},MetricOn->All];
EinsteinLin$Φ$o=MakeRule[{Φ$o[],-1/2*𝒽$ab$p[a,-a]},MetricOn->All];

EinsteinLin$Total=Flatten[{EinsteinLin$Φ$ab$p,EinsteinLin$Φ$a$p,EinsteinLin$Φ$a$m,EinsteinLin$Φ$p,EinsteinLin$Φ$m,EinsteinLin$Φ$o}];

(*((Pert$Einstein$ab[-a,-b]/.Eval$Pert$Einstein$Total)/.EinsteinLin$Total//ToCanonical)==0
((Pert$Einstein$aZA[-a]/.Eval$Pert$Einstein$Total)/.EinsteinLin$Total//ToCanonical)==0
((Pert$Einstein$aXA[-a]/.Eval$Pert$Einstein$Total)/.EinsteinLin$Total//ToCanonical)==0
((Pert$Einstein$UAB[]/.Eval$Pert$Einstein$Total)/.EinsteinLin$Total//ToCanonical)==0
((Pert$Einstein$VAB[]/.Eval$Pert$Einstein$Total)/.EinsteinLin$Total//ToCanonical)==0
((Pert$Einstein$WAB[]/.Eval$Pert$Einstein$Total)/.EinsteinLin$Total//ToCanonical)==0*)

Substitution rules Version 2

The linearised Einstein equations can be used to eliminate the explicit occurrences of metric perturbations in the curvature wave equations. Here we introduce a set of replacement rules that are useful in doing so. First we give the rules, then the derivation:

Einstein$Rep$Trace$𝒽$ab$p=MakeRule[{𝒽$ab$p[a,-a],-2*Φ$o[]},MetricOn->All];
Einstein$Rep$Lorentz$𝒽$ab$p=MakeRule[{CD1[b][𝒽$ab$p[-a,-b]],2*Φ$a$p[-a]-(2*𝒽$p[]*CD1[-a][r[]])/r[]^3+(2*Φ$o[]*CD1[-a][r[]])/r[]+CD1[-a][𝒽$p[]]/r[]^2-2*CD1[-a][Φ$o[]]},MetricOn->All];
Einstein$Rep$Box$𝒽$p=MakeRule[{CD1[a][CD1[-a][𝒽$p[]]],(Λ^2*𝒽$p[])/r[]^2-2*r[]^2*𝒽$ab$p[a,b]*Τ[-a,-b]+𝒽$p[]*Τ[a,-a]+r[]^2*Φ$ab$p[a,-a]-2*Φ$o[]-Λ^2*Φ$o[]-2*r[]^2*Τ[a,-a]*Φ$o[]+4*r[]*Φ$a$p[a]*CD1[-a][r[]]-(4*𝒽$p[]*CD1[-a][r[]]*CD1[a][r[]])/r[]^2+6*Φ$o[]*CD1[-a][r[]]*CD1[a][r[]]+(2*CD1[-a][𝒽$p[]]*CD1[a][r[]])/r[]-2*r[]*CD1[-a][Φ$o[]]*CD1[a][r[]]+2*𝒽$ab$p[-a,-b]*CD1[a][r[]]*CD1[b][r[]]},MetricOn->All];
Einstein$Rep$Div$𝒽$a$m=MakeRule[{CD1[a][𝒽$a$m[-a]],Φ$m[]},MetricOn->All];
Einstein$Rep$Box$𝒽$a$m=MakeRule[{CD1[-b][CD1[b][𝒽$a$m[-a]]],(Λ^2*𝒽$a$m[-a])/r[]^2+𝒽$a$m[-a]*𝒫[]+𝒽$a$m[b]*Τ[-a,-b]+(𝒽$a$m[-a]*Τ[b,-b])/2-2*Φ$a$m[-a]-(2*Φ$m[]*CD1[-a][r[]])/r[]+CD1[-a][Φ$m[]]-(2*𝒽$a$m[b]*CD1[-a][r[]]*CD1[-b][r[]])/r[]^2+(2*CD1[-a][𝒽$a$m[-b]]*CD1[b][r[]])/r[]},MetricOn->All];
Einstein$Rep$rarb𝒽$ab$p=MakeRule[{𝒽$ab$p[-a,-b]*CD1[a][r[]]*CD1[b][r[]],-1/2*(Λ^2*𝒽$p[])/r[]^2+r[]^2*𝒽$ab$p[a,b]*Τ[-a,-b]-(𝒽$p[]*Τ[a,-a])/2-(r[]^2*Φ$ab$p[a,-a])/2+Φ$o[]+(Λ^2*Φ$o[])/2+r[]^2*Τ[a,-a]*Φ$o[]-2*r[]*Φ$a$p[a]*CD1[-a][r[]]+CD1[-a][CD1[a][𝒽$p[]]]/2+(2*𝒽$p[]*CD1[-a][r[]]*CD1[a][r[]])/r[]^2-3*Φ$o[]*CD1[-a][r[]]*CD1[a][r[]]-(CD1[-a][𝒽$p[]]*CD1[a][r[]])/r[]+r[]*CD1[-a][Φ$o[]]*CD1[a][r[]]},MetricOn->All];
Einstein$Rep$DD𝒽$p=MakeRule[{CD1[-b][CD1[-a][𝒽$p[]]],𝒽$ab$p[-a,-b]+(Λ^2*𝒽$ab$p[-a,-b])/2+(Λ^2*g1[-a,-b]*𝒽$p[])/(2*r[]^2)-𝒽$p[]*Τ[-a,-b]-g1[-a,-b]*r[]^2*𝒽$ab$p[c,d]*Τ[-c,-d]+r[]^2*𝒽$ab$p[-a,-b]*Τ[c,-c]+g1[-a,-b]*𝒽$p[]*Τ[c,-c]-r[]^2*Φ$ab$p[-a,-b]+g1[-a,-b]*r[]^2*Φ$ab$p[c,-c]-(2*𝒽$p[]*CD1[-a][r[]]*CD1[-b][r[]])/r[]^2+(CD1[-a][𝒽$p[]]*CD1[-b][r[]])/r[]+(CD1[-a][r[]]*CD1[-b][𝒽$p[]])/r[]+r[]*CD1[-a][𝒽$ab$p[-b,-c]]*CD1[c][r[]]+r[]*CD1[-b][𝒽$ab$p[-a,-c]]*CD1[c][r[]]-𝒽$ab$p[-a,-b]*CD1[-c][r[]]*CD1[c][r[]]+(g1[-a,-b]*𝒽$p[]*CD1[-c][r[]]*CD1[c][r[]])/r[]^2-r[]*CD1[-c][𝒽$ab$p[-a,-b]]*CD1[c][r[]]-(g1[-a,-b]*CD1[-c][𝒽$p[]]*CD1[c][r[]])/r[]+g1[-a,-b]*𝒽$ab$p[-c,-d]*CD1[c][r[]]*CD1[d][r[]]},MetricOn->All];

(*((Pert$Einstein$VAB[]/.Eval$Pert$Einstein$Total)/.Einstein$Rep$Trace$𝒽$ab$p//ToCanonical)==0
(((Pert$Einstein$aZA[-a]/.Eval$Pert$Einstein$Total)/.Einstein$Rep$Lorentz$𝒽$ab$p)/.Einstein$Rep$Trace$𝒽$ab$p//ToCanonical)==0
((Pert$Einstein$WAB[]/.Eval$Pert$Einstein$Total)/.Einstein$Rep$Div$𝒽$a$m//ToCanonical)==0
g1[a,b](Pert$Einstein$ab[-a,-b]/.Eval$Pert$Einstein$Total)//ContractMetric//ToCanonical;
%/.Einstein$Rep$Box$𝒽$p//ContractMetric//ToCanonical;
%/.Flatten[{Einstein$Rep$Lorentz$𝒽$ab$p,Einstein$Rep$Trace$𝒽$ab$p}]//ContractMetric//ToCanonical;
ToCanonical[%==0]
ToCanonical[((CD1[a][CD1[-a][𝒽$p[]]]-(CD1[a][CD1[-a][𝒽$p[]]]/.Einstein$Rep$Box$𝒽$p))/.Einstein$Rep$rarb𝒽$ab$p//ContractMetric//ToCanonical)==0]*)

1.4. Conservation equations for δ (Subscript[𝒯, μ ν])

Test passed: several, including explicit comparison with literature

DefTensor[𝒯$Cons$a[-a],M1]
DefTensor[𝒯$Cons$ZA[],M1]
DefTensor[𝒯$Cons$XA[],M1]

a

Eval$𝒯$Cons$a=MakeRule[{𝒯$Cons$a[-a],-((Λ^2*Y[]*Φ$a$p[-a])/r[]^2)+(4*Y[]*𝒽$p[]*𝒫[]*CD1[-a][r[]])/r[]^3-(2*Y[]*Φ$p[]*CD1[-a][r[]])/r[]^3-(Y[]*Τ[b,c]*CD1[-a][𝒽$ab$p[-b,-c]])/2-(Y[]*𝒫[]*CD1[-a][𝒽$p[]])/r[]^2+(Y[]*Τ[-a,b]*CD1[-b][𝒽$ab$p[c,-c]])/2+Y[]*CD1[-b][Φ$ab$p[-a,b]]-(2*Y[]*𝒽$p[]*Τ[-a,-b]*CD1[b][r[]])/r[]^3-(2*Y[]*𝒽$ab$p[-b,c]*Τ[-a,-c]*CD1[b][r[]])/r[]+(2*Y[]*Φ$ab$p[-a,-b]*CD1[b][r[]])/r[]+(Y[]*Τ[-a,-b]*CD1[b][𝒽$p[]])/r[]^2-Y[]*Τ[-a,b]*CD1[-c][𝒽$ab$p[-b,c]]-Y[]*𝒽$ab$p[b,c]*CD1[-c][Τ[-a,-b]]},MetricOn->All];

(*Pert$Exp$Sep[CD3[-μ][𝒯[μ,-ν]],{ν},{a},{𝒽$Harm$Rep,𝒯$Harm$Rep}];
%/.Dir$Harm$Def$Total//ContractMetric//ToCanonical;
%//.Harm$EQ$Y$Total//ContractMetric//ToCanonical;
ToCanonical[%-(𝒯$Cons$a[-a]/.Eval$𝒯$Cons$a)]*)

ZA and XA

Eval$𝒯$Cons$ZA=MakeRule[{𝒯$Cons$ZA[],(𝒽$ab$p[a,-a]*𝒫[])/2-(𝒽$p[]*𝒫[])/r[]^2-(𝒽$ab$p[a,b]*Τ[-a,-b])/2+Φ$o[]/r[]^2-(Λ^2*Φ$o[])/(2*r[]^2)+Φ$p[]/r[]^2+(2*Φ$a$p[a]*CD1[-a][r[]])/r[]+CD1[-a][Φ$a$p[a]]},MetricOn->All];
Eval$𝒯$Cons$XA=MakeRule[{𝒯$Cons$XA[],Φ$m[]/r[]^2-(Λ^2*Φ$m[])/(2*r[]^2)-(2*𝒽$a$m[a]*𝒫[]*CD1[-a][r[]])/r[]+(2*Φ$a$m[a]*CD1[-a][r[]])/r[]-𝒫[]*CD1[-a][𝒽$a$m[a]]-𝒽$a$m[a]*CD1[-a][𝒫[]]+CD1[-a][Φ$a$m[a]]},MetricOn->All];

(*Pert$Exp$Sep[CD3[-μ][𝒯[μ,-ν]],{ν},{P},{𝒽$Harm$Rep,𝒯$Harm$Rep}];
%/.Dir$Harm$Def$Total//ContractMetric//ToCanonical;
Dum=%//.Harm$EQ$Y$Total//ContractMetric//ToCanonical;
ZA$Component[Dum];
ToCanonical[%-(𝒯$Cons$ZA[]/.Eval$𝒯$Cons$ZA)]
XA$Component[Dum];
ToCanonical[%-(𝒯$Cons$XA[]/.Eval$𝒯$Cons$XA)]*)

Total

Eval$𝒯$Cons$Total=Flatten[{Eval$𝒯$Cons$a,Eval$𝒯$Cons$ZA,Eval$𝒯$Cons$XA}];

Substitution Rules

𝒯$Cons$Φ$ab$p=MakeRule[{CD1[-b][Φ$ab$p[-a,b]],-1/2*(-2*Λ^2*r[]*Φ$a$p[-a]+(8*𝒽$p[]*𝒫[]-4*Φ$p[])*CD1[-a][r[]]-r[]^3*Τ[b,c]*CD1[-a][𝒽$ab$p[-b,-c]]-2*r[]*𝒫[]*CD1[-a][𝒽$p[]]+r[]^3*Τ[-a,b]*CD1[-b][𝒽$ab$p[c,-c]]-4*𝒽$p[]*Τ[-a,-b]*CD1[b][r[]]-4*r[]^2*𝒽$ab$p[-b,c]*Τ[-a,-c]*CD1[b][r[]]+4*r[]^2*Φ$ab$p[-a,-b]*CD1[b][r[]]+2*r[]*Τ[-a,-b]*CD1[b][𝒽$p[]]-2*r[]^3*Τ[-a,b]*CD1[-c][𝒽$ab$p[-b,c]]-2*r[]^3*𝒽$ab$p[b,c]*CD1[-c][Τ[-a,-b]])/r[]^3},MetricOn->All];
𝒯$Cons$Φ$a$p=MakeRule[{CD1[-a][Φ$a$p[a]],-1/2*(r[]^2*𝒽$ab$p[a,-a]*𝒫[]-2*𝒽$p[]*𝒫[]-r[]^2*𝒽$ab$p[a,b]*Τ[-a,-b]+2*Φ$o[]-Λ^2*Φ$o[]+2*Φ$p[]+4*r[]*Φ$a$p[a]*CD1[-a][r[]])/r[]^2},MetricOn->All];
𝒯$Cons$Φ$a$m=MakeRule[{CD1[-a][Φ$a$m[a]],-(Φ$m[]/r[]^2)+(Λ^2*Φ$m[])/(2*r[]^2)-(2*Φ$a$m[a]*CD1[-a][r[]])/r[]+𝒫[]*CD1[-a][𝒽$a$m[a]]+(𝒽$a$m[a]*(2*𝒫[]*CD1[-a][r[]]+r[]*CD1[-a][𝒫[]]))/r[]},MetricOn->All];

𝒯$Cons$Total=Flatten[{𝒯$Cons$Φ$ab$p,𝒯$Cons$Φ$a$p,𝒯$Cons$Φ$a$m}];

(*ToCanonical[(𝒯$Cons$a[-a]/.Eval$𝒯$Cons$Total)/.𝒯$Cons$Φ$ab$p]==0
ToCanonical[(𝒯$Cons$ZA[]/.Eval$𝒯$Cons$Total)/.𝒯$Cons$Φ$a$p]==0
ToCanonical[(𝒯$Cons$XA[]/.Eval$𝒯$Cons$Total)/.𝒯$Cons$Φ$a$m]==0*)

1.5. Linearised algebraic identities of δ (Subscript[ℂ , μ ν ρ σ])

(*Trace identity tensors*)

DefTensor[ℂ$Trace$ab[-a,-b],M1]
DefTensor[ℂ$Trace$aZA[-a],M1]
DefTensor[ℂ$Trace$aXA[-a],M1]
DefTensor[ℂ$Trace$UAB[],M1]
DefTensor[ℂ$Trace$VAB[],M1]
DefTensor[ℂ$Trace$ϵAB[],M1]
DefTensor[ℂ$Trace$WAB[],M1]

(*Self-duality identity tensors*)

DefTensor[ℂ$SD$abcd[],M1]
DefTensor[ℂ$SD$Aabc$ZA[-a],M1]
DefTensor[ℂ$SD$Aabc$XA[-a],M1]
DefTensor[ℂ$SD$aAbB$UAB[-a,-b],M1]
DefTensor[ℂ$SD$aAbB$VAB[-a,-b],M1]
DefTensor[ℂ$SD$aAbB$WAB[-a,-b],M1]
DefTensor[ℂ$SD$aAbB$ϵAB[-a,-b],M1]
DefTensor[ℂ$SD$abAB[],M1]
DefTensor[ℂ$SD$aABF$ZA[-a],M1]
DefTensor[ℂ$SD$aABF$XA[-a],M1]
DefTensor[ℂ$SD$ABFG[],M1]

Trace identities

δ ((Subscript[(ℂ^μ), a μ b]))=0

Eval$Pert$ℂ$Trace$ab=MakeRule[{ℂ$Trace$ab[-a,-b],𝒽$ab$p[-a,-b]/r[]^2-(g1[-a,-b]*𝒽$ab$p[c,-c])/r[]^2+(g1[-a,-b]*𝒽$p[])/r[]^4-(𝒽$ab$p[-a,-b]*𝒫[])/3+(g1[-a,-b]*𝒽$ab$p[c,-c]*𝒫[])/3-(g1[-a,-b]*𝒽$p[]*𝒫[])/(3*r[]^2)+(𝒽$ab$p[-a,-b]*Τ[c,-c])/3+(g1[-a,-b]*𝒽$p[]*Τ[c,-c])/(3*r[]^2)-(g1[-a,-b]*𝒽$ab$p[c,-c]*Τ[d,-d])/3+(2*Ψ$ab$p[-a,-b])/r[]^2-g1[-a,-b]*Ψ$p[]-(𝒽$ab$p[-a,-b]*CD1[-c][r[]]*CD1[c][r[]])/r[]^2+(g1[-a,-b]*𝒽$ab$p[d,-d]*CD1[-c][r[]]*CD1[c][r[]])/r[]^2-(g1[-a,-b]*𝒽$p[]*CD1[-c][r[]]*CD1[c][r[]])/r[]^4},MetricOn->All]

(*Pert$Exp$Sep[CC[μ,-ν,-μ,-ρ],{ν,ρ},{a,b},{𝒽$Harm$Rep,ℂ$Harm$Rep}];
%/.Dir$Harm$Def$Total//ContractMetric//ToCanonical;
%//.Harm$EQ$Y$Total//ContractMetric//ToCanonical;
ToCanonical[((ℂ$Trace$ab[-a,-b]/.Eval$Pert$ℂ$Trace$ab)-(%/Y[]))==0]*)

δ ((Subscript[(ℂ^μ), a μ P]))=0

Eval$Pert$ℂ$Trace$aZA=MakeRule[{ℂ$Trace$aZA[-a],(((-3*I)/2)*epsilong1[-a,-b]*𝒽$a$m[b])/r[]^2+(I/2)*epsilong1[-a,-b]*𝒽$a$m[b]*𝒫[]+Ξ$a$p[-a]/r[]^2-(I/2)*epsilong1[-a,-b]*𝒽$a$m[b]*Τ[c,-c]+epsilong1[-a,-b]*Ψ$a$p[b]+(((3*I)/2)*epsilong1[-a,-b]*𝒽$a$m[b]*CD1[-c][r[]]*CD1[c][r[]])/r[]^2},MetricOn->All];

(*Pert$Exp$Sep[CC[μ,-ν,-μ,-ρ],{ν,ρ},{a,P},{𝒽$Harm$Rep,ℂ$Harm$Rep}];
%/.Dir$Harm$Def$Total//ContractMetric//ToCanonical;
%//.Harm$EQ$Y$Total//ContractMetric//ToCanonical;
ToCanonical[(ℂ$Trace$aZA[-a]/.Eval$Pert$ℂ$Trace$aZA)-ZA$Component[%]]*)

Eval$Pert$ℂ$Trace$aXA=MakeRule[{ℂ$Trace$aXA[-a],-1/2*𝒽$a$m[-a]/r[]^2+(𝒽$a$m[-a]*𝒫[])/6-Ξ$a$m[-a]/r[]^2-(𝒽$a$m[-a]*Τ[b,-b])/6+epsilong1[-a,-b]*Ψ$a$m[b]+(𝒽$a$m[-a]*CD1[-b][r[]]*CD1[b][r[]])/(2*r[]^2)},MetricOn->All];

(*Pert$Exp$Sep[CC[μ,-ν,-μ,-ρ],{ν,ρ},{a,P},{𝒽$Harm$Rep,ℂ$Harm$Rep}];
%/.Dir$Harm$Def$Total//ContractMetric//ToCanonical;
%//.Harm$EQ$Y$Total//ContractMetric//ToCanonical;
ToCanonical[(ℂ$Trace$aXA[-a]/.Eval$Pert$ℂ$Trace$aXA)-XA$Component[%]]*)

δ ((Subscript[(ℂ^μ), P μ a]))=0

(*Pert$Exp$Sep[CC[μ,-ν,-μ,-ρ],{ν,ρ},{P,a},{𝒽$Harm$Rep,ℂ$Harm$Rep}];
%/.Dir$Harm$Def$Total//ContractMetric//ToCanonical;
%//.Harm$EQ$Y$Total//ContractMetric//ToCanonical;
ZA$Component[%];
ToCanonical[%-ℂ$Trace$aZA]*)

(*Pert$Exp$Sep[CC[μ,-ν,-μ,-ρ],{ν,ρ},{P,a},{𝒽$Harm$Rep,ℂ$Harm$Rep}];
%/.Dir$Harm$Def$Total//ContractMetric//ToCanonical;
%//.Harm$EQ$Y$Total//ContractMetric//ToCanonical;
XA$Component[%];
ToCanonical[%-ℂ$Trace$aXA]*)

δ ((Subscript[(ℂ^μ), P μ Q]))=0

Eval$Pert$ℂ$Trace$UAB=MakeRule[{ℂ$Trace$UAB[],𝒽$ab$p[a,-a]/2-𝒽$p[]/r[]^2-(r[]^2*𝒽$ab$p[a,-a]*𝒫[])/6+(𝒽$p[]*𝒫[])/3+Ξ$p[]/r[]^2-(𝒽$p[]*Τ[a,-a])/3+(r[]^2*𝒽$ab$p[a,-a]*Τ[b,-b])/6+Ψ$ab$p[a,-a]-(𝒽$ab$p[b,-b]*CD1[-a][r[]]*CD1[a][r[]])/2+(𝒽$p[]*CD1[-a][r[]]*CD1[a][r[]])/r[]^2},MetricOn->All];
Eval$Pert$ℂ$Trace$VAB=MakeRule[{ℂ$Trace$VAB[],Ψ$ab$o[a,-a]},MetricOn->All];
Eval$Pert$ℂ$Trace$WAB=MakeRule[{ℂ$Trace$WAB[],Ψ$ab$m[a,-a]},MetricOn->All];

(*Pert$Exp$Sep[CC[μ,-ν,-μ,-ρ],{ν,ρ},{P,Q},{𝒽$Harm$Rep,ℂ$Harm$Rep}];
%/.Dir$Harm$Def$Total//ContractMetric//ToCanonical;
DUM=%//.Harm$EQ$Y$Total//ContractMetric//ToCanonical;
(UAB$Component[DUM]-(ℂ$Trace$UAB[]/.Eval$Pert$ℂ$Trace$UAB))//ContractMetric//ToCanonical
(VAB$Component[DUM]-(ℂ$Trace$VAB[]/.Eval$Pert$ℂ$Trace$VAB))//ContractMetric//ToCanonical//Simplify
(WAB$Component[DUM]-(ℂ$Trace$WAB[]/.Eval$Pert$ℂ$Trace$WAB))//ContractMetric//ToCanonical//Simplify
(ϵAB$Component[DUM])//ContractMetric//ToCanonical//Simplify*)

Total

Eval$Pert$ℂ$Trace$Total=Flatten[{Eval$Pert$ℂ$Trace$ab,Eval$Pert$ℂ$Trace$aZA,Eval$Pert$ℂ$Trace$aXA,Eval$Pert$ℂ$Trace$UAB,Eval$Pert$ℂ$Trace$VAB,Eval$Pert$ℂ$Trace$WAB}];

Self-duality identities

δ((Subscript[ℂ, abcd])+i *(Subscript[ℂ, abcd]))=0

Eval$Pert$ℂ$SD$abcd=MakeRule[{ℂ$SD$abcd[],𝒽$ab$p[a,-a]/(2*r[]^2)-𝒽$p[]/r[]^4-(𝒽$ab$p[a,-a]*𝒫[])/6+(𝒽$p[]*𝒫[])/(3*r[]^2)-(𝒽$p[]*Τ[a,-a])/(3*r[]^2)+(𝒽$ab$p[a,-a]*Τ[b,-b])/6+(I*Ψ$m[])/r[]^2+Ψ$p[]-(𝒽$ab$p[b,-b]*CD1[-a][r[]]*CD1[a][r[]])/(2*r[]^2)+(𝒽$p[]*CD1[-a][r[]]*CD1[a][r[]])/r[]^4},MetricOn->All];

(*Pert$Exp$Sep[CC[-μ,-ν,-ρ,-σ]+(I/2)epsilong3[-μ,-ν,α,β]CC[-α,-β,-ρ,-σ],{μ,ν,ρ,σ},{a,b,c,d},{𝒽$Harm$Rep,ℂ$Harm$Rep}];
%/.Dir$Harm$Def$Total//ContractMetric//ToCanonical;
%//.Harm$EQ$Y$Total//ContractMetric//ToCanonical;
ToCanonical[(ℂ$SD$abcd[]/.Eval$Pert$ℂ$SD$abcd)-(((1/4)epsilong1[a,b]epsilong1[c,d]%//ContractMetric//ToCanonical)/Y[])]*)

δ((Subscript[ℂ, Pabc])+i *(Subscript[ℂ, Pabc]))=0

Eval$Pert$ℂ$SD$Aabc$ZA=MakeRule[{ℂ$SD$Aabc$ZA[-a],((-I)*𝒽$a$m[-a])/r[]^2+(I/3)*𝒽$a$m[-a]*𝒫[]-(I/3)*𝒽$a$m[-a]*Τ[b,-b]+I*epsilong1[-a,-b]*Ψ$a$m[b]+Ψ$a$p[-a]+(I*𝒽$a$m[-a]*CD1[-b][r[]]*CD1[b][r[]])/r[]^2},MetricOn->All];
Eval$Pert$ℂ$SD$Aabc$XA=MakeRule[{ℂ$SD$Aabc$XA[-a],-((epsilong1[-a,-b]*𝒽$a$m[b])/r[]^2)+(epsilong1[-a,-b]*𝒽$a$m[b]*𝒫[])/3-(epsilong1[-a,-b]*𝒽$a$m[b]*Τ[c,-c])/3+Ψ$a$m[-a]-I*epsilong1[-a,-b]*Ψ$a$p[b]+(epsilong1[-a,-b]*𝒽$a$m[b]*CD1[-c][r[]]*CD1[c][r[]])/r[]^2},MetricOn->All];

(*Pert$Exp$Sep[CC[-μ,-ν,-ρ,-σ]+(I/2)epsilong3[-μ,-ν,α,β]CC[-α,-β,-ρ,-σ],{μ,ν,ρ,σ},{P,a,b,c},{𝒽$Harm$Rep,ℂ$Harm$Rep}];
%/.Dir$Harm$Def$Total//ContractMetric//ToCanonical;
%//.Harm$EQ$Y$Total//ContractMetric//ToCanonical;
Dum=ToCanonical[((-1/2)epsilong1[b,c]%//ContractMetric//ToCanonical)];
ToCanonical[(ℂ$SD$Aabc$ZA[-a]/.Eval$Pert$ℂ$SD$Aabc$ZA)-ZA$Component[Dum]]
ToCanonical[(ℂ$SD$Aabc$XA[-a]/.Eval$Pert$ℂ$SD$Aabc$XA)-XA$Component[Dum]]*)

δ((Subscript[ℂ, PaQb])+i *(Subscript[ℂ, PaQb]))=0

Eval$Pert$ℂ$SD$aAbB$UAB=MakeRule[{ℂ$SD$aAbB$UAB[-a,-b],𝒽$ab$p[-a,-b]/2-(g1[-a,-b]*𝒽$ab$p[c,-c])/4-(r[]^2*𝒽$ab$p[-a,-b]*𝒫[])/6+(g1[-a,-b]*r[]^2*𝒽$ab$p[c,-c]*𝒫[])/12+(r[]^2*𝒽$ab$p[-a,-b]*Τ[c,-c])/6-(g1[-a,-b]*r[]^2*𝒽$ab$p[c,-c]*Τ[d,-d])/12+Ψ$ab$p[-a,-b]+(I/2)*g1[-a,-b]*Ψ$m[]-(𝒽$ab$p[-a,-b]*CD1[-c][r[]]*CD1[c][r[]])/2+(g1[-a,-b]*𝒽$ab$p[d,-d]*CD1[-c][r[]]*CD1[c][r[]])/4},MetricOn->All];
Eval$Pert$ℂ$SD$aAbB$VAB=MakeRule[{ℂ$SD$aAbB$VAB[-a,-b],I*epsilong1[-a,c]*Ψ$ab$m[-b,-c]+Ψ$ab$o[-a,-b]},MetricOn->All];
Eval$Pert$ℂ$SD$aAbB$WAB=MakeRule[{ℂ$SD$aAbB$WAB[-a,-b],Ψ$ab$m[-a,-b]-I*epsilong1[-a,c]*Ψ$ab$o[-b,-c]},MetricOn->All];
Eval$Pert$ℂ$SD$aAbB$ϵAB=MakeRule[{ℂ$SD$aAbB$ϵAB[-a,-b],(-1/2*I)*epsilong1[-a,c]*𝒽$ab$p[-b,-c]+(I/4)*epsilong1[-a,-b]*𝒽$ab$p[c,-c]+(I/6)*epsilong1[-a,c]*r[]^2*𝒽$ab$p[-b,-c]*𝒫[]-(I/12)*epsilong1[-a,-b]*r[]^2*𝒽$ab$p[c,-c]*𝒫[]-(I/6)*epsilong1[-a,c]*r[]^2*𝒽$ab$p[-b,-c]*Τ[d,-d]+(I/12)*epsilong1[-a,-b]*r[]^2*𝒽$ab$p[c,-c]*Τ[d,-d]-I*epsilong1[-a,c]*Ψ$ab$p[-b,-c]+(epsilong1[-a,-b]*Ψ$m[])/2+(I/2)*epsilong1[-a,d]*𝒽$ab$p[-b,-d]*CD1[-c][r[]]*CD1[c][r[]]-(I/4)*epsilong1[-a,-b]*𝒽$ab$p[d,-d]*CD1[-c][r[]]*CD1[c][r[]]},MetricOn->All];

(*Pert$Exp$Sep[CC[-μ,-ν,-ρ,-σ]+(I/2)epsilong3[-μ,-ν,α,β]CC[-α,-β,-ρ,-σ],{μ,ν,ρ,σ},{P,a,Q,b},{𝒽$Harm$Rep,ℂ$Harm$Rep}];
%/.Dir$Harm$Def$Total//ContractMetric//ToCanonical;
Dum=%//.Harm$EQ$Y$Total//ContractMetric//ToCanonical;
(ℂ$SD$aAbB$UAB[-a,-b]/.Eval$Pert$ℂ$SD$aAbB$UAB)-UAB$Component[Dum]//ToCanonical//Simplify
(ℂ$SD$aAbB$VAB[-a,-b]/.Eval$Pert$ℂ$SD$aAbB$VAB)-VAB$Component[Dum]//ToCanonical//Simplify
(ℂ$SD$aAbB$WAB[-a,-b]/.Eval$Pert$ℂ$SD$aAbB$WAB)-WAB$Component[Dum]//ToCanonical//Simplify
(ℂ$SD$aAbB$ϵAB[-a,-b]/.Eval$Pert$ℂ$SD$aAbB$ϵAB)-ϵAB$Component[Dum]//ToCanonical//Simplify*)

δ((Subscript[ℂ, abAB])+i *(Subscript[ℂ, abAB]))=0

Eval$Pert$ℂ$SD$abAB=MakeRule[{ℂ$SD$abAB[],(I/2)*𝒽$ab$p[a,-a]-(I*𝒽$p[])/r[]^2-(I/6)*r[]^2*𝒽$ab$p[a,-a]*𝒫[]+(I/3)*𝒽$p[]*𝒫[]+(I*Ξ$p[])/r[]^2-(I/3)*𝒽$p[]*Τ[a,-a]+(I/6)*r[]^2*𝒽$ab$p[a,-a]*Τ[b,-b]+Ψ$m[]-(I/2)*𝒽$ab$p[b,-b]*CD1[-a][r[]]*CD1[a][r[]]+(I*𝒽$p[]*CD1[-a][r[]]*CD1[a][r[]])/r[]^2},MetricOn->All];

(*Pert$Exp$Sep[CC[-μ,-ν,-ρ,-σ]+(I/2)epsilong3[-μ,-ν,α,β]CC[-α,-β,-ρ,-σ],{μ,ν,ρ,σ},{a,b,A,B},{𝒽$Harm$Rep,ℂ$Harm$Rep}];
%/.Dir$Harm$Def$Total//ContractMetric//ToCanonical;
(-1/4)epsilong1[a,b]epsilong2[A,B]%//ContractMetric//ToCanonical;
(ℂ$SD$abAB[]/.Eval$Pert$ℂ$SD$abAB)-ToCanonical[(%//.Harm$EQ$Y$Total//ContractMetric//ToCanonical)/Y[]]//ToCanonical*)

δ((Subscript[ℂ, ABPd])+i *(Subscript[ℂ, ABPd]))=0

Eval$Pert$ℂ$SD$aABF$ZA=MakeRule[{ℂ$SD$aABF$ZA[-a],-𝒽$a$m[-a]+(r[]^2*𝒽$a$m[-a]*𝒫[])/3-Ξ$a$m[-a]-I*epsilong1[-a,-b]*Ξ$a$p[b]-(r[]^2*𝒽$a$m[-a]*Τ[b,-b])/3+𝒽$a$m[-a]*CD1[-b][r[]]*CD1[b][r[]]},MetricOn->All];
Eval$Pert$ℂ$SD$aABF$XA=MakeRule[{ℂ$SD$aABF$XA[-a],I*epsilong1[-a,-b]*𝒽$a$m[b]-(I/3)*epsilong1[-a,-b]*r[]^2*𝒽$a$m[b]*𝒫[]+I*epsilong1[-a,-b]*Ξ$a$m[b]-Ξ$a$p[-a]+(I/3)*epsilong1[-a,-b]*r[]^2*𝒽$a$m[b]*Τ[c,-c]-I*epsilong1[-a,-b]*𝒽$a$m[b]*CD1[-c][r[]]*CD1[c][r[]]},MetricOn->All];

(*Pert$Exp$Sep[CC[-μ,-ν,-ρ,-σ]+(I/2)epsilong3[-μ,-ν,α,β]CC[-α,-β,-ρ,-σ],{μ,ν,ρ,σ},{a,P,A,B},{𝒽$Harm$Rep,ℂ$Harm$Rep}];
%/.Dir$Harm$Def$Total//ContractMetric//ToCanonical;
(1/2)epsilong2[A,B]%//ContractMetric//ToCanonical;
Dum=%//.Harm$EQ$Y$Total//ContractMetric//ToCanonical;
(ℂ$SD$aABF$ZA[-a]/.Eval$Pert$ℂ$SD$aABF$ZA)-ZA$Component[Dum]//ToCanonical
(ℂ$SD$aABF$XA[-a]/.Eval$Pert$ℂ$SD$aABF$XA)-XA$Component[Dum]//ToCanonical*)

δ((Subscript[ℂ, ABPQ])+i *(Subscript[ℂ, ABPQ]))=0

Eval$Pert$ℂ$SD$ABFG=MakeRule[{ℂ$SD$ABFG[],(r[]^2*𝒽$ab$p[a,-a])/2-𝒽$p[]-(r[]^4*𝒽$ab$p[a,-a]*𝒫[])/6+(r[]^2*𝒽$p[]*𝒫[])/3+Ξ$p[]-(r[]^2*𝒽$p[]*Τ[a,-a])/3+(r[]^4*𝒽$ab$p[a,-a]*Τ[b,-b])/6-I*r[]^2*Ψ$m[]-(r[]^2*𝒽$ab$p[b,-b]*CD1[-a][r[]]*CD1[a][r[]])/2+𝒽$p[]*CD1[-a][r[]]*CD1[a][r[]]},MetricOn->All];

(*Pert$Exp$Sep[CC[-μ,-ν,-ρ,-σ]+(I/2)epsilong3[-μ,-ν,α,β]CC[-α,-β,-ρ,-σ],{μ,ν,ρ,σ},{A,B,P,Q},{𝒽$Harm$Rep,ℂ$Harm$Rep}];
%/.Dir$Harm$Def$Total//ContractMetric//ToCanonical;
(1/4)epsilong2[A,B]epsilong2[P,Q]%//ContractMetric//ToCanonical;
(ℂ$SD$ABFG[]/.Eval$Pert$ℂ$SD$ABFG)-ToCanonical[(%//.Harm$EQ$Y$Total//ContractMetric//ToCanonical)/Y[]]//ToCanonical*)

Total

Eval$Pert$ℂ$SD$Total=Flatten[{Eval$Pert$ℂ$SD$abcd,Eval$Pert$ℂ$SD$ABFG,Eval$Pert$ℂ$SD$abAB,Eval$Pert$ℂ$SD$aAbB$UAB,Eval$Pert$ℂ$SD$aAbB$VAB,Eval$Pert$ℂ$SD$aAbB$WAB,Eval$Pert$ℂ$SD$aAbB$ϵAB,Eval$Pert$ℂ$SD$aABF$ZA,Eval$Pert$ℂ$SD$aABF$XA,Eval$Pert$ℂ$SD$Aabc$ZA,Eval$Pert$ℂ$SD$Aabc$XA}];

Total

Eval$Pert$ℂ$Algebraic=Flatten[{Eval$Pert$ℂ$Trace$Total,Eval$Pert$ℂ$SD$Total}];

Substitution Rules

From the above, we notice the following important substitution rules (the derivations are straightforward, given in a previous .nb)

(Subscript[(SuperMinus[Ψ]), a b]) and (Subscript[(Ψ^⊕), a b]): they are the dual of each other, and traceless

ℂ$Alg$Ψ$ab$o$To$Ψ$ab$m=MakeRule[{Ψ$ab$o[-a,-b],(-I)*epsilong1[-a,c]*Ψ$ab$m[-b,-c]},MetricOn->All];
(*ToCanonical[(ℂ$Algebraic[[11]]/.ℂ$Alg$Ψ$ab$o$To$Ψ$ab$m//ContractMetric//ToCanonical)==0]*)
ℂ$Alg$Trace$Ψ$ab$m=MakeRule[{Ψ$ab$m[a,-a],0}];
(*ToCanonical[(ℂ$Algebraic[[6]]/.ℂ$Alg$Trace$Ψ$ab$m//ContractMetric//ToCanonical)==0]*)
ℂ$Alg$Ψ$ab$m$To$Ψ$ab$o=MakeRule[{Ψ$ab$m[-a,-b],I*epsilong1[-a,c]*Ψ$ab$o[-b,-c]},MetricOn->All];
(*ToCanonical[(ℂ$Algebraic[[12]]/.ℂ$Alg$Ψ$ab$m$To$Ψ$ab$o//ContractMetric//ToCanonical)==0]*)
ℂ$Alg$Trace$Ψ$ab$o=MakeRule[{Ψ$ab$o[a,-a],0},MetricOn->All];
(*ToCanonical[(ℂ$Algebraic[[5]]/.ℂ$Alg$Trace$Ψ$ab$o//ContractMetric//ToCanonical)==0]*)

(SuperPlus[Ψ]), (SuperMinus[Ψ]), (SuperPlus[Ξ]),(SuperPlus[𝒽]),(Subscript[(h^a), a]): only three are independent. I write some representative substitution rules.

ℂ$Alg$Ψ$m$To$Ψ$p$𝒽$p=MakeRule[{Ψ$m[],(I/2)*𝒽$ab$p[a,-a]-(I*𝒽$p[])/r[]^2-(I/6)*r[]^2*𝒽$ab$p[a,-a]*𝒫[]+(I/3)*𝒽$p[]*𝒫[]-(I/3)*𝒽$p[]*Τ[a,-a]+(I/6)*r[]^2*𝒽$ab$p[a,-a]*Τ[b,-b]+I*r[]^2*Ψ$p[]-(I/2)*𝒽$ab$p[b,-b]*CD1[-a][r[]]*CD1[a][r[]]+(I*𝒽$p[]*CD1[-a][r[]]*CD1[a][r[]])/r[]^2},MetricOn->All];
(*ToCanonical[(ℂ$Algebraic[[7]]/.ℂ$Alg$Ψ$m$To$Ψ$p$𝒽$p//ContractMetric//ToCanonical)==0]*)
ℂ$Alg$Ξ$p$To$Ψ$p$𝒽$p=MakeRule[{Ξ$p[],-(r[]^2*𝒽$ab$p[a,-a])+2*𝒽$p[]+(r[]^4*𝒽$ab$p[a,-a]*𝒫[])/3-(2*r[]^2*𝒽$p[]*𝒫[])/3+(2*r[]^2*𝒽$p[]*Τ[a,-a])/3-(r[]^4*𝒽$ab$p[a,-a]*Τ[b,-b])/3-r[]^4*Ψ$p[]+r[]^2*𝒽$ab$p[b,-b]*CD1[-a][r[]]*CD1[a][r[]]-2*𝒽$p[]*CD1[-a][r[]]*CD1[a][r[]]},MetricOn->All];
(*Column[(ℂ$Algebraic/.Flatten[{ℂ$Alg$Ξ$p$To$Ψ$p$𝒽$p,ℂ$Alg$Ψ$m$To$Ψ$p$𝒽$p}]//ContractMetric//ToCanonical),Dividers->All]*)
ℂ$Alg$Ψ$p$To$Ξ$p$𝒽$p=MakeRule[{Ψ$p[],-(𝒽$ab$p[a,-a]/r[]^2)+(2*𝒽$p[])/r[]^4+(𝒽$ab$p[a,-a]*𝒫[])/3-(2*𝒽$p[]*𝒫[])/(3*r[]^2)-Ξ$p[]/r[]^4+(2*𝒽$p[]*Τ[a,-a])/(3*r[]^2)-(𝒽$ab$p[a,-a]*Τ[b,-b])/3+(𝒽$ab$p[b,-b]*CD1[-a][r[]]*CD1[a][r[]])/r[]^2-(2*𝒽$p[]*CD1[-a][r[]]*CD1[a][r[]])/r[]^4},MetricOn->All];
(*(Ξ$p[]-(Ξ$p[]/.ℂ$Alg$Ξ$p$To$Ψ$p$𝒽$p))/.ℂ$Alg$Ψ$p$To$Ξ$p$𝒽$p//ToCanonical*)
ℂ$Alg$Ψ$p$To$Ψ$m$Ξ$p=MakeRule[{Ψ$p[],Ξ$p[]/r[]^4-((2*I)*Ψ$m[])/r[]^2},MetricOn->All];
(*((((r[])^4)ℂ$SD$abcd-ℂ$SD$ABFG)//ContractMetric//ToCanonical)
%/.ℂ$Alg$Ψ$p$To$Ψ$m$Ξ$p//ContractMetric//ToCanonical*)
ℂ$Alg$Ψ$m$To$Ψ$p$Ξ$p=MakeRule[{Ψ$m[],((I/2)*(-Ξ$p[]+r[]^4*Ψ$p[]))/r[]^2},MetricOn->All];
(*(Ψ$p[]-(Ψ$p[]/.ℂ$Alg$Ψ$p$To$Ψ$m$Ξ$p))
%/.ℂ$Alg$Ψ$m$To$Ψ$p$Ξ$p//ToCanonical*)
(**)
ℂ$Alg$𝒽$p$To$Ψ$m$Ξ$p=MakeRule[{𝒽$p[],(3*r[]^2*𝒽$ab$p[a,-a])/(2*𝒲[])-(r[]^4*𝒽$ab$p[a,-a]*𝒫[])/(2*𝒲[])+(3*Ξ$p[])/𝒲[]+(r[]^4*𝒽$ab$p[a,-a]*Τ[b,-b])/(2*𝒲[])-((3*I)*r[]^2*Ψ$m[])/𝒲[]-(3*r[]^2*𝒽$ab$p[b,-b]*CD1[-a][r[]]*CD1[a][r[]])/(2*𝒲[])},MetricOn->All]
ℂ$Alg$𝒽$p$To$Ψ$p$Ξ$p=MakeRule[{𝒽$p[],-1/2*(-3*r[]^2*𝒽$ab$p[a,-a]+r[]^4*𝒽$ab$p[a,-a]*𝒫[]-3*Ξ$p[]-r[]^4*𝒽$ab$p[a,-a]*Τ[b,-b]-3*r[]^4*Ψ$p[]+3*r[]^2*𝒽$ab$p[b,-b]*CD1[-a][r[]]*CD1[a][r[]])/𝒲[]},MetricOn->All];
ℂ$Alg$Ξ$p$To$Ψ$p$Ψ$m=MakeRule[{Ξ$p[],(2*I)*r[]^2*Ψ$m[]+r[]^4*Ψ$p[]},MetricOn->All]
ℂ$Alg$𝒽$p$To$Ψ$p$Ψ$m=MakeRule[{𝒽$p[],(3*r[]^2*𝒽$ab$p[a,-a])/(2*𝒲[])-(r[]^4*𝒽$ab$p[a,-a]*𝒫[])/(2*𝒲[])+(r[]^4*𝒽$ab$p[a,-a]*Τ[b,-b])/(2*𝒲[])+((3*I)*r[]^2*Ψ$m[])/𝒲[]+(3*r[]^4*Ψ$p[])/𝒲[]-(3*r[]^2*𝒽$ab$p[b,-b]*CD1[-a][r[]]*CD1[a][r[]])/(2*𝒲[])},MetricOn->All]

((Subscript[(SuperPlus[Ψ]), a]),(Subscript[(SuperMinus[Ψ]), a]),(Subscript[(SuperPlus[Ξ]), a]), (Subscript[(SuperMinus[Ξ]), a]),(Subscript[(SuperMinus[𝒽]), a])): only 2 are independent. I write some representative substitution rules.

(*To Ψ$a$p and 𝒽$a$m*)
ℂ$Alg$Ψ$a$m$To$Ψ$a$p$𝒽$a$m=MakeRule[{Ψ$a$m[-d],(epsilong1[-d,-a]*𝒽$a$m[a])/r[]^2-(epsilong1[-d,-a]*𝒽$a$m[a]*𝒫[])/3+(epsilong1[-d,-a]*𝒽$a$m[a]*Τ[b,-b])/3+I*epsilong1[-d,-a]*Ψ$a$p[a]-(epsilong1[-d,-a]*𝒽$a$m[a]*CD1[-b][r[]]*CD1[b][r[]])/r[]^2},MetricOn->All];
(*Column[ℂ$Algebraic/.ℂ$Alg$Ψ$a$m$To$Ψ$a$p$𝒽$a$m//ContractMetric//ToCanonical,Dividers->All]*)
ℂ$Alg$Ξ$a$p$To$Ψ$a$p$𝒽$a$m=MakeRule[{Ξ$a$p[-a],((3*I)/2)*epsilong1[-a,-b]*𝒽$a$m[b]-(I/2)*epsilong1[-a,-b]*r[]^2*𝒽$a$m[b]*𝒫[]+(I/2)*epsilong1[-a,-b]*r[]^2*𝒽$a$m[b]*Τ[c,-c]-epsilong1[-a,-b]*r[]^2*Ψ$a$p[b]-((3*I)/2)*epsilong1[-a,-b]*𝒽$a$m[b]*CD1[-c][r[]]*CD1[c][r[]]},MetricOn->All];
(*Column[ℂ$Algebraic/.ℂ$Alg$Ξ$a$p$To$Ψ$a$p$𝒽$a$m//ContractMetric//ToCanonical,Dividers->All]*)
ℂ$Alg$Ξ$a$m$To$Ψ$a$p$𝒽$a$m=MakeRule[{Ξ$a$m[-a],I*r[]^2*Ψ$a$p[-a]-(𝒽$a$m[-a]*(-3+r[]^2*𝒫[]-r[]^2*Τ[b,-b]+3*CD1[-b][r[]]*CD1[b][r[]]))/6},MetricOn->All];
(*Column[ℂ$Algebraic/.Flatten[{ℂ$Alg$Ξ$a$m$To$Ψ$a$p$𝒽$a$m,ℂ$Alg$Ξ$a$p$To$Ψ$a$p$𝒽$a$m}]//ContractMetric//ToCanonical,Dividers->All]*)
(*To Ψ$a$m and 𝒽$a$m*)
ℂ$Alg$Ψ$a$p$To$Ψ$a$m$𝒽$a$m=MakeRule[{Ψ$a$p[-a],((I/3)*(3*𝒽$a$m[-a]-r[]^2*𝒽$a$m[-a]*𝒫[]+r[]^2*𝒽$a$m[-a]*Τ[b,-b]-3*epsilong1[-a,-b]*r[]^2*Ψ$a$m[b]-3*𝒽$a$m[-a]*CD1[-b][r[]]*CD1[b][r[]]))/r[]^2},MetricOn->All];
(*Column[ℂ$Algebraic/.ℂ$Alg$Ψ$a$p$To$Ψ$a$m$𝒽$a$m//ContractMetric//ToCanonical,Dividers->All]*)
ℂ$Alg$Ξ$a$p$To$Ψ$a$m$𝒽$a$m=MakeRule[{Ξ$a$p[-c],(I/6)*(3*epsilong1[-c,-a]*𝒽$a$m[a]-epsilong1[-c,-a]*r[]^2*𝒽$a$m[a]*𝒫[]+epsilong1[-c,-a]*r[]^2*𝒽$a$m[a]*Τ[b,-b]+6*r[]^2*Ψ$a$m[-c]-3*epsilong1[-c,-a]*𝒽$a$m[a]*CD1[-b][r[]]*CD1[b][r[]])},MetricOn->All];
(*Column[ℂ$Algebraic/.Flatten[{ℂ$Alg$Ψ$a$p$To$Ψ$a$m$𝒽$a$m,ℂ$Alg$Ξ$a$p$To$Ψ$a$m$𝒽$a$m}]//ContractMetric//ToCanonical,Dividers->All]*)
ℂ$Alg$Ξ$a$m$To$Ψ$a$m$𝒽$a$m=MakeRule[{Ξ$a$m[-a],-1/2*𝒽$a$m[-a]+(r[]^2*𝒽$a$m[-a]*𝒫[])/6-(r[]^2*𝒽$a$m[-a]*Τ[b,-b])/6+epsilong1[-a,-b]*r[]^2*Ψ$a$m[b]+(𝒽$a$m[-a]*CD1[-b][r[]]*CD1[b][r[]])/2},MetricOn->All];
(*Column[ℂ$Algebraic/.Flatten[{ℂ$Alg$Ξ$a$m$To$Ψ$a$m$𝒽$a$m}]//ContractMetric//ToCanonical,Dividers->All]*)
(*To Ψ$a$p and Ψ$a$m*)
ℂ$Alg$𝒽$a$m$To$Ψ$a$p$Ψ$a$m=MakeRule[{𝒽$a$m[-a],-((-3*epsilong1[-a,-b]*r[]^2*Ψ$a$m[b]+(3*I)*r[]^2*Ψ$a$p[-a])/𝒲[])},MetricOn->All];
ℂ$Alg$Ξ$a$p$To$Ψ$a$p$Ψ$a$m=MakeRule[{Ξ$a$p[-b],-1/2*(r[]^2*(I*Ψ$a$m[-b]*(-9+3*r[]^2*𝒫[]-2*r[]^2*Τ[a,-a]-r[]^2*Τ[c,-c]+6*CD1[-a][r[]]*CD1[a][r[]]+3*CD1[-c][r[]]*CD1[c][r[]])+epsilong1[-b,-c]*Ψ$a$p[c]*(-3+r[]^2*𝒫[]-r[]^2*Τ[d,-d]+3*CD1[-d][r[]]*CD1[d][r[]])))/𝒲[]},MetricOn->All];
ℂ$Alg$Ξ$a$m$To$Ψ$a$p$Ψ$a$m=MakeRule[{Ξ$a$m[-b],-1/2*(r[]^2*(I*Ψ$a$p[-b]*(-3+r[]^2*𝒫[]-2*r[]^2*Τ[a,-a]+r[]^2*Τ[c,-c]+6*CD1[-a][r[]]*CD1[a][r[]]-3*CD1[-c][r[]]*CD1[c][r[]])+epsilong1[-b,-c]*Ψ$a$m[c]*(-3+r[]^2*𝒫[]-r[]^2*Τ[d,-d]+3*CD1[-d][r[]]*CD1[d][r[]])))/𝒲[]},MetricOn->All];
(*To Ξ$a$m and 𝒽$a$m*)
ℂ$Alg$Ψ$a$m$To$Ξ$a$m$𝒽$a$m=MakeRule[{Ψ$a$m[-b],(epsilong1[-b,-a]*𝒽$a$m[a])/(2*r[]^2)-(epsilong1[-b,-a]*𝒽$a$m[a]*𝒫[])/6+(epsilong1[-b,-a]*Ξ$a$m[a])/r[]^2+(epsilong1[-b,-a]*𝒽$a$m[a]*Τ[c,-c])/6-(epsilong1[-b,-a]*𝒽$a$m[a]*CD1[-c][r[]]*CD1[c][r[]])/(2*r[]^2)},MetricOn->All];
(*Column[ℂ$Algebraic/.Flatten[{ℂ$Alg$Ψ$a$m$To$Ξ$a$m$𝒽$a$m}]//ContractMetric//ToCanonical,Dividers->All]*)
ℂ$Alg$Ψ$a$p$To$Ξ$a$m$𝒽$a$m=MakeRule[{Ψ$a$p[-a],((I/2)*𝒽$a$m[-a])/r[]^2-(I/6)*𝒽$a$m[-a]*𝒫[]-(I*Ξ$a$m[-a])/r[]^2+(I/6)*𝒽$a$m[-a]*Τ[b,-b]-((I/2)*𝒽$a$m[-a]*CD1[-b][r[]]*CD1[b][r[]])/r[]^2},MetricOn->All];
(*Column[ℂ$Algebraic/.Flatten[{ℂ$Alg$Ψ$a$p$To$Ξ$a$m$𝒽$a$m,ℂ$Alg$Ψ$a$m$To$Ξ$a$m$𝒽$a$m}]//ContractMetric//ToCanonical,Dividers->All]*)
ℂ$Alg$Ξ$a$p$To$Ξ$a$m$𝒽$a$m=MakeRule[{Ξ$a$p[-a],I*epsilong1[-a,-b]*𝒽$a$m[b]-(I/3)*epsilong1[-a,-b]*r[]^2*𝒽$a$m[b]*𝒫[]+I*epsilong1[-a,-b]*Ξ$a$m[b]+(I/3)*epsilong1[-a,-b]*r[]^2*𝒽$a$m[b]*Τ[c,-c]-I*epsilong1[-a,-b]*𝒽$a$m[b]*CD1[-c][r[]]*CD1[c][r[]]},MetricOn->All];
(*Column[ℂ$Algebraic/.Flatten[{ℂ$Alg$Ξ$a$p$To$Ξ$a$m$𝒽$a$m}]//ContractMetric//ToCanonical,Dividers->All]*)
(*To Ξ$a$p and 𝒽$a$m*)
ℂ$Alg$Ψ$a$p$To$Ξ$a$p$𝒽$a$m=MakeRule[{Ψ$a$p[-d],(((3*I)/2)*𝒽$a$m[-d])/r[]^2-(I/2)*𝒽$a$m[-d]*𝒫[]-(epsilong1[-d,-a]*Ξ$a$p[a])/r[]^2+(I/2)*𝒽$a$m[-d]*Τ[a,-a]-(((3*I)/2)*𝒽$a$m[-d]*CD1[-a][r[]]*CD1[a][r[]])/r[]^2},MetricOn->All];
(*Column[ℂ$Algebraic/.Flatten[{ℂ$Alg$Ψ$a$p$To$Ξ$a$p$𝒽$a$m}]//ContractMetric//ToCanonical,Dividers->All]*)
ℂ$Alg$Ψ$a$m$To$Ξ$a$p$𝒽$a$m=MakeRule[{Ψ$a$m[-a],-1/2*(epsilong1[-a,-b]*𝒽$a$m[b])/r[]^2+(epsilong1[-a,-b]*𝒽$a$m[b]*𝒫[])/6-(I*Ξ$a$p[-a])/r[]^2-(epsilong1[-a,-b]*𝒽$a$m[b]*Τ[c,-c])/6+(epsilong1[-a,-b]*𝒽$a$m[b]*CD1[-c][r[]]*CD1[c][r[]])/(2*r[]^2)},MetricOn->All];
(*Column[ℂ$Algebraic/.Flatten[{ℂ$Alg$Ψ$a$m$To$Ξ$a$p$𝒽$a$m,ℂ$Alg$Ψ$a$p$To$Ξ$a$p$𝒽$a$m}]//ContractMetric//ToCanonical,Dividers->All]*)
ℂ$Alg$Ξ$a$m$To$Ξ$a$p$𝒽$a$m=MakeRule[{Ξ$a$m[-a],-𝒽$a$m[-a]+(r[]^2*𝒽$a$m[-a]*𝒫[])/3-I*epsilong1[-a,-b]*Ξ$a$p[b]-(r[]^2*𝒽$a$m[-a]*Τ[b,-b])/3+𝒽$a$m[-a]*CD1[-b][r[]]*CD1[b][r[]]},MetricOn->All];
(*Column[ℂ$Algebraic/.Flatten[{ℂ$Alg$Ξ$a$m$To$Ξ$a$p$𝒽$a$m}]//ContractMetric//ToCanonical,Dividers->All]*)

((Subscript[(SuperPlus[Ψ]), a b]), (Subscript[Η, a b]), (SuperMinus[Ψ])): (Subscript[(SuperPlus[Ψ]), a b]) and (SuperMinus[Ψ]) give (Subscript[(SuperPlus[Η]), a b]) algebraically, but (Subscript[((SuperPlus[𝒽])^a), a]) remains free.

ℂ$Alg$Ψ$ab$p$To$Ψ$m=MakeRule[{Ψ$ab$p[-a,-b],-1/2*𝒽$ab$p[-a,-b]+(g1[-a,-b]*𝒽$ab$p[c,-c])/4+(r[]^2*𝒽$ab$p[-a,-b]*𝒫[])/6-(g1[-a,-b]*r[]^2*𝒽$ab$p[c,-c]*𝒫[])/12-(r[]^2*𝒽$ab$p[-a,-b]*Τ[c,-c])/6+(g1[-a,-b]*r[]^2*𝒽$ab$p[c,-c]*Τ[d,-d])/12-(I/2)*g1[-a,-b]*Ψ$m[]+(𝒽$ab$p[-a,-b]*CD1[-c][r[]]*CD1[c][r[]])/2-(g1[-a,-b]*𝒽$ab$p[d,-d]*CD1[-c][r[]]*CD1[c][r[]])/4},MetricOn->All];
(*Column[ℂ$Algebraic/.Flatten[{ℂ$Alg$Ψ$ab$p$To$Ψ$m}]//ContractMetric//ToCanonical,Dividers->All]*)
ℂ$Alg$Trace$Ψ$ab$p=MakeRule[{Ψ$ab$p[a,-a],(-I)*Ψ$m[]},MetricOn->All];
ℂ$Alg$Η$To$Ψ$ab$p=MakeRule[{Η[-a,-b],-1/2*(12*r[]^2*Ψ$ab$p[-a,-b]+3*g1[-a,-b]*(Ξ$p[]-r[]^4*Ψ$p[]))/(r[]^2*𝒲[])},MetricOn->All];

1.6. Linearised algebraic identities of δ (Subscript[𝕁 , μ ν ρ ])

DefTensor[𝕁$Trace$a[-a],M1]
DefTensor[𝕁$Trace$ZA[],M1]
DefTensor[𝕁$Trace$XA[],M1]
DefTensor[𝕁$SD$abc[-a],M1]
DefTensor[𝕁$SD$abP$ZA[-a,-b],M1]
DefTensor[𝕁$SD$abP$XA[-a,-b],M1]
DefTensor[𝕁$SD$aPQ[-a],M1]
DefTensor[𝕁$SD$Pab$ZA[],M1]
DefTensor[𝕁$SD$Pab$XA[],M1]
DefTensor[𝕁$SD$PQa$UAB[-a],M1]
DefTensor[𝕁$SD$PQa$VAB[-a],M1]
DefTensor[𝕁$SD$PQa$WAB[-a],M1]
DefTensor[𝕁$SD$PQa$ϵAB[-a],M1]
DefTensor[𝕁$SD$PAB$ZA[],M1]
DefTensor[𝕁$SD$PAB$XA[],M1]

Trace identities

(Subscript[(𝕁^μ), μ a])

Eval$Pert$𝕁$Trace$a=MakeRule[{𝕁$Trace$a[-a],(2*Γ$a$p[-a])/r[]^2-epsilong1[-a,-b]*Π$a$p[b]+(𝒽$p[]*𝒫[]*CD1[-a][r[]])/r[]^3-(𝒽$ab$p[b,-b]*CD1[-a][𝒫[]])/3+(𝒽$p[]*CD1[-a][𝒫[]])/(3*r[]^2)+(𝒽$ab$p[b,c]*CD1[-a][Τ[-b,-c]])/2-(𝒽$p[]*CD1[-a][Τ[b,-b]])/(3*r[]^2)-(𝒽$ab$p[b,-b]*CD1[-a][Τ[c,-c]])/6+(𝒽$ab$p[-a,b]*CD1[-b][Τ[c,-c]])/6-(𝒽$p[]*Τ[-a,-b]*CD1[b][r[]])/r[]^3+(𝒽$ab$p[-a,-b]*CD1[b][𝒫[]])/3-(𝒽$ab$p[b,c]*CD1[-c][Τ[-a,-b]])/2},MetricOn->All];

(*Pert$Exp$Sep[𝕁[μ,-μ,-α],{α},{a},{𝕁$Harm$Rep,𝒽$Harm$Rep}];
%/.Dir$Harm$Def$Total//ContractMetric//ToCanonical;
%//.Harm$EQ$Y$Total//ContractMetric//ToCanonical;
ToCanonical[(𝕁$Trace$a[-a]/.Eval$Pert$𝕁$Trace$a)-(%/Y[])]*)

(Subscript[(𝕁^μ), μ A])

Eval$Pert$𝕁$Trace$ZA=MakeRule[{𝕁$Trace$ZA[],Γ$p[]/r[]^2+Π$ab$p[a,-a]+((I/2)*epsilong1[-a,-b]*𝒽$a$m[a]*𝒫[]*CD1[b][r[]])/r[]-((I/2)*epsilong1[-a,c]*𝒽$a$m[a]*Τ[-b,-c]*CD1[b][r[]])/r[]+(I/2)*epsilong1[-a,-b]*𝒽$a$m[a]*CD1[b][𝒫[]]-(I/2)*epsilong1[b,c]*𝒽$a$m[a]*CD1[-c][Τ[-a,-b]]},MetricOn->All];
Eval$Pert$𝕁$Trace$XA=MakeRule[{𝕁$Trace$XA[],-(Γ$m[]/r[]^2)+Π$ab$m[a,-a]-(𝒽$a$m[a]*𝒫[]*CD1[-a][r[]])/(2*r[])-(𝒽$a$m[a]*CD1[-a][𝒫[]])/6+(𝒽$a$m[a]*CD1[-a][Τ[b,-b]])/6+(𝒽$a$m[a]*Τ[-a,-b]*CD1[b][r[]])/(2*r[])},MetricOn->All];

(*Pert$Exp$Sep[𝕁[μ,-μ,-α],{α},{P},{𝕁$Harm$Rep,𝒽$Harm$Rep}];
%/.Dir$Harm$Def$Total//ContractMetric//ToCanonical;
Dum=%//.Harm$EQ$Y$Total//ContractMetric//ToCanonical;
((𝕁$Trace$ZA[]/.Eval$Pert$𝕁$Trace$ZA)-ZA$Component[Dum]//ContractMetric//ToCanonical)//ToCanonical
((𝕁$Trace$XA[]/.Eval$Pert$𝕁$Trace$XA)-XA$Component[Dum]//ContractMetric//ToCanonical)//ToCanonical*)

Self-duality relations

(Subscript[𝕁, a b c])+i𝕁(Subscript[*, a b c])

Eval$Pert$𝕁$SD$abc=MakeRule[{𝕁$SD$abc[-a],((-2*I)*Π$a$m[-a])/r[]^2+Π$a$p[-a]-(epsilong1[-a,b]*𝒽$p[]*CD1[-b][Τ[c,-c]])/(6*r[]^2)+(epsilong1[-a,b]*𝒽$ab$p[c,-c]*CD1[-b][Τ[d,-d]])/12+(epsilong1[-a,-b]*𝒽$ab$p[c,-c]*CD1[b][𝒫[]])/6-(epsilong1[-a,-b]*𝒽$p[]*CD1[b][𝒫[]])/(3*r[]^2)-(epsilong1[b,c]*𝒽$ab$p[d,-d]*CD1[-c][Τ[-a,-b]])/4+(epsilong1[b,c]*𝒽$p[]*CD1[-c][Τ[-a,-b]])/(2*r[]^2)},MetricOn->All];

(*Pert$Exp$Sep[𝕁[-μ,-ν,-ρ]+(I/2)𝕁[-μ,α,β]epsilong3[-α,-β,-ν,-ρ],{μ,ν,ρ},{a,b,c},{𝕁$Harm$Rep,𝒽$Harm$Rep}];
%/.Dir$Harm$Def$Total//ContractMetric//ToCanonical;
((-1/2)epsilong1[b,c]%//.Harm$EQ$Y$Total//ContractMetric//ToCanonical);
((𝕁$SD$abc[-a]/.Eval$Pert$𝕁$SD$abc)-(%/Y[]))//ToCanonical*)

(Subscript[𝕁, a b P])+i𝕁(Subscript[*, a b P])

Eval$Pert$𝕁$SD$abP$ZA=MakeRule[{𝕁$SD$abP$ZA[-a,-b],I*epsilong1[-b,c]*Π$ab$m[-a,-c]+Π$ab$p[-a,-b]-I*g1[-a,-b]*Π$m[]+epsilong1[-a,-b]*Π$p[]+(I/3)*epsilong1[-a,-b]*𝒽$a$m[c]*CD1[-c][𝒫[]]+(I/2)*epsilong1[-b,d]*𝒽$a$m[c]*CD1[-c][Τ[-a,-d]]+(I/6)*epsilong1[-b,c]*𝒽$a$m[-a]*CD1[-c][Τ[d,-d]]+(I/6)*epsilong1[-a,-b]*𝒽$a$m[c]*CD1[-c][Τ[d,-d]]+(I/3)*epsilong1[-b,-c]*𝒽$a$m[-a]*CD1[c][𝒫[]]-(I/2)*epsilong1[-b,d]*𝒽$a$m[c]*CD1[-d][Τ[-a,-c]]},MetricOn->All];
Eval$Pert$𝕁$SD$abP$XA=MakeRule[{𝕁$SD$abP$XA[-a,-b],Π$ab$m[-a,-b]-I*epsilong1[-b,c]*Π$ab$p[-a,-c]+epsilong1[-a,-b]*Π$m[]+I*g1[-a,-b]*Π$p[]+(𝒽$a$m[-a]*CD1[-b][𝒫[]])/3-(𝒽$a$m[c]*CD1[-b][Τ[-a,-c]])/2+(𝒽$a$m[-a]*CD1[-b][Τ[c,-c]])/6-(g1[-a,-b]*𝒽$a$m[c]*CD1[-c][𝒫[]])/3+(𝒽$a$m[c]*CD1[-c][Τ[-a,-b]])/2-(g1[-a,-b]*𝒽$a$m[c]*CD1[-c][Τ[d,-d]])/6},MetricOn->All];

(*Pert$Exp$Sep[𝕁[-μ,-ν,-ρ]+(I/2)𝕁[-μ,α,β]epsilong3[-α,-β,-ν,-ρ],{μ,ν,ρ},{a,b,P},{𝕁$Harm$Rep,𝒽$Harm$Rep}];
%/.Dir$Harm$Def$Total//ContractMetric//ToCanonical;
Dum=%//.Harm$EQ$Y$Total//ContractMetric//ToCanonical;
ToCanonical[ZA$Component[Dum]-𝕁$SD$abP$ZA[-a,-b]/.Eval$Pert$𝕁$SD$abP$ZA]
ToCanonical[XA$Component[Dum]-𝕁$SD$abP$XA[-a,-b]/.Eval$Pert$𝕁$SD$abP$XA]*)

(Subscript[𝕁, a P Q])+i𝕁(Subscript[*, a P Q])

Eval$Pert$𝕁$SD$aPQ=MakeRule[{𝕁$SD$aPQ[-a],-2*Π$a$m[-a]-I*r[]^2*Π$a$p[-a]+(I/6)*epsilong1[-a,b]*𝒽$p[]*CD1[-b][Τ[c,-c]]+(I/12)*epsilong1[-a,b]*r[]^2*𝒽$ab$p[c,-c]*CD1[-b][Τ[d,-d]]+(I/3)*epsilong1[-b,c]*r[]^2*𝒽$ab$p[-a,-c]*CD1[b][𝒫[]]-(I/3)*epsilong1[-a,c]*r[]^2*𝒽$ab$p[-b,-c]*CD1[b][𝒫[]]+(I/6)*epsilong1[-a,-b]*r[]^2*𝒽$ab$p[c,-c]*CD1[b][𝒫[]]+(I/3)*epsilong1[-a,-b]*𝒽$p[]*CD1[b][𝒫[]]-(I/4)*epsilong1[b,c]*r[]^2*𝒽$ab$p[d,-d]*CD1[-c][Τ[-a,-b]]-(I/2)*epsilong1[b,c]*𝒽$p[]*CD1[-c][Τ[-a,-b]]+(I/2)*epsilong1[b,c]*r[]^2*𝒽$ab$p[-b,d]*CD1[-c][Τ[-a,-d]]-(I/6)*epsilong1[b,c]*r[]^2*𝒽$ab$p[-a,-b]*CD1[-c][Τ[d,-d]]-(I/6)*epsilong1[-a,b]*r[]^2*𝒽$ab$p[-b,c]*CD1[-c][Τ[d,-d]]-(I/2)*epsilong1[b,c]*r[]^2*𝒽$ab$p[-b,d]*CD1[-d][Τ[-a,-c]]},MetricOn->All];

(*Pert$Exp$Sep[𝕁[-μ,-ν,-ρ]+(I/2)𝕁[-μ,α,β]epsilong3[-α,-β,-ν,-ρ],{μ,ν,ρ},{a,P,Q},{𝕁$Harm$Rep,𝒽$Harm$Rep}];
(1/2)epsilong2[P,Q]%/.Dir$Harm$Def$Total//ContractMetric//ToCanonical;
ToCanonical[(𝕁$SD$aPQ[-a]/.Eval$Pert$𝕁$SD$aPQ)-((%//.Harm$EQ$Y$Total//ContractMetric//ToCanonical)/Y[])]*)

(Subscript[𝕁, P a b])+i𝕁(Subscript[*, P a b])

Eval$Pert$𝕁$SD$Pab$ZA=MakeRule[{𝕁$SD$Pab$ZA[],(I*Γ$m[])/r[]^2-2*Π$p[]-((I/2)*𝒽$a$m[a]*𝒫[]*CD1[-a][r[]])/r[]-(I/6)*𝒽$a$m[a]*CD1[-a][𝒫[]]+(I/6)*𝒽$a$m[a]*CD1[-a][Τ[b,-b]]+((I/2)*𝒽$a$m[a]*Τ[-a,-b]*CD1[b][r[]])/r[]},MetricOn->All];
Eval$Pert$𝕁$SD$Pab$XA=MakeRule[{𝕁$SD$Pab$XA[],(I*Γ$p[])/r[]^2-2*Π$m[]+(epsilong1[-a,b]*𝒽$a$m[a]*CD1[-b][Τ[c,-c]])/6-(epsilong1[-a,-b]*𝒽$a$m[a]*𝒫[]*CD1[b][r[]])/(2*r[])+(epsilong1[-a,c]*𝒽$a$m[a]*Τ[-b,-c]*CD1[b][r[]])/(2*r[])-(epsilong1[-a,-b]*𝒽$a$m[a]*CD1[b][𝒫[]])/6},MetricOn->All];

(*Pert$Exp$Sep[𝕁[-μ,-ν,-ρ]+(I/2)𝕁[-μ,α,β]epsilong3[-α,-β,-ν,-ρ],{μ,ν,ρ},{P,a,b},{𝕁$Harm$Rep,𝒽$Harm$Rep}];
%/.Dir$Harm$Def$Total//ContractMetric//ToCanonical;
%//.Harm$EQ$Y$Total//ContractMetric//ToCanonical;
Dum=(-1/2)epsilong1[a,b]%//ContractMetric//ToCanonical;
ToCanonical[ZA$Component[Dum]-(𝕁$SD$Pab$ZA[]/.Eval$Pert$𝕁$SD$Pab$ZA)]
ToCanonical[XA$Component[Dum]-(𝕁$SD$Pab$XA[]/.Eval$Pert$𝕁$SD$Pab$XA)]*)

(Subscript[𝕁,  P Q a])+i𝕁(Subscript[*, P Q a])

Eval$Pert$𝕁$SD$PQa$UAB=MakeRule[{𝕁$SD$PQa$UAB[-a],Γ$a$p[-a]-I*epsilong1[-a,-b]*Π$a$m[b]-(r[]*𝒽$ab$p[b,-b]*𝒫[]*CD1[-a][r[]])/4-(r[]^2*𝒽$ab$p[b,-b]*CD1[-a][𝒫[]])/12+(r[]^2*𝒽$ab$p[b,-b]*CD1[-a][Τ[c,-c]])/12-(r[]^2*𝒽$ab$p[-a,b]*CD1[-b][Τ[c,-c]])/6+(r[]*𝒽$ab$p[-a,-b]*𝒫[]*CD1[b][r[]])/2+(r[]*𝒽$ab$p[c,-c]*Τ[-a,-b]*CD1[b][r[]])/4-(r[]*𝒽$ab$p[-a,c]*Τ[-b,-c]*CD1[b][r[]])/2+(r[]^2*𝒽$ab$p[-a,-b]*CD1[b][𝒫[]])/6},MetricOn->All];
Eval$Pert$𝕁$SD$PQa$VAB=MakeRule[{𝕁$SD$PQa$VAB[-a],I*epsilong1[-a,-b]*Γ$a$m[b]+Π$a$o[-a]},MetricOn->All];
Eval$Pert$𝕁$SD$PQa$WAB=MakeRule[{𝕁$SD$PQa$WAB[-a],Γ$a$m[-a]-I*epsilong1[-a,-b]*Π$a$o[b]},MetricOn->All];
Eval$Pert$𝕁$SD$PQa$ϵAB=MakeRule[{𝕁$SD$PQa$ϵAB[-a],I*epsilong1[-a,-b]*Γ$a$p[b]+Π$a$m[-a]+(I/12)*epsilong1[-a,b]*r[]^2*𝒽$ab$p[c,-c]*CD1[-b][Τ[d,-d]]+(I/2)*epsilong1[-a,c]*r[]*𝒽$ab$p[-b,-c]*𝒫[]*CD1[b][r[]]-(I/4)*epsilong1[-a,-b]*r[]*𝒽$ab$p[c,-c]*𝒫[]*CD1[b][r[]]+(I/4)*epsilong1[-a,c]*r[]*𝒽$ab$p[d,-d]*Τ[-b,-c]*CD1[b][r[]]-(I/2)*epsilong1[-a,c]*r[]*𝒽$ab$p[-c,d]*Τ[-b,-d]*CD1[b][r[]]+(I/6)*epsilong1[-a,c]*r[]^2*𝒽$ab$p[-b,-c]*CD1[b][𝒫[]]-(I/12)*epsilong1[-a,-b]*r[]^2*𝒽$ab$p[c,-c]*CD1[b][𝒫[]]-(I/6)*epsilong1[-a,b]*r[]^2*𝒽$ab$p[-b,c]*CD1[-c][Τ[d,-d]]},MetricOn->All];

(*Pert$Exp$Sep[𝕁[-μ,-ν,-ρ]+(I/2)𝕁[-μ,α,β]epsilong3[-α,-β,-ν,-ρ],{μ,ν,ρ},{P,Q,a},{𝕁$Harm$Rep,𝒽$Harm$Rep}];
%/.Dir$Harm$Def$Total//ContractMetric//ToCanonical;
Dum=%//.Harm$EQ$Y$Total//ContractMetric//ToCanonical;
ToCanonical[UAB$Component[Dum]-(𝕁$SD$PQa$UAB[-a]/.Eval$Pert$𝕁$SD$PQa$UAB)]//Simplify
ToCanonical[VAB$Component[Dum]-(𝕁$SD$PQa$VAB[-a]/.Eval$Pert$𝕁$SD$PQa$VAB)]//Simplify
ToCanonical[WAB$Component[Dum]-(𝕁$SD$PQa$WAB[-a]/.Eval$Pert$𝕁$SD$PQa$WAB)]//Simplify
ToCanonical[ϵAB$Component[Dum]-(𝕁$SD$PQa$ϵAB[-a]/.Eval$Pert$𝕁$SD$PQa$ϵAB)]//Simplify*)

(Subscript[𝕁,  P A B])+i𝕁(Subscript[*, P A B])

Eval$Pert$𝕁$SD$PAB$ZA=MakeRule[{𝕁$SD$PAB$ZA[],Γ$m[]+(2*I)*r[]^2*Π$p[]-(r[]*𝒽$a$m[a]*𝒫[]*CD1[-a][r[]])/2-(r[]^2*𝒽$a$m[a]*CD1[-a][𝒫[]])/6+(r[]^2*𝒽$a$m[a]*CD1[-a][Τ[b,-b]])/6+(r[]*𝒽$a$m[a]*Τ[-a,-b]*CD1[b][r[]])/2},MetricOn->All];
Eval$Pert$𝕁$SD$PAB$XA=MakeRule[{𝕁$SD$PAB$XA[],Γ$p[]+(2*I)*r[]^2*Π$m[]-(I/6)*epsilong1[-a,b]*r[]^2*𝒽$a$m[a]*CD1[-b][Τ[c,-c]]+(I/2)*epsilong1[-a,-b]*r[]*𝒽$a$m[a]*𝒫[]*CD1[b][r[]]-(I/2)*epsilong1[-a,c]*r[]*𝒽$a$m[a]*Τ[-b,-c]*CD1[b][r[]]+(I/6)*epsilong1[-a,-b]*r[]^2*𝒽$a$m[a]*CD1[b][𝒫[]]},MetricOn->All];

(*Pert$Exp$Sep[𝕁[-μ,-ν,-ρ]+(I/2)𝕁[-μ,α,β]epsilong3[-α,-β,-ν,-ρ],{μ,ν,ρ},{P,A,B},{𝕁$Harm$Rep,𝒽$Harm$Rep}];
(1/2)epsilong2[A,B]%/.Dir$Harm$Def$Total//ContractMetric//ToCanonical;
Dum=%//.Harm$EQ$Y$Total//ContractMetric//ToCanonical;
ToCanonical[ZA$Component[Dum]-(𝕁$SD$PAB$ZA[]/.Eval$Pert$𝕁$SD$PAB$ZA)]
ToCanonical[XA$Component[Dum]-(𝕁$SD$PAB$XA[]/.Eval$Pert$𝕁$SD$PAB$XA)]*)

Total

Eval$Pert$𝕁$Algebraic=Flatten[{Eval$Pert$𝕁$Trace$a,Eval$Pert$𝕁$Trace$ZA,Eval$Pert$𝕁$Trace$XA,Eval$Pert$𝕁$SD$abc,Eval$Pert$𝕁$SD$abP$ZA,Eval$Pert$𝕁$SD$abP$XA,Eval$Pert$𝕁$SD$aPQ,Eval$Pert$𝕁$SD$Pab$ZA,Eval$Pert$𝕁$SD$Pab$XA,Eval$Pert$𝕁$SD$PQa$UAB,Eval$Pert$𝕁$SD$PQa$VAB,Eval$Pert$𝕁$SD$PQa$WAB,Eval$Pert$𝕁$SD$PQa$ϵAB,Eval$Pert$𝕁$SD$PAB$ZA,Eval$Pert$𝕁$SD$PAB$XA}];

1.7. Linearised first-derivative equations for δ (Subscript[ℂ , μ ν ρ σ]) and δ (Subscript[𝕁 , μ ν ρ ])

DefTensor[Dℂ$𝕁$abc[-a],M1]
DefTensor[Dℂ$𝕁$abP$ZA[-a,-b],M1]
DefTensor[Dℂ$𝕁$abP$XA[-a,-b],M1]
DefTensor[Dℂ$𝕁$aPQ[-a],M1]
DefTensor[Dℂ$𝕁$Pab$ZA[],M1]
DefTensor[Dℂ$𝕁$Pab$XA[],M1]
DefTensor[Dℂ$𝕁$PQa$UAB[-a],M1]
DefTensor[Dℂ$𝕁$PQa$VAB[-a],M1]
DefTensor[Dℂ$𝕁$PQa$WAB[-a],M1]
DefTensor[Dℂ$𝕁$PQa$ϵAB[-a],M1]
DefTensor[Dℂ$𝕁$PAB$ZA[],M1]
DefTensor[Dℂ$𝕁$PAB$XA[],M1]

Dℂ$𝕁$Total={Dℂ$𝕁$abc[-a],Dℂ$𝕁$abP$ZA[-a,-b],Dℂ$𝕁$abP$XA[-a,-b],Dℂ$𝕁$aPQ[-a],Dℂ$𝕁$Pab$ZA[],Dℂ$𝕁$Pab$XA[],Dℂ$𝕁$PQa$UAB[-a],Dℂ$𝕁$PQa$VAB[-a],Dℂ$𝕁$PQa$WAB[-a],Dℂ$𝕁$PQa$ϵAB[-a],Dℂ$𝕁$PAB$ZA[],Dℂ$𝕁$PAB$XA[]};

((Subscript[ℂ, μ a b c ])^(;μ))-(Subscript[𝕁, a b c])=0

Eval$Pert$Dℂ$𝕁$abc=MakeRule[{Dℂ$𝕁$abc[-a],(((3*I)/2)*Λ^2*𝒽$a$m[-a])/r[]^4-((I/2)*Λ^2*𝒽$a$m[-a]*𝒫[])/r[]^2-Π$a$p[-a]+((I/2)*Λ^2*𝒽$a$m[-a]*Τ[b,-b])/r[]^2-(Λ^2*Ψ$a$p[-a])/r[]^2+(epsilong1[-a,c]*𝒽$ab$p[-b,-c]*CD1[b][r[]])/r[]^3-(4*epsilong1[-a,-b]*𝒽$p[]*CD1[b][r[]])/r[]^5+(2*epsilong1[-a,c]*𝒽$ab$p[-b,-c]*𝒫[]*CD1[b][r[]])/(3*r[])+(4*epsilong1[-a,-b]*𝒽$p[]*𝒫[]*CD1[b][r[]])/(3*r[]^3)-(epsilong1[-a,c]*𝒽$ab$p[-c,d]*Τ[-b,-d]*CD1[b][r[]])/r[]-(4*epsilong1[-a,-b]*𝒽$p[]*Τ[c,-c]*CD1[b][r[]])/(3*r[]^3)+(epsilong1[-a,c]*𝒽$ab$p[-b,-c]*Τ[d,-d]*CD1[b][r[]])/(3*r[])+(2*epsilong1[-b,c]*Ψ$ab$p[-a,-c]*CD1[b][r[]])/r[]^3-(2*epsilong1[-a,-b]*Ψ$p[]*CD1[b][r[]])/r[]-(((3*I)/2)*Λ^2*𝒽$a$m[-a]*CD1[-b][r[]]*CD1[b][r[]])/r[]^4+(3*epsilong1[-a,-b]*CD1[b][𝒽$p[]])/(2*r[]^4)-(epsilong1[-a,-b]*𝒫[]*CD1[b][𝒽$p[]])/(2*r[]^2)+(epsilong1[-a,-b]*Τ[c,-c]*CD1[b][𝒽$p[]])/(2*r[]^2)+(epsilong1[-a,c]*𝒽$ab$p[-b,-c]*CD1[b][𝒫[]])/3-epsilong1[-a,-b]*CD1[b][Ψ$p[]]-(epsilong1[b,c]*CD1[-c][𝒽$ab$p[-a,-b]])/r[]^2+(epsilong1[b,c]*𝒫[]*CD1[-c][𝒽$ab$p[-a,-b]])/3-(epsilong1[b,c]*Τ[d,-d]*CD1[-c][𝒽$ab$p[-a,-b]])/3-(epsilong1[-a,b]*CD1[-c][𝒽$ab$p[-b,c]])/r[]^2+(epsilong1[-a,b]*𝒫[]*CD1[-c][𝒽$ab$p[-b,c]])/3-(epsilong1[-a,b]*𝒽$ab$p[-b,c]*CD1[-c][Τ[d,-d]])/3-(epsilong1[-a,d]*𝒽$ab$p[-c,-d]*CD1[-b][r[]]*CD1[b][r[]]*CD1[c][r[]])/r[]^3+(4*epsilong1[-a,-c]*𝒽$p[]*CD1[-b][r[]]*CD1[b][r[]]*CD1[c][r[]])/r[]^5-(3*epsilong1[-a,-c]*CD1[-b][r[]]*CD1[b][r[]]*CD1[c][𝒽$p[]])/(2*r[]^4)+(epsilong1[c,d]*CD1[-b][r[]]*CD1[b][r[]]*CD1[-d][𝒽$ab$p[-a,-c]])/r[]^2-(epsilong1[-a,b]*Τ[c,-c]*CD1[-d][𝒽$ab$p[-b,d]])/3+(epsilong1[-a,c]*CD1[-b][r[]]*CD1[b][r[]]*CD1[-d][𝒽$ab$p[-c,d]])/r[]^2},MetricOn->All];

(*Pert$Exp$Sep[CD3[μ][CC[-μ,-ν,-α,-β]]-𝕁[-ν,-α,-β],{ν,α,β},{a,b,c},{𝒽$Harm$Rep,ℂ$Harm$Rep,𝕁$Harm$Rep}];
%/.Dir$Harm$Def$Total//ContractMetric//ToCanonical;
((-1/2)epsilong1[b,c]%//ContractMetric//ToCanonical)//.Harm$EQ$Y$Total//ContractMetric//ToCanonical;
(%/Y[])//ToCanonical;
ToCanonical[%-(Dℂ$𝕁$abc[-a]/.Eval$Pert$Dℂ$𝕁$abc)]*)

(*Test: it should be possible to show the last expression vanishes by using 2D identities and the conservation of T*)

(*epsilong1[a,-f]Dℂ$𝕁$abc//.Flatten[{Pert$ℂ$To$Met,Pert𝕁$To$metTmn,BG$1}]//ContractMetric//ToCanonical;
%/.Flatten[{CommCovs$𝒽$ab$p$FreeFirst,CommCovs$𝒽$a$m$FreeFirst,BG$1}]//ContractMetric//ToCanonical;
%//.Flatten[{Einstein$Rep$DD𝒽$p,Einstein$Rep$Lorentz$𝒽$ab$p,Einstein$Rep$Trace$𝒽$ab$p,Einstein$Rep$Box$𝒽$a$m,Einstein$Rep$Div$𝒽$a$m,BG$1}]//ContractMetric//ToCanonical;
%/.FirstOrder$Gauss$Bonnet$Rule//ContractMetric//ToCanonical;
%//.Flatten[{Einstein$Rep$DD𝒽$p,Einstein$Rep$Lorentz$𝒽$ab$p,Einstein$Rep$Trace$𝒽$ab$p,Einstein$Rep$Box$𝒽$a$m,Einstein$Rep$Div$𝒽$a$m,BG$1}]//ContractMetric//ToCanonical;
%//.𝒯$Cons//ContractMetric//ToCanonical;
%//.Flatten[{Einstein$Rep$DD𝒽$p,Einstein$Rep$Lorentz$𝒽$ab$p,Einstein$Rep$Trace$𝒽$ab$p,Einstein$Rep$Box$𝒽$a$m,Einstein$Rep$Div$𝒽$a$m,BG$1}]//ContractMetric//ToCanonical;
epsilong1[f,-d]%//ContractMetric//ToCanonical;
%/.Flatten[{CommCovs$𝒽$a$m$FreeFirst,BG$1}]//ContractMetric//ToCanonical;
%//.Flatten[{Einstein$Rep$DD𝒽$p,Einstein$Rep$Lorentz$𝒽$ab$p,Einstein$Rep$Trace$𝒽$ab$p,Einstein$Rep$Box$𝒽$a$m,Einstein$Rep$Div$𝒽$a$m,BG$1}]//ContractMetric//ToCanonical;
epsilong1[d,-c]%//ContractMetric//ToCanonical;
%/.𝒽$To$Η//ContractMetric//ToCanonical;
%//.Flatten[{Einstein$Rep$DD𝒽$p,Einstein$Rep$Lorentz$𝒽$ab$p,Einstein$Rep$Trace$𝒽$ab$p,Einstein$Rep$Box$𝒽$a$m,Einstein$Rep$Div$𝒽$a$m,BG$1}]//ContractMetric//ToCanonical;
2((r[])^2)%/.BG$T$Cons//ContractMetric//ToCanonical*)

((Subscript[ℂ, μ a b P ])^(;μ))-(Subscript[𝕁, a b P])=0

Eval$Pert$Dℂ$𝕁$abP$ZA=MakeRule[{Dℂ$𝕁$abP$ZA[-a,-b],(-5*𝒽$ab$p[-a,-b])/(4*r[]^2)+(3*g1[-a,-b]*𝒽$ab$p[c,-c])/(4*r[]^2)-(g1[-a,-b]*𝒽$p[])/(2*r[]^4)+(5*𝒽$ab$p[-a,-b]*𝒫[])/12-(g1[-a,-b]*𝒽$ab$p[c,-c]*𝒫[])/4+(g1[-a,-b]*𝒽$p[]*𝒫[])/(6*r[]^2)-Π$ab$p[-a,-b]-epsilong1[-a,-b]*Π$p[]-(5*𝒽$ab$p[-a,-b]*Τ[c,-c])/12-(g1[-a,-b]*𝒽$p[]*Τ[c,-c])/(6*r[]^2)+(g1[-a,-b]*𝒽$ab$p[c,-c]*Τ[d,-d])/4-Ψ$ab$o[-a,-b]/r[]^2+(Λ^2*Ψ$ab$o[-a,-b])/(2*r[]^2)-Ψ$ab$p[-a,-b]/r[]^2+(((3*I)/2)*epsilong1[-a,-c]*𝒽$a$m[c]*CD1[-b][r[]])/r[]^3-((I/2)*epsilong1[-a,-c]*𝒽$a$m[c]*𝒫[]*CD1[-b][r[]])/r[]-(Ξ$a$p[-a]*CD1[-b][r[]])/r[]^3+((I/2)*epsilong1[-a,-c]*𝒽$a$m[c]*Τ[d,-d]*CD1[-b][r[]])/r[]-(((3*I)/4)*epsilong1[-a,-c]*CD1[-b][𝒽$a$m[c]])/r[]^2+(I/4)*epsilong1[-a,-c]*𝒫[]*CD1[-b][𝒽$a$m[c]]-(I/4)*epsilong1[-a,-c]*Τ[d,-d]*CD1[-b][𝒽$a$m[c]]+(((3*I)/2)*epsilong1[-a,-b]*𝒽$a$m[c]*CD1[-c][r[]])/r[]^3+((I/2)*epsilong1[-a,-b]*𝒽$a$m[c]*Τ[d,-d]*CD1[-c][r[]])/r[]-((I/2)*epsilong1[-a,-b]*CD1[-c][𝒽$a$m[c]])/r[]^2+(I/6)*epsilong1[-a,-b]*𝒫[]*CD1[-c][𝒽$a$m[c]]-(I/6)*epsilong1[-a,-b]*Τ[d,-d]*CD1[-c][𝒽$a$m[c]]+(I/6)*epsilong1[-a,-b]*𝒽$a$m[c]*CD1[-c][𝒫[]]-(I/6)*epsilong1[-a,-b]*𝒽$a$m[c]*CD1[-c][Τ[d,-d]]+(epsilong1[-b,-c]*Ψ$a$p[-a]*CD1[c][r[]])/r[]+(epsilong1[-a,-c]*Ψ$a$p[-b]*CD1[c][r[]])/r[]+(5*𝒽$ab$p[-a,-b]*CD1[-c][r[]]*CD1[c][r[]])/(4*r[]^2)-(3*g1[-a,-b]*𝒽$ab$p[d,-d]*CD1[-c][r[]]*CD1[c][r[]])/(4*r[]^2)+(g1[-a,-b]*𝒽$p[]*CD1[-c][r[]]*CD1[c][r[]])/(2*r[]^4)+(((3*I)/4)*epsilong1[-a,-d]*CD1[-b][𝒽$a$m[d]]*CD1[-c][r[]]*CD1[c][r[]])/r[]^2-((I/2)*epsilong1[-b,-c]*CD1[c][𝒽$a$m[-a]])/r[]^2+(I/6)*epsilong1[-b,-c]*𝒫[]*CD1[c][𝒽$a$m[-a]]-(I/6)*epsilong1[-b,-c]*Τ[d,-d]*CD1[c][𝒽$a$m[-a]]-((I/4)*epsilong1[-a,-c]*CD1[c][𝒽$a$m[-b]])/r[]^2+(I/12)*epsilong1[-a,-c]*𝒫[]*CD1[c][𝒽$a$m[-b]]-(I/12)*epsilong1[-a,-c]*Τ[d,-d]*CD1[c][𝒽$a$m[-b]]+epsilong1[-a,-c]*CD1[c][Ψ$a$p[-b]]+((I/2)*epsilong1[-a,-b]*CD1[-c][r[]]*CD1[c][r[]]*CD1[-d][𝒽$a$m[d]])/r[]^2-((I/2)*epsilong1[-a,-b]*𝒽$a$m[c]*Τ[-c,-d]*CD1[d][r[]])/r[]-(((3*I)/2)*epsilong1[-a,-c]*𝒽$a$m[c]*CD1[-b][r[]]*CD1[-d][r[]]*CD1[d][r[]])/r[]^3-(((3*I)/2)*epsilong1[-a,-b]*𝒽$a$m[c]*CD1[-c][r[]]*CD1[-d][r[]]*CD1[d][r[]])/r[]^3+((I/2)*epsilong1[-b,-d]*CD1[-c][r[]]*CD1[c][r[]]*CD1[d][𝒽$a$m[-a]])/r[]^2+((I/4)*epsilong1[-a,-d]*CD1[-c][r[]]*CD1[c][r[]]*CD1[d][𝒽$a$m[-b]])/r[]^2},MetricOn->All];
Eval$Pert$Dℂ$𝕁$abP$XA=MakeRule[{Dℂ$𝕁$abP$XA[-a,-b],((-1/2*I)*epsilong1[-b,c]*𝒽$ab$p[-a,-c])/r[]^2-((I/4)*epsilong1[-a,c]*𝒽$ab$p[-b,-c])/r[]^2-((I/4)*epsilong1[-a,-b]*𝒽$ab$p[c,-c])/r[]^2+((I/2)*epsilong1[-a,-b]*𝒽$p[])/r[]^4+(I/6)*epsilong1[-b,c]*𝒽$ab$p[-a,-c]*𝒫[]+(I/12)*epsilong1[-a,c]*𝒽$ab$p[-b,-c]*𝒫[]+(I/12)*epsilong1[-a,-b]*𝒽$ab$p[c,-c]*𝒫[]-((I/6)*epsilong1[-a,-b]*𝒽$p[]*𝒫[])/r[]^2-Π$ab$m[-a,-b]-epsilong1[-a,-b]*Π$m[]+((I/6)*epsilong1[-a,-b]*𝒽$p[]*Τ[c,-c])/r[]^2-(I/6)*epsilong1[-b,c]*𝒽$ab$p[-a,-c]*Τ[d,-d]-(I/12)*epsilong1[-a,c]*𝒽$ab$p[-b,-c]*Τ[d,-d]-(I/12)*epsilong1[-a,-b]*𝒽$ab$p[c,-c]*Τ[d,-d]-Ψ$ab$m[-a,-b]/r[]^2+(Λ^2*Ψ$ab$m[-a,-b])/(2*r[]^2)+(epsilong1[-a,-b]*Ψ$m[])/(2*r[]^2)-(𝒽$a$m[-b]*CD1[-a][r[]])/r[]^3+(𝒽$a$m[-b]*𝒫[]*CD1[-a][r[]])/(3*r[])-(𝒽$a$m[-b]*Τ[c,-c]*CD1[-a][r[]])/(3*r[])+(3*CD1[-a][𝒽$a$m[-b]])/(4*r[]^2)-(𝒫[]*CD1[-a][𝒽$a$m[-b]])/4+(Τ[c,-c]*CD1[-a][𝒽$a$m[-b]])/4+(3*𝒽$a$m[-a]*CD1[-b][r[]])/(2*r[]^3)-(𝒽$a$m[-a]*𝒫[]*CD1[-b][r[]])/(2*r[])+(Ξ$a$m[-a]*CD1[-b][r[]])/r[]^3+(𝒽$a$m[-a]*Τ[c,-c]*CD1[-b][r[]])/(2*r[])-CD1[-b][𝒽$a$m[-a]]/(4*r[]^2)+(𝒫[]*CD1[-b][𝒽$a$m[-a]])/12-(Τ[c,-c]*CD1[-b][𝒽$a$m[-a]])/12+(g1[-a,-b]*𝒽$a$m[c]*CD1[-c][r[]])/(2*r[]^3)+(g1[-a,-b]*𝒽$a$m[c]*𝒫[]*CD1[-c][r[]])/(3*r[])+(g1[-a,-b]*𝒽$a$m[c]*Τ[d,-d]*CD1[-c][r[]])/(6*r[])-(g1[-a,-b]*CD1[-c][𝒽$a$m[c]])/(2*r[]^2)+(g1[-a,-b]*𝒫[]*CD1[-c][𝒽$a$m[c]])/6-(g1[-a,-b]*Τ[d,-d]*CD1[-c][𝒽$a$m[c]])/6+(g1[-a,-b]*𝒽$a$m[c]*CD1[-c][𝒫[]])/6-(g1[-a,-b]*𝒽$a$m[c]*CD1[-c][Τ[d,-d]])/6+(epsilong1[-b,-c]*Ψ$a$m[-a]*CD1[c][r[]])/r[]+(epsilong1[-a,-c]*Ψ$a$m[-b]*CD1[c][r[]])/r[]+((I/2)*epsilong1[-b,d]*𝒽$ab$p[-a,-d]*CD1[-c][r[]]*CD1[c][r[]])/r[]^2+((I/4)*epsilong1[-a,d]*𝒽$ab$p[-b,-d]*CD1[-c][r[]]*CD1[c][r[]])/r[]^2+((I/4)*epsilong1[-a,-b]*𝒽$ab$p[d,-d]*CD1[-c][r[]]*CD1[c][r[]])/r[]^2-((I/2)*epsilong1[-a,-b]*𝒽$p[]*CD1[-c][r[]]*CD1[c][r[]])/r[]^4+(𝒽$a$m[-b]*CD1[-a][r[]]*CD1[-c][r[]]*CD1[c][r[]])/r[]^3-(3*CD1[-a][𝒽$a$m[-b]]*CD1[-c][r[]]*CD1[c][r[]])/(4*r[]^2)-(3*𝒽$a$m[-a]*CD1[-b][r[]]*CD1[-c][r[]]*CD1[c][r[]])/(2*r[]^3)+(CD1[-b][𝒽$a$m[-a]]*CD1[-c][r[]]*CD1[c][r[]])/(4*r[]^2)+epsilong1[-a,-c]*CD1[c][Ψ$a$m[-b]]+(g1[-a,-b]*CD1[-c][r[]]*CD1[c][r[]]*CD1[-d][𝒽$a$m[d]])/(2*r[]^2)-(g1[-a,-b]*𝒽$a$m[c]*Τ[-c,-d]*CD1[d][r[]])/(2*r[])-(g1[-a,-b]*𝒽$a$m[c]*CD1[-c][r[]]*CD1[-d][r[]]*CD1[d][r[]])/(2*r[]^3)},MetricOn->All];

(*Pert$Exp$Sep[CD3[μ][CC[-μ,-ν,-α,-β]]-𝕁[-ν,-α,-β],{ν,α,β},{a,b,P},{𝒽$Harm$Rep,ℂ$Harm$Rep,𝕁$Harm$Rep}];
%/.Dir$Harm$Def$Total//ContractMetric//ToCanonical;
Dum=%//.Harm$EQ$Y$Total//ContractMetric//ToCanonical;
ToCanonical[(Dℂ$𝕁$abP$ZA[-a,-b]/.Eval$Pert$Dℂ$𝕁$abP$ZA)-ZA$Component[Dum]]
ToCanonical[(Dℂ$𝕁$abP$XA[-a,-b]/.Eval$Pert$Dℂ$𝕁$abP$XA)-XA$Component[Dum]]*)

((Subscript[ℂ, μ a P Q ])^(;μ))-(Subscript[𝕁, a P Q])=0

Eval$Pert$Dℂ$𝕁$aPQ=MakeRule[{Dℂ$𝕁$aPQ[-a],-((Λ^2*𝒽$a$m[-a])/r[]^2)+(Λ^2*𝒽$a$m[-a]*𝒫[])/3-(Λ^2*Ξ$a$m[-a])/r[]^2+2*Π$a$m[-a]-(Λ^2*𝒽$a$m[-a]*Τ[b,-b])/3+(I/2)*epsilong1[-a,b]*CD1[-b][𝒽$ab$p[c,-c]]-(I/6)*epsilong1[-a,b]*r[]^2*𝒫[]*CD1[-b][𝒽$ab$p[c,-c]]+(I/6)*epsilong1[-a,b]*r[]^2*Τ[c,-c]*CD1[-b][𝒽$ab$p[d,-d]]-(I*epsilong1[-a,-b]*𝒽$p[]*CD1[b][r[]])/r[]^3+I*epsilong1[-a,c]*r[]*𝒽$ab$p[-b,-c]*𝒫[]*CD1[b][r[]]+((I/3)*epsilong1[-a,-b]*𝒽$p[]*𝒫[]*CD1[b][r[]])/r[]-I*epsilong1[-a,c]*r[]*𝒽$ab$p[-c,d]*Τ[-b,-d]*CD1[b][r[]]-((I/3)*epsilong1[-a,-b]*𝒽$p[]*Τ[c,-c]*CD1[b][r[]])/r[]-(epsilong1[-a,-b]*Ψ$m[]*CD1[b][r[]])/r[]+(Λ^2*𝒽$a$m[-a]*CD1[-b][r[]]*CD1[b][r[]])/r[]^2+((I/2)*epsilong1[-a,-b]*CD1[b][𝒽$p[]])/r[]^2-(I/6)*epsilong1[-a,-b]*𝒫[]*CD1[b][𝒽$p[]]+(I/6)*epsilong1[-a,-b]*Τ[c,-c]*CD1[b][𝒽$p[]]+(I/3)*epsilong1[-a,c]*r[]^2*𝒽$ab$p[-b,-c]*CD1[b][𝒫[]]-epsilong1[-a,-b]*CD1[b][Ψ$m[]]-I*epsilong1[b,c]*CD1[-c][𝒽$ab$p[-a,-b]]+(I/3)*epsilong1[b,c]*r[]^2*𝒫[]*CD1[-c][𝒽$ab$p[-a,-b]]-(I/3)*epsilong1[b,c]*r[]^2*Τ[d,-d]*CD1[-c][𝒽$ab$p[-a,-b]]-I*epsilong1[-a,b]*CD1[-c][𝒽$ab$p[-b,c]]+(I/3)*epsilong1[-a,b]*r[]^2*𝒫[]*CD1[-c][𝒽$ab$p[-b,c]]-(I/2)*epsilong1[-a,c]*CD1[-b][r[]]*CD1[b][r[]]*CD1[-c][𝒽$ab$p[d,-d]]-(I/3)*epsilong1[-a,b]*r[]^2*𝒽$ab$p[-b,c]*CD1[-c][Τ[d,-d]]+(I*epsilong1[-a,-c]*𝒽$p[]*CD1[-b][r[]]*CD1[b][r[]]*CD1[c][r[]])/r[]^3-((I/2)*epsilong1[-a,-c]*CD1[-b][r[]]*CD1[b][r[]]*CD1[c][𝒽$p[]])/r[]^2+I*epsilong1[c,d]*CD1[-b][r[]]*CD1[b][r[]]*CD1[-d][𝒽$ab$p[-a,-c]]-(I/3)*epsilong1[-a,b]*r[]^2*Τ[c,-c]*CD1[-d][𝒽$ab$p[-b,d]]+I*epsilong1[-a,c]*CD1[-b][r[]]*CD1[b][r[]]*CD1[-d][𝒽$ab$p[-c,d]]},MetricOn->All];

(*Pert$Exp$Sep[CD3[μ][CC[-μ,-ν,-α,-β]]-𝕁[-ν,-α,-β],{ν,α,β},{a,P,Q},{𝒽$Harm$Rep,ℂ$Harm$Rep,𝕁$Harm$Rep}];
%/.Dir$Harm$Def$Total//ContractMetric//ToCanonical;
((1/2)epsilong2[P,Q]%//ContractMetric//ToCanonical)//.Harm$EQ$Y$Total//ContractMetric//ToCanonical;
ToCanonical[(%/Y[])-(Dℂ$𝕁$aPQ[-a]/.Eval$Pert$Dℂ$𝕁$aPQ)]*)

(*Tests*)

(*Dℂ$𝕁$aPQ//.Flatten[{Pert$ℂ$To$Met,Pert𝕁$To$metTmn,BG$1}]//ContractMetric//ToCanonical;
%//.Flatten[{Einstein$Rep$DD𝒽$p,Einstein$Rep$Lorentz$𝒽$ab$p,Einstein$Rep$Trace$𝒽$ab$p,Einstein$Rep$Box$𝒽$a$m,Einstein$Rep$Div$𝒽$a$m,BG$1}]//ContractMetric//ToCanonical;
%/.CommCovs$𝒽$ab$p$FreeFirst//ContractMetric//ToCanonical;
%//.Flatten[{Einstein$Rep$DD𝒽$p,Einstein$Rep$Lorentz$𝒽$ab$p,Einstein$Rep$Trace$𝒽$ab$p,Einstein$Rep$Box$𝒽$a$m,Einstein$Rep$Div$𝒽$a$m,BG$1}]//ContractMetric//ToCanonical;
%/.𝒯$Cons//ContractMetric//ToCanonical;
%//.Flatten[{Einstein$Rep$DD𝒽$p,Einstein$Rep$Lorentz$𝒽$ab$p,Einstein$Rep$Trace$𝒽$ab$p,Einstein$Rep$Box$𝒽$a$m,Einstein$Rep$Div$𝒽$a$m,BG$1}]//ContractMetric//ToCanonical;
%/.CommCovs$𝒽$a$m$FreeFirst//ContractMetric//ToCanonical;
epsilong1[a,-f]%//.Flatten[{Einstein$Rep$DD𝒽$p,Einstein$Rep$Lorentz$𝒽$ab$p,Einstein$Rep$Trace$𝒽$ab$p,Einstein$Rep$Box$𝒽$a$m,Einstein$Rep$Div$𝒽$a$m,BG$1}]//ContractMetric//ToCanonical;
%//.Flatten[{Einstein$Rep$DD𝒽$p,Einstein$Rep$Lorentz$𝒽$ab$p,Einstein$Rep$Trace$𝒽$ab$p,Einstein$Rep$Box$𝒽$a$m,Einstein$Rep$Div$𝒽$a$m,BG$1}]//ContractMetric//ToCanonical*)

((Subscript[ℂ, μ P a b ])^(;μ))-(Subscript[𝕁, P a b])=0

Eval$Pert$Dℂ$𝕁$Pab$ZA=MakeRule[{Dℂ$𝕁$Pab$ZA[],2*Π$p[]-(((9*I)/2)*𝒽$a$m[a]*CD1[-a][r[]])/r[]^3+((I/2)*𝒽$a$m[a]*𝒫[]*CD1[-a][r[]])/r[]-(((3*I)/2)*𝒽$a$m[a]*Τ[b,-b]*CD1[-a][r[]])/r[]+(((3*I)/2)*CD1[-a][𝒽$a$m[a]])/r[]^2-(I/2)*𝒫[]*CD1[-a][𝒽$a$m[a]]+(I/2)*Τ[b,-b]*CD1[-a][𝒽$a$m[a]]-(I/3)*𝒽$a$m[a]*CD1[-a][𝒫[]]+(I/3)*𝒽$a$m[a]*CD1[-a][Τ[b,-b]]-CD1[-a][Ψ$a$p[a]]-(((3*I)/2)*CD1[-a][r[]]*CD1[a][r[]]*CD1[-b][𝒽$a$m[b]])/r[]^2-(epsilong1[-a,-b]*Ξ$a$p[a]*CD1[b][r[]])/r[]^3+(I*𝒽$a$m[a]*Τ[-a,-b]*CD1[b][r[]])/r[]+(((9*I)/2)*𝒽$a$m[a]*CD1[-a][r[]]*CD1[-b][r[]]*CD1[b][r[]])/r[]^3},MetricOn->All];
Eval$Pert$Dℂ$𝕁$Pab$XA=MakeRule[{Dℂ$𝕁$Pab$XA[],((I/4)*𝒽$ab$p[a,-a])/r[]^2-(I*𝒽$p[])/r[]^4-(I/12)*𝒽$ab$p[a,-a]*𝒫[]+((I/3)*𝒽$p[]*𝒫[])/r[]^2+2*Π$m[]-((I/3)*𝒽$p[]*Τ[a,-a])/r[]^2+(I/12)*𝒽$ab$p[a,-a]*Τ[b,-b]-Ψ$m[]/r[]^2-CD1[-a][Ψ$a$m[a]]-((I/4)*𝒽$ab$p[b,-b]*CD1[-a][r[]]*CD1[a][r[]])/r[]^2+(I*𝒽$p[]*CD1[-a][r[]]*CD1[a][r[]])/r[]^4+(5*epsilong1[-a,-b]*𝒽$a$m[a]*CD1[b][r[]])/(2*r[]^3)-(5*epsilong1[-a,-b]*𝒽$a$m[a]*𝒫[]*CD1[b][r[]])/(6*r[])+(epsilong1[-a,-b]*Ξ$a$m[a]*CD1[b][r[]])/r[]^3+(5*epsilong1[-a,-b]*𝒽$a$m[a]*Τ[c,-c]*CD1[b][r[]])/(6*r[])-(epsilong1[-a,-b]*CD1[b][𝒽$a$m[a]])/r[]^2+(epsilong1[-a,-b]*𝒫[]*CD1[b][𝒽$a$m[a]])/3-(epsilong1[-a,-b]*Τ[c,-c]*CD1[b][𝒽$a$m[a]])/3-(5*epsilong1[-a,-c]*𝒽$a$m[a]*CD1[-b][r[]]*CD1[b][r[]]*CD1[c][r[]])/(2*r[]^3)+(epsilong1[-b,-c]*CD1[-a][r[]]*CD1[a][r[]]*CD1[c][𝒽$a$m[b]])/r[]^2},MetricOn->All];

(*Pert$Exp$Sep[CD3[μ][CC[-μ,-ν,-α,-β]]-𝕁[-ν,-α,-β],{ν,α,β},{P,a,b},{𝒽$Harm$Rep,ℂ$Harm$Rep,𝕁$Harm$Rep}];
%/.Dir$Harm$Def$Total//ContractMetric//ToCanonical;
Dum=((-1/2)epsilong1[a,b]%//ContractMetric//ToCanonical)//.Harm$EQ$Y$Total//ContractMetric//ToCanonical;
ToCanonical[(Dℂ$𝕁$Pab$ZA[]/.Eval$Pert$Dℂ$𝕁$Pab$ZA)-ZA$Component[Dum]]
ToCanonical[(Dℂ$𝕁$Pab$XA[]/.Eval$Pert$Dℂ$𝕁$Pab$XA)-XA$Component[Dum]]*)

((Subscript[ℂ, μ P Q a ])^(;μ))-(Subscript[𝕁, P Q a])=0

Eval$Pert$Dℂ$𝕁$PQa$UAB=MakeRule[{Dℂ$𝕁$PQa$UAB[-a],-Γ$a$p[-a]+(Λ^2*Ξ$a$p[-a])/(2*r[]^2)-(3*𝒽$p[]*CD1[-a][r[]])/(2*r[]^3)+(𝒽$p[]*𝒫[]*CD1[-a][r[]])/(2*r[])+(Ξ$p[]*CD1[-a][r[]])/r[]^3-(𝒽$p[]*Τ[b,-b]*CD1[-a][r[]])/(2*r[])+CD1[-a][𝒽$p[]]/(4*r[]^2)-(𝒫[]*CD1[-a][𝒽$p[]])/12+(Τ[b,-b]*CD1[-a][𝒽$p[]])/12-CD1[-b][𝒽$ab$p[-a,b]]/2+(r[]^2*𝒫[]*CD1[-b][𝒽$ab$p[-a,b]])/6-(r[]^2*𝒽$ab$p[-a,b]*CD1[-b][Τ[c,-c]])/6-CD1[-b][Ψ$ab$p[-a,b]]+(𝒽$ab$p[-a,-b]*CD1[b][r[]])/r[]+(r[]*𝒽$ab$p[-a,-b]*𝒫[]*CD1[b][r[]])/6-(r[]*𝒽$ab$p[-a,c]*Τ[-b,-c]*CD1[b][r[]])/2+(r[]*𝒽$ab$p[-a,-b]*Τ[c,-c]*CD1[b][r[]])/3+(Ψ$ab$p[-a,-b]*CD1[b][r[]])/r[]+(3*𝒽$p[]*CD1[-a][r[]]*CD1[-b][r[]]*CD1[b][r[]])/(2*r[]^3)-(CD1[-a][𝒽$p[]]*CD1[-b][r[]]*CD1[b][r[]])/(4*r[]^2)+(r[]^2*𝒽$ab$p[-a,-b]*CD1[b][𝒫[]])/6-(r[]^2*Τ[b,-b]*CD1[-c][𝒽$ab$p[-a,c]])/6+(CD1[-b][r[]]*CD1[b][r[]]*CD1[-c][𝒽$ab$p[-a,c]])/2-(𝒽$ab$p[-a,-c]*CD1[-b][r[]]*CD1[b][r[]]*CD1[c][r[]])/r[]},MetricOn->All];
Eval$Pert$Dℂ$𝕁$PQa$VAB=MakeRule[{Dℂ$𝕁$PQa$VAB[-a],(((-3*I)/2)*epsilong1[-a,-b]*𝒽$a$m[b])/((1-Λ^2/2)*r[]^2)+(((3*I)/4)*Λ^2*epsilong1[-a,-b]*𝒽$a$m[b])/((1-Λ^2/2)*r[]^2)+((I/2)*epsilong1[-a,-b]*𝒽$a$m[b]*𝒫[])/(1-Λ^2/2)-((I/4)*Λ^2*epsilong1[-a,-b]*𝒽$a$m[b]*𝒫[])/(1-Λ^2/2)+Ξ$a$p[-a]/((1-Λ^2/2)*r[]^2)-(Λ^2*Ξ$a$p[-a])/(2*(1-Λ^2/2)*r[]^2)-Π$a$o[-a]/(1-Λ^2/2)+(Λ^2*Π$a$o[-a])/(2*(1-Λ^2/2))-((I/2)*epsilong1[-a,-b]*𝒽$a$m[b]*Τ[c,-c])/(1-Λ^2/2)+((I/4)*Λ^2*epsilong1[-a,-b]*𝒽$a$m[b]*Τ[c,-c])/(1-Λ^2/2)-CD1[-b][Ψ$ab$o[-a,b]]/(1-Λ^2/2)+(Λ^2*CD1[-b][Ψ$ab$o[-a,b]])/(2*(1-Λ^2/2))+(Ψ$ab$o[-a,-b]*CD1[b][r[]])/((1-Λ^2/2)*r[])-(Λ^2*Ψ$ab$o[-a,-b]*CD1[b][r[]])/(2*(1-Λ^2/2)*r[])+(((3*I)/2)*epsilong1[-a,-b]*𝒽$a$m[b]*CD1[-c][r[]]*CD1[c][r[]])/((1-Λ^2/2)*r[]^2)-(((3*I)/4)*Λ^2*epsilong1[-a,-b]*𝒽$a$m[b]*CD1[-c][r[]]*CD1[c][r[]])/((1-Λ^2/2)*r[]^2)},MetricOn->All];
Eval$Pert$Dℂ$𝕁$PQa$WAB=MakeRule[{Dℂ$𝕁$PQa$WAB[-a],(𝒽$a$m[-a]*(3-r[]^2*𝒫[]+r[]^2*Τ[b,-b]-3*CD1[-b][r[]]*CD1[b][r[]])-6*(Ξ$a$m[-a]+r[]*(r[]*Γ$a$m[-a]+r[]*CD1[-b][Ψ$ab$m[-a,b]]-Ψ$ab$m[-a,-b]*CD1[b][r[]])))/(6*r[]^2)},MetricOn->All];
Eval$Pert$Dℂ$𝕁$PQa$ϵAB=MakeRule[{Dℂ$𝕁$PQa$ϵAB[-a],(Λ^2*𝒽$a$m[-a])/(2*r[]^2)-(Λ^2*𝒽$a$m[-a]*𝒫[])/6+(Λ^2*Ξ$a$m[-a])/(2*r[]^2)-Π$a$m[-a]+(Λ^2*𝒽$a$m[-a]*Τ[b,-b])/6-(I/4)*epsilong1[-a,b]*CD1[-b][𝒽$ab$p[c,-c]]+(I/12)*epsilong1[-a,b]*r[]^2*𝒫[]*CD1[-b][𝒽$ab$p[c,-c]]-(I/12)*epsilong1[-a,b]*r[]^2*Τ[c,-c]*CD1[-b][𝒽$ab$p[d,-d]]+((I/2)*epsilong1[-a,-b]*𝒽$p[]*CD1[b][r[]])/r[]^3-(I/2)*epsilong1[-a,c]*r[]*𝒽$ab$p[-b,-c]*𝒫[]*CD1[b][r[]]-((I/6)*epsilong1[-a,-b]*𝒽$p[]*𝒫[]*CD1[b][r[]])/r[]+(I/2)*epsilong1[-a,c]*r[]*𝒽$ab$p[-c,d]*Τ[-b,-d]*CD1[b][r[]]+((I/6)*epsilong1[-a,-b]*𝒽$p[]*Τ[c,-c]*CD1[b][r[]])/r[]+(epsilong1[-a,-b]*Ψ$m[]*CD1[b][r[]])/(2*r[])-(Λ^2*𝒽$a$m[-a]*CD1[-b][r[]]*CD1[b][r[]])/(2*r[]^2)-((I/4)*epsilong1[-a,-b]*CD1[b][𝒽$p[]])/r[]^2+(I/12)*epsilong1[-a,-b]*𝒫[]*CD1[b][𝒽$p[]]-(I/12)*epsilong1[-a,-b]*Τ[c,-c]*CD1[b][𝒽$p[]]-(I/6)*epsilong1[-a,c]*r[]^2*𝒽$ab$p[-b,-c]*CD1[b][𝒫[]]+(epsilong1[-a,-b]*CD1[b][Ψ$m[]])/2+(I/2)*epsilong1[b,c]*CD1[-c][𝒽$ab$p[-a,-b]]-(I/6)*epsilong1[b,c]*r[]^2*𝒫[]*CD1[-c][𝒽$ab$p[-a,-b]]+(I/6)*epsilong1[b,c]*r[]^2*Τ[d,-d]*CD1[-c][𝒽$ab$p[-a,-b]]+(I/2)*epsilong1[-a,b]*CD1[-c][𝒽$ab$p[-b,c]]-(I/6)*epsilong1[-a,b]*r[]^2*𝒫[]*CD1[-c][𝒽$ab$p[-b,c]]+(I/4)*epsilong1[-a,c]*CD1[-b][r[]]*CD1[b][r[]]*CD1[-c][𝒽$ab$p[d,-d]]+(I/6)*epsilong1[-a,b]*r[]^2*𝒽$ab$p[-b,c]*CD1[-c][Τ[d,-d]]-((I/2)*epsilong1[-a,-c]*𝒽$p[]*CD1[-b][r[]]*CD1[b][r[]]*CD1[c][r[]])/r[]^3+((I/4)*epsilong1[-a,-c]*CD1[-b][r[]]*CD1[b][r[]]*CD1[c][𝒽$p[]])/r[]^2-(I/2)*epsilong1[c,d]*CD1[-b][r[]]*CD1[b][r[]]*CD1[-d][𝒽$ab$p[-a,-c]]+(I/6)*epsilong1[-a,b]*r[]^2*Τ[c,-c]*CD1[-d][𝒽$ab$p[-b,d]]-(I/2)*epsilong1[-a,c]*CD1[-b][r[]]*CD1[b][r[]]*CD1[-d][𝒽$ab$p[-c,d]]},MetricOn->All];

(*Pert$Exp$Sep[CD3[μ][CC[-μ,-ν,-α,-β]]-𝕁[-ν,-α,-β],{ν,α,β},{P,Q,a},{𝒽$Harm$Rep,ℂ$Harm$Rep,𝕁$Harm$Rep}];
%/.Dir$Harm$Def$Total//ContractMetric//ToCanonical;
Dum=%//.Harm$EQ$Y$Total//ContractMetric//ToCanonical
ToCanonical[(Dℂ$𝕁$PQa$UAB[-a]/.Eval$Pert$Dℂ$𝕁$PQa$UAB)-UAB$Component[Dum]]//Simplify
ToCanonical[(Dℂ$𝕁$PQa$VAB[-a]/.Eval$Pert$Dℂ$𝕁$PQa$VAB)-VAB$Component[Dum]]//Simplify
ToCanonical[(Dℂ$𝕁$PQa$WAB[-a]/.Eval$Pert$Dℂ$𝕁$PQa$WAB)-WAB$Component[Dum]]//Simplify
ToCanonical[(Dℂ$𝕁$PQa$ϵAB[-a]/.Eval$Pert$Dℂ$𝕁$PQa$ϵAB)-ϵAB$Component[Dum]]//Simplify*)

(*epsilong1[a,-f]Dℂ$𝕁$PQa$VAB//.Flatten[{Pert$ℂ$To$Met,Pert𝕁$To$metTmn,BG$1}]//ContractMetric//ToCanonical//Simplify;
%/.CommCovs$𝒽$a$m$FreeFirst;
%//.Flatten[{Einstein$Rep$DD𝒽$p,Einstein$Rep$Lorentz$𝒽$ab$p,Einstein$Rep$Trace$𝒽$ab$p,Einstein$Rep$Box$𝒽$a$m,Einstein$Rep$Div$𝒽$a$m,BG$1}]//ContractMetric//ToCanonical*)

((Subscript[ℂ, μ P A B ])^(;μ))-(Subscript[𝕁, P A B])=0

Eval$Pert$Dℂ$𝕁$PAB$ZA=MakeRule[{Dℂ$𝕁$PAB$ZA[],-Γ$m[]+(2*𝒽$a$m[a]*CD1[-a][r[]])/r[]+(r[]*𝒽$a$m[a]*𝒫[]*CD1[-a][r[]])/3+(Ξ$a$m[a]*CD1[-a][r[]])/r[]+(2*r[]*𝒽$a$m[a]*Τ[b,-b]*CD1[-a][r[]])/3-CD1[-a][𝒽$a$m[a]]+(r[]^2*𝒫[]*CD1[-a][𝒽$a$m[a]])/3-(r[]^2*Τ[b,-b]*CD1[-a][𝒽$a$m[a]])/3+(r[]^2*𝒽$a$m[a]*CD1[-a][𝒫[]])/3-CD1[-a][Ξ$a$m[a]]-(r[]^2*𝒽$a$m[a]*CD1[-a][Τ[b,-b]])/3+CD1[-a][r[]]*CD1[a][r[]]*CD1[-b][𝒽$a$m[b]]-r[]*𝒽$a$m[a]*Τ[-a,-b]*CD1[b][r[]]-(2*𝒽$a$m[a]*CD1[-a][r[]]*CD1[-b][r[]]*CD1[b][r[]])/r[]},MetricOn->All];
Eval$Pert$Dℂ$𝕁$PAB$XA=MakeRule[{Dℂ$𝕁$PAB$XA[],(-3*𝒽$ab$p[a,-a])/4+(2*𝒽$p[])/r[]^2+(r[]^2*𝒽$ab$p[a,-a]*𝒫[])/4-(2*𝒽$p[]*𝒫[])/3-Γ$p[]-Ξ$p[]/r[]^2+(2*𝒽$p[]*Τ[a,-a])/3-(r[]^2*𝒽$ab$p[a,-a]*Τ[b,-b])/4+(Ξ$a$p[a]*CD1[-a][r[]])/r[]-CD1[-a][Ξ$a$p[a]]+(3*𝒽$ab$p[b,-b]*CD1[-a][r[]]*CD1[a][r[]])/4-(2*𝒽$p[]*CD1[-a][r[]]*CD1[a][r[]])/r[]^2+((3*I)*epsilong1[-a,-b]*𝒽$a$m[a]*CD1[b][r[]])/r[]-I*epsilong1[-a,-b]*r[]*𝒽$a$m[a]*𝒫[]*CD1[b][r[]]+I*epsilong1[-a,-b]*r[]*𝒽$a$m[a]*Τ[c,-c]*CD1[b][r[]]-((3*I)/2)*epsilong1[-a,-b]*CD1[b][𝒽$a$m[a]]+(I/2)*epsilong1[-a,-b]*r[]^2*𝒫[]*CD1[b][𝒽$a$m[a]]-(I/2)*epsilong1[-a,-b]*r[]^2*Τ[c,-c]*CD1[b][𝒽$a$m[a]]-((3*I)*epsilong1[-a,-c]*𝒽$a$m[a]*CD1[-b][r[]]*CD1[b][r[]]*CD1[c][r[]])/r[]+((3*I)/2)*epsilong1[-b,-c]*CD1[-a][r[]]*CD1[a][r[]]*CD1[c][𝒽$a$m[b]]},MetricOn->All];

(*Pert$Exp$Sep[CD3[μ][CC[-μ,-ν,-α,-β]]-𝕁[-ν,-α,-β],{ν,α,β},{P,A,B},{𝒽$Harm$Rep,ℂ$Harm$Rep,𝕁$Harm$Rep}];
%/.Dir$Harm$Def$Total//ContractMetric//ToCanonical;
Dum=((1/2)epsilong2[A,B]%//ContractMetric//ToCanonical)//.Harm$EQ$Y$Total//ContractMetric//ToCanonical;
ToCanonical[(Dℂ$𝕁$PAB$ZA[]/.Eval$Pert$Dℂ$𝕁$PAB$ZA)-ZA$Component[Dum]]
ToCanonical[(Dℂ$𝕁$PAB$XA[]/.Eval$Pert$Dℂ$𝕁$PAB$XA)-XA$Component[Dum]]*)

Total

Eval$Pert$Dℂ$𝕁=Flatten[{Eval$Pert$Dℂ$𝕁$abc,Eval$Pert$Dℂ$𝕁$abP$ZA,Eval$Pert$Dℂ$𝕁$abP$XA,Eval$Pert$Dℂ$𝕁$aPQ,Eval$Pert$Dℂ$𝕁$Pab$ZA,Eval$Pert$Dℂ$𝕁$Pab$XA,Eval$Pert$Dℂ$𝕁$PQa$UAB,Eval$Pert$Dℂ$𝕁$PQa$VAB,Eval$Pert$Dℂ$𝕁$PQa$WAB,Eval$Pert$Dℂ$𝕁$PQa$ϵAB,Eval$Pert$Dℂ$𝕁$PAB$ZA,Eval$Pert$Dℂ$𝕁$PAB$XA}];

1.8. Linearised second-derivative equations for δ (Subscript[ℂ , μ ν ρ σ]) and δ (Subscript[𝕁 , μ ν ρ ])

We write left and right hand sides of the curvature equation:

LHS=ToCanonical[CD3[-α][CD3[α][CC[-μ,-ν,-ρ,-σ]]]+4*SkewSymetrise[SkewSymetrise[Symetrise[CC[-μ,α,β,-ρ],{α,β}]CC[-σ,-α,-β,-ν],{-μ,-ν}],{-ρ,-σ}]+SkewSymetrise[SkewSymetrise[CC[α,β,γ,-ρ]g3[-σ,-μ]CC[-ν,-γ,-α,-β],{-μ,-ν}],{-ρ,-σ}]]
RHS=ToCanonical[SkewSymetrise[2CD3[-μ][𝕁[-ν,-ρ,-σ]],{-μ,-ν}]-I epsilong3[-μ,-ν,α,β]CD3[-α][𝕁[-β,-ρ,-σ]]-SkewSymetrise[𝒯[γ,-ρ]CC[-σ,-γ,-μ,-ν],{-ρ,-σ}]-SkewSymetrise[𝒯[γ,-μ]CC[-ν,-γ,-ρ,-σ],{-μ,-ν}]-𝒯[γ,-γ]CC[-μ,-ν,-ρ,-σ]+SkewSymetrise[SkewSymetrise[2𝒯[α,β]g3[-ρ,-μ]CC[-σ,-α,-β,-ν],{-μ,-ν}],{-ρ,-σ}]]

DefTensor[Δℂ$D𝕁$abcd[],M1]
DefTensor[Δℂ$D𝕁$ZComp$Pabc[-a],M1]
DefTensor[Δℂ$D𝕁$XComp$Pabc[-a],M1]
DefTensor[Δℂ$D𝕁$abPQ[],M1]
DefTensor[Δℂ$D𝕁$UComp$PaQb[-a,-b],M1]
DefTensor[Δℂ$D𝕁$VComp$PaQb[-a,-b],M1]
DefTensor[Δℂ$D𝕁$WComp$PaQb[-a,-b],M1]
DefTensor[Δℂ$D𝕁$ϵComp$PaQb[-a,-b],M1]
DefTensor[Δℂ$D𝕁$ϵCompSkew$PaQb[],M1]
DefTensor[Δℂ$D𝕁$ZComp$ABPa[-a],M1]
DefTensor[Δℂ$D𝕁$XComp$ABPa[-a],M1]
DefTensor[Δℂ$D𝕁$ABPQ[],M1]

δ((Subscript[Δℂ, a b c d])-((Subscript[D, a])-i ⋆(Subscript[D, a]))(Subscript[𝕁, b c d]))=0

Eval$Pert$Δℂ$D𝕁$abcd=MakeRule[{Δℂ$D𝕁$abcd[],(-3*𝒽$ab$p[a,-a])/(2*r[]^4)-(Λ^2*𝒽$ab$p[a,-a])/r[]^4+(6*𝒽$p[])/r[]^6+(𝒽$ab$p[a,-a]*𝒫[])/(2*r[]^2)+(Λ^2*𝒽$ab$p[a,-a]*𝒫[])/(3*r[]^2)-(4*𝒽$p[]*𝒫[])/r[]^4-(𝒽$ab$p[a,-a]*𝒫[]^2)/6+(2*𝒽$p[]*𝒫[]^2)/(3*r[]^2)-((2*I)*Λ^2*Π$m[])/r[]^2+(5*𝒽$ab$p[a,b]*Τ[-a,-b])/(2*r[]^2)-(𝒽$ab$p[a,b]*𝒫[]*Τ[-a,-b])/6+(4*𝒽$p[]*Τ[a,-a])/r[]^4-(4*𝒽$p[]*𝒫[]*Τ[a,-a])/(3*r[]^2)-(𝒽$ab$p[a,b]*Τ[-a,c]*Τ[-b,-c])/2-(2*𝒽$ab$p[a,-a]*Τ[b,-b])/r[]^2-(Λ^2*𝒽$ab$p[a,-a]*Τ[b,-b])/(3*r[]^2)+(𝒽$ab$p[a,-a]*𝒫[]*Τ[b,-b])/6+(2*𝒽$p[]*Τ[a,-a]*Τ[b,-b])/(3*r[]^2)+(7*𝒽$ab$p[a,b]*Τ[-a,-b]*Τ[c,-c])/6-(𝒽$ab$p[a,-a]*Τ[b,-b]*Τ[c,-c])/2-Φ$ab$p[a,-a]/(2*r[]^2)+(𝒫[]*Φ$ab$p[a,-a])/6-(Τ[b,-b]*Φ$ab$p[a,-a])/6-Φ$p[]/r[]^4+(𝒫[]*Φ$p[])/(3*r[]^2)-(Τ[a,-a]*Φ$p[])/(3*r[]^2)+(3*Ψ$ab$p[a,-a])/r[]^4-(2*𝒫[]*Ψ$ab$p[a,-a])/r[]^2+(Τ[b,-b]*Ψ$ab$p[a,-a])/r[]^2-((3*I)*Ψ$m[])/r[]^4+(I*𝒫[]*Ψ$m[])/r[]^2-(I*Τ[a,-a]*Ψ$m[])/r[]^2-(Λ^2*Ψ$p[])/r[]^2+2*𝒫[]*Ψ$p[]+(Τ[a,-a]*Ψ$p[])/2+CD1[-a][CD1[a][Ψ$p[]]]+(3*𝒽$ab$p[b,-b]*CD1[-a][r[]]*CD1[a][r[]])/r[]^4+(Λ^2*𝒽$ab$p[b,-b]*CD1[-a][r[]]*CD1[a][r[]])/r[]^4-(32*𝒽$p[]*CD1[-a][r[]]*CD1[a][r[]])/r[]^6+(20*𝒽$p[]*𝒫[]*CD1[-a][r[]]*CD1[a][r[]])/(3*r[]^4)-(5*𝒽$ab$p[b,c]*Τ[-b,-c]*CD1[-a][r[]]*CD1[a][r[]])/(2*r[]^2)-(32*𝒽$p[]*Τ[b,-b]*CD1[-a][r[]]*CD1[a][r[]])/(3*r[]^4)+(2*𝒽$ab$p[b,-b]*Τ[c,-c]*CD1[-a][r[]]*CD1[a][r[]])/r[]^2+(Φ$ab$p[b,-b]*CD1[-a][r[]]*CD1[a][r[]])/(2*r[]^2)+(Φ$p[]*CD1[-a][r[]]*CD1[a][r[]])/r[]^4-(7*Ψ$ab$p[b,-b]*CD1[-a][r[]]*CD1[a][r[]])/r[]^4+((3*I)*Ψ$m[]*CD1[-a][r[]]*CD1[a][r[]])/r[]^4-(4*Ψ$p[]*CD1[-a][r[]]*CD1[a][r[]])/r[]^2-(5*CD1[-a][𝒽$ab$p[b,-b]]*CD1[a][r[]])/(2*r[]^3)-(2*𝒫[]*CD1[-a][𝒽$ab$p[b,-b]]*CD1[a][r[]])/(3*r[])-(5*Τ[b,-b]*CD1[-a][𝒽$ab$p[c,-c]]*CD1[a][r[]])/(6*r[])+(9*CD1[-a][𝒽$p[]]*CD1[a][r[]])/r[]^5-(3*𝒫[]*CD1[-a][𝒽$p[]]*CD1[a][r[]])/(2*r[]^3)+(3*Τ[b,-b]*CD1[-a][𝒽$p[]]*CD1[a][r[]])/r[]^3+(𝒽$ab$p[b,-b]*CD1[-a][𝒫[]]*CD1[a][r[]])/(6*r[])-(4*𝒽$p[]*CD1[-a][𝒫[]]*CD1[a][r[]])/(3*r[]^3)+(4*𝒽$p[]*CD1[-a][Τ[b,-b]]*CD1[a][r[]])/(3*r[]^3)-(𝒽$ab$p[b,-b]*CD1[-a][Τ[c,-c]]*CD1[a][r[]])/(6*r[])+(2*CD1[-a][Ψ$p[]]*CD1[a][r[]])/r[]+(CD1[-a][𝒫[]]*CD1[a][𝒽$p[]])/(2*r[]^2)-(CD1[-a][Τ[b,-b]]*CD1[a][𝒽$p[]])/(2*r[]^2)-(CD1[-a][𝒽$ab$p[b,-b]]*CD1[a][𝒫[]])/3-(3*CD1[a][r[]]*CD1[-b][𝒽$ab$p[-a,b]])/r[]^3-(CD1[a][𝒫[]]*CD1[-b][𝒽$ab$p[-a,b]])/3+(3*Τ[-a,b]*CD1[a][r[]]*CD1[-b][𝒽$ab$p[c,-c]])/(2*r[])+((I/6)*Λ^2*epsilong1[-a,b]*𝒽$a$m[a]*CD1[-b][Τ[c,-c]])/r[]^2-(CD1[-a][𝒽$ab$p[a,b]]*CD1[-b][Τ[c,-c]])/6-(𝒽$ab$p[-a,b]*CD1[a][r[]]*CD1[-b][Τ[c,-c]])/(3*r[])-(𝒽$ab$p[a,b]*CD1[-b][CD1[-a][𝒫[]]])/3+(𝒽$ab$p[a,b]*CD1[-b][CD1[-a][Τ[c,-c]]])/3+CD1[-b][CD1[b][𝒽$ab$p[a,-a]]]/r[]^2-(𝒫[]*CD1[-b][CD1[b][𝒽$ab$p[a,-a]]])/3-((6*I)*Λ^2*epsilong1[-a,-b]*𝒽$a$m[a]*CD1[b][r[]])/r[]^5+(((3*I)/2)*Λ^2*epsilong1[-a,-b]*𝒽$a$m[a]*𝒫[]*CD1[b][r[]])/r[]^3+((2*I)*epsilong1[-a,-b]*Π$a$m[a]*CD1[b][r[]])/r[]^3+((I/2)*Λ^2*epsilong1[-a,c]*𝒽$a$m[a]*Τ[-b,-c]*CD1[b][r[]])/r[]^3-((2*I)*Λ^2*epsilong1[-a,-b]*𝒽$a$m[a]*Τ[c,-c]*CD1[b][r[]])/r[]^3+(4*Λ^2*epsilong1[-a,-b]*Ψ$a$p[a]*CD1[b][r[]])/r[]^3+(2*𝒽$ab$p[-a,-b]*CD1[a][r[]]*CD1[b][r[]])/r[]^4+(4*𝒽$ab$p[-a,-b]*𝒫[]*CD1[a][r[]]*CD1[b][r[]])/(3*r[]^2)-(𝒽$ab$p[c,-c]*Τ[-a,-b]*CD1[a][r[]]*CD1[b][r[]])/(2*r[]^2)+(4*𝒽$p[]*Τ[-a,-b]*CD1[a][r[]]*CD1[b][r[]])/r[]^4-(2*𝒽$ab$p[-a,c]*Τ[-b,-c]*CD1[a][r[]]*CD1[b][r[]])/r[]^2+(2*𝒽$ab$p[-a,-b]*Τ[c,-c]*CD1[a][r[]]*CD1[b][r[]])/(3*r[]^2)+(4*Ψ$ab$p[-a,-b]*CD1[a][r[]]*CD1[b][r[]])/r[]^4-(3*𝒽$ab$p[c,-c]*CD1[-a][r[]]*CD1[a][r[]]*CD1[-b][r[]]*CD1[b][r[]])/(2*r[]^4)+(26*𝒽$p[]*CD1[-a][r[]]*CD1[a][r[]]*CD1[-b][r[]]*CD1[b][r[]])/r[]^6+(5*CD1[-a][r[]]*CD1[a][r[]]*CD1[-b][𝒽$ab$p[c,-c]]*CD1[b][r[]])/(2*r[]^3)-(9*CD1[-a][r[]]*CD1[a][r[]]*CD1[-b][𝒽$p[]]*CD1[b][r[]])/r[]^5+(5*CD1[-b][Τ[c,-c]]*CD1[b][𝒽$ab$p[a,-a]])/6-(3*Τ[-a,-b]*CD1[a][r[]]*CD1[b][𝒽$p[]])/(2*r[]^3)-((I/6)*Λ^2*epsilong1[-a,-b]*𝒽$a$m[a]*CD1[b][𝒫[]])/r[]^2-(2*𝒽$ab$p[-a,-b]*CD1[a][r[]]*CD1[b][𝒫[]])/(3*r[])-epsilong1[-a,-b]*CD1[b][Π$a$p[a]]-(Τ[b,-b]*CD1[a][r[]]*CD1[-c][𝒽$ab$p[-a,c]])/r[]+(Τ[-a,b]*CD1[a][r[]]*CD1[-c][𝒽$ab$p[-b,c]])/r[]+(3*CD1[-a][r[]]*CD1[a][r[]]*CD1[b][r[]]*CD1[-c][𝒽$ab$p[-b,c]])/r[]^3+(𝒽$ab$p[b,c]*CD1[a][r[]]*CD1[-c][Τ[-a,-b]])/r[]+(CD1[-a][𝒽$ab$p[a,b]]*CD1[-c][Τ[-b,c]])/2-(CD1[b][𝒽$ab$p[a,-a]]*CD1[-c][Τ[-b,c]])/4+(Τ[a,-a]*CD1[-c][CD1[c][𝒽$ab$p[b,-b]]])/3-(CD1[-a][r[]]*CD1[a][r[]]*CD1[-c][CD1[c][𝒽$ab$p[b,-b]]])/r[]^2-(2*𝒽$ab$p[-b,-c]*CD1[-a][r[]]*CD1[a][r[]]*CD1[b][r[]]*CD1[c][r[]])/r[]^4+((6*I)*Λ^2*epsilong1[-a,-c]*𝒽$a$m[a]*CD1[-b][r[]]*CD1[b][r[]]*CD1[c][r[]])/r[]^5+(CD1[-b][Τ[-a,-c]]*CD1[c][𝒽$ab$p[a,b]])/2-(CD1[-c][Τ[-a,-b]]*CD1[c][𝒽$ab$p[a,b]])/2},MetricOn->All];

(*{LHS$abcd$time,LHS$abcd$result}=AbsoluteTiming[LHSstep1=Pert$Exp$Sep[LHS,{μ,ν,ρ,σ},{a,b,c,d},{ℂ$Harm$Rep,𝒽$Harm$Rep}];
LHSstep2=LHSstep1/.Dir$Harm$Def$Total//ContractMetric//ToCanonical;
LHSstep3=LHSstep2//.Harm$EQ$Y$Total//ContractMetric//ToCanonical;
LHSstep4=((LHSstep3((1/4)epsilong1[a,b]epsilong1[c,d])//ContractMetric//ToCanonical));
LHSstep4];
Print["Computing Time:"]
LHS$abcd$time
Print["Result:"]
LHS$abcd$result
{RHS$abcd$time,RHS$abcd$result}=AbsoluteTiming[RHSstep1=Pert$Exp$Sep[RHS,{μ,ν,ρ,σ},{a,b,c,d},{ℂ$Harm$Rep,𝒽$Harm$Rep,𝕁$Harm$Rep,𝒯$Harm$Rep}];
RHSstep2=RHSstep1/.Dir$Harm$Def$Total//ContractMetric//ToCanonical;
RHSstep3=RHSstep2//.Harm$EQ$Y$Total//ContractMetric//ToCanonical;
RHSstep4=((RHSstep3((1/4)epsilong1[a,b]epsilong1[c,d])//ContractMetric//ToCanonical));
RHSstep4];
Print["Computing Time:"]
RHS$abcd$time
Print["Result:"]
RHS$abcd$result
Print["Match with saved result:"]
Simplify[ToCanonical[(Δℂ$D𝕁$abcd[]/.Eval$Pert$Δℂ$D𝕁$abcd)-((LHS$abcd$result-RHS$abcd$result)/Y[])]==0]*)

Z-Component of δ((Subscript[Δℂ, P a b c])-((Subscript[D, P])-i ⋆(Subscript[D, P]))(Subscript[𝕁, a b c]))=0

Eval$Pert$Δℂ$D𝕁$ZComp$Pabc=MakeRule[{Δℂ$D𝕁$ZComp$Pabc[-a],((3*I)*𝒽$a$m[-a])/r[]^4+(((3*I)/4)*Λ^2*𝒽$a$m[-a])/r[]^4-(((5*I)/2)*𝒽$a$m[-a]*𝒫[])/r[]^2-((I/4)*Λ^2*𝒽$a$m[-a]*𝒫[])/r[]^2+(I/2)*𝒽$a$m[-a]*𝒫[]^2-(((3*I)/4)*Ξ$a$m[-a])/r[]^4+((I/4)*𝒫[]*Ξ$a$m[-a])/r[]^2-(3*epsilong1[-a,-b]*Ξ$a$p[b])/(4*r[]^4)+(3*epsilong1[-a,-b]*𝒫[]*Ξ$a$p[b])/(4*r[]^2)-Π$a$p[-a]-(((3*I)/4)*𝒽$a$m[b]*Τ[-a,-b])/r[]^2+(I/4)*𝒽$a$m[b]*𝒫[]*Τ[-a,-b]+(((5*I)/2)*𝒽$a$m[-a]*Τ[b,-b])/r[]^2+((I/4)*Λ^2*𝒽$a$m[-a]*Τ[b,-b])/r[]^2-I*𝒽$a$m[-a]*𝒫[]*Τ[b,-b]-((I/4)*Ξ$a$m[-a]*Τ[b,-b])/r[]^2-(epsilong1[-a,-b]*Ξ$a$p[b]*Τ[c,-c])/(4*r[]^2)-(I/4)*𝒽$a$m[b]*Τ[-a,-b]*Τ[c,-c]+(I/2)*𝒽$a$m[-a]*Τ[b,-b]*Τ[c,-c]-(((3*I)/4)*epsilong1[-a,-b]*Ψ$a$m[b])/r[]^2+(I/4)*epsilong1[-a,-b]*𝒫[]*Ψ$a$m[b]-(I/4)*epsilong1[-a,-b]*Τ[c,-c]*Ψ$a$m[b]+(3*Ψ$a$p[-a])/(4*r[]^2)-(Λ^2*Ψ$a$p[-a])/r[]^2+(5*𝒫[]*Ψ$a$p[-a])/4+(Τ[b,-b]*Ψ$a$p[-a])/4+(I*Π$ab$m[b,-b]*CD1[-a][r[]])/r[]-(Π$p[]*CD1[-a][r[]])/r[]-2*CD1[-a][Π$p[]]-(((27*I)/2)*𝒽$a$m[b]*CD1[-a][r[]]*CD1[-b][r[]])/r[]^4+(((3*I)/2)*𝒽$a$m[b]*𝒫[]*CD1[-a][r[]]*CD1[-b][r[]])/r[]^2-(((9*I)/2)*𝒽$a$m[b]*Τ[c,-c]*CD1[-a][r[]]*CD1[-b][r[]])/r[]^2+(2*Ψ$a$p[b]*CD1[-a][r[]]*CD1[-b][r[]])/r[]^2-((I/3)*𝒽$a$m[b]*CD1[-a][𝒫[]]*CD1[-b][r[]])/r[]-((I/6)*𝒽$a$m[b]*CD1[-a][Τ[c,-c]]*CD1[-b][r[]])/r[]+(((3*I)/2)*CD1[-a][r[]]*CD1[-b][𝒽$a$m[b]])/r[]^3+((I/2)*Τ[c,-c]*CD1[-a][r[]]*CD1[-b][𝒽$a$m[b]])/r[]+(I/6)*CD1[-a][𝒫[]]*CD1[-b][𝒽$a$m[b]]-(I/6)*CD1[-a][Τ[c,-c]]*CD1[-b][𝒽$a$m[b]]-(I*𝒽$a$m[b]*CD1[-a][r[]]*CD1[-b][𝒫[]])/r[]+(I/3)*CD1[-a][𝒽$a$m[b]]*CD1[-b][𝒫[]]+(((3*I)/2)*𝒽$a$m[b]*CD1[-a][r[]]*CD1[-b][Τ[c,-c]])/r[]-((7*I)/12)*CD1[-a][𝒽$a$m[b]]*CD1[-b][Τ[c,-c]]-(((3*I)/4)*CD1[-b][CD1[-a][𝒽$a$m[b]]])/r[]^2+(I/4)*𝒫[]*CD1[-b][CD1[-a][𝒽$a$m[b]]]-(I/4)*Τ[c,-c]*CD1[-b][CD1[-a][𝒽$a$m[b]]]-(I/2)*𝒽$a$m[b]*CD1[-b][CD1[-a][Τ[c,-c]]]-(((3*I)/4)*CD1[-b][CD1[b][𝒽$a$m[-a]]])/r[]^2+(I/4)*𝒫[]*CD1[-b][CD1[b][𝒽$a$m[-a]]]-(I/4)*Τ[c,-c]*CD1[-b][CD1[b][𝒽$a$m[-a]]]+(I/3)*𝒽$a$m[-a]*CD1[-b][CD1[b][𝒫[]]]+CD1[-b][CD1[b][Ψ$a$p[-a]]]+(I/2)*𝒽$a$m[b]*CD1[-b][CD1[-c][Τ[-a,c]]]+(3*epsilong1[-b,c]*𝒽$ab$p[-a,-c]*CD1[b][r[]])/r[]^3+(epsilong1[-a,c]*𝒽$ab$p[-b,-c]*CD1[b][r[]])/r[]^3+(3*epsilong1[-a,-b]*𝒽$ab$p[c,-c]*CD1[b][r[]])/(4*r[]^3)-(5*epsilong1[-a,-b]*𝒽$p[]*CD1[b][r[]])/(2*r[]^5)-(5*epsilong1[-b,c]*𝒽$ab$p[-a,-c]*𝒫[]*CD1[b][r[]])/(4*r[])+(11*epsilong1[-a,c]*𝒽$ab$p[-b,-c]*𝒫[]*CD1[b][r[]])/(12*r[])-(epsilong1[-a,-b]*𝒽$ab$p[c,-c]*𝒫[]*CD1[b][r[]])/(4*r[])+(5*epsilong1[-a,-b]*𝒽$p[]*𝒫[]*CD1[b][r[]])/(6*r[]^3)-(I*Π$ab$m[-a,-b]*CD1[b][r[]])/r[]+(epsilong1[-b,c]*Π$ab$p[-a,-c]*CD1[b][r[]])/r[]+(I*epsilong1[-a,-b]*Π$m[]*CD1[b][r[]])/r[]-(epsilong1[c,d]*𝒽$ab$p[-a,-c]*Τ[-b,-d]*CD1[b][r[]])/(4*r[])-(5*epsilong1[-a,c]*𝒽$ab$p[-c,d]*Τ[-b,-d]*CD1[b][r[]])/(4*r[])-(5*epsilong1[-a,-b]*𝒽$p[]*Τ[c,-c]*CD1[b][r[]])/(6*r[]^3)+(epsilong1[-b,c]*𝒽$ab$p[-a,-c]*Τ[d,-d]*CD1[b][r[]])/r[]+(epsilong1[-a,c]*𝒽$ab$p[-b,-c]*Τ[d,-d]*CD1[b][r[]])/(3*r[])+(epsilong1[-a,-b]*𝒽$ab$p[c,-c]*Τ[d,-d]*CD1[b][r[]])/(4*r[])+(2*epsilong1[-b,c]*Ψ$ab$o[-a,-c]*CD1[b][r[]])/r[]^3-(Λ^2*epsilong1[-b,c]*Ψ$ab$o[-a,-c]*CD1[b][r[]])/r[]^3+(2*epsilong1[-b,c]*Ψ$ab$p[-a,-c]*CD1[b][r[]])/r[]^3-(2*epsilong1[-a,-b]*Ψ$p[]*CD1[b][r[]])/r[]+(((9*I)/2)*CD1[-a][𝒽$a$m[-b]]*CD1[b][r[]])/r[]^3-((I/2)*𝒫[]*CD1[-a][𝒽$a$m[-b]]*CD1[b][r[]])/r[]+(((3*I)/2)*Τ[c,-c]*CD1[-a][𝒽$a$m[-b]]*CD1[b][r[]])/r[]-(I*Τ[-b,-c]*CD1[-a][𝒽$a$m[c]]*CD1[b][r[]])/r[]-((6*I)*𝒽$a$m[-a]*CD1[-b][r[]]*CD1[b][r[]])/r[]^4-(((3*I)/4)*Λ^2*𝒽$a$m[-a]*CD1[-b][r[]]*CD1[b][r[]])/r[]^4+(((5*I)/2)*𝒽$a$m[-a]*𝒫[]*CD1[-b][r[]]*CD1[b][r[]])/r[]^2+(((3*I)/4)*Ξ$a$m[-a]*CD1[-b][r[]]*CD1[b][r[]])/r[]^4-(((5*I)/2)*𝒽$a$m[-a]*Τ[c,-c]*CD1[-b][r[]]*CD1[b][r[]])/r[]^2-(19*Ψ$a$p[-a]*CD1[-b][r[]]*CD1[b][r[]])/(4*r[]^2)+((3*I)*CD1[-b][𝒽$a$m[-a]]*CD1[b][r[]])/r[]^3+(I*Τ[c,-c]*CD1[-b][𝒽$a$m[-a]]*CD1[b][r[]])/r[]+(I/3)*CD1[-b][𝒫[]]*CD1[b][𝒽$a$m[-a]]-(I/12)*CD1[-b][Τ[c,-c]]*CD1[b][𝒽$a$m[-a]]+(3*epsilong1[-a,-b]*CD1[b][𝒽$p[]])/(4*r[]^4)-(epsilong1[-a,-b]*𝒫[]*CD1[b][𝒽$p[]])/(4*r[]^2)+(epsilong1[-a,-b]*Τ[c,-c]*CD1[b][𝒽$p[]])/(4*r[]^2)+(epsilong1[-b,c]*𝒽$ab$p[-a,-c]*CD1[b][𝒫[]])/12+(epsilong1[-a,c]*𝒽$ab$p[-b,-c]*CD1[b][𝒫[]])/4+(2*I)*epsilong1[-a,-b]*CD1[b][Π$m[]]-(3*epsilong1[-a,b]*CD1[-c][𝒽$ab$p[-b,c]])/(4*r[]^2)+(epsilong1[-a,b]*𝒫[]*CD1[-c][𝒽$ab$p[-b,c]])/4-((I/2)*Τ[-a,-b]*CD1[b][r[]]*CD1[-c][𝒽$a$m[c]])/r[]-(((3*I)/2)*CD1[-a][r[]]*CD1[-b][r[]]*CD1[b][r[]]*CD1[-c][𝒽$a$m[c]])/r[]^3+(epsilong1[b,c]*𝒽$ab$p[-b,d]*CD1[-c][Τ[-a,-d]])/4-((I/2)*𝒽$a$m[b]*CD1[-a][r[]]*CD1[-c][Τ[-b,c]])/r[]+(I/4)*CD1[-a][𝒽$a$m[b]]*CD1[-c][Τ[-b,c]]-(I/4)*CD1[b][𝒽$a$m[-a]]*CD1[-c][Τ[-b,c]]-(epsilong1[b,c]*𝒽$ab$p[-a,-b]*CD1[-c][Τ[d,-d]])/6-(epsilong1[-a,b]*𝒽$ab$p[-b,c]*CD1[-c][Τ[d,-d]])/2+(((3*I)/4)*CD1[-b][r[]]*CD1[b][r[]]*CD1[-c][CD1[-a][𝒽$a$m[c]]])/r[]^2+(I/2)*𝒽$a$m[b]*CD1[-c][CD1[-a][Τ[-b,c]]]+(((3*I)/4)*CD1[-b][r[]]*CD1[b][r[]]*CD1[-c][CD1[c][𝒽$a$m[-a]]])/r[]^2-(I/2)*𝒽$a$m[b]*CD1[-c][CD1[c][Τ[-a,-b]]]+(I/6)*𝒽$a$m[-a]*CD1[-c][CD1[c][Τ[b,-b]]]-(2*epsilong1[-b,-c]*Ξ$a$p[b]*CD1[-a][r[]]*CD1[c][r[]])/r[]^4+((3*I)*𝒽$a$m[b]*Τ[-b,-c]*CD1[-a][r[]]*CD1[c][r[]])/r[]^2+((I/2)*𝒽$a$m[b]*CD1[-a][Τ[-b,-c]]*CD1[c][r[]])/r[]-((I/2)*𝒽$a$m[b]*CD1[-b][Τ[-a,-c]]*CD1[c][r[]])/r[]-(3*epsilong1[-c,d]*𝒽$ab$p[-a,-d]*CD1[-b][r[]]*CD1[b][r[]]*CD1[c][r[]])/r[]^3-(epsilong1[-a,d]*𝒽$ab$p[-c,-d]*CD1[-b][r[]]*CD1[b][r[]]*CD1[c][r[]])/r[]^3-(3*epsilong1[-a,-c]*𝒽$ab$p[d,-d]*CD1[-b][r[]]*CD1[b][r[]]*CD1[c][r[]])/(4*r[]^3)+(5*epsilong1[-a,-c]*𝒽$p[]*CD1[-b][r[]]*CD1[b][r[]]*CD1[c][r[]])/(2*r[]^5)-(((9*I)/2)*CD1[-a][𝒽$a$m[-c]]*CD1[-b][r[]]*CD1[b][r[]]*CD1[c][r[]])/r[]^3+(3*epsilong1[-a,-b]*Ξ$a$p[b]*CD1[-c][r[]]*CD1[c][r[]])/(4*r[]^4)+(((3*I)/4)*𝒽$a$m[b]*Τ[-a,-b]*CD1[-c][r[]]*CD1[c][r[]])/r[]^2+(((3*I)/4)*epsilong1[-a,-b]*Ψ$a$m[b]*CD1[-c][r[]]*CD1[c][r[]])/r[]^2+(((27*I)/2)*𝒽$a$m[b]*CD1[-a][r[]]*CD1[-b][r[]]*CD1[-c][r[]]*CD1[c][r[]])/r[]^4+((3*I)*𝒽$a$m[-a]*CD1[-b][r[]]*CD1[b][r[]]*CD1[-c][r[]]*CD1[c][r[]])/r[]^4-((3*I)*CD1[-b][r[]]*CD1[b][r[]]*CD1[-c][𝒽$a$m[-a]]*CD1[c][r[]])/r[]^3-(I*Τ[-b,-c]*CD1[b][r[]]*CD1[c][𝒽$a$m[-a]])/r[]+(I/4)*CD1[-b][Τ[-a,-c]]*CD1[c][𝒽$a$m[b]]-(I/4)*CD1[-c][Τ[-a,-b]]*CD1[c][𝒽$a$m[b]]-(3*epsilong1[-a,-c]*CD1[-b][r[]]*CD1[b][r[]]*CD1[c][𝒽$p[]])/(4*r[]^4)-(epsilong1[-a,b]*Τ[c,-c]*CD1[-d][𝒽$ab$p[-b,d]])/4+(3*epsilong1[-a,c]*CD1[-b][r[]]*CD1[b][r[]]*CD1[-d][𝒽$ab$p[-c,d]])/(4*r[]^2)-(epsilong1[b,c]*𝒽$ab$p[-b,d]*CD1[-d][Τ[-a,-c]])/4},MetricOn->All];

(*{LHS$ZComp$Pabc$time,LHS$ZComp$Pabc$result}=AbsoluteTiming[LHSstep1=Pert$Exp$Sep[LHS,{μ,ν,ρ,σ},{P,a,b,c},{ℂ$Harm$Rep,𝒽$Harm$Rep}];
LHSstep2=LHSstep1/.Dir$Harm$Def$Total//ContractMetric//ToCanonical;
LHSstep3=((-1/2)epsilong1[b,c]LHSstep2//ContractMetric//ToCanonical)//.Harm$EQ$Y$Total//ContractMetric//ToCanonical;
LHSstep4=ZA$Component[LHSstep3]//ContractMetric//ToCanonical;
LHSstep4
];
Print["Computing Time:"]
LHS$ZComp$Pabc$time
Print["Result:"]
LHS$ZComp$Pabc$result
{RHS$ZComp$Pabc$time,RHS$ZComp$Pabc$result}=AbsoluteTiming[RHSstep1=Pert$Exp$Sep[RHS,{μ,ν,ρ,σ},{P,a,b,c},{ℂ$Harm$Rep,𝒽$Harm$Rep,𝕁$Harm$Rep,𝒯$Harm$Rep}];
RHSstep2=RHSstep1/.Dir$Harm$Def$Total//ContractMetric//ToCanonical;
RHSstep3=((-1/2)epsilong1[b,c]RHSstep2//ContractMetric//ToCanonical)//.Harm$EQ$Y$Total//ContractMetric//ToCanonical;
RHSstep4=ZA$Component[RHSstep3]//ContractMetric//ToCanonical;
RHSstep4
];
Print["Computing Time:"]
RHS$ZComp$Pabc$time
Print["Result:"]
RHS$ZComp$Pabc$result
Print["Match with saved result:"]
Simplify[ToCanonical[(Δℂ$D𝕁$ZComp$Pabc[-a]/.Eval$Pert$Δℂ$D𝕁$ZComp$Pabc)-(LHS$ZComp$Pabc$result-RHS$ZComp$Pabc$result)]==0]*)

X-Component of δ((Subscript[Δℂ, P a b c])-((Subscript[D, P])-i ⋆(Subscript[D, P]))(Subscript[𝕁, a b c]))=0

Eval$Pert$Δℂ$D𝕁$XComp$Pabc=MakeRule[{Δℂ$D𝕁$XComp$Pabc[-a],(Λ^2*epsilong1[-a,-b]*𝒽$a$m[b])/(4*r[]^4)-(epsilong1[-a,-b]*𝒽$a$m[b]*𝒫[])/(2*r[]^2)-(Λ^2*epsilong1[-a,-b]*𝒽$a$m[b]*𝒫[])/(12*r[]^2)+(epsilong1[-a,-b]*𝒽$a$m[b]*𝒫[]^2)/6+(3*epsilong1[-a,-b]*Ξ$a$m[b])/(4*r[]^4)-(3*epsilong1[-a,-b]*𝒫[]*Ξ$a$m[b])/(4*r[]^2)-(((3*I)/4)*Ξ$a$p[-a])/r[]^4+((I/4)*𝒫[]*Ξ$a$p[-a])/r[]^2-I*epsilong1[-a,-b]*Π$a$p[b]+(epsilong1[-b,c]*𝒽$a$m[b]*Τ[-a,-c])/(2*r[]^2)-(epsilong1[-b,c]*𝒽$a$m[b]*𝒫[]*Τ[-a,-c])/6+(epsilong1[-a,c]*𝒽$a$m[b]*Τ[-b,-c])/(4*r[]^2)-(epsilong1[-a,c]*𝒽$a$m[b]*𝒫[]*Τ[-b,-c])/12-((I/4)*Ξ$a$p[-a]*Τ[b,-b])/r[]^2+(epsilong1[-a,-b]*𝒽$a$m[b]*Τ[c,-c])/(2*r[]^2)+(Λ^2*epsilong1[-a,-b]*𝒽$a$m[b]*Τ[c,-c])/(12*r[]^2)-(epsilong1[-a,-b]*𝒽$a$m[b]*𝒫[]*Τ[c,-c])/3+(epsilong1[-a,-b]*Ξ$a$m[b]*Τ[c,-c])/(4*r[]^2)+(epsilong1[-b,c]*𝒽$a$m[b]*Τ[-a,-c]*Τ[d,-d])/6+(epsilong1[-a,c]*𝒽$a$m[b]*Τ[-b,-c]*Τ[d,-d])/12+(epsilong1[-a,-b]*𝒽$a$m[b]*Τ[c,-c]*Τ[d,-d])/6+(3*Ψ$a$m[-a])/(4*r[]^2)-(Λ^2*Ψ$a$m[-a])/r[]^2+(5*𝒫[]*Ψ$a$m[-a])/4+(Τ[b,-b]*Ψ$a$m[-a])/4+(((3*I)/4)*epsilong1[-a,-b]*Ψ$a$p[b])/r[]^2-(I/4)*epsilong1[-a,-b]*𝒫[]*Ψ$a$p[b]+(I/4)*epsilong1[-a,-b]*Τ[c,-c]*Ψ$a$p[b]-(((3*I)/4)*𝒽$ab$p[b,-b]*CD1[-a][r[]])/r[]^3-(((9*I)/2)*𝒽$p[]*CD1[-a][r[]])/r[]^5+((I/2)*𝒽$ab$p[b,-b]*𝒫[]*CD1[-a][r[]])/r[]+(((3*I)/2)*𝒽$p[]*𝒫[]*CD1[-a][r[]])/r[]^3-(I*Π$ab$p[b,-b]*CD1[-a][r[]])/r[]-(Π$m[]*CD1[-a][r[]])/r[]-(((3*I)/2)*𝒽$p[]*Τ[b,-b]*CD1[-a][r[]])/r[]^3-((I/4)*𝒽$ab$p[b,-b]*Τ[c,-c]*CD1[-a][r[]])/r[]-(3*Ψ$m[]*CD1[-a][r[]])/r[]^3+(((3*I)/4)*CD1[-a][𝒽$p[]])/r[]^4-((I/4)*𝒫[]*CD1[-a][𝒽$p[]])/r[]^2+((I/4)*Τ[b,-b]*CD1[-a][𝒽$p[]])/r[]^2-(I/12)*𝒽$ab$p[b,-b]*CD1[-a][𝒫[]]-2*CD1[-a][Π$m[]]+(I/4)*𝒽$ab$p[b,c]*CD1[-a][Τ[-b,-c]]-(I/6)*𝒽$ab$p[b,-b]*CD1[-a][Τ[c,-c]]+(2*Ψ$a$m[b]*CD1[-a][r[]]*CD1[-b][r[]])/r[]^2-(((3*I)/4)*CD1[-b][𝒽$ab$p[-a,b]])/r[]^2+(I/4)*𝒫[]*CD1[-b][𝒽$ab$p[-a,b]]-(I/12)*𝒽$ab$p[-a,b]*CD1[-b][Τ[c,-c]]+CD1[-b][CD1[b][Ψ$a$m[-a]]]+(3*epsilong1[-a,-c]*CD1[-b][CD1[c][𝒽$a$m[b]]])/(4*r[]^2)-(epsilong1[-a,-c]*𝒫[]*CD1[-b][CD1[c][𝒽$a$m[b]]])/4+(epsilong1[-a,-c]*Τ[d,-d]*CD1[-b][CD1[c][𝒽$a$m[b]]])/4+((3*I)*𝒽$ab$p[-a,-b]*CD1[b][r[]])/r[]^3+(epsilong1[-b,c]*Π$ab$m[-a,-c]*CD1[b][r[]])/r[]+(I*Π$ab$p[-a,-b]*CD1[b][r[]])/r[]-(I*epsilong1[-a,-b]*Π$p[]*CD1[b][r[]])/r[]-((I/4)*𝒽$ab$p[c,-c]*Τ[-a,-b]*CD1[b][r[]])/r[]-(I*𝒽$ab$p[-a,c]*Τ[-b,-c]*CD1[b][r[]])/r[]+(I*𝒽$ab$p[-a,-b]*Τ[c,-c]*CD1[b][r[]])/r[]+(2*epsilong1[-b,c]*Ψ$ab$m[-a,-c]*CD1[b][r[]])/r[]^3-(Λ^2*epsilong1[-b,c]*Ψ$ab$m[-a,-c]*CD1[b][r[]])/r[]^3+(epsilong1[-b,-c]*𝒫[]*CD1[-a][𝒽$a$m[c]]*CD1[b][r[]])/(4*r[])+(epsilong1[-c,d]*Τ[-b,-d]*CD1[-a][𝒽$a$m[c]]*CD1[b][r[]])/(4*r[])+(((3*I)/4)*Ξ$a$p[-a]*CD1[-b][r[]]*CD1[b][r[]])/r[]^4-(19*Ψ$a$m[-a]*CD1[-b][r[]]*CD1[b][r[]])/(4*r[]^2)+(((3*I)/4)*𝒽$ab$p[c,-c]*CD1[-a][r[]]*CD1[-b][r[]]*CD1[b][r[]])/r[]^3+(((9*I)/2)*𝒽$p[]*CD1[-a][r[]]*CD1[-b][r[]]*CD1[b][r[]])/r[]^5-(((3*I)/4)*CD1[-a][𝒽$p[]]*CD1[-b][r[]]*CD1[b][r[]])/r[]^4+(3*epsilong1[-a,-c]*CD1[-b][𝒽$a$m[c]]*CD1[b][r[]])/r[]^3-(epsilong1[-a,-c]*𝒫[]*CD1[-b][𝒽$a$m[c]]*CD1[b][r[]])/(4*r[])+(epsilong1[-a,-c]*Τ[d,-d]*CD1[-b][𝒽$a$m[c]]*CD1[b][r[]])/r[]+(I/3)*𝒽$ab$p[-a,-b]*CD1[b][𝒫[]]-(2*I)*epsilong1[-a,-b]*CD1[b][Π$p[]]-(I/4)*Τ[b,-b]*CD1[-c][𝒽$ab$p[-a,c]]+(((3*I)/4)*CD1[-b][r[]]*CD1[b][r[]]*CD1[-c][𝒽$ab$p[-a,c]])/r[]^2-(3*epsilong1[-a,-b]*CD1[b][r[]]*CD1[-c][𝒽$a$m[c]])/(2*r[]^3)+(epsilong1[-a,-b]*𝒫[]*CD1[b][r[]]*CD1[-c][𝒽$a$m[c]])/(2*r[])-(epsilong1[-a,-b]*Τ[d,-d]*CD1[b][r[]]*CD1[-c][𝒽$a$m[c]])/(2*r[])-(I/4)*𝒽$ab$p[b,c]*CD1[-c][Τ[-a,-b]]-(I/4)*𝒽$ab$p[-a,b]*CD1[-c][Τ[-b,c]]+(epsilong1[-b,c]*CD1[-a][𝒽$a$m[b]]*CD1[-c][Τ[d,-d]])/6+(epsilong1[-b,c]*𝒽$a$m[-a]*CD1[b][r[]]*CD1[-c][Τ[d,-d]])/(6*r[])-(epsilong1[-a,-b]*CD1[-c][CD1[c][𝒽$a$m[b]]])/(4*r[]^2)+(epsilong1[-a,-b]*𝒫[]*CD1[-c][CD1[c][𝒽$a$m[b]]])/12-(epsilong1[-a,-b]*Τ[d,-d]*CD1[-c][CD1[c][𝒽$a$m[b]]])/12-(3*epsilong1[-a,-d]*CD1[-b][r[]]*CD1[b][r[]]*CD1[-c][CD1[d][𝒽$a$m[c]]])/(4*r[]^2)+(3*epsilong1[-b,-c]*𝒽$a$m[b]*CD1[-a][r[]]*CD1[c][r[]])/r[]^4+(2*epsilong1[-b,-c]*Ξ$a$m[b]*CD1[-a][r[]]*CD1[c][r[]])/r[]^4+(epsilong1[-b,-c]*𝒽$a$m[b]*Τ[d,-d]*CD1[-a][r[]]*CD1[c][r[]])/r[]^2+(epsilong1[-b,-c]*𝒽$a$m[b]*CD1[-a][𝒫[]]*CD1[c][r[]])/(3*r[])-(epsilong1[-b,-c]*𝒽$a$m[b]*CD1[-a][Τ[d,-d]]*CD1[c][r[]])/(3*r[])+(17*epsilong1[-a,-c]*𝒽$a$m[b]*CD1[-b][r[]]*CD1[c][r[]])/(2*r[]^4)-(5*epsilong1[-a,-c]*𝒽$a$m[b]*𝒫[]*CD1[-b][r[]]*CD1[c][r[]])/(6*r[]^2)+(17*epsilong1[-a,-c]*𝒽$a$m[b]*Τ[d,-d]*CD1[-b][r[]]*CD1[c][r[]])/(6*r[]^2)+(epsilong1[-a,-c]*𝒽$a$m[b]*CD1[-b][𝒫[]]*CD1[c][r[]])/r[]+(epsilong1[-c,d]*𝒽$a$m[b]*CD1[-b][Τ[-a,-d]]*CD1[c][r[]])/(2*r[])-(epsilong1[-a,-c]*𝒽$a$m[b]*CD1[-b][Τ[d,-d]]*CD1[c][r[]])/(2*r[])-((3*I)*𝒽$ab$p[-a,-c]*CD1[-b][r[]]*CD1[b][r[]]*CD1[c][r[]])/r[]^3-(6*epsilong1[-a,-b]*𝒽$a$m[b]*CD1[-c][r[]]*CD1[c][r[]])/r[]^4-(Λ^2*epsilong1[-a,-b]*𝒽$a$m[b]*CD1[-c][r[]]*CD1[c][r[]])/(4*r[]^4)+(epsilong1[-a,-b]*𝒽$a$m[b]*𝒫[]*CD1[-c][r[]]*CD1[c][r[]])/(2*r[]^2)-(3*epsilong1[-a,-b]*Ξ$a$m[b]*CD1[-c][r[]]*CD1[c][r[]])/(4*r[]^4)-(epsilong1[-b,d]*𝒽$a$m[b]*Τ[-a,-d]*CD1[-c][r[]]*CD1[c][r[]])/(2*r[]^2)-(epsilong1[-a,d]*𝒽$a$m[b]*Τ[-b,-d]*CD1[-c][r[]]*CD1[c][r[]])/(4*r[]^2)-(5*epsilong1[-a,-b]*𝒽$a$m[b]*Τ[d,-d]*CD1[-c][r[]]*CD1[c][r[]])/(2*r[]^2)-(((3*I)/4)*epsilong1[-a,-b]*Ψ$a$p[b]*CD1[-c][r[]]*CD1[c][r[]])/r[]^2-(3*epsilong1[-a,-d]*CD1[-b][r[]]*CD1[b][r[]]*CD1[-c][𝒽$a$m[d]]*CD1[c][r[]])/r[]^3-(2*epsilong1[-a,-b]*𝒽$a$m[b]*CD1[-c][𝒫[]]*CD1[c][r[]])/(3*r[])+(2*epsilong1[-a,-b]*𝒽$a$m[b]*CD1[-c][Τ[d,-d]]*CD1[c][r[]])/(3*r[])+(epsilong1[-b,-c]*𝒫[]*CD1[b][r[]]*CD1[c][𝒽$a$m[-a]])/(4*r[])+(epsilong1[-c,d]*Τ[-b,-d]*CD1[b][r[]]*CD1[c][𝒽$a$m[-a]])/(4*r[])-(9*epsilong1[-a,-c]*CD1[b][r[]]*CD1[c][𝒽$a$m[-b]])/(2*r[]^3)+(epsilong1[-a,-c]*𝒫[]*CD1[b][r[]]*CD1[c][𝒽$a$m[-b]])/(4*r[])-(3*epsilong1[-a,-c]*Τ[d,-d]*CD1[b][r[]]*CD1[c][𝒽$a$m[-b]])/(2*r[])-(7*epsilong1[-a,-c]*CD1[-b][𝒫[]]*CD1[c][𝒽$a$m[b]])/12-(epsilong1[-c,d]*CD1[-b][Τ[-a,-d]]*CD1[c][𝒽$a$m[b]])/4+(epsilong1[-a,-c]*CD1[-b][Τ[d,-d]]*CD1[c][𝒽$a$m[b]])/3+(5*epsilong1[-a,-b]*CD1[-c][𝒫[]]*CD1[c][𝒽$a$m[b]])/12+(epsilong1[-b,d]*CD1[-c][Τ[-a,-d]]*CD1[c][𝒽$a$m[b]])/4-(epsilong1[-a,-b]*CD1[-c][Τ[d,-d]]*CD1[c][𝒽$a$m[b]])/6+(epsilong1[-b,-c]*CD1[-a][𝒽$a$m[b]]*CD1[c][𝒫[]])/12+(epsilong1[-b,-c]*𝒽$a$m[-a]*CD1[b][r[]]*CD1[c][𝒫[]])/(3*r[])-(epsilong1[-b,-c]*CD1[b][𝒽$a$m[-a]]*CD1[c][𝒫[]])/4+(3*epsilong1[-a,-c]*CD1[-b][r[]]*CD1[b][r[]]*CD1[c][r[]]*CD1[-d][𝒽$a$m[d]])/(2*r[]^3)-(epsilong1[-c,d]*𝒽$a$m[b]*CD1[c][r[]]*CD1[-d][Τ[-a,-b]])/(2*r[])+(epsilong1[-c,d]*CD1[c][𝒽$a$m[b]]*CD1[-d][Τ[-a,-b]])/4-(epsilong1[-b,d]*CD1[c][𝒽$a$m[b]]*CD1[-d][Τ[-a,-c]])/4+(epsilong1[-a,-c]*CD1[-b][r[]]*CD1[b][r[]]*CD1[-d][CD1[d][𝒽$a$m[c]]])/(4*r[]^2)-(epsilong1[-b,-c]*𝒽$a$m[b]*Τ[-a,-d]*CD1[c][r[]]*CD1[d][r[]])/r[]^2-(2*epsilong1[-a,-c]*𝒽$a$m[b]*Τ[-b,-d]*CD1[c][r[]]*CD1[d][r[]])/r[]^2+(2*epsilong1[-a,-b]*𝒽$a$m[b]*Τ[-c,-d]*CD1[c][r[]]*CD1[d][r[]])/r[]^2-(3*epsilong1[-b,-d]*𝒽$a$m[b]*CD1[-a][r[]]*CD1[-c][r[]]*CD1[c][r[]]*CD1[d][r[]])/r[]^4-(17*epsilong1[-a,-d]*𝒽$a$m[b]*CD1[-b][r[]]*CD1[-c][r[]]*CD1[c][r[]]*CD1[d][r[]])/(2*r[]^4)+(6*epsilong1[-a,-b]*𝒽$a$m[b]*CD1[-c][r[]]*CD1[c][r[]]*CD1[-d][r[]]*CD1[d][r[]])/r[]^4+(9*epsilong1[-a,-d]*CD1[-b][r[]]*CD1[b][r[]]*CD1[c][r[]]*CD1[d][𝒽$a$m[-c]])/(2*r[]^3)+(5*epsilong1[-a,-d]*Τ[-b,-c]*CD1[b][r[]]*CD1[d][𝒽$a$m[c]])/(4*r[])-(3*epsilong1[-a,-c]*Τ[-b,-d]*CD1[b][r[]]*CD1[d][𝒽$a$m[c]])/(4*r[])},MetricOn->All];

(*{LHS$XComp$Pabc$time,LHS$XComp$Pabc$result}=AbsoluteTiming[LHSstep1=Pert$Exp$Sep[LHS,{μ,ν,ρ,σ},{P,a,b,c},{ℂ$Harm$Rep,𝒽$Harm$Rep}];
LHSstep2=LHSstep1/.Dir$Harm$Def$Total//ContractMetric//ToCanonical;
LHSstep3=((-1/2)epsilong1[b,c]LHSstep2//ContractMetric//ToCanonical)//.Harm$EQ$Y$Total//ContractMetric//ToCanonical;
LHSstep4=XA$Component[LHSstep3]//ContractMetric//ToCanonical;
LHSstep4
];
Print["Computing Time:"]
LHS$XComp$Pabc$time
Print["Result:"]
LHS$XComp$Pabc$result
{RHS$XComp$Pabc$time,RHS$XComp$Pabc$result}=AbsoluteTiming[RHSstep1=Pert$Exp$Sep[RHS,{μ,ν,ρ,σ},{P,a,b,c},{ℂ$Harm$Rep,𝒽$Harm$Rep,𝕁$Harm$Rep,𝒯$Harm$Rep}];
RHSstep2=RHSstep1/.Dir$Harm$Def$Total//ContractMetric//ToCanonical;
RHSstep3=((-1/2)epsilong1[b,c]RHSstep2//ContractMetric//ToCanonical)//.Harm$EQ$Y$Total//ContractMetric//ToCanonical;
RHSstep4=XA$Component[RHSstep3]//ContractMetric//ToCanonical;
RHSstep4
];
Print["Computing Time:"]
RHS$XComp$Pabc$time
Print["Result:"]
RHS$XComp$Pabc$result
Print["Match with saved result:"]
Simplify[ToCanonical[(Δℂ$D𝕁$XComp$Pabc[-a]/.Eval$Pert$Δℂ$D𝕁$XComp$Pabc)-(LHS$XComp$Pabc$result-RHS$XComp$Pabc$result)]==0]*)

δ((Subscript[Δℂ, a b P Q])-((Subscript[D, a])-i ⋆(Subscript[D, a]))(Subscript[𝕁, b P Q]))=0

Eval$Pert$Δℂ$D𝕁$abPQ=MakeRule[{Δℂ$D𝕁$abPQ[],((-1/2*I)*Λ^2*𝒽$ab$p[a,-a])/r[]^2+(I*𝒽$p[])/r[]^4-(I*Λ^2*𝒽$p[])/r[]^4-I*𝒽$ab$p[a,-a]*𝒫[]+(I/6)*Λ^2*𝒽$ab$p[a,-a]*𝒫[]-((I/3)*𝒽$p[]*𝒫[])/r[]^2+((I/3)*Λ^2*𝒽$p[]*𝒫[])/r[]^2+(I/6)*r[]^2*𝒽$ab$p[a,-a]*𝒫[]^2+(I*Λ^2*Γ$p[])/r[]^2+((5*I)/2)*𝒽$ab$p[a,b]*Τ[-a,-b]-(I/6)*r[]^2*𝒽$ab$p[a,b]*𝒫[]*Τ[-a,-b]+((I/3)*𝒽$p[]*Τ[a,-a])/r[]^2-((I/3)*Λ^2*𝒽$p[]*Τ[a,-a])/r[]^2-(I/2)*r[]^2*𝒽$ab$p[a,b]*Τ[-a,c]*Τ[-b,-c]-I*𝒽$ab$p[a,-a]*Τ[b,-b]-(I/6)*Λ^2*𝒽$ab$p[a,-a]*Τ[b,-b]-(I/3)*r[]^2*𝒽$ab$p[a,-a]*𝒫[]*Τ[b,-b]+((7*I)/6)*r[]^2*𝒽$ab$p[a,b]*Τ[-a,-b]*Τ[c,-c]-(I/3)*r[]^2*𝒽$ab$p[a,-a]*Τ[b,-b]*Τ[c,-c]-(I/2)*Φ$ab$p[a,-a]+(I/6)*r[]^2*𝒫[]*Φ$ab$p[a,-a]-(I/6)*r[]^2*Τ[b,-b]*Φ$ab$p[a,-a]-(I*Φ$p[])/r[]^2+(I/3)*𝒫[]*Φ$p[]-(I/3)*Τ[a,-a]*Φ$p[]+((3*I)*Ψ$ab$p[a,-a])/r[]^2-I*𝒫[]*Ψ$ab$p[a,-a]+I*Τ[b,-b]*Ψ$ab$p[a,-a]+Ψ$m[]/r[]^2-(Λ^2*Ψ$m[])/r[]^2+(Τ[a,-a]*Ψ$m[])/2-((2*I)*Γ$a$p[a]*CD1[-a][r[]])/r[]-(2*Λ^2*Ψ$a$m[a]*CD1[-a][r[]])/r[]+(I*CD1[-a][CD1[a][𝒽$p[]]])/r[]^2-(I/3)*𝒫[]*CD1[-a][CD1[a][𝒽$p[]]]+CD1[-a][CD1[a][Ψ$m[]]]+((I/2)*Λ^2*𝒽$ab$p[b,-b]*CD1[-a][r[]]*CD1[a][r[]])/r[]^2-((6*I)*𝒽$p[]*CD1[-a][r[]]*CD1[a][r[]])/r[]^4+(I*Λ^2*𝒽$p[]*CD1[-a][r[]]*CD1[a][r[]])/r[]^4+((3*I)/2)*𝒽$ab$p[b,-b]*𝒫[]*CD1[-a][r[]]*CD1[a][r[]]+(((8*I)/3)*𝒽$p[]*𝒫[]*CD1[-a][r[]]*CD1[a][r[]])/r[]^2-((5*I)/2)*𝒽$ab$p[b,c]*Τ[-b,-c]*CD1[-a][r[]]*CD1[a][r[]]-(((5*I)/3)*𝒽$p[]*Τ[b,-b]*CD1[-a][r[]]*CD1[a][r[]])/r[]^2+I*𝒽$ab$p[b,-b]*Τ[c,-c]*CD1[-a][r[]]*CD1[a][r[]]+(I/2)*Φ$ab$p[b,-b]*CD1[-a][r[]]*CD1[a][r[]]+(I*Φ$p[]*CD1[-a][r[]]*CD1[a][r[]])/r[]^2-((3*I)*Ψ$ab$p[b,-b]*CD1[-a][r[]]*CD1[a][r[]])/r[]^2-(5*Ψ$m[]*CD1[-a][r[]]*CD1[a][r[]])/r[]^2-((I/2)*CD1[-a][𝒽$ab$p[b,-b]]*CD1[a][r[]])/r[]-(I/3)*r[]*𝒫[]*CD1[-a][𝒽$ab$p[b,-b]]*CD1[a][r[]]-(I/6)*r[]*Τ[b,-b]*CD1[-a][𝒽$ab$p[c,-c]]*CD1[a][r[]]+(I*CD1[-a][𝒽$p[]]*CD1[a][r[]])/r[]^3-(((5*I)/6)*𝒫[]*CD1[-a][𝒽$p[]]*CD1[a][r[]])/r[]+((I/3)*Τ[b,-b]*CD1[-a][𝒽$p[]]*CD1[a][r[]])/r[]+(I/6)*r[]*𝒽$ab$p[b,-b]*CD1[-a][𝒫[]]*CD1[a][r[]]-((I/3)*𝒽$p[]*CD1[-a][𝒫[]]*CD1[a][r[]])/r[]+((I/3)*𝒽$p[]*CD1[-a][Τ[b,-b]]*CD1[a][r[]])/r[]-(I/6)*r[]*𝒽$ab$p[b,-b]*CD1[-a][Τ[c,-c]]*CD1[a][r[]]-(2*CD1[-a][Ψ$m[]]*CD1[a][r[]])/r[]+(I/6)*CD1[-a][𝒫[]]*CD1[a][𝒽$p[]]-(I/6)*CD1[-a][Τ[b,-b]]*CD1[a][𝒽$p[]]-(I/6)*r[]^2*CD1[-a][𝒽$ab$p[b,-b]]*CD1[a][𝒫[]]-((3*I)*CD1[a][r[]]*CD1[-b][𝒽$ab$p[-a,b]])/r[]-(I/3)*r[]^2*CD1[a][𝒫[]]*CD1[-b][𝒽$ab$p[-a,b]]+(I/2)*r[]*Τ[-a,b]*CD1[a][r[]]*CD1[-b][𝒽$ab$p[c,-c]]-(I*𝒽$p[]*CD1[a][r[]]*CD1[-b][Τ[-a,b]])/r[]+(I/2)*CD1[a][𝒽$p[]]*CD1[-b][Τ[-a,b]]+(I/3)*r[]^2*CD1[-a][𝒽$ab$p[a,b]]*CD1[-b][Τ[c,-c]]-(I/3)*r[]^2*𝒽$ab$p[a,b]*CD1[-b][CD1[-a][𝒫[]]]+(I/3)*r[]^2*𝒽$ab$p[a,b]*CD1[-b][CD1[-a][Τ[c,-c]]]+(I/2)*CD1[-b][CD1[b][𝒽$ab$p[a,-a]]]-(I/6)*r[]^2*𝒫[]*CD1[-b][CD1[b][𝒽$ab$p[a,-a]]]+(I/3)*Τ[a,-a]*CD1[-b][CD1[b][𝒽$p[]]]-(I*CD1[-a][r[]]*CD1[a][r[]]*CD1[-b][CD1[b][𝒽$p[]]])/r[]^2+(3*Λ^2*epsilong1[-a,-b]*𝒽$a$m[a]*CD1[b][r[]])/r[]^3-(Λ^2*epsilong1[-a,-b]*𝒽$a$m[a]*𝒫[]*CD1[b][r[]])/r[]+(2*Λ^2*epsilong1[-a,-b]*Ξ$a$m[a]*CD1[b][r[]])/r[]^3-(4*epsilong1[-a,-b]*Π$a$m[a]*CD1[b][r[]])/r[]+(Λ^2*epsilong1[-a,-b]*𝒽$a$m[a]*Τ[c,-c]*CD1[b][r[]])/r[]+I*𝒽$ab$p[-a,-b]*𝒫[]*CD1[a][r[]]*CD1[b][r[]]-(I/2)*𝒽$ab$p[c,-c]*Τ[-a,-b]*CD1[a][r[]]*CD1[b][r[]]-(I*𝒽$p[]*Τ[-a,-b]*CD1[a][r[]]*CD1[b][r[]])/r[]^2-I*𝒽$ab$p[-a,c]*Τ[-b,-c]*CD1[a][r[]]*CD1[b][r[]]+((5*I)*𝒽$p[]*CD1[-a][r[]]*CD1[a][r[]]*CD1[-b][r[]]*CD1[b][r[]])/r[]^4+((I/2)*CD1[-a][r[]]*CD1[a][r[]]*CD1[-b][𝒽$ab$p[c,-c]]*CD1[b][r[]])/r[]-(I*CD1[-a][r[]]*CD1[a][r[]]*CD1[-b][𝒽$p[]]*CD1[b][r[]])/r[]^3+(I/6)*r[]^2*CD1[-b][Τ[c,-c]]*CD1[b][𝒽$ab$p[a,-a]]+((I/2)*Τ[-a,-b]*CD1[a][r[]]*CD1[b][𝒽$p[]])/r[]-I*r[]*𝒽$ab$p[-a,-b]*CD1[a][r[]]*CD1[b][𝒫[]]+2*epsilong1[-a,-b]*CD1[b][Π$a$m[a]]-I*r[]*Τ[b,-b]*CD1[a][r[]]*CD1[-c][𝒽$ab$p[-a,c]]+I*r[]*Τ[-a,b]*CD1[a][r[]]*CD1[-c][𝒽$ab$p[-b,c]]+((3*I)*CD1[-a][r[]]*CD1[a][r[]]*CD1[b][r[]]*CD1[-c][𝒽$ab$p[-b,c]])/r[]+I*r[]*𝒽$ab$p[b,c]*CD1[a][r[]]*CD1[-c][Τ[-a,-b]]+(I/6)*r[]^2*Τ[a,-a]*CD1[-c][CD1[c][𝒽$ab$p[b,-b]]]-(I/2)*CD1[-a][r[]]*CD1[a][r[]]*CD1[-c][CD1[c][𝒽$ab$p[b,-b]]]-(3*Λ^2*epsilong1[-a,-c]*𝒽$a$m[a]*CD1[-b][r[]]*CD1[b][r[]]*CD1[c][r[]])/r[]^3},MetricOn->All];

(*{LHS$abPQ$time,LHS$abPQ$result}=AbsoluteTiming[LHSstep1=Pert$Exp$Sep[LHS,{μ,ν,ρ,σ},{a,b,P,Q},{ℂ$Harm$Rep,𝒽$Harm$Rep}];
LHSstep2=LHSstep1/.Dir$Harm$Def$Total//ContractMetric//ToCanonical;
LHSstep3=((-1/2)epsilong1[a,b]LHSstep2//ContractMetric//ToCanonical)//.Harm$EQ$Y$Total//ContractMetric//ToCanonical;
LHSstep4=ϵAB$Component[LHSstep3]//ContractMetric//ToCanonical;
LHSstep4
];
Print["Computing Time:"]
LHS$abPQ$time
Print["Result:"]
LHS$abPQ$result
{RHS$abPQ$time,RHS$abPQ$result}=AbsoluteTiming[RHSstep1=Pert$Exp$Sep[RHS,{μ,ν,ρ,σ},{a,b,P,Q},{ℂ$Harm$Rep,𝒽$Harm$Rep,𝕁$Harm$Rep,𝒯$Harm$Rep}];
RHSstep2=RHSstep1/.Dir$Harm$Def$Total//ContractMetric//ToCanonical;
RHSstep3=((-1/2)epsilong1[a,b]RHSstep2//ContractMetric//ToCanonical)//.Harm$EQ$Y$Total//ContractMetric//ToCanonical;
RHSstep4=ϵAB$Component[RHSstep3]//ContractMetric//ToCanonical;
RHSstep4
];
Print["Computing Time:"]
RHS$abPQ$time
Print["Result:"]
RHS$abPQ$result
Print["Match with saved result:"]
Simplify[ToCanonical[(Δℂ$D𝕁$abPQ[]/.Eval$Pert$Δℂ$D𝕁$abPQ)-(LHS$abPQ$result-RHS$abPQ$result)]==0]*)

U-Component of δ((Subscript[Δℂ, P a Q b])-((Subscript[D, P])-i ⋆(Subscript[D, P]))(Subscript[𝕁, a Q b]))=0

Eval$Pert$Δℂ$D𝕁$UComp$PaQb=MakeRule[{Δℂ$D𝕁$UComp$PaQb[-a,-b],(-3*𝒽$ab$p[-a,-b])/(2*r[]^2)-(Λ^2*𝒽$ab$p[-a,-b])/(2*r[]^2)+(3*g1[-a,-b]*𝒽$ab$p[c,-c])/(4*r[]^2)+(g1[-a,-b]*𝒽$p[])/(2*r[]^4)-(Λ^2*g1[-a,-b]*𝒽$p[])/(2*r[]^4)+(5*𝒽$ab$p[-a,-b]*𝒫[])/4+(Λ^2*𝒽$ab$p[-a,-b]*𝒫[])/6-(7*g1[-a,-b]*𝒽$ab$p[c,-c]*𝒫[])/8-(2*g1[-a,-b]*𝒽$p[]*𝒫[])/(3*r[]^2)+(Λ^2*g1[-a,-b]*𝒽$p[]*𝒫[])/(6*r[]^2)-(r[]^2*𝒽$ab$p[-a,-b]*𝒫[]^2)/3+(g1[-a,-b]*r[]^2*𝒽$ab$p[c,-c]*𝒫[]^2)/6+(g1[-a,-b]*𝒽$p[]*𝒫[]^2)/6+(g1[-a,-b]*𝒫[]*Ξ$p[])/(2*r[]^2)-(I/2)*Λ^2*epsilong1[-a,c]*Π$ab$m[-b,-c]-(Λ^2*Π$ab$p[-a,-b])/2-(I/2)*Λ^2*g1[-a,-b]*Π$m[]-(Λ^2*epsilong1[-a,-b]*Π$p[])/2-(𝒽$ab$p[c,-c]*Τ[-a,-b])/8-(𝒽$p[]*Τ[-a,-b])/(2*r[]^2)-(r[]^2*𝒽$ab$p[c,-c]*𝒫[]*Τ[-a,-b])/8+(𝒽$p[]*𝒫[]*Τ[-a,-b])/6+(𝒽$ab$p[-b,c]*Τ[-a,-c])/2+(r[]^2*𝒽$ab$p[-b,c]*𝒫[]*Τ[-a,-c])/6+(𝒽$ab$p[-a,c]*Τ[-b,-c])/4-(r[]^2*𝒽$ab$p[-a,c]*𝒫[]*Τ[-b,-c])/12+(r[]^2*𝒽$ab$p[c,-c]*Τ[-a,d]*Τ[-b,-d])/8+(3*g1[-a,-b]*𝒽$ab$p[c,d]*Τ[-c,-d])/4-(g1[-a,-b]*r[]^2*𝒽$ab$p[c,d]*𝒫[]*Τ[-c,-d])/4-(r[]^2*𝒽$ab$p[-b,c]*Τ[-a,d]*Τ[-c,-d])/4-(5*𝒽$ab$p[-a,-b]*Τ[c,-c])/4-(Λ^2*𝒽$ab$p[-a,-b]*Τ[c,-c])/6+(2*g1[-a,-b]*𝒽$p[]*Τ[c,-c])/(3*r[]^2)-(Λ^2*g1[-a,-b]*𝒽$p[]*Τ[c,-c])/(6*r[]^2)+(r[]^2*𝒽$ab$p[-a,-b]*𝒫[]*Τ[c,-c])/3-(g1[-a,-b]*𝒽$p[]*𝒫[]*Τ[c,-c])/3-(𝒽$p[]*Τ[-a,-b]*Τ[c,-c])/6+(g1[-a,-b]*𝒽$ab$p[c,-c]*Τ[d,-d])/8-(g1[-a,-b]*r[]^2*𝒽$ab$p[c,-c]*𝒫[]*Τ[d,-d])/12-(r[]^2*𝒽$ab$p[c,-c]*Τ[-a,-b]*Τ[d,-d])/8+(r[]^2*𝒽$ab$p[-b,c]*Τ[-a,-c]*Τ[d,-d])/3+(r[]^2*𝒽$ab$p[-a,c]*Τ[-b,-c]*Τ[d,-d])/12-(r[]^2*𝒽$ab$p[-a,-b]*Τ[c,-c]*Τ[d,-d])/4+(g1[-a,-b]*𝒽$p[]*Τ[c,-c]*Τ[d,-d])/6-(g1[-a,-b]*r[]^2*𝒽$ab$p[c,-c]*Τ[-d,-e]*Τ[d,e])/8+(g1[-a,-b]*r[]^2*𝒽$ab$p[c,d]*Τ[-c,-d]*Τ[e,-e])/4+(g1[-a,-b]*r[]^2*𝒽$ab$p[c,-c]*Τ[d,-d]*Τ[e,-e])/24-(g1[-a,-b]*Φ$ab$p[c,-c])/4+(g1[-a,-b]*r[]^2*𝒫[]*Φ$ab$p[c,-c])/12-(g1[-a,-b]*r[]^2*Τ[d,-d]*Φ$ab$p[c,-c])/12-(g1[-a,-b]*Φ$p[])/(2*r[]^2)+(g1[-a,-b]*𝒫[]*Φ$p[])/6-(g1[-a,-b]*Τ[c,-c]*Φ$p[])/6-(2*Ψ$ab$p[-a,-b])/r[]^2-(Λ^2*Ψ$ab$p[-a,-b])/r[]^2+2*𝒫[]*Ψ$ab$p[-a,-b]-(Τ[-b,-c]*Ψ$ab$p[-a,c])/2-(Τ[-a,-c]*Ψ$ab$p[-b,c])/2+(3*g1[-a,-b]*Ψ$ab$p[c,-c])/(2*r[]^2)-(g1[-a,-b]*𝒫[]*Ψ$ab$p[c,-c])/2+(g1[-a,-b]*Τ[d,-d]*Ψ$ab$p[c,-c])/2+(g1[-a,-b]*Τ[-c,-d]*Ψ$ab$p[c,d])/2-(((3*I)/2)*g1[-a,-b]*Ψ$m[])/r[]^2+(I/2)*g1[-a,-b]*𝒫[]*Ψ$m[]-(I/2)*g1[-a,-b]*Τ[c,-c]*Ψ$m[]+(r[]^2*Τ[-a,-b]*Ψ$p[])/2-(g1[-a,-b]*r[]^2*Τ[c,-c]*Ψ$p[])/2-(Γ$a$p[-b]*CD1[-a][r[]])/r[]+(Λ^2*Ξ$a$p[-b]*CD1[-a][r[]])/r[]^3+CD1[-a][Γ$a$p[-b]]+((I/4)*Λ^2*epsilong1[-a,-c]*𝒽$a$m[c]*𝒫[]*CD1[-b][r[]])/r[]+(Λ^2*Ξ$a$p[-a]*CD1[-b][r[]])/r[]^3-((2*I)*epsilong1[-a,-c]*Π$a$m[c]*CD1[-b][r[]])/r[]-(4*𝒽$p[]*CD1[-a][r[]]*CD1[-b][r[]])/r[]^4+(4*𝒽$p[]*𝒫[]*CD1[-a][r[]]*CD1[-b][r[]])/(3*r[]^2)+(2*Ξ$p[]*CD1[-a][r[]]*CD1[-b][r[]])/r[]^4-(4*𝒽$p[]*Τ[c,-c]*CD1[-a][r[]]*CD1[-b][r[]])/(3*r[]^2)+2*Ψ$p[]*CD1[-a][r[]]*CD1[-b][r[]]+(𝒫[]*CD1[-a][𝒽$p[]]*CD1[-b][r[]])/(4*r[])-(r[]*𝒽$ab$p[c,-c]*CD1[-a][𝒫[]]*CD1[-b][r[]])/12-(𝒽$p[]*CD1[-a][𝒫[]]*CD1[-b][r[]])/(6*r[])+(r[]*𝒽$ab$p[c,d]*CD1[-a][Τ[-c,-d]]*CD1[-b][r[]])/2+(𝒽$p[]*CD1[-a][Τ[c,-c]]*CD1[-b][r[]])/(6*r[])-(5*r[]*𝒽$ab$p[c,-c]*CD1[-a][Τ[d,-d]]*CD1[-b][r[]])/12-(r[]*𝒫[]*CD1[-a][r[]]*CD1[-b][𝒽$ab$p[c,-c]])/4-(r[]^2*CD1[-a][𝒫[]]*CD1[-b][𝒽$ab$p[c,-c]])/12+(r[]^2*CD1[-a][Τ[d,-d]]*CD1[-b][𝒽$ab$p[c,-c]])/12-(𝒫[]*CD1[-a][r[]]*CD1[-b][𝒽$p[]])/(4*r[])+(CD1[-a][𝒫[]]*CD1[-b][𝒽$p[]])/12-(CD1[-a][Τ[c,-c]]*CD1[-b][𝒽$p[]])/12+(I/12)*Λ^2*epsilong1[-a,-c]*𝒽$a$m[c]*CD1[-b][𝒫[]]+(r[]*𝒽$ab$p[c,-c]*CD1[-a][r[]]*CD1[-b][𝒫[]])/4-(𝒽$p[]*CD1[-a][r[]]*CD1[-b][𝒫[]])/(6*r[])-(CD1[-a][𝒽$p[]]*CD1[-b][𝒫[]])/12+(𝒽$p[]*CD1[-a][r[]]*CD1[-b][Τ[c,-c]])/(6*r[])-(CD1[-a][𝒽$p[]]*CD1[-b][Τ[c,-c]])/6-(I/12)*Λ^2*epsilong1[-a,-c]*𝒽$a$m[c]*CD1[-b][Τ[d,-d]]+(r[]^2*𝒽$ab$p[c,-c]*CD1[-b][CD1[-a][𝒫[]]])/12-(r[]^2*𝒽$ab$p[c,-c]*CD1[-b][CD1[-a][Τ[d,-d]]])/12-(𝒽$p[]*CD1[-b][r[]]*CD1[-c][Τ[-a,c]])/(2*r[])+(CD1[-b][𝒽$p[]]*CD1[-c][Τ[-a,c]])/4+(I/12)*Λ^2*epsilong1[-b,c]*𝒽$a$m[-a]*CD1[-c][Τ[d,-d]]-(I/12)*Λ^2*epsilong1[-a,c]*𝒽$a$m[-b]*CD1[-c][Τ[d,-d]]+(r[]*𝒽$ab$p[-a,c]*CD1[-b][r[]]*CD1[-c][Τ[d,-d]])/6-(r[]^2*CD1[-b][𝒽$ab$p[-a,c]]*CD1[-c][Τ[d,-d]])/6-(r[]^2*𝒽$ab$p[-b,c]*CD1[-c][CD1[-a][𝒫[]]])/6+(r[]^2*𝒽$ab$p[-b,c]*CD1[-c][CD1[-a][Τ[d,-d]]])/6+CD1[-c][CD1[c][𝒽$ab$p[-a,-b]]]/2-(r[]^2*𝒫[]*CD1[-c][CD1[c][𝒽$ab$p[-a,-b]]])/6+(g1[-a,-b]*CD1[-c][CD1[c][𝒽$p[]]])/(2*r[]^2)-(g1[-a,-b]*𝒫[]*CD1[-c][CD1[c][𝒽$p[]]])/6+CD1[-c][CD1[c][Ψ$ab$p[-a,-b]]]-(((3*I)/2)*Λ^2*epsilong1[-b,-c]*𝒽$a$m[-a]*CD1[c][r[]])/r[]^3-(((3*I)/2)*Λ^2*epsilong1[-a,-c]*𝒽$a$m[-b]*CD1[c][r[]])/r[]^3+((I/4)*Λ^2*epsilong1[-b,-c]*𝒽$a$m[-a]*𝒫[]*CD1[c][r[]])/r[]+((I/2)*Λ^2*epsilong1[-a,-c]*𝒽$a$m[-b]*𝒫[]*CD1[c][r[]])/r[]+(I*epsilong1[-a,-c]*Π$a$m[-b]*CD1[c][r[]])/r[]+epsilong1[-b,-c]*r[]*Π$a$p[-a]*CD1[c][r[]]+((I/4)*Λ^2*epsilong1[-b,d]*𝒽$a$m[-a]*Τ[-c,-d]*CD1[c][r[]])/r[]-((I/2)*Λ^2*epsilong1[-b,-c]*𝒽$a$m[-a]*Τ[d,-d]*CD1[c][r[]])/r[]-((I/2)*Λ^2*epsilong1[-a,-c]*𝒽$a$m[-b]*Τ[d,-d]*CD1[c][r[]])/r[]+(Λ^2*epsilong1[-b,-c]*Ψ$a$p[-a]*CD1[c][r[]])/r[]+(Λ^2*epsilong1[-a,-c]*Ψ$a$p[-b]*CD1[c][r[]])/r[]+(2*𝒽$ab$p[-b,-c]*CD1[-a][r[]]*CD1[c][r[]])/r[]^2-(2*𝒽$ab$p[-b,-c]*𝒫[]*CD1[-a][r[]]*CD1[c][r[]])/3+(𝒽$p[]*Τ[-b,-c]*CD1[-a][r[]]*CD1[c][r[]])/(2*r[]^2)+(2*𝒽$ab$p[-b,-c]*Τ[d,-d]*CD1[-a][r[]]*CD1[c][r[]])/3-(Τ[-b,-c]*CD1[-a][𝒽$p[]]*CD1[c][r[]])/(4*r[])-(r[]*𝒽$ab$p[-b,-c]*CD1[-a][𝒫[]]*CD1[c][r[]])/6+(r[]*𝒽$ab$p[-b,-c]*CD1[-a][Τ[d,-d]]*CD1[c][r[]])/6+(2*𝒽$ab$p[-a,-c]*CD1[-b][r[]]*CD1[c][r[]])/r[]^2-(2*𝒽$ab$p[-a,-c]*𝒫[]*CD1[-b][r[]]*CD1[c][r[]])/3-(𝒽$p[]*Τ[-a,-c]*CD1[-b][r[]]*CD1[c][r[]])/(2*r[]^2)+(2*𝒽$ab$p[-a,-c]*Τ[d,-d]*CD1[-b][r[]]*CD1[c][r[]])/3+(r[]*𝒫[]*CD1[-b][𝒽$ab$p[-a,-c]]*CD1[c][r[]])/2-(r[]*Τ[-c,d]*CD1[-b][𝒽$ab$p[-a,-d]]*CD1[c][r[]])/2+(r[]*Τ[-a,-c]*CD1[-b][𝒽$ab$p[d,-d]]*CD1[c][r[]])/4+(Τ[-a,-c]*CD1[-b][𝒽$p[]]*CD1[c][r[]])/(4*r[])+(r[]*𝒽$ab$p[-a,-c]*CD1[-b][𝒫[]]*CD1[c][r[]])/3-(r[]*𝒽$ab$p[d,-d]*CD1[-b][Τ[-a,-c]]*CD1[c][r[]])/4-(r[]*𝒽$ab$p[-c,d]*CD1[-b][Τ[-a,-d]]*CD1[c][r[]])/2+(r[]*𝒽$ab$p[-a,-c]*CD1[-b][Τ[d,-d]]*CD1[c][r[]])/6+(3*𝒽$ab$p[-a,-b]*CD1[-c][r[]]*CD1[c][r[]])/r[]^2+(Λ^2*𝒽$ab$p[-a,-b]*CD1[-c][r[]]*CD1[c][r[]])/(2*r[]^2)-(3*g1[-a,-b]*𝒽$ab$p[d,-d]*CD1[-c][r[]]*CD1[c][r[]])/(2*r[]^2)-(g1[-a,-b]*𝒽$p[]*CD1[-c][r[]]*CD1[c][r[]])/r[]^4+(Λ^2*g1[-a,-b]*𝒽$p[]*CD1[-c][r[]]*CD1[c][r[]])/(2*r[]^4)-(5*𝒽$ab$p[-a,-b]*𝒫[]*CD1[-c][r[]]*CD1[c][r[]])/4+(7*g1[-a,-b]*𝒽$ab$p[d,-d]*𝒫[]*CD1[-c][r[]]*CD1[c][r[]])/8+(7*g1[-a,-b]*𝒽$p[]*𝒫[]*CD1[-c][r[]]*CD1[c][r[]])/(6*r[]^2)+(𝒽$ab$p[d,-d]*Τ[-a,-b]*CD1[-c][r[]]*CD1[c][r[]])/8+(𝒽$p[]*Τ[-a,-b]*CD1[-c][r[]]*CD1[c][r[]])/(2*r[]^2)-(𝒽$ab$p[-b,d]*Τ[-a,-d]*CD1[-c][r[]]*CD1[c][r[]])/2-(𝒽$ab$p[-a,d]*Τ[-b,-d]*CD1[-c][r[]]*CD1[c][r[]])/4-(3*g1[-a,-b]*𝒽$ab$p[d,e]*Τ[-d,-e]*CD1[-c][r[]]*CD1[c][r[]])/4+(5*𝒽$ab$p[-a,-b]*Τ[d,-d]*CD1[-c][r[]]*CD1[c][r[]])/4-(2*g1[-a,-b]*𝒽$p[]*Τ[d,-d]*CD1[-c][r[]]*CD1[c][r[]])/(3*r[]^2)-(g1[-a,-b]*𝒽$ab$p[d,-d]*Τ[e,-e]*CD1[-c][r[]]*CD1[c][r[]])/8+(g1[-a,-b]*Φ$ab$p[d,-d]*CD1[-c][r[]]*CD1[c][r[]])/4+(g1[-a,-b]*Φ$p[]*CD1[-c][r[]]*CD1[c][r[]])/(2*r[]^2)+(2*Ψ$ab$p[-a,-b]*CD1[-c][r[]]*CD1[c][r[]])/r[]^2-(3*g1[-a,-b]*Ψ$ab$p[d,-d]*CD1[-c][r[]]*CD1[c][r[]])/(2*r[]^2)+(((3*I)/2)*g1[-a,-b]*Ψ$m[]*CD1[-c][r[]]*CD1[c][r[]])/r[]^2-2*g1[-a,-b]*Ψ$p[]*CD1[-c][r[]]*CD1[c][r[]]+(4*𝒽$p[]*CD1[-a][r[]]*CD1[-b][r[]]*CD1[-c][r[]]*CD1[c][r[]])/r[]^4-(2*CD1[-c][𝒽$ab$p[-a,-b]]*CD1[c][r[]])/r[]-(r[]*𝒫[]*CD1[-c][𝒽$ab$p[-a,-b]]*CD1[c][r[]])/3-(2*r[]*Τ[d,-d]*CD1[-c][𝒽$ab$p[-a,-b]]*CD1[c][r[]])/3+(3*g1[-a,-b]*CD1[-c][𝒽$ab$p[d,-d]]*CD1[c][r[]])/(4*r[])+(g1[-a,-b]*r[]*Τ[d,-d]*CD1[-c][𝒽$ab$p[e,-e]]*CD1[c][r[]])/4+(g1[-a,-b]*CD1[-c][𝒽$p[]]*CD1[c][r[]])/(2*r[]^3)-(5*g1[-a,-b]*𝒫[]*CD1[-c][𝒽$p[]]*CD1[c][r[]])/(12*r[])+(g1[-a,-b]*Τ[d,-d]*CD1[-c][𝒽$p[]]*CD1[c][r[]])/(6*r[])-(g1[-a,-b]*r[]*𝒽$ab$p[d,-d]*CD1[-c][𝒫[]]*CD1[c][r[]])/3+(g1[-a,-b]*𝒽$p[]*CD1[-c][𝒫[]]*CD1[c][r[]])/(6*r[])-(g1[-a,-b]*𝒽$p[]*CD1[-c][Τ[d,-d]]*CD1[c][r[]])/(6*r[])+(g1[-a,-b]*r[]*𝒽$ab$p[d,-d]*CD1[-c][Τ[e,-e]]*CD1[c][r[]])/12-(2*CD1[-c][Ψ$ab$p[-a,-b]]*CD1[c][r[]])/r[]+(r[]^2*CD1[-c][Τ[d,-d]]*CD1[c][𝒽$ab$p[-a,-b]])/3+(CD1[-b][Τ[-a,-c]]*CD1[c][𝒽$p[]])/4+(g1[-a,-b]*CD1[-c][𝒫[]]*CD1[c][𝒽$p[]])/12-(CD1[-c][Τ[-a,-b]]*CD1[c][𝒽$p[]])/4+(g1[-a,-b]*CD1[-c][Τ[d,-d]]*CD1[c][𝒽$p[]])/6-(I/12)*Λ^2*epsilong1[-b,-c]*𝒽$a$m[-a]*CD1[c][𝒫[]]-(I/6)*Λ^2*epsilong1[-a,-c]*𝒽$a$m[-b]*CD1[c][𝒫[]]-(r[]*𝒽$ab$p[-b,-c]*CD1[-a][r[]]*CD1[c][𝒫[]])/2+(r[]*𝒽$ab$p[-a,-c]*CD1[-b][r[]]*CD1[c][𝒫[]])/3+(r[]^2*CD1[-b][𝒽$ab$p[-a,-c]]*CD1[c][𝒫[]])/6-(r[]^2*CD1[-c][𝒽$ab$p[-a,-b]]*CD1[c][𝒫[]])/3+(g1[-a,-b]*r[]^2*CD1[-c][𝒽$ab$p[d,-d]]*CD1[c][𝒫[]])/12-I*epsilong1[-a,-c]*CD1[c][Π$a$m[-b]]+r[]*Τ[-c,d]*CD1[c][r[]]*CD1[-d][𝒽$ab$p[-a,-b]]-(3*g1[-a,-b]*CD1[c][r[]]*CD1[-d][𝒽$ab$p[-c,d]])/(2*r[])-(g1[-a,-b]*r[]^2*CD1[c][𝒫[]]*CD1[-d][𝒽$ab$p[-c,d]])/6-(g1[-a,-b]*r[]*Τ[-c,d]*CD1[c][r[]]*CD1[-d][𝒽$ab$p[e,-e]])/4+(r[]*𝒽$ab$p[-c,d]*CD1[c][r[]]*CD1[-d][Τ[-a,-b]])/2+(I/4)*Λ^2*epsilong1[c,d]*𝒽$a$m[-b]*CD1[-d][Τ[-a,-c]]-(r[]*𝒽$ab$p[c,d]*CD1[-b][r[]]*CD1[-d][Τ[-a,-c]])/2+(r[]*𝒽$ab$p[-b,d]*CD1[c][r[]]*CD1[-d][Τ[-a,-c]])/2+(r[]*𝒽$ab$p[c,-c]*CD1[-b][r[]]*CD1[-d][Τ[-a,d]])/4+(g1[-a,-b]*r[]^2*CD1[-c][𝒽$ab$p[c,d]]*CD1[-d][Τ[e,-e]])/6-(g1[-a,-b]*r[]*𝒽$ab$p[-c,d]*CD1[c][r[]]*CD1[-d][Τ[e,-e]])/2+(r[]^2*Τ[c,-c]*CD1[-d][CD1[d][𝒽$ab$p[-a,-b]]])/6-(CD1[-c][r[]]*CD1[c][r[]]*CD1[-d][CD1[d][𝒽$ab$p[-a,-b]]])/2+(g1[-a,-b]*Τ[c,-c]*CD1[-d][CD1[d][𝒽$p[]]])/6-(g1[-a,-b]*CD1[-c][r[]]*CD1[c][r[]]*CD1[-d][CD1[d][𝒽$p[]]])/(2*r[]^2)-(g1[-a,-b]*r[]^2*𝒽$ab$p[c,-c]*CD1[-d][CD1[d][𝒫[]]])/12-((I/4)*Λ^2*epsilong1[-a,-c]*𝒽$a$m[c]*Τ[-b,-d]*CD1[d][r[]])/r[]-(2*g1[-a,-b]*𝒽$ab$p[-c,-d]*CD1[c][r[]]*CD1[d][r[]])/r[]^2+(5*g1[-a,-b]*𝒽$ab$p[-c,-d]*𝒫[]*CD1[c][r[]]*CD1[d][r[]])/3-(g1[-a,-b]*𝒽$p[]*Τ[-c,-d]*CD1[c][r[]]*CD1[d][r[]])/(2*r[]^2)-g1[-a,-b]*𝒽$ab$p[-c,e]*Τ[-d,-e]*CD1[c][r[]]*CD1[d][r[]]-(2*g1[-a,-b]*𝒽$ab$p[-c,-d]*Τ[e,-e]*CD1[c][r[]]*CD1[d][r[]])/3+(((3*I)/2)*Λ^2*epsilong1[-b,-d]*𝒽$a$m[-a]*CD1[-c][r[]]*CD1[c][r[]]*CD1[d][r[]])/r[]^3+(((3*I)/2)*Λ^2*epsilong1[-a,-d]*𝒽$a$m[-b]*CD1[-c][r[]]*CD1[c][r[]]*CD1[d][r[]])/r[]^3-(2*𝒽$ab$p[-b,-d]*CD1[-a][r[]]*CD1[-c][r[]]*CD1[c][r[]]*CD1[d][r[]])/r[]^2-(2*𝒽$ab$p[-a,-d]*CD1[-b][r[]]*CD1[-c][r[]]*CD1[c][r[]]*CD1[d][r[]])/r[]^2-(3*𝒽$ab$p[-a,-b]*CD1[-c][r[]]*CD1[c][r[]]*CD1[-d][r[]]*CD1[d][r[]])/(2*r[]^2)+(3*g1[-a,-b]*𝒽$ab$p[e,-e]*CD1[-c][r[]]*CD1[c][r[]]*CD1[-d][r[]]*CD1[d][r[]])/(4*r[]^2)+(g1[-a,-b]*𝒽$p[]*CD1[-c][r[]]*CD1[c][r[]]*CD1[-d][r[]]*CD1[d][r[]])/(2*r[]^4)+(2*CD1[-c][r[]]*CD1[c][r[]]*CD1[-d][𝒽$ab$p[-a,-b]]*CD1[d][r[]])/r[]-(3*g1[-a,-b]*CD1[-c][r[]]*CD1[c][r[]]*CD1[-d][𝒽$ab$p[e,-e]]*CD1[d][r[]])/(4*r[])-(g1[-a,-b]*CD1[-c][r[]]*CD1[c][r[]]*CD1[-d][𝒽$p[]]*CD1[d][r[]])/(2*r[]^3)-(g1[-a,-b]*r[]^2*CD1[-d][Τ[e,-e]]*CD1[d][𝒽$ab$p[c,-c]])/12+(g1[-a,-b]*Τ[-c,-d]*CD1[c][r[]]*CD1[d][𝒽$p[]])/(4*r[])-(g1[-a,-b]*r[]*Τ[d,-d]*CD1[c][r[]]*CD1[-e][𝒽$ab$p[-c,e]])/2+(g1[-a,-b]*r[]*Τ[-c,d]*CD1[c][r[]]*CD1[-e][𝒽$ab$p[-d,e]])/2+(3*g1[-a,-b]*CD1[-c][r[]]*CD1[c][r[]]*CD1[d][r[]]*CD1[-e][𝒽$ab$p[-d,e]])/(2*r[])+(g1[-a,-b]*r[]*𝒽$ab$p[d,-d]*CD1[c][r[]]*CD1[-e][Τ[-c,e]])/4+(g1[-a,-b]*r[]^2*𝒽$ab$p[c,-c]*CD1[-e][CD1[e][Τ[d,-d]]])/12+(2*g1[-a,-b]*𝒽$ab$p[-d,-e]*CD1[-c][r[]]*CD1[c][r[]]*CD1[d][r[]]*CD1[e][r[]])/r[]^2},MetricOn->All];

(*{LHS$UComp$PaQb$time,LHS$UComp$PaQb$result}=AbsoluteTiming[LHSstep1=Pert$Exp$Sep[LHS,{μ,ν,ρ,σ},{P,a,Q,b},{ℂ$Harm$Rep,𝒽$Harm$Rep}];
LHSstep2=LHSstep1/.Dir$Harm$Def$Total//ContractMetric//ToCanonical;
LHSstep3=LHSstep2//.Harm$EQ$Y$Total//ContractMetric//ToCanonical;
LHSstep4=UAB$Component[LHSstep3]//ContractMetric//ToCanonical;
LHSstep4
];
Print["Computing Time:"]
LHS$UComp$PaQb$time
Print["Result:"]
LHS$UComp$PaQb$result
{RHS$UComp$PaQb$time,RHS$UComp$PaQb$result}=AbsoluteTiming[RHSstep1=Pert$Exp$Sep[RHS,{μ,ν,ρ,σ},{P,a,Q,b},{ℂ$Harm$Rep,𝒽$Harm$Rep,𝕁$Harm$Rep,𝒯$Harm$Rep}];
RHSstep2=RHSstep1/.Dir$Harm$Def$Total//ContractMetric//ToCanonical;
RHSstep3=RHSstep2//.Harm$EQ$Y$Total//ContractMetric//ToCanonical;
RHSstep4=UAB$Component[RHSstep3]//ContractMetric//ToCanonical;
RHSstep4
];
Print["Computing Time:"]
RHS$UComp$PaQb$time
Print["Result:"]
RHS$UComp$PaQb$result
Print["Match with saved result:"]
Simplify[ToCanonical[(Δℂ$D𝕁$UComp$PaQb[-a,-b]/.Eval$Pert$Δℂ$D𝕁$UComp$PaQb)-(LHS$UComp$PaQb$result-RHS$UComp$PaQb$result)]==0]*)

V-Component of δ((Subscript[Δℂ, P a Q b])-((Subscript[D, P])-i ⋆(Subscript[D, P]))(Subscript[𝕁, a Q b]))=0

Eval$Pert$Δℂ$D𝕁$VComp$PaQb=MakeRule[{Δℂ$D𝕁$VComp$PaQb[-a,-b],((-I)*epsilong1[-a,c]*Π$ab$m[-b,-c])/(1-Λ^2/2)+((I/2)*Λ^2*epsilong1[-a,c]*Π$ab$m[-b,-c])/(1-Λ^2/2)+Π$ab$p[-a,-b]/(1-Λ^2/2)-(Λ^2*Π$ab$p[-a,-b])/(2*(1-Λ^2/2))-(I*g1[-a,-b]*Π$m[])/(1-Λ^2/2)+((I/2)*Λ^2*g1[-a,-b]*Π$m[])/(1-Λ^2/2)+(epsilong1[-a,-b]*Π$p[])/(1-Λ^2/2)-(Λ^2*epsilong1[-a,-b]*Π$p[])/(2*(1-Λ^2/2))+(((3*I)/2)*epsilong1[-b,c]*Ψ$ab$m[-a,-c])/((1-Λ^2/2)*r[]^2)-(((3*I)/4)*Λ^2*epsilong1[-b,c]*Ψ$ab$m[-a,-c])/((1-Λ^2/2)*r[]^2)-((I/2)*epsilong1[-b,c]*𝒫[]*Ψ$ab$m[-a,-c])/(1-Λ^2/2)+((I/4)*Λ^2*epsilong1[-b,c]*𝒫[]*Ψ$ab$m[-a,-c])/(1-Λ^2/2)+((I/2)*epsilong1[-b,c]*Τ[d,-d]*Ψ$ab$m[-a,-c])/(1-Λ^2/2)-((I/4)*Λ^2*epsilong1[-b,c]*Τ[d,-d]*Ψ$ab$m[-a,-c])/(1-Λ^2/2)+(((3*I)/2)*epsilong1[-a,c]*Ψ$ab$m[-b,-c])/((1-Λ^2/2)*r[]^2)-(((3*I)/4)*Λ^2*epsilong1[-a,c]*Ψ$ab$m[-b,-c])/((1-Λ^2/2)*r[]^2)-((I/2)*epsilong1[-a,c]*𝒫[]*Ψ$ab$m[-b,-c])/(1-Λ^2/2)+((I/4)*Λ^2*epsilong1[-a,c]*𝒫[]*Ψ$ab$m[-b,-c])/(1-Λ^2/2)+((I/2)*epsilong1[-a,c]*Τ[d,-d]*Ψ$ab$m[-b,-c])/(1-Λ^2/2)-((I/4)*Λ^2*epsilong1[-a,c]*Τ[d,-d]*Ψ$ab$m[-b,-c])/(1-Λ^2/2)-Ψ$ab$o[-a,-b]/((1-Λ^2/2)*r[]^2)-(Λ^2*Ψ$ab$o[-a,-b])/(2*(1-Λ^2/2)*r[]^2)+(Λ^4*Ψ$ab$o[-a,-b])/(2*(1-Λ^2/2)*r[]^2)+(2*𝒫[]*Ψ$ab$o[-a,-b])/(1-Λ^2/2)-(Λ^2*𝒫[]*Ψ$ab$o[-a,-b])/(1-Λ^2/2)-(Τ[c,-c]*Ψ$ab$o[-a,-b])/(1-Λ^2/2)+(Λ^2*Τ[c,-c]*Ψ$ab$o[-a,-b])/(2*(1-Λ^2/2))-(Τ[-b,-c]*Ψ$ab$o[-a,c])/(2*(1-Λ^2/2))+(Λ^2*Τ[-b,-c]*Ψ$ab$o[-a,c])/(4*(1-Λ^2/2))-(Τ[-a,-c]*Ψ$ab$o[-b,c])/(2*(1-Λ^2/2))+(Λ^2*Τ[-a,-c]*Ψ$ab$o[-b,c])/(4*(1-Λ^2/2))+(3*g1[-a,-b]*Ψ$ab$o[c,-c])/(2*(1-Λ^2/2)*r[]^2)-(3*Λ^2*g1[-a,-b]*Ψ$ab$o[c,-c])/(4*(1-Λ^2/2)*r[]^2)-(g1[-a,-b]*𝒫[]*Ψ$ab$o[c,-c])/(2*(1-Λ^2/2))+(Λ^2*g1[-a,-b]*𝒫[]*Ψ$ab$o[c,-c])/(4*(1-Λ^2/2))+(g1[-a,-b]*Τ[d,-d]*Ψ$ab$o[c,-c])/(2*(1-Λ^2/2))-(Λ^2*g1[-a,-b]*Τ[d,-d]*Ψ$ab$o[c,-c])/(4*(1-Λ^2/2))+(g1[-a,-b]*Τ[-c,-d]*Ψ$ab$o[c,d])/(2*(1-Λ^2/2))-(Λ^2*g1[-a,-b]*Τ[-c,-d]*Ψ$ab$o[c,d])/(4*(1-Λ^2/2))-((3*I)*epsilong1[-b,-c]*𝒽$a$m[c]*CD1[-a][r[]])/((1-Λ^2/2)*r[]^3)+(((3*I)/2)*Λ^2*epsilong1[-b,-c]*𝒽$a$m[c]*CD1[-a][r[]])/((1-Λ^2/2)*r[]^3)+(I*epsilong1[-b,-c]*𝒽$a$m[c]*𝒫[]*CD1[-a][r[]])/((1-Λ^2/2)*r[])-((I/2)*Λ^2*epsilong1[-b,-c]*𝒽$a$m[c]*𝒫[]*CD1[-a][r[]])/((1-Λ^2/2)*r[])+(2*Ξ$a$p[-b]*CD1[-a][r[]])/((1-Λ^2/2)*r[]^3)-(Λ^2*Ξ$a$p[-b]*CD1[-a][r[]])/((1-Λ^2/2)*r[]^3)-(Π$a$o[-b]*CD1[-a][r[]])/((1-Λ^2/2)*r[])+(Λ^2*Π$a$o[-b]*CD1[-a][r[]])/(2*(1-Λ^2/2)*r[])-(I*epsilong1[-b,-c]*𝒽$a$m[c]*Τ[d,-d]*CD1[-a][r[]])/((1-Λ^2/2)*r[])+((I/2)*Λ^2*epsilong1[-b,-c]*𝒽$a$m[c]*Τ[d,-d]*CD1[-a][r[]])/((1-Λ^2/2)*r[])+CD1[-a][Π$a$o[-b]]/(1-Λ^2/2)-(Λ^2*CD1[-a][Π$a$o[-b]])/(2*(1-Λ^2/2))-((3*I)*epsilong1[-a,-c]*𝒽$a$m[c]*CD1[-b][r[]])/((1-Λ^2/2)*r[]^3)+(((3*I)/2)*Λ^2*epsilong1[-a,-c]*𝒽$a$m[c]*CD1[-b][r[]])/((1-Λ^2/2)*r[]^3)+(I*epsilong1[-a,-c]*𝒽$a$m[c]*𝒫[]*CD1[-b][r[]])/((1-Λ^2/2)*r[])-((I/2)*Λ^2*epsilong1[-a,-c]*𝒽$a$m[c]*𝒫[]*CD1[-b][r[]])/((1-Λ^2/2)*r[])+(2*Ξ$a$p[-a]*CD1[-b][r[]])/((1-Λ^2/2)*r[]^3)-(Λ^2*Ξ$a$p[-a]*CD1[-b][r[]])/((1-Λ^2/2)*r[]^3)-(I*epsilong1[-a,-c]*𝒽$a$m[c]*Τ[d,-d]*CD1[-b][r[]])/((1-Λ^2/2)*r[])+((I/2)*Λ^2*epsilong1[-a,-c]*𝒽$a$m[c]*Τ[d,-d]*CD1[-b][r[]])/((1-Λ^2/2)*r[])-((I/3)*epsilong1[-a,-c]*𝒽$a$m[c]*CD1[-b][𝒫[]])/(1-Λ^2/2)+((I/6)*Λ^2*epsilong1[-a,-c]*𝒽$a$m[c]*CD1[-b][𝒫[]])/(1-Λ^2/2)+((I/2)*epsilong1[-a,d]*𝒽$a$m[c]*CD1[-b][Τ[-c,-d]])/(1-Λ^2/2)-((I/4)*Λ^2*epsilong1[-a,d]*𝒽$a$m[c]*CD1[-b][Τ[-c,-d]])/(1-Λ^2/2)-((I/6)*epsilong1[-a,-c]*𝒽$a$m[c]*CD1[-b][Τ[d,-d]])/(1-Λ^2/2)+((I/12)*Λ^2*epsilong1[-a,-c]*𝒽$a$m[c]*CD1[-b][Τ[d,-d]])/(1-Λ^2/2)+((I/3)*epsilong1[-a,-b]*𝒽$a$m[c]*CD1[-c][𝒫[]])/(1-Λ^2/2)-((I/6)*Λ^2*epsilong1[-a,-b]*𝒽$a$m[c]*CD1[-c][𝒫[]])/(1-Λ^2/2)-((I/2)*epsilong1[-a,d]*𝒽$a$m[c]*CD1[-c][Τ[-b,-d]])/(1-Λ^2/2)+((I/4)*Λ^2*epsilong1[-a,d]*𝒽$a$m[c]*CD1[-c][Τ[-b,-d]])/(1-Λ^2/2)+((I/6)*epsilong1[-a,-b]*𝒽$a$m[c]*CD1[-c][Τ[d,-d]])/(1-Λ^2/2)-((I/12)*Λ^2*epsilong1[-a,-b]*𝒽$a$m[c]*CD1[-c][Τ[d,-d]])/(1-Λ^2/2)+CD1[-c][CD1[c][Ψ$ab$o[-a,-b]]]/(1-Λ^2/2)-(Λ^2*CD1[-c][CD1[c][Ψ$ab$o[-a,-b]]])/(2*(1-Λ^2/2))+(I*epsilong1[-a,-c]*Γ$a$m[-b]*CD1[c][r[]])/((1-Λ^2/2)*r[])-((I/2)*Λ^2*epsilong1[-a,-c]*Γ$a$m[-b]*CD1[c][r[]])/((1-Λ^2/2)*r[])-(2*epsilong1[-b,-c]*Ψ$a$p[-a]*CD1[c][r[]])/((1-Λ^2/2)*r[])+(Λ^2*epsilong1[-b,-c]*Ψ$a$p[-a]*CD1[c][r[]])/((1-Λ^2/2)*r[])-(2*epsilong1[-a,-c]*Ψ$a$p[-b]*CD1[c][r[]])/((1-Λ^2/2)*r[])+(Λ^2*epsilong1[-a,-c]*Ψ$a$p[-b]*CD1[c][r[]])/((1-Λ^2/2)*r[])-(((3*I)/2)*epsilong1[-b,d]*Ψ$ab$m[-a,-d]*CD1[-c][r[]]*CD1[c][r[]])/((1-Λ^2/2)*r[]^2)+(((3*I)/4)*Λ^2*epsilong1[-b,d]*Ψ$ab$m[-a,-d]*CD1[-c][r[]]*CD1[c][r[]])/((1-Λ^2/2)*r[]^2)-(((3*I)/2)*epsilong1[-a,d]*Ψ$ab$m[-b,-d]*CD1[-c][r[]]*CD1[c][r[]])/((1-Λ^2/2)*r[]^2)+(((3*I)/4)*Λ^2*epsilong1[-a,d]*Ψ$ab$m[-b,-d]*CD1[-c][r[]]*CD1[c][r[]])/((1-Λ^2/2)*r[]^2)+(5*Ψ$ab$o[-a,-b]*CD1[-c][r[]]*CD1[c][r[]])/((1-Λ^2/2)*r[]^2)-(5*Λ^2*Ψ$ab$o[-a,-b]*CD1[-c][r[]]*CD1[c][r[]])/(2*(1-Λ^2/2)*r[]^2)-(3*g1[-a,-b]*Ψ$ab$o[d,-d]*CD1[-c][r[]]*CD1[c][r[]])/(2*(1-Λ^2/2)*r[]^2)+(3*Λ^2*g1[-a,-b]*Ψ$ab$o[d,-d]*CD1[-c][r[]]*CD1[c][r[]])/(4*(1-Λ^2/2)*r[]^2)-(2*CD1[-c][Ψ$ab$o[-a,-b]]*CD1[c][r[]])/((1-Λ^2/2)*r[])+(Λ^2*CD1[-c][Ψ$ab$o[-a,-b]]*CD1[c][r[]])/((1-Λ^2/2)*r[])-(I*epsilong1[-a,-c]*CD1[c][Γ$a$m[-b]])/(1-Λ^2/2)+((I/2)*Λ^2*epsilong1[-a,-c]*CD1[c][Γ$a$m[-b]])/(1-Λ^2/2)+((3*I)*epsilong1[-b,-c]*𝒽$a$m[c]*CD1[-a][r[]]*CD1[-d][r[]]*CD1[d][r[]])/((1-Λ^2/2)*r[]^3)-(((3*I)/2)*Λ^2*epsilong1[-b,-c]*𝒽$a$m[c]*CD1[-a][r[]]*CD1[-d][r[]]*CD1[d][r[]])/((1-Λ^2/2)*r[]^3)+((3*I)*epsilong1[-a,-c]*𝒽$a$m[c]*CD1[-b][r[]]*CD1[-d][r[]]*CD1[d][r[]])/((1-Λ^2/2)*r[]^3)-(((3*I)/2)*Λ^2*epsilong1[-a,-c]*𝒽$a$m[c]*CD1[-b][r[]]*CD1[-d][r[]]*CD1[d][r[]])/((1-Λ^2/2)*r[]^3)},MetricOn->All];

(*{LHS$VComp$PaQb$time,LHS$VComp$PaQb$result}=AbsoluteTiming[LHSstep1=Pert$Exp$Sep[LHS,{μ,ν,ρ,σ},{P,a,Q,b},{ℂ$Harm$Rep,𝒽$Harm$Rep}];
LHSstep2=LHSstep1/.Dir$Harm$Def$Total//ContractMetric//ToCanonical;
LHSstep3=LHSstep2//.Harm$EQ$Y$Total//ContractMetric//ToCanonical;
LHSstep4=VAB$Component[LHSstep3]//ContractMetric//ToCanonical;
LHSstep4
];
Print["Computing Time:"]
LHS$VComp$PaQb$time
Print["Result:"]
LHS$VComp$PaQb$result
{RHS$VComp$PaQb$time,RHS$VComp$PaQb$result}=AbsoluteTiming[RHSstep1=Pert$Exp$Sep[RHS,{μ,ν,ρ,σ},{P,a,Q,b},{ℂ$Harm$Rep,𝒽$Harm$Rep,𝕁$Harm$Rep,𝒯$Harm$Rep}];
RHSstep2=RHSstep1/.Dir$Harm$Def$Total//ContractMetric//ToCanonical;
RHSstep3=RHSstep2//.Harm$EQ$Y$Total//ContractMetric//ToCanonical;
RHSstep4=VAB$Component[RHSstep3]//ContractMetric//ToCanonical;
RHSstep4
];
Print["Computing Time:"]
RHS$VComp$PaQb$time
Print["Result:"]
RHS$VComp$PaQb$result
Print["Match with saved result:"]
Simplify[ToCanonical[(Δℂ$D𝕁$VComp$PaQb[-a,-b]/.Eval$Pert$Δℂ$D𝕁$VComp$PaQb)-(Simplify[LHS$VComp$PaQb$result-RHS$VComp$PaQb$result])]==0]*)

W-Component of δ((Subscript[Δℂ, P a Q b])-((Subscript[D, P])-i ⋆(Subscript[D, P]))(Subscript[𝕁, a Q b]))=0

Eval$Pert$Δℂ$D𝕁$WComp$PaQb=MakeRule[{Δℂ$D𝕁$WComp$PaQb[-a,-b],Π$ab$m[-a,-b]/(1-Λ^2/2)-(Λ^2*Π$ab$m[-a,-b])/(2*(1-Λ^2/2))+(I*epsilong1[-a,c]*Π$ab$p[-b,-c])/(1-Λ^2/2)-((I/2)*Λ^2*epsilong1[-a,c]*Π$ab$p[-b,-c])/(1-Λ^2/2)+(epsilong1[-a,-b]*Π$m[])/(1-Λ^2/2)-(Λ^2*epsilong1[-a,-b]*Π$m[])/(2*(1-Λ^2/2))+(I*g1[-a,-b]*Π$p[])/(1-Λ^2/2)-((I/2)*Λ^2*g1[-a,-b]*Π$p[])/(1-Λ^2/2)-Ψ$ab$m[-a,-b]/((1-Λ^2/2)*r[]^2)-(Λ^2*Ψ$ab$m[-a,-b])/(2*(1-Λ^2/2)*r[]^2)+(Λ^4*Ψ$ab$m[-a,-b])/(2*(1-Λ^2/2)*r[]^2)+(2*𝒫[]*Ψ$ab$m[-a,-b])/(1-Λ^2/2)-(Λ^2*𝒫[]*Ψ$ab$m[-a,-b])/(1-Λ^2/2)-(Τ[c,-c]*Ψ$ab$m[-a,-b])/(1-Λ^2/2)+(Λ^2*Τ[c,-c]*Ψ$ab$m[-a,-b])/(2*(1-Λ^2/2))-(Τ[-b,-c]*Ψ$ab$m[-a,c])/(2*(1-Λ^2/2))+(Λ^2*Τ[-b,-c]*Ψ$ab$m[-a,c])/(4*(1-Λ^2/2))-(Τ[-a,-c]*Ψ$ab$m[-b,c])/(2*(1-Λ^2/2))+(Λ^2*Τ[-a,-c]*Ψ$ab$m[-b,c])/(4*(1-Λ^2/2))+(3*g1[-a,-b]*Ψ$ab$m[c,-c])/(2*(1-Λ^2/2)*r[]^2)-(3*Λ^2*g1[-a,-b]*Ψ$ab$m[c,-c])/(4*(1-Λ^2/2)*r[]^2)-(g1[-a,-b]*𝒫[]*Ψ$ab$m[c,-c])/(2*(1-Λ^2/2))+(Λ^2*g1[-a,-b]*𝒫[]*Ψ$ab$m[c,-c])/(4*(1-Λ^2/2))+(g1[-a,-b]*Τ[d,-d]*Ψ$ab$m[c,-c])/(2*(1-Λ^2/2))-(Λ^2*g1[-a,-b]*Τ[d,-d]*Ψ$ab$m[c,-c])/(4*(1-Λ^2/2))+(g1[-a,-b]*Τ[-c,-d]*Ψ$ab$m[c,d])/(2*(1-Λ^2/2))-(Λ^2*g1[-a,-b]*Τ[-c,-d]*Ψ$ab$m[c,d])/(4*(1-Λ^2/2))-(((3*I)/2)*epsilong1[-b,c]*Ψ$ab$o[-a,-c])/((1-Λ^2/2)*r[]^2)+(((3*I)/4)*Λ^2*epsilong1[-b,c]*Ψ$ab$o[-a,-c])/((1-Λ^2/2)*r[]^2)+((I/2)*epsilong1[-b,c]*𝒫[]*Ψ$ab$o[-a,-c])/(1-Λ^2/2)-((I/4)*Λ^2*epsilong1[-b,c]*𝒫[]*Ψ$ab$o[-a,-c])/(1-Λ^2/2)-((I/2)*epsilong1[-b,c]*Τ[d,-d]*Ψ$ab$o[-a,-c])/(1-Λ^2/2)+((I/4)*Λ^2*epsilong1[-b,c]*Τ[d,-d]*Ψ$ab$o[-a,-c])/(1-Λ^2/2)-(((3*I)/2)*epsilong1[-a,c]*Ψ$ab$o[-b,-c])/((1-Λ^2/2)*r[]^2)+(((3*I)/4)*Λ^2*epsilong1[-a,c]*Ψ$ab$o[-b,-c])/((1-Λ^2/2)*r[]^2)+((I/2)*epsilong1[-a,c]*𝒫[]*Ψ$ab$o[-b,-c])/(1-Λ^2/2)-((I/4)*Λ^2*epsilong1[-a,c]*𝒫[]*Ψ$ab$o[-b,-c])/(1-Λ^2/2)-((I/2)*epsilong1[-a,c]*Τ[d,-d]*Ψ$ab$o[-b,-c])/(1-Λ^2/2)+((I/4)*Λ^2*epsilong1[-a,c]*Τ[d,-d]*Ψ$ab$o[-b,-c])/(1-Λ^2/2)+(3*𝒽$a$m[-b]*CD1[-a][r[]])/((1-Λ^2/2)*r[]^3)-(3*Λ^2*𝒽$a$m[-b]*CD1[-a][r[]])/(2*(1-Λ^2/2)*r[]^3)-(𝒽$a$m[-b]*𝒫[]*CD1[-a][r[]])/((1-Λ^2/2)*r[])+(Λ^2*𝒽$a$m[-b]*𝒫[]*CD1[-a][r[]])/(2*(1-Λ^2/2)*r[])-(Γ$a$m[-b]*CD1[-a][r[]])/((1-Λ^2/2)*r[])+(Λ^2*Γ$a$m[-b]*CD1[-a][r[]])/(2*(1-Λ^2/2)*r[])-(2*Ξ$a$m[-b]*CD1[-a][r[]])/((1-Λ^2/2)*r[]^3)+(Λ^2*Ξ$a$m[-b]*CD1[-a][r[]])/((1-Λ^2/2)*r[]^3)+(𝒽$a$m[-b]*Τ[c,-c]*CD1[-a][r[]])/((1-Λ^2/2)*r[])-(Λ^2*𝒽$a$m[-b]*Τ[c,-c]*CD1[-a][r[]])/(2*(1-Λ^2/2)*r[])+CD1[-a][Γ$a$m[-b]]/(1-Λ^2/2)-(Λ^2*CD1[-a][Γ$a$m[-b]])/(2*(1-Λ^2/2))+(3*𝒽$a$m[-a]*CD1[-b][r[]])/((1-Λ^2/2)*r[]^3)-(3*Λ^2*𝒽$a$m[-a]*CD1[-b][r[]])/(2*(1-Λ^2/2)*r[]^3)-(𝒽$a$m[-a]*𝒫[]*CD1[-b][r[]])/((1-Λ^2/2)*r[])+(Λ^2*𝒽$a$m[-a]*𝒫[]*CD1[-b][r[]])/(2*(1-Λ^2/2)*r[])-(2*Ξ$a$m[-a]*CD1[-b][r[]])/((1-Λ^2/2)*r[]^3)+(Λ^2*Ξ$a$m[-a]*CD1[-b][r[]])/((1-Λ^2/2)*r[]^3)+(𝒽$a$m[-a]*Τ[c,-c]*CD1[-b][r[]])/((1-Λ^2/2)*r[])-(Λ^2*𝒽$a$m[-a]*Τ[c,-c]*CD1[-b][r[]])/(2*(1-Λ^2/2)*r[])+(𝒽$a$m[-a]*CD1[-b][𝒫[]])/(3*(1-Λ^2/2))-(Λ^2*𝒽$a$m[-a]*CD1[-b][𝒫[]])/(6*(1-Λ^2/2))-(𝒽$a$m[c]*CD1[-b][Τ[-a,-c]])/(2*(1-Λ^2/2))+(Λ^2*𝒽$a$m[c]*CD1[-b][Τ[-a,-c]])/(4*(1-Λ^2/2))+(𝒽$a$m[-a]*CD1[-b][Τ[c,-c]])/(6*(1-Λ^2/2))-(Λ^2*𝒽$a$m[-a]*CD1[-b][Τ[c,-c]])/(12*(1-Λ^2/2))-(4*g1[-a,-b]*𝒽$a$m[c]*CD1[-c][r[]])/((1-Λ^2/2)*r[]^3)+(2*Λ^2*g1[-a,-b]*𝒽$a$m[c]*CD1[-c][r[]])/((1-Λ^2/2)*r[]^3)+(4*g1[-a,-b]*𝒽$a$m[c]*𝒫[]*CD1[-c][r[]])/(3*(1-Λ^2/2)*r[])-(2*Λ^2*g1[-a,-b]*𝒽$a$m[c]*𝒫[]*CD1[-c][r[]])/(3*(1-Λ^2/2)*r[])-(4*g1[-a,-b]*𝒽$a$m[c]*Τ[d,-d]*CD1[-c][r[]])/(3*(1-Λ^2/2)*r[])+(2*Λ^2*g1[-a,-b]*𝒽$a$m[c]*Τ[d,-d]*CD1[-c][r[]])/(3*(1-Λ^2/2)*r[])-(g1[-a,-b]*𝒽$a$m[c]*CD1[-c][𝒫[]])/(3*(1-Λ^2/2))+(Λ^2*g1[-a,-b]*𝒽$a$m[c]*CD1[-c][𝒫[]])/(6*(1-Λ^2/2))+(𝒽$a$m[c]*CD1[-c][Τ[-a,-b]])/(2*(1-Λ^2/2))-(Λ^2*𝒽$a$m[c]*CD1[-c][Τ[-a,-b]])/(4*(1-Λ^2/2))-(g1[-a,-b]*𝒽$a$m[c]*CD1[-c][Τ[d,-d]])/(6*(1-Λ^2/2))+(Λ^2*g1[-a,-b]*𝒽$a$m[c]*CD1[-c][Τ[d,-d]])/(12*(1-Λ^2/2))+CD1[-c][CD1[c][Ψ$ab$m[-a,-b]]]/(1-Λ^2/2)-(Λ^2*CD1[-c][CD1[c][Ψ$ab$m[-a,-b]]])/(2*(1-Λ^2/2))-(I*epsilong1[-a,-c]*Π$a$o[-b]*CD1[c][r[]])/((1-Λ^2/2)*r[])+((I/2)*Λ^2*epsilong1[-a,-c]*Π$a$o[-b]*CD1[c][r[]])/((1-Λ^2/2)*r[])-(2*epsilong1[-b,-c]*Ψ$a$m[-a]*CD1[c][r[]])/((1-Λ^2/2)*r[])+(Λ^2*epsilong1[-b,-c]*Ψ$a$m[-a]*CD1[c][r[]])/((1-Λ^2/2)*r[])-(2*epsilong1[-a,-c]*Ψ$a$m[-b]*CD1[c][r[]])/((1-Λ^2/2)*r[])+(Λ^2*epsilong1[-a,-c]*Ψ$a$m[-b]*CD1[c][r[]])/((1-Λ^2/2)*r[])+(5*Ψ$ab$m[-a,-b]*CD1[-c][r[]]*CD1[c][r[]])/((1-Λ^2/2)*r[]^2)-(5*Λ^2*Ψ$ab$m[-a,-b]*CD1[-c][r[]]*CD1[c][r[]])/(2*(1-Λ^2/2)*r[]^2)-(3*g1[-a,-b]*Ψ$ab$m[d,-d]*CD1[-c][r[]]*CD1[c][r[]])/(2*(1-Λ^2/2)*r[]^2)+(3*Λ^2*g1[-a,-b]*Ψ$ab$m[d,-d]*CD1[-c][r[]]*CD1[c][r[]])/(4*(1-Λ^2/2)*r[]^2)+(((3*I)/2)*epsilong1[-b,d]*Ψ$ab$o[-a,-d]*CD1[-c][r[]]*CD1[c][r[]])/((1-Λ^2/2)*r[]^2)-(((3*I)/4)*Λ^2*epsilong1[-b,d]*Ψ$ab$o[-a,-d]*CD1[-c][r[]]*CD1[c][r[]])/((1-Λ^2/2)*r[]^2)+(((3*I)/2)*epsilong1[-a,d]*Ψ$ab$o[-b,-d]*CD1[-c][r[]]*CD1[c][r[]])/((1-Λ^2/2)*r[]^2)-(((3*I)/4)*Λ^2*epsilong1[-a,d]*Ψ$ab$o[-b,-d]*CD1[-c][r[]]*CD1[c][r[]])/((1-Λ^2/2)*r[]^2)-(3*𝒽$a$m[-b]*CD1[-a][r[]]*CD1[-c][r[]]*CD1[c][r[]])/((1-Λ^2/2)*r[]^3)+(3*Λ^2*𝒽$a$m[-b]*CD1[-a][r[]]*CD1[-c][r[]]*CD1[c][r[]])/(2*(1-Λ^2/2)*r[]^3)-(3*𝒽$a$m[-a]*CD1[-b][r[]]*CD1[-c][r[]]*CD1[c][r[]])/((1-Λ^2/2)*r[]^3)+(3*Λ^2*𝒽$a$m[-a]*CD1[-b][r[]]*CD1[-c][r[]]*CD1[c][r[]])/(2*(1-Λ^2/2)*r[]^3)-(2*CD1[-c][Ψ$ab$m[-a,-b]]*CD1[c][r[]])/((1-Λ^2/2)*r[])+(Λ^2*CD1[-c][Ψ$ab$m[-a,-b]]*CD1[c][r[]])/((1-Λ^2/2)*r[])+(I*epsilong1[-a,-c]*CD1[c][Π$a$o[-b]])/(1-Λ^2/2)-((I/2)*Λ^2*epsilong1[-a,-c]*CD1[c][Π$a$o[-b]])/(1-Λ^2/2)+(4*g1[-a,-b]*𝒽$a$m[c]*CD1[-c][r[]]*CD1[-d][r[]]*CD1[d][r[]])/((1-Λ^2/2)*r[]^3)-(2*Λ^2*g1[-a,-b]*𝒽$a$m[c]*CD1[-c][r[]]*CD1[-d][r[]]*CD1[d][r[]])/((1-Λ^2/2)*r[]^3)},MetricOn->All];

(*{LHS$WComp$PaQb$time,LHS$WComp$PaQb$result}=AbsoluteTiming[LHSstep1=Pert$Exp$Sep[LHS,{μ,ν,ρ,σ},{P,a,Q,b},{ℂ$Harm$Rep,𝒽$Harm$Rep}];
LHSstep2=LHSstep1/.Dir$Harm$Def$Total//ContractMetric//ToCanonical;
LHSstep3=LHSstep2//.Harm$EQ$Y$Total//ContractMetric//ToCanonical;
LHSstep4=WAB$Component[LHSstep3]//ContractMetric//ToCanonical;
LHSstep4
];
Print["Computing Time:"]
LHS$WComp$PaQb$time
Print["Result:"]
LHS$WComp$PaQb$result
{RHS$WComp$PaQb$time,RHS$WComp$PaQb$result}=AbsoluteTiming[RHSstep1=Pert$Exp$Sep[RHS,{μ,ν,ρ,σ},{P,a,Q,b},{ℂ$Harm$Rep,𝒽$Harm$Rep,𝕁$Harm$Rep,𝒯$Harm$Rep}];
RHSstep2=RHSstep1/.Dir$Harm$Def$Total//ContractMetric//ToCanonical;
RHSstep3=RHSstep2//.Harm$EQ$Y$Total//ContractMetric//ToCanonical;
RHSstep4=WAB$Component[RHSstep3]//ContractMetric//ToCanonical;
RHSstep4
];
Print["Computing Time:"]
RHS$WComp$PaQb$time
Print["Result:"]
RHS$WComp$PaQb$result
Print["Match with saved result:"]
Simplify[ToCanonical[(Δℂ$D𝕁$WComp$PaQb[-a,-b]/.Eval$Pert$Δℂ$D𝕁$WComp$PaQb)-(LHS$WComp$PaQb$result-RHS$WComp$PaQb$result)]==0]*)

ϵ-Component of δ((Subscript[Δℂ, P a Q b])-((Subscript[D, P])-i ⋆(Subscript[D, P]))(Subscript[𝕁, a Q b]))=0

Eval$Pert$Δℂ$D𝕁$ϵCompSkew$PaQb=MakeRule[{Δℂ$D𝕁$ϵCompSkew$PaQb[],((-1/4*I)*Λ^2*𝒽$ab$p[a,-a])/r[]^2+((I/2)*𝒽$p[])/r[]^4-((I/2)*Λ^2*𝒽$p[])/r[]^4-(I/2)*𝒽$ab$p[a,-a]*𝒫[]+(I/12)*Λ^2*𝒽$ab$p[a,-a]*𝒫[]-((I/6)*𝒽$p[]*𝒫[])/r[]^2+((I/6)*Λ^2*𝒽$p[]*𝒫[])/r[]^2+(I/12)*r[]^2*𝒽$ab$p[a,-a]*𝒫[]^2-(I/4)*Λ^2*Π$ab$p[a,-a]+(Λ^2*Π$m[])/2+((9*I)/8)*𝒽$ab$p[a,b]*Τ[-a,-b]-((5*I)/24)*r[]^2*𝒽$ab$p[a,b]*𝒫[]*Τ[-a,-b]+((I/6)*𝒽$p[]*Τ[a,-a])/r[]^2-((I/6)*Λ^2*𝒽$p[]*Τ[a,-a])/r[]^2-(I/8)*r[]^2*𝒽$ab$p[a,b]*Τ[-a,c]*Τ[-b,-c]-((7*I)/16)*𝒽$ab$p[a,-a]*Τ[b,-b]-(I/12)*Λ^2*𝒽$ab$p[a,-a]*Τ[b,-b]-((5*I)/48)*r[]^2*𝒽$ab$p[a,-a]*𝒫[]*Τ[b,-b]-(I/16)*r[]^2*𝒽$ab$p[a,-a]*Τ[-b,-c]*Τ[b,c]+((11*I)/24)*r[]^2*𝒽$ab$p[a,b]*Τ[-a,-b]*Τ[c,-c]-((5*I)/48)*r[]^2*𝒽$ab$p[a,-a]*Τ[b,-b]*Τ[c,-c]-(I/4)*Φ$ab$p[a,-a]+(I/12)*r[]^2*𝒫[]*Φ$ab$p[a,-a]-(I/12)*r[]^2*Τ[b,-b]*Φ$ab$p[a,-a]-((I/2)*Φ$p[])/r[]^2+(I/6)*𝒫[]*Φ$p[]-(I/6)*Τ[a,-a]*Φ$p[]+(((3*I)/2)*Ψ$ab$p[a,-a])/r[]^2-(I/2)*𝒫[]*Ψ$ab$p[a,-a]+(I/2)*Τ[b,-b]*Ψ$ab$p[a,-a]+Ψ$m[]/(2*r[]^2)-(Λ^2*Ψ$m[])/(2*r[]^2)+(Τ[a,-a]*Ψ$m[])/4-((I/2)*Γ$a$p[a]*CD1[-a][r[]])/r[]-(Λ^2*Ψ$a$m[a]*CD1[-a][r[]])/r[]+(I/2)*CD1[-a][Γ$a$p[a]]+((I/2)*CD1[-a][CD1[a][𝒽$p[]]])/r[]^2-(I/6)*𝒫[]*CD1[-a][CD1[a][𝒽$p[]]]+CD1[-a][CD1[a][Ψ$m[]]]/2+((I/4)*Λ^2*𝒽$ab$p[b,-b]*CD1[-a][r[]]*CD1[a][r[]])/r[]^2-((3*I)*𝒽$p[]*CD1[-a][r[]]*CD1[a][r[]])/r[]^4+((I/2)*Λ^2*𝒽$p[]*CD1[-a][r[]]*CD1[a][r[]])/r[]^4+(I/2)*𝒽$ab$p[b,-b]*𝒫[]*CD1[-a][r[]]*CD1[a][r[]]+(((4*I)/3)*𝒽$p[]*𝒫[]*CD1[-a][r[]]*CD1[a][r[]])/r[]^2-((9*I)/8)*𝒽$ab$p[b,c]*Τ[-b,-c]*CD1[-a][r[]]*CD1[a][r[]]-(((5*I)/6)*𝒽$p[]*Τ[b,-b]*CD1[-a][r[]]*CD1[a][r[]])/r[]^2+((7*I)/16)*𝒽$ab$p[b,-b]*Τ[c,-c]*CD1[-a][r[]]*CD1[a][r[]]+(I/4)*Φ$ab$p[b,-b]*CD1[-a][r[]]*CD1[a][r[]]+((I/2)*Φ$p[]*CD1[-a][r[]]*CD1[a][r[]])/r[]^2-(((3*I)/2)*Ψ$ab$p[b,-b]*CD1[-a][r[]]*CD1[a][r[]])/r[]^2-(5*Ψ$m[]*CD1[-a][r[]]*CD1[a][r[]])/(2*r[]^2)-((I/4)*CD1[-a][𝒽$ab$p[b,-b]]*CD1[a][r[]])/r[]-((7*I)/24)*r[]*𝒫[]*CD1[-a][𝒽$ab$p[b,-b]]*CD1[a][r[]]-(I/12)*r[]*Τ[b,-b]*CD1[-a][𝒽$ab$p[c,-c]]*CD1[a][r[]]+((I/2)*CD1[-a][𝒽$p[]]*CD1[a][r[]])/r[]^3-(((5*I)/12)*𝒫[]*CD1[-a][𝒽$p[]]*CD1[a][r[]])/r[]+((I/6)*Τ[b,-b]*CD1[-a][𝒽$p[]]*CD1[a][r[]])/r[]-(I/4)*r[]*𝒽$ab$p[b,-b]*CD1[-a][𝒫[]]*CD1[a][r[]]+(I/4)*r[]*𝒽$ab$p[b,c]*CD1[-a][Τ[-b,-c]]*CD1[a][r[]]-(I/8)*r[]*𝒽$ab$p[b,-b]*CD1[-a][Τ[c,-c]]*CD1[a][r[]]-(CD1[-a][Ψ$m[]]*CD1[a][r[]])/r[]+(I/12)*CD1[-a][𝒫[]]*CD1[a][𝒽$p[]]-(I/12)*CD1[-a][Τ[b,-b]]*CD1[a][𝒽$p[]]-(I/8)*r[]^2*CD1[-a][𝒽$ab$p[b,-b]]*CD1[a][𝒫[]]-(((3*I)/2)*CD1[a][r[]]*CD1[-b][𝒽$ab$p[-a,b]])/r[]+(I/4)*r[]*𝒫[]*CD1[a][r[]]*CD1[-b][𝒽$ab$p[-a,b]]-(I/12)*r[]^2*CD1[a][𝒫[]]*CD1[-b][𝒽$ab$p[-a,b]]+((3*I)/8)*r[]*Τ[-a,b]*CD1[a][r[]]*CD1[-b][𝒽$ab$p[c,-c]]-((I/4)*𝒽$p[]*CD1[a][r[]]*CD1[-b][Τ[-a,b]])/r[]+(I/4)*CD1[a][𝒽$p[]]*CD1[-b][Τ[-a,b]]-(Λ^2*epsilong1[-a,b]*𝒽$a$m[a]*CD1[-b][Τ[c,-c]])/6+(I/12)*r[]^2*CD1[-a][𝒽$ab$p[a,b]]*CD1[-b][Τ[c,-c]]-(I/12)*r[]^2*𝒽$ab$p[a,b]*CD1[-b][CD1[-a][𝒫[]]]+(I/12)*r[]^2*𝒽$ab$p[a,b]*CD1[-b][CD1[-a][Τ[c,-c]]]+(I/4)*CD1[-b][CD1[b][𝒽$ab$p[a,-a]]]-(I/12)*r[]^2*𝒫[]*CD1[-b][CD1[b][𝒽$ab$p[a,-a]]]+(I/6)*Τ[a,-a]*CD1[-b][CD1[b][𝒽$p[]]]-((I/2)*CD1[-a][r[]]*CD1[a][r[]]*CD1[-b][CD1[b][𝒽$p[]]])/r[]^2-(I/24)*r[]^2*𝒽$ab$p[a,-a]*CD1[-b][CD1[b][𝒫[]]]+(3*Λ^2*epsilong1[-a,-b]*𝒽$a$m[a]*CD1[b][r[]])/(2*r[]^3)-(Λ^2*epsilong1[-a,-b]*𝒽$a$m[a]*𝒫[]*CD1[b][r[]])/(4*r[])+(Λ^2*epsilong1[-a,-b]*Ξ$a$m[a]*CD1[b][r[]])/r[]^3-(3*epsilong1[-a,-b]*Π$a$m[a]*CD1[b][r[]])/(2*r[])+(I/2)*epsilong1[-a,-b]*r[]*Π$a$p[a]*CD1[b][r[]]-(Λ^2*epsilong1[-a,c]*𝒽$a$m[a]*Τ[-b,-c]*CD1[b][r[]])/(4*r[])+(Λ^2*epsilong1[-a,-b]*𝒽$a$m[a]*Τ[c,-c]*CD1[b][r[]])/(2*r[])+I*𝒽$ab$p[-a,-b]*𝒫[]*CD1[a][r[]]*CD1[b][r[]]-((I/2)*𝒽$p[]*Τ[-a,-b]*CD1[a][r[]]*CD1[b][r[]])/r[]^2-I*𝒽$ab$p[-a,c]*Τ[-b,-c]*CD1[a][r[]]*CD1[b][r[]]+(((5*I)/2)*𝒽$p[]*CD1[-a][r[]]*CD1[a][r[]]*CD1[-b][r[]]*CD1[b][r[]])/r[]^4+((I/4)*CD1[-a][r[]]*CD1[a][r[]]*CD1[-b][𝒽$ab$p[c,-c]]*CD1[b][r[]])/r[]-((I/2)*CD1[-a][r[]]*CD1[a][r[]]*CD1[-b][𝒽$p[]]*CD1[b][r[]])/r[]^3+(I/8)*r[]^2*CD1[-b][Τ[c,-c]]*CD1[b][𝒽$ab$p[a,-a]]+((I/4)*Τ[-a,-b]*CD1[a][r[]]*CD1[b][𝒽$p[]])/r[]+(Λ^2*epsilong1[-a,-b]*𝒽$a$m[a]*CD1[b][𝒫[]])/6+(epsilong1[-a,-b]*CD1[b][Π$a$m[a]])/2-(I/2)*r[]*Τ[b,-b]*CD1[a][r[]]*CD1[-c][𝒽$ab$p[-a,c]]+(I/4)*r[]*Τ[-a,b]*CD1[a][r[]]*CD1[-c][𝒽$ab$p[-b,c]]+(((3*I)/2)*CD1[-a][r[]]*CD1[a][r[]]*CD1[b][r[]]*CD1[-c][𝒽$ab$p[-b,c]])/r[]+(I/4)*r[]*𝒽$ab$p[b,-b]*CD1[a][r[]]*CD1[-c][Τ[-a,c]]+(Λ^2*epsilong1[-a,b]*𝒽$a$m[a]*CD1[-c][Τ[-b,c]])/8-(I/4)*r[]*𝒽$ab$p[-a,b]*CD1[a][r[]]*CD1[-c][Τ[-b,c]]+(I/12)*r[]^2*Τ[a,-a]*CD1[-c][CD1[c][𝒽$ab$p[b,-b]]]-(I/4)*CD1[-a][r[]]*CD1[a][r[]]*CD1[-c][CD1[c][𝒽$ab$p[b,-b]]]+(I/24)*r[]^2*𝒽$ab$p[a,-a]*CD1[-c][CD1[c][Τ[b,-b]]]-(3*Λ^2*epsilong1[-a,-c]*𝒽$a$m[a]*CD1[-b][r[]]*CD1[b][r[]]*CD1[c][r[]])/(2*r[]^3)},MetricOn->All];
Eval$Pert$Δℂ$D𝕁$ϵComp$PaQb=MakeRule[{Δℂ$D𝕁$ϵComp$PaQb[-a,-b],(((-3*I)/4)*epsilong1[-b,c]*𝒽$ab$p[-a,-c])/r[]^2+((I/4)*Λ^2*epsilong1[-b,c]*𝒽$ab$p[-a,-c])/r[]^2+(((3*I)/4)*epsilong1[-a,c]*𝒽$ab$p[-b,-c])/r[]^2-((I/4)*Λ^2*epsilong1[-a,c]*𝒽$ab$p[-b,-c])/r[]^2-(((3*I)/4)*epsilong1[-a,-b]*𝒽$ab$p[c,-c])/r[]^2+((I/2)*epsilong1[-a,-b]*𝒽$p[])/r[]^4-((I/2)*Λ^2*epsilong1[-a,-b]*𝒽$p[])/r[]^4+(I/2)*epsilong1[-b,c]*𝒽$ab$p[-a,-c]*𝒫[]-(I/12)*Λ^2*epsilong1[-b,c]*𝒽$ab$p[-a,-c]*𝒫[]-(I/4)*epsilong1[-a,c]*𝒽$ab$p[-b,-c]*𝒫[]+(I/12)*Λ^2*epsilong1[-a,c]*𝒽$ab$p[-b,-c]*𝒫[]-(I/8)*epsilong1[-a,-b]*𝒽$ab$p[c,-c]*𝒫[]-((I/6)*epsilong1[-a,-b]*𝒽$p[]*𝒫[])/r[]^2+((I/6)*Λ^2*epsilong1[-a,-b]*𝒽$p[]*𝒫[])/r[]^2-(I/12)*epsilong1[-b,c]*r[]^2*𝒽$ab$p[-a,-c]*𝒫[]^2+(I/12)*epsilong1[-a,c]*r[]^2*𝒽$ab$p[-b,-c]*𝒫[]^2+(Λ^2*Π$ab$m[-a,-b])/2-(I/2)*Λ^2*epsilong1[-a,c]*Π$ab$p[-b,-c]+(Λ^2*epsilong1[-a,-b]*Π$m[])/2-(I/2)*Λ^2*g1[-a,-b]*Π$p[]+(I/4)*epsilong1[-b,c]*𝒽$ab$p[-c,d]*Τ[-a,-d]-(I/12)*epsilong1[-b,c]*r[]^2*𝒽$ab$p[-c,d]*𝒫[]*Τ[-a,-d]+(I/8)*epsilong1[-a,c]*𝒽$ab$p[d,-d]*Τ[-b,-c]+(I/8)*epsilong1[-a,c]*r[]^2*𝒽$ab$p[d,-d]*𝒫[]*Τ[-b,-c]-(I/2)*epsilong1[-a,c]*𝒽$ab$p[-c,d]*Τ[-b,-d]-(I/6)*epsilong1[-a,c]*r[]^2*𝒽$ab$p[-c,d]*𝒫[]*Τ[-b,-d]+((3*I)/2)*epsilong1[-a,-b]*𝒽$ab$p[c,d]*Τ[-c,-d]-(I/6)*epsilong1[-a,-b]*r[]^2*𝒽$ab$p[c,d]*𝒫[]*Τ[-c,-d]-(I/8)*epsilong1[-a,c]*r[]^2*𝒽$ab$p[d,-d]*Τ[-b,e]*Τ[-c,-e]+((I/6)*epsilong1[-a,-b]*𝒽$p[]*Τ[c,-c])/r[]^2-((I/6)*Λ^2*epsilong1[-a,-b]*𝒽$p[]*Τ[c,-c])/r[]^2+(I/4)*epsilong1[-a,c]*r[]^2*𝒽$ab$p[-c,d]*Τ[-b,e]*Τ[-d,-e]-(I/4)*epsilong1[-a,-b]*r[]^2*𝒽$ab$p[c,d]*Τ[-c,e]*Τ[-d,-e]-(I/2)*epsilong1[-b,c]*𝒽$ab$p[-a,-c]*Τ[d,-d]+(I/12)*Λ^2*epsilong1[-b,c]*𝒽$ab$p[-a,-c]*Τ[d,-d]+(I/2)*epsilong1[-a,c]*𝒽$ab$p[-b,-c]*Τ[d,-d]-(I/12)*Λ^2*epsilong1[-a,c]*𝒽$ab$p[-b,-c]*Τ[d,-d]-I*epsilong1[-a,-b]*𝒽$ab$p[c,-c]*Τ[d,-d]+(I/6)*epsilong1[-b,c]*r[]^2*𝒽$ab$p[-a,-c]*𝒫[]*Τ[d,-d]+(I/12)*epsilong1[-a,c]*r[]^2*𝒽$ab$p[-b,-c]*𝒫[]*Τ[d,-d]-(I/8)*epsilong1[-a,-b]*r[]^2*𝒽$ab$p[c,-c]*𝒫[]*Τ[d,-d]+(I/12)*epsilong1[-b,c]*r[]^2*𝒽$ab$p[-c,d]*Τ[-a,-d]*Τ[e,-e]+(I/8)*epsilong1[-a,c]*r[]^2*𝒽$ab$p[d,-d]*Τ[-b,-c]*Τ[e,-e]-(I/3)*epsilong1[-a,c]*r[]^2*𝒽$ab$p[-c,d]*Τ[-b,-d]*Τ[e,-e]+((2*I)/3)*epsilong1[-a,-b]*r[]^2*𝒽$ab$p[c,d]*Τ[-c,-d]*Τ[e,-e]-(I/12)*epsilong1[-b,c]*r[]^2*𝒽$ab$p[-a,-c]*Τ[d,-d]*Τ[e,-e]+(I/12)*epsilong1[-a,c]*r[]^2*𝒽$ab$p[-b,-c]*Τ[d,-d]*Τ[e,-e]-(I/4)*epsilong1[-a,-b]*r[]^2*𝒽$ab$p[c,-c]*Τ[d,-d]*Τ[e,-e]-(I/4)*epsilong1[-b,c]*Φ$ab$p[-a,-c]+(I/12)*epsilong1[-b,c]*r[]^2*𝒫[]*Φ$ab$p[-a,-c]-(I/12)*epsilong1[-b,c]*r[]^2*Τ[d,-d]*Φ$ab$p[-a,-c]+(I/4)*epsilong1[-a,c]*Φ$ab$p[-b,-c]-(I/12)*epsilong1[-a,c]*r[]^2*𝒫[]*Φ$ab$p[-b,-c]+(I/12)*epsilong1[-a,c]*r[]^2*Τ[d,-d]*Φ$ab$p[-b,-c]-(I/2)*epsilong1[-a,-b]*Φ$ab$p[c,-c]+(I/6)*epsilong1[-a,-b]*r[]^2*𝒫[]*Φ$ab$p[c,-c]-(I/6)*epsilong1[-a,-b]*r[]^2*Τ[d,-d]*Φ$ab$p[c,-c]-((I/2)*epsilong1[-a,-b]*Φ$p[])/r[]^2+(I/6)*epsilong1[-a,-b]*𝒫[]*Φ$p[]-(I/6)*epsilong1[-a,-b]*Τ[c,-c]*Φ$p[]-(((3*I)/2)*epsilong1[-b,c]*Ψ$ab$p[-a,-c])/r[]^2+(I/2)*epsilong1[-b,c]*𝒫[]*Ψ$ab$p[-a,-c]-(I/2)*epsilong1[-b,c]*Τ[d,-d]*Ψ$ab$p[-a,-c]+(((3*I)/2)*epsilong1[-a,c]*Ψ$ab$p[-b,-c])/r[]^2-(I/2)*epsilong1[-a,c]*𝒫[]*Ψ$ab$p[-b,-c]+(I/2)*epsilong1[-a,c]*Τ[d,-d]*Ψ$ab$p[-b,-c]+(epsilong1[-a,-b]*Ψ$m[])/(2*r[]^2)-(Λ^2*epsilong1[-a,-b]*Ψ$m[])/(2*r[]^2)+(epsilong1[-b,c]*Τ[-a,-c]*Ψ$m[])/4-(epsilong1[-a,c]*Τ[-b,-c]*Ψ$m[])/4+(epsilong1[-a,-b]*Τ[c,-c]*Ψ$m[])/2+(3*Λ^2*𝒽$a$m[-b]*CD1[-a][r[]])/(2*r[]^3)-(Λ^2*𝒽$a$m[-b]*𝒫[]*CD1[-a][r[]])/(4*r[])+(Λ^2*Ξ$a$m[-b]*CD1[-a][r[]])/r[]^3-(Π$a$m[-b]*CD1[-a][r[]])/r[]+I*r[]*Π$a$p[-b]*CD1[-a][r[]]+(Λ^2*𝒽$a$m[-b]*Τ[c,-c]*CD1[-a][r[]])/(2*r[])+(Λ^2*𝒽$a$m[-b]*CD1[-a][𝒫[]])/4+CD1[-a][Π$a$m[-b]]-(Λ^2*𝒽$a$m[-b]*CD1[-a][Τ[c,-c]])/4-(3*Λ^2*𝒽$a$m[-a]*CD1[-b][r[]])/(2*r[]^3)+(Λ^2*𝒽$a$m[-a]*𝒫[]*CD1[-b][r[]])/(4*r[])-(Λ^2*Ξ$a$m[-a]*CD1[-b][r[]])/r[]^3+(2*Π$a$m[-a]*CD1[-b][r[]])/r[]-(Λ^2*𝒽$a$m[-a]*Τ[c,-c]*CD1[-b][r[]])/(2*r[])-(Λ^2*𝒽$a$m[-a]*CD1[-b][𝒫[]])/12+(Λ^2*𝒽$a$m[-a]*CD1[-b][Τ[c,-c]])/12-(Λ^2*g1[-a,-b]*𝒽$a$m[c]*𝒫[]*CD1[-c][r[]])/(4*r[])-I*g1[-a,-b]*r[]*Π$a$p[c]*CD1[-c][r[]]-(Λ^2*g1[-a,-b]*𝒽$a$m[c]*CD1[-c][𝒫[]])/12+(Λ^2*𝒽$a$m[-b]*CD1[-c][Τ[-a,c]])/4+(Λ^2*g1[-a,-b]*𝒽$a$m[c]*CD1[-c][Τ[d,-d]])/12-((I/6)*epsilong1[-b,c]*𝒽$p[]*CD1[-a][r[]]*CD1[-c][Τ[d,-d]])/r[]+(I/12)*epsilong1[-b,c]*CD1[-a][𝒽$p[]]*CD1[-c][Τ[d,-d]]-((I/6)*epsilong1[-a,c]*𝒽$p[]*CD1[-b][r[]]*CD1[-c][Τ[d,-d]])/r[]+(I/12)*epsilong1[-a,c]*CD1[-b][𝒽$p[]]*CD1[-c][Τ[d,-d]]-(I/12)*epsilong1[-a,c]*r[]^2*𝒽$ab$p[d,-d]*CD1[-c][CD1[-b][𝒫[]]]+(I/12)*epsilong1[-a,c]*r[]^2*𝒽$ab$p[d,-d]*CD1[-c][CD1[-b][Τ[e,-e]]]+((I/2)*epsilong1[-a,-b]*CD1[-c][CD1[c][𝒽$p[]]])/r[]^2-(I/6)*epsilong1[-a,-b]*𝒫[]*CD1[-c][CD1[c][𝒽$p[]]]+(epsilong1[-a,-b]*CD1[-c][CD1[c][Ψ$m[]]])/2-(I*epsilong1[-a,-c]*Γ$a$p[-b]*CD1[c][r[]])/r[]-(Λ^2*𝒽$a$m[-b]*Τ[-a,-c]*CD1[c][r[]])/(4*r[])+(Λ^2*𝒽$a$m[-a]*Τ[-b,-c]*CD1[c][r[]])/(4*r[])+(Λ^2*epsilong1[-b,-c]*Ψ$a$m[-a]*CD1[c][r[]])/r[]-(Λ^2*epsilong1[-a,-c]*Ψ$a$m[-b]*CD1[c][r[]])/r[]+((2*I)*epsilong1[-b,d]*𝒽$ab$p[-c,-d]*CD1[-a][r[]]*CD1[c][r[]])/r[]^2+((4*I)*epsilong1[-b,-c]*𝒽$p[]*CD1[-a][r[]]*CD1[c][r[]])/r[]^4-((2*I)/3)*epsilong1[-b,d]*𝒽$ab$p[-c,-d]*𝒫[]*CD1[-a][r[]]*CD1[c][r[]]-(((5*I)/6)*epsilong1[-b,-c]*𝒽$p[]*𝒫[]*CD1[-a][r[]]*CD1[c][r[]])/r[]^2-((I/2)*epsilong1[-b,d]*𝒽$p[]*Τ[-c,-d]*CD1[-a][r[]]*CD1[c][r[]])/r[]^2+(((4*I)/3)*epsilong1[-b,-c]*𝒽$p[]*Τ[d,-d]*CD1[-a][r[]]*CD1[c][r[]])/r[]^2+((2*I)/3)*epsilong1[-b,d]*𝒽$ab$p[-c,-d]*Τ[e,-e]*CD1[-a][r[]]*CD1[c][r[]]+(2*epsilong1[-b,-c]*Ψ$m[]*CD1[-a][r[]]*CD1[c][r[]])/r[]^2+(I/4)*epsilong1[-c,d]*r[]*𝒫[]*CD1[-a][𝒽$ab$p[-b,-d]]*CD1[c][r[]]-(I/4)*epsilong1[d,e]*r[]*Τ[-c,-d]*CD1[-a][𝒽$ab$p[-b,-e]]*CD1[c][r[]]+(I*epsilong1[-b,d]*CD1[-a][𝒽$ab$p[-c,-d]]*CD1[c][r[]])/r[]+(I/6)*epsilong1[-b,d]*r[]*𝒫[]*CD1[-a][𝒽$ab$p[-c,-d]]*CD1[c][r[]]+(I/3)*epsilong1[-b,d]*r[]*Τ[e,-e]*CD1[-a][𝒽$ab$p[-c,-d]]*CD1[c][r[]]-(I/2)*epsilong1[-b,d]*r[]*Τ[-c,e]*CD1[-a][𝒽$ab$p[-d,-e]]*CD1[c][r[]]-(I*epsilong1[-b,-c]*CD1[-a][𝒽$p[]]*CD1[c][r[]])/r[]^3+((I/12)*epsilong1[-b,-c]*𝒫[]*CD1[-a][𝒽$p[]]*CD1[c][r[]])/r[]+((I/4)*epsilong1[-b,d]*Τ[-c,-d]*CD1[-a][𝒽$p[]]*CD1[c][r[]])/r[]-((I/3)*epsilong1[-b,-c]*Τ[d,-d]*CD1[-a][𝒽$p[]]*CD1[c][r[]])/r[]-((2*I)*epsilong1[-a,d]*𝒽$ab$p[-c,-d]*CD1[-b][r[]]*CD1[c][r[]])/r[]^2-((4*I)*epsilong1[-a,-c]*𝒽$p[]*CD1[-b][r[]]*CD1[c][r[]])/r[]^4+((2*I)/3)*epsilong1[-a,d]*𝒽$ab$p[-c,-d]*𝒫[]*CD1[-b][r[]]*CD1[c][r[]]+(((5*I)/6)*epsilong1[-a,-c]*𝒽$p[]*𝒫[]*CD1[-b][r[]]*CD1[c][r[]])/r[]^2-(((4*I)/3)*epsilong1[-a,-c]*𝒽$p[]*Τ[d,-d]*CD1[-b][r[]]*CD1[c][r[]])/r[]^2-((2*I)/3)*epsilong1[-a,d]*𝒽$ab$p[-c,-d]*Τ[e,-e]*CD1[-b][r[]]*CD1[c][r[]]-(2*epsilong1[-a,-c]*Ψ$m[]*CD1[-b][r[]]*CD1[c][r[]])/r[]^2+(I/4)*epsilong1[-c,d]*r[]*𝒫[]*CD1[-b][𝒽$ab$p[-a,-d]]*CD1[c][r[]]-(I/4)*epsilong1[d,e]*r[]*Τ[-c,-d]*CD1[-b][𝒽$ab$p[-a,-e]]*CD1[c][r[]]-(I*epsilong1[-a,d]*CD1[-b][𝒽$ab$p[-c,-d]]*CD1[c][r[]])/r[]+(I/12)*epsilong1[-a,d]*r[]*𝒫[]*CD1[-b][𝒽$ab$p[-c,-d]]*CD1[c][r[]]-(I/3)*epsilong1[-a,d]*r[]*Τ[e,-e]*CD1[-b][𝒽$ab$p[-c,-d]]*CD1[c][r[]]+(I/4)*epsilong1[-a,d]*r[]*Τ[-c,e]*CD1[-b][𝒽$ab$p[-d,-e]]*CD1[c][r[]]+(I*epsilong1[-a,-c]*CD1[-b][𝒽$p[]]*CD1[c][r[]])/r[]^3-((I/3)*epsilong1[-a,-c]*𝒫[]*CD1[-b][𝒽$p[]]*CD1[c][r[]])/r[]+((I/3)*epsilong1[-a,-c]*Τ[d,-d]*CD1[-b][𝒽$p[]]*CD1[c][r[]])/r[]+((5*I)/6)*epsilong1[-a,d]*r[]*𝒽$ab$p[-c,-d]*CD1[-b][𝒫[]]*CD1[c][r[]]-(I/4)*epsilong1[-a,-c]*r[]*𝒽$ab$p[d,-d]*CD1[-b][𝒫[]]*CD1[c][r[]]-((I/6)*epsilong1[-a,-c]*𝒽$p[]*CD1[-b][𝒫[]]*CD1[c][r[]])/r[]+(I/4)*epsilong1[-a,d]*r[]*𝒽$ab$p[e,-e]*CD1[-b][Τ[-c,-d]]*CD1[c][r[]]-(I/2)*epsilong1[-a,d]*r[]*𝒽$ab$p[-d,e]*CD1[-b][Τ[-c,-e]]*CD1[c][r[]]-(I/2)*epsilong1[-a,d]*r[]*𝒽$ab$p[-c,e]*CD1[-b][Τ[-d,-e]]*CD1[c][r[]]+((I/6)*epsilong1[-a,-c]*𝒽$p[]*CD1[-b][Τ[d,-d]]*CD1[c][r[]])/r[]+(I/6)*epsilong1[-a,d]*r[]*𝒽$ab$p[-c,-d]*CD1[-b][Τ[e,-e]]*CD1[c][r[]]+(((3*I)/2)*epsilong1[-b,d]*𝒽$ab$p[-a,-d]*CD1[-c][r[]]*CD1[c][r[]])/r[]^2-((I/4)*Λ^2*epsilong1[-b,d]*𝒽$ab$p[-a,-d]*CD1[-c][r[]]*CD1[c][r[]])/r[]^2-(((3*I)/2)*epsilong1[-a,d]*𝒽$ab$p[-b,-d]*CD1[-c][r[]]*CD1[c][r[]])/r[]^2+((I/4)*Λ^2*epsilong1[-a,d]*𝒽$ab$p[-b,-d]*CD1[-c][r[]]*CD1[c][r[]])/r[]^2+(((3*I)/2)*epsilong1[-a,-b]*𝒽$ab$p[d,-d]*CD1[-c][r[]]*CD1[c][r[]])/r[]^2+(I*epsilong1[-a,-b]*𝒽$p[]*CD1[-c][r[]]*CD1[c][r[]])/r[]^4+((I/2)*Λ^2*epsilong1[-a,-b]*𝒽$p[]*CD1[-c][r[]]*CD1[c][r[]])/r[]^4-(I/2)*epsilong1[-b,d]*𝒽$ab$p[-a,-d]*𝒫[]*CD1[-c][r[]]*CD1[c][r[]]+(I/4)*epsilong1[-a,d]*𝒽$ab$p[-b,-d]*𝒫[]*CD1[-c][r[]]*CD1[c][r[]]+(I/8)*epsilong1[-a,-b]*𝒽$ab$p[d,-d]*𝒫[]*CD1[-c][r[]]*CD1[c][r[]]+((I/2)*epsilong1[-a,-b]*𝒽$p[]*𝒫[]*CD1[-c][r[]]*CD1[c][r[]])/r[]^2-(I/4)*epsilong1[-b,d]*𝒽$ab$p[-d,e]*Τ[-a,-e]*CD1[-c][r[]]*CD1[c][r[]]-(I/8)*epsilong1[-a,d]*𝒽$ab$p[e,-e]*Τ[-b,-d]*CD1[-c][r[]]*CD1[c][r[]]+(I/2)*epsilong1[-a,d]*𝒽$ab$p[-d,e]*Τ[-b,-e]*CD1[-c][r[]]*CD1[c][r[]]-((3*I)/2)*epsilong1[-a,-b]*𝒽$ab$p[d,e]*Τ[-d,-e]*CD1[-c][r[]]*CD1[c][r[]]+((I/2)*epsilong1[-a,-b]*𝒽$p[]*Τ[d,-d]*CD1[-c][r[]]*CD1[c][r[]])/r[]^2+(I/2)*epsilong1[-b,d]*𝒽$ab$p[-a,-d]*Τ[e,-e]*CD1[-c][r[]]*CD1[c][r[]]-(I/2)*epsilong1[-a,d]*𝒽$ab$p[-b,-d]*Τ[e,-e]*CD1[-c][r[]]*CD1[c][r[]]+I*epsilong1[-a,-b]*𝒽$ab$p[d,-d]*Τ[e,-e]*CD1[-c][r[]]*CD1[c][r[]]+(I/4)*epsilong1[-b,d]*Φ$ab$p[-a,-d]*CD1[-c][r[]]*CD1[c][r[]]-(I/4)*epsilong1[-a,d]*Φ$ab$p[-b,-d]*CD1[-c][r[]]*CD1[c][r[]]+(I/2)*epsilong1[-a,-b]*Φ$ab$p[d,-d]*CD1[-c][r[]]*CD1[c][r[]]+((I/2)*epsilong1[-a,-b]*Φ$p[]*CD1[-c][r[]]*CD1[c][r[]])/r[]^2+(((3*I)/2)*epsilong1[-b,d]*Ψ$ab$p[-a,-d]*CD1[-c][r[]]*CD1[c][r[]])/r[]^2-(((3*I)/2)*epsilong1[-a,d]*Ψ$ab$p[-b,-d]*CD1[-c][r[]]*CD1[c][r[]])/r[]^2-(epsilong1[-a,-b]*Ψ$m[]*CD1[-c][r[]]*CD1[c][r[]])/(2*r[]^2)-(3*Λ^2*𝒽$a$m[-b]*CD1[-a][r[]]*CD1[-c][r[]]*CD1[c][r[]])/(2*r[]^3)+(3*Λ^2*𝒽$a$m[-a]*CD1[-b][r[]]*CD1[-c][r[]]*CD1[c][r[]])/(2*r[]^3)+(I*epsilong1[-b,d]*CD1[-c][𝒽$ab$p[-a,-d]]*CD1[c][r[]])/r[]+(I/6)*epsilong1[-b,d]*r[]*𝒫[]*CD1[-c][𝒽$ab$p[-a,-d]]*CD1[c][r[]]+(I/3)*epsilong1[-b,d]*r[]*Τ[e,-e]*CD1[-c][𝒽$ab$p[-a,-d]]*CD1[c][r[]]-(I*epsilong1[-a,d]*CD1[-c][𝒽$ab$p[-b,-d]]*CD1[c][r[]])/r[]-((5*I)/12)*epsilong1[-a,d]*r[]*𝒫[]*CD1[-c][𝒽$ab$p[-b,-d]]*CD1[c][r[]]-(I/3)*epsilong1[-a,d]*r[]*Τ[e,-e]*CD1[-c][𝒽$ab$p[-b,-d]]*CD1[c][r[]]+(((3*I)/4)*epsilong1[-a,-b]*CD1[-c][𝒽$ab$p[d,-d]]*CD1[c][r[]])/r[]+(I/4)*epsilong1[-a,-b]*r[]*Τ[d,-d]*CD1[-c][𝒽$ab$p[e,-e]]*CD1[c][r[]]-(((3*I)/2)*epsilong1[-a,-b]*CD1[-c][𝒽$p[]]*CD1[c][r[]])/r[]^3-((I/2)*epsilong1[-a,-b]*Τ[d,-d]*CD1[-c][𝒽$p[]]*CD1[c][r[]])/r[]-(I/3)*epsilong1[-a,d]*r[]*𝒽$ab$p[-b,-d]*CD1[-c][𝒫[]]*CD1[c][r[]]+(I/6)*epsilong1[-a,-b]*r[]*𝒽$ab$p[d,-d]*CD1[-c][𝒫[]]*CD1[c][r[]]+((I/3)*epsilong1[-a,-b]*𝒽$p[]*CD1[-c][𝒫[]]*CD1[c][r[]])/r[]-(I/4)*epsilong1[-a,d]*r[]*𝒽$ab$p[e,-e]*CD1[-c][Τ[-b,-d]]*CD1[c][r[]]+(I/2)*epsilong1[-a,d]*r[]*𝒽$ab$p[-d,e]*CD1[-c][Τ[-b,-e]]*CD1[c][r[]]-((I/3)*epsilong1[-a,-b]*𝒽$p[]*CD1[-c][Τ[d,-d]]*CD1[c][r[]])/r[]-(I/6)*epsilong1[-a,d]*r[]*𝒽$ab$p[-b,-d]*CD1[-c][Τ[e,-e]]*CD1[c][r[]]+(I/12)*epsilong1[-a,-b]*r[]*𝒽$ab$p[d,-d]*CD1[-c][Τ[e,-e]]*CD1[c][r[]]-(epsilong1[-a,-b]*CD1[-c][Ψ$m[]]*CD1[c][r[]])/r[]-(I*epsilong1[-b,-c]*CD1[-a][r[]]*CD1[c][𝒽$p[]])/r[]^3+((I/3)*epsilong1[-b,-c]*𝒫[]*CD1[-a][r[]]*CD1[c][𝒽$p[]])/r[]-((I/3)*epsilong1[-b,-c]*Τ[d,-d]*CD1[-a][r[]]*CD1[c][𝒽$p[]])/r[]+(I*epsilong1[-a,-c]*CD1[-b][r[]]*CD1[c][𝒽$p[]])/r[]^3-((I/12)*epsilong1[-a,-c]*𝒫[]*CD1[-b][r[]]*CD1[c][𝒽$p[]])/r[]+((I/3)*epsilong1[-a,-c]*Τ[d,-d]*CD1[-b][r[]]*CD1[c][𝒽$p[]])/r[]-(I/12)*epsilong1[-a,-c]*CD1[-b][𝒫[]]*CD1[c][𝒽$p[]]+(I/4)*epsilong1[-a,d]*CD1[-b][Τ[-c,-d]]*CD1[c][𝒽$p[]]-(I/6)*epsilong1[-a,-c]*CD1[-b][Τ[d,-d]]*CD1[c][𝒽$p[]]-(I/4)*epsilong1[-a,d]*CD1[-c][Τ[-b,-d]]*CD1[c][𝒽$p[]]+(I/4)*epsilong1[-a,-b]*CD1[-c][Τ[d,-d]]*CD1[c][𝒽$p[]]+((I/6)*epsilong1[-b,-c]*𝒽$p[]*CD1[-a][r[]]*CD1[c][𝒫[]])/r[]+(I/12)*epsilong1[-c,d]*r[]^2*CD1[-a][𝒽$ab$p[-b,-d]]*CD1[c][𝒫[]]+(I/6)*epsilong1[-b,d]*r[]^2*CD1[-a][𝒽$ab$p[-c,-d]]*CD1[c][𝒫[]]-(I/12)*epsilong1[-b,-c]*CD1[-a][𝒽$p[]]*CD1[c][𝒫[]]+(I/2)*epsilong1[-a,d]*r[]*𝒽$ab$p[-c,-d]*CD1[-b][r[]]*CD1[c][𝒫[]]-(I/4)*epsilong1[-a,-c]*r[]*𝒽$ab$p[d,-d]*CD1[-b][r[]]*CD1[c][𝒫[]]-((I/3)*epsilong1[-a,-c]*𝒽$p[]*CD1[-b][r[]]*CD1[c][𝒫[]])/r[]+(I/12)*epsilong1[-c,d]*r[]^2*CD1[-b][𝒽$ab$p[-a,-d]]*CD1[c][𝒫[]]-(I/12)*epsilong1[-a,d]*r[]^2*CD1[-b][𝒽$ab$p[-c,-d]]*CD1[c][𝒫[]]+(I/6)*epsilong1[-a,-c]*CD1[-b][𝒽$p[]]*CD1[c][𝒫[]]+(I/6)*epsilong1[-b,d]*r[]^2*CD1[-c][𝒽$ab$p[-a,-d]]*CD1[c][𝒫[]]-(I/4)*epsilong1[-a,d]*r[]^2*CD1[-c][𝒽$ab$p[-b,-d]]*CD1[c][𝒫[]]+(I/12)*epsilong1[-a,-b]*r[]^2*CD1[-c][𝒽$ab$p[d,-d]]*CD1[c][𝒫[]]+I*epsilong1[-a,-c]*CD1[c][Γ$a$p[-b]]-(I/4)*epsilong1[-c,d]*r[]*𝒫[]*CD1[c][r[]]*CD1[-d][𝒽$ab$p[-a,-b]]-(I/12)*epsilong1[-c,d]*r[]^2*CD1[c][𝒫[]]*CD1[-d][𝒽$ab$p[-a,-b]]-(I*epsilong1[-b,d]*CD1[c][r[]]*CD1[-d][𝒽$ab$p[-a,-c]])/r[]-(I/6)*epsilong1[-b,d]*r[]*𝒫[]*CD1[c][r[]]*CD1[-d][𝒽$ab$p[-a,-c]]-(I/3)*epsilong1[-b,d]*r[]*Τ[e,-e]*CD1[c][r[]]*CD1[-d][𝒽$ab$p[-a,-c]]-(I/6)*epsilong1[-b,d]*r[]^2*CD1[c][𝒫[]]*CD1[-d][𝒽$ab$p[-a,-c]]+(I/2)*epsilong1[-b,d]*r[]*Τ[-c,e]*CD1[c][r[]]*CD1[-d][𝒽$ab$p[-a,-e]]+(I*epsilong1[-a,d]*CD1[c][r[]]*CD1[-d][𝒽$ab$p[-b,-c]])/r[]+((5*I)/12)*epsilong1[-a,d]*r[]*𝒫[]*CD1[c][r[]]*CD1[-d][𝒽$ab$p[-b,-c]]+(I/3)*epsilong1[-a,d]*r[]*Τ[e,-e]*CD1[c][r[]]*CD1[-d][𝒽$ab$p[-b,-c]]+(I/4)*epsilong1[-a,d]*r[]^2*CD1[c][𝒫[]]*CD1[-d][𝒽$ab$p[-b,-c]]-((3*I)/4)*epsilong1[-a,d]*r[]*Τ[-c,e]*CD1[c][r[]]*CD1[-d][𝒽$ab$p[-b,-e]]-(((3*I)/2)*epsilong1[-a,-b]*CD1[c][r[]]*CD1[-d][𝒽$ab$p[-c,d]])/r[]-(I/6)*epsilong1[-a,-b]*r[]^2*CD1[c][𝒫[]]*CD1[-d][𝒽$ab$p[-c,d]]-(I/4)*epsilong1[-a,-b]*r[]*Τ[-c,d]*CD1[c][r[]]*CD1[-d][𝒽$ab$p[e,-e]]+((I/2)*epsilong1[c,d]*𝒽$p[]*CD1[-b][r[]]*CD1[-d][Τ[-a,-c]])/r[]-(I/4)*epsilong1[c,d]*CD1[-b][𝒽$p[]]*CD1[-d][Τ[-a,-c]]+(I/4)*epsilong1[-a,d]*r[]*𝒽$ab$p[e,-e]*CD1[c][r[]]*CD1[-d][Τ[-b,-c]]+(I/12)*epsilong1[c,d]*r[]^2*CD1[-a][𝒽$ab$p[-b,-c]]*CD1[-d][Τ[e,-e]]-(I/6)*epsilong1[-b,c]*r[]^2*CD1[-a][𝒽$ab$p[-c,d]]*CD1[-d][Τ[e,-e]]+(I/12)*epsilong1[c,d]*r[]^2*CD1[-b][𝒽$ab$p[-a,-c]]*CD1[-d][Τ[e,-e]]+(I/12)*epsilong1[-a,c]*r[]^2*CD1[-b][𝒽$ab$p[-c,d]]*CD1[-d][Τ[e,-e]]-(I/12)*epsilong1[c,d]*r[]^2*CD1[-c][𝒽$ab$p[-a,-b]]*CD1[-d][Τ[e,-e]]+(I/6)*epsilong1[-b,c]*r[]^2*CD1[-c][𝒽$ab$p[-a,d]]*CD1[-d][Τ[e,-e]]-(I/4)*epsilong1[-a,c]*r[]^2*CD1[-c][𝒽$ab$p[-b,d]]*CD1[-d][Τ[e,-e]]+(I/6)*epsilong1[-a,-b]*r[]^2*CD1[-c][𝒽$ab$p[c,d]]*CD1[-d][Τ[e,-e]]-(I/3)*epsilong1[-a,-b]*r[]*𝒽$ab$p[-c,d]*CD1[c][r[]]*CD1[-d][Τ[e,-e]]-(I/4)*epsilong1[-b,c]*CD1[-d][CD1[-a][𝒽$ab$p[-c,d]]]+(I/12)*epsilong1[-b,c]*r[]^2*𝒫[]*CD1[-d][CD1[-a][𝒽$ab$p[-c,d]]]+(I/4)*epsilong1[-a,c]*CD1[-d][CD1[-b][𝒽$ab$p[-c,d]]]-(I/12)*epsilong1[-a,c]*r[]^2*𝒫[]*CD1[-d][CD1[-b][𝒽$ab$p[-c,d]]]+(I/6)*epsilong1[-a,c]*r[]^2*𝒽$ab$p[-c,d]*CD1[-d][CD1[-b][𝒫[]]]-(I/6)*epsilong1[-a,c]*r[]^2*𝒽$ab$p[-c,d]*CD1[-d][CD1[-b][Τ[e,-e]]]+(I/4)*epsilong1[-b,c]*CD1[-d][CD1[-c][𝒽$ab$p[-a,d]]]-(I/12)*epsilong1[-b,c]*r[]^2*𝒫[]*CD1[-d][CD1[-c][𝒽$ab$p[-a,d]]]-(I/4)*epsilong1[-a,c]*CD1[-d][CD1[-c][𝒽$ab$p[-b,d]]]+(I/12)*epsilong1[-a,c]*r[]^2*𝒫[]*CD1[-d][CD1[-c][𝒽$ab$p[-b,d]]]-(I/6)*epsilong1[-a,-b]*r[]^2*𝒽$ab$p[c,d]*CD1[-d][CD1[-c][𝒫[]]]+(I/6)*epsilong1[-a,-b]*r[]^2*𝒽$ab$p[c,d]*CD1[-d][CD1[-c][Τ[e,-e]]]-(I/4)*epsilong1[-b,c]*CD1[-d][CD1[d][𝒽$ab$p[-a,-c]]]+(I/12)*epsilong1[-b,c]*r[]^2*𝒫[]*CD1[-d][CD1[d][𝒽$ab$p[-a,-c]]]+(I/4)*epsilong1[-a,c]*CD1[-d][CD1[d][𝒽$ab$p[-b,-c]]]-(I/12)*epsilong1[-a,c]*r[]^2*𝒫[]*CD1[-d][CD1[d][𝒽$ab$p[-b,-c]]]+(I/6)*epsilong1[-a,-b]*Τ[c,-c]*CD1[-d][CD1[d][𝒽$p[]]]-((I/2)*epsilong1[-a,-b]*CD1[-c][r[]]*CD1[c][r[]]*CD1[-d][CD1[d][𝒽$p[]]])/r[]^2+(Λ^2*g1[-a,-b]*𝒽$a$m[c]*Τ[-c,-d]*CD1[d][r[]])/(4*r[])+((2*I)*epsilong1[-a,-b]*𝒽$ab$p[-c,-d]*CD1[c][r[]]*CD1[d][r[]])/r[]^2+(I/3)*epsilong1[-a,-b]*𝒽$ab$p[-c,-d]*𝒫[]*CD1[c][r[]]*CD1[d][r[]]+((I/2)*epsilong1[-a,-c]*𝒽$p[]*Τ[-b,-d]*CD1[c][r[]]*CD1[d][r[]])/r[]^2-(I*epsilong1[-a,-b]*𝒽$p[]*Τ[-c,-d]*CD1[c][r[]]*CD1[d][r[]])/r[]^2-I*epsilong1[-a,-b]*𝒽$ab$p[-c,e]*Τ[-d,-e]*CD1[c][r[]]*CD1[d][r[]]+((2*I)/3)*epsilong1[-a,-b]*𝒽$ab$p[-c,-d]*Τ[e,-e]*CD1[c][r[]]*CD1[d][r[]]-((2*I)*epsilong1[-b,e]*𝒽$ab$p[-d,-e]*CD1[-a][r[]]*CD1[-c][r[]]*CD1[c][r[]]*CD1[d][r[]])/r[]^2-((4*I)*epsilong1[-b,-d]*𝒽$p[]*CD1[-a][r[]]*CD1[-c][r[]]*CD1[c][r[]]*CD1[d][r[]])/r[]^4-(I*epsilong1[-b,e]*CD1[-a][𝒽$ab$p[-d,-e]]*CD1[-c][r[]]*CD1[c][r[]]*CD1[d][r[]])/r[]+(I*epsilong1[-b,-d]*CD1[-a][𝒽$p[]]*CD1[-c][r[]]*CD1[c][r[]]*CD1[d][r[]])/r[]^3+((2*I)*epsilong1[-a,e]*𝒽$ab$p[-d,-e]*CD1[-b][r[]]*CD1[-c][r[]]*CD1[c][r[]]*CD1[d][r[]])/r[]^2+((4*I)*epsilong1[-a,-d]*𝒽$p[]*CD1[-b][r[]]*CD1[-c][r[]]*CD1[c][r[]]*CD1[d][r[]])/r[]^4+(I*epsilong1[-a,e]*CD1[-b][𝒽$ab$p[-d,-e]]*CD1[-c][r[]]*CD1[c][r[]]*CD1[d][r[]])/r[]-(I*epsilong1[-a,-d]*CD1[-b][𝒽$p[]]*CD1[-c][r[]]*CD1[c][r[]]*CD1[d][r[]])/r[]^3-(((3*I)/4)*epsilong1[-b,e]*𝒽$ab$p[-a,-e]*CD1[-c][r[]]*CD1[c][r[]]*CD1[-d][r[]]*CD1[d][r[]])/r[]^2+(((3*I)/4)*epsilong1[-a,e]*𝒽$ab$p[-b,-e]*CD1[-c][r[]]*CD1[c][r[]]*CD1[-d][r[]]*CD1[d][r[]])/r[]^2-(((3*I)/4)*epsilong1[-a,-b]*𝒽$ab$p[e,-e]*CD1[-c][r[]]*CD1[c][r[]]*CD1[-d][r[]]*CD1[d][r[]])/r[]^2-(((3*I)/2)*epsilong1[-a,-b]*𝒽$p[]*CD1[-c][r[]]*CD1[c][r[]]*CD1[-d][r[]]*CD1[d][r[]])/r[]^4-(I*epsilong1[-b,e]*CD1[-c][r[]]*CD1[c][r[]]*CD1[-d][𝒽$ab$p[-a,-e]]*CD1[d][r[]])/r[]+(I*epsilong1[-a,e]*CD1[-c][r[]]*CD1[c][r[]]*CD1[-d][𝒽$ab$p[-b,-e]]*CD1[d][r[]])/r[]-(((3*I)/4)*epsilong1[-a,-b]*CD1[-c][r[]]*CD1[c][r[]]*CD1[-d][𝒽$ab$p[e,-e]]*CD1[d][r[]])/r[]+(((3*I)/2)*epsilong1[-a,-b]*CD1[-c][r[]]*CD1[c][r[]]*CD1[-d][𝒽$p[]]*CD1[d][r[]])/r[]^3-(I/6)*epsilong1[-b,c]*r[]^2*CD1[-d][Τ[e,-e]]*CD1[d][𝒽$ab$p[-a,-c]]+(I/4)*epsilong1[-a,c]*r[]^2*CD1[-d][Τ[e,-e]]*CD1[d][𝒽$ab$p[-b,-c]]-(I/12)*epsilong1[-a,-b]*r[]^2*CD1[-d][Τ[e,-e]]*CD1[d][𝒽$ab$p[c,-c]]-((I/4)*epsilong1[-a,-d]*Τ[-b,-c]*CD1[c][r[]]*CD1[d][𝒽$p[]])/r[]+((I/2)*epsilong1[-a,-b]*Τ[-c,-d]*CD1[c][r[]]*CD1[d][𝒽$p[]])/r[]+(I*epsilong1[-b,-d]*CD1[-a][r[]]*CD1[-c][r[]]*CD1[c][r[]]*CD1[d][𝒽$p[]])/r[]^3-(I*epsilong1[-a,-d]*CD1[-b][r[]]*CD1[-c][r[]]*CD1[c][r[]]*CD1[d][𝒽$p[]])/r[]^3-((2*I)/3)*epsilong1[-a,-b]*r[]*𝒽$ab$p[-c,-d]*CD1[c][r[]]*CD1[d][𝒫[]]+(I/4)*epsilong1[d,e]*r[]*Τ[-c,-d]*CD1[c][r[]]*CD1[-e][𝒽$ab$p[-a,-b]]-(I/2)*epsilong1[-b,d]*r[]*Τ[-c,e]*CD1[c][r[]]*CD1[-e][𝒽$ab$p[-a,-d]]+(I*epsilong1[-b,e]*CD1[-c][r[]]*CD1[c][r[]]*CD1[d][r[]]*CD1[-e][𝒽$ab$p[-a,-d]])/r[]+((3*I)/4)*epsilong1[-a,d]*r[]*Τ[-c,e]*CD1[c][r[]]*CD1[-e][𝒽$ab$p[-b,-d]]-(I*epsilong1[-a,e]*CD1[-c][r[]]*CD1[c][r[]]*CD1[d][r[]]*CD1[-e][𝒽$ab$p[-b,-d]])/r[]-(I/2)*epsilong1[-a,-b]*r[]*Τ[d,-d]*CD1[c][r[]]*CD1[-e][𝒽$ab$p[-c,e]]+(I/2)*epsilong1[-a,-b]*r[]*Τ[-c,d]*CD1[c][r[]]*CD1[-e][𝒽$ab$p[-d,e]]+(((3*I)/2)*epsilong1[-a,-b]*CD1[-c][r[]]*CD1[c][r[]]*CD1[d][r[]]*CD1[-e][𝒽$ab$p[-d,e]])/r[]-(I/2)*epsilong1[-a,d]*r[]*𝒽$ab$p[-d,e]*CD1[c][r[]]*CD1[-e][Τ[-b,-c]]+(I/2)*epsilong1[-a,d]*r[]*𝒽$ab$p[-c,e]*CD1[c][r[]]*CD1[-e][Τ[-b,-d]]+(I/2)*epsilong1[-a,-b]*r[]*𝒽$ab$p[d,e]*CD1[c][r[]]*CD1[-e][Τ[-c,-d]]-(I/12)*epsilong1[-b,c]*r[]^2*Τ[d,-d]*CD1[-e][CD1[-a][𝒽$ab$p[-c,e]]]+(I/4)*epsilong1[-b,d]*CD1[-c][r[]]*CD1[c][r[]]*CD1[-e][CD1[-a][𝒽$ab$p[-d,e]]]+(I/12)*epsilong1[-a,c]*r[]^2*Τ[d,-d]*CD1[-e][CD1[-b][𝒽$ab$p[-c,e]]]-(I/4)*epsilong1[-a,d]*CD1[-c][r[]]*CD1[c][r[]]*CD1[-e][CD1[-b][𝒽$ab$p[-d,e]]]+(I/12)*epsilong1[-b,c]*r[]^2*Τ[d,-d]*CD1[-e][CD1[-c][𝒽$ab$p[-a,e]]]-(I/12)*epsilong1[-a,c]*r[]^2*Τ[d,-d]*CD1[-e][CD1[-c][𝒽$ab$p[-b,e]]]-(I/4)*epsilong1[-b,d]*CD1[-c][r[]]*CD1[c][r[]]*CD1[-e][CD1[-d][𝒽$ab$p[-a,e]]]+(I/4)*epsilong1[-a,d]*CD1[-c][r[]]*CD1[c][r[]]*CD1[-e][CD1[-d][𝒽$ab$p[-b,e]]]-(I/12)*epsilong1[-b,c]*r[]^2*Τ[d,-d]*CD1[-e][CD1[e][𝒽$ab$p[-a,-c]]]+(I/4)*epsilong1[-b,d]*CD1[-c][r[]]*CD1[c][r[]]*CD1[-e][CD1[e][𝒽$ab$p[-a,-d]]]+(I/12)*epsilong1[-a,c]*r[]^2*Τ[d,-d]*CD1[-e][CD1[e][𝒽$ab$p[-b,-c]]]-(I/4)*epsilong1[-a,d]*CD1[-c][r[]]*CD1[c][r[]]*CD1[-e][CD1[e][𝒽$ab$p[-b,-d]]]-((2*I)*epsilong1[-a,-b]*𝒽$ab$p[-d,-e]*CD1[-c][r[]]*CD1[c][r[]]*CD1[d][r[]]*CD1[e][r[]])/r[]^2},MetricOn->All];

(*{LHS$ϵComp$PaQb$time,LHS$ϵComp$PaQb$result}=AbsoluteTiming[LHSstep1=Pert$Exp$Sep[LHS,{μ,ν,ρ,σ},{P,a,Q,b},{ℂ$Harm$Rep,𝒽$Harm$Rep}];
LHSstep2=LHSstep1/.Dir$Harm$Def$Total//ContractMetric//ToCanonical;
LHSstep3=LHSstep2//.Harm$EQ$Y$Total//ContractMetric//ToCanonical;
LHSstep4=ϵAB$Component[LHSstep3]//ContractMetric//ToCanonical;
LHSstep4
];
Print["Computing Time:"]
LHS$ϵComp$PaQb$time
Print["Result:"]
LHS$ϵComp$PaQb$result
{RHS$ϵComp$PaQb$time,RHS$ϵComp$PaQb$result}=AbsoluteTiming[RHSstep1=Pert$Exp$Sep[RHS,{μ,ν,ρ,σ},{P,a,Q,b},{ℂ$Harm$Rep,𝒽$Harm$Rep,𝕁$Harm$Rep,𝒯$Harm$Rep}];
RHSstep2=RHSstep1/.Dir$Harm$Def$Total//ContractMetric//ToCanonical;
RHSstep3=RHSstep2//.Harm$EQ$Y$Total//ContractMetric//ToCanonical;
RHSstep4=ϵAB$Component[RHSstep3]//ContractMetric//ToCanonical;
RHSstep4
];
Print["Computing Time:"]
RHS$ϵComp$PaQb$time
Print["Result:"]
RHS$ϵComp$PaQb$result
Print["Match with saved result:"]
Simplify[ToCanonical[(Δℂ$D𝕁$ϵComp$PaQb[-a,-b]/.Eval$Pert$Δℂ$D𝕁$ϵComp$PaQb)-(LHS$ϵComp$PaQb$result-RHS$ϵComp$PaQb$result)]==0]
Simplify[(ToCanonical[((-1/2)epsilong1[a,b](LHS$ϵComp$PaQb$result-RHS$ϵComp$PaQb$result)//ContractMetric//ToCanonical)-(Δℂ$D𝕁$ϵCompSkew$PaQb[]/.Eval$Pert$Δℂ$D𝕁$ϵCompSkew$PaQb)])==0]*)

Z-Component of δ((Subscript[Δℂ, A B P a])-((Subscript[D, A])-i ⋆(Subscript[D, A]))(Subscript[𝕁, B P a]))=0

Eval$Pert$Δℂ$D𝕁$ZComp$ABPa=MakeRule[{Δℂ$D𝕁$ZComp$ABPa[-a],(-3*𝒽$a$m[-a])/(2*r[]^2)-(Λ^2*𝒽$a$m[-a])/(4*r[]^2)+𝒽$a$m[-a]*𝒫[]+(Λ^2*𝒽$a$m[-a]*𝒫[])/12-(r[]^2*𝒽$a$m[-a]*𝒫[]^2)/12+Γ$a$m[-a]-(Λ^2*Γ$a$m[-a])/2-(5*Ξ$a$m[-a])/(4*r[]^2)-(Λ^2*Ξ$a$m[-a])/r[]^2+(3*𝒫[]*Ξ$a$m[-a])/4-(((3*I)/4)*epsilong1[-a,-b]*Ξ$a$p[b])/r[]^2+(I/4)*epsilong1[-a,-b]*𝒫[]*Ξ$a$p[b]+Π$a$m[-a]+(3*𝒽$a$m[b]*Τ[-a,-b])/4+(r[]^2*𝒽$a$m[b]*𝒫[]*Τ[-a,-b])/12-(Ξ$a$m[b]*Τ[-a,-b])/2-(r[]^2*𝒽$a$m[b]*Τ[-a,c]*Τ[-b,-c])/4-(5*𝒽$a$m[-a]*Τ[b,-b])/4-(Λ^2*𝒽$a$m[-a]*Τ[b,-b])/12+(r[]^2*𝒽$a$m[-a]*𝒫[]*Τ[b,-b])/4-(Ξ$a$m[-a]*Τ[b,-b])/4+(r[]^2*𝒽$a$m[-a]*Τ[-b,-c]*Τ[b,c])/4-(I/4)*epsilong1[-a,-b]*Ξ$a$p[b]*Τ[c,-c]+(5*r[]^2*𝒽$a$m[b]*Τ[-a,-b]*Τ[c,-c])/12-(5*r[]^2*𝒽$a$m[-a]*Τ[b,-b]*Τ[c,-c])/12+(3*epsilong1[-a,-b]*Ψ$a$m[b])/4-(epsilong1[-a,-b]*r[]^2*𝒫[]*Ψ$a$m[b])/4-(epsilong1[-a,c]*r[]^2*Τ[-b,-c]*Ψ$a$m[b])/2+(epsilong1[-a,-b]*r[]^2*Τ[c,-c]*Ψ$a$m[b])/4+((3*I)/4)*Ψ$a$p[-a]-(I/4)*r[]^2*𝒫[]*Ψ$a$p[-a]+(I/4)*r[]^2*Τ[b,-b]*Ψ$a$p[-a]-(Γ$m[]*CD1[-a][r[]])/r[]+I*r[]*Π$p[]*CD1[-a][r[]]-I*r[]^2*CD1[-a][Π$p[]]+(I/4)*epsilong1[b,c]*r[]^2*𝒽$ab$p[-b,d]*CD1[-a][Τ[-c,-d]]+(19*𝒽$a$m[b]*CD1[-a][r[]]*CD1[-b][r[]])/(2*r[]^2)+(5*𝒽$a$m[b]*𝒫[]*CD1[-a][r[]]*CD1[-b][r[]])/6-(2*Ξ$a$m[b]*CD1[-a][r[]]*CD1[-b][r[]])/r[]^2+(19*𝒽$a$m[b]*Τ[c,-c]*CD1[-a][r[]]*CD1[-b][r[]])/6-(r[]*𝒽$a$m[b]*CD1[-a][𝒫[]]*CD1[-b][r[]])/3-(r[]*𝒽$a$m[b]*CD1[-a][Τ[c,-c]]*CD1[-b][r[]])/6-(3*CD1[-a][r[]]*CD1[-b][𝒽$a$m[b]])/(2*r[])+(r[]*𝒫[]*CD1[-a][r[]]*CD1[-b][𝒽$a$m[b]])/2-(r[]*Τ[c,-c]*CD1[-a][r[]]*CD1[-b][𝒽$a$m[b]])/2+(2*r[]*𝒽$a$m[b]*CD1[-a][r[]]*CD1[-b][𝒫[]])/3-(5*r[]^2*CD1[-a][𝒽$a$m[b]]*CD1[-b][𝒫[]])/12+(I/6)*epsilong1[-a,b]*𝒽$p[]*CD1[-b][Τ[c,-c]]-(2*r[]*𝒽$a$m[b]*CD1[-a][r[]]*CD1[-b][Τ[c,-c]])/3+(5*r[]^2*CD1[-a][𝒽$a$m[b]]*CD1[-b][Τ[c,-c]])/12+(3*CD1[-b][CD1[-a][𝒽$a$m[b]]])/4-(r[]^2*𝒫[]*CD1[-b][CD1[-a][𝒽$a$m[b]]])/4+(r[]^2*Τ[c,-c]*CD1[-b][CD1[-a][𝒽$a$m[b]]])/4-(r[]^2*𝒽$a$m[b]*CD1[-b][CD1[-a][𝒫[]]])/6+(r[]^2*𝒽$a$m[b]*CD1[-b][CD1[-a][Τ[c,-c]]])/6+CD1[-b][CD1[b][𝒽$a$m[-a]]]/4-(r[]^2*𝒫[]*CD1[-b][CD1[b][𝒽$a$m[-a]]])/12+(r[]^2*Τ[c,-c]*CD1[-b][CD1[b][𝒽$a$m[-a]]])/12+(r[]^2*𝒽$a$m[-a]*CD1[-b][CD1[b][𝒫[]]])/6+CD1[-b][CD1[b][Ξ$a$m[-a]]]+(((3*I)/2)*epsilong1[-b,c]*𝒽$ab$p[-a,-c]*CD1[b][r[]])/r[]+(((3*I)/2)*epsilong1[-a,c]*𝒽$ab$p[-b,-c]*CD1[b][r[]])/r[]+(((3*I)/4)*epsilong1[-a,-b]*𝒽$ab$p[c,-c]*CD1[b][r[]])/r[]-(((9*I)/2)*epsilong1[-a,-b]*𝒽$p[]*CD1[b][r[]])/r[]^3-(I/4)*epsilong1[-b,c]*r[]*𝒽$ab$p[-a,-c]*𝒫[]*CD1[b][r[]]+I*epsilong1[-a,c]*r[]*𝒽$ab$p[-b,-c]*𝒫[]*CD1[b][r[]]-(I/4)*epsilong1[-a,-b]*r[]*𝒽$ab$p[c,-c]*𝒫[]*CD1[b][r[]]+(I*epsilong1[-a,-b]*𝒽$p[]*𝒫[]*CD1[b][r[]])/r[]+I*epsilong1[-b,c]*r[]*Π$ab$p[-a,-c]*CD1[b][r[]]+2*epsilong1[-a,-b]*r[]*Π$m[]*CD1[b][r[]]+((I/2)*epsilong1[-a,c]*𝒽$p[]*Τ[-b,-c]*CD1[b][r[]])/r[]+(I/4)*epsilong1[c,d]*r[]*𝒽$ab$p[-a,-c]*Τ[-b,-d]*CD1[b][r[]]-((3*I)/2)*epsilong1[-a,c]*r[]*𝒽$ab$p[-c,d]*Τ[-b,-d]*CD1[b][r[]]-(((3*I)/2)*epsilong1[-a,-b]*𝒽$p[]*Τ[c,-c]*CD1[b][r[]])/r[]+(I/2)*epsilong1[-b,c]*r[]*𝒽$ab$p[-a,-c]*Τ[d,-d]*CD1[b][r[]]+(I/2)*epsilong1[-a,c]*r[]*𝒽$ab$p[-b,-c]*Τ[d,-d]*CD1[b][r[]]+(I/4)*epsilong1[-a,-b]*r[]*𝒽$ab$p[c,-c]*Τ[d,-d]*CD1[b][r[]]+(2*Ψ$ab$m[-a,-b]*CD1[b][r[]])/r[]-(Λ^2*Ψ$ab$m[-a,-b]*CD1[b][r[]])/r[]-(3*epsilong1[-a,-b]*Ψ$m[]*CD1[b][r[]])/r[]-(9*CD1[-a][𝒽$a$m[-b]]*CD1[b][r[]])/(2*r[])-(r[]*𝒫[]*CD1[-a][𝒽$a$m[-b]]*CD1[b][r[]])/4-(3*r[]*Τ[c,-c]*CD1[-a][𝒽$a$m[-b]]*CD1[b][r[]])/2+(7*r[]*Τ[-b,-c]*CD1[-a][𝒽$a$m[c]]*CD1[b][r[]])/4+(Λ^2*𝒽$a$m[-a]*CD1[-b][r[]]*CD1[b][r[]])/(4*r[]^2)-𝒽$a$m[-a]*𝒫[]*CD1[-b][r[]]*CD1[b][r[]]+(21*Ξ$a$m[-a]*CD1[-b][r[]]*CD1[b][r[]])/(4*r[]^2)+(𝒽$a$m[-a]*Τ[c,-c]*CD1[-b][r[]]*CD1[b][r[]])/4-((3*I)/4)*Ψ$a$p[-a]*CD1[-b][r[]]*CD1[b][r[]]-(r[]*𝒫[]*CD1[-b][𝒽$a$m[-a]]*CD1[b][r[]])/4+(r[]*𝒽$a$m[-a]*CD1[-b][𝒫[]]*CD1[b][r[]])/3-(4*CD1[-b][Ξ$a$m[-a]]*CD1[b][r[]])/r[]+(r[]*𝒽$a$m[-a]*CD1[-b][Τ[c,-c]]*CD1[b][r[]])/6+(r[]^2*CD1[-b][𝒫[]]*CD1[b][𝒽$a$m[-a]])/12-(r[]^2*CD1[-b][Τ[c,-c]]*CD1[b][𝒽$a$m[-a]])/12+(((3*I)/4)*epsilong1[-a,-b]*CD1[b][𝒽$p[]])/r[]^2-(I/4)*epsilong1[-a,-b]*𝒫[]*CD1[b][𝒽$p[]]+(I/4)*epsilong1[-a,-b]*Τ[c,-c]*CD1[b][𝒽$p[]]+(I/12)*epsilong1[-b,c]*r[]^2*𝒽$ab$p[-a,-c]*CD1[b][𝒫[]]+(I/3)*epsilong1[-a,c]*r[]^2*𝒽$ab$p[-b,-c]*CD1[b][𝒫[]]-(I/6)*epsilong1[-a,-b]*𝒽$p[]*CD1[b][𝒫[]]-((3*I)/4)*epsilong1[-a,b]*CD1[-c][𝒽$ab$p[-b,c]]+(I/4)*epsilong1[-a,b]*r[]^2*𝒫[]*CD1[-c][𝒽$ab$p[-b,c]]+(3*CD1[-a][r[]]*CD1[-b][r[]]*CD1[b][r[]]*CD1[-c][𝒽$a$m[c]])/(2*r[])+I*epsilong1[b,c]*r[]^2*CD1[-c][Π$ab$p[-a,-b]]-(r[]*𝒽$a$m[b]*CD1[-a][r[]]*CD1[-c][Τ[-b,c]])/2+(r[]^2*CD1[-a][𝒽$a$m[b]]*CD1[-c][Τ[-b,c]])/4-(r[]*𝒽$a$m[-a]*CD1[b][r[]]*CD1[-c][Τ[-b,c]])/2+(r[]^2*CD1[b][𝒽$a$m[-a]]*CD1[-c][Τ[-b,c]])/4+(I/12)*epsilong1[b,c]*r[]^2*𝒽$ab$p[-a,-b]*CD1[-c][Τ[d,-d]]-((7*I)/12)*epsilong1[-a,b]*r[]^2*𝒽$ab$p[-b,c]*CD1[-c][Τ[d,-d]]-(3*CD1[-b][r[]]*CD1[b][r[]]*CD1[-c][CD1[-a][𝒽$a$m[c]]])/4-(CD1[-b][r[]]*CD1[b][r[]]*CD1[-c][CD1[c][𝒽$a$m[-a]]])/4-(r[]^2*𝒽$a$m[-a]*CD1[-c][CD1[c][Τ[b,-b]]])/6-(7*𝒽$a$m[b]*Τ[-b,-c]*CD1[-a][r[]]*CD1[c][r[]])/2+(r[]*𝒽$a$m[b]*CD1[-a][Τ[-b,-c]]*CD1[c][r[]])/2-(𝒽$a$m[b]*Τ[-a,-c]*CD1[-b][r[]]*CD1[c][r[]])/2-2*epsilong1[-a,-c]*Ψ$a$m[b]*CD1[-b][r[]]*CD1[c][r[]]+𝒽$a$m[-a]*Τ[-b,-c]*CD1[b][r[]]*CD1[c][r[]]-(((3*I)/2)*epsilong1[-c,d]*𝒽$ab$p[-a,-d]*CD1[-b][r[]]*CD1[b][r[]]*CD1[c][r[]])/r[]-(((3*I)/2)*epsilong1[-a,d]*𝒽$ab$p[-c,-d]*CD1[-b][r[]]*CD1[b][r[]]*CD1[c][r[]])/r[]-(((3*I)/4)*epsilong1[-a,-c]*𝒽$ab$p[d,-d]*CD1[-b][r[]]*CD1[b][r[]]*CD1[c][r[]])/r[]+(((9*I)/2)*epsilong1[-a,-c]*𝒽$p[]*CD1[-b][r[]]*CD1[b][r[]]*CD1[c][r[]])/r[]^3+(9*CD1[-a][𝒽$a$m[-c]]*CD1[-b][r[]]*CD1[b][r[]]*CD1[c][r[]])/(2*r[])+(((3*I)/4)*epsilong1[-a,-b]*Ξ$a$p[b]*CD1[-c][r[]]*CD1[c][r[]])/r[]^2-(3*𝒽$a$m[b]*Τ[-a,-b]*CD1[-c][r[]]*CD1[c][r[]])/4-(3*epsilong1[-a,-b]*Ψ$a$m[b]*CD1[-c][r[]]*CD1[c][r[]])/4-(19*𝒽$a$m[b]*CD1[-a][r[]]*CD1[-b][r[]]*CD1[-c][r[]]*CD1[c][r[]])/(2*r[]^2)+(3*𝒽$a$m[-a]*CD1[-b][r[]]*CD1[b][r[]]*CD1[-c][r[]]*CD1[c][r[]])/(2*r[]^2)+(r[]*Τ[-b,-c]*CD1[b][r[]]*CD1[c][𝒽$a$m[-a]])/4-(((3*I)/4)*epsilong1[-a,-c]*CD1[-b][r[]]*CD1[b][r[]]*CD1[c][𝒽$p[]])/r[]^2-(I/4)*epsilong1[-a,b]*r[]^2*Τ[c,-c]*CD1[-d][𝒽$ab$p[-b,d]]+((3*I)/4)*epsilong1[-a,c]*CD1[-b][r[]]*CD1[b][r[]]*CD1[-d][𝒽$ab$p[-c,d]]-(I/4)*epsilong1[b,c]*r[]^2*𝒽$ab$p[-b,d]*CD1[-d][Τ[-a,-c]]},MetricOn->All];

(*{LHS$ZComp$ABPa$time,LHS$ZComp$ABPa$result}=AbsoluteTiming[LHSstep1=Pert$Exp$Sep[LHS,{μ,ν,ρ,σ},{A,B,P,a},{ℂ$Harm$Rep,𝒽$Harm$Rep}];
LHSstep2=LHSstep1/.Dir$Harm$Def$Total//ContractMetric//ToCanonical;
LHSstep3=((1/2)epsilong2[A,B]LHSstep2//ContractMetric//ToCanonical)//.Harm$EQ$Y$Total//ContractMetric//ToCanonical;
LHSstep4=ZA$Component[LHSstep3]//ContractMetric//ToCanonical;
LHSstep4
];
Print["Computing Time:"]
LHS$ZComp$ABPa$time
Print["Result:"]
LHS$ZComp$ABPa$result
{RHS$ZComp$ABPa$time,RHS$ZComp$ABPa$result}=AbsoluteTiming[RHSstep1=Pert$Exp$Sep[RHS,{μ,ν,ρ,σ},{A,B,P,a},{ℂ$Harm$Rep,𝒽$Harm$Rep,𝕁$Harm$Rep,𝒯$Harm$Rep}];
RHSstep2=RHSstep1/.Dir$Harm$Def$Total//ContractMetric//ToCanonical;
RHSstep3=((1/2)epsilong2[A,B]RHSstep2//ContractMetric//ToCanonical)//.Harm$EQ$Y$Total//ContractMetric//ToCanonical;
RHSstep4=ZA$Component[RHSstep3]//ContractMetric//ToCanonical;
RHSstep4
];
Print["Computing Time:"]
RHS$ZComp$ABPa$time
Print["Result:"]
RHS$ZComp$ABPa$result
Print["Mathc with saved result"]
Simplify[ToCanonical[(Δℂ$D𝕁$ZComp$ABPa[-a]/.Eval$Pert$Δℂ$D𝕁$ZComp$ABPa)-(LHS$ZComp$ABPa$result-RHS$ZComp$ABPa$result)]==0]*)

X-Component of δ((Subscript[Δℂ, A B P a])-((Subscript[D, A])-i ⋆(Subscript[D, A]))(Subscript[𝕁, B P a]))=0

Eval$Pert$Δℂ$D𝕁$XComp$ABPa=MakeRule[{Δℂ$D𝕁$XComp$ABPa[-a],(((3*I)/2)*epsilong1[-a,-b]*𝒽$a$m[b])/r[]^2+(((3*I)/4)*Λ^2*epsilong1[-a,-b]*𝒽$a$m[b])/r[]^2-I*epsilong1[-a,-b]*𝒽$a$m[b]*𝒫[]-(I/4)*Λ^2*epsilong1[-a,-b]*𝒽$a$m[b]*𝒫[]+(I/4)*epsilong1[-a,-b]*r[]^2*𝒽$a$m[b]*𝒫[]^2+Γ$a$p[-a]+(((3*I)/4)*epsilong1[-a,-b]*Ξ$a$m[b])/r[]^2-(I/4)*epsilong1[-a,-b]*𝒫[]*Ξ$a$m[b]-(5*Ξ$a$p[-a])/(4*r[]^2)-(Λ^2*Ξ$a$p[-a])/r[]^2+(3*𝒫[]*Ξ$a$p[-a])/4-Π$a$o[-a]+(Λ^2*Π$a$o[-a])/2-(Ξ$a$p[b]*Τ[-a,-b])/2+(I/4)*epsilong1[-b,c]*𝒽$a$m[b]*Τ[-a,-c]+(I/4)*epsilong1[-b,c]*r[]^2*𝒽$a$m[b]*𝒫[]*Τ[-a,-c]-(Ξ$a$p[-a]*Τ[b,-b])/4-(I/4)*epsilong1[-b,c]*r[]^2*𝒽$a$m[b]*Τ[-a,d]*Τ[-c,-d]+((5*I)/4)*epsilong1[-a,-b]*𝒽$a$m[b]*Τ[c,-c]+(I/4)*Λ^2*epsilong1[-a,-b]*𝒽$a$m[b]*Τ[c,-c]-(I/4)*epsilong1[-a,-b]*r[]^2*𝒽$a$m[b]*𝒫[]*Τ[c,-c]+(I/4)*epsilong1[-a,-b]*Ξ$a$m[b]*Τ[c,-c]+(I/4)*epsilong1[-b,c]*r[]^2*𝒽$a$m[b]*Τ[-a,-c]*Τ[d,-d]+(I/4)*epsilong1[-a,-b]*r[]^2*𝒽$a$m[b]*Τ[c,-c]*Τ[d,-d]+((3*I)/4)*Ψ$a$m[-a]-(I/4)*r[]^2*𝒫[]*Ψ$a$m[-a]+(I/4)*r[]^2*Τ[b,-b]*Ψ$a$m[-a]-(3*epsilong1[-a,-b]*Ψ$a$p[b])/4+(epsilong1[-a,-b]*r[]^2*𝒫[]*Ψ$a$p[b])/4+(epsilong1[-a,c]*r[]^2*Τ[-b,-c]*Ψ$a$p[b])/2-(epsilong1[-a,-b]*r[]^2*Τ[c,-c]*Ψ$a$p[b])/4-(3*𝒽$ab$p[b,-b]*CD1[-a][r[]])/(4*r[])+(13*𝒽$p[]*CD1[-a][r[]])/(2*r[]^3)+(r[]*𝒽$ab$p[b,-b]*𝒫[]*CD1[-a][r[]])/4-(5*𝒽$p[]*𝒫[]*CD1[-a][r[]])/(3*r[])-(Γ$p[]*CD1[-a][r[]])/r[]-(2*Ξ$p[]*CD1[-a][r[]])/r[]^3+I*r[]*Π$m[]*CD1[-a][r[]]+(13*𝒽$p[]*Τ[b,-b]*CD1[-a][r[]])/(6*r[])-(r[]*𝒽$ab$p[b,-b]*Τ[c,-c]*CD1[-a][r[]])/4-(3*CD1[-a][𝒽$p[]])/(4*r[]^2)+(𝒫[]*CD1[-a][𝒽$p[]])/4-(Τ[b,-b]*CD1[-a][𝒽$p[]])/4+(𝒽$p[]*CD1[-a][𝒫[]])/6-I*r[]^2*CD1[-a][Π$m[]]-(𝒽$p[]*CD1[-a][Τ[b,-b]])/6-(2*Ξ$a$p[b]*CD1[-a][r[]]*CD1[-b][r[]])/r[]^2+(3*CD1[-b][𝒽$ab$p[-a,b]])/4-(r[]^2*𝒫[]*CD1[-b][𝒽$ab$p[-a,b]])/4+(r[]^2*𝒽$ab$p[-a,b]*CD1[-b][Τ[c,-c]])/4+CD1[-b][CD1[b][Ξ$a$p[-a]]]+((3*I)/4)*epsilong1[-a,-c]*CD1[-b][CD1[c][𝒽$a$m[b]]]-(I/4)*epsilong1[-a,-c]*r[]^2*𝒫[]*CD1[-b][CD1[c][𝒽$a$m[b]]]+(I/4)*epsilong1[-a,-c]*r[]^2*Τ[d,-d]*CD1[-b][CD1[c][𝒽$a$m[b]]]-(2*𝒽$ab$p[-a,-b]*CD1[b][r[]])/r[]-(7*r[]*𝒽$ab$p[-a,-b]*𝒫[]*CD1[b][r[]])/12+I*epsilong1[-b,c]*r[]*Π$ab$m[-a,-c]*CD1[b][r[]]-2*epsilong1[-a,-b]*r[]*Π$p[]*CD1[b][r[]]-(𝒽$p[]*Τ[-a,-b]*CD1[b][r[]])/(2*r[])+(5*r[]*𝒽$ab$p[-a,c]*Τ[-b,-c]*CD1[b][r[]])/4-(2*r[]*𝒽$ab$p[-a,-b]*Τ[c,-c]*CD1[b][r[]])/3-(2*Ψ$ab$o[-a,-b]*CD1[b][r[]])/r[]+(Λ^2*Ψ$ab$o[-a,-b]*CD1[b][r[]])/r[]+(2*Ψ$ab$p[-a,-b]*CD1[b][r[]])/r[]+(((3*I)/2)*epsilong1[-b,-c]*CD1[-a][𝒽$a$m[c]]*CD1[b][r[]])/r[]-(I/4)*epsilong1[-b,-c]*r[]*𝒫[]*CD1[-a][𝒽$a$m[c]]*CD1[b][r[]]+(I/4)*epsilong1[-c,d]*r[]*Τ[-b,-d]*CD1[-a][𝒽$a$m[c]]*CD1[b][r[]]+(I/2)*epsilong1[-b,-c]*r[]*Τ[d,-d]*CD1[-a][𝒽$a$m[c]]*CD1[b][r[]]+(21*Ξ$a$p[-a]*CD1[-b][r[]]*CD1[b][r[]])/(4*r[]^2)-((3*I)/4)*Ψ$a$m[-a]*CD1[-b][r[]]*CD1[b][r[]]+(3*𝒽$ab$p[c,-c]*CD1[-a][r[]]*CD1[-b][r[]]*CD1[b][r[]])/(4*r[])-(13*𝒽$p[]*CD1[-a][r[]]*CD1[-b][r[]]*CD1[b][r[]])/(2*r[]^3)+(3*CD1[-a][𝒽$p[]]*CD1[-b][r[]]*CD1[b][r[]])/(4*r[]^2)+(((9*I)/2)*epsilong1[-a,-c]*CD1[-b][𝒽$a$m[c]]*CD1[b][r[]])/r[]+((3*I)/2)*epsilong1[-a,-c]*r[]*Τ[d,-d]*CD1[-b][𝒽$a$m[c]]*CD1[b][r[]]-(4*CD1[-b][Ξ$a$p[-a]]*CD1[b][r[]])/r[]-(r[]^2*𝒽$ab$p[-a,-b]*CD1[b][𝒫[]])/4+(r[]^2*Τ[b,-b]*CD1[-c][𝒽$ab$p[-a,c]])/4-(3*CD1[-b][r[]]*CD1[b][r[]]*CD1[-c][𝒽$ab$p[-a,c]])/4-(((3*I)/2)*epsilong1[-a,-b]*CD1[b][r[]]*CD1[-c][𝒽$a$m[c]])/r[]+(I/2)*epsilong1[-a,-b]*r[]*𝒫[]*CD1[b][r[]]*CD1[-c][𝒽$a$m[c]]-(I/2)*epsilong1[-a,-b]*r[]*Τ[d,-d]*CD1[b][r[]]*CD1[-c][𝒽$a$m[c]]+I*epsilong1[b,c]*r[]^2*CD1[-c][Π$ab$m[-a,-b]]+(r[]^2*𝒽$ab$p[-a,b]*CD1[-c][Τ[-b,c]])/4-(I/6)*epsilong1[-b,c]*r[]*𝒽$a$m[b]*CD1[-a][r[]]*CD1[-c][Τ[d,-d]]+(I/12)*epsilong1[-b,c]*r[]^2*CD1[-a][𝒽$a$m[b]]*CD1[-c][Τ[d,-d]]+(I/6)*epsilong1[-a,c]*r[]*𝒽$a$m[b]*CD1[-b][r[]]*CD1[-c][Τ[d,-d]]-(I/12)*epsilong1[-b,c]*r[]^2*CD1[b][𝒽$a$m[-a]]*CD1[-c][Τ[d,-d]]-(I/6)*epsilong1[-b,c]*r[]^2*𝒽$a$m[b]*CD1[-c][CD1[-a][𝒫[]]]+(I/6)*epsilong1[-b,c]*r[]^2*𝒽$a$m[b]*CD1[-c][CD1[-a][Τ[d,-d]]]-((3*I)/4)*epsilong1[-a,-b]*CD1[-c][CD1[c][𝒽$a$m[b]]]+(I/4)*epsilong1[-a,-b]*r[]^2*𝒫[]*CD1[-c][CD1[c][𝒽$a$m[b]]]-(I/4)*epsilong1[-a,-b]*r[]^2*Τ[d,-d]*CD1[-c][CD1[c][𝒽$a$m[b]]]-((3*I)/4)*epsilong1[-a,-d]*CD1[-b][r[]]*CD1[b][r[]]*CD1[-c][CD1[d][𝒽$a$m[c]]]+((3*I)*epsilong1[-b,-c]*𝒽$a$m[b]*CD1[-a][r[]]*CD1[c][r[]])/r[]^2-(I/2)*epsilong1[-b,-c]*𝒽$a$m[b]*𝒫[]*CD1[-a][r[]]*CD1[c][r[]]-(I/2)*epsilong1[-b,d]*𝒽$a$m[b]*Τ[-c,-d]*CD1[-a][r[]]*CD1[c][r[]]+I*epsilong1[-b,-c]*𝒽$a$m[b]*Τ[d,-d]*CD1[-a][r[]]*CD1[c][r[]]-((5*I)/6)*epsilong1[-b,-c]*r[]*𝒽$a$m[b]*CD1[-a][𝒫[]]*CD1[c][r[]]-(I/2)*epsilong1[-c,d]*r[]*𝒽$a$m[b]*CD1[-a][Τ[-b,-d]]*CD1[c][r[]]+(I/2)*epsilong1[-b,d]*r[]*𝒽$a$m[b]*CD1[-a][Τ[-c,-d]]*CD1[c][r[]]-(I/6)*epsilong1[-b,-c]*r[]*𝒽$a$m[b]*CD1[-a][Τ[d,-d]]*CD1[c][r[]]+(((9*I)/2)*epsilong1[-a,-c]*𝒽$a$m[b]*CD1[-b][r[]]*CD1[c][r[]])/r[]^2+I*epsilong1[-a,-c]*𝒽$a$m[b]*𝒫[]*CD1[-b][r[]]*CD1[c][r[]]+(I/2)*epsilong1[-a,d]*𝒽$a$m[b]*Τ[-c,-d]*CD1[-b][r[]]*CD1[c][r[]]+((3*I)/2)*epsilong1[-a,-c]*𝒽$a$m[b]*Τ[d,-d]*CD1[-b][r[]]*CD1[c][r[]]+2*epsilong1[-a,-c]*Ψ$a$p[b]*CD1[-b][r[]]*CD1[c][r[]]+((4*I)/3)*epsilong1[-a,-c]*r[]*𝒽$a$m[b]*CD1[-b][𝒫[]]*CD1[c][r[]]+(I/2)*epsilong1[-c,d]*r[]*𝒽$a$m[b]*CD1[-b][Τ[-a,-d]]*CD1[c][r[]]-((5*I)/6)*epsilong1[-a,-c]*r[]*𝒽$a$m[b]*CD1[-b][Τ[d,-d]]*CD1[c][r[]]+(2*𝒽$ab$p[-a,-c]*CD1[-b][r[]]*CD1[b][r[]]*CD1[c][r[]])/r[]-(((3*I)/2)*epsilong1[-c,-d]*CD1[-a][𝒽$a$m[d]]*CD1[-b][r[]]*CD1[b][r[]]*CD1[c][r[]])/r[]-((12*I)*epsilong1[-a,-b]*𝒽$a$m[b]*CD1[-c][r[]]*CD1[c][r[]])/r[]^2-(((3*I)/4)*Λ^2*epsilong1[-a,-b]*𝒽$a$m[b]*CD1[-c][r[]]*CD1[c][r[]])/r[]^2+I*epsilong1[-a,-b]*𝒽$a$m[b]*𝒫[]*CD1[-c][r[]]*CD1[c][r[]]-(((3*I)/4)*epsilong1[-a,-b]*Ξ$a$m[b]*CD1[-c][r[]]*CD1[c][r[]])/r[]^2-(I/4)*epsilong1[-b,d]*𝒽$a$m[b]*Τ[-a,-d]*CD1[-c][r[]]*CD1[c][r[]]-((17*I)/4)*epsilong1[-a,-b]*𝒽$a$m[b]*Τ[d,-d]*CD1[-c][r[]]*CD1[c][r[]]+(3*epsilong1[-a,-b]*Ψ$a$p[b]*CD1[-c][r[]]*CD1[c][r[]])/4-(((9*I)/2)*epsilong1[-a,-d]*CD1[-b][r[]]*CD1[b][r[]]*CD1[-c][𝒽$a$m[d]]*CD1[c][r[]])/r[]-((4*I)/3)*epsilong1[-a,-b]*r[]*𝒽$a$m[b]*CD1[-c][𝒫[]]*CD1[c][r[]]-(I/2)*epsilong1[-b,d]*r[]*𝒽$a$m[b]*CD1[-c][Τ[-a,-d]]*CD1[c][r[]]+((5*I)/6)*epsilong1[-a,-b]*r[]*𝒽$a$m[b]*CD1[-c][Τ[d,-d]]*CD1[c][r[]]-(((3*I)/2)*epsilong1[-b,-c]*CD1[b][r[]]*CD1[c][𝒽$a$m[-a]])/r[]+(I/4)*epsilong1[-b,-c]*r[]*𝒫[]*CD1[b][r[]]*CD1[c][𝒽$a$m[-a]]-(I/4)*epsilong1[-c,d]*r[]*Τ[-b,-d]*CD1[b][r[]]*CD1[c][𝒽$a$m[-a]]-(I/2)*epsilong1[-b,-c]*r[]*Τ[d,-d]*CD1[b][r[]]*CD1[c][𝒽$a$m[-a]]-((3*I)*epsilong1[-a,-c]*CD1[b][r[]]*CD1[c][𝒽$a$m[-b]])/r[]-(I/2)*epsilong1[-a,-c]*r[]*𝒫[]*CD1[b][r[]]*CD1[c][𝒽$a$m[-b]]-I*epsilong1[-a,-c]*r[]*Τ[d,-d]*CD1[b][r[]]*CD1[c][𝒽$a$m[-b]]+(I/3)*epsilong1[-b,-c]*r[]^2*CD1[-a][𝒫[]]*CD1[c][𝒽$a$m[b]]+(I/4)*epsilong1[-c,d]*r[]^2*CD1[-a][Τ[-b,-d]]*CD1[c][𝒽$a$m[b]]-(I/4)*epsilong1[-b,d]*r[]^2*CD1[-a][Τ[-c,-d]]*CD1[c][𝒽$a$m[b]]+(I/6)*epsilong1[-b,-c]*r[]^2*CD1[-a][Τ[d,-d]]*CD1[c][𝒽$a$m[b]]-((2*I)/3)*epsilong1[-a,-c]*r[]^2*CD1[-b][𝒫[]]*CD1[c][𝒽$a$m[b]]-(I/4)*epsilong1[-c,d]*r[]^2*CD1[-b][Τ[-a,-d]]*CD1[c][𝒽$a$m[b]]+((5*I)/12)*epsilong1[-a,-c]*r[]^2*CD1[-b][Τ[d,-d]]*CD1[c][𝒽$a$m[b]]+((2*I)/3)*epsilong1[-a,-b]*r[]^2*CD1[-c][𝒫[]]*CD1[c][𝒽$a$m[b]]+(I/4)*epsilong1[-b,d]*r[]^2*CD1[-c][Τ[-a,-d]]*CD1[c][𝒽$a$m[b]]-((5*I)/12)*epsilong1[-a,-b]*r[]^2*CD1[-c][Τ[d,-d]]*CD1[c][𝒽$a$m[b]]-(I/3)*epsilong1[-b,-c]*r[]*𝒽$a$m[b]*CD1[-a][r[]]*CD1[c][𝒫[]]-(I/12)*epsilong1[-b,-c]*r[]^2*CD1[-a][𝒽$a$m[b]]*CD1[c][𝒫[]]-(I/6)*epsilong1[-a,-c]*r[]*𝒽$a$m[b]*CD1[-b][r[]]*CD1[c][𝒫[]]+(I/12)*epsilong1[-b,-c]*r[]^2*CD1[b][𝒽$a$m[-a]]*CD1[c][𝒫[]]+(((3*I)/2)*epsilong1[-a,-c]*CD1[-b][r[]]*CD1[b][r[]]*CD1[c][r[]]*CD1[-d][𝒽$a$m[d]])/r[]+(I/2)*epsilong1[-b,d]*r[]*𝒽$a$m[b]*CD1[c][r[]]*CD1[-d][Τ[-a,-c]]+((3*I)/4)*epsilong1[-a,-c]*CD1[-b][r[]]*CD1[b][r[]]*CD1[-d][CD1[d][𝒽$a$m[c]]]-(3*I)*epsilong1[-a,-c]*𝒽$a$m[b]*Τ[-b,-d]*CD1[c][r[]]*CD1[d][r[]]+(3*I)*epsilong1[-a,-b]*𝒽$a$m[b]*Τ[-c,-d]*CD1[c][r[]]*CD1[d][r[]]-((3*I)*epsilong1[-b,-d]*𝒽$a$m[b]*CD1[-a][r[]]*CD1[-c][r[]]*CD1[c][r[]]*CD1[d][r[]])/r[]^2-(((9*I)/2)*epsilong1[-a,-d]*𝒽$a$m[b]*CD1[-b][r[]]*CD1[-c][r[]]*CD1[c][r[]]*CD1[d][r[]])/r[]^2+(((21*I)/2)*epsilong1[-a,-b]*𝒽$a$m[b]*CD1[-c][r[]]*CD1[c][r[]]*CD1[-d][r[]]*CD1[d][r[]])/r[]^2+(((3*I)/2)*epsilong1[-c,-d]*CD1[-b][r[]]*CD1[b][r[]]*CD1[c][r[]]*CD1[d][𝒽$a$m[-a]])/r[]+((3*I)*epsilong1[-a,-d]*CD1[-b][r[]]*CD1[b][r[]]*CD1[c][r[]]*CD1[d][𝒽$a$m[-c]])/r[]+((3*I)/2)*epsilong1[-a,-d]*r[]*Τ[-b,-c]*CD1[b][r[]]*CD1[d][𝒽$a$m[c]]-((3*I)/2)*epsilong1[-a,-c]*r[]*Τ[-b,-d]*CD1[b][r[]]*CD1[d][𝒽$a$m[c]]},MetricOn->All];

(*{LHS$XComp$ABPa$time,LHS$XComp$ABPa$result}=AbsoluteTiming[LHSstep1=Pert$Exp$Sep[LHS,{μ,ν,ρ,σ},{A,B,P,a},{ℂ$Harm$Rep,𝒽$Harm$Rep}];
LHSstep2=LHSstep1/.Dir$Harm$Def$Total//ContractMetric//ToCanonical;
LHSstep3=((1/2)epsilong2[A,B]LHSstep2//ContractMetric//ToCanonical)//.Harm$EQ$Y$Total//ContractMetric//ToCanonical;
LHSstep4=XA$Component[LHSstep3]//ContractMetric//ToCanonical;
LHSstep4
];
Print["Computing Time:"]
LHS$XComp$ABPa$time
Print["Result:"]
LHS$XComp$ABPa$result
{RHS$XComp$ABPa$time,RHS$XComp$ABPa$result}=AbsoluteTiming[RHSstep1=Pert$Exp$Sep[RHS,{μ,ν,ρ,σ},{A,B,P,a},{ℂ$Harm$Rep,𝒽$Harm$Rep,𝕁$Harm$Rep,𝒯$Harm$Rep}];
RHSstep2=RHSstep1/.Dir$Harm$Def$Total//ContractMetric//ToCanonical;
RHSstep3=((1/2)epsilong2[A,B]RHSstep2//ContractMetric//ToCanonical)//.Harm$EQ$Y$Total//ContractMetric//ToCanonical;
RHSstep4=XA$Component[RHSstep3]//ContractMetric//ToCanonical;
RHSstep4
];
Print["Computing Time:"]
RHS$XComp$ABPa$time
Print["Result:"]
RHS$XComp$ABPa$result
Print["Match with saved result"]
Simplify[ToCanonical[(Δℂ$D𝕁$XComp$ABPa[-a]/.Eval$Pert$Δℂ$D𝕁$XComp$ABPa)-(LHS$XComp$ABPa$result-RHS$XComp$ABPa$result)]==0]*)

δ((Subscript[Δℂ, A B P Q])-((Subscript[D, A])-i ⋆(Subscript[D, A]))(Subscript[𝕁, B P Q]))=0

Eval$Pert$Δℂ$D𝕁$ABPQ=MakeRule[{Δℂ$D𝕁$ABPQ[],(-3*𝒽$ab$p[a,-a])/2+(4*𝒽$p[])/r[]^2+(2*Λ^2*𝒽$p[])/r[]^2+2*r[]^2*𝒽$ab$p[a,-a]*𝒫[]-(7*𝒽$p[]*𝒫[])/3-(2*Λ^2*𝒽$p[]*𝒫[])/3-(r[]^4*𝒽$ab$p[a,-a]*𝒫[]^2)/3+(r[]^2*𝒽$p[]*𝒫[]^2)/3-Λ^2*Γ$p[]-(4*Ξ$p[])/r[]^2-(Λ^2*Ξ$p[])/r[]^2+𝒫[]*Ξ$p[]-2*r[]^2*𝒽$ab$p[a,b]*Τ[-a,-b]+(17*𝒽$p[]*Τ[a,-a])/6+(2*Λ^2*𝒽$p[]*Τ[a,-a])/3-(5*r[]^2*𝒽$p[]*𝒫[]*Τ[a,-a])/6-Ξ$p[]*Τ[a,-a]+(r[]^4*𝒽$ab$p[a,b]*Τ[-a,c]*Τ[-b,-c])/2+(2*r[]^4*𝒽$ab$p[a,-a]*𝒫[]*Τ[b,-b])/3+(r[]^2*𝒽$p[]*Τ[a,-a]*Τ[b,-b])/2-r[]^4*𝒽$ab$p[a,b]*Τ[-a,-b]*Τ[c,-c]+(r[]^4*𝒽$ab$p[a,-a]*Τ[b,-b]*Τ[c,-c])/6+(r[]^2*Φ$ab$p[a,-a])/2-(r[]^4*𝒫[]*Φ$ab$p[a,-a])/6+(r[]^4*Τ[b,-b]*Φ$ab$p[a,-a])/6+Φ$p[]-(r[]^2*𝒫[]*Φ$p[])/3+(r[]^2*Τ[a,-a]*Φ$p[])/3-3*Ψ$ab$p[a,-a]+r[]^2*𝒫[]*Ψ$ab$p[a,-a]-r[]^2*Τ[b,-b]*Ψ$ab$p[a,-a]+r[]^2*Τ[-a,-b]*Ψ$ab$p[a,b]+(3*I)*Ψ$m[]-I*r[]^2*𝒫[]*Ψ$m[]+I*r[]^2*Τ[a,-a]*Ψ$m[]+2*r[]*Γ$a$p[a]*CD1[-a][r[]]-(4*Λ^2*Ξ$a$p[a]*CD1[-a][r[]])/r[]-2*CD1[-a][CD1[a][𝒽$p[]]]+(2*r[]^2*𝒫[]*CD1[-a][CD1[a][𝒽$p[]]])/3+(r[]^2*𝒽$p[]*CD1[-a][CD1[a][𝒫[]]])/3+CD1[-a][CD1[a][Ξ$p[]]]+3*𝒽$ab$p[b,-b]*CD1[-a][r[]]*CD1[a][r[]]-(20*𝒽$p[]*CD1[-a][r[]]*CD1[a][r[]])/r[]^2-(2*Λ^2*𝒽$p[]*CD1[-a][r[]]*CD1[a][r[]])/r[]^2-2*r[]^2*𝒽$ab$p[b,-b]*𝒫[]*CD1[-a][r[]]*CD1[a][r[]]+(𝒽$p[]*𝒫[]*CD1[-a][r[]]*CD1[a][r[]])/3+(12*Ξ$p[]*CD1[-a][r[]]*CD1[a][r[]])/r[]^2+2*r[]^2*𝒽$ab$p[b,c]*Τ[-b,-c]*CD1[-a][r[]]*CD1[a][r[]]-(41*𝒽$p[]*Τ[b,-b]*CD1[-a][r[]]*CD1[a][r[]])/6-(r[]^2*Φ$ab$p[b,-b]*CD1[-a][r[]]*CD1[a][r[]])/2-Φ$p[]*CD1[-a][r[]]*CD1[a][r[]]+3*Ψ$ab$p[b,-b]*CD1[-a][r[]]*CD1[a][r[]]-(3*I)*Ψ$m[]*CD1[-a][r[]]*CD1[a][r[]]-(3*r[]*CD1[-a][𝒽$ab$p[b,-b]]*CD1[a][r[]])/2-(r[]^3*Τ[b,-b]*CD1[-a][𝒽$ab$p[c,-c]]*CD1[a][r[]])/2+(7*CD1[-a][𝒽$p[]]*CD1[a][r[]])/r[]+(r[]*𝒫[]*CD1[-a][𝒽$p[]]*CD1[a][r[]])/6+(7*r[]*Τ[b,-b]*CD1[-a][𝒽$p[]]*CD1[a][r[]])/3-(4*r[]*𝒽$p[]*CD1[-a][𝒫[]]*CD1[a][r[]])/3-(6*CD1[-a][Ξ$p[]]*CD1[a][r[]])/r[]+(4*r[]*𝒽$p[]*CD1[-a][Τ[b,-b]]*CD1[a][r[]])/3+(r[]^2*CD1[-a][𝒫[]]*CD1[a][𝒽$p[]])/2-(r[]^2*CD1[-a][Τ[b,-b]]*CD1[a][𝒽$p[]])/2-(r[]^4*CD1[-a][𝒽$ab$p[b,-b]]*CD1[a][𝒫[]])/6+3*r[]*CD1[a][r[]]*CD1[-b][𝒽$ab$p[-a,b]]+(r[]^4*CD1[a][𝒫[]]*CD1[-b][𝒽$ab$p[-a,b]])/3+(r[]^3*Τ[-a,b]*CD1[a][r[]]*CD1[-b][𝒽$ab$p[c,-c]])/2+r[]*𝒽$p[]*CD1[a][r[]]*CD1[-b][Τ[-a,b]]-(r[]^2*CD1[a][𝒽$p[]]*CD1[-b][Τ[-a,b]])/2-(r[]^4*CD1[-a][𝒽$ab$p[a,b]]*CD1[-b][Τ[c,-c]])/3+(r[]^4*𝒽$ab$p[a,b]*CD1[-b][CD1[-a][𝒫[]]])/3+(r[]^2*𝒽$p[]*CD1[-b][CD1[-a][Τ[a,b]]])/2+(r[]^4*𝒽$ab$p[a,b]*CD1[-b][CD1[-a][Τ[c,-c]]])/6-(2*r[]^2*Τ[a,-a]*CD1[-b][CD1[b][𝒽$p[]]])/3+2*CD1[-a][r[]]*CD1[a][r[]]*CD1[-b][CD1[b][𝒽$p[]]]-(r[]^4*𝒽$ab$p[a,-a]*CD1[-b][CD1[b][𝒫[]]])/6-(r[]^2*𝒽$p[]*CD1[-b][CD1[b][Τ[a,-a]]])/3-(r[]^4*𝒽$ab$p[a,b]*CD1[-b][CD1[-c][Τ[-a,c]]])/2-(4*I)*epsilong1[-a,-b]*r[]*Π$a$m[a]*CD1[b][r[]]+2*𝒽$ab$p[-a,-b]*CD1[a][r[]]*CD1[b][r[]]-(5*r[]^2*𝒽$ab$p[-a,-b]*𝒫[]*CD1[a][r[]]*CD1[b][r[]])/3+6*𝒽$p[]*Τ[-a,-b]*CD1[a][r[]]*CD1[b][r[]]+r[]^2*𝒽$ab$p[-a,c]*Τ[-b,-c]*CD1[a][r[]]*CD1[b][r[]]+(2*r[]^2*𝒽$ab$p[-a,-b]*Τ[c,-c]*CD1[a][r[]]*CD1[b][r[]])/3+4*Ψ$ab$p[-a,-b]*CD1[a][r[]]*CD1[b][r[]]-(3*𝒽$ab$p[c,-c]*CD1[-a][r[]]*CD1[a][r[]]*CD1[-b][r[]]*CD1[b][r[]])/2+(16*𝒽$p[]*CD1[-a][r[]]*CD1[a][r[]]*CD1[-b][r[]]*CD1[b][r[]])/r[]^2+(3*r[]*CD1[-a][r[]]*CD1[a][r[]]*CD1[-b][𝒽$ab$p[c,-c]]*CD1[b][r[]])/2-(7*CD1[-a][r[]]*CD1[a][r[]]*CD1[-b][𝒽$p[]]*CD1[b][r[]])/r[]+(r[]^4*CD1[-b][Τ[c,-c]]*CD1[b][𝒽$ab$p[a,-a]])/6-(5*r[]*Τ[-a,-b]*CD1[a][r[]]*CD1[b][𝒽$p[]])/2+r[]^3*𝒽$ab$p[-a,-b]*CD1[a][r[]]*CD1[b][𝒫[]]+(2*I)*epsilong1[-a,-b]*r[]^2*CD1[b][Π$a$m[a]]+r[]^3*Τ[b,-b]*CD1[a][r[]]*CD1[-c][𝒽$ab$p[-a,c]]-r[]^3*Τ[-a,b]*CD1[a][r[]]*CD1[-c][𝒽$ab$p[-b,c]]-3*r[]*CD1[-a][r[]]*CD1[a][r[]]*CD1[b][r[]]*CD1[-c][𝒽$ab$p[-b,c]]-r[]^3*𝒽$ab$p[b,c]*CD1[a][r[]]*CD1[-c][Τ[-a,-b]]-(r[]^4*𝒽$ab$p[a,b]*CD1[-c][CD1[-b][Τ[-a,c]]])/2+(r[]^4*𝒽$ab$p[a,-a]*CD1[-c][CD1[-b][Τ[b,c]]])/4+(r[]^4*𝒽$ab$p[a,b]*CD1[-c][CD1[c][Τ[-a,-b]]])/2-(r[]^4*𝒽$ab$p[a,-a]*CD1[-c][CD1[c][Τ[b,-b]]])/3-2*𝒽$ab$p[-b,-c]*CD1[-a][r[]]*CD1[a][r[]]*CD1[b][r[]]*CD1[c][r[]]},MetricOn->All];

(*{LHS$ABPQ$time,LHS$ABPQ$result}=AbsoluteTiming[LHSstep1=Pert$Exp$Sep[LHS,{μ,ν,ρ,σ},{A,B,P,Q},{ℂ$Harm$Rep,𝒽$Harm$Rep}];
LHSstep2=LHSstep1/.Dir$Harm$Def$Total//ContractMetric//ToCanonical;
LHSstep3=((1/4)epsilong2[A,B]epsilong2[P,Q]LHSstep2//ContractMetric//ToCanonical)//.Harm$EQ$Y$Total//ContractMetric//ToCanonical;
LHSstep3];
Print["Computing Time:"]
LHS$ABPQ$time
Print["Result:"]
LHS$ABPQ$result
{RHS$ABPQ$time,RHS$ABPQ$result}=AbsoluteTiming[RHSstep1=Pert$Exp$Sep[RHS,{μ,ν,ρ,σ},{A,B,P,Q},{ℂ$Harm$Rep,𝒽$Harm$Rep,𝕁$Harm$Rep,𝒯$Harm$Rep}];
RHSstep2=RHSstep1/.Dir$Harm$Def$Total//ContractMetric//ToCanonical;
RHSstep3=((1/4)epsilong2[A,B]epsilong2[P,Q]RHSstep2//ContractMetric//ToCanonical)//.Harm$EQ$Y$Total//ContractMetric//ToCanonical;
RHSstep3];
Print["Computing Time:"]
RHS$ABPQ$time
Print["Result:"]
RHS$ABPQ$result
Print["Match with saved result:"]
Simplify[ToCanonical[(Δℂ$D𝕁$ABPQ[]/.Eval$Pert$Δℂ$D𝕁$ABPQ)-((LHS$ABPQ$result-RHS$ABPQ$result)/Y[])]==0]*)

Total

Eval$Pert$Δℂ$D𝕁$Total=Flatten[{Eval$Pert$Δℂ$D𝕁$abcd,Eval$Pert$Δℂ$D𝕁$ZComp$Pabc,Eval$Pert$Δℂ$D𝕁$XComp$Pabc,Eval$Pert$Δℂ$D𝕁$abPQ,Eval$Pert$Δℂ$D𝕁$UComp$PaQb,Eval$Pert$Δℂ$D𝕁$VComp$PaQb,Eval$Pert$Δℂ$D𝕁$WComp$PaQb,Eval$Pert$Δℂ$D𝕁$ϵComp$PaQb,Eval$Pert$Δℂ$D𝕁$ϵCompSkew$PaQb,Eval$Pert$Δℂ$D𝕁$ZComp$ABPa,Eval$Pert$Δℂ$D𝕁$XComp$ABPa,Eval$Pert$Δℂ$D𝕁$ABPQ}];

1.9. Frame Perturbations

(Subscript[(OverDot[l]), a]):

Pert$l$1=MakeRule[{(dℓ[a]*l[-a]),(l[a]*l[b]*Y[]*𝒽$ab$p[-a,-b])/2}];
Pert$l$2=MakeRule[{dℓ[a]*n[-a],-(d𝓃[a]*l[-a])+l[a]*n[b]*Y[]*𝒽$ab$p[-a,-b]}];
Pert$l$3=MakeRule[{(dℓ[A]*m[-A]),-Sqrt[2]*d𝓂[a]*l[-a]*r[]+l[a]*m[A]*X[-A]*𝒽$a$m[-a]}];
Pert$l$4=MakeRule[{(dℓ[A]*mC[-A]),-Sqrt[2]*d𝓂C[a]*l[-a]*r[]+l[a]*mC[A]*X[-A]*𝒽$a$m[-a]}];
Pert$l=Flatten[{Pert$l$1,Pert$l$2,Pert$l$3,Pert$l$4}];
(*(Perturbation[ℓ[-μ]]ℓ[μ]+ℓ[-μ]Perturbation[ℓ[μ]]);
%//ExpandPerturbation//SeparateMetric[];
%/.Pert$Frame//TraceProductDummy;
ExpandProductMetric[%,g3];
%/.Flatten[{𝒽$Harm$Rep,FourD$To$TwoD}];
%//ContractMetric//ToCanonical;
(Perturbation[ℓ[-μ]]𝓃[μ]+ℓ[-μ]Perturbation[𝓃[μ]]);
%//ExpandPerturbation//SeparateMetric[];
%/.Pert$Frame//TraceProductDummy;
ExpandProductMetric[%,g3];
%/.Flatten[{𝒽$Harm$Rep,FourD$To$TwoD}];
%//ContractMetric//ToCanonical;
(Perturbation[ℓ[-μ]]𝓂[μ]+ℓ[-μ]Perturbation[𝓂[μ]]);
%//ExpandPerturbation//SeparateMetric[];
%/.Pert$Frame//TraceProductDummy;
ExpandProductMetric[%,g3];
%/.Flatten[{𝒽$Harm$Rep,FourD$To$TwoD}];
%//ContractMetric//ToCanonical;
(Perturbation[ℓ[-μ]]𝓂C[μ]+ℓ[-μ]Perturbation[𝓂C[μ]]);
%//ExpandPerturbation//SeparateMetric[];
%/.Pert$Frame//TraceProductDummy;
ExpandProductMetric[%,g3];
%/.Flatten[{𝒽$Harm$Rep,FourD$To$TwoD}];
%//ContractMetric//ToCanonical;*)

(Subscript[(OverDot[n]), a]):

Pert$n$1=MakeRule[{d𝓃[a]*n[-a],+(n[a]*n[b]*Y[]*𝒽$ab$p[-a,-b])/2}];
Pert$n$2=MakeRule[{d𝓃[A]*m[-A],-Sqrt[2]*d𝓂[a]*n[-a]*r[]+m[A]*n[a]*X[-A]*𝒽$a$m[-a]}];
Pert$n$3=MakeRule[{d𝓃[A]*mC[-A],-Sqrt[2]*d𝓂C[a]*n[-a]*r[]+mC[A]*n[a]*X[-A]*𝒽$a$m[-a]}];
Pert$n=Flatten[{Pert$n$1,Pert$n$2,Pert$n$3}];

(*(Perturbation[𝓃[-μ]]𝓃[μ]+𝓃[-μ]Perturbation[𝓃[μ]])
%//ExpandPerturbation//SeparateMetric[];
%/.Pert$Frame//TraceProductDummy;
ExpandProductMetric[%,g3];
%/.Flatten[{h$Harm$Rep,FourD$To$TwoD}];
%//ContractMetric//ToCanonical
(Perturbation[𝓃[-μ]]𝓂[μ]+𝓃[-μ]Perturbation[𝓂[μ]])
%//ExpandPerturbation//SeparateMetric[];
%/.Pert$Frame//TraceProductDummy;
ExpandProductMetric[%,g3];
%/.Flatten[{h$Harm$Rep,FourD$To$TwoD}];
%//ContractMetric//ToCanonical
(Perturbation[𝓃[-μ]]𝓂C[μ]+𝓃[-μ]Perturbation[𝓂C[μ]])
%//ExpandPerturbation//SeparateMetric[];
%/.Pert$Frame//TraceProductDummy;
ExpandProductMetric[%,g3];
%/.Flatten[{h$Harm$Rep,FourD$To$TwoD}];
%//ContractMetric//ToCanonical*)

(Subscript[(OverDot[m]), a]):

Pert$m$1=MakeRule[{d𝓂[A]*m[-A],+(m[A]*m[B]*U[-A,-B]*𝒽$p[])/(2*Sqrt[2]*r[])}];
Pert$m$2=MakeRule[{d𝓂[A]*mC[-A],-(d𝓂C[A]*m[-A])+(m[A]*mC[B]*U[-A,-B]*𝒽$p[])/(Sqrt[2]*r[])}];
Pert$m=Flatten[{Pert$m$1,Pert$m$2}];

(*(Perturbation[𝓂[-μ]]𝓂[μ]+𝓂[-μ]Perturbation[𝓂[μ]])
%//ExpandPerturbation//SeparateMetric[];
%/.Pert$Frame//TraceProductDummy;
ExpandProductMetric[%,g3];
%/.Flatten[{h$Harm$Rep,FourD$To$TwoD}];
%//ContractMetric//ToCanonical
(Perturbation[𝓂[-μ]]𝓂C[μ]+𝓂[-μ]Perturbation[𝓂C[μ]])
%//ExpandPerturbation//SeparateMetric[];
%/.Pert$Frame//TraceProductDummy;
ExpandProductMetric[%,g3];
%/.Flatten[{h$Harm$Rep,FourD$To$TwoD}];
%//ContractMetric//ToCanonical*)

(Subscript[(OverDot[(OverBar[m])]), a]):

Pert$mC=MakeRule[{d𝓂C[A]*mC[-A],0}]

(*(Perturbation[𝓂C[-μ]]𝓂C[μ]+𝓂C[-μ]Perturbation[𝓂C[μ]])
%//ExpandPerturbation//SeparateMetric[];
%/.Pert$Frame//TraceProductDummy;
ExpandProductMetric[%,g3];
%/.Flatten[{𝒽$Harm$Rep,FourD$To$TwoD}];
%//ContractMetric//ToCanonical*)

1.10. Relation to GHP Weyl curvature scalars

Working in the sign conventions of the Stephani’s book (that work in the same signature conventions etc as us):

(Subscript[Ψ, i]) in terms of ℂ

Ψ0=(1/2)CC[-μ,-ν,-ρ,-σ]ℓ[μ]𝓂[ν]ℓ[ρ]𝓂[σ];
(*((Ψ0/.ℂ$To$Weyl)/.Vol$To$Frame)//ContractMetric//ToCanonical*)
Ψ1=CC[-μ,-ν,-ρ,-σ](ℓ[μ]𝓃[ν]ℓ[ρ]𝓂[σ]-(1/2)ℓ[μ]𝓂[ν]𝓂C[ρ]𝓂[σ]);
(*((Ψ1/.ℂ$To$Weyl)/.Vol$To$Frame)//ContractMetric//ToCanonical*)
Ψ2=(-1/2)CC[-μ,-ν,-ρ,-σ]ℓ[μ]𝓂[ν]𝓃[ρ]𝓂C[σ];
(*((Ψ2/.ℂ$To$Weyl)/.Vol$To$Frame)//ContractMetric//ToCanonical*)
Ψ3=CC[-μ,-ν,-ρ,-σ](𝓃[μ]ℓ[ν]𝓃[ρ]𝓂C[σ]-(1/2)𝓃[μ]𝓂C[ν]𝓂[ρ]𝓂C[σ]);
(*((Ψ3/.ℂ$To$Weyl)/.Vol$To$Frame)//ContractMetric//ToCanonical*)
Ψ4=(1/2)CC[-μ,-ν,-ρ,-σ]𝓃[μ]𝓂C[ν]𝓃[ρ]𝓂C[σ];
(*((Ψ4/.ℂ$To$Weyl)/.Vol$To$Frame)//ContractMetric//ToCanonical*)

We note that the background is always a type-D spacetime:

(*Print["(Subscript[Ψ, 0]):"]
Ψ0//SeparateMetric[];
%//TraceProductDummy;
ExpandProductMetric[%,g3];
%/.BG$Total;
%//ContractMetric//ToCanonical
Print["(Subscript[Ψ, 1]):"]
Ψ1//SeparateMetric[];
%//TraceProductDummy;
ExpandProductMetric[%,g3];
%/.BG$Total;
%//ContractMetric//ToCanonical
Print["(Subscript[Ψ, 2]):"]
Ψ2//SeparateMetric[];
%//TraceProductDummy;
ExpandProductMetric[%,g3];
%/.BG$Total;
%//ContractMetric//ToCanonical
Print["(Subscript[Ψ, 3]):"]
Ψ3//SeparateMetric[];
%//TraceProductDummy;
ExpandProductMetric[%,g3];
%/.BG$Total;
%//ContractMetric//ToCanonical
Print["(Subscript[Ψ, 4]):"]
Ψ4//SeparateMetric[];
%//TraceProductDummy;
ExpandProductMetric[%,g3];
%/.BG$Total;
%//ContractMetric//ToCanonical*)

Relation of Weyl scalar perturbations and our curvature components

dΨ0$To$Components=(Sqrt[(-1+LL)*LL*(1+LL)*(2+LL)]*p2Y[]*Ψ$ll$o[])/(2*r[]^2);
dΨ1$To$Components=(-3*dℓ[A]*m[-A])/(2*Sqrt[2]*r[]^3)+(((3*I)/2)*Λ*p1Y[]*𝒽$l$m[])/(Sqrt[2]*r[]^3)+(dℓ[A]*m[-A]*𝒫[])/(2*Sqrt[2]*r[])-((I/2)*Λ*p1Y[]*𝒽$l$m[]*𝒫[])/(Sqrt[2]*r[])-((I/2)*Λ*p1Y[]*Ξ$l$m[])/(Sqrt[2]*r[]^3)-(Λ*p1Y[]*Ξ$l$p[])/(2*Sqrt[2]*r[]^3)-(dℓ[A]*m[-A]*Τ[a,-a])/(2*Sqrt[2]*r[])+((I/2)*Λ*p1Y[]*𝒽$l$m[]*Τ[a,-a])/(Sqrt[2]*r[])+(I*Λ*p1Y[]*Ψ$l$m[])/(Sqrt[2]*r[])-(Λ*p1Y[]*Ψ$l$p[])/(Sqrt[2]*r[])+(3*dℓ[A]*m[-A]*CD1[-a][r[]]*CD1[a][r[]])/(2*Sqrt[2]*r[]^3)-(((3*I)/2)*Λ*p1Y[]*𝒽$l$m[]*CD1[-a][r[]]*CD1[a][r[]])/(Sqrt[2]*r[]^3);
dΨ2$To$Components=-((Y[]*𝒽$ln$p[])/r[]^2)+(Y[]*𝒽$ln$p[]*𝒫[])/3-(Y[]*𝒽$ln$p[]*Τ[a,-a])/3+(Y[]*Ψ$p[])/2+(Y[]*𝒽$ln$p[]*CD1[-a][r[]]*CD1[a][r[]])/r[]^2;
dΨ3$To$Components=(-3*d𝓃[A]*mC[-A])/(2*Sqrt[2]*r[]^3)+(((3*I)/2)*Λ*m1Y[]*𝒽$n$m[])/(Sqrt[2]*r[]^3)+(d𝓃[A]*mC[-A]*𝒫[])/(2*Sqrt[2]*r[])-((I/2)*Λ*m1Y[]*𝒽$n$m[]*𝒫[])/(Sqrt[2]*r[])-((I/2)*Λ*m1Y[]*Ξ$n$m[])/(Sqrt[2]*r[]^3)+(Λ*m1Y[]*Ξ$n$p[])/(2*Sqrt[2]*r[]^3)-(d𝓃[A]*mC[-A]*Τ[a,-a])/(2*Sqrt[2]*r[])+((I/2)*Λ*m1Y[]*𝒽$n$m[]*Τ[a,-a])/(Sqrt[2]*r[])-(I*Λ*m1Y[]*Ψ$n$m[])/(Sqrt[2]*r[])-(Λ*m1Y[]*Ψ$n$p[])/(Sqrt[2]*r[])+(3*d𝓃[A]*mC[-A]*CD1[-a][r[]]*CD1[a][r[]])/(2*Sqrt[2]*r[]^3)-(((3*I)/2)*Λ*m1Y[]*𝒽$n$m[]*CD1[-a][r[]]*CD1[a][r[]])/(Sqrt[2]*r[]^3);
dΨ4$To$Components=(Sqrt[(-1+LL)*LL*(1+LL)*(2+LL)]*m2Y[]*Ψ$nn$o[])/(2*r[]^2);

(*Perturbation[Ψ0]//ExpandPerturbation//SeparateMetric[];
%/.Pert$Frame;
%//TraceProductDummy;
ExpandProductMetric[%,g3];
%/.Flatten[{𝒽$Harm$Rep,ℂ$Harm$Rep}];
%/.BG$Total;
%//ContractMetric//ToCanonical;
%/.Dir$Harm$Def$Total;
%/.DDY$To$sY;
%/.DY$To$sY;
%//ContractMetric//ToCanonical;
%/.Flatten[{ℂ$Alg$Ψ$ab$m$To$Ψ$ab$o,Λ$To$L}]//ContractMetric//ToCanonical;
%/.Ψ$ab$o$ToFrame//ContractMetric//ToCanonical;
Simplify[ToCanonical[%-dΨ0$To$Components],Assumptions->LL>0]
Perturbation[Ψ1]//ExpandPerturbation//SeparateMetric[];
%/.Pert$Frame;
%//TraceProductDummy;
ExpandProductMetric[%,g3];
%/.Flatten[{𝒽$Harm$Rep,ℂ$Harm$Rep}];
%/.BG$Total;
%//ContractMetric//ToCanonical;
%/.Dir$Harm$Def$Total;
%/.DDY$To$sY;
%/.DY$To$sY;
%//ContractMetric//ToCanonical;
%/.Flatten[{𝒽$a$m$ToFrame,Ψ$a$p$ToFrame,Ψ$a$m$ToFrame,Ξ$a$p$ToFrame,Ξ$a$m$ToFrame}]//ContractMetric//ToCanonical;
Simplify[ToCanonical[%-dΨ1$To$Components],Assumptions->LL>0]
Perturbation[Ψ2]//ExpandPerturbation//SeparateMetric[];
%/.Pert$Frame;
%//TraceProductDummy;
ExpandProductMetric[%,g3];
%/.Flatten[{𝒽$Harm$Rep,ℂ$Harm$Rep}];
%/.BG$Total;
%//ContractMetric//ToCanonical;
%/.Dir$Harm$Def$Total;
%/.DDY$To$sY;
%/.DY$To$sY;
%//ContractMetric//ToCanonical;
%/.Pert$l//ContractMetric//ToCanonical;
%/.Pert$m//ContractMetric//ToCanonical;
%/.Dir$Harm$Def$Total//ContractMetric//ToCanonical;
%//.Flatten[{ℂ$Alg$Ψ$ab$p$To$Ψ$m ,ℂ$Alg$Ψ$m$To$Ψ$p$𝒽$p}]//ContractMetric//ToCanonical;
%/.Flatten[{𝒽$a$m$ToFrame,𝒽$ab$p$ToFrame}]//ContractMetric//ToCanonical;
Simplify[ToCanonical[%-dΨ2$To$Components],Assumptions->LL>0]
Perturbation[Ψ3]//ExpandPerturbation//SeparateMetric[];
%/.Pert$Frame;
%//TraceProductDummy;
ExpandProductMetric[%,g3];
%/.Flatten[{𝒽$Harm$Rep,ℂ$Harm$Rep}];
%/.BG$Total;
%//ContractMetric//ToCanonical;
%/.Dir$Harm$Def$Total;
%/.DDY$To$sY;
%/.DY$To$sY;
%//ContractMetric//ToCanonical;
%/.Flatten[{𝒽$a$m$ToFrame,Ψ$a$p$ToFrame,Ψ$a$m$ToFrame,Ξ$a$p$ToFrame,Ξ$a$m$ToFrame}]//ContractMetric//ToCanonical;
Simplify[ToCanonical[%-dΨ3$To$Components],Assumptions->LL>0]
Perturbation[Ψ4]//ExpandPerturbation//SeparateMetric[];
%/.Pert$Frame;
%//TraceProductDummy;
ExpandProductMetric[%,g3];
%/.Flatten[{𝒽$Harm$Rep,ℂ$Harm$Rep}];
%/.BG$Total;
%//ContractMetric//ToCanonical;
%/.Dir$Harm$Def$Total;
%/.DDY$To$sY;
%/.DY$To$sY;
%//ContractMetric//ToCanonical;
%/.ℂ$Alg$Ψ$ab$m$To$Ψ$ab$o//ContractMetric//ToCanonical;
%/.Flatten[{Ψ$ab$o$ToFrame,Λ$To$L}]//ContractMetric//ToCanonical;
Simplify[ToCanonical[%-dΨ4$To$Components],Assumptions->LL>0]*)

2. Examples

2.0. Linearised Einstein Equations

Say we want to compute the harmonic components of the linearisation of the Einstein tensor δ((Subscript[R, μ ν ])-(1/2)(Subscript[g, μ ν ])R)=δ (Subscript[T, μ ν]). We proceed as follows, for the index cases (μ ,ν)=(a, b), (μ ,ν)=(a, P),  (μ ,ν)=(P,Q):

(μ ,ν)=(a, b)

We apply Pert$Exp$Sep to the total Einstein equation (in the total manifold M3), replacing (μ ,ν)=(a, b):

RicciCD3[-μ,-ν]-(1/2)RicciScalarCD3[]g3[-μ,-ν]-𝒯[-μ,-ν]

Pert$Exp$Sep[RicciCD3[-μ,-ν]-(1/2)RicciScalarCD3[]g3[-μ,-ν]-𝒯[-μ,-ν],{μ,ν},{a,b},{𝒽$Harm$Rep,𝒯$Harm$Rep}]
%/.Dir$Harm$Def$Total//ContractMetric//ToCanonical 
%//.Harm$EQ$Y$Total//ContractMetric//ToCanonical 
%/.{Y[]->1}

If restricted to vacuum, this is the form one can find in the literature e.g. [0502028]. However, using that the Einstein tensor of a 2d manifold vanishes identically, there are additional identities that second derivatives of (Subscript[𝒽, a b]) satisfy, and that can be used to remove entirely second derivatives of  (Subscript[𝒽, a b]) above. These relations are implemented in this code and can be used as follows

%/.CommCovs$𝒽$ab$p$FreeFirst//ContractMetric//ToCanonical(*We first commute second derivatives of terms like ({{𝒽, ({{ , c,   ,   }, {b,  , :a, :c}})}}) so that the free derivative index is the first one, i.e. the last derivative applied*)
%/.FirstOrder$Gauss$Bonnet$Rule //ContractMetric//ToCanonical(*We apply the special second-derivative identities due to 2-dimensions*)

ToCanonical[%-(Pert$Einstein$ab[-a,-b]/.Eval$Pert$Einstein$Total)]

and indeed second derivatives of (Subscript[𝒽, a b]) disappear. This formulation of the equations was given in [2302.10942].

(μ ,ν)=(a, P)

Very much as before. However, the free spherical indices have to be either P, if there is only one, or P and Q if there are two, so that the harmonic component functions have the correct input:

Pert$Exp$Sep[RicciCD3[-μ,-ν]-(1/2)RicciScalarCD3[]g3[-μ,-ν]-𝒯[-μ,-ν],{μ,ν},{a,P},{𝒽$Harm$Rep,𝒯$Harm$Rep}] 
%/.Dir$Harm$Def$Total//ContractMetric//ToCanonical 
Dum=%//.Harm$EQ$Y$Total//ContractMetric//ToCanonical

Now we can simply use the projection functions, that give back the even and odd components of the equation:

ZA$Component[Dum]

ToCanonical[%-(Pert$Einstein$aZA[-a]/.Eval$Pert$Einstein$Total)]

XA$Component[Dum]

ToCanonical[%-(Pert$Einstein$aXA[-a]/.Eval$Pert$Einstein$Total)]

(μ ,ν)=( P,Q)

Very much as before. However, the free spherical indices have to be either P, if there is only one, or P and Q if there are two, so that the harmonic component functions have the correct input:

Pert$Exp$Sep[RicciCD3[-μ,-ν]-(1/2)RicciScalarCD3[]g3[-μ,-ν]-𝒯[-μ,-ν],{μ,ν},{P,Q},{𝒽$Harm$Rep,𝒯$Harm$Rep}] 
%/.Dir$Harm$Def$Total//ContractMetric//ToCanonical 
Dum=%//.Harm$EQ$Y$Total//ContractMetric//ToCanonical

Now we can simply use the harmonic functions

UAB$Component[Dum]//ContractMetric//ToCanonical
(ToCanonical[%-(Pert$Einstein$UAB[]/.Eval$Pert$Einstein$Total)]==0)//Simplify
VAB$Component[Dum]//ContractMetric//ToCanonical//Simplify
(ToCanonical[%-(Pert$Einstein$VAB[]/.Eval$Pert$Einstein$Total)]==0)//Simplify
WAB$Component[Dum]//ContractMetric//ToCanonical//Simplify
(ToCanonical[%-(Pert$Einstein$WAB[]/.Eval$Pert$Einstein$Total)]==0)//Simplify
ϵAB$Component[Dum]//ContractMetric//ToCanonical
(ToCanonical[%]==0)//Simplify

2.1. Some components of the curvature wave equation

Proceeding similarly as before we can obtain the pre-master equation, which is one of the main results. We focus on the purely-lorentzian component of such equation, that is, δ((Subscript[Δℂ, a b c d]))=δ(((Subscript[D, a])-i ⋆(Subscript[D, a]))(Subscript[𝕁, b c d])). For clarity we can work out the LHS and RHS separately:

LHS=ToCanonical[CD3[-α][CD3[α][CC[-μ,-ν,-ρ,-σ]]]+4*SkewSymetrise[SkewSymetrise[Symetrise[CC[-μ,α,β,-ρ],{α,β}]CC[-σ,-α,-β,-ν],{-μ,-ν}],{-ρ,-σ}]+SkewSymetrise[SkewSymetrise[CC[α,β,γ,-ρ]g3[-σ,-μ]CC[-ν,-γ,-α,-β],{-μ,-ν}],{-ρ,-σ}]]

{LHS$time,LHS$result}=AbsoluteTiming[LHSstep1=Pert$Exp$Sep[LHS,{μ,ν,ρ,σ},{a,b,c,d},{ℂ$Harm$Rep,𝒽$Harm$Rep}];
LHSstep2=LHSstep1/.Dir$Harm$Def$Total//ContractMetric//ToCanonical;
LHSstep3=LHSstep2//.Harm$EQ$Y$Total//ContractMetric//ToCanonical;
LHSstep4=((LHSstep3((1/4)epsilong1[a,b]epsilong1[c,d])//ContractMetric//ToCanonical));
LHSstep4];
Print["Computing Time:"]
LHS$time
Print["Result:"]
LHS$result

RHS=ToCanonical[SkewSymetrise[2CD3[-μ][𝕁[-ν,-ρ,-σ]],{-μ,-ν}]-I epsilong3[-μ,-ν,α,β]CD3[-α][𝕁[-β,-ρ,-σ]]-SkewSymetrise[𝒯[γ,-ρ]CC[-σ,-γ,-μ,-ν],{-ρ,-σ}]-SkewSymetrise[𝒯[γ,-μ]CC[-ν,-γ,-ρ,-σ],{-μ,-ν}]-𝒯[γ,-γ]CC[-μ,-ν,-ρ,-σ]+SkewSymetrise[SkewSymetrise[2𝒯[α,β]g3[-ρ,-μ]CC[-σ,-α,-β,-ν],{-μ,-ν}],{-ρ,-σ}]]

{RHS$time,RHS$result}=AbsoluteTiming[RHSstep1=Pert$Exp$Sep[RHS,{μ,ν,ρ,σ},{a,b,c,d},{ℂ$Harm$Rep,𝒽$Harm$Rep,𝕁$Harm$Rep,𝒯$Harm$Rep}];
RHSstep2=RHSstep1/.Dir$Harm$Def$Total//ContractMetric//ToCanonical;
RHSstep3=RHSstep2//.Harm$EQ$Y$Total//ContractMetric//ToCanonical;
RHSstep4=((RHSstep3((1/4)epsilong1[a,b]epsilong1[c,d])//ContractMetric//ToCanonical));
RHSstep4];
Print["Computing Time:"]
RHS$time
Print["Result:"]
RHS$result

We can now combine LHS and RHS to get the purely lorentzian component of the curvature equation, and verify that the equation obtained coincides with the one implemented in the code before hand:

Δℂ$D𝕁$abcs$Example=((LHS$result-RHS$result)/Y[])//ToCanonical
ToCanonical[Δℂ$D𝕁$abcs$Example-(Δℂ$D𝕁$abcd[]/.Eval$Pert$Δℂ$D𝕁$abcd)]==0

Applications

1. Vacuum GR with sources

1.0. Basic relations and definitions

We first introduce the vacuum rules, to impose that background  correspond to the vacuum of GR:

Vacuum$GR=Flatten[{MakeRule[{Τ[-a,-b],0}],MakeRule[{𝒫[],0}]}];

Zero$Source=Flatten[{MakeRule[{Φ$ab$p[-a,-b],0}],MakeRule[{Φ$a$p[-a],0}],MakeRule[{Φ$p[],0}],MakeRule[{Φ$o[],0}],MakeRule[{Φ$a$m[-a],0}],MakeRule[{Φ$m[],0}],MakeRule[{Π$ab$p[-a,-b],0}],MakeRule[{Π$ab$m[-a,-b],0}],MakeRule[{Π$a$p[-a],0}],MakeRule[{Π$a$m[-a],0}],MakeRule[{Π$a$o[-a],0}],MakeRule[{Π$p[],0}],MakeRule[{Π$m[],0}],MakeRule[{Γ$a$p[-a],0}],MakeRule[{Γ$a$m[-a],0}],MakeRule[{Γ$p[],0}],MakeRule[{Γ$m[],0}]}]

Working on the Kinnersly tetrad corresponds to choosing l affine, while in the horizon frame it is n that is affine:

l$Affine=MakeRule[{ϵ$sc[],0}]
n$Affine=MakeRule[{ϵp$sc[],0}]

DefConstantSymbol[Μ]
DefTensor[𝒻[],M1]
DrDr$To$𝒻=MakeRule[{CD1[a][r[]]CD1[-a][r[]],𝒻[]}]
𝒻$To$DrDr=MakeRule[{𝒻[],CD1[a][r[]]CD1[-a][r[]]}]
𝒻$To$Schwarzschild=MakeRule[{𝒻[],1-((2 Μ)/(r[]))}]

In tetrad where l is affine:

ϵp$l$Affine$ToM=MakeRule[{ϵp$sc[],-(Μ/(2 ((r[])^2)))}]
ρ$l$Affine$ToM=MakeRule[{ρ$sc[], ((-1)/(r[]))}]
ρp$l$Affine$ToM=MakeRule[{ρp$sc[], ((-2Μ+r[])/(2 ((r[])^2)))}]

In tetrad where n is affine:

ϵ$n$Affine$ToM=MakeRule[{ϵ$sc[],(Μ/(2 ((r[])^2)))}]
ρ$n$Affine$ToM=MakeRule[{ρ$sc[],- ((-2Μ+r[])/(2 ((r[])^2)))}]
ρp$n$Affine$ToM=MakeRule[{ρp$sc[], (1/(r[]))}]

Finally we add some relation about another choice of frame

tata$To$𝒻=MakeRule[{𝓉[a]𝓉[-a],-𝒻[]},MetricOn->All]

DefTensor[𝒽$tr$p[],M1,PrintAs->"(Subscript[(SuperPlus[𝒽]), tr])"]
DefTensor[𝒽$rr$p[],M1,PrintAs->"(Subscript[(SuperPlus[𝒽]), rr])"]

𝒽$ab$p$To$tr$Frame=MakeRule[{𝒽$ab$p[-a,-b],(𝒽$rr$p[]-((𝒽$ab$p[c,-c])/(𝒻[])))𝓉[-a]𝓉[-b]+𝒽$rr$p[]CD1[-a][r[]]CD1[-b][r[]]-((𝒽$tr$p[])/(𝒻[]))(𝓉[-a]CD1[-b][r[]]+𝓉[-b]CD1[-a][r[]])},MetricOn->All]

DefTensor[𝒽$t$m[],M1,PrintAs->"(Subscript[(SuperMinus[𝒽]), t])"]
DefTensor[𝒽$r$m[],M1,PrintAs->"(Subscript[(SuperMinus[𝒽]), r])"]

𝒽$a$m$To$tr$Frame=MakeRule[{𝒽$a$m[-a],-((𝒽$t$m[])/(𝒻[]))𝓉[-a]+𝒽$r$m[]CD1[-a][r[]]},MetricOn->All]

Energy-momentum conservation laws (in vacuum)

We define some replacement rules in the vacuum background:

𝒯$Cons$Rep$Div$Φ$ab$p=MakeRule[{CD1[-b][Φ$ab$p[-a,b]],(Λ^2*Φ$a$p[-a])/r[]^2+(2*Φ$p[]*CD1[-a][r[]])/r[]^3-(2*Φ$ab$p[-a,-b]*CD1[b][r[]])/r[]},MetricOn->All];
(*-𝒯$Cons$a[-a]//.Flatten[{Eval$𝒯$Cons$Total,Vacuum$GR}];
%/Y[]//ContractMetric//ToCanonical;*)
𝒯$Cons$Rep$Div$Φ$a$p=MakeRule[{CD1[-a][Φ$a$p[a]],-(Φ$o[]/r[]^2)+(Λ^2*Φ$o[])/(2*r[]^2)-Φ$p[]/r[]^2-(2*Φ$a$p[a]*CD1[-a][r[]])/r[]},MetricOn->All];
(*-𝒯$Cons$ZA[]//.Flatten[{Eval$𝒯$Cons$Total,Vacuum$GR}];
%//ContractMetric//ToCanonical*)
𝒯$Cons$Rep$Div$Φ$a$m=MakeRule[{CD1[-a][Φ$a$m[a]],-(Φ$m[]/r[]^2)+(Λ^2*Φ$m[])/(2*r[]^2)-(2*Φ$a$m[a]*CD1[-a][r[]])/r[]},MetricOn->All];
(*-𝒯$Cons$XA[]//.Flatten[{Eval$𝒯$Cons$Total,Vacuum$GR}];
%//ContractMetric//ToCanonical;*)

δ𝕁 in terms of δ𝒯 (in vacuum)

Π$ab$p$To$VacuumSource=MakeRule[{Π$ab$p[-a,-b],-1/2*Φ$ab$p[-a,-b]+(g1[-a,-b]*Φ$ab$p[c,-c])/6+(g1[-a,-b]*Φ$p[])/(3*r[]^2)-((I/4)*epsilong1[-b,-c]*Φ$a$m[c]*CD1[-a][r[]])/r[]+(Φ$a$p[-b]*CD1[-a][r[]])/(4*r[])+CD1[-a][Φ$a$p[-b]]/4-((I/4)*epsilong1[-a,-c]*Φ$a$m[c]*CD1[-b][r[]])/r[]+(Φ$a$p[-a]*CD1[-b][r[]])/(4*r[])+CD1[-b][Φ$a$p[-a]]/4-(I/4)*epsilong1[-b,-c]*CD1[c][Φ$a$m[-a]]-(I/4)*epsilong1[-a,-c]*CD1[c][Φ$a$m[-b]]},MetricOn->All];
Π$ab$m$To$VacuumSource=MakeRule[{Π$ab$m[-a,-b],(-1/4*I)*epsilong1[-b,c]*Φ$ab$p[-a,-c]-(I/4)*epsilong1[-a,c]*Φ$ab$p[-b,-c]+(Φ$a$m[-b]*CD1[-a][r[]])/(4*r[])+((I/4)*epsilong1[-b,-c]*Φ$a$p[c]*CD1[-a][r[]])/r[]+CD1[-a][Φ$a$m[-b]]/4+(Φ$a$m[-a]*CD1[-b][r[]])/(4*r[])+((I/4)*epsilong1[-a,-c]*Φ$a$p[c]*CD1[-b][r[]])/r[]+CD1[-b][Φ$a$m[-a]]/4+(I/4)*epsilong1[-b,-c]*CD1[c][Φ$a$p[-a]]+(I/4)*epsilong1[-a,-c]*CD1[c][Φ$a$p[-b]]},MetricOn->All];
Π$a$p$To$VacuumSource=MakeRule[{Π$a$p[-a],((-1/2*I)*Λ^2*Φ$a$m[-a])/r[]^2-(epsilong1[-a,b]*CD1[-b][Φ$ab$p[c,-c]])/6+(2*epsilong1[-a,-b]*Φ$p[]*CD1[b][r[]])/(3*r[]^3)-(epsilong1[-a,-b]*CD1[b][Φ$p[]])/(3*r[]^2)+(epsilong1[b,c]*CD1[-c][Φ$ab$p[-a,-b]])/2},MetricOn->All];
Π$a$m$To$VacuumSource=MakeRule[{Π$a$m[-a],-1/4*(Λ^2*Φ$a$m[-a])+(I/4)*Λ^2*epsilong1[-a,-b]*Φ$a$p[b]-(I/6)*epsilong1[-a,b]*r[]^2*CD1[-b][Φ$ab$p[c,-c]]-(I/2)*epsilong1[-a,c]*r[]*Φ$ab$p[-b,-c]*CD1[b][r[]]+((I/6)*epsilong1[-a,-b]*Φ$p[]*CD1[b][r[]])/r[]+(I/6)*epsilong1[-a,-b]*CD1[b][Φ$p[]]},MetricOn->All];
Π$a$o$To$VacuumSource=MakeRule[{Π$a$o[-a],(-1/2*I)*epsilong1[-a,-b]*Φ$a$m[b]+Φ$a$p[-a]/2+(Φ$o[]*CD1[-a][r[]])/(2*r[])-CD1[-a][Φ$o[]]/2-((I/2)*epsilong1[-a,-b]*Φ$m[]*CD1[b][r[]])/r[]+(I/2)*epsilong1[-a,-b]*CD1[b][Φ$m[]]},MetricOn->All];
Π$p$To$VacuumSource=MakeRule[{Π$p[],((I/4)*Φ$a$m[a]*CD1[-a][r[]])/r[]+(I/4)*CD1[-a][Φ$a$m[a]]+(epsilong1[-a,-b]*Φ$a$p[a]*CD1[b][r[]])/(4*r[])-(epsilong1[-a,-b]*CD1[b][Φ$a$p[a]])/4},MetricOn->All];
Π$m$To$VacuumSource=MakeRule[{Π$m[],(I/12)*Φ$ab$p[a,-a]-((I/3)*Φ$p[])/r[]^2-((I/4)*Φ$a$p[a]*CD1[-a][r[]])/r[]-(I/4)*CD1[-a][Φ$a$p[a]]+(epsilong1[-a,-b]*Φ$a$m[a]*CD1[b][r[]])/(4*r[])-(epsilong1[-a,-b]*CD1[b][Φ$a$m[a]])/4},MetricOn->All];
Γ$a$p$To$VacuumSource=MakeRule[{Γ$a$p[-a],(-1/4*I)*Λ^2*epsilong1[-a,-b]*Φ$a$m[b]-(Λ^2*Φ$a$p[-a])/4-(Φ$p[]*CD1[-a][r[]])/(6*r[])+(r[]^2*CD1[-a][Φ$ab$p[b,-b]])/6-CD1[-a][Φ$p[]]/6+(r[]*Φ$ab$p[-a,-b]*CD1[b][r[]])/2},MetricOn->All];
Γ$a$m$To$VacuumSource=MakeRule[{Γ$a$m[-a],Φ$a$m[-a]/2+(I/2)*epsilong1[-a,-b]*Φ$a$p[b]+(Φ$m[]*CD1[-a][r[]])/(2*r[])-CD1[-a][Φ$m[]]/2+((I/2)*epsilong1[-a,-b]*Φ$o[]*CD1[b][r[]])/r[]-(I/2)*epsilong1[-a,-b]*CD1[b][Φ$o[]]},MetricOn->All];
Γ$p$To$VacuumSource=MakeRule[{Γ$p[],(r[]^2*Φ$ab$p[a,-a])/6+Φ$o[]/2-(Λ^2*Φ$o[])/4-Φ$p[]/6+(r[]*Φ$a$p[a]*CD1[-a][r[]])/2-(I/2)*epsilong1[-a,-b]*r[]*Φ$a$m[a]*CD1[b][r[]]+(I/2)*epsilong1[-a,-b]*r[]^2*CD1[b][Φ$a$m[a]]},MetricOn->All];
Γ$m$To$VacuumSource=MakeRule[{Γ$m[],-1/2*Φ$m[]+(Λ^2*Φ$m[])/4-(r[]*Φ$a$m[a]*CD1[-a][r[]])/2-(I/2)*epsilong1[-a,-b]*r[]*Φ$a$p[a]*CD1[b][r[]]+(I/2)*epsilong1[-a,-b]*r[]^2*CD1[b][Φ$a$p[a]]},MetricOn->All];
(*Π$ab$p[-a,-b]/.Pert𝕁$To$metTmn;
%//.Vacuum$GR//ContractMetric//ToCanonical;
Π$ab$m[-a,-b]/.Pert𝕁$To$metTmn;
%//.Vacuum$GR//ContractMetric//ToCanonical;
etc;*)

Vacuum$Source$GR=Flatten[{Π$ab$p$To$VacuumSource,Π$ab$m$To$VacuumSource,Π$a$p$To$VacuumSource,Π$a$m$To$VacuumSource,Π$a$o$To$VacuumSource,Π$p$To$VacuumSource,Π$m$To$VacuumSource,Γ$a$p$To$VacuumSource,Γ$a$m$To$VacuumSource,Γ$p$To$VacuumSource,Γ$m$To$VacuumSource}];

1.1. The RW equation for both sectors

Deriving the equation

We first evaluate a particular component of the curvature wave equation:

(Δℂ$D𝕁$abPQ[]/.Eval$Pert$Δℂ$D𝕁$abPQ)//.Flatten[{Vacuum$GR,Einstein$Rep$Lorentz$𝒽$ab$p,Einstein$Rep$Trace$𝒽$ab$p}]//ContractMetric//ToCanonical;
step1=%//.Flatten[{DrDr$To$𝒻,𝒻$To$Schwarzschild}]//ContractMetric//ToCanonical

-step1/.Zero$Source
TexPrint[%]

Ψ$a$m[-a]/.ℂ$Alg$Ψ$a$m$To$Ξ$a$m$𝒽$a$m;
%//.Flatten[{Vacuum$GR,Einstein$Rep$Lorentz$𝒽$ab$p,Einstein$Rep$Trace$𝒽$ab$p}]//ContractMetric//ToCanonical;
%//.Flatten[{DrDr$To$𝒻,𝒻$To$Schwarzschild}]//ContractMetric//ToCanonical
TexPrint[%]

Ψ$ab$p[-a,-b]/.ℂ$Alg$Ψ$ab$p$To$Ψ$m;
%//.Flatten[{Vacuum$GR,Einstein$Rep$Lorentz$𝒽$ab$p,Einstein$Rep$Trace$𝒽$ab$p}]//ContractMetric//ToCanonical;
%//.Flatten[{Zero$Source,DrDr$To$𝒻,𝒻$To$Schwarzschild}]//ContractMetric//ToCanonical
TexPrint[%]

(step1/.Flatten[{ℂ$Alg$Ψ$a$m$To$Ξ$a$m$𝒽$a$m,ℂ$Alg$Ψ$ab$p$To$Ψ$m}])//.Flatten[{Vacuum$GR,Einstein$Rep$Lorentz$𝒽$ab$p,Einstein$Rep$Trace$𝒽$ab$p}]//ContractMetric//ToCanonical;
step2=%//.Flatten[{DrDr$To$𝒻,𝒻$To$Schwarzschild}]//ContractMetric//ToCanonical

Column[(Dℂ$𝕁$Total/.Eval$Pert$Dℂ$𝕁//ScreenDollarIndices)//.Flatten[{Vacuum$GR,Einstein$Rep$Lorentz$𝒽$ab$p,Einstein$Rep$Trace$𝒽$ab$p}]//ContractMetric//ToCanonical//Simplify,Frame->All]

((((r[])^2)/(Λ^2)))((Dℂ$𝕁$Total[[4]]/.Eval$Pert$Dℂ$𝕁//ScreenDollarIndices)//.Flatten[{Vacuum$GR,Einstein$Rep$Lorentz$𝒽$ab$p,Einstein$Rep$Trace$𝒽$ab$p}]//ContractMetric//ToCanonical//Simplify)//ToCanonical;
%//.Flatten[{DrDr$To$𝒻,𝒻$To$Schwarzschild}]//ContractMetric//ToCanonical

InputForm[%//ScreenDollarIndices]

Ξ$a$m$To$DΨ$m=MakeRule[{Ξ$a$m[-a],(-2*Μ*𝒽$a$m[-a])/r[]+(2*r[]^2*Π$a$m[-a])/Λ^2-((4*I)*Μ*epsilong1[-a,-b]*r[]*Φ$a$p[b])/Λ^2+((2*I)*Μ*epsilong1[-a,-b]*𝒽$p[]*CD1[b][r[]])/(Λ^2*r[]^2)-((4*I)*Μ*epsilong1[-a,-b]*Φ$o[]*CD1[b][r[]])/Λ^2-(epsilong1[-a,-b]*r[]*Ψ$m[]*CD1[b][r[]])/Λ^2-(I*Μ*epsilong1[-a,-b]*CD1[b][𝒽$p[]])/(Λ^2*r[])+((2*I)*Μ*epsilong1[-a,-b]*r[]*CD1[b][Φ$o[]])/Λ^2-(epsilong1[-a,-b]*r[]^2*CD1[b][Ψ$m[]])/Λ^2-((2*I)*Μ*epsilong1[b,c]*r[]*CD1[-c][𝒽$ab$p[-a,-b]])/Λ^2},MetricOn->All]

epsilong1[-a,-b]Ξ$a$m[b]/.Ξ$a$m$To$DΨ$m//ContractMetric//ToCanonical;
%//.Flatten[{Zero$Source,Vacuum$GR,Einstein$Rep$Trace$𝒽$ab$p,Einstein$Rep$Lorentz$𝒽$ab$p}]//ContractMetric//ToCanonical
TexPrint[%]

(step2/.Ξ$a$m$To$DΨ$m//ContractMetric//ToCanonical)//.Flatten[{Vacuum$GR,Einstein$Rep$Lorentz$𝒽$ab$p,Einstein$Rep$Trace$𝒽$ab$p,ℂ$Alg$Ψ$ab$p$To$Ψ$m}]//ContractMetric//ToCanonical;
step3=%//.Flatten[{DrDr$To$𝒻,𝒻$To$Schwarzschild}]//ContractMetric//ToCanonical

Collect[-step3/.Zero$Source,{Ψ$m[],𝒽$p[]}]
TexPrint[%]

In a single move, trading (SuperMinus[Ψ]) by (SuperPlus[Ψ]) and (SuperPlus[𝒽]), we get a decoupled equation for (SuperPlus[Ψ]):

((((-I)/((r[])^2)))step3/.ℂ$Alg$Ψ$m$To$Ψ$p$𝒽$p)//.Flatten[{Vacuum$GR,Einstein$Rep$Lorentz$𝒽$ab$p,Einstein$Rep$Trace$𝒽$ab$p,BG$1,BG$2}]//ContractMetric//ToCanonical;
step4=%//.Flatten[{DrDr$To$𝒻,𝒻$To$Schwarzschild}]//ToCanonical

Collect[step4/.Zero$Source//ToCanonical,{Ψ$p[]}]
TexPrint[%]

InputForm[%//ScreenDollarIndices]

DefTensor[Master$Ψ$p[],M1]

Eval$Master$Ψ$p=MakeRule[{Master$Ψ$p[],(Λ^2*Γ$p[])/r[]^4-(Μ*Φ$ab$p[a,-a])/r[]^3+(12*Μ^2*Φ$o[])/r[]^6-(12*Μ*Φ$o[])/r[]^5+(4*Λ^2*Μ*Φ$o[])/r[]^5-(2*Μ*Φ$p[])/r[]^5+(6*Ψ$p[])/r[]^2-(Λ^2*Ψ$p[])/r[]^2-(2*Γ$a$p[a]*CD1[-a][r[]])/r[]^3-(12*Μ*Φ$a$p[a]*CD1[-a][r[]])/r[]^4-(4*Μ*CD1[-a][CD1[a][Φ$o[]]])/r[]^3+CD1[-a][CD1[a][Ψ$p[]]]+(6*Μ*CD1[-a][Φ$o[]]*CD1[a][r[]])/r[]^4+(6*CD1[-a][Ψ$p[]]*CD1[a][r[]])/r[]-((4*I)*epsilong1[-a,-b]*Π$a$m[a]*CD1[b][r[]])/r[]^3-((2*I)*epsilong1[-a,-b]*CD1[b][Π$a$m[a]])/r[]^2}]

We can put it in canonical form using the equations of the Schwarzschild background:

DefTensor[Ψ$RW[],M1,PrintAs->"(Subscript[Ψ, RW])"]

Ψ$p$To$Ψ$RW=MakeRule[{Ψ$p[],((r[])^(-3))Ψ$RW[]}]
Ψ$RW$To$Ψ$p=MakeRule[{Ψ$RW[],((r[])^3)Ψ$p[]}]

Master$Ψ$p[]/.Eval$Master$Ψ$p;
%//.Flatten[{Ψ$p[]->((r[])^(-3))Ψ$RW[],Vacuum$GR,BG$1,BG$2}]//ContractMetric//ToCanonical;
Collect[(((r[])^3)%)//.Flatten[{DrDr$To$𝒻,𝒻$To$Schwarzschild}]//ContractMetric//ToCanonical,{Ψ$RW[]}]

InputForm[%//ScreenDollarIndices]

DefTensor[Master$ΨRW[],M1]

Eval$Master$ΨRW=MakeRule[{Master$ΨRW[],(Λ^2*Γ$p[])/r[]-Μ*Φ$ab$p[a,-a]+(12*Μ^2*Φ$o[])/r[]^3-(12*Μ*Φ$o[])/r[]^2+(4*Λ^2*Μ*Φ$o[])/r[]^2-(2*Μ*Φ$p[])/r[]^2+((6*Μ)/r[]^3-Λ^2/r[]^2)*Ψ$RW[]-2*Γ$a$p[a]*CD1[-a][r[]]-(12*Μ*Φ$a$p[a]*CD1[-a][r[]])/r[]-4*Μ*CD1[-a][CD1[a][Φ$o[]]]+CD1[-a][CD1[a][Ψ$RW[]]]+(6*Μ*CD1[-a][Φ$o[]]*CD1[a][r[]])/r[]-(4*I)*epsilong1[-a,-b]*Π$a$m[a]*CD1[b][r[]]-(2*I)*epsilong1[-a,-b]*r[]*CD1[b][Π$a$m[a]]}]

DefTensor[LHS$RW[],M1]
DefTensor[RHS$RW$SelfDual[],M1]
DefTensor[RHS$RW$NonSelfDual[],M1]

Eval$LHS$RW=MakeRule[{LHS$RW[], CD1[-a][CD1[a][Ψ$RW[]]] + ((6*Μ)/r[]^3 - Λ^2/r[]^2)*Ψ$RW[]}]
Eval$RHS$RW$SelfDual=MakeRule[{RHS$RW$SelfDual[],-( - (4*I)*epsilong1[-a, -b]*Π$a$m[a]*CD1[b][r[]] 
- (2*I)*epsilong1[-a, -b]*r[]*CD1[b][Π$a$m[a]]
- 2*Γ$a$p[a]*CD1[-a][r[]] 
+(Λ^2*Γ$p[])/r[] )}]
Eval$RHS$RW$NonSelfDual=MakeRule[{RHS$RW$NonSelfDual[],-(- Μ*Φ$ab$p[a, -a] 
+ ((12*Μ^2)/r[]^3 - (12*Μ)/r[]^2 + (4*Λ^2*Μ)/r[]^2)*Φ$o[] 
-  (2*Μ*Φ$p[])/r[]^2 
- (12*Μ*Φ$a$p[a]*CD1[-a][r[]])/r[] 
-  4*Μ*CD1[-a][CD1[a][Φ$o[]]] 
+  (6*Μ*CD1[-a][Φ$o[]]*CD1[a][r[]])/r[] )}]

(Master$ΨRW[]-(LHS$RW[]-RHS$RW$SelfDual[]-RHS$RW$NonSelfDual[]))/.Flatten[{Eval$Master$ΨRW,Eval$LHS$RW,Eval$RHS$RW$SelfDual,Eval$RHS$RW$NonSelfDual}]//ToCanonical

Result

DefTensor[Master$Ψ$p[],M1]

Eval$Master$Ψ$p=MakeRule[{Master$Ψ$p[],(Λ^2*Γ$p[])/r[]^4-(Μ*Φ$ab$p[a,-a])/r[]^3+(12*Μ^2*Φ$o[])/r[]^6-(12*Μ*Φ$o[])/r[]^5+(4*Λ^2*Μ*Φ$o[])/r[]^5-(2*Μ*Φ$p[])/r[]^5+(6*Ψ$p[])/r[]^2-(Λ^2*Ψ$p[])/r[]^2-(2*Γ$a$p[a]*CD1[-a][r[]])/r[]^3-(12*Μ*Φ$a$p[a]*CD1[-a][r[]])/r[]^4-(4*Μ*CD1[-a][CD1[a][Φ$o[]]])/r[]^3+CD1[-a][CD1[a][Ψ$p[]]]+(6*Μ*CD1[-a][Φ$o[]]*CD1[a][r[]])/r[]^4+(6*CD1[-a][Ψ$p[]]*CD1[a][r[]])/r[]-((4*I)*epsilong1[-a,-b]*Π$a$m[a]*CD1[b][r[]])/r[]^3-((2*I)*epsilong1[-a,-b]*CD1[b][Π$a$m[a]])/r[]^2}]

DefTensor[Ψ$RW[],M1,PrintAs->"(Subscript[Ψ, RW])"]

Ψ$p$To$Ψ$RW=MakeRule[{Ψ$p[],((r[])^(-3))Ψ$RW[]}]
Ψ$RW$To$Ψ$p=MakeRule[{Ψ$RW[],((r[])^3)Ψ$p[]}]

DefTensor[Master$ΨRW[],M1]

Eval$Master$ΨRW=MakeRule[{Master$ΨRW[],(Λ^2*Γ$p[])/r[]-Μ*Φ$ab$p[a,-a]+(12*Μ^2*Φ$o[])/r[]^3-(12*Μ*Φ$o[])/r[]^2+(4*Λ^2*Μ*Φ$o[])/r[]^2-(2*Μ*Φ$p[])/r[]^2+((6*Μ)/r[]^3-Λ^2/r[]^2)*Ψ$RW[]-2*Γ$a$p[a]*CD1[-a][r[]]-(12*Μ*Φ$a$p[a]*CD1[-a][r[]])/r[]-4*Μ*CD1[-a][CD1[a][Φ$o[]]]+CD1[-a][CD1[a][Ψ$RW[]]]+(6*Μ*CD1[-a][Φ$o[]]*CD1[a][r[]])/r[]-(4*I)*epsilong1[-a,-b]*Π$a$m[a]*CD1[b][r[]]-(2*I)*epsilong1[-a,-b]*r[]*CD1[b][Π$a$m[a]]}]

DefTensor[LHS$RW[],M1]
DefTensor[RHS$RW$SelfDual[],M1]
DefTensor[RHS$RW$NonSelfDual[],M1]

Eval$LHS$RW=MakeRule[{LHS$RW[], CD1[-a][CD1[a][Ψ$RW[]]] + ((6*Μ)/r[]^3 - Λ^2/r[]^2)*Ψ$RW[]}]
Eval$RHS$RW$SelfDual=MakeRule[{RHS$RW$SelfDual[],-( - (4*I)*epsilong1[-a, -b]*Π$a$m[a]*CD1[b][r[]] 
- (2*I)*epsilong1[-a, -b]*r[]*CD1[b][Π$a$m[a]]
- 2*Γ$a$p[a]*CD1[-a][r[]] 
+(Λ^2*Γ$p[])/r[] )}]
Eval$RHS$RW$NonSelfDual=MakeRule[{RHS$RW$NonSelfDual[],-(- Μ*Φ$ab$p[a, -a] 
+ ((12*Μ^2)/r[]^3 - (12*Μ)/r[]^2 + (4*Λ^2*Μ)/r[]^2)*Φ$o[] 
-  (2*Μ*Φ$p[])/r[]^2 
- (12*Μ*Φ$a$p[a]*CD1[-a][r[]])/r[] 
-  4*Μ*CD1[-a][CD1[a][Φ$o[]]] 
+  (6*Μ*CD1[-a][Φ$o[]]*CD1[a][r[]])/r[] )}]

Relation to variables in the literature

As a reference we use [2306.17847]

DefTensor[Ψ$p$PP[],M1,PrintAs->"(SubPlus[((SuperPlus[Ψ]))])"]
DefTensor[Ψ$p$MP[],M1,PrintAs->"((SuperPlus[Ψ]))_(-)"]

DefTensor[Ψ$RW$PP[],M1,PrintAs->"((Subscript[Ψ, RW]))_(+)"]
DefTensor[Ψ$RW$MP[],M1,PrintAs->"((Subscript[Ψ, RW]))_(-)"]

Ψ$p$PP$To$Ψ$RW$PP=MakeRule[{Ψ$p$PP[],((r[])^(-3))Ψ$RW$PP[]}];
Ψ$RW$PP$To$Ψ$p$PP=MakeRule[{Ψ$RW$PP[],((r[])^3)Ψ$p$PP[]}];
Ψ$p$MP$To$Ψ$RW$MP=MakeRule[{Ψ$p$MP[],((r[])^(-3))Ψ$RW$MP[]}];
Ψ$RW$MP$To$Ψ$p$MP=MakeRule[{Ψ$RW$MP[],((r[])^3)Ψ$p$MP[]}];

Ψ$p$PP$ToMet=MakeRule[{Ψ$p$PP[],(-4*𝒽$ab$p[a,-a])/(3*r[]^2)+(Λ^2*𝒽$ab$p[a,-a])/(6*r[]^2)+(4*𝒽$p[])/(3*r[]^4)-(Λ^2*𝒽$p[])/(3*r[]^4)+(𝒽$ab$p[a,-a]*𝒫[])/3-(𝒽$ab$p[a,b]*Τ[-a,-b])/3+(𝒽$p[]*Τ[a,-a])/(3*r[]^2)-(𝒽$ab$p[a,-a]*Τ[b,-b])/6-CD1[-a][CD1[a][𝒽$p[]]]/(3*r[]^2)+(4*𝒽$ab$p[b,-b]*CD1[-a][r[]]*CD1[a][r[]])/(3*r[]^2)-(8*𝒽$p[]*CD1[-a][r[]]*CD1[a][r[]])/(3*r[]^4)-(CD1[-a][𝒽$ab$p[b,-b]]*CD1[a][r[]])/(3*r[])+(4*CD1[-a][𝒽$p[]]*CD1[a][r[]])/(3*r[]^3)+(2*CD1[a][r[]]*CD1[-b][𝒽$ab$p[-a,b]])/(3*r[])-CD1[-b][CD1[-a][𝒽$ab$p[a,b]]]/3+CD1[-b][CD1[b][𝒽$ab$p[a,-a]]]/3-(2*𝒽$ab$p[-a,-b]*CD1[a][r[]]*CD1[b][r[]])/(3*r[]^2)}];
Ψ$p$MP$ToMet=MakeRule[{Ψ$p$MP[],((-2*I)*Λ^2*epsilong1[-a,-b]*𝒽$a$m[a]*CD1[b][r[]])/r[]^3+(I*Λ^2*epsilong1[-a,-b]*CD1[b][𝒽$a$m[a]])/r[]^2}];
(*Ψ$p[]/.Ψ$p$To$Met;
%/.Minus$Part;*)

The relation to the odd variable is direct (taken from [2306.17847]), but taking into account the change in sign in the convention for the odd vector harmonic:

DefTensor[Ψ$odd[],M1,PrintAs->"(Subscript[Ψ, odd])"]

Ψ$odd$To$Met=MakeRule[{Ψ$odd[],-((2r[])/((LL+2)(LL-1)))epsilong1[a,b](CD1[-a][𝒽$a$m[-b]]-(2/(r[]))CD1[-a][r[]]𝒽$a$m[-b ])}]

DefConstantSymbol[Cons]

(Ψ$RW$MP[]-Cons Ψ$odd[])//.Flatten[{Ψ$RW$MP$To$Ψ$p$MP,Ψ$p$MP$ToMet,Ψ$odd$To$Met}];
%//ContractMetric//ToCanonical;
%//.Flatten[{Cons->I (((LL+2)(LL+1)LL(LL-1))/2),Λ$To$L}]

Ψ$RW$MP$To$Ψ$odd=MakeRule[{Ψ$RW$MP[],I (((LL+2)(LL+1)LL(LL-1))/2)Ψ$odd[]}]

The even variable is new and related to the even variable (taken from [2306.17847]) through (here there is no change of sign due to conventions):

DefTensor[Ψ$even[],M1,PrintAs->"(Subscript[Ψ, even])"]

Ψ$even$To$Met=MakeRule[{Ψ$even[],((2r[])/(Λ^2))(((r[])^(-2))𝒽$p[]+(((2r[])/(r[](LL+2)(LL-1)+6Μ)))(CD1[a][r[]]CD1[b][r[]]𝒽$ab$p[-a,-b]-r[]CD1[a][r[]]CD1[-a][((r[])^(-2))𝒽$p[]]))}]

(Ψ$RW$PP[]+(((LL(LL+1))/2)( (LL+2)(LL-1)+((6Μ)/(r[])))Ψ$even[]-((6Μ)/((r[])^2))𝒽$p[]))//.Flatten[{Ψ$RW$PP$To$Ψ$p$PP,Ψ$p$PP$ToMet,Ψ$even$To$Met,Vacuum$GR}];
%//.Flatten[{BG$1,Einstein$Rep$DD𝒽$p,Einstein$Rep$Lorentz$𝒽$ab$p,Einstein$Rep$Trace$𝒽$ab$p,Vacuum$GR,DrDr$To$𝒻,𝒻$To$Schwarzschild,Λ$To$L}]//ContractMetric//ToCanonical;
%//.Flatten[{DrDr$To$𝒻,𝒻$To$Schwarzschild,𝒯$Cons$Φ$a$p,Λ$To$L}];
%//ContractMetric//ToCanonical//Simplify

(Ψ$RW$PP[]+(((LL(LL+1))/2)( (LL+2)(LL-1)+((6Μ)/(r[])))Ψ$even[]-((6Μ)/((r[])^2))𝒽$p[]))//.Flatten[{Ψ$RW$PP$To$Ψ$p$PP,Ψ$p$PP$ToMet,Ψ$even$To$Met,Vacuum$GR}];
%//.Flatten[{BG$1,Einstein$Rep$DD𝒽$p,Einstein$Rep$Lorentz$𝒽$ab$p,Einstein$Rep$Trace$𝒽$ab$p,Vacuum$GR,DrDr$To$𝒻,𝒻$To$Schwarzschild,Λ$To$L}]//ContractMetric//ToCanonical;
%//.Flatten[{DrDr$To$𝒻,𝒻$To$Schwarzschild,𝒯$Cons$Rep$Div$Φ$a$p,Λ$To$L}];
%//ContractMetric//ToCanonical//Simplify

InputForm[%//ScreenDollarIndices]

We notice the relation through the source term.

Ψ$RW$PP$To$Ψ$even=MakeRule[{Ψ$RW$PP[],(2*(-(r[]^3*Φ$ab$p[a,-a])+12*Μ*Φ$o[]+r[]*Φ$p[]))/3-(((LL(LL+1))/2)( (LL+2)(LL-1)+((6Μ)/(r[])))Ψ$even[]-((6Μ)/((r[])^2))𝒽$p[])}]

It will also be useful to relate our (Subscript[Ψ, RW +]) to the (Subscript[Ψ, MP]) used by Poisson in [2501.12377]:

DefTensor[Ψ$MP[],M1,PrintAs->"(Subscript[Ψ, MP])"]

Ψ$MP$To$Met=MakeRule[{Ψ$MP[],((LL(LL+1))/2)( (LL+2)(LL-1)+((6Μ)/(r[])))Ψ$even[]-((6Μ)/((r[])^2))𝒽$p[]}]

Ψ$RW$PP$To$Ψ$even=MakeRule[{Ψ$RW$PP[],(2*(-(r[]^3*Φ$ab$p[a,-a])+12*Μ*Φ$o[]+r[]*Φ$p[]))/3-(((LL(LL+1))/2)( (LL+2)(LL-1)+((6Μ)/(r[])))Ψ$even[]-((6Μ)/((r[])^2))𝒽$p[])}]

Ψ$RW$PP$To$Ψ$MP=MakeRule[{Ψ$RW$PP[],(2*(-(r[]^3*Φ$ab$p[a,-a])+12*Μ*Φ$o[]+r[]*Φ$p[]))/3-Ψ$MP[]}]

Ψ$RW$PP[]/.Ψ$RW$PP$To$Ψ$MP
TexPrint[%]

Ψ$RW$PP[]/.Ψ$RW$PP$To$Ψ$MP
%//.Flatten[{Ψ$MP$To$Met,Ψ$even$To$Met,Λ$To$L}]//ContractMetric//ToCanonical//Simplify;
Collect[%//.Flatten[{DrDr$To$𝒻,𝒻$To$Schwarzschild}]//ToCanonical,{𝒽$p[]}]
TexPrint[%]

Comparing source terms with the literature

As a nontrivial check we compare our source terms with those of the literature. We do so sector by sector.

(RHS$RW$SelfDual[]+RHS$RW$NonSelfDual[])/.Flatten[{Eval$RHS$RW$SelfDual,Eval$RHS$RW$NonSelfDual}]//ContractMetric//ToCanonical;
%//.Flatten[{Pert𝕁$To$metTmn,BG$1,DrDr$To$𝒻,𝒻$To$Schwarzschild,Vacuum$GR}]//ContractMetric//ToCanonical;
%//.Flatten[{𝒯$Cons$Φ$ab$p,𝒯$Cons$Φ$a$p,Vacuum$GR}]//ContractMetric//ToCanonical;
%//.Flatten[{DrDr$To$𝒻,𝒻$To$Schwarzschild}]//ContractMetric//ToCanonical;
Collect[%,{Φ$o[],Φ$a$p[a]}]

Odd source

The source term in the literature is, taking into account the different convention for the signs:

DefTensor[S$odd[],M1,PrintAs->"(Subscript[S, odd])"]

S$odd$To$Φ=MakeRule[{S$odd[],((4r[])/((LL+2)(LL-1)))epsilong1[a,b]CD1[-a][Φ$a$m[-b]]}]

Ψ$RW$MP$To$Ψ$odd

Collect[(2/(I(LL+2)(LL+1)LL(LL-1)))(Ψ$RW$MP[]/.Ψ$RW$MP$To$Ψ$odd),{Ψ$odd[]}]//Simplify

(((2/(I(LL+2)(LL+1)LL(LL-1))))%932)/.Minus$Part//ContractMetric//ToCanonical;
%/.Λ$To$L//ContractMetric//ToCanonical
ToCanonical[%-(S$odd[]/.S$odd$To$Φ)]

Match.

Even

Here we first write the source term for (SubPlus[((Subscript[Ψ, RW]))]) explicitly:

DefTensor[S$Ψ$RW$PP[],M1,PrintAs->"(Subscript[S, RW+])"]

%932/.Plus$Part//ContractMetric//ToCanonical;
Collect[%,{Φ$o[],Φ$p[],Φ$a$p[a]CD1[-a][r[]],Φ$ab$p[a,-a]}]

InputForm[%//ScreenDollarIndices]

S$Ψ$RW$PP$To$Φ=MakeRule[{S$Ψ$RW$PP[],-1/3*(Λ^2*r[]*Φ$ab$p[a,-a])+((-24*Μ^2)/r[]^3+(24*Μ)/r[]^2-(8*Λ^2*Μ)/r[]^2-(2*Λ^2)/r[]+Λ^4/r[])*Φ$o[]+((40*Μ)/(3*r[]^2)-4/r[]-(2*Λ^2)/(3*r[]))*Φ$p[]-4*Λ^2*Φ$a$p[a]*CD1[-a][r[]]+(24*Μ*Φ$a$p[a]*CD1[-a][r[]])/r[]+8*Μ*CD1[-a][CD1[a][Φ$o[]]]+(2*r[]*CD1[-a][CD1[a][Φ$p[]]])/3-2*r[]^2*CD1[-a][Φ$ab$p[b,-b]]*CD1[a][r[]]-(12*Μ*CD1[-a][Φ$o[]]*CD1[a][r[]])/r[]+(4*CD1[-a][Φ$p[]]*CD1[a][r[]])/3-(2*r[]^3*CD1[-b][CD1[b][Φ$ab$p[a,-a]]])/3}]

The source term for (Subscript[Ψ, MP]) was obtained by Poisson in [2501.12377], and is given by (recalling the relation between he’s Q’s and our Φ’s, which is (Subscript[Q, ab])=(Subscript[(SuperPlus[Φ]), ab]), (Subscript[Q, a])=2(Subscript[(SuperPlus[Φ]), a]), (Q^𝒷)=(2/(r^2))(SuperPlus[Φ]), (Q^(#))=2(Φ^⊕)), his source term is

DefTensor[S$Ψ$MP[],M1,PrintAs->"(Subscript[S, MP])"]

S$Ψ$MP$To$Φ=MakeRule[{S$Ψ$MP[],-2 ((r[])^2)CD1[a][r[]]CD1[-a][Φ$ab$p[c,-c]]+((LL^2)+LL-4)r[]Φ$ab$p[c,-c]+6(Μ/(r[]))CD1[a][r[]]CD1[-a][2Φ$o[]]-(1/(r[]))((1/2)(LL-1)(LL)(LL+1)(LL+2)+12(Μ/(r[]))-36 ((Μ^2)/((r[])^2)))2Φ$o[]+2(LL(LL+1)-6(Μ/(r[])))CD1[-a][r[]]2Φ$a$p[a]+2r[]𝒻[](2/((r[])^2))Φ$p[]}]

Given the relation betwee the MP variable and  (SubPlus[((Subscript[Ψ, RW]))]), we know that their source terms must be related by

Ψ$RW$PP$To$Ψ$MP

InputForm[%//ScreenDollarIndices]

S$Ψ$MP[]-(-S$Ψ$RW$PP[]+CD1[c][CD1[-c][((-2*r[]^3*Φ$ab$p[a,-a])/3+8*Μ*Φ$o[]+(2*r[]*Φ$p[])/3)]]+(((6Μ)/((r[])^3))-((Λ^2)/((r[])^2)))((-2*r[]^3*Φ$ab$p[a,-a])/3+8*Μ*Φ$o[]+(2*r[]*Φ$p[])/3));
%//.Flatten[{BG$1,Vacuum$GR,S$Ψ$MP$To$Φ,S$Ψ$RW$PP$To$Φ,Λ$To$L,DrDr$To$𝒻,𝒻$To$Schwarzschild}]//ContractMetric//ToCanonical

Match.

1.3. Teukolsky equations

Deriving the equation

We first derive a decoupled equation for the tensor (Subscript[(Ψ^⊕), a b]), which is already an interesting result in itself:

(Δℂ$D𝕁$VComp$PaQb[-a,-b]/.Eval$Pert$Δℂ$D𝕁$VComp$PaQb)//.Flatten[{Vacuum$GR}]//ContractMetric//ToCanonical//Simplify;
step1=%//.Flatten[{DrDr$To$𝒻,𝒻$To$Schwarzschild}]//ContractMetric//ToCanonical

step1/.Zero$Source

ℂ$Alg$Ψ$ab$m$To$Ψ$ab$o
ℂ$Alg$Trace$Ψ$ab$o

(step1//.Flatten[{ℂ$Alg$Ψ$ab$m$To$Ψ$ab$o,ℂ$Alg$Trace$Ψ$ab$o}]//ContractMetric//ToCanonical)//.Flatten[{ℂ$Alg$Ψ$ab$m$To$Ψ$ab$o,ℂ$Alg$Trace$Ψ$ab$o}];
step2=%//.Flatten[{DrDr$To$𝒻,𝒻$To$Schwarzschild}]//ContractMetric//ToCanonical

(Ψ$a$p[-a]/.ℂ$Alg$Ψ$a$p$To$Ξ$a$p$𝒽$a$m)//.Flatten[{Vacuum$GR,Einstein$Rep$Lorentz$𝒽$ab$p,Einstein$Rep$Trace$𝒽$ab$p}];
%//.Flatten[{DrDr$To$𝒻,𝒻$To$Schwarzschild}]//ContractMetric//ToCanonical

((step2/.ℂ$Alg$Ψ$a$p$To$Ξ$a$p$𝒽$a$m)//ContractMetric//ToCanonical)//.Flatten[{Vacuum$GR,Einstein$Rep$Lorentz$𝒽$ab$p,Einstein$Rep$Trace$𝒽$ab$p}];
step3=%//.Flatten[{DrDr$To$𝒻,𝒻$To$Schwarzschild}]//ContractMetric//ToCanonical

-((r[])^2)(((Dℂ$𝕁$Total[[8]]/.Eval$Pert$Dℂ$𝕁)//.Flatten[{Vacuum$GR,Einstein$Rep$Lorentz$𝒽$ab$p,Einstein$Rep$Trace$𝒽$ab$p}]//ToCanonical//Simplify))//ToCanonical;
%//.Flatten[{DrDr$To$𝒻,𝒻$To$Schwarzschild}]//ContractMetric//ToCanonical

InputForm[%//ScreenDollarIndices]

Ξ$a$p$To$DΨ$ab$o=MakeRule[{Ξ$a$p[-a],((3*I)*Μ*epsilong1[-a,-b]*𝒽$a$m[b])/r[]+r[]^2*Π$a$o[-a]+r[]^2*CD1[-b][Ψ$ab$o[-a,b]]-r[]*Ψ$ab$o[-a,-b]*CD1[b][r[]]}]

(step3/.Ξ$a$p$To$DΨ$ab$o//ContractMetric//ToCanonical)//.Flatten[{Vacuum$GR,Einstein$Rep$Lorentz$𝒽$ab$p,Einstein$Rep$Trace$𝒽$ab$p}]//ContractMetric//ToCanonical;
step4=%//.Flatten[{DrDr$To$𝒻,𝒻$To$Schwarzschild}]//ContractMetric//ToCanonical

InputForm[%//ScreenDollarIndices]

DefTensor[EQ$Ψ$ab$o$1[-a,-b],M1]
DefTensor[EQ$Ψ$ab$o$2[-a,-b],M1]
DefTensor[EQ$Ψ$ab$o[-a,-b],M1]

Eval$EQ$Ψ$ab$o$1=MakeRule[{EQ$Ψ$ab$o$1[-a,-b],(-I)*epsilong1[-a,c]*Π$ab$m[-b,-c]+Π$ab$p[-a,-b]-I*g1[-a,-b]*Π$m[]+epsilong1[-a,-b]*Π$p[]-(16*Μ*Ψ$ab$o[-a,-b])/r[]^3+(4*Ψ$ab$o[-a,-b])/r[]^2-(Λ^2*Ψ$ab$o[-a,-b])/r[]^2+((6*I)*Μ*epsilong1[-b,-c]*𝒽$a$m[c]*CD1[-a][r[]])/r[]^4+(3*Π$a$o[-b]*CD1[-a][r[]])/r[]+CD1[-a][Π$a$o[-b]]+((6*I)*Μ*epsilong1[-a,-c]*𝒽$a$m[c]*CD1[-b][r[]])/r[]^4+(4*Π$a$o[-a]*CD1[-b][r[]])/r[]-(4*g1[-a,-b]*Π$a$o[c]*CD1[-c][r[]])/r[]+(4*CD1[-b][r[]]*CD1[-c][Ψ$ab$o[-a,c]])/r[]+(4*CD1[-a][r[]]*CD1[-c][Ψ$ab$o[-b,c]])/r[]+CD1[-c][CD1[c][Ψ$ab$o[-a,-b]]]-((6*I)*Μ*epsilong1[-b,-c]*𝒽$a$m[-a]*CD1[c][r[]])/r[]^4-((6*I)*Μ*epsilong1[-a,-c]*𝒽$a$m[-b]*CD1[c][r[]])/r[]^4+(I*epsilong1[-a,-c]*Γ$a$m[-b]*CD1[c][r[]])/r[]-(4*Ψ$ab$o[-b,-c]*CD1[-a][r[]]*CD1[c][r[]])/r[]^2-(4*Ψ$ab$o[-a,-c]*CD1[-b][r[]]*CD1[c][r[]])/r[]^2-(2*CD1[-c][Ψ$ab$o[-a,-b]]*CD1[c][r[]])/r[]-I*epsilong1[-a,-c]*CD1[c][Γ$a$m[-b]]-(4*g1[-a,-b]*CD1[c][r[]]*CD1[-d][Ψ$ab$o[-c,d]])/r[]+((12*I)*Μ*epsilong1[-c,-d]*g1[-a,-b]*𝒽$a$m[c]*CD1[d][r[]])/r[]^4+(4*g1[-a,-b]*Ψ$ab$o[-c,-d]*CD1[c][r[]]*CD1[d][r[]])/r[]^2},MetricOn->All]

(epsilong1[a,-c]EQ$Ψ$ab$o$1[-a,-b]/.Eval$EQ$Ψ$ab$o$1)//ContractMetric//ToCanonical

InputForm[%//ScreenDollarIndices]

Eval$EQ$Ψ$ab$o$2=MakeRule[{EQ$Ψ$ab$o$2[-b,-c],I*Π$ab$m[-b,-c]-epsilong1[-c,a]*Π$ab$p[-b,-a]-I*epsilong1[-b,-c]*Π$m[]-g1[-b,-c]*Π$p[]+(16*Μ*epsilong1[-c,a]*Ψ$ab$o[-b,-a])/r[]^3-(4*epsilong1[-c,a]*Ψ$ab$o[-b,-a])/r[]^2+(Λ^2*epsilong1[-c,a]*Ψ$ab$o[-b,-a])/r[]^2-(4*epsilong1[-b,-c]*Π$a$o[a]*CD1[-a][r[]])/r[]-(3*epsilong1[-c,-a]*Π$a$o[-b]*CD1[a][r[]])/r[]+(2*epsilong1[-c,d]*CD1[-a][Ψ$ab$o[-b,-d]]*CD1[a][r[]])/r[]-epsilong1[-c,-a]*CD1[a][Π$a$o[-b]]-(4*epsilong1[-c,-a]*Π$a$o[a]*CD1[-b][r[]])/r[]+(4*epsilong1[-c,d]*Ψ$ab$o[-a,-d]*CD1[a][r[]]*CD1[-b][r[]])/r[]^2-(I*Γ$a$m[-b]*CD1[-c][r[]])/r[]+I*CD1[-c][Γ$a$m[-b]]-(4*epsilong1[-b,-c]*CD1[a][r[]]*CD1[-d][Ψ$ab$o[-a,d]])/r[]-(4*epsilong1[-c,a]*CD1[-b][r[]]*CD1[-d][Ψ$ab$o[-a,d]])/r[]-(4*epsilong1[-c,-a]*CD1[a][r[]]*CD1[-d][Ψ$ab$o[-b,d]])/r[]-epsilong1[-c,a]*CD1[-d][CD1[d][Ψ$ab$o[-b,-a]]]+(4*epsilong1[-b,-c]*Ψ$ab$o[-a,-d]*CD1[a][r[]]*CD1[d][r[]])/r[]^2+(4*epsilong1[-c,-a]*Ψ$ab$o[-b,-d]*CD1[a][r[]]*CD1[d][r[]])/r[]^2},MetricOn->All]

(epsilong1[c,-b]EQ$Ψ$ab$o$2[-a,-c])/.Eval$EQ$Ψ$ab$o$2//ContractMetric//ToCanonical

InputForm[%//ScreenDollarIndices]

Eval$EQ$Ψ$ab$o=MakeRule[{EQ$Ψ$ab$o[-a,-b],(-I)*epsilong1[-b,c]*Π$ab$m[-a,-c]+Π$ab$p[-a,-b]-I*g1[-a,-b]*Π$m[]-epsilong1[-a,-b]*Π$p[]-(16*Μ*Ψ$ab$o[-a,-b])/r[]^3+(4*Ψ$ab$o[-a,-b])/r[]^2-(Λ^2*Ψ$ab$o[-a,-b])/r[]^2+(4*Π$a$o[-b]*CD1[-a][r[]])/r[]+(3*Π$a$o[-a]*CD1[-b][r[]])/r[]+CD1[-b][Π$a$o[-a]]-(4*g1[-a,-b]*Π$a$o[c]*CD1[-c][r[]])/r[]+(4*CD1[-b][r[]]*CD1[-c][Ψ$ab$o[-a,c]])/r[]+(4*CD1[-a][r[]]*CD1[-c][Ψ$ab$o[-b,c]])/r[]+CD1[-c][CD1[c][Ψ$ab$o[-a,-b]]]+(I*epsilong1[-b,-c]*Γ$a$m[-a]*CD1[c][r[]])/r[]-(4*Ψ$ab$o[-b,-c]*CD1[-a][r[]]*CD1[c][r[]])/r[]^2-(4*Ψ$ab$o[-a,-c]*CD1[-b][r[]]*CD1[c][r[]])/r[]^2-(2*CD1[-c][Ψ$ab$o[-a,-b]]*CD1[c][r[]])/r[]-I*epsilong1[-b,-c]*CD1[c][Γ$a$m[-a]]-(4*g1[-a,-b]*CD1[c][r[]]*CD1[-d][Ψ$ab$o[-c,d]])/r[]+(4*g1[-a,-b]*Ψ$ab$o[-c,-d]*CD1[c][r[]]*CD1[d][r[]])/r[]^2}]

We split into LHS and RHS

DefTensor[EQ$Ψ$ab$o$LHS[-a,-b],M1]
DefTensor[EQ$Ψ$ab$o$RHS[-a,-b],M1]

-((EQ$Ψ$ab$o[-a,-b]/.Eval$EQ$Ψ$ab$o)-((EQ$Ψ$ab$o[-a,-b]/.Eval$EQ$Ψ$ab$o)/.Zero$Source))//ContractMetric//ToCanonical

InputForm[%//ScreenDollarIndices]

(EQ$Ψ$ab$o[-a,-b]/.Eval$EQ$Ψ$ab$o)/.Zero$Source

InputForm[%//ScreenDollarIndices]

Eval$EQ$Ψ$ab$o$LHS=MakeRule[{EQ$Ψ$ab$o$LHS[-a,-b],(-4*Ψ$ab$o[-a,-b])/r[]^2-(Λ^2*Ψ$ab$o[-a,-b])/r[]^2+(4*CD1[-b][r[]]*CD1[-c][Ψ$ab$o[-a,c]])/r[]+(4*CD1[-a][r[]]*CD1[-c][Ψ$ab$o[-b,c]])/r[]+CD1[-c][CD1[c][Ψ$ab$o[-a,-b]]]-(4*Ψ$ab$o[-b,-c]*CD1[-a][r[]]*CD1[c][r[]])/r[]^2-(4*Ψ$ab$o[-a,-c]*CD1[-b][r[]]*CD1[c][r[]])/r[]^2+(8*Ψ$ab$o[-a,-b]*CD1[-c][r[]]*CD1[c][r[]])/r[]^2-(2*CD1[-c][Ψ$ab$o[-a,-b]]*CD1[c][r[]])/r[]-(4*g1[-b,-a]*CD1[c][r[]]*CD1[-d][Ψ$ab$o[-c,d]])/r[]+(4*g1[-b,-a]*Ψ$ab$o[-c,-d]*CD1[c][r[]]*CD1[d][r[]])/r[]^2},MetricOn->All]
Eval$EQ$Ψ$ab$o$RHS=MakeRule[{EQ$Ψ$ab$o$RHS[-a,-b],I*epsilong1[-b,c]*Π$ab$m[-a,-c]-Π$ab$p[-a,-b]+I*g1[-a,-b]*Π$m[]+epsilong1[-a,-b]*Π$p[]-(4*Π$a$o[-b]*CD1[-a][r[]])/r[]-(3*Π$a$o[-a]*CD1[-b][r[]])/r[]-CD1[-b][Π$a$o[-a]]+(4*g1[-a,-b]*Π$a$o[c]*CD1[-c][r[]])/r[]-(I*epsilong1[-b,-c]*Γ$a$m[-a]*CD1[c][r[]])/r[]+I*epsilong1[-b,-c]*CD1[c][Γ$a$m[-a]]},MetricOn->All]

ToCanonical[(EQ$Ψ$ab$o[-a,-b]/.Eval$EQ$Ψ$ab$o)-(EQ$Ψ$ab$o$LHS[-a,-b]-EQ$Ψ$ab$o$RHS[-a,-b])/.Flatten[{Eval$EQ$Ψ$ab$o$LHS,Eval$EQ$Ψ$ab$o$RHS}]]

Result

DefTensor[EQ$Ψ$ab$o$LHS[-a,-b],M1]
DefTensor[EQ$Ψ$ab$o$RHS[-a,-b],M1]

Eval$EQ$Ψ$ab$o$LHS=MakeRule[{EQ$Ψ$ab$o$LHS[-a,-b],(-4*Ψ$ab$o[-a,-b])/r[]^2-(Λ^2*Ψ$ab$o[-a,-b])/r[]^2+(4*CD1[-b][r[]]*CD1[-c][Ψ$ab$o[-a,c]])/r[]+(4*CD1[-a][r[]]*CD1[-c][Ψ$ab$o[-b,c]])/r[]+CD1[-c][CD1[c][Ψ$ab$o[-a,-b]]]-(4*Ψ$ab$o[-b,-c]*CD1[-a][r[]]*CD1[c][r[]])/r[]^2-(4*Ψ$ab$o[-a,-c]*CD1[-b][r[]]*CD1[c][r[]])/r[]^2+(8*Ψ$ab$o[-a,-b]*CD1[-c][r[]]*CD1[c][r[]])/r[]^2-(2*CD1[-c][Ψ$ab$o[-a,-b]]*CD1[c][r[]])/r[]-(4*g1[-b,-a]*CD1[c][r[]]*CD1[-d][Ψ$ab$o[-c,d]])/r[]+(4*g1[-b,-a]*Ψ$ab$o[-c,-d]*CD1[c][r[]]*CD1[d][r[]])/r[]^2},MetricOn->All]
Eval$EQ$Ψ$ab$o$RHS=MakeRule[{EQ$Ψ$ab$o$RHS[-a,-b],I*epsilong1[-b,c]*Π$ab$m[-a,-c]-Π$ab$p[-a,-b]+I*g1[-a,-b]*Π$m[]+epsilong1[-a,-b]*Π$p[]-(4*Π$a$o[-b]*CD1[-a][r[]])/r[]-(3*Π$a$o[-a]*CD1[-b][r[]])/r[]-CD1[-b][Π$a$o[-a]]+(4*g1[-a,-b]*Π$a$o[c]*CD1[-c][r[]])/r[]-(I*epsilong1[-b,-c]*Γ$a$m[-a]*CD1[c][r[]])/r[]+I*epsilong1[-b,-c]*CD1[c][Γ$a$m[-a]]},MetricOn->All]

Equation for (Subscript[(Ψ^⊕), ll])

LHS:

With ℓ affine:

l[a]l[b](EQ$Ψ$ab$o$LHS[-a,-b]/.Eval$EQ$Ψ$ab$o$LHS);
%/.Ψ$ab$o$ToFrame//ContractMetric//ToCanonical;
%//.Flatten[{Dr$ToFrame,Dl$ToFrame,Dn$ToFrame}]//ContractMetric//ToCanonical
%//.Flatten[{l$Affine,Dr$ToFrame,ϵp$l$Affine$ToM,ρp$l$Affine$ToM,ρ$l$Affine$ToM}]//ContractMetric//ToCanonical

With n affine:

l[a]l[b](EQ$Ψ$ab$o$LHS[-a,-b]/.Eval$EQ$Ψ$ab$o$LHS);
%/.Ψ$ab$o$ToFrame//ContractMetric//ToCanonical;
%//.Flatten[{Dr$ToFrame,Dl$ToFrame,Dn$ToFrame}]//ContractMetric//ToCanonical
%//.Flatten[{n$Affine,Dr$ToFrame,ϵ$n$Affine$ToM,ρp$n$Affine$ToM,ρ$n$Affine$ToM}]//ContractMetric//ToCanonical

RHS:

With ℓ affine:

l[a]l[b](EQ$Ψ$ab$o$RHS[-a,-b]/.Eval$EQ$Ψ$ab$o$RHS);
%/.Flatten[{Π$ab$p$ToFrame,Π$ab$m$ToFrame,Π$a$o$ToFrame,Γ$a$m$ToFrame,Γ$a$p$ToFrame}]//ContractMetric//ToCanonical;
%//.Flatten[{Dr$ToFrame,Dl$ToFrame,Dn$ToFrame}]//ContractMetric//ToCanonical;
%//.Flatten[{l$Affine,Dr$ToFrame,ϵp$l$Affine$ToM,ρp$l$Affine$ToM,ρ$l$Affine$ToM}]//ContractMetric//ToCanonical

With 𝓃 affine:

l[a]l[b](EQ$Ψ$ab$o$RHS[-a,-b]/.Eval$EQ$Ψ$ab$o$RHS);
%/.Flatten[{Π$ab$p$ToFrame,Π$ab$m$ToFrame,Π$a$o$ToFrame,Γ$a$m$ToFrame,Γ$a$p$ToFrame}]//ContractMetric//ToCanonical;
%//.Flatten[{Dr$ToFrame,Dl$ToFrame,Dn$ToFrame}]//ContractMetric//ToCanonical
%//.Flatten[{n$Affine,Dr$ToFrame,ϵ$n$Affine$ToM,ρp$n$Affine$ToM,ρ$n$Affine$ToM}]//ContractMetric//ToCanonical

Equation for (Subscript[(Ψ^⊕), 𝓃𝓃])

LHS:

With ℓ affine:

n[a]n[b](EQ$Ψ$ab$o$LHS[-a,-b]/.Eval$EQ$Ψ$ab$o$LHS);
%/.Ψ$ab$o$ToFrame//ContractMetric//ToCanonical;
%//.Flatten[{Dr$ToFrame,Dl$ToFrame,Dn$ToFrame}]//ContractMetric//ToCanonical;
%//.Flatten[{l$Affine,Dr$ToFrame,ϵp$l$Affine$ToM,ρp$l$Affine$ToM,ρ$l$Affine$ToM}]//ContractMetric//ToCanonical

With n affine:

n[a]n[b](EQ$Ψ$ab$o$LHS[-a,-b]/.Eval$EQ$Ψ$ab$o$LHS);
%/.Ψ$ab$o$ToFrame//ContractMetric//ToCanonical;
%//.Flatten[{Dr$ToFrame,Dl$ToFrame,Dn$ToFrame}]//ContractMetric//ToCanonical;
%//.Flatten[{n$Affine,Dr$ToFrame,ϵ$n$Affine$ToM,ρp$n$Affine$ToM,ρ$n$Affine$ToM}]//ContractMetric//ToCanonical

RHS:

With ℓ affine:

n[a]n[b](EQ$Ψ$ab$o$RHS[-a,-b]/.Eval$EQ$Ψ$ab$o$RHS);
%/.Flatten[{Π$ab$p$ToFrame,Π$ab$m$ToFrame,Π$a$o$ToFrame,Γ$a$m$ToFrame,Γ$a$p$ToFrame}]//ContractMetric//ToCanonical;
%//.Flatten[{Dr$ToFrame,Dl$ToFrame,Dn$ToFrame}]//ContractMetric//ToCanonical;
%//.Flatten[{l$Affine,Dr$ToFrame,ϵp$l$Affine$ToM,ρp$l$Affine$ToM,ρ$l$Affine$ToM}]//ContractMetric//ToCanonical

With 𝓃 affine:

n[a]n[b](EQ$Ψ$ab$o$RHS[-a,-b]/.Eval$EQ$Ψ$ab$o$RHS);
%/.Flatten[{Π$ab$p$ToFrame,Π$ab$m$ToFrame,Π$a$o$ToFrame,Γ$a$m$ToFrame,Γ$a$p$ToFrame}]//ContractMetric//ToCanonical;
%//.Flatten[{Dr$ToFrame,Dl$ToFrame,Dn$ToFrame}]//ContractMetric//ToCanonical;
%//.Flatten[{n$Affine,Dr$ToFrame,ϵ$n$Affine$ToM,ρp$n$Affine$ToM,ρ$n$Affine$ToM}]//ContractMetric//ToCanonical

A GHP-covariant form of the equations

Here we take a really artesanal approach to rewrite the equations in a 2D GHP-covariant form

(Subscript[(Ψ^⊕), ll]):

DefTensor[Θ$l$Ψ$ll$o[],M1,PrintAs->"(Subscript[Θ, ℓ])((Subscript[(Ψ^⊕), ℓℓ]))"]
DefTensor[Θ$n$Ψ$ll$o[],M1,PrintAs->"(Subscript[Θ, 𝓃])((Subscript[(Ψ^⊕), ℓℓ]))"]
DefTensor[Box$Θ$Ψ$ll$o[],M1,PrintAs->"(Θ^2)((Subscript[(Ψ^⊕), ℓℓ]))"]

DlΨ$ll$o$ToGHP=MakeRule[{l[a]CD1[-a][Ψ$ll$o[]],Θ$l$Ψ$ll$o[]+2l[a]ω$a[-a]Ψ$ll$o[]},MetricOn->All]
DnΨ$ll$o$ToGHP=MakeRule[{n[a]CD1[-a][Ψ$ll$o[]],Θ$n$Ψ$ll$o[]+2n[a]ω$a[-a]Ψ$ll$o[]},MetricOn->All]
Box$Ψ$ll$o$ToGHP=MakeRule[{CD1[a][CD1[-a][Ψ$ll$o[]]],Box$Θ$Ψ$ll$o[]-(4 ω$a[-a]ω$a[a]-2CD1[a][ω$a[-a]])Ψ$ll$o[]+4ω$a[a]CD1[-a][Ψ$ll$o[]]}]

Hence the LHS is:

l[a]l[b](EQ$Ψ$ab$o$LHS[-a,-b]/.Eval$EQ$Ψ$ab$o$LHS);
%/.Ψ$ab$o$ToFrame//ContractMetric//ToCanonical;
%//.Flatten[{Dr$ToFrame,Dl$ToFrame,Dn$ToFrame}]//ContractMetric//ToCanonical;
%//.Flatten[{DlΨ$ll$o$ToGHP,DnΨ$ll$o$ToGHP,Box$Ψ$ll$o$ToGHP}]//ContractMetric//ToCanonical;
%//.Flatten[{ω$a$ToFrame,Dl$ToFrame,Dn$ToFrame,DlΨ$ll$o$ToGHP,DnΨ$ll$o$ToGHP}]//ContractMetric//ToCanonical;
%//.Flatten[{ω$a$ToFrame,Dl$ToFrame,Dn$ToFrame,DlΨ$ll$o$ToGHP,DnΨ$ll$o$ToGHP}]//ContractMetric//ToCanonical;
Collect[%,{Ψ$ll$o[]}]

(Subscript[(Ψ^⊕), nn]):

DefTensor[Θ$l$Ψ$nn$o[],M1,PrintAs->"(Subscript[Θ, ℓ])((Subscript[(Ψ^⊕), 𝓃𝓃]))"]
DefTensor[Θ$n$Ψ$nn$o[],M1,PrintAs->"(Subscript[Θ, 𝓃])((Subscript[(Ψ^⊕), 𝓃𝓃]))"]
DefTensor[Box$Θ$Ψ$nn$o[],M1,PrintAs->"(Θ^2)((Subscript[(Ψ^⊕), 𝓃𝓃]))"]

DlΨ$nn$o$ToGHP=MakeRule[{l[a]CD1[-a][Ψ$nn$o[]],Θ$l$Ψ$nn$o[]-2l[a]ω$a[-a]Ψ$nn$o[]},MetricOn->All]
DnΨ$nn$o$ToGHP=MakeRule[{n[a]CD1[-a][Ψ$nn$o[]],Θ$n$Ψ$nn$o[]-2n[a]ω$a[-a]Ψ$nn$o[]},MetricOn->All]
Box$Ψ$nn$o$ToGHP=MakeRule[{CD1[a][CD1[-a][Ψ$nn$o[]]],Box$Θ$Ψ$nn$o[]-(4 ω$a[-a]ω$a[a]+2CD1[a][ω$a[-a]])Ψ$nn$o[]-4ω$a[a]CD1[-a][Ψ$nn$o[]]}]

Hence the LHS is:

n[a]n[b](EQ$Ψ$ab$o$LHS[-a,-b]/.Eval$EQ$Ψ$ab$o$LHS);
%/.Ψ$ab$o$ToFrame//ContractMetric//ToCanonical;
%//.Flatten[{Dr$ToFrame,Dl$ToFrame,Dn$ToFrame}]//ContractMetric//ToCanonical;
%//.Flatten[{DlΨ$nn$o$ToGHP,DnΨ$nn$o$ToGHP,Box$Ψ$nn$o$ToGHP}]//ContractMetric//ToCanonical;
%//.Flatten[{ω$a$ToFrame,Dl$ToFrame,Dn$ToFrame,DlΨ$nn$o$ToGHP,DnΨ$nn$o$ToGHP}]//ContractMetric//ToCanonical;
%//.Flatten[{ω$a$ToFrame,Dl$ToFrame,Dn$ToFrame,DlΨ$nn$o$ToGHP,DnΨ$nn$o$ToGHP}]//ContractMetric//ToCanonical;
Collect[%,{Ψ$nn$o[]}]

(Subscript[(Π^⊕), a]), (Subscript[(SuperMinus[Γ]), a]):

DefTensor[Θ$l$Π$l$o[],M1,PrintAs->"(Subscript[Θ, ℓ])((Subscript[(Π^⊕), ℓ]))"]
DefTensor[Θ$n$Π$l$o[],M1,PrintAs->"(Subscript[Θ, 𝓃])((Subscript[(Π^⊕), ℓ]))"]
DefTensor[Θ$l$Π$n$o[],M1,PrintAs->"(Subscript[Θ, ℓ])((Subscript[(Π^⊕), 𝓃]))"]
DefTensor[Θ$n$Π$n$o[],M1,PrintAs->"(Subscript[Θ, 𝓃])((Subscript[(Π^⊕), 𝓃]))"]
DefTensor[Θ$l$Γ$l$m[],M1,PrintAs->"(Subscript[Θ, ℓ])((Subscript[(SuperMinus[Γ]), ℓ]))"]
DefTensor[Θ$n$Γ$l$m[],M1,PrintAs->"(Subscript[Θ, 𝓃])((Subscript[(SuperMinus[Γ]), ℓ]))"]
DefTensor[Θ$l$Γ$n$m[],M1,PrintAs->"(Subscript[Θ, ℓ])((Subscript[(SuperMinus[Γ]), 𝓃]))"]
DefTensor[Θ$n$Γ$n$m[],M1,PrintAs->"(Subscript[Θ, 𝓃])((Subscript[(SuperMinus[Γ]), 𝓃]))"]

DlΠ$l$o$ToGHP=MakeRule[{l[a]CD1[-a][Π$l$o[]],Θ$l$Π$l$o[]+l[a]ω$a[-a]Π$l$o[]},MetricOn->All];
DnΠ$l$o$ToGHP=MakeRule[{n[a]CD1[-a][Π$l$o[]],Θ$n$Π$l$o[]+n[a]ω$a[-a]Π$l$o[]},MetricOn->All];
DlΠ$n$o$ToGHP=MakeRule[{l[a]CD1[-a][Π$n$o[]],Θ$l$Π$n$o[]-l[a]ω$a[-a]Π$n$o[]},MetricOn->All];
DnΠ$n$o$ToGHP=MakeRule[{n[a]CD1[-a][Π$n$o[]],Θ$n$Π$n$o[]-n[a]ω$a[-a]Π$n$o[]},MetricOn->All];
DlΓ$l$m$ToGHP=MakeRule[{l[a]CD1[-a][Γ$l$m[]],Θ$l$Γ$l$m[]+l[a]ω$a[-a]Γ$l$m[]},MetricOn->All];
DnΓ$l$m$ToGHP=MakeRule[{n[a]CD1[-a][Γ$l$m[]],Θ$n$Γ$l$m[]+n[a]ω$a[-a]Γ$l$m[]},MetricOn->All];
DlΓ$n$m$ToGHP=MakeRule[{l[a]CD1[-a][Γ$n$m[]],Θ$l$Γ$n$m[]-l[a]ω$a[-a]Γ$n$m[]},MetricOn->All];
DnΓ$n$m$ToGHP=MakeRule[{n[a]CD1[-a][Γ$n$m[]],Θ$n$Γ$n$m[]-n[a]ω$a[-a]Γ$n$m[]},MetricOn->All];

Hence the RHS for (Subscript[(Ψ^⊕), ll]) is:

l[a]l[b](EQ$Ψ$ab$o$RHS[-a,-b]/.Eval$EQ$Ψ$ab$o$RHS);
%/.Flatten[{Π$ab$p$ToFrame,Π$ab$m$ToFrame,Π$a$o$ToFrame,Γ$a$m$ToFrame,Γ$a$p$ToFrame}]//ContractMetric//ToCanonical;
%//.Flatten[{Dr$ToFrame,Dl$ToFrame,Dn$ToFrame}]//ContractMetric//ToCanonical;
%//.Flatten[{DlΠ$l$o$ToGHP,DlΓ$l$m$ToGHP,ω$a$ToFrame}]//ContractMetric//ToCanonical

Hence the RHS for (Subscript[(Ψ^⊕), 𝓃𝓃]) is:

n[a]n[b](EQ$Ψ$ab$o$RHS[-a,-b]/.Eval$EQ$Ψ$ab$o$RHS);
%/.Flatten[{Π$ab$p$ToFrame,Π$ab$m$ToFrame,Π$a$o$ToFrame,Γ$a$m$ToFrame,Γ$a$p$ToFrame}]//ContractMetric//ToCanonical;
%//.Flatten[{Dr$ToFrame,Dl$ToFrame,Dn$ToFrame}]//ContractMetric//ToCanonical;
%//.Flatten[{DnΠ$n$o$ToGHP,DnΓ$n$m$ToGHP,ω$a$ToFrame}]//ContractMetric//ToCanonical

(Subscript[(SuperPlus[Φ]), a]), (Subscript[(SuperMinus[Φ]), a]):

DefTensor[Θ$l$Φ$l$p[],M1,PrintAs->"(Subscript[Θ, ℓ])((Subscript[(SuperPlus[Φ]), ℓ]))"]
DefTensor[Θ$n$Φ$l$p[],M1,PrintAs->"(Subscript[Θ, 𝓃])((Subscript[(SuperPlus[Φ]), ℓ]))"]
DefTensor[Θ$l$Φ$n$p[],M1,PrintAs->"(Subscript[Θ, ℓ])((Subscript[(SuperPlus[Φ]), 𝓃]))"]
DefTensor[Θ$n$Φ$n$p[],M1,PrintAs->"(Subscript[Θ, 𝓃])((Subscript[(SuperPlus[Φ]), 𝓃]))"]
DefTensor[Θ$l$Φ$l$m[],M1,PrintAs->"(Subscript[Θ, ℓ])((Subscript[(SuperMinus[Φ]), ℓ]))"]
DefTensor[Θ$n$Φ$l$m[],M1,PrintAs->"(Subscript[Θ, 𝓃])((Subscript[(SuperMinus[Φ]), ℓ]))"]
DefTensor[Θ$l$Φ$n$m[],M1,PrintAs->"(Subscript[Θ, ℓ])((Subscript[(SuperMinus[Φ]), 𝓃]))"]
DefTensor[Θ$n$Φ$n$m[],M1,PrintAs->"(Subscript[Θ, 𝓃])((Subscript[(SuperMinus[Φ]), 𝓃]))"]

DlΦ$l$p$ToGHP=MakeRule[{l[a]CD1[-a][Φ$l$p[]],Θ$l$Φ$l$p[]+l[a]ω$a[-a]Φ$l$p[]},MetricOn->All];
DnΦ$l$p$ToGHP=MakeRule[{n[a]CD1[-a][Φ$l$p[]],Θ$n$Φ$l$p[]+n[a]ω$a[-a]Φ$l$p[]},MetricOn->All];
DlΦ$n$p$ToGHP=MakeRule[{l[a]CD1[-a][Φ$n$p[]],Θ$l$Φ$n$p[]-l[a]ω$a[-a]Φ$n$p[]},MetricOn->All];
DnΦ$n$p$ToGHP=MakeRule[{n[a]CD1[-a][Φ$n$p[]],Θ$n$Φ$n$p[]-n[a]ω$a[-a]Φ$n$p[]},MetricOn->All];
DlΦ$l$m$ToGHP=MakeRule[{l[a]CD1[-a][Φ$l$m[]],Θ$l$Φ$l$m[]+l[a]ω$a[-a]Φ$l$m[]},MetricOn->All];
DnΦ$l$m$ToGHP=MakeRule[{n[a]CD1[-a][Φ$l$m[]],Θ$n$Φ$l$m[]+n[a]ω$a[-a]Φ$l$m[]},MetricOn->All];
DlΦ$n$m$ToGHP=MakeRule[{l[a]CD1[-a][Φ$n$m[]],Θ$l$Φ$n$m[]-l[a]ω$a[-a]Φ$n$m[]},MetricOn->All];
DnΦ$n$m$ToGHP=MakeRule[{n[a]CD1[-a][Φ$n$m[]],Θ$n$Φ$n$m[]-n[a]ω$a[-a]Φ$n$m[]},MetricOn->All];

Notice that the corresponding source terms have the expected symmetry (invariance under 2d-priming plus complex conjugation):

l[a]l[b]EQ$Ψ$ab$o$RHS[-a,-b]/.Eval$EQ$Ψ$ab$o$RHS;
%//.Flatten[{Vacuum$Source$GR,BG$1,Vacuum$GR,DrDr$To$𝒻,𝒻$To$Schwarzschild}]//ContractMetric//ToCanonical;
%//.Flatten[{DrDr$To$𝒻,𝒻$To$Schwarzschild}]//ContractMetric//ToCanonical;
%//.Flatten[{Dr$ToFrame,Φ$ab$p$ToFrame,Φ$a$p$ToFrame,Φ$a$m$ToFrame,Dl$ToFrame,Dn$ToFrame}]//ContractMetric//ToCanonical;
%//.Flatten[{DlΦ$l$m$ToGHP,DlΦ$l$p$ToGHP,ω$a$ToFrame}]//ContractMetric//ToCanonical

InputForm[%//ScreenDollarIndices]

DefTensor[S$Ψ$ll$o[],M1,PrintAs->"(Subscript[𝒮, (Subscript[(Ψ^⊕), ℓℓ])])"]

Eval$S$Ψ$ll$o=MakeRule[{S$Ψ$ll$o[],(2*I)*Θ$l$Φ$l$m[]-2*Θ$l$Φ$l$p[]+Φ$ll$p[]-(4*I)*ρ$sc[]*Φ$l$m[]+4*ρ$sc[]*Φ$l$p[]+(2*I)*ρ$sc[]^2*Φ$m[]-2*ρ$sc[]^2*Φ$o[]+(2*I)*l[a]*ρ$sc[]*CD1[-a][Φ$m[]]-2*l[a]*ρ$sc[]*CD1[-a][Φ$o[]]-I*l[a]*l[b]*CD1[-b][CD1[-a][Φ$m[]]]+l[a]*l[b]*CD1[-b][CD1[-a][Φ$o[]]]}];

n[a]n[b]EQ$Ψ$ab$o$RHS[-a,-b]/.Eval$EQ$Ψ$ab$o$RHS;
%//.Flatten[{Vacuum$Source$GR,BG$1,Vacuum$GR,DrDr$To$𝒻,𝒻$To$Schwarzschild}]//ContractMetric//ToCanonical;
%//.Flatten[{DrDr$To$𝒻,𝒻$To$Schwarzschild}]//ContractMetric//ToCanonical;
%//.Flatten[{Dr$ToFrame,Φ$ab$p$ToFrame,Φ$a$p$ToFrame,Φ$a$m$ToFrame,Dl$ToFrame,Dn$ToFrame}]//ContractMetric//ToCanonical;
%//.Flatten[{DnΦ$n$m$ToGHP,DnΦ$n$p$ToGHP,ω$a$ToFrame}]//ContractMetric//ToCanonical

InputForm[%//ScreenDollarIndices]

DefTensor[S$Ψ$nn$o[],M1,PrintAs->"(Subscript[𝒮, (Subscript[(Ψ^⊕), 𝓃𝓃])])"]

Eval$S$Ψ$nn$o=MakeRule[{S$Ψ$nn$o[],(-2*I)*Θ$n$Φ$n$m[]-2*Θ$n$Φ$n$p[]-(2*I)*ρp$sc[]^2*Φ$m[]+Φ$nn$p[]+(4*I)*ρp$sc[]*Φ$n$m[]+4*ρp$sc[]*Φ$n$p[]-2*ρp$sc[]^2*Φ$o[]-(2*I)*n[a]*ρp$sc[]*CD1[-a][Φ$m[]]-2*n[a]*ρp$sc[]*CD1[-a][Φ$o[]]+I*n[a]*n[b]*CD1[-b][CD1[-a][Φ$m[]]]+n[a]*n[b]*CD1[-b][CD1[-a][Φ$o[]]]}];

Matching source terms to literature

First we write the source terms in the literature for the Teukolsky equations [2306.17847]:

(Subscript[𝒮, 0]):

The first task is to compute the spin-weighted harmonic components of each  piece in the source term, since this is what enters in the expressions of [2306.17847]:

DefTensor[S$ll[],M1,PrintAs->"(Subscript[S, ℓℓ])"]
DefTensor[S$lm[],M1,PrintAs->"(Subscript[S, ℓ𝓂])"]
DefTensor[S$mm[],M1,PrintAs->"(Subscript[S, 𝓂𝓂])"]

S$ll$To𝒯=MakeRule[{S$ll[],Φ$ll$p[]}]
(*ℓ[μ]ℓ[ν]Perturbation[𝒯[-μ,-ν]]//SeparateMetric[]//TraceProductDummy;
ExpandProductMetric[%,g3];
%/.𝒯$Harm$Rep;
%/. BG$Total;
%//ContractMetric//ToCanonical;
%/Y[]//ToCanonical;
%/.Φ$ab$p$ToFrame//ContractMetric//ToCanonical*)

S$lm$To𝒯=MakeRule[{S$lm[],(I*Λ*Φ$l$m[])/(Sqrt[2]*r[])-(Λ*Φ$l$p[])/(Sqrt[2]*r[])}]
(*ℓ[μ]𝓂[ν]Perturbation[𝒯[-μ,-ν]]//SeparateMetric[]//TraceProductDummy;
ExpandProductMetric[%,g3];
%/.𝒯$Harm$Rep;
%/. BG$Total;
%//ContractMetric//ToCanonical;
%/.Dir$Harm$Def$Total;
%/.DY$To$sY//ContractMetric//ToCanonical;
%/p1Y[]//ToCanonical;
%/.Flatten[{Φ$a$p$ToFrame,Φ$a$m$ToFrame}]//ContractMetric//ToCanonical;*)

S$mm$To𝒯=MakeRule[{S$mm[],((-1/2*I)*Sqrt[(-1+LL)*(2+LL)]*Λ*Φ$m[])/r[]^2+(Sqrt[(-1+LL)*(2+LL)]*Λ*Φ$o[])/(2*r[]^2)}]
(*𝓂[μ]𝓂[ν]Perturbation[𝒯[-μ,-ν]]//SeparateMetric[]//TraceProductDummy;
ExpandProductMetric[%,g3];
%/.𝒯$Harm$Rep;
%/. BG$Total;
%//ContractMetric//ToCanonical;
%/.Dir$Harm$Def$Total//ContractMetric//ToCanonical;
%/.DDY$To$sY//ContractMetric//ToCanonical;
%/p2Y[]//ToCanonical*)

Now with these source terms we can compute (Subscript[𝒮, 0]):

DefTensor[S$0[],M1,PrintAs->"(Subscript[𝒮, 0])"]

Eval$S$0=MakeRule[{S$0[],((-(Sqrt[(LL+2)(LL+1)LL(LL-1)]))/(4 ((r[])^2)))S$ll[]-((Sqrt[(LL+2)(LL-1)])/(r[](Sqrt[2])))(l[a]CD1[-a][S$lm[]]-l[a]ω$a[-a]S$lm[]-3ρ$sc[]S$lm[])-(1/2)(l[a]CD1[-a][l[b]CD1[-b][S$mm[]]-ρ$sc[]S$mm[]]-l[a]ω$a[-a](l[b]CD1[-b][S$mm[]]-ρ$sc[]S$mm[])-5ρ$sc[](l[b]CD1[-b][S$mm[]]-ρ$sc[]S$mm[]))}]

dΨ0$To$Components
dΨ4$To$Components

-((4((r[])^2))/(Sqrt[(LL+2)(LL+1)LL(LL-1)]))S$0[]/.Eval$S$0;
%/.Flatten[{ω$a$ToFrame,S$ll$To𝒯,S$lm$To𝒯,S$mm$To𝒯}]//ContractMetric//ToCanonical;
%//.Flatten[{Dr$ToFrame,Dl$ToFrame,Dn$ToFrame,Λ$To$L}]//ContractMetric//ToCanonical;
Simplify[%,Assumptions->{LL>1}]//ToCanonical;
%//.Flatten[{ω$a$ToFrame,DlΦ$l$m$ToGHP,DlΦ$l$p$ToGHP}]//ContractMetric//ToCanonical

CD1[-a][ρ$sc[]]/.ρ$sc$To$Dr;
%//.Flatten[{Dl$ToFrame,BG$1,Vacuum$GR}]//ContractMetric//ToCanonical;
%//.Flatten[{Dr$ToFrame}]//ContractMetric//ToCanonical

InputForm[%]

Dρ$ToFrame=MakeRule[{CD1[-a][ρ$sc[]],-1/2*l[-a]/r[]^2+2*l[-a]*ϵp$sc[]*ρ$sc[]-2*n[-a]*ϵ$sc[]*ρ$sc[]-2*l[-a]*ρp$sc[]*ρ$sc[]-n[-a]*ρ$sc[]^2},MetricOn->All]

-((4((r[])^2))/(Sqrt[(LL+2)(LL+1)LL(LL-1)]))S$0[]/.Eval$S$0;
%/.Flatten[{ω$a$ToFrame,S$ll$To𝒯,S$lm$To𝒯,S$mm$To𝒯}]//ContractMetric//ToCanonical;
%//.Flatten[{Dr$ToFrame,Dl$ToFrame,Dn$ToFrame,Λ$To$L}]//ContractMetric//ToCanonical;
Simplify[%,Assumptions->{LL>1}]//ToCanonical;
%//.Flatten[{ω$a$ToFrame,DlΦ$l$m$ToGHP,DlΦ$l$p$ToGHP}]//ContractMetric//ToCanonical;
%/.Dρ$ToFrame//ContractMetric//ToCanonical
ToCanonical[%-(S$Ψ$ll$o[]/.Eval$S$Ψ$ll$o)]

Match.

(Subscript[𝒮, 4]):

Now same but for 𝓃𝓃:

DefTensor[S$nn[],M1,PrintAs->"(Subscript[S, 𝓃𝓃])"]
DefTensor[S$nmC[],M1,PrintAs->"(Subscript[S, 𝓃 (OverBar[𝓂])])"]
DefTensor[S$mCmC[],M1,PrintAs->"(Subscript[S, (OverBar[𝓂]) (OverBar[𝓂])])"]

S$nn$To𝒯=MakeRule[{S$nn[],Φ$nn$p[]}]
(*𝓃[μ]𝓃[ν]Perturbation[𝒯[-μ,-ν]]//SeparateMetric[]//TraceProductDummy;
ExpandProductMetric[%,g3];
%/.𝒯$Harm$Rep;
%/. BG$Total;
%//ContractMetric//ToCanonical;
%/Y[]//ToCanonical;
%/.Φ$ab$p$ToFrame//ContractMetric//ToCanonical;*)

S$nmC$To𝒯=MakeRule[{S$nmC[],(I*Λ*Φ$n$m[])/(Sqrt[2]*r[])+(Λ*Φ$n$p[])/(Sqrt[2]*r[])}]
(*𝓃[μ]𝓂C[ν]Perturbation[𝒯[-μ,-ν]]//SeparateMetric[]//TraceProductDummy;
ExpandProductMetric[%,g3];
%/.𝒯$Harm$Rep;
%/. BG$Total;
%//ContractMetric//ToCanonical;
%/.Dir$Harm$Def$Total;
%/.DY$To$sY//ContractMetric//ToCanonical;
%/m1Y[]//ToCanonical;
%/.Flatten[{Φ$a$p$ToFrame,Φ$a$m$ToFrame}]//ContractMetric//ToCanonical*)

S$mCmC$To𝒯=MakeRule[{S$mCmC[],((I/2)*Sqrt[(-1+LL)*(2+LL)]*Λ*Φ$m[])/r[]^2+(Sqrt[(-1+LL)*(2+LL)]*Λ*Φ$o[])/(2*r[]^2)}]
(*𝓂C[μ]𝓂C[ν]Perturbation[𝒯[-μ,-ν]]//SeparateMetric[]//TraceProductDummy;
ExpandProductMetric[%,g3];
%/.𝒯$Harm$Rep;
%/. BG$Total;
%//ContractMetric//ToCanonical;
%/.Dir$Harm$Def$Total//ContractMetric//ToCanonical;
%/.DDY$To$sY//ContractMetric//ToCanonical;
%/m2Y[]//ToCanonical*)

We now introduce the literature’s source term from the same reference:

DefTensor[S$4[],M1,PrintAs->"(Subscript[𝒮, 4])"]

Eval$S$4=MakeRule[{S$4[],((-(Sqrt[(LL+2)(LL+1)LL(LL-1)]))/(4 ((r[])^2)))S$nn[]+((Sqrt[(LL+2)(LL-1)])/(r[](Sqrt[2])))(n[a]CD1[-a][S$nmC[]]+n[a]ω$a[-a]S$nmC[]-3ρp$sc[]S$nmC[])-(1/2)(n[a]CD1[-a][n[b]CD1[-b][S$mCmC[]]-ρp$sc[]S$mCmC[]]+n[a]ω$a[-a](n[b]CD1[-b][S$mCmC[]]-ρp$sc[]S$mCmC[])-5ρp$sc[](n[b]CD1[-b][S$mCmC[]]-ρp$sc[]S$mCmC[]))}]

It should be related to our source term precisely by the same factor as in the ℓℓ case:

-((4((r[])^2))/(Sqrt[(LL+2)(LL+1)LL(LL-1)]))S$4[]/.Eval$S$4;
%/.Flatten[{ω$a$ToFrame,S$nn$To𝒯,S$nmC$To𝒯,S$mCmC$To𝒯}]//ContractMetric//ToCanonical;
%//.Flatten[{Dr$ToFrame,Dl$ToFrame,Dn$ToFrame,Λ$To$L}]//ContractMetric//ToCanonical;
Simplify[%,Assumptions->{LL>1}]//ToCanonical;
%//.Flatten[{ω$a$ToFrame,DnΦ$n$m$ToGHP,DnΦ$n$p$ToGHP}]//ContractMetric//ToCanonical

CD1[-a][ρp$sc[]]/.ρp$sc$To$Dr;
%//.Flatten[{Dn$ToFrame,BG$1,Vacuum$GR}]//ContractMetric//ToCanonical;
%//.Flatten[{Dr$ToFrame}]//ContractMetric//ToCanonical

InputForm[%]

Dρp$ToFrame=MakeRule[{CD1[-a][ρp$sc[]],-1/2*n[-a]/r[]^2-2*l[-a]*ϵp$sc[]*ρp$sc[]+2*n[-a]*ϵ$sc[]*ρp$sc[]-l[-a]*ρp$sc[]^2-2*n[-a]*ρp$sc[]*ρ$sc[]},MetricOn->All]

-((4((r[])^2))/(Sqrt[(LL+2)(LL+1)LL(LL-1)]))S$4[]/.Eval$S$4;
%/.Flatten[{ω$a$ToFrame,S$nn$To𝒯,S$nmC$To𝒯,S$mCmC$To𝒯}]//ContractMetric//ToCanonical;
%//.Flatten[{Dr$ToFrame,Dl$ToFrame,Dn$ToFrame,Λ$To$L}]//ContractMetric//ToCanonical;
Simplify[%,Assumptions->{LL>1}]//ToCanonical;
%//.Flatten[{ω$a$ToFrame,DnΦ$n$m$ToGHP,DnΦ$n$p$ToGHP}]//ContractMetric//ToCanonical;
%/.Dρp$ToFrame//ContractMetric//ToCanonical
ToCanonical[%-(S$Ψ$nn$o[]/.Eval$S$Ψ$nn$o)]

Match.

1.4. The Zerilli equation for the even sector

Derivation

(Δℂ$D𝕁$abPQ[]/.Eval$Pert$Δℂ$D𝕁$abPQ)//.Flatten[{Vacuum$GR,Einstein$Rep$Lorentz$𝒽$ab$p,Einstein$Rep$Trace$𝒽$ab$p,DrDr$To$𝒻,𝒻$To$Schwarzschild}]//ContractMetric//ToCanonical;
step1=%//.Flatten[{DrDr$To$𝒻,𝒻$To$Schwarzschild}]//ContractMetric//ToCanonical

step2=(step1/.Flatten[{ℂ$Alg$Ψ$a$m$To$Ξ$a$m$𝒽$a$m,ℂ$Alg$Ψ$ab$p$To$Ψ$m}])//.Flatten[{Vacuum$GR,Einstein$Rep$Lorentz$𝒽$ab$p,Einstein$Rep$Trace$𝒽$ab$p,DrDr$To$𝒻,𝒻$To$Schwarzschild}]//ContractMetric//ToCanonical

From the RW derivation we had:

Ξ$a$m$To$DΨ$m=MakeRule[{Ξ$a$m[-a],(-2*Μ*𝒽$a$m[-a])/r[]+(2*r[]^2*Π$a$m[-a])/Λ^2-((4*I)*Μ*epsilong1[-a,-b]*r[]*Φ$a$p[b])/Λ^2+((2*I)*Μ*epsilong1[-a,-b]*𝒽$p[]*CD1[b][r[]])/(Λ^2*r[]^2)-((4*I)*Μ*epsilong1[-a,-b]*Φ$o[]*CD1[b][r[]])/Λ^2-(epsilong1[-a,-b]*r[]*Ψ$m[]*CD1[b][r[]])/Λ^2-(I*Μ*epsilong1[-a,-b]*CD1[b][𝒽$p[]])/(Λ^2*r[])+((2*I)*Μ*epsilong1[-a,-b]*r[]*CD1[b][Φ$o[]])/Λ^2-(epsilong1[-a,-b]*r[]^2*CD1[b][Ψ$m[]])/Λ^2-((2*I)*Μ*epsilong1[b,c]*r[]*CD1[-c][𝒽$ab$p[-a,-b]])/Λ^2},MetricOn->All]

(I step2/.Ξ$a$m$To$DΨ$m//ContractMetric//ToCanonical)//.Flatten[{Vacuum$GR,Einstein$Rep$Lorentz$𝒽$ab$p,Einstein$Rep$Trace$𝒽$ab$p,DrDr$To$𝒻,𝒻$To$Schwarzschild}]//ContractMetric//ToCanonical;
step3=%//.Flatten[{DrDr$To$𝒻,𝒻$To$Schwarzschild}]//ContractMetric//ToCanonical

step3/.Zero$Source

Convert now into an eq for (SuperPlus[Ξ])

(step3//.Flatten[{ℂ$Alg$Ψ$m$To$Ψ$p$Ξ$p,ℂ$Alg$Ψ$p$To$Ξ$p$𝒽$p}]//ContractMetric//ToCanonical)//.Flatten[{BG$1,Vacuum$GR,Einstein$Rep$Lorentz$𝒽$ab$p,Einstein$Rep$Trace$𝒽$ab$p,DrDr$To$𝒻,𝒻$To$Schwarzschild}]//ContractMetric//ToCanonical;
step4=%//.Flatten[{DrDr$To$𝒻,𝒻$To$Schwarzschild}]//ContractMetric//ToCanonical

Now, the gauge transformations of the non-gauge-compensated versions of (SuperPlus[Ξ]) and (SuperPlus[𝒽]) are

(SuperPlus[Ξ])

LieD[ξ[α],CD3][CC[-μ,-ν,-ρ,-σ]]//SeparateMetric[];
%/.{μ->A,ν->B,ρ->P,σ->Q}//TraceProductDummy;
ExpandProductMetric[%,g3]/.ξ$Harm$Rep;
%/.BG$Total//CheckZeroDerivative;
(1/4)epsilong2[A,B]epsilong2[P,Q]%//.Flatten[{BG$1,Vacuum$GR}]//ContractMetric//ToCanonical;
%/.Dir$Harm$Def$Total//ContractMetric//ToCanonical;
%//.Harm$EQ$Y$Total//ContractMetric//ToCanonical;
%//.Flatten[{DrDr$To$𝒻,𝒻$To$Schwarzschild}]//ContractMetric//ToCanonical;
%/Y[]//ToCanonical

(SuperPlus[𝒽])

LieD[ξ[α],CD3][g3[-μ,-ν]]//SeparateMetric[];
%/.{μ->P,ν->Q}//TraceProductDummy;
ExpandProductMetric[%,g3]/.ξ$Harm$Rep;
%//CheckZeroDerivative;
%/.Dir$Harm$Def$Total//ContractMetric//ToCanonical;
UAB$Component[%]//ContractMetric//ToCanonical

We thus build the gauge-invariant combination (Subscript[Ψ, Z])=(SuperPlus[Ξ])- (Μ/r) (SuperPlus[𝒽]). Let us evaluate it in terms of the metric

DefTensor[Ψ$Z[],M1]

Ψ$Z$To$Ξ$p=MakeRule[{Ψ$Z[],((Ξ$p[])/(r[]))-(Μ/((r[])^2))𝒽$p[]}]
Ξ$p$To$Ψ$Z=MakeRule[{Ξ$p[],r[]Ψ$Z[]+(Μ/(r[]))𝒽$p[]}]

Ψ$Z[]/.Ψ$Z$To$Ξ$p;
%/.Pert$ℂ$To$Met;
%//.Flatten[{BG$1,Vacuum$GR,Einstein$Rep$Lorentz$𝒽$ab$p,Einstein$Rep$Trace$𝒽$ab$p,Einstein$Rep$Box$𝒽$p}]//ContractMetric//ToCanonical
%/.Plus$Part//ContractMetric//ToCanonical

InputForm[%//ScreenDollarIndices]

DefTensor[Ψ$Z$PP[],M1]

Ψ$Z$PP$To$Met=MakeRule[{Ψ$Z$PP[],(-2*Μ*𝒽$p[])/r[]^2+(2*𝒽$p[])/r[]+(Λ^2*𝒽$p[])/r[]+(2*r[]^3*Φ$ab$p[a,-a])/3+(2*r[]*Φ$o[])/3-(Λ^2*r[]*Φ$o[])/3+(4*r[]^2*Φ$a$p[a]*CD1[-a][r[]])/3+(2*r[]^3*CD1[-a][Φ$a$p[a]])/3-2*CD1[-a][𝒽$p[]]*CD1[a][r[]]+2*r[]*𝒽$ab$p[-a,-b]*CD1[a][r[]]*CD1[b][r[]]},MetricOn->All]

We first derive Zerilli’s equation for a variable that is natural in our set up, and at the end we fix its prefactors to match exactly the literature:

Following the vacuum case above, we first relate Ψ$Z$PP[] (the ZM conformal variable) to radial contractions of (Subscript[(SuperPlus[𝒽]), a b]) and (SuperPlus[𝒽]):

((1/(2 r[])))(Ψ$Z$PP[]-(Ψ$Z$PP[]/.Ψ$Z$PP$To$Met))//ToCanonical;
%//.Flatten[{𝒯$Cons$Φ$a$p,Vacuum$GR}]//ContractMetric//ToCanonical

InputForm[%//ScreenDollarIndices]

𝒽$ab$rara$To$ΨZ$PP=MakeRule[{𝒽$ab$p[-a,-b]*CD1[a][r[]]*CD1[b][r[]],(Μ*𝒽$p[])/r[]^3-𝒽$p[]/r[]^2-(Λ^2*𝒽$p[])/(2*r[]^2)-(r[]^2*Φ$ab$p[a,-a])/3+Φ$p[]/3+Ψ$Z$PP[]/(2*r[])+(CD1[-a][𝒽$p[]]*CD1[a][r[]])/r[]},MetricOn->All];

CD1[a][r[]]CD1[-a][Ψ$Z$PP[]]//.Flatten[{Ψ$Z$PP$To$Met,BG$1,Vacuum$GR,Einstein$Rep$Lorentz$𝒽$ab$p,Einstein$Rep$Trace$𝒽$ab$p,Einstein$Rep$DD𝒽$p}]//ContractMetric//ToCanonical;
%//.Flatten[{𝒽$ab$rara$To$ΨZ$PP,DrDr$To$𝒻,𝒻$To$Schwarzschild}]//ContractMetric//ToCanonical;
%//.Flatten[{BG$1,Vacuum$GR,𝒯$Cons$Φ$a$p,DrDr$To$𝒻,𝒻$To$Schwarzschild}]//ContractMetric//ToCanonical;
%//.Flatten[{DrDr$To$𝒻,𝒻$To$Schwarzschild}]//ContractMetric//ToCanonical;
Collect[%,{𝒽$p[]}]

InputForm[%//ScreenDollarIndices]

(3*Λ^2*Μ)/r[]^3-Λ^2/r[]^2+Λ^4/(2*r[]^2)//Simplify

InputForm[%]

𝒽$To$ΨZ$PP=MakeRule[{𝒽$p[],((1/(((Λ^2*(6*Μ+(-2+Λ^2)*r[]))/(2*r[]^3)))))(CD1[a][r[]]CD1[-a][Ψ$Z$PP[]]-((Λ^2*r[]^2*Φ$ab$p[a,-a])/3-(2*Φ$p[])/3-(Λ^2*Φ$p[])/3+(4*Μ*Φ$p[])/(3*r[])-(Λ^2*Ψ$Z$PP[])/(2*r[])+(2*r[]^3*CD1[-a][Φ$ab$p[b,-b]]*CD1[a][r[]])/3-(2*r[]*CD1[-a][Φ$p[]]*CD1[a][r[]])/3+2*r[]^2*Φ$ab$p[-a,-b]*CD1[a][r[]]*CD1[b][r[]]))},MetricOn->All]

step4
%//.Flatten[{Vacuum$Source$GR,BG$1,Vacuum$GR,DrDr$To$𝒻,𝒻$To$Schwarzschild}]//ContractMetric//ToCanonical;
%//.Flatten[{Vacuum$GR,𝒯$Cons$Φ$ab$p,𝒯$Cons$Φ$a$p}]//ContractMetric//ToCanonical;
step5=%//.Flatten[{DrDr$To$𝒻,𝒻$To$Schwarzschild}]//ContractMetric//ToCanonical

r[]step5//.Flatten[{Ξ$p$To$Ψ$Z,BG$1,Vacuum$GR,Einstein$Rep$Box$𝒽$p,DrDr$To$𝒻,𝒻$To$Schwarzschild}]//ContractMetric//ToCanonical;
%//.Flatten[{DrDr$To$𝒻,𝒻$To$Schwarzschild}]//ContractMetric//ToCanonical;
%/.Flatten[{Plus$Part,Ψ$Z[]->Ψ$Z$PP[]}];
step6=%//.Flatten[{DrDr$To$𝒻,𝒻$To$Schwarzschild,𝒽$ab$rara$To$ΨZ$PP,𝒽$To$ΨZ$PP}]//ContractMetric//ToCanonical

Rescaling the variable as

DefTensor[Ψ$Zer[],M1,PrintAs->"(Subscript[Ψ, ZM])"]

ΨZ$To$ΨZer=MakeRule[{Ψ$Z$PP[],((Λ^2)-2+6(Μ/(r[])))Ψ$Zer[]},MetricOn->All]
ΨZer$To$ΨZ=MakeRule[{Ψ$Zer[],((Ψ$Z$PP[])/(((Λ^2)-2+6(Μ/(r[])))))},MetricOn->All]

Ψ$Zer[]/.ΨZer$To$ΨZ//ToCanonical
TexPrint[%]

-((1/(((Λ^2)-2+6(Μ/(r[]))))))step6/.ΨZ$To$ΨZer;
%//.Flatten[{BG$1,Vacuum$GR}]//ContractMetric//ToCanonical;
step7=-%//.Flatten[{DrDr$To$𝒻,𝒻$To$Schwarzschild}]//ContractMetric//ToCanonical

Let us split this into LHS and RHS:

step7/.Zero$Source//ToCanonical//Simplify
TexPrint[%//ScreenDollarIndices]

InputForm[%//ScreenDollarIndices]

DefTensor[EQ$ΨZer$LHS[],M1]

Eval$EQ$ΨZer$LHS=MakeRule[{EQ$ΨZer$LHS[],-(((72*Μ^3+36*(-2+Λ^2)*Μ^2*r[]+6*(-2+Λ^2)^2*Μ*r[]^2+Λ^2*(-2+Λ^2)^2*r[]^3)*Ψ$Zer[])/(r[]^3*(6*Μ+(-2+Λ^2)*r[])^2))+CD1[-a][CD1[a][Ψ$Zer[]]]},MetricOn->All]

We doublecheck again this is zerilli’s potential:

V$Zerilli=-(1/((((LL-1)(LL+2)+((6 Μ)/(r[]))))^2))(((((LL-1)(LL+2)))^2)(((((LL-1)(LL+2)+2))/((r[])^2))+6 (Μ/((r[])^3)))+((36((Μ^2)))/((r[])^4))((LL-1)(LL+2)+((2 Μ)/(r[]))));

ToCanonical[(-(((72*Μ^3+36*(-2+Λ^2)*Μ^2*r[]+6*(-2+Λ^2)^2*Μ*r[]^2+Λ^2*(-2+Λ^2)^2*r[]^3))/(r[]^3*(6*Μ+(-2+Λ^2)*r[])^2)))-V$Zerilli];
%/.Λ$To$L//ToCanonical//Simplify

-(step7-(EQ$ΨZer$LHS[]/.Eval$EQ$ΨZer$LHS))//ToCanonical//Simplify
(*%/.Flatten[{MakeRule[{Φ$a$p[-a],0}],MakeRule[{Φ$p[],0}],MakeRule[{Φ$o[],0}]}]//ToCanonical;
%//Simplify*)

InputForm[%//ScreenDollarIndices]

DefTensor[S$ΨZer$Φ$o[],M1,PrintAs->"(Subscript[(S^(Φ^⊕)), (Subscript[Ψ, Z])])"]
DefTensor[S$ΨZer$Φ$p[],M1,PrintAs->"(Subscript[(S^(SuperPlus[Φ])), (Subscript[Ψ, Z])])"]
DefTensor[S$ΨZer$Φ$a$p[],M1,PrintAs->"(Subscript[(S^(Subscript[(SuperPlus[Φ]), a])), (Subscript[Ψ, Z])])"]
DefTensor[S$ΨZer$Φ$ab$p[],M1,PrintAs->"(Subscript[(S^(Subscript[(SuperPlus[Φ]), ab])), (Subscript[Ψ, Z])])"]

Eval$S$ΨZer$Φ$o=MakeRule[{S$ΨZer$Φ$o[],-((Λ^2*Φ$o[])/r[])}];
Eval$S$ΨZer$Φ$p=MakeRule[{S$ΨZer$Φ$p[],(-2*((60*Μ^2+2*(-26+7*Λ^2)*Μ*r[]-(-12+4*Λ^2+Λ^4)*r[]^2)*Φ$p[]+r[]^3*(6*Μ+(-2+Λ^2)*r[])*CD1[-a][CD1[a][Φ$p[]]]+2*r[]^2*(12*Μ+(-2+Λ^2)*r[])*CD1[-a][Φ$p[]]*CD1[a][r[]]))/(3*r[]*(6*Μ+(-2+Λ^2)*r[])^2)}];
Eval$S$ΨZer$Φ$a$p=MakeRule[{S$ΨZer$Φ$a$p[],(4*Λ^2*r[]*Φ$a$p[a]*CD1[-a][r[]])/(6*Μ+(-2+Λ^2)*r[])}];
Eval$S$ΨZer$Φ$ab$p=MakeRule[{S$ΨZer$Φ$ab$p[],(r[]*((36*Μ^2+12*(-1+2*Λ^2)*Μ*r[]+Λ^2*(-2+Λ^2)*r[]^2)*Φ$ab$p[a,-a]+2*r[]*(3*r[]*(10*Μ+(-2+Λ^2)*r[])*CD1[-a][Φ$ab$p[b,-b]]*CD1[a][r[]]+r[]^2*(6*Μ+(-2+Λ^2)*r[])*CD1[-b][CD1[b][Φ$ab$p[a,-a]]]+36*Μ*Φ$ab$p[-a,-b]*CD1[a][r[]]*CD1[b][r[]])))/(3*(6*Μ+(-2+Λ^2)*r[])^2)}];

DefTensor[EQ$ΨZer$RHS[],M1]

Eval$EQ$ΨZer$RHS=MakeRule[{EQ$ΨZer$RHS[],(r[]^2*(36*Μ^2 + 12*(-1 + 2*Λ^2)*Μ*r[] + Λ^2*(-2 + Λ^2)*r[]^2)*
   Φ$ab$p[a, -a] - 108*Λ^2*Μ^2*Φ$o[] + 72*Λ^2*Μ*r[]*Φ$o[] - 
  36*Λ^4*Μ*r[]*Φ$o[] - 12*Λ^2*r[]^2*Φ$o[] + 12*Λ^4*r[]^2*Φ$o[] - 
  3*Λ^6*r[]^2*Φ$o[] - 120*Μ^2*Φ$p[] + 104*Μ*r[]*Φ$p[] - 
  28*Λ^2*Μ*r[]*Φ$p[] - 24*r[]^2*Φ$p[] + 8*Λ^2*r[]^2*Φ$p[] + 
  2*Λ^4*r[]^2*Φ$p[] + 12*Λ^2*r[]^2*(6*Μ + (-2 + Λ^2)*r[])*Φ$a$p[a]*
   CD1[-a][r[]] - 12*Μ*r[]^3*CD1[-a][CD1[a][Φ$p[]]] + 
  4*r[]^4*CD1[-a][CD1[a][Φ$p[]]] - 2*Λ^2*r[]^4*CD1[-a][CD1[a][Φ$p[]]] + 
  60*Μ*r[]^4*CD1[-a][Φ$ab$p[b, -b]]*CD1[a][r[]] - 
  12*r[]^5*CD1[-a][Φ$ab$p[b, -b]]*CD1[a][r[]] + 
  6*Λ^2*r[]^5*CD1[-a][Φ$ab$p[b, -b]]*CD1[a][r[]] - 
  48*Μ*r[]^2*CD1[-a][Φ$p[]]*CD1[a][r[]] + 8*r[]^3*CD1[-a][Φ$p[]]*
   CD1[a][r[]] - 4*Λ^2*r[]^3*CD1[-a][Φ$p[]]*CD1[a][r[]] + 
  12*Μ*r[]^5*CD1[-b][CD1[b][Φ$ab$p[a, -a]]] - 
  4*r[]^6*CD1[-b][CD1[b][Φ$ab$p[a, -a]]] + 
  2*Λ^2*r[]^6*CD1[-b][CD1[b][Φ$ab$p[a, -a]]] + 
  72*Μ*r[]^3*Φ$ab$p[-a, -b]*CD1[a][r[]]*CD1[b][r[]])/
 (3*r[]*(6*Μ + (-2 + Λ^2)*r[])^2)}];

Matching variables and source terms to the literature

Next we check it matches the literature. For that we need to relate our variable to exactly the usual even variable.

DefTensor[Ψ$even[],M1,PrintAs->"(Subscript[Ψ, even])"]

Ψ$even$ToMet=MakeRule[{Ψ$even[],2((r[])/(Λ^2))(((𝒽$p[])/((r[])^2))+((2r[])/(((LL+2)(LL-1)r[]+6Μ)))(CD1[a][r[]]CD1[b][r[]]𝒽$ab$p[-a,-b]-r[]CD1[a][r[]]CD1[-a][((𝒽$p[])/((r[])^2))]))}]

Ψ$Zer[]-((LL*(1+LL))/2) Ψ$even[];
%//.Flatten[{ΨZer$To$ΨZ,Ψ$Z$PP$To$Met,Ψ$even$ToMet,Λ$To$L,DrDr$To$𝒻,𝒻$To$Schwarzschild}]//ContractMetric//ToCanonical//Simplify

InputForm[%//ScreenDollarIndices]

Ψ$Zer$To$Ψ$even=MakeRule[{Ψ$Zer[],((LL*(1+LL))/2) Ψ$even[]+(r[]^2*(2*r[]^2*Φ$ab$p[a,-a]+2*Φ$o[]-LL*Φ$o[]-LL^2*Φ$o[]+4*r[]*Φ$a$p[a]*CD1[-a][r[]]+2*r[]^2*CD1[-a][Φ$a$p[a]]))/(3*(6*Μ+(-2+LL+LL^2)*r[]))}]

Last step is to copy the even source term and compare it to the one arising from our equation after accounting for the relation between our Ψ$Zer and Ψ$even

DefTensor[S$even[],M1,PrintAs->"(Subscript[S, even])"]

DefTensor[ΛΛ[],M1,PrintAs->"(Subscript[λ, ℓ])"]

ΛΛ$To$r=MakeRule[{ΛΛ[],(LL+2)(LL-1)+((6Μ)/(r[]))}]

Eval$S$even=MakeRule[{S$even[],(8/(ΛΛ[]))CD1[a][r[]]Φ$a$p[-a]-(2/(r[]))Φ$o[]+(2/((Λ^2)ΛΛ[]))(((24Μ)/(ΛΛ[]))CD1[a][r[]]CD1[b][r[]]Φ$ab$p[-a,-b]-2((r[])^2)CD1[a][r[]]CD1[-a][Φ$ab$p[b,-b]]+4((𝒻[])/(r[]))Φ$p[]+((r[])/(ΛΛ[]))((LL+2)(LL-1)((LL+2)(LL-1)-2)+12(Μ/(r[]))((LL+2)(LL-1)-3)+84((Μ^2)/((r[])^2)))Φ$ab$p[b,-b])}]

The source terms should match as follows:

Ψ$Zer[]/.Ψ$Zer$To$Ψ$even;
Collect[%,{Ψ$even[]}]
%/.{Ψ$even[]->0}//ToCanonical

InputForm[%//ScreenDollarIndices]

S$even[]-((2/(LL(LL+1))))(EQ$ΨZer$RHS[]-(CD1[c][CD1[-c][(2*r[]^4*Φ$ab$p[a,-a])/(3*(6*Μ+(-2+LL+LL^2)*r[]))+(2*r[]^2*Φ$o[])/(3*(6*Μ+(-2+LL+LL^2)*r[]))-(LL*r[]^2*Φ$o[])/(3*(6*Μ+(-2+LL+LL^2)*r[]))-(LL^2*r[]^2*Φ$o[])/(3*(6*Μ+(-2+LL+LL^2)*r[]))+(4*r[]^3*Φ$a$p[a]*CD1[-a][r[]])/(3*(6*Μ+(-2+LL+LL^2)*r[]))+(2*r[]^4*CD1[-a][Φ$a$p[a]])/(3*(6*Μ+(-2+LL+LL^2)*r[]))]]+V$Zerilli ((2*r[]^4*Φ$ab$p[a,-a])/(3*(6*Μ+(-2+LL+LL^2)*r[]))+(2*r[]^2*Φ$o[])/(3*(6*Μ+(-2+LL+LL^2)*r[]))-(LL*r[]^2*Φ$o[])/(3*(6*Μ+(-2+LL+LL^2)*r[]))-(LL^2*r[]^2*Φ$o[])/(3*(6*Μ+(-2+LL+LL^2)*r[]))+(4*r[]^3*Φ$a$p[a]*CD1[-a][r[]])/(3*(6*Μ+(-2+LL+LL^2)*r[]))+(2*r[]^4*CD1[-a][Φ$a$p[a]])/(3*(6*Μ+(-2+LL+LL^2)*r[])))));
%//.Flatten[{Eval$S$even,Eval$EQ$ΨZer$RHS,ΛΛ$To$r,Λ$To$L}];
%//.Flatten[{BG$1,Vacuum$GR,DrDr$To$𝒻,𝒻$To$Schwarzschild,Λ$To$L}]//ContractMetric//ToCanonical;
%//.Flatten[{BG$1,Vacuum$GR,DrDr$To$𝒻,𝒻$To$Schwarzschild,Λ$To$L}]//ContractMetric//ToCanonical//Simplify;
%//.Flatten[{BG$1,Vacuum$GR,𝒯$Cons$Rep$Div$Φ$a$p,DrDr$To$𝒻,𝒻$To$Schwarzschild,Λ$To$L}]//ContractMetric//ToCanonical//Simplify;
%//.Flatten[{BG$1,Vacuum$GR,𝒯$Cons$Rep$Div$Φ$a$p,DrDr$To$𝒻,𝒻$To$Schwarzschild,Λ$To$L}]//ContractMetric//ToCanonical//Simplify

Match.

1.5. Darboux Transformations

General expressions and algebraically special transformations

DefTensor[W$Darb[],M1,PrintAs->"(Subscript[W, 𝒟])"]

V$RW=(6*Μ)/r[]^3-Λ^2/r[]^2

Ψ$Dual=CD1[a][r[]]CD1[-a][Ψ$RW[]]+W$Darb[] Ψ$RW[];
V$Dual=V$RW-(2/(𝒻[]))CD1[a][r[]]CD1[-a][W$Darb[]];

NN=(1/2)(LL-1)(LL+2);
ω$Alg=I (((NN(NN+1)))/(3 Μ));
V$RW=(6*Μ)/r[]^3-Λ^2/r[]^2;
Ψ$Alg=Exp[I ω$Alg r[]](((((r[])/(2Μ))-1))^(I 2 Μ ω$Alg))(((2 Μ)/(r[])))(1+r[] (((LL(LL+1)-2))/(6 Μ)));
W$Alg=((-CD1[a][r[]]((CD1[-a][Ψ$Alg])/(Ψ$Alg)))//ToCanonical)//.Flatten[{DrDr$To$𝒻,𝒻$To$Schwarzschild}]//ToCanonical//Simplify;

From Zerilli to (SubPlus[((SuperPlus[Ψ]))])

Ψ$RW$PP$To$Ψ$Z$PP=MakeRule[{Ψ$RW$PP[],(6*Μ*𝒽$p[])/r[]^2-Ψ$Z$PP[]}]

Collect[((Λ^2)/(12 Μ))Ψ$RW$PP[]//.Flatten[{Ψ$RW$PP$To$Ψ$Z$PP,ΨZ$To$ΨZer}]//ToCanonical,{Ψ$Zer[]}]
%/.𝒽$To$ΨZ$PP//ContractMetric//ToCanonical;
%//.Flatten[{ΨZ$To$ΨZer,DrDr$To$𝒻,Vacuum$GR,Zero$Source}]//ContractMetric//ToCanonical;
%//.Flatten[{ΨZ$To$ΨZer,DrDr$To$𝒻,𝒻$To$Schwarzschild,(*Λ$To$L,*)Vacuum$GR,Zero$Source}]//ContractMetric//ToCanonical//Simplify;
Collect[%,{Ψ$Zer[]}]

TexPrint[%]

W$Alg

Comments

0. Products of epsilons not handling index conflicts

If there is a conflict of indices involving a product of epsilons, instead of giving a warning the package returns the expression assuming the contraction is amongst the epsilon tensors, which might not be the case in the original equation where there is the conflict:

DefTensor[V$test[-a],M1]

epsilong1[a,-d]epsilong1[c,d]V$test[-d]

epsilong1[a,-f]epsilong1[c,d]V$test[-d]//ContractMetric//ToCanonical