Strain tensor

ClearAll["`*"]

Simplify[LegendreP[2,2,Cos[θ]]]

Simplify[SphericalHarmonicY[2,2,θ,ϕ]/LegendreP[2,2,Cos[θ]]]

SphericalHarmonicY[2,2,θ,ϕ]

1. Spacetime

$Assumptions={r≥0,θ≥0,θ≤π,r>2 M[r]&&r>0,ν[r]∈Reals};
x={t,r,θ,ϕ};
δ={{1,0,0,0},{0,1,0,0},{0,0,1,0},{0,0,0,1}};
dim=4;
order =1;
P2[θ_]:=3X[θ]*c0;
(*c0=(1/4) *(Sqrt[(5/(6 π))])*)
Y2[ϕ_]:=(1-2*Y[ϕ]);
g0={{-Exp[ν[r]],0,0,0},{0,Exp[λ[r]],0,0},{0,0,(r^2),0},{0,0,0,(r^2) X[θ]}};
rule1[Z_]:=Z/.Union[{((X'[θ])^2)->4 X[θ](1-X[θ]),X''[θ]->2 (1-2X[θ]),((Y'[ϕ])^2)->4 Y[ϕ](1-Y[ϕ]),Y''[ϕ]->2 (1-2Y[ϕ])}]
ξ0= (r^2){0,W[r](r^(-1))*Exp[-λ[r]/2]*P2[θ]*Y2[ϕ] *ϵ,-V[r](r^(-2))D[P2[θ],θ]*Y2[ϕ]*ϵ,-V[r](r^(-2))/X[θ]*P2[θ] *D[Y2[ϕ],ϕ]*ϵ}Exp[I*ω*t];
h=-(r^2){{H0[r]*Exp[ν[r]],I*ω*r*H1[r],0,0},{I*ω*r*H1[r],H2[r]*Exp[λ[r]],0,0},{0,0,(r^2)*K[r],0},{0,0,0,(r^2)*X[θ]*K[r]}}*P2[θ]Exp[I*ω*t]*Y2[ϕ]*ϵ;
gdd = g0 +h;
guu = Simplify[rule1[Normal[Series[Inverse[gdd],{ϵ,0,order}]]]];
Do[ξ[i]=ξ0[[i]],{i,1,dim}];
u0={Exp[-ν[r]/2],0,0,0};
ud0={-Exp[ν[r]/2],0,0,0};
Do[u[i]=u0[[i]],{i,1,dim}];
Do[ud[i]=ud0[[i]],{i,1,dim}];
Do[g[i,j]=Simplify[gdd[[i]][[j]]];,{i,1,dim},{j,1,dim}];
Do[ig[i,j]=Simplify[guu[[i]][[j]]];,{i,1,dim},{j,1,dim}];
Do[δg[i,j]=Simplify[h[[i]][[j]]];,{i,1,dim},{j,1,dim}];
Do[ds[i]=x[[i]];,{i,1,dim}];
Do[gud[i,j]=Simplify[Normal[Series[Sum[g[i,k]*ig[k,j],{k,1,dim}],{ϵ,0,order}]]];,{i,1,dim},{j,1,dim}];
Do[Chr[i,j,k]=Simplify[rule1[Normal[Series[(1/2)*Sum[ig[i,s]*(-D[g[j,k],ds[s]]+D[g[j,s],ds[k]]+D[g[s,k],ds[j]]),{s,1,dim}],{ϵ,0,order}]]]];,{i,1,dim},{j,1,dim},{k,1,dim}];
Do[Ruddd[i,k,l,m]= Simplify[rule1[Normal[Series[D[Chr[i, k, m], ds[l]]
       - D[Chr[i, k, l], ds[m]]
       + Sum[Chr[i, s, l]*Chr[s, k, m]-Chr[i, s, m]*Chr[s, k, l], {s, 1, dim}], {ϵ, 0, order}]]]]; , {i, 1, dim}, {k, 1, dim}, {l, 1, dim}, {m, 1, dim}];

Do[Ricci[i,k]=Simplify[rule1[Normal[Series[Sum[Ruddd[l,i,l,k],{l,1,dim}],{ϵ,0,order}]]]];,{i,1,dim},{k,1,dim}];
Do[Ruu[i,k]=Simplify[rule1[Normal[Series[Sum[Ricci[l,m]*ig[i,l]*ig[k,m],{l,1,dim},{m,1,dim}],{ϵ,0,order}]]]];,{i,1,dim},{k,1,dim}];
Rs=Simplify[rule1[Normal[Series[Sum[ig[i,j]*Ricci[i,j],{i,1,dim},{j,1,dim}],{ϵ,0,order}]]]];
Do[Gdd[i,j]=Simplify[rule1[Normal[Series[Ricci[i,j]-(1/2) g[i,j]Rs,{ϵ,0,order}]]]];,{i,1,dim},{j,1,dim}];
Do[Gud[i,k]=Simplify[rule1[Normal[Series[Sum[Gdd[i,j]*ig[j,k],{j,1,dim}],{ϵ,0,order}]]]];,{i,1,4},{k,1,4}]
Do[Lie[i]=rule1[Normal[Series[Sum[u[j]*D[ξ[i],ds[j]]-ξ[j]*D[u[i],ds[j]],{j,1,dim}],{ϵ,0,order}]]];,{i,1,dim}]

Do[q[i,j]=ig[i,j]+u[i]*u[j];,{i,1,dim},{j,1,dim}];
Do[ξd[i]=Simplify[rule1[Normal[Series[Sum[g[i,j]*ξ[j],{j,1,dim}],{ϵ,0,order}]]]];,{i,1,dim}];
(*Lagrangian perturbation of the metric*)
Do[Δg[i,j]=Simplify[rule1[Normal[Series[δg[i,j]+ D[ξd[i],ds[j]]+ D[ξd[j],ds[i]] -2Sum[Chr[k,i,j]*ξd[k],{k,1,dim}],{ϵ,0,order}]]]];,{i,1,dim},{j,1,dim}];
(*restmass fraction Lagrangian perturbation*)
Δnb=FullSimplify[rule1[Normal[Series[((-1)/2)*Sum[q[i,j]*Δg[i,j],{i,1,dim},{j,1,dim}],{ϵ,0,order}]]]];
(*perutrbation of four-velocity*)
Do[qud[i,j]=Simplify[rule1[Normal[Series[Sum[g[i,k]*q[k,j],{k,1,dim}],{ϵ,0,order}]]]];,{i,1,dim},{j,1,dim}];
Do[δu[i]=Simplify[rule1[Normal[Series[Sum[qud[i,j]*Lie[j],{j,1,dim}],{ϵ,0,order}]]+Normal[Series[Sum[(1/2)*u[i]*u[j]*u[k]*δg[j,k],{j,1,dim},{k,1,dim}],{ϵ,0,order}]]]];,{i,1,dim}];
Do[δud[i]=Simplify[Normal[Series[Sum[g[i,j]*δu[j],{j,1,dim}],{ϵ,0,order}]]];,{i,1,dim}];
Do[δ1[i,j]=δ[[i]][[j]];,{i,1,dim},{j,1,dim}];
Do[δsud[i,j]=Simplify[rule1[Normal[Series[Sum[(1/2)(qud[i,k]*q[l,j]-(1/3)qud[i,j]*q[k,l])*Δg[k,l],{k,1,dim},{l,1,dim}],{ϵ,0,order}]]]];,{i,1,dim},{j,1,dim}]
Do[δπud[i,j]=-2*μ[r]*δsud[i,j];,{i,1,dim},{j,1,dim}]

T1=Simplify[δπud[2,2]/(P2[θ]*Y2[ϕ]*Exp[I*ω*t]*ϵ)]

T2 = Simplify[r*δπud[2,3]/(D[P2[θ],θ]*Y2[ϕ]*Exp[I*ω*t]*ϵ)]

Simplify[δsud[2,2]/T1]

Simplify[δsud[2,3]/T2]

Simplify[δsud[2,4]/T2]

Simplify[δsud[4,3]]

FullSimplify[δsud[3,3]+(T1/((-2*μ[r])2)) * (P2[θ]Y2[ϕ]*Exp[I*ω*t]*ϵ)]

FullSimplify[δsud[4,4]+(T1/((-2*μ[r])2)) * (P2[θ]Y2[ϕ]*Exp[I*ω*t]*ϵ)]

δsud[3,4]

δsud[4,2]/δsud[2,4]

T2 = Simplify[r*δπud[2,3]/(3*X'[θ]*c0*(1-2Y[ϕ])*Exp[I*ω*t]*ϵ)]

Simplify[δsud[2,2]/(T1/(-2μ[r]))]

δsud[2,3]/T2

FullSimplify[δsud[2,2]]

δπud[2,3]

sud[1,1]=0;
sud[1,2]=0;
sud[1,3]=0;
sud[1,4]=0;
sud[2,1]=0;
sud[3,1]=0;
sud[4,1]=0;
sud[2,2]=t1[r]*(((3 c0 (E^(I t ω)) ϵ X[θ] (-1+2 Y[ϕ]))/(2 μ[r])));
sud[2,3] = t2[r]*(((3 c0 (E^(I t ω)) ϵ (-1+2 Y[ϕ]) X'[θ])/(2 r μ[r])));
sud[2,4]=t2[r]*(((3 c0 (E^(I t ω)) ϵ Y'[ϕ])/(r μ[r])));
sud[3,2]=(E^(-λ[r])) (r^2) *sud[2,3];
sud[3,3] =- ((t1[r])/((-2*μ[r])2)) * (3*X[θ]*c0*(1-2Y[ϕ])*Exp[I*ω*t]*ϵ)-3 c0 (E^(I t ω)) ϵ V[r] (-2+X[θ]) (-1+2 Y[ϕ]);
sud[3,4]=((3 c0 (E^(I t ω)) ϵ V[r] X'[θ] Y'[ϕ])/(X[θ]));
sud[4,2]=(E^(-λ[r])) (r^2) X[θ]*sud[2,4];
sud[4,3]=3 c0 (E^(I t ω)) ϵ V[r] X'[θ] Y'[ϕ];
sud[4,4] = - ((t1[r])/((-2*μ[r])2))  * (3*X[θ]*c0*(1-2Y[ϕ])*Exp[I*ω*t]*ϵ)+3 c0 (E^(I t ω)) ϵ V[r] (-2+X[θ]) (-1+2 Y[ϕ]);

s2=FullSimplify[3/2 * rule1[Sum[sud[i,j]*sud[j,i],{j,1,dim},{i,1,dim}]]/.{c0->(1/4) *(Sqrt[(5/(6 π))]),X[θ]->((Sin[θ])^2),Y[ϕ]->((Sin[ϕ])^2)}]

Test tides: PHYSICAL REVIEW D 101, 103025 (2020)

ClearAll["`*"]

$Assumptions={r≥0,θ≥0,θ≤π,r>2 M[r]&&r>0,ν[r]∈Reals};
x={t,r,θ,ϕ};
δ={{1,0,0,0},{0,1,0,0},{0,0,1,0},{0,0,0,1}};
dim=4;
order =1;
P2[θ_]:=(((3(1-X[θ])-1)/2))*c0;
(*c0=(Sqrt[(5/(4 π))])*)
g0={{-Exp[ν[r]],0,0,0},{0,Exp[λ[r]],0,0},{0,0,(r^2),0},{0,0,0,(r^2) X[θ]}};
rule1[Z_]:=Z/.Union[{((X'[θ])^2)->4 X[θ](1-X[θ]),X''[θ]->2 (1-2X[θ])}]
ξ0= {0,W[r](r^(-1))*P2[θ] *ϵ,V[r](r^(-2))D[P2[θ],θ]*ϵ,0}Exp[I*ω*t];
h={{H0[r]*Exp[ν[r]],I*ω*r*H1[r],0,0},{I*ω*r*H1[r],H2[r]*Exp[λ[r]],0,0},{0,0,(r^2)*K[r],0},{0,0,0,(r^2)*X[θ]*K[r]}}*P2[θ]Exp[I*ω*t]*ϵ;
gdd = g0 +h;
guu = Simplify[rule1[Normal[Series[Inverse[gdd],{ϵ,0,order}]]]];
Do[ξ[i]=ξ0[[i]],{i,1,dim}];
u0={Exp[-ν[r]/2],0,0,0};
ud0={-Exp[ν[r]/2],0,0,0};
Do[u[i]=u0[[i]],{i,1,dim}];
Do[ud[i]=ud0[[i]],{i,1,dim}];
Do[g[i,j]=Simplify[gdd[[i]][[j]]];,{i,1,dim},{j,1,dim}];
Do[ig[i,j]=Simplify[guu[[i]][[j]]];,{i,1,dim},{j,1,dim}];
Do[δg[i,j]=Simplify[h[[i]][[j]]];,{i,1,dim},{j,1,dim}];
Do[ds[i]=x[[i]];,{i,1,dim}];
Do[gud[i,j]=Simplify[Normal[Series[Sum[g[i,k]*ig[k,j],{k,1,dim}],{ϵ,0,order}]]];,{i,1,dim},{j,1,dim}];
Do[Chr[i,j,k]=Simplify[rule1[Normal[Series[(1/2)*Sum[ig[i,s]*(-D[g[j,k],ds[s]]+D[g[j,s],ds[k]]+D[g[s,k],ds[j]]),{s,1,dim}],{ϵ,0,order}]]]];,{i,1,dim},{j,1,dim},{k,1,dim}];
Do[Ruddd[i,k,l,m]= Simplify[rule1[Normal[Series[D[Chr[i, k, m], ds[l]]
       - D[Chr[i, k, l], ds[m]]
       + Sum[Chr[i, s, l]*Chr[s, k, m]-Chr[i, s, m]*Chr[s, k, l], {s, 1, dim}], {ϵ, 0, order}]]]]; , {i, 1, dim}, {k, 1, dim}, {l, 1, dim}, {m, 1, dim}];

Do[Ricci[i,k]=Simplify[rule1[Normal[Series[Sum[Ruddd[l,i,l,k],{l,1,dim}],{ϵ,0,order}]]]];,{i,1,dim},{k,1,dim}];
Do[Ruu[i,k]=Simplify[rule1[Normal[Series[Sum[Ricci[l,m]*ig[i,l]*ig[k,m],{l,1,dim},{m,1,dim}],{ϵ,0,order}]]]];,{i,1,dim},{k,1,dim}];
Rs=Simplify[rule1[Normal[Series[Sum[ig[i,j]*Ricci[i,j],{i,1,dim},{j,1,dim}],{ϵ,0,order}]]]];
Do[Gdd[i,j]=Simplify[rule1[Normal[Series[Ricci[i,j]-(1/2) g[i,j]Rs,{ϵ,0,order}]]]];,{i,1,dim},{j,1,dim}];
Do[Gud[i,k]=Simplify[rule1[Normal[Series[Sum[Gdd[i,j]*ig[j,k],{j,1,dim}],{ϵ,0,order}]]]];,{i,1,4},{k,1,4}]
Do[Lie[i]=rule1[Normal[Series[Sum[u[j]*D[ξ[i],ds[j]]-ξ[j]*D[u[i],ds[j]],{j,1,dim}],{ϵ,0,order}]]];,{i,1,dim}]

Do[q[i,j]=ig[i,j]+u[i]*u[j];,{i,1,dim},{j,1,dim}];
Do[ξd[i]=Simplify[rule1[Normal[Series[Sum[g[i,j]*ξ[j],{j,1,dim}],{ϵ,0,order}]]]];,{i,1,dim}];
(*Lagrangian perturbation of the metric*)
Do[Δg[i,j]=Simplify[rule1[Normal[Series[δg[i,j]+ D[ξd[i],ds[j]]+ D[ξd[j],ds[i]] -2Sum[Chr[k,i,j]*ξd[k],{k,1,dim}],{ϵ,0,order}]]]];,{i,1,dim},{j,1,dim}];

(*restmass fraction Lagrangian perturbation*)
Δnb=FullSimplify[rule1[Normal[Series[((-1)/2)*Sum[q[i,j]*Δg[i,j],{i,1,dim},{j,1,dim}],{ϵ,0,order}]]]];
(*perutrbation of four-velocity*)
Do[qud[i,j]=Simplify[rule1[Normal[Series[Sum[g[i,k]*q[k,j],{k,1,dim}],{ϵ,0,order}]]]];,{i,1,dim},{j,1,dim}];
Do[δu[i]=Simplify[rule1[Normal[Series[Sum[qud[i,j]*Lie[j],{j,1,dim}],{ϵ,0,order}]]+Normal[Series[Sum[(1/2)*u[i]*u[j]*u[k]*δg[j,k],{j,1,dim},{k,1,dim}],{ϵ,0,order}]]]];,{i,1,dim}];
Do[δud[i]=Simplify[Normal[Series[Sum[g[i,j]*δu[j],{j,1,dim}],{ϵ,0,order}]]];,{i,1,dim}];
Do[δ1[i,j]=δ[[i]][[j]];,{i,1,dim},{j,1,dim}];
Do[δsud[i,j]=Simplify[rule1[Normal[Series[Sum[(1/2)(qud[i,k]*q[l,j]-(1/3)qud[i,j]*q[k,l])*Δg[k,l],{k,1,dim},{l,1,dim}],{ϵ,0,order}]]]];,{i,1,dim},{j,1,dim}]
Do[δπud[i,j]=-2*μ[r]*δsud[i,j];,{i,1,dim},{j,1,dim}]

T1=Simplify[(r^2)*δπud[2,2]/P2[θ]/(E^(I t ω)) / ϵ]

T2 = (r^3)* Simplify[δπud[2,3]/(D[P2[θ],θ]*Exp[I*ω*t]*ϵ)]

Simplify[δsud[3,3]+T1*(P2[θ](E^(I t ω))  ϵ)/(-2*μ[r])/2/(r^2)]

Simplify[δsud[4,4]+T1*(P2[θ](E^(I t ω))  ϵ)/(-2*μ[r])/2/(r^2)]

sud[1,1]=0;
sud[1,2]=0;
sud[1,3]=0;
sud[1,4]=0;
sud[2,1]=0;
sud[3,1]=0;
sud[4,1]=0;
sud[2,2]=t1[r]*((c0 (-2+3 X[θ]))/(4 (r^2) μ[r]));
sud[2,3] = t2[r]*((3 c0  X'[θ])/(4 (r^3) μ[r]));
sud[2,4]=0;
sud[3,2]=(E^(-λ[r])) (r^2) *sud[2,3];
sud[3,3] =- t1[r]*(P2[θ])/(-2*μ[r])/2/(r^2)+((3 c0 V[r] X[θ])/(2 (r^2)));
sud[3,4]=0;
sud[4,2]=0;
sud[4,3]=0;
sud[4,4] = - t1[r]*(P2[θ])/(-2*μ[r])/2/(r^2)-((3 c0 V[r] X[θ])/(2 (r^2)));

s2=FullSimplify[rule1[Sum[sud[i,j]*sud[j,i],{j,1,dim},{i,1,dim}]]/.{c0->(Sqrt[(5/(4π))]),X[θ]->((Sin[θ])^2)}]*(3/2)

ClearAll["`*"]
$Assumptions={r≥0,θ≥0,θ≤π,r>2 M[r]&&r>0,ν[r]∈Reals};
x={t,r,θ,ϕ};
δ={{1,0,0,0},{0,1,0,0},{0,0,1,0},{0,0,0,1}};
dim=4;
order =1;
P2[θ_]:=3X[θ]*c0;
(*c0=(1/4) *(Sqrt[(5/(6 π))])*)
Y2[ϕ_]:=Exp[2I*ϕ];
g0={{-Exp[ν[r]],0,0,0},{0,Exp[λ[r]],0,0},{0,0,(r^2),0},{0,0,0,(r^2) X[θ]}};
rule1[Z_]:=Z/.Union[{((X'[θ])^2)->4 X[θ](1-X[θ]),X''[θ]->2 (1-2X[θ])}]
ξ0= (r^2){0,W[r](r^(-1))*Exp[-λ[r]/2]*P2[θ]*Y2[ϕ] *ϵ,-V[r](r^(-2))D[P2[θ],θ]*Y2[ϕ]*ϵ,-V[r](r^(-2))/X[θ]*P2[θ] *D[Y2[ϕ],ϕ]*ϵ}Exp[I*ω*t];
h=-(r^2){{H0[r]*Exp[ν[r]],I*ω*r*H1[r],0,0},{I*ω*r*H1[r],H2[r]*Exp[λ[r]],0,0},{0,0,(r^2)*K[r],0},{0,0,0,(r^2)*X[θ]*K[r]}}*P2[θ]Exp[I*ω*t]*Y2[ϕ]*ϵ;
gdd = g0 +h;
guu = Simplify[rule1[Normal[Series[Inverse[gdd],{ϵ,0,order}]]]];
Do[ξ[i]=ξ0[[i]],{i,1,dim}];
u0={Exp[-ν[r]/2],0,0,0};
ud0={-Exp[ν[r]/2],0,0,0};
Do[u[i]=u0[[i]],{i,1,dim}];
Do[ud[i]=ud0[[i]],{i,1,dim}];
Do[g[i,j]=Simplify[gdd[[i]][[j]]];,{i,1,dim},{j,1,dim}];
Do[ig[i,j]=Simplify[guu[[i]][[j]]];,{i,1,dim},{j,1,dim}];
Do[δg[i,j]=Simplify[h[[i]][[j]]];,{i,1,dim},{j,1,dim}];
Do[ds[i]=x[[i]];,{i,1,dim}];
Do[gud[i,j]=Simplify[Normal[Series[Sum[g[i,k]*ig[k,j],{k,1,dim}],{ϵ,0,order}]]];,{i,1,dim},{j,1,dim}];
Do[Chr[i,j,k]=Simplify[rule1[Normal[Series[(1/2)*Sum[ig[i,s]*(-D[g[j,k],ds[s]]+D[g[j,s],ds[k]]+D[g[s,k],ds[j]]),{s,1,dim}],{ϵ,0,order}]]]];,{i,1,dim},{j,1,dim},{k,1,dim}];
Do[Ruddd[i,k,l,m]= Simplify[rule1[Normal[Series[D[Chr[i, k, m], ds[l]]
       - D[Chr[i, k, l], ds[m]]
       + Sum[Chr[i, s, l]*Chr[s, k, m]-Chr[i, s, m]*Chr[s, k, l], {s, 1, dim}], {ϵ, 0, order}]]]]; , {i, 1, dim}, {k, 1, dim}, {l, 1, dim}, {m, 1, dim}];

Do[Ricci[i,k]=Simplify[rule1[Normal[Series[Sum[Ruddd[l,i,l,k],{l,1,dim}],{ϵ,0,order}]]]];,{i,1,dim},{k,1,dim}];
Do[Ruu[i,k]=Simplify[rule1[Normal[Series[Sum[Ricci[l,m]*ig[i,l]*ig[k,m],{l,1,dim},{m,1,dim}],{ϵ,0,order}]]]];,{i,1,dim},{k,1,dim}];
Rs=Simplify[rule1[Normal[Series[Sum[ig[i,j]*Ricci[i,j],{i,1,dim},{j,1,dim}],{ϵ,0,order}]]]];
Do[Gdd[i,j]=Simplify[rule1[Normal[Series[Ricci[i,j]-(1/2) g[i,j]Rs,{ϵ,0,order}]]]];,{i,1,dim},{j,1,dim}];
Do[Gud[i,k]=Simplify[rule1[Normal[Series[Sum[Gdd[i,j]*ig[j,k],{j,1,dim}],{ϵ,0,order}]]]];,{i,1,4},{k,1,4}]
Do[Lie[i]=rule1[Normal[Series[Sum[u[j]*D[ξ[i],ds[j]]-ξ[j]*D[u[i],ds[j]],{j,1,dim}],{ϵ,0,order}]]];,{i,1,dim}]

Do[q[i,j]=ig[i,j]+u[i]*u[j];,{i,1,dim},{j,1,dim}];
Do[ξd[i]=Simplify[rule1[Normal[Series[Sum[g[i,j]*ξ[j],{j,1,dim}],{ϵ,0,order}]]]];,{i,1,dim}];
(*Lagrangian perturbation of the metric*)
Do[Δg[i,j]=Simplify[rule1[Normal[Series[δg[i,j]+ D[ξd[i],ds[j]]+ D[ξd[j],ds[i]] -2Sum[Chr[k,i,j]*ξd[k],{k,1,dim}],{ϵ,0,order}]]]];,{i,1,dim},{j,1,dim}];

(*restmass fraction Lagrangian perturbation*)
Δnb=FullSimplify[rule1[Normal[Series[((-1)/2)*Sum[q[i,j]*Δg[i,j],{i,1,dim},{j,1,dim}],{ϵ,0,order}]]]];
(*perutrbation of four-velocity*)
Do[qud[i,j]=Simplify[rule1[Normal[Series[Sum[g[i,k]*q[k,j],{k,1,dim}],{ϵ,0,order}]]]];,{i,1,dim},{j,1,dim}];
Do[δu[i]=Simplify[rule1[Normal[Series[Sum[qud[i,j]*Lie[j],{j,1,dim}],{ϵ,0,order}]]+Normal[Series[Sum[(1/2)*u[i]*u[j]*u[k]*δg[j,k],{j,1,dim},{k,1,dim}],{ϵ,0,order}]]]];,{i,1,dim}];
Do[δud[i]=Simplify[Normal[Series[Sum[g[i,j]*δu[j],{j,1,dim}],{ϵ,0,order}]]];,{i,1,dim}];
Do[δ1[i,j]=δ[[i]][[j]];,{i,1,dim},{j,1,dim}];
Do[δsud[i,j]=Simplify[rule1[Normal[Series[Sum[(1/2)(qud[i,k]*q[l,j]-(1/3)qud[i,j]*q[k,l])*Δg[k,l],{k,1,dim},{l,1,dim}],{ϵ,0,order}]]]];,{i,1,dim},{j,1,dim}]
Do[δπud[i,j]=-2*μ[r]*δsud[i,j];,{i,1,dim},{j,1,dim}]

T1 =δπud[2,2]/((E^(2 I ϕ+I*t*ω))*P2[θ]ϵ)

T2 =Simplify[ δπud[2,3]*r/D[P2[θ],θ]/(E^(2 I ϕ+I*t*ω))/ϵ]

Simplify[δsud[2,2]/T1]

Simplify[δsud[2,3]/T2]

Simplify[δsud[2,4]/T2]

Simplify[δsud[3,2]/δsud[2,3]]

Simplify[δsud[3,3]+T1/(-2*μ[r])/2*(E^(2 I ϕ+I*t*ω))*P2[θ]ϵ]

δsud[3,4]

δsud[4,3]

Simplify[δsud[4,2]/δsud[2,4]]

δsud[4,3]

Simplify[δsud[4,4]+T1/(-2*μ[r])/2*(E^(2 I ϕ+I*t*ω))*P2[θ]ϵ]

sud[1,1]=0;
sud[1,2]=0;
sud[1,3]=0;
sud[1,4]=0;
sud[2,1]=0;
sud[3,1]=0;
sud[4,1]=0;
sud[2,2]=t1[r]*(((3 c0 (E^(I t ω)) ϵ X[θ] (-1+2 Y[ϕ]))/(2 μ[r])));
sud[2,3] = t2[r]*(((3 c0 (E^(I t ω)) ϵ (-1+2 Y[ϕ]) X'[θ])/(2 r μ[r])));
sud[2,4]=t2[r]*(((3 c0 (E^(I t ω)) ϵ Y'[ϕ])/(r μ[r])));
sud[3,2]=(E^(-λ[r])) (r^2) *sud[2,3];
sud[3,3] =- ((t1[r])/((-2*μ[r])2)) * (3*X[θ]*c0*(1-2Y[ϕ])*Exp[I*ω*t]*ϵ)-3 c0 (E^(I t ω)) ϵ V[r] (-2+X[θ]) (-1+2 Y[ϕ]);
sud[3,4]=((3 c0 (E^(I t ω)) ϵ V[r] X'[θ] Y'[ϕ])/(X[θ]));
sud[4,2]=(E^(-λ[r])) (r^2) X[θ]*sud[2,4];
sud[4,3]=3 c0 (E^(I t ω)) ϵ V[r] X'[θ] Y'[ϕ];
sud[4,4] = - ((t1[r])/((-2*μ[r])2))  * (3*X[θ]*c0*(1-2Y[ϕ])*Exp[I*ω*t]*ϵ)+3 c0 (E^(I t ω)) ϵ V[r] (-2+X[θ]) (-1+2 Y[ϕ]);

Simplify[sud[2,2]+sud[3,3]+sud[4,4]]

rule1[Z_]:=Z/.Union[{((X'[θ])^2)->4 X[θ](1-X[θ]),X''[θ]->2 (1-2X[θ]),((Y'[ϕ])^2)->4 Y[ϕ](1-Y[ϕ]),Y''[ϕ]->2 (1-2Y[ϕ])}]
ss=FullSimplify[rule1[Sum[sud[i,j]*sud[j,i],{j,1,4},{i,1,4}]]/.{c0->(1/4) (Sqrt[(5/(6 π))]),X[θ]->((Sin[θ])^2),ω->0,Y[ϕ]->((Sin[ϕ])^2),ϵ->1}]

Integrate[Integrate[ss,{ϕ,0,2π}],{θ,0,π}]

int1 =Integrate[Integrate[ss,{ϕ,0,2π}],{θ,-π/2,π/2}]

sud[1,1]=0;
sud[1,2]=0;
sud[1,3]=0;
sud[1,4]=0;
sud[2,1]=0;
sud[3,1]=0;
sud[4,1]=0;
sud[2,2]=t1[r]*(-((3 c0 (E^(2 I ϕ+I t ω)) ϵ X[θ])/(2 μ[r])));
sud[2,3] = t2[r]*(-((3 c0 (E^(I (2 ϕ+t ω))) ϵ X'[θ])/(2 r μ[r])));
sud[2,4]=t2[r]*(-((3 I c0 (E^(I (2 ϕ+t ω))) ϵ)/(r μ[r])));
sud[3,2]=(E^(-λ[r])) (r^2) *sud[2,3];
sud[3,3] =- t1[r]/(-2*μ[r])/2*(E^(2 I ϕ+I*t*ω))*P2[θ]ϵ+3 c0 (E^(I (2 ϕ+t ω))) ϵ V[r] (-2+X[θ]);
sud[3,4]=-((3 I c0 (E^(I (2 ϕ+t ω))) ϵ V[r] X'[θ])/(X[θ]));
sud[4,2]=(E^(-λ[r])) (r^2) X[θ]*sud[2,4];
sud[4,3]=-3 I c0 (E^(I (2 ϕ+t ω))) ϵ V[r] X'[θ];
sud[4,4] = - t1[r]/(-2*μ[r])/2*(E^(2 I ϕ+I*t*ω))*P2[θ]ϵ-3 c0 (E^(I (2 ϕ+t ω))) ϵ V[r] (-2+X[θ]);

dim=4;
rule1[Z_]:=Z/.Union[{((X'[θ])^2)->4 X[θ](1-X[θ]),X''[θ]->2 (1-2X[θ])}];
$Assumptions={r≥0,θ≥0,θ≤π,r>2 M[r]&&r>0,ν[r]∈Reals,t1[r]∈Reals,t2[r]∈Reals,μ[r]∈Reals,V[r]∈Reals,ε∈Reals,ω∈Reals,c0∈Reals,λ[r]∈Reals,ϵ∈Reals,t∈Reals,ϕ∈Reals,X[θ]∈Reals};
s2 =rule1[FullSimplify[rule1[Sum[Conjugate[3/2 * sud[i,j]]*sud[j,i],{j,1,dim},{i,1,dim}]/. {c0->1/4 Sqrt[5/(6 π)],X[θ]->Sin[θ]^2}]]]/.X[θ]->Sin[θ]^2

(*s2=FullSimplify[(3/2)*rule1[Sum[Conjugate[sud[i,j]]*sud[j,i],{j,1,dim},{i,1,dim}]]/. {c0->1/4 Sqrt[5/(6 π)],X[θ]->Sin[θ]^2}]*)
Integrate[s2,{θ,0,π}]

Simplify[4 (4 ((Sin[θ])^2)+4 ((Sin[θ])^2) (1-((Sin[θ])^2)))]

int1

FortranForm[((15 (E^(-λ))  (80 (T2^2)+(E^λ) (9 (T1^2)+560 (V^2) (μ^2))))/(2048 (μ^2)))]