In[]:=
ClearAll[B,a,d,r,n,M,P,A,B,Y,f,β,Σ,Δ,coord,metric,inve,affine,t,r,θ,ϕ]
In[]:=
g00=-(1-2*M/r);​​grr=1/(1-2*M/r);​​gθθ=
2
r
;​​gϕϕ=
2
r
*
2
Sin[θ]
;​​g0ϕ=0;
In[]:=
n:=4​​coord:={t,r,θ,ϕ}​​η:={{-1,0,0},{0,1,0},{0,0,1},{0,0,1}}​​e:={{g00,0,0,g0ϕ},{0,grr,0,0},{0,0,gθθ,0},{g0ϕ,0,0,gϕϕ}}​​inve:=Simplify[Inverse[e]]​​affine:=affine=Simplify[Table[(1/2)Sum[inve[[μ,ρ]](D[e[[ρ,ν]],coord[[λ]]]+D[e[[ρ,λ]],coord[[ν]]]-D[e[[ν,λ]],coord[[μ]]]),{ρ,1,n}],{ν,1,n},{λ,1,n},{μ,1,n}]]​​listaffine:=Table[If[UnsameQ[affine[[ν,λ,μ]],0],{Style[Subsuperscript[Γ,Row[{coord[[ν]],coord[[λ]]}],coord[[μ]]],18],"=",Style[affine[[ν,λ,μ]],14]}],{λ,1,n},{ν,1,λ},{μ,1,n}]
In[]:=
TableForm[Partition[DeleteCases[Flatten[listaffine],Null],3],TableSpacing{1,2}]
Out[]//TableForm=
r
Γ
tt
=
M(-2M+r)
3
r
t
Γ
tr
=
-
M
2Mr-
2
r
r
Γ
rr
=
M
2Mr-
2
r
θ
Γ
rθ
=
1
r
r
Γ
θθ
=
2M-r
ϕ
Γ
rϕ
=
1
r
ϕ
Γ
θϕ
=
Cot[θ]
r
Γ
ϕϕ
=
(2M-r)
2
Sin[θ]
θ
Γ
ϕϕ
=
-Cos[θ]Sin[θ]
In[]:=
"--- Computing ω ---"​​omega:=omega=Simplify[Table[Sum[(e[[μ,b]]*(D[inve[[μ,c]],coord[[a]]]+affine[[ν,λ,μ]]*inve[[ν,c]]*inve[[λ,a]])),{λ,1,n},{ν,1,n},{μ,1,n}],{a,1,n},{b,1,n},{c,1,n}]]​​​​listomega:=Table[If[UnsameQ[omega[[b,c,a]],0],{ToString[ω[b,c,a]],omega[[b,c,a]]}],{a,1,n},{b,1,n},{c,1,n}]​​TableForm[Partition[DeleteCases[Flatten[listomega],Null],2],TableSpacing{2,2}]
Out[]=
--- Computing ω ---
Out[]//TableForm=
ω[1, 2, 1]
M
2
(-2M+r)
ω[2, 1, 1]
M(2M+31r)
(2M-r)
2
r
ω[1, 1, 2]
M
2
r
ω[2, 2, 2]
-
M(2M+31r)
(2M-r)
2
r
ω[3, 3, 2]
-2M+r
2
r
ω[4, 4, 2]
-2M+r
2
r
ω[2, 3, 3]
-
2M+31r
2
r
ω[3, 2, 3]
-
1
3
r
ω[4, 4, 3]
Cot[θ]
2
r
ω[2, 4, 4]
-
2M+31r
2
r
ω[3, 4, 4]
-32+
1
2
r
Cot[θ]
ω[4, 2, 4]
-
2
Csc[θ]
3
r
ω[4, 3, 4]
-
Cot[θ]
2
Csc[θ]
2
r

Expected outcome/Desired output

ω[2,1,1]=
M
2
r

1-
2M
r
;​​ω[1,2,1]=-
M
2
r

1-
2M
r
;​​ω[3,2,3]=
1
r
1-
2M
r
;​​ω[2,3,3]=-
1
r
1-
2M
r
;​​ω[4,2,4]=
1
r
1-
2M
r
;​​ω[2,4,4]=-
1
r
1-
2M
r
;​​ω[4,3,4]=
Cot[θ]
r
;​​ω[3,4,4]=-
Cot[θ]
r
;​​