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(* | Mathematica Syntax | GEODESIC SOLVER | geodesics.yukterez.net | Version 13.09.2018 | *)
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ClearAll["Global`*"]

                                               (* Input: kovariante metrische Komponenten *)
gtt=1-(2r-℧^2)/Σ;
grr=-Σ/Δ;
gθθ=-Σ;
gφφ=-Χ/Σ Sin[θ]^2;
gtφ=a (2r-℧^2) Sin[θ]^2/Σ;

                                                                           (* Abkürzungen *)
Σ=r^2+a^2 Cos[θ]^2;
Δ=r^2-2 r+a^2+℧^2;
Χ=(r^2+a^2)^2-a^2 Sin[θ]^2 Δ;
щ=(q ℧ r (a^2+r^2))/(Δ Σ);

              (* Koordinaten, Dimensionen, magnetisches Monopol, elektrische Ladung, Spin *)
x={t, r, θ, φ}; n=4; P=0; Ω=℧; ℧=℧; a=a;

                                                                         "Metrischer Tensor"
metrik={{gtt, 0, 0, gtφ}, {0, grr, 0, 0}, {0, 0, gθθ, 0}, {gtφ, 0, 0, gφφ}};
Subscript["g", μσ] -> MatrixForm[metrik]
inversemetrik=Simplify[Inverse[metrik]];
"g"^μσ -> MatrixForm[inversemetrik]

                                                                            "Maxwell Tensor"
A={(Ω r)/Σ+P/Σ Cos[θ] a, 0, 0, -(Ω r/Σ) Sin[θ]^2 a-P/Σ Cos[θ](r^2+a^2)};
F=Simplify[Table[((D[A[[j]], x[[k]]]-D[A[[k]], x[[j]]])), {j, 1, 4}, {k, 1, 4}], Reals];
Subscript["F", μσ] -> MatrixForm[F]
f=FullSimplify[Table[Sum[
inversemetrik[[i, k]] inversemetrik[[j, l]] F[[k, l]], 
{k, 1, 4}, {l, 1, 4}], {i, 1, 4}, {j, 1, 4}], Reals];
"F"^μσ -> MatrixForm[f]

                                                                        "Christoffelsymbole"
christoffel=Simplify[Table[(1/2)Sum[(inversemetrik[[i, s]])
(D[metrik[[s, j]], x[[k]]]+D[metrik[[s, k]], x[[j]]] -D[metrik[[j, k]], x[[s]]]), {s, 1, n}], 
{i, 1, n}, {j, 1, n}, {k, 1, n}]];
Christoffel=Table[If[UnsameQ[christoffel[[i, j, k]], 0], 
{ToString[Γ[i, j, k]], christoffel[[i, j, k]]}], {i, 1, n}, {j, 1, n}, {k, 1, j}];
TableForm[Partition[DeleteCases[Flatten[Christoffel], Null], 2]]
                                                
                                                                            "Riemann Tensor"
riemann=FullSimplify[Table[
D[christoffel[[i, j, l]], x[[k]]] - D[christoffel[[i, j, k]], x[[l]]] + 
Sum[christoffel[[s, j, l]] christoffel[[i, k, s]] - 
christoffel[[s, j, k]] christoffel[[i, l, s]],
{s, 1, n}], {i, 1, n}, {j, 1, n}, {k, 1, n}, {l, 1, n}]];
Riemann=Table[If[UnsameQ[riemann[[i, j, k, l]], 0],
{ToString[R[i, j, k, l]], riemann[[i, j, k, l]]}],
{i, 1, n}, {j, 1, n}, {k, 1, n}, {l, 1, k - 1}];
TableForm[Partition[DeleteCases[Flatten[Riemann], Null], 2]]
 
                                                                              "Ricci Tensor"
ricci=FullSimplify[Table[
Sum[riemann[[i, j, i, l]], {i, 1, n}], {j, 1, n}, {l, 1, n}]];
Subscript["Ř", μσ] -> MatrixForm[ricci]
Ricci=FullSimplify[Table[Sum[
inversemetrik[[i, k]] inversemetrik[[j, l]] ricci[[k, l]], {k, 1, 4}, {l, 1, 4}], 
{i, 1, 4}, {j, 1, 4}], Reals];
"Ř"^μσ -> MatrixForm[Ricci]

                                                                              "Ricci Skalar"
Ř=Simplify[Sum[inversemetrik[[i, j]] ricci[[i, j]], {i, 1, n}, {j, 1, n}]]; "Ř"->Ř


rplc[y_]:=(((((((y/.t->t[τ])/.r->r[τ])/.θ->θ[τ])/.φ->φ[τ])/.Derivative[1][t[τ]]->
t'[τ])/.Derivative[1][r[τ]]->r'[τ])/.Derivative[1][θ[τ]]->θ'[τ])/.Derivative[1][φ[τ]]->φ'[τ];
rple[y_]:=(((((((y/.t->t[τ])/.r->r[τ])/.θ->θ[τ])/.φ->φ[τ])/.Derivative[1][t[τ]]-> 
t'[τ])/. r\.b4->r'[τ])/. θ\.b4->θ'[τ])/. φ\.b4->φ'[τ];
list[y_]:=y[[1]]==y[[2]];

                                                                      "Bewegungsgleichungen"
geodäsie=Simplify[Table[-Sum[
christoffel[[i, j, k]] x[[j]]' x[[k]]'+q f[[i, k]] x[[j]]' metrik[[j, k]], 
{j, 1, n}, {k, 1, n}], {i, 1, n}]];

bewegungsgleichung=Table[{x[[i]]''[τ]==FullSimplify[rplc[geodäsie[[i]]], Reals]}, {i, 1, n}];

geodesic1=bewegungsgleichung[[1]][[1]]
geodesic2=bewegungsgleichung[[2]][[1]]
geodesic3=bewegungsgleichung[[3]][[1]]
geodesic4=bewegungsgleichung[[4]][[1]]

                                                                            "Zeitdilatation"
ṫ=rple[Simplify[Normal[Solve[
-μ==gtt ť^2+grr r\.b4^2+gθθ θ\.b4^2+gφφ φ\.b4^2 + 2 gtφ ť φ\.b4, ť, Reals]]][[2, 1, 2]]-щ];
Derivative[1][t][τ]==ṫ