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WeylTensor.wl
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(* ::Package:: *)
WeylTensor[(metricTensor_)[matrixRepresentation_List, coordinates_List, index1_, index2_]] :=
WeylTensor[ResourceFunction["MetricTensor"][matrixRepresentation, coordinates, index1, index2], True, True, True,
True] /; SymbolName[metricTensor] === "MetricTensor" && Length[Dimensions[matrixRepresentation]] == 2 &&
Length[coordinates] == Length[matrixRepresentation] && BooleanQ[index1] && BooleanQ[index2]
WeylTensor[(metricTensor_)[matrixRepresentation_List, coordinates_List, index1_, index2_], newCoordinates_List] :=
WeylTensor[ResourceFunction["MetricTensor"][matrixRepresentation /. Thread[coordinates -> newCoordinates],
newCoordinates, index1, index2], True, True, True, True] /; SymbolName[metricTensor] === "MetricTensor" &&
Length[Dimensions[matrixRepresentation]] == 2 && Length[coordinates] == Length[matrixRepresentation] &&
Length[newCoordinates] == Length[matrixRepresentation] && BooleanQ[index1] && BooleanQ[index2]
WeylTensor[(metricTensor_)[matrixRepresentation_List, coordinates_List, metricIndex1_, metricIndex2_],
newCoordinates_List, index1_, index2_, index3_, index4_] :=
WeylTensor[ResourceFunction["MetricTensor"][matrixRepresentation /. Thread[coordinates -> newCoordinates],
newCoordinates, metricIndex1, metricIndex2], index1, index2, index3, index4] /;
SymbolName[metricTensor] === "MetricTensor" && Length[Dimensions[matrixRepresentation]] == 2 &&
Length[coordinates] == Length[matrixRepresentation] && Length[newCoordinates] == Length[matrixRepresentation] &&
BooleanQ[metricIndex1] && BooleanQ[metricIndex2] && BooleanQ[index1] && BooleanQ[index2] && BooleanQ[index3] &&
BooleanQ[index4]
WeylTensor[(metricTensor_)[matrixRepresentation_List, coordinates_List, metricIndex1_, metricIndex2_], index1_, index2_,
index3_, index4_]["TensorRepresentation"] :=
Module[{newMatrixRepresentation, newCoordinates, christoffelSymbols, riemannTensor, covariantRiemannTensor,
ricciTensor, ricciScalar, weylTensor}, newMatrixRepresentation = matrixRepresentation /.
(#1 -> ToExpression[#1] & ) /@ Select[coordinates, StringQ];
newCoordinates = coordinates /. (#1 -> ToExpression[#1] & ) /@ Select[coordinates, StringQ];
christoffelSymbols = Normal[SparseArray[
(Module[{index = #1}, index -> Total[((1/2)*Inverse[newMatrixRepresentation][[index[[1]],#1]]*
(D[newMatrixRepresentation[[#1,index[[3]]]], newCoordinates[[index[[2]]]]] + D[newMatrixRepresentation[[
index[[2]],#1]], newCoordinates[[index[[3]]]]] - D[newMatrixRepresentation[[index[[2]],index[[3]]]],
newCoordinates[[#1]]]) & ) /@ Range[Length[newMatrixRepresentation]]]] & ) /@
Tuples[Range[Length[newMatrixRepresentation]], 3]]];
riemannTensor = Normal[SparseArray[(Module[{index = #1}, index -> D[christoffelSymbols[[index[[1]],index[[2]],
index[[4]]]], newCoordinates[[index[[3]]]]] - D[christoffelSymbols[[index[[1]],index[[2]],index[[3]]]],
newCoordinates[[index[[4]]]]] + Total[(christoffelSymbols[[index[[1]],#1,index[[3]]]]*christoffelSymbols[[
#1,index[[2]],index[[4]]]] & ) /@ Range[Length[newMatrixRepresentation]]] -
Total[(christoffelSymbols[[index[[1]],#1,index[[4]]]]*christoffelSymbols[[#1,index[[2]],index[[3]]]] & ) /@
Range[Length[newMatrixRepresentation]]]] & ) /@ Tuples[Range[Length[newMatrixRepresentation]], 4]]] /.
(ToExpression[#1] -> #1 & ) /@ Select[coordinates, StringQ]; covariantRiemannTensor =
Normal[SparseArray[(Module[{index = #1}, index -> Total[(matrixRepresentation[[index[[1]],#1]]*riemannTensor[[#1,
index[[2]],index[[3]],index[[4]]]] & ) /@ Range[Length[matrixRepresentation]]]] & ) /@
Tuples[Range[Length[matrixRepresentation]], 4]]];
ricciTensor = Normal[SparseArray[(Module[{index = #1}, index -> Total[(riemannTensor[[#1,First[index],#1,
Last[index]]] & ) /@ Range[Length[matrixRepresentation]]]] & ) /@
Tuples[Range[Length[matrixRepresentation]], 2]]];
ricciScalar = Total[(Inverse[matrixRepresentation][[First[#1],Last[#1]]]*ricciTensor[[First[#1],Last[#1]]] & ) /@
Tuples[Range[Length[matrixRepresentation]], 2]];
weylTensor = Normal[SparseArray[(Module[{index = #1}, index -> covariantRiemannTensor[[index[[1]],index[[2]],
index[[3]],index[[4]]]] + (1/(Length[matrixRepresentation] - 2))*(ricciTensor[[index[[1]],index[[4]]]]*
matrixRepresentation[[index[[2]],index[[3]]]] - ricciTensor[[index[[1]],index[[3]]]]*
matrixRepresentation[[index[[2]],index[[4]]]] + ricciTensor[[index[[2]],index[[3]]]]*
matrixRepresentation[[index[[1]],index[[4]]]] - ricciTensor[[index[[2]],index[[4]]]]*
matrixRepresentation[[index[[1]],index[[3]]]]) + (1/((Length[matrixRepresentation] - 1)*
(Length[matrixRepresentation] - 2)))*(ricciScalar*(matrixRepresentation[[index[[1]],index[[3]]]]*
matrixRepresentation[[index[[2]],index[[4]]]] - matrixRepresentation[[index[[1]],index[[4]]]]*
matrixRepresentation[[index[[2]],index[[3]]]]))] & ) /@ Tuples[Range[Length[matrixRepresentation]],
4]]]; If[index1 === True && index2 === True && index3 === True && index4 === True, weylTensor,
If[index1 === False && index2 === False && index3 === False && index4 === False,
Normal[SparseArray[(Module[{index = #1}, index -> Total[(Inverse[matrixRepresentation][[index[[1]],#1[[1]]]]*
Inverse[matrixRepresentation][[index[[2]],#1[[2]]]]*Inverse[matrixRepresentation][[#1[[3]],index[[3]]]]*
Inverse[matrixRepresentation][[#1[[4]],index[[4]]]]*weylTensor[[#1[[1]],#1[[2]],#1[[3]],#1[[4]]]] & ) /@
Tuples[Range[Length[matrixRepresentation]], 4]]] & ) /@ Tuples[Range[Length[matrixRepresentation]], 4]]],
If[index1 === True && index2 === False && index3 === False && index4 === False,
Normal[SparseArray[(Module[{index = #1}, index -> Total[(Inverse[matrixRepresentation][[index[[2]],#1[[1]]]]*
Inverse[matrixRepresentation][[#1[[2]],index[[3]]]]*Inverse[matrixRepresentation][[#1[[3]],index[[4]]]]*
weylTensor[[index[[1]],#1[[1]],#1[[2]],#1[[3]]]] & ) /@ Tuples[Range[Length[matrixRepresentation]],
3]]] & ) /@ Tuples[Range[Length[matrixRepresentation]], 4]]],
If[index1 === False && index2 === True && index3 === False && index4 === False,
Normal[SparseArray[(Module[{index = #1}, index -> Total[(Inverse[matrixRepresentation][[index[[1]],#1[[1]]]]*
Inverse[matrixRepresentation][[#1[[2]],index[[3]]]]*Inverse[matrixRepresentation][[#1[[3]],
index[[4]]]]*weylTensor[[#1[[1]],index[[2]],#1[[2]],#1[[3]]]] & ) /@ Tuples[
Range[Length[matrixRepresentation]], 3]]] & ) /@ Tuples[Range[Length[matrixRepresentation]], 4]]],
If[index1 === False && index2 === False && index3 === True && index4 === False,
Normal[SparseArray[(Module[{index = #1}, index -> Total[(Inverse[matrixRepresentation][[index[[1]],#1[[1]]]]*
Inverse[matrixRepresentation][[index[[2]],#1[[2]]]]*Inverse[matrixRepresentation][[#1[[3]],
index[[4]]]]*weylTensor[[#1[[1]],#1[[2]],index[[3]],#1[[3]]]] & ) /@ Tuples[
Range[Length[matrixRepresentation]], 3]]] & ) /@ Tuples[Range[Length[matrixRepresentation]], 4]]],
If[index1 === False && index2 === False && index3 === False && index4 === True,
Normal[SparseArray[(Module[{index = #1}, index -> Total[(Inverse[matrixRepresentation][[index[[1]],#1[[1]]]]*
Inverse[matrixRepresentation][[index[[2]],#1[[2]]]]*Inverse[matrixRepresentation][[#1[[3]],
index[[3]]]]*weylTensor[[#1[[1]],#1[[2]],#1[[3]],index[[4]]]] & ) /@ Tuples[
Range[Length[matrixRepresentation]], 3]]] & ) /@ Tuples[Range[Length[matrixRepresentation]], 4]]],
If[index1 === True && index2 === True && index3 === False && index4 === False,
Normal[SparseArray[(Module[{index = #1}, index -> Total[(Inverse[matrixRepresentation][[#1[[1]],index[[3]]]]*
Inverse[matrixRepresentation][[#1[[2]],index[[4]]]]*weylTensor[[index[[1]],index[[2]],#1[[1]],
#1[[2]]]] & ) /@ Tuples[Range[Length[matrixRepresentation]], 2]]] & ) /@ Tuples[
Range[Length[matrixRepresentation]], 4]]], If[index1 === True && index2 === False && index3 === True &&
index4 === False, Normal[SparseArray[(Module[{index = #1}, index -> Total[(Inverse[matrixRepresentation][[
index[[2]],#1[[1]]]]*Inverse[matrixRepresentation][[#1[[2]],index[[4]]]]*weylTensor[[index[[1]],
#1[[1]],index[[3]],#1[[2]]]] & ) /@ Tuples[Range[Length[matrixRepresentation]], 2]]] & ) /@
Tuples[Range[Length[matrixRepresentation]], 4]]], If[index1 === True && index2 === False && index3 ===
False && index4 === True, Normal[SparseArray[(Module[{index = #1}, index -> Total[
(Inverse[matrixRepresentation][[index[[2]],#1[[1]]]]*Inverse[matrixRepresentation][[#1[[2]],
index[[3]]]]*weylTensor[[index[[1]],#1[[1]],#1[[2]],index[[4]]]] & ) /@ Tuples[
Range[Length[matrixRepresentation]], 2]]] & ) /@ Tuples[Range[Length[matrixRepresentation]], 4]]],
If[index1 === False && index2 === True && index3 === True && index4 === False, Normal[
SparseArray[(Module[{index = #1}, index -> Total[(Inverse[matrixRepresentation][[index[[1]],#1[[1]]]]*
Inverse[matrixRepresentation][[#1[[2]],index[[4]]]]*weylTensor[[#1[[1]],index[[2]],index[[3]],
#1[[2]]]] & ) /@ Tuples[Range[Length[matrixRepresentation]], 2]]] & ) /@
Tuples[Range[Length[matrixRepresentation]], 4]]], If[index1 === False && index2 === True &&
index3 === False && index4 === True, Normal[SparseArray[(Module[{index = #1}, index ->
Total[(Inverse[matrixRepresentation][[index[[1]],#1[[1]]]]*Inverse[matrixRepresentation][[#1[[2]],
index[[3]]]]*weylTensor[[#1[[1]],index[[2]],#1[[2]],index[[4]]]] & ) /@ Tuples[
Range[Length[matrixRepresentation]], 2]]] & ) /@ Tuples[Range[Length[matrixRepresentation]],
4]]], If[index1 === False && index2 === False && index3 === True && index4 === True,
Normal[SparseArray[(Module[{index = #1}, index -> Total[(Inverse[matrixRepresentation][[index[[1]],
#1[[1]]]]*Inverse[matrixRepresentation][[index[[2]],#1[[2]]]]*weylTensor[[#1[[1]],#1[[2]],
index[[3]],index[[4]]]] & ) /@ Tuples[Range[Length[matrixRepresentation]], 2]]] & ) /@
Tuples[Range[Length[matrixRepresentation]], 4]]], If[index1 === True && index2 === True &&
index3 === True && index4 === False, Normal[SparseArray[(Module[{index = #1}, index ->
Total[(Inverse[matrixRepresentation][[#1,index[[4]]]]*weylTensor[[index[[1]],index[[2]],
index[[3]],#1]] & ) /@ Range[Length[matrixRepresentation]]]] & ) /@ Tuples[
Range[Length[matrixRepresentation]], 4]]], If[index1 === True && index2 === True &&
index3 === False && index4 === True, Normal[SparseArray[(Module[{index = #1}, index ->
Total[(Inverse[matrixRepresentation][[#1,index[[3]]]]*weylTensor[[index[[1]],index[[2]],#1,
index[[4]]]] & ) /@ Range[Length[matrixRepresentation]]]] & ) /@ Tuples[
Range[Length[matrixRepresentation]], 4]]], If[index1 === True && index2 === False &&
index3 === True && index4 === True, Normal[SparseArray[(Module[{index = #1}, index ->
Total[(Inverse[matrixRepresentation][[index[[2]],#1]]*weylTensor[[index[[1]],#1,index[[3]],
index[[4]]]] & ) /@ Range[Length[matrixRepresentation]]]] & ) /@ Tuples[Range[
Length[matrixRepresentation]], 4]]], If[index1 === False && index2 === True && index3 ===
True && index4 === True, Normal[SparseArray[(Module[{index = #1}, index -> Total[
(Inverse[matrixRepresentation][[index[[1]],#1]]*weylTensor[[#1,index[[2]],index[[3]],
index[[4]]]] & ) /@ Range[Length[matrixRepresentation]]]] & ) /@ Tuples[Range[
Length[matrixRepresentation]], 4]]], Indeterminate]]]]]]]]]]]]]]]]] /;
SymbolName[metricTensor] === "MetricTensor" && Length[Dimensions[matrixRepresentation]] == 2 &&
Length[coordinates] == Length[matrixRepresentation] && BooleanQ[metricIndex1] && BooleanQ[metricIndex2] &&
BooleanQ[index1] && BooleanQ[index2] && BooleanQ[index3] && BooleanQ[index4]
WeylTensor[(metricTensor_)[matrixRepresentation_List, coordinates_List, metricIndex1_, metricIndex2_], index1_, index2_,
index3_, index4_]["ReducedTensorRepresentation"] :=
Module[{newMatrixRepresentation, newCoordinates, christoffelSymbols, riemannTensor, covariantRiemannTensor,
ricciTensor, ricciScalar, weylTensor}, newMatrixRepresentation = matrixRepresentation /.
(#1 -> ToExpression[#1] & ) /@ Select[coordinates, StringQ];
newCoordinates = coordinates /. (#1 -> ToExpression[#1] & ) /@ Select[coordinates, StringQ];
christoffelSymbols = Normal[SparseArray[
(Module[{index = #1}, index -> Total[((1/2)*Inverse[newMatrixRepresentation][[index[[1]],#1]]*
(D[newMatrixRepresentation[[#1,index[[3]]]], newCoordinates[[index[[2]]]]] + D[newMatrixRepresentation[[
index[[2]],#1]], newCoordinates[[index[[3]]]]] - D[newMatrixRepresentation[[index[[2]],index[[3]]]],
newCoordinates[[#1]]]) & ) /@ Range[Length[newMatrixRepresentation]]]] & ) /@
Tuples[Range[Length[newMatrixRepresentation]], 3]]];
riemannTensor = Normal[SparseArray[(Module[{index = #1}, index -> D[christoffelSymbols[[index[[1]],index[[2]],
index[[4]]]], newCoordinates[[index[[3]]]]] - D[christoffelSymbols[[index[[1]],index[[2]],index[[3]]]],
newCoordinates[[index[[4]]]]] + Total[(christoffelSymbols[[index[[1]],#1,index[[3]]]]*christoffelSymbols[[
#1,index[[2]],index[[4]]]] & ) /@ Range[Length[newMatrixRepresentation]]] -
Total[(christoffelSymbols[[index[[1]],#1,index[[4]]]]*christoffelSymbols[[#1,index[[2]],index[[3]]]] & ) /@
Range[Length[newMatrixRepresentation]]]] & ) /@ Tuples[Range[Length[newMatrixRepresentation]], 4]]] /.
(ToExpression[#1] -> #1 & ) /@ Select[coordinates, StringQ]; covariantRiemannTensor =
Normal[SparseArray[(Module[{index = #1}, index -> Total[(matrixRepresentation[[index[[1]],#1]]*riemannTensor[[#1,
index[[2]],index[[3]],index[[4]]]] & ) /@ Range[Length[matrixRepresentation]]]] & ) /@
Tuples[Range[Length[matrixRepresentation]], 4]]];
ricciTensor = Normal[SparseArray[(Module[{index = #1}, index -> Total[(riemannTensor[[#1,First[index],#1,
Last[index]]] & ) /@ Range[Length[matrixRepresentation]]]] & ) /@
Tuples[Range[Length[matrixRepresentation]], 2]]];
ricciScalar = Total[(Inverse[matrixRepresentation][[First[#1],Last[#1]]]*ricciTensor[[First[#1],Last[#1]]] & ) /@
Tuples[Range[Length[matrixRepresentation]], 2]];
weylTensor = Normal[SparseArray[(Module[{index = #1}, index -> covariantRiemannTensor[[index[[1]],index[[2]],
index[[3]],index[[4]]]] + (1/(Length[matrixRepresentation] - 2))*(ricciTensor[[index[[1]],index[[4]]]]*
matrixRepresentation[[index[[2]],index[[3]]]] - ricciTensor[[index[[1]],index[[3]]]]*
matrixRepresentation[[index[[2]],index[[4]]]] + ricciTensor[[index[[2]],index[[3]]]]*
matrixRepresentation[[index[[1]],index[[4]]]] - ricciTensor[[index[[2]],index[[4]]]]*
matrixRepresentation[[index[[1]],index[[3]]]]) + (1/((Length[matrixRepresentation] - 1)*
(Length[matrixRepresentation] - 2)))*(ricciScalar*(matrixRepresentation[[index[[1]],index[[3]]]]*
matrixRepresentation[[index[[2]],index[[4]]]] - matrixRepresentation[[index[[1]],index[[4]]]]*
matrixRepresentation[[index[[2]],index[[3]]]]))] & ) /@ Tuples[Range[Length[matrixRepresentation]],
4]]]; If[index1 === True && index2 === True && index3 === True && index4 === True, FullSimplify[weylTensor],
If[index1 === False && index2 === False && index3 === False && index4 === False,
FullSimplify[Normal[SparseArray[(Module[{index = #1}, index -> Total[(Inverse[matrixRepresentation][[index[[1]],
#1[[1]]]]*Inverse[matrixRepresentation][[index[[2]],#1[[2]]]]*Inverse[matrixRepresentation][[#1[[3]],
index[[3]]]]*Inverse[matrixRepresentation][[#1[[4]],index[[4]]]]*weylTensor[[#1[[1]],#1[[2]],#1[[3]],
#1[[4]]]] & ) /@ Tuples[Range[Length[matrixRepresentation]], 4]]] & ) /@
Tuples[Range[Length[matrixRepresentation]], 4]]]], If[index1 === True && index2 === False &&
index3 === False && index4 === False, FullSimplify[
Normal[SparseArray[(Module[{index = #1}, index -> Total[(Inverse[matrixRepresentation][[index[[2]],#1[[1]]]]*
Inverse[matrixRepresentation][[#1[[2]],index[[3]]]]*Inverse[matrixRepresentation][[#1[[3]],
index[[4]]]]*weylTensor[[index[[1]],#1[[1]],#1[[2]],#1[[3]]]] & ) /@ Tuples[
Range[Length[matrixRepresentation]], 3]]] & ) /@ Tuples[Range[Length[matrixRepresentation]], 4]]]],
If[index1 === False && index2 === True && index3 === False && index4 === False,
FullSimplify[Normal[SparseArray[(Module[{index = #1}, index -> Total[(Inverse[matrixRepresentation][[index[[1]],
#1[[1]]]]*Inverse[matrixRepresentation][[#1[[2]],index[[3]]]]*Inverse[matrixRepresentation][[#1[[3]],
index[[4]]]]*weylTensor[[#1[[1]],index[[2]],#1[[2]],#1[[3]]]] & ) /@ Tuples[
Range[Length[matrixRepresentation]], 3]]] & ) /@ Tuples[Range[Length[matrixRepresentation]], 4]]]],
If[index1 === False && index2 === False && index3 === True && index4 === False,
FullSimplify[Normal[SparseArray[(Module[{index = #1}, index -> Total[(Inverse[matrixRepresentation][[index[[1]],
#1[[1]]]]*Inverse[matrixRepresentation][[index[[2]],#1[[2]]]]*Inverse[matrixRepresentation][[
#1[[3]],index[[4]]]]*weylTensor[[#1[[1]],#1[[2]],index[[3]],#1[[3]]]] & ) /@
Tuples[Range[Length[matrixRepresentation]], 3]]] & ) /@ Tuples[Range[Length[matrixRepresentation]],
4]]]], If[index1 === False && index2 === False && index3 === False && index4 === True,
FullSimplify[Normal[SparseArray[(Module[{index = #1}, index -> Total[(Inverse[matrixRepresentation][[
index[[1]],#1[[1]]]]*Inverse[matrixRepresentation][[index[[2]],#1[[2]]]]*
Inverse[matrixRepresentation][[#1[[3]],index[[3]]]]*weylTensor[[#1[[1]],#1[[2]],#1[[3]],
index[[4]]]] & ) /@ Tuples[Range[Length[matrixRepresentation]], 3]]] & ) /@ Tuples[
Range[Length[matrixRepresentation]], 4]]]], If[index1 === True && index2 === True && index3 === False &&
index4 === False, FullSimplify[Normal[SparseArray[(Module[{index = #1}, index ->
Total[(Inverse[matrixRepresentation][[#1[[1]],index[[3]]]]*Inverse[matrixRepresentation][[#1[[2]],
index[[4]]]]*weylTensor[[index[[1]],index[[2]],#1[[1]],#1[[2]]]] & ) /@ Tuples[
Range[Length[matrixRepresentation]], 2]]] & ) /@ Tuples[Range[Length[matrixRepresentation]], 4]]]],
If[index1 === True && index2 === False && index3 === True && index4 === False,
FullSimplify[Normal[SparseArray[(Module[{index = #1}, index -> Total[(Inverse[matrixRepresentation][[
index[[2]],#1[[1]]]]*Inverse[matrixRepresentation][[#1[[2]],index[[4]]]]*weylTensor[[index[[1]],
#1[[1]],index[[3]],#1[[2]]]] & ) /@ Tuples[Range[Length[matrixRepresentation]], 2]]] & ) /@
Tuples[Range[Length[matrixRepresentation]], 4]]]], If[index1 === True && index2 === False && index3 ===
False && index4 === True, FullSimplify[Normal[SparseArray[(Module[{index = #1}, index ->
Total[(Inverse[matrixRepresentation][[index[[2]],#1[[1]]]]*Inverse[matrixRepresentation][[#1[[2]],
index[[3]]]]*weylTensor[[index[[1]],#1[[1]],#1[[2]],index[[4]]]] & ) /@ Tuples[
Range[Length[matrixRepresentation]], 2]]] & ) /@ Tuples[Range[Length[matrixRepresentation]],
4]]]], If[index1 === False && index2 === True && index3 === True && index4 === False, FullSimplify[
Normal[SparseArray[(Module[{index = #1}, index -> Total[(Inverse[matrixRepresentation][[index[[1]],
#1[[1]]]]*Inverse[matrixRepresentation][[#1[[2]],index[[4]]]]*weylTensor[[#1[[1]],index[[2]],
index[[3]],#1[[2]]]] & ) /@ Tuples[Range[Length[matrixRepresentation]], 2]]] & ) /@
Tuples[Range[Length[matrixRepresentation]], 4]]]], If[index1 === False && index2 === True &&
index3 === False && index4 === True, FullSimplify[Normal[SparseArray[(Module[{index = #1},
index -> Total[(Inverse[matrixRepresentation][[index[[1]],#1[[1]]]]*Inverse[matrixRepresentation][[
#1[[2]],index[[3]]]]*weylTensor[[#1[[1]],index[[2]],#1[[2]],index[[4]]]] & ) /@
Tuples[Range[Length[matrixRepresentation]], 2]]] & ) /@ Tuples[Range[Length[
matrixRepresentation]], 4]]]], If[index1 === False && index2 === False && index3 === True &&
index4 === True, FullSimplify[Normal[SparseArray[(Module[{index = #1}, index -> Total[
(Inverse[matrixRepresentation][[index[[1]],#1[[1]]]]*Inverse[matrixRepresentation][[index[[2]],
#1[[2]]]]*weylTensor[[#1[[1]],#1[[2]],index[[3]],index[[4]]]] & ) /@ Tuples[Range[
Length[matrixRepresentation]], 2]]] & ) /@ Tuples[Range[Length[matrixRepresentation]], 4]]]],
If[index1 === True && index2 === True && index3 === True && index4 === False, FullSimplify[
Normal[SparseArray[(Module[{index = #1}, index -> Total[(Inverse[matrixRepresentation][[#1,
index[[4]]]]*weylTensor[[index[[1]],index[[2]],index[[3]],#1]] & ) /@ Range[Length[
matrixRepresentation]]]] & ) /@ Tuples[Range[Length[matrixRepresentation]], 4]]]],
If[index1 === True && index2 === True && index3 === False && index4 === True, FullSimplify[
Normal[SparseArray[(Module[{index = #1}, index -> Total[(Inverse[matrixRepresentation][[#1,
index[[3]]]]*weylTensor[[index[[1]],index[[2]],#1,index[[4]]]] & ) /@ Range[Length[
matrixRepresentation]]]] & ) /@ Tuples[Range[Length[matrixRepresentation]], 4]]]],
If[index1 === True && index2 === False && index3 === True && index4 === True, FullSimplify[
Normal[SparseArray[(Module[{index = #1}, index -> Total[(Inverse[matrixRepresentation][[index[[2]],
#1]]*weylTensor[[index[[1]],#1,index[[3]],index[[4]]]] & ) /@ Range[Length[
matrixRepresentation]]]] & ) /@ Tuples[Range[Length[matrixRepresentation]], 4]]]],
If[index1 === False && index2 === True && index3 === True && index4 === True, FullSimplify[
Normal[SparseArray[(Module[{index = #1}, index -> Total[(Inverse[matrixRepresentation][[index[[1]],
#1]]*weylTensor[[#1,index[[2]],index[[3]],index[[4]]]] & ) /@ Range[Length[
matrixRepresentation]]]] & ) /@ Tuples[Range[Length[matrixRepresentation]], 4]]]],
Indeterminate]]]]]]]]]]]]]]]]] /; SymbolName[metricTensor] === "MetricTensor" &&
Length[Dimensions[matrixRepresentation]] == 2 && Length[coordinates] == Length[matrixRepresentation] &&
BooleanQ[metricIndex1] && BooleanQ[metricIndex2] && BooleanQ[index1] && BooleanQ[index2] && BooleanQ[index3] &&
BooleanQ[index4]
WeylTensor[(metricTensor_)[matrixRepresentation_List, coordinates_List, metricIndex1_, metricIndex2_], index1_, index2_,
index3_, index4_]["SymbolicTensorRepresentation"] :=
Module[{newMatrixRepresentation, newCoordinates, christoffelSymbols, riemannTensor, covariantRiemannTensor,
ricciTensor, ricciScalar, weylTensor}, newMatrixRepresentation = matrixRepresentation /.
(#1 -> ToExpression[#1] & ) /@ Select[coordinates, StringQ];
newCoordinates = coordinates /. (#1 -> ToExpression[#1] & ) /@ Select[coordinates, StringQ];
christoffelSymbols = Normal[SparseArray[
(Module[{index = #1}, index -> Total[((1/2)*Inverse[newMatrixRepresentation][[index[[1]],#1]]*
(Inactive[D][newMatrixRepresentation[[#1,index[[3]]]], newCoordinates[[index[[2]]]]] +
Inactive[D][newMatrixRepresentation[[index[[2]],#1]], newCoordinates[[index[[3]]]]] -
Inactive[D][newMatrixRepresentation[[index[[2]],index[[3]]]], newCoordinates[[#1]]]) & ) /@
Range[Length[newMatrixRepresentation]]]] & ) /@ Tuples[Range[Length[newMatrixRepresentation]], 3]]];
riemannTensor = Normal[SparseArray[(Module[{index = #1}, index -> Inactive[D][christoffelSymbols[[index[[1]],
index[[2]],index[[4]]]], newCoordinates[[index[[3]]]]] - Inactive[D][christoffelSymbols[[index[[1]],
index[[2]],index[[3]]]], newCoordinates[[index[[4]]]]] + Total[(christoffelSymbols[[index[[1]],#1,
index[[3]]]]*christoffelSymbols[[#1,index[[2]],index[[4]]]] & ) /@
Range[Length[newMatrixRepresentation]]] - Total[(christoffelSymbols[[index[[1]],#1,index[[4]]]]*
christoffelSymbols[[#1,index[[2]],index[[3]]]] & ) /@ Range[Length[newMatrixRepresentation]]]] & ) /@
Tuples[Range[Length[newMatrixRepresentation]], 4]]] /. (ToExpression[#1] -> #1 & ) /@
Select[coordinates, StringQ]; covariantRiemannTensor =
Normal[SparseArray[(Module[{index = #1}, index -> Total[(matrixRepresentation[[index[[1]],#1]]*riemannTensor[[#1,
index[[2]],index[[3]],index[[4]]]] & ) /@ Range[Length[matrixRepresentation]]]] & ) /@
Tuples[Range[Length[matrixRepresentation]], 4]]];
ricciTensor = Normal[SparseArray[(Module[{index = #1}, index -> Total[(riemannTensor[[#1,First[index],#1,
Last[index]]] & ) /@ Range[Length[matrixRepresentation]]]] & ) /@
Tuples[Range[Length[matrixRepresentation]], 2]]];
ricciScalar = Total[(Inverse[matrixRepresentation][[First[#1],Last[#1]]]*ricciTensor[[First[#1],Last[#1]]] & ) /@
Tuples[Range[Length[matrixRepresentation]], 2]];
weylTensor = Normal[SparseArray[(Module[{index = #1}, index -> covariantRiemannTensor[[index[[1]],index[[2]],
index[[3]],index[[4]]]] + (1/(Length[matrixRepresentation] - 2))*(ricciTensor[[index[[1]],index[[4]]]]*
matrixRepresentation[[index[[2]],index[[3]]]] - ricciTensor[[index[[1]],index[[3]]]]*
matrixRepresentation[[index[[2]],index[[4]]]] + ricciTensor[[index[[2]],index[[3]]]]*
matrixRepresentation[[index[[1]],index[[4]]]] - ricciTensor[[index[[2]],index[[4]]]]*
matrixRepresentation[[index[[1]],index[[3]]]]) + (1/((Length[matrixRepresentation] - 1)*
(Length[matrixRepresentation] - 2)))*(ricciScalar*(matrixRepresentation[[index[[1]],index[[3]]]]*
matrixRepresentation[[index[[2]],index[[4]]]] - matrixRepresentation[[index[[1]],index[[4]]]]*
matrixRepresentation[[index[[2]],index[[3]]]]))] & ) /@ Tuples[Range[Length[matrixRepresentation]],
4]]]; If[index1 === True && index2 === True && index3 === True && index4 === True, weylTensor,
If[index1 === False && index2 === False && index3 === False && index4 === False,
Normal[SparseArray[(Module[{index = #1}, index -> Total[(Inverse[matrixRepresentation][[index[[1]],#1[[1]]]]*
Inverse[matrixRepresentation][[index[[2]],#1[[2]]]]*Inverse[matrixRepresentation][[#1[[3]],index[[3]]]]*
Inverse[matrixRepresentation][[#1[[4]],index[[4]]]]*weylTensor[[#1[[1]],#1[[2]],#1[[3]],#1[[4]]]] & ) /@
Tuples[Range[Length[matrixRepresentation]], 4]]] & ) /@ Tuples[Range[Length[matrixRepresentation]], 4]]],
If[index1 === True && index2 === False && index3 === False && index4 === False,
Normal[SparseArray[(Module[{index = #1}, index -> Total[(Inverse[matrixRepresentation][[index[[2]],#1[[1]]]]*
Inverse[matrixRepresentation][[#1[[2]],index[[3]]]]*Inverse[matrixRepresentation][[#1[[3]],index[[4]]]]*
weylTensor[[index[[1]],#1[[1]],#1[[2]],#1[[3]]]] & ) /@ Tuples[Range[Length[matrixRepresentation]],
3]]] & ) /@ Tuples[Range[Length[matrixRepresentation]], 4]]],
If[index1 === False && index2 === True && index3 === False && index4 === False,
Normal[SparseArray[(Module[{index = #1}, index -> Total[(Inverse[matrixRepresentation][[index[[1]],#1[[1]]]]*
Inverse[matrixRepresentation][[#1[[2]],index[[3]]]]*Inverse[matrixRepresentation][[#1[[3]],
index[[4]]]]*weylTensor[[#1[[1]],index[[2]],#1[[2]],#1[[3]]]] & ) /@ Tuples[
Range[Length[matrixRepresentation]], 3]]] & ) /@ Tuples[Range[Length[matrixRepresentation]], 4]]],
If[index1 === False && index2 === False && index3 === True && index4 === False,
Normal[SparseArray[(Module[{index = #1}, index -> Total[(Inverse[matrixRepresentation][[index[[1]],#1[[1]]]]*
Inverse[matrixRepresentation][[index[[2]],#1[[2]]]]*Inverse[matrixRepresentation][[#1[[3]],
index[[4]]]]*weylTensor[[#1[[1]],#1[[2]],index[[3]],#1[[3]]]] & ) /@ Tuples[
Range[Length[matrixRepresentation]], 3]]] & ) /@ Tuples[Range[Length[matrixRepresentation]], 4]]],
If[index1 === False && index2 === False && index3 === False && index4 === True,
Normal[SparseArray[(Module[{index = #1}, index -> Total[(Inverse[matrixRepresentation][[index[[1]],#1[[1]]]]*
Inverse[matrixRepresentation][[index[[2]],#1[[2]]]]*Inverse[matrixRepresentation][[#1[[3]],
index[[3]]]]*weylTensor[[#1[[1]],#1[[2]],#1[[3]],index[[4]]]] & ) /@ Tuples[
Range[Length[matrixRepresentation]], 3]]] & ) /@ Tuples[Range[Length[matrixRepresentation]], 4]]],
If[index1 === True && index2 === True && index3 === False && index4 === False,
Normal[SparseArray[(Module[{index = #1}, index -> Total[(Inverse[matrixRepresentation][[#1[[1]],index[[3]]]]*
Inverse[matrixRepresentation][[#1[[2]],index[[4]]]]*weylTensor[[index[[1]],index[[2]],#1[[1]],
#1[[2]]]] & ) /@ Tuples[Range[Length[matrixRepresentation]], 2]]] & ) /@ Tuples[
Range[Length[matrixRepresentation]], 4]]], If[index1 === True && index2 === False && index3 === True &&
index4 === False, Normal[SparseArray[(Module[{index = #1}, index -> Total[(Inverse[matrixRepresentation][[
index[[2]],#1[[1]]]]*Inverse[matrixRepresentation][[#1[[2]],index[[4]]]]*weylTensor[[index[[1]],
#1[[1]],index[[3]],#1[[2]]]] & ) /@ Tuples[Range[Length[matrixRepresentation]], 2]]] & ) /@
Tuples[Range[Length[matrixRepresentation]], 4]]], If[index1 === True && index2 === False && index3 ===
False && index4 === True, Normal[SparseArray[(Module[{index = #1}, index -> Total[
(Inverse[matrixRepresentation][[index[[2]],#1[[1]]]]*Inverse[matrixRepresentation][[#1[[2]],
index[[3]]]]*weylTensor[[index[[1]],#1[[1]],#1[[2]],index[[4]]]] & ) /@ Tuples[
Range[Length[matrixRepresentation]], 2]]] & ) /@ Tuples[Range[Length[matrixRepresentation]], 4]]],
If[index1 === False && index2 === True && index3 === True && index4 === False, Normal[
SparseArray[(Module[{index = #1}, index -> Total[(Inverse[matrixRepresentation][[index[[1]],#1[[1]]]]*
Inverse[matrixRepresentation][[#1[[2]],index[[4]]]]*weylTensor[[#1[[1]],index[[2]],index[[3]],
#1[[2]]]] & ) /@ Tuples[Range[Length[matrixRepresentation]], 2]]] & ) /@
Tuples[Range[Length[matrixRepresentation]], 4]]], If[index1 === False && index2 === True &&
index3 === False && index4 === True, Normal[SparseArray[(Module[{index = #1}, index ->
Total[(Inverse[matrixRepresentation][[index[[1]],#1[[1]]]]*Inverse[matrixRepresentation][[#1[[2]],
index[[3]]]]*weylTensor[[#1[[1]],index[[2]],#1[[2]],index[[4]]]] & ) /@ Tuples[
Range[Length[matrixRepresentation]], 2]]] & ) /@ Tuples[Range[Length[matrixRepresentation]],
4]]], If[index1 === False && index2 === False && index3 === True && index4 === True,
Normal[SparseArray[(Module[{index = #1}, index -> Total[(Inverse[matrixRepresentation][[index[[1]],
#1[[1]]]]*Inverse[matrixRepresentation][[index[[2]],#1[[2]]]]*weylTensor[[#1[[1]],#1[[2]],
index[[3]],index[[4]]]] & ) /@ Tuples[Range[Length[matrixRepresentation]], 2]]] & ) /@
Tuples[Range[Length[matrixRepresentation]], 4]]], If[index1 === True && index2 === True &&
index3 === True && index4 === False, Normal[SparseArray[(Module[{index = #1}, index ->
Total[(Inverse[matrixRepresentation][[#1,index[[4]]]]*weylTensor[[index[[1]],index[[2]],
index[[3]],#1]] & ) /@ Range[Length[matrixRepresentation]]]] & ) /@ Tuples[
Range[Length[matrixRepresentation]], 4]]], If[index1 === True && index2 === True &&
index3 === False && index4 === True, Normal[SparseArray[(Module[{index = #1}, index ->
Total[(Inverse[matrixRepresentation][[#1,index[[3]]]]*weylTensor[[index[[1]],index[[2]],#1,
index[[4]]]] & ) /@ Range[Length[matrixRepresentation]]]] & ) /@ Tuples[
Range[Length[matrixRepresentation]], 4]]], If[index1 === True && index2 === False &&
index3 === True && index4 === True, Normal[SparseArray[(Module[{index = #1}, index ->
Total[(Inverse[matrixRepresentation][[index[[2]],#1]]*weylTensor[[index[[1]],#1,index[[3]],
index[[4]]]] & ) /@ Range[Length[matrixRepresentation]]]] & ) /@ Tuples[Range[
Length[matrixRepresentation]], 4]]], If[index1 === False && index2 === True && index3 ===
True && index4 === True, Normal[SparseArray[(Module[{index = #1}, index -> Total[
(Inverse[matrixRepresentation][[index[[1]],#1]]*weylTensor[[#1,index[[2]],index[[3]],
index[[4]]]] & ) /@ Range[Length[matrixRepresentation]]]] & ) /@ Tuples[Range[
Length[matrixRepresentation]], 4]]], Indeterminate]]]]]]]]]]]]]]]]] /;
SymbolName[metricTensor] === "MetricTensor" && Length[Dimensions[matrixRepresentation]] == 2 &&
Length[coordinates] == Length[matrixRepresentation] && BooleanQ[metricIndex1] && BooleanQ[metricIndex2] &&
BooleanQ[index1] && BooleanQ[index2] && BooleanQ[index3] && BooleanQ[index4]
WeylTensor[(metricTensor_)[matrixRepresentation_List, coordinates_List, metricIndex1_, metricIndex2_], index1_, index2_,
index3_, index4_]["FirstPrincipalInvariant"] :=
Module[{newMatrixRepresentation, newCoordinates, christoffelSymbols, riemannTensor, covariantRiemannTensor,
ricciTensor, ricciScalar, weylTensor, contravariantWeylTensor},
newMatrixRepresentation = matrixRepresentation /. (#1 -> ToExpression[#1] & ) /@ Select[coordinates, StringQ];
newCoordinates = coordinates /. (#1 -> ToExpression[#1] & ) /@ Select[coordinates, StringQ];
christoffelSymbols = Normal[SparseArray[
(Module[{index = #1}, index -> Total[((1/2)*Inverse[newMatrixRepresentation][[index[[1]],#1]]*
(D[newMatrixRepresentation[[#1,index[[3]]]], newCoordinates[[index[[2]]]]] + D[newMatrixRepresentation[[
index[[2]],#1]], newCoordinates[[index[[3]]]]] - D[newMatrixRepresentation[[index[[2]],index[[3]]]],
newCoordinates[[#1]]]) & ) /@ Range[Length[newMatrixRepresentation]]]] & ) /@
Tuples[Range[Length[newMatrixRepresentation]], 3]]];
riemannTensor = Normal[SparseArray[(Module[{index = #1}, index -> D[christoffelSymbols[[index[[1]],index[[2]],
index[[4]]]], newCoordinates[[index[[3]]]]] - D[christoffelSymbols[[index[[1]],index[[2]],index[[3]]]],
newCoordinates[[index[[4]]]]] + Total[(christoffelSymbols[[index[[1]],#1,index[[3]]]]*christoffelSymbols[[
#1,index[[2]],index[[4]]]] & ) /@ Range[Length[newMatrixRepresentation]]] -
Total[(christoffelSymbols[[index[[1]],#1,index[[4]]]]*christoffelSymbols[[#1,index[[2]],index[[3]]]] & ) /@
Range[Length[newMatrixRepresentation]]]] & ) /@ Tuples[Range[Length[newMatrixRepresentation]], 4]]] /.
(ToExpression[#1] -> #1 & ) /@ Select[coordinates, StringQ]; covariantRiemannTensor =
Normal[SparseArray[(Module[{index = #1}, index -> Total[(matrixRepresentation[[index[[1]],#1]]*riemannTensor[[#1,
index[[2]],index[[3]],index[[4]]]] & ) /@ Range[Length[matrixRepresentation]]]] & ) /@
Tuples[Range[Length[matrixRepresentation]], 4]]];
ricciTensor = Normal[SparseArray[(Module[{index = #1}, index -> Total[(riemannTensor[[#1,First[index],#1,
Last[index]]] & ) /@ Range[Length[matrixRepresentation]]]] & ) /@
Tuples[Range[Length[matrixRepresentation]], 2]]];
ricciScalar = Total[(Inverse[matrixRepresentation][[First[#1],Last[#1]]]*ricciTensor[[First[#1],Last[#1]]] & ) /@
Tuples[Range[Length[matrixRepresentation]], 2]];
weylTensor = Normal[SparseArray[(Module[{index = #1}, index -> covariantRiemannTensor[[index[[1]],index[[2]],
index[[3]],index[[4]]]] + (1/(Length[matrixRepresentation] - 2))*(ricciTensor[[index[[1]],index[[4]]]]*
matrixRepresentation[[index[[2]],index[[3]]]] - ricciTensor[[index[[1]],index[[3]]]]*
matrixRepresentation[[index[[2]],index[[4]]]] + ricciTensor[[index[[2]],index[[3]]]]*
matrixRepresentation[[index[[1]],index[[4]]]] - ricciTensor[[index[[2]],index[[4]]]]*
matrixRepresentation[[index[[1]],index[[3]]]]) + (1/((Length[matrixRepresentation] - 1)*
(Length[matrixRepresentation] - 2)))*(ricciScalar*(matrixRepresentation[[index[[1]],index[[3]]]]*
matrixRepresentation[[index[[2]],index[[4]]]] - matrixRepresentation[[index[[1]],index[[4]]]]*
matrixRepresentation[[index[[2]],index[[3]]]]))] & ) /@ Tuples[Range[Length[matrixRepresentation]],
4]]]; contravariantWeylTensor =
Normal[SparseArray[(Module[{index = #1}, index -> Total[(Inverse[matrixRepresentation][[index[[1]],#1[[1]]]]*
Inverse[matrixRepresentation][[index[[2]],#1[[2]]]]*Inverse[matrixRepresentation][[#1[[3]],index[[3]]]]*
Inverse[matrixRepresentation][[#1[[4]],index[[4]]]]*weylTensor[[#1[[1]],#1[[2]],#1[[3]],#1[[4]]]] & ) /@
Tuples[Range[Length[matrixRepresentation]], 4]]] & ) /@ Tuples[Range[Length[matrixRepresentation]], 4]]];
Total[(weylTensor[[#1[[1]],#1[[2]],#1[[3]],#1[[4]]]]*contravariantWeylTensor[[#1[[1]],#1[[2]],#1[[3]],
#1[[4]]]] & ) /@ Tuples[Range[Length[matrixRepresentation]], 4]]] /;
SymbolName[metricTensor] === "MetricTensor" && Length[Dimensions[matrixRepresentation]] == 2 &&
Length[coordinates] == Length[matrixRepresentation] && BooleanQ[metricIndex1] && BooleanQ[metricIndex2] &&
BooleanQ[index1] && BooleanQ[index2] && BooleanQ[index3] && BooleanQ[index4]
WeylTensor[(metricTensor_)[matrixRepresentation_List, coordinates_List, metricIndex1_, metricIndex2_], index1_, index2_,
index3_, index4_]["ReducedFirstPrincipalInvariant"] :=
Module[{newMatrixRepresentation, newCoordinates, christoffelSymbols, riemannTensor, covariantRiemannTensor,
ricciTensor, ricciScalar, weylTensor, contravariantWeylTensor},
newMatrixRepresentation = matrixRepresentation /. (#1 -> ToExpression[#1] & ) /@ Select[coordinates, StringQ];
newCoordinates = coordinates /. (#1 -> ToExpression[#1] & ) /@ Select[coordinates, StringQ];
christoffelSymbols = Normal[SparseArray[
(Module[{index = #1}, index -> Total[((1/2)*Inverse[newMatrixRepresentation][[index[[1]],#1]]*
(D[newMatrixRepresentation[[#1,index[[3]]]], newCoordinates[[index[[2]]]]] + D[newMatrixRepresentation[[
index[[2]],#1]], newCoordinates[[index[[3]]]]] - D[newMatrixRepresentation[[index[[2]],index[[3]]]],
newCoordinates[[#1]]]) & ) /@ Range[Length[newMatrixRepresentation]]]] & ) /@
Tuples[Range[Length[newMatrixRepresentation]], 3]]];
riemannTensor = Normal[SparseArray[(Module[{index = #1}, index -> D[christoffelSymbols[[index[[1]],index[[2]],
index[[4]]]], newCoordinates[[index[[3]]]]] - D[christoffelSymbols[[index[[1]],index[[2]],index[[3]]]],
newCoordinates[[index[[4]]]]] + Total[(christoffelSymbols[[index[[1]],#1,index[[3]]]]*christoffelSymbols[[
#1,index[[2]],index[[4]]]] & ) /@ Range[Length[newMatrixRepresentation]]] -
Total[(christoffelSymbols[[index[[1]],#1,index[[4]]]]*christoffelSymbols[[#1,index[[2]],index[[3]]]] & ) /@
Range[Length[newMatrixRepresentation]]]] & ) /@ Tuples[Range[Length[newMatrixRepresentation]], 4]]] /.
(ToExpression[#1] -> #1 & ) /@ Select[coordinates, StringQ]; covariantRiemannTensor =
Normal[SparseArray[(Module[{index = #1}, index -> Total[(matrixRepresentation[[index[[1]],#1]]*riemannTensor[[#1,
index[[2]],index[[3]],index[[4]]]] & ) /@ Range[Length[matrixRepresentation]]]] & ) /@
Tuples[Range[Length[matrixRepresentation]], 4]]];
ricciTensor = Normal[SparseArray[(Module[{index = #1}, index -> Total[(riemannTensor[[#1,First[index],#1,
Last[index]]] & ) /@ Range[Length[matrixRepresentation]]]] & ) /@
Tuples[Range[Length[matrixRepresentation]], 2]]];
ricciScalar = Total[(Inverse[matrixRepresentation][[First[#1],Last[#1]]]*ricciTensor[[First[#1],Last[#1]]] & ) /@
Tuples[Range[Length[matrixRepresentation]], 2]];
weylTensor = Normal[SparseArray[(Module[{index = #1}, index -> covariantRiemannTensor[[index[[1]],index[[2]],
index[[3]],index[[4]]]] + (1/(Length[matrixRepresentation] - 2))*(ricciTensor[[index[[1]],index[[4]]]]*
matrixRepresentation[[index[[2]],index[[3]]]] - ricciTensor[[index[[1]],index[[3]]]]*
matrixRepresentation[[index[[2]],index[[4]]]] + ricciTensor[[index[[2]],index[[3]]]]*
matrixRepresentation[[index[[1]],index[[4]]]] - ricciTensor[[index[[2]],index[[4]]]]*
matrixRepresentation[[index[[1]],index[[3]]]]) + (1/((Length[matrixRepresentation] - 1)*
(Length[matrixRepresentation] - 2)))*(ricciScalar*(matrixRepresentation[[index[[1]],index[[3]]]]*
matrixRepresentation[[index[[2]],index[[4]]]] - matrixRepresentation[[index[[1]],index[[4]]]]*
matrixRepresentation[[index[[2]],index[[3]]]]))] & ) /@ Tuples[Range[Length[matrixRepresentation]],
4]]]; contravariantWeylTensor =
Normal[SparseArray[(Module[{index = #1}, index -> Total[(Inverse[matrixRepresentation][[index[[1]],#1[[1]]]]*
Inverse[matrixRepresentation][[index[[2]],#1[[2]]]]*Inverse[matrixRepresentation][[#1[[3]],index[[3]]]]*
Inverse[matrixRepresentation][[#1[[4]],index[[4]]]]*weylTensor[[#1[[1]],#1[[2]],#1[[3]],#1[[4]]]] & ) /@
Tuples[Range[Length[matrixRepresentation]], 4]]] & ) /@ Tuples[Range[Length[matrixRepresentation]], 4]]];
FullSimplify[Total[(weylTensor[[#1[[1]],#1[[2]],#1[[3]],#1[[4]]]]*contravariantWeylTensor[[#1[[1]],#1[[2]],#1[[3]],
#1[[4]]]] & ) /@ Tuples[Range[Length[matrixRepresentation]], 4]]]] /;
SymbolName[metricTensor] === "MetricTensor" && Length[Dimensions[matrixRepresentation]] == 2 &&
Length[coordinates] == Length[matrixRepresentation] && BooleanQ[metricIndex1] && BooleanQ[metricIndex2] &&
BooleanQ[index1] && BooleanQ[index2] && BooleanQ[index3] && BooleanQ[index4]
WeylTensor[(metricTensor_)[matrixRepresentation_List, coordinates_List, metricIndex1_, metricIndex2_], index1_, index2_,
index3_, index4_]["SymbolicFirstPrincipalInvariant"] :=
Module[{newMatrixRepresentation, newCoordinates, christoffelSymbols, riemannTensor, covariantRiemannTensor,
ricciTensor, ricciScalar, weylTensor, contravariantWeylTensor},
newMatrixRepresentation = matrixRepresentation /. (#1 -> ToExpression[#1] & ) /@ Select[coordinates, StringQ];
newCoordinates = coordinates /. (#1 -> ToExpression[#1] & ) /@ Select[coordinates, StringQ];
christoffelSymbols = Normal[SparseArray[
(Module[{index = #1}, index -> Total[((1/2)*Inverse[newMatrixRepresentation][[index[[1]],#1]]*
(Inactive[D][newMatrixRepresentation[[#1,index[[3]]]], newCoordinates[[index[[2]]]]] +
Inactive[D][newMatrixRepresentation[[index[[2]],#1]], newCoordinates[[index[[3]]]]] -
Inactive[D][newMatrixRepresentation[[index[[2]],index[[3]]]], newCoordinates[[#1]]]) & ) /@
Range[Length[newMatrixRepresentation]]]] & ) /@ Tuples[Range[Length[newMatrixRepresentation]], 3]]];
riemannTensor = Normal[SparseArray[(Module[{index = #1}, index -> Inactive[D][christoffelSymbols[[index[[1]],
index[[2]],index[[4]]]], newCoordinates[[index[[3]]]]] - Inactive[D][christoffelSymbols[[index[[1]],
index[[2]],index[[3]]]], newCoordinates[[index[[4]]]]] + Total[(christoffelSymbols[[index[[1]],#1,
index[[3]]]]*christoffelSymbols[[#1,index[[2]],index[[4]]]] & ) /@
Range[Length[newMatrixRepresentation]]] - Total[(christoffelSymbols[[index[[1]],#1,index[[4]]]]*
christoffelSymbols[[#1,index[[2]],index[[3]]]] & ) /@ Range[Length[newMatrixRepresentation]]]] & ) /@
Tuples[Range[Length[newMatrixRepresentation]], 4]]] /. (ToExpression[#1] -> #1 & ) /@
Select[coordinates, StringQ]; covariantRiemannTensor =
Normal[SparseArray[(Module[{index = #1}, index -> Total[(matrixRepresentation[[index[[1]],#1]]*riemannTensor[[#1,
index[[2]],index[[3]],index[[4]]]] & ) /@ Range[Length[matrixRepresentation]]]] & ) /@
Tuples[Range[Length[matrixRepresentation]], 4]]];
ricciTensor = Normal[SparseArray[(Module[{index = #1}, index -> Total[(riemannTensor[[#1,First[index],#1,
Last[index]]] & ) /@ Range[Length[matrixRepresentation]]]] & ) /@
Tuples[Range[Length[matrixRepresentation]], 2]]];
ricciScalar = Total[(Inverse[matrixRepresentation][[First[#1],Last[#1]]]*ricciTensor[[First[#1],Last[#1]]] & ) /@
Tuples[Range[Length[matrixRepresentation]], 2]];
weylTensor = Normal[SparseArray[(Module[{index = #1}, index -> covariantRiemannTensor[[index[[1]],index[[2]],
index[[3]],index[[4]]]] + (1/(Length[matrixRepresentation] - 2))*(ricciTensor[[index[[1]],index[[4]]]]*
matrixRepresentation[[index[[2]],index[[3]]]] - ricciTensor[[index[[1]],index[[3]]]]*
matrixRepresentation[[index[[2]],index[[4]]]] + ricciTensor[[index[[2]],index[[3]]]]*
matrixRepresentation[[index[[1]],index[[4]]]] - ricciTensor[[index[[2]],index[[4]]]]*
matrixRepresentation[[index[[1]],index[[3]]]]) + (1/((Length[matrixRepresentation] - 1)*
(Length[matrixRepresentation] - 2)))*(ricciScalar*(matrixRepresentation[[index[[1]],index[[3]]]]*
matrixRepresentation[[index[[2]],index[[4]]]] - matrixRepresentation[[index[[1]],index[[4]]]]*
matrixRepresentation[[index[[2]],index[[3]]]]))] & ) /@ Tuples[Range[Length[matrixRepresentation]],
4]]]; contravariantWeylTensor =
Normal[SparseArray[(Module[{index = #1}, index -> Total[(Inverse[matrixRepresentation][[index[[1]],#1[[1]]]]*
Inverse[matrixRepresentation][[index[[2]],#1[[2]]]]*Inverse[matrixRepresentation][[#1[[3]],index[[3]]]]*
Inverse[matrixRepresentation][[#1[[4]],index[[4]]]]*weylTensor[[#1[[1]],#1[[2]],#1[[3]],#1[[4]]]] & ) /@
Tuples[Range[Length[matrixRepresentation]], 4]]] & ) /@ Tuples[Range[Length[matrixRepresentation]], 4]]];
Total[(weylTensor[[#1[[1]],#1[[2]],#1[[3]],#1[[4]]]]*contravariantWeylTensor[[#1[[1]],#1[[2]],#1[[3]],
#1[[4]]]] & ) /@ Tuples[Range[Length[matrixRepresentation]], 4]]] /;
SymbolName[metricTensor] === "MetricTensor" && Length[Dimensions[matrixRepresentation]] == 2 &&
Length[coordinates] == Length[matrixRepresentation] && BooleanQ[metricIndex1] && BooleanQ[metricIndex2] &&
BooleanQ[index1] && BooleanQ[index2] && BooleanQ[index3] && BooleanQ[index4]
WeylTensor[(metricTensor_)[matrixRepresentation_List, coordinates_List, metricIndex1_, metricIndex2_], index1_, index2_,
index3_, index4_]["SecondPrincipalInvariant"] :=
Module[{newMatrixRepresentation, newCoordinates, christoffelSymbols, riemannTensor, covariantRiemannTensor,
ricciTensor, ricciScalar, weylTensor, mixedWeylTensor, leviCivitaTensor, contravariantLeviCivitaTensor},
If[Length[matrixRepresentation] == 4, newMatrixRepresentation = matrixRepresentation /.
(#1 -> ToExpression[#1] & ) /@ Select[coordinates, StringQ];
newCoordinates = coordinates /. (#1 -> ToExpression[#1] & ) /@ Select[coordinates, StringQ];
christoffelSymbols = Normal[SparseArray[
(Module[{index = #1}, index -> Total[((1/2)*Inverse[newMatrixRepresentation][[index[[1]],#1]]*
(D[newMatrixRepresentation[[#1,index[[3]]]], newCoordinates[[index[[2]]]]] + D[newMatrixRepresentation[[
index[[2]],#1]], newCoordinates[[index[[3]]]]] - D[newMatrixRepresentation[[index[[2]],index[[3]]]],
newCoordinates[[#1]]]) & ) /@ Range[Length[newMatrixRepresentation]]]] & ) /@
Tuples[Range[Length[newMatrixRepresentation]], 3]]]; riemannTensor =
Normal[SparseArray[(Module[{index = #1}, index -> D[christoffelSymbols[[index[[1]],index[[2]],index[[4]]]],
newCoordinates[[index[[3]]]]] - D[christoffelSymbols[[index[[1]],index[[2]],index[[3]]]],
newCoordinates[[index[[4]]]]] + Total[(christoffelSymbols[[index[[1]],#1,index[[3]]]]*christoffelSymbols[[
#1,index[[2]],index[[4]]]] & ) /@ Range[Length[newMatrixRepresentation]]] - Total[
(christoffelSymbols[[index[[1]],#1,index[[4]]]]*christoffelSymbols[[#1,index[[2]],index[[3]]]] & ) /@
Range[Length[newMatrixRepresentation]]]] & ) /@ Tuples[Range[Length[newMatrixRepresentation]], 4]]] /.
(ToExpression[#1] -> #1 & ) /@ Select[coordinates, StringQ]; covariantRiemannTensor =
Normal[SparseArray[(Module[{index = #1}, index -> Total[(matrixRepresentation[[index[[1]],#1]]*riemannTensor[[#1,
index[[2]],index[[3]],index[[4]]]] & ) /@ Range[Length[matrixRepresentation]]]] & ) /@
Tuples[Range[Length[matrixRepresentation]], 4]]]; ricciTensor =
Normal[SparseArray[(Module[{index = #1}, index -> Total[(riemannTensor[[#1,First[index],#1,Last[index]]] & ) /@
Range[Length[matrixRepresentation]]]] & ) /@ Tuples[Range[Length[matrixRepresentation]], 2]]];
ricciScalar = Total[(Inverse[matrixRepresentation][[First[#1],Last[#1]]]*ricciTensor[[First[#1],Last[#1]]] & ) /@
Tuples[Range[Length[matrixRepresentation]], 2]];
weylTensor = Normal[SparseArray[(Module[{index = #1}, index -> covariantRiemannTensor[[index[[1]],index[[2]],index[[
3]],index[[4]]]] + (1/(Length[matrixRepresentation] - 2))*(ricciTensor[[index[[1]],index[[4]]]]*
matrixRepresentation[[index[[2]],index[[3]]]] - ricciTensor[[index[[1]],index[[3]]]]*
matrixRepresentation[[index[[2]],index[[4]]]] + ricciTensor[[index[[2]],index[[3]]]]*
matrixRepresentation[[index[[1]],index[[4]]]] - ricciTensor[[index[[2]],index[[4]]]]*
matrixRepresentation[[index[[1]],index[[3]]]]) + (1/((Length[matrixRepresentation] - 1)*
(Length[matrixRepresentation] - 2)))*(ricciScalar*(matrixRepresentation[[index[[1]],index[[3]]]]*
matrixRepresentation[[index[[2]],index[[4]]]] - matrixRepresentation[[index[[1]],index[[4]]]]*
matrixRepresentation[[index[[2]],index[[3]]]]))] & ) /@ Tuples[Range[Length[matrixRepresentation]],
4]]]; mixedWeylTensor = Normal[SparseArray[
(Module[{index = #1}, index -> Total[(Inverse[matrixRepresentation][[index[[1]],#1[[1]]]]*
Inverse[matrixRepresentation][[index[[2]],#1[[2]]]]*weylTensor[[#1[[1]],#1[[2]],index[[3]],
index[[4]]]] & ) /@ Tuples[Range[Length[matrixRepresentation]], 2]]] & ) /@
Tuples[Range[Length[matrixRepresentation]], 4]]]; leviCivitaTensor = Normal[LeviCivitaTensor[4]];
contravariantLeviCivitaTensor = Normal[SparseArray[
(Module[{index = #1}, index -> Total[(Inverse[matrixRepresentation][[index[[1]],#1[[1]]]]*
Inverse[matrixRepresentation][[index[[2]],#1[[2]]]]*Inverse[matrixRepresentation][[#1[[3]],index[[3]]]]*
Inverse[matrixRepresentation][[#1[[4]],index[[4]]]]*leviCivitaTensor[[#1[[1]],#1[[2]],#1[[3]],
#1[[4]]]] & ) /@ Tuples[Range[Length[matrixRepresentation]], 4]]] & ) /@
Tuples[Range[Length[matrixRepresentation]], 4]]];
Total[(weylTensor[[#1[[1]],#1[[2]],#1[[3]],#1[[4]]]]*contravariantLeviCivitaTensor[[#1[[1]],#1[[2]],#1[[5]],
#1[[6]]]]*mixedWeylTensor[[#1[[3]],#1[[4]],#1[[5]],#1[[6]]]] & ) /@
Tuples[Range[Length[matrixRepresentation]], 6]], Indeterminate]] /;
SymbolName[metricTensor] === "MetricTensor" && Length[Dimensions[matrixRepresentation]] == 2 &&
Length[coordinates] == Length[matrixRepresentation] && BooleanQ[metricIndex1] && BooleanQ[metricIndex2] &&
BooleanQ[index1] && BooleanQ[index2] && BooleanQ[index3] && BooleanQ[index4]
WeylTensor[(metricTensor_)[matrixRepresentation_List, coordinates_List, metricIndex1_, metricIndex2_], index1_, index2_,
index3_, index4_]["ReducedSecondPrincipalInvariant"] :=
Module[{newMatrixRepresentation, newCoordinates, christoffelSymbols, riemannTensor, covariantRiemannTensor,
ricciTensor, ricciScalar, weylTensor, mixedWeylTensor, leviCivitaTensor, contravariantLeviCivitaTensor},
If[Length[matrixRepresentation] == 4, newMatrixRepresentation = matrixRepresentation /.
(#1 -> ToExpression[#1] & ) /@ Select[coordinates, StringQ];
newCoordinates = coordinates /. (#1 -> ToExpression[#1] & ) /@ Select[coordinates, StringQ];
christoffelSymbols = Normal[SparseArray[
(Module[{index = #1}, index -> Total[((1/2)*Inverse[newMatrixRepresentation][[index[[1]],#1]]*
(D[newMatrixRepresentation[[#1,index[[3]]]], newCoordinates[[index[[2]]]]] + D[newMatrixRepresentation[[
index[[2]],#1]], newCoordinates[[index[[3]]]]] - D[newMatrixRepresentation[[index[[2]],index[[3]]]],
newCoordinates[[#1]]]) & ) /@ Range[Length[newMatrixRepresentation]]]] & ) /@
Tuples[Range[Length[newMatrixRepresentation]], 3]]]; riemannTensor =
Normal[SparseArray[(Module[{index = #1}, index -> D[christoffelSymbols[[index[[1]],index[[2]],index[[4]]]],
newCoordinates[[index[[3]]]]] - D[christoffelSymbols[[index[[1]],index[[2]],index[[3]]]],
newCoordinates[[index[[4]]]]] + Total[(christoffelSymbols[[index[[1]],#1,index[[3]]]]*christoffelSymbols[[
#1,index[[2]],index[[4]]]] & ) /@ Range[Length[newMatrixRepresentation]]] - Total[
(christoffelSymbols[[index[[1]],#1,index[[4]]]]*christoffelSymbols[[#1,index[[2]],index[[3]]]] & ) /@
Range[Length[newMatrixRepresentation]]]] & ) /@ Tuples[Range[Length[newMatrixRepresentation]], 4]]] /.
(ToExpression[#1] -> #1 & ) /@ Select[coordinates, StringQ]; covariantRiemannTensor =
Normal[SparseArray[(Module[{index = #1}, index -> Total[(matrixRepresentation[[index[[1]],#1]]*riemannTensor[[#1,
index[[2]],index[[3]],index[[4]]]] & ) /@ Range[Length[matrixRepresentation]]]] & ) /@
Tuples[Range[Length[matrixRepresentation]], 4]]]; ricciTensor =
Normal[SparseArray[(Module[{index = #1}, index -> Total[(riemannTensor[[#1,First[index],#1,Last[index]]] & ) /@
Range[Length[matrixRepresentation]]]] & ) /@ Tuples[Range[Length[matrixRepresentation]], 2]]];
ricciScalar = Total[(Inverse[matrixRepresentation][[First[#1],Last[#1]]]*ricciTensor[[First[#1],Last[#1]]] & ) /@
Tuples[Range[Length[matrixRepresentation]], 2]];
weylTensor = Normal[SparseArray[(Module[{index = #1}, index -> covariantRiemannTensor[[index[[1]],index[[2]],index[[
3]],index[[4]]]] + (1/(Length[matrixRepresentation] - 2))*(ricciTensor[[index[[1]],index[[4]]]]*
matrixRepresentation[[index[[2]],index[[3]]]] - ricciTensor[[index[[1]],index[[3]]]]*
matrixRepresentation[[index[[2]],index[[4]]]] + ricciTensor[[index[[2]],index[[3]]]]*
matrixRepresentation[[index[[1]],index[[4]]]] - ricciTensor[[index[[2]],index[[4]]]]*
matrixRepresentation[[index[[1]],index[[3]]]]) + (1/((Length[matrixRepresentation] - 1)*
(Length[matrixRepresentation] - 2)))*(ricciScalar*(matrixRepresentation[[index[[1]],index[[3]]]]*
matrixRepresentation[[index[[2]],index[[4]]]] - matrixRepresentation[[index[[1]],index[[4]]]]*
matrixRepresentation[[index[[2]],index[[3]]]]))] & ) /@ Tuples[Range[Length[matrixRepresentation]],
4]]]; mixedWeylTensor = Normal[SparseArray[
(Module[{index = #1}, index -> Total[(Inverse[matrixRepresentation][[index[[1]],#1[[1]]]]*
Inverse[matrixRepresentation][[index[[2]],#1[[2]]]]*weylTensor[[#1[[1]],#1[[2]],index[[3]],
index[[4]]]] & ) /@ Tuples[Range[Length[matrixRepresentation]], 2]]] & ) /@
Tuples[Range[Length[matrixRepresentation]], 4]]]; leviCivitaTensor = Normal[LeviCivitaTensor[4]];
contravariantLeviCivitaTensor = Normal[SparseArray[
(Module[{index = #1}, index -> Total[(Inverse[matrixRepresentation][[index[[1]],#1[[1]]]]*
Inverse[matrixRepresentation][[index[[2]],#1[[2]]]]*Inverse[matrixRepresentation][[#1[[3]],index[[3]]]]*
Inverse[matrixRepresentation][[#1[[4]],index[[4]]]]*leviCivitaTensor[[#1[[1]],#1[[2]],#1[[3]],
#1[[4]]]] & ) /@ Tuples[Range[Length[matrixRepresentation]], 4]]] & ) /@
Tuples[Range[Length[matrixRepresentation]], 4]]];
FullSimplify[Total[(weylTensor[[#1[[1]],#1[[2]],#1[[3]],#1[[4]]]]*contravariantLeviCivitaTensor[[#1[[1]],#1[[2]],
#1[[5]],#1[[6]]]]*mixedWeylTensor[[#1[[3]],#1[[4]],#1[[5]],#1[[6]]]] & ) /@
Tuples[Range[Length[matrixRepresentation]], 6]]], Indeterminate]] /;
SymbolName[metricTensor] === "MetricTensor" && Length[Dimensions[matrixRepresentation]] == 2 &&
Length[coordinates] == Length[matrixRepresentation] && BooleanQ[metricIndex1] && BooleanQ[metricIndex2] &&
BooleanQ[index1] && BooleanQ[index2] && BooleanQ[index3] && BooleanQ[index4]
WeylTensor[(metricTensor_)[matrixRepresentation_List, coordinates_List, metricIndex1_, metricIndex2_], index1_, index2_,
index3_, index4_]["SymbolicSecondPrincipalInvariant"] :=
Module[{newMatrixRepresentation, newCoordinates, christoffelSymbols, riemannTensor, covariantRiemannTensor,
ricciTensor, ricciScalar, weylTensor, mixedWeylTensor, leviCivitaTensor, contravariantLeviCivitaTensor},
If[Length[matrixRepresentation] == 4, newMatrixRepresentation = matrixRepresentation /.
(#1 -> ToExpression[#1] & ) /@ Select[coordinates, StringQ];
newCoordinates = coordinates /. (#1 -> ToExpression[#1] & ) /@ Select[coordinates, StringQ];
christoffelSymbols = Normal[SparseArray[
(Module[{index = #1}, index -> Total[((1/2)*Inverse[newMatrixRepresentation][[index[[1]],#1]]*
(Inactive[D][newMatrixRepresentation[[#1,index[[3]]]], newCoordinates[[index[[2]]]]] +
Inactive[D][newMatrixRepresentation[[index[[2]],#1]], newCoordinates[[index[[3]]]]] -
Inactive[D][newMatrixRepresentation[[index[[2]],index[[3]]]], newCoordinates[[#1]]]) & ) /@ Range[
Length[newMatrixRepresentation]]]] & ) /@ Tuples[Range[Length[newMatrixRepresentation]], 3]]];
riemannTensor = Normal[SparseArray[(Module[{index = #1}, index -> Inactive[D][christoffelSymbols[[index[[1]],
index[[2]],index[[4]]]], newCoordinates[[index[[3]]]]] - Inactive[D][christoffelSymbols[[index[[1]],
index[[2]],index[[3]]]], newCoordinates[[index[[4]]]]] + Total[(christoffelSymbols[[index[[1]],#1,
index[[3]]]]*christoffelSymbols[[#1,index[[2]],index[[4]]]] & ) /@ Range[Length[
newMatrixRepresentation]]] - Total[(christoffelSymbols[[index[[1]],#1,index[[4]]]]*christoffelSymbols[[
#1,index[[2]],index[[3]]]] & ) /@ Range[Length[newMatrixRepresentation]]]] & ) /@
Tuples[Range[Length[newMatrixRepresentation]], 4]]] /. (ToExpression[#1] -> #1 & ) /@
Select[coordinates, StringQ]; covariantRiemannTensor =
Normal[SparseArray[(Module[{index = #1}, index -> Total[(matrixRepresentation[[index[[1]],#1]]*riemannTensor[[#1,
index[[2]],index[[3]],index[[4]]]] & ) /@ Range[Length[matrixRepresentation]]]] & ) /@
Tuples[Range[Length[matrixRepresentation]], 4]]]; ricciTensor =
Normal[SparseArray[(Module[{index = #1}, index -> Total[(riemannTensor[[#1,First[index],#1,Last[index]]] & ) /@
Range[Length[matrixRepresentation]]]] & ) /@ Tuples[Range[Length[matrixRepresentation]], 2]]];
ricciScalar = Total[(Inverse[matrixRepresentation][[First[#1],Last[#1]]]*ricciTensor[[First[#1],Last[#1]]] & ) /@
Tuples[Range[Length[matrixRepresentation]], 2]];
weylTensor = Normal[SparseArray[(Module[{index = #1}, index -> covariantRiemannTensor[[index[[1]],index[[2]],index[[
3]],index[[4]]]] + (1/(Length[matrixRepresentation] - 2))*(ricciTensor[[index[[1]],index[[4]]]]*
matrixRepresentation[[index[[2]],index[[3]]]] - ricciTensor[[index[[1]],index[[3]]]]*
matrixRepresentation[[index[[2]],index[[4]]]] + ricciTensor[[index[[2]],index[[3]]]]*
matrixRepresentation[[index[[1]],index[[4]]]] - ricciTensor[[index[[2]],index[[4]]]]*
matrixRepresentation[[index[[1]],index[[3]]]]) + (1/((Length[matrixRepresentation] - 1)*
(Length[matrixRepresentation] - 2)))*(ricciScalar*(matrixRepresentation[[index[[1]],index[[3]]]]*
matrixRepresentation[[index[[2]],index[[4]]]] - matrixRepresentation[[index[[1]],index[[4]]]]*
matrixRepresentation[[index[[2]],index[[3]]]]))] & ) /@ Tuples[Range[Length[matrixRepresentation]],
4]]]; mixedWeylTensor = Normal[SparseArray[
(Module[{index = #1}, index -> Total[(Inverse[matrixRepresentation][[index[[1]],#1[[1]]]]*
Inverse[matrixRepresentation][[index[[2]],#1[[2]]]]*weylTensor[[#1[[1]],#1[[2]],index[[3]],
index[[4]]]] & ) /@ Tuples[Range[Length[matrixRepresentation]], 2]]] & ) /@
Tuples[Range[Length[matrixRepresentation]], 4]]]; leviCivitaTensor = Normal[LeviCivitaTensor[4]];
contravariantLeviCivitaTensor = Normal[SparseArray[
(Module[{index = #1}, index -> Total[(Inverse[matrixRepresentation][[index[[1]],#1[[1]]]]*
Inverse[matrixRepresentation][[index[[2]],#1[[2]]]]*Inverse[matrixRepresentation][[#1[[3]],index[[3]]]]*
Inverse[matrixRepresentation][[#1[[4]],index[[4]]]]*leviCivitaTensor[[#1[[1]],#1[[2]],#1[[3]],
#1[[4]]]] & ) /@ Tuples[Range[Length[matrixRepresentation]], 4]]] & ) /@
Tuples[Range[Length[matrixRepresentation]], 4]]];
Total[(weylTensor[[#1[[1]],#1[[2]],#1[[3]],#1[[4]]]]*contravariantLeviCivitaTensor[[#1[[1]],#1[[2]],#1[[5]],
#1[[6]]]]*mixedWeylTensor[[#1[[3]],#1[[4]],#1[[5]],#1[[6]]]] & ) /@
Tuples[Range[Length[matrixRepresentation]], 6]], Indeterminate]] /;
SymbolName[metricTensor] === "MetricTensor" && Length[Dimensions[matrixRepresentation]] == 2 &&
Length[coordinates] == Length[matrixRepresentation] && BooleanQ[metricIndex1] && BooleanQ[metricIndex2] &&
BooleanQ[index1] && BooleanQ[index2] && BooleanQ[index3] && BooleanQ[index4]
WeylTensor[(metricTensor_)[matrixRepresentation_List, coordinates_List, metricIndex1_, metricIndex2_], index1_, index2_,
index3_, index4_]["MetricTensor"] := ResourceFunction["MetricTensor"][matrixRepresentation, coordinates,
metricIndex1, metricIndex2] /; SymbolName[metricTensor] === "MetricTensor" &&
Length[Dimensions[matrixRepresentation]] == 2 && Length[coordinates] == Length[matrixRepresentation] &&
BooleanQ[metricIndex1] && BooleanQ[metricIndex2] && BooleanQ[index1] && BooleanQ[index2] && BooleanQ[index3] &&
BooleanQ[index4]
WeylTensor[(metricTensor_)[matrixRepresentation_List, coordinates_List, metricIndex1_, metricIndex2_], index1_, index2_,
index3_, index4_]["Coordinates"] := coordinates /; SymbolName[metricTensor] === "MetricTensor" &&
Length[Dimensions[matrixRepresentation]] == 2 && Length[coordinates] == Length[matrixRepresentation] &&
BooleanQ[metricIndex1] && BooleanQ[metricIndex2] && BooleanQ[index1] && BooleanQ[index2] && BooleanQ[index3] &&
BooleanQ[index4]
WeylTensor[(metricTensor_)[matrixRepresentation_List, coordinates_List, metricIndex1_, metricIndex2_], index1_, index2_,
index3_, index4_]["CoordinateOneForms"] :=
(If[Head[#1] === Subscript, Subscript[StringJoin["\[FormalD]", ToString[First[#1]]], ToString[Last[#1]]],
If[Head[#1] === Superscript, Superscript[StringJoin["\[FormalD]", ToString[First[#1]]], ToString[Last[#1]]],
StringJoin["\[FormalD]", ToString[#1]]]] & ) /@ coordinates /; SymbolName[metricTensor] === "MetricTensor" &&
Length[Dimensions[matrixRepresentation]] == 2 && Length[coordinates] == Length[matrixRepresentation] &&
BooleanQ[metricIndex1] && BooleanQ[metricIndex2] && BooleanQ[index1] && BooleanQ[index2] && BooleanQ[index3] &&
BooleanQ[index4]
WeylTensor[(metricTensor_)[matrixRepresentation_List, coordinates_List, metricIndex1_, metricIndex2_], index1_, index2_,
index3_, index4_]["Symbol"] := If[index1 === True && index2 === True && index3 === True && index4 === True,
Subscript["\[FormalCapitalC]", "\[FormalRho]\[FormalSigma]\[FormalMu]\[FormalNu]"], If[index1 === False && index2 === False && index3 === False && index4 === False,
Superscript["\[FormalCapitalC]", "\[FormalRho]\[FormalSigma]\[FormalMu]\[FormalNu]"], If[index1 === True && index2 === False && index3 === False && index4 === False,
Subsuperscript["\[FormalCapitalC]", "\[FormalRho]", "\[FormalSigma]\[FormalMu]\[FormalNu]"], If[index1 === False && index2 === True && index3 === False &&
index4 === False, Subsuperscript["\[FormalCapitalC]", "\[FormalSigma]", "\[FormalRho]\[FormalMu]\[FormalNu]"], If[index1 === False && index2 === False &&
index3 === True && index4 === False, Subsuperscript["\[FormalCapitalC]", "\[FormalMu]", "\[FormalRho]\[FormalSigma]\[FormalNu]"],
If[index1 === False && index2 === False && index3 === False && index4 === True,
Subsuperscript["\[FormalCapitalC]", "\[FormalNu]", "\[FormalRho]\[FormalSigma]\[FormalMu]"], If[index1 === True && index2 === True && index3 === False &&
index4 === False, Subsuperscript["\[FormalCapitalC]", "\[FormalRho]\[FormalSigma]", "\[FormalMu]\[FormalNu]"], If[index1 === True && index2 === False &&
index3 === True && index4 === False, Subsuperscript["\[FormalCapitalC]", "\[FormalRho]\[FormalMu]", "\[FormalSigma]\[FormalNu]"],
If[index1 === True && index2 === False && index3 === False && index4 === True, Subsuperscript["\[FormalCapitalC]", "\[FormalRho]\[FormalNu]",
"\[FormalSigma]\[FormalMu]"], If[index1 === False && index2 === True && index3 === True && index4 === False,
Subsuperscript["\[FormalCapitalC]", "\[FormalSigma]\[FormalMu]", "\[FormalRho]\[FormalNu]"], If[index1 === False && index2 === True && index3 === False &&
index4 === True, Subsuperscript["\[FormalCapitalC]", "\[FormalSigma]\[FormalNu]", "\[FormalRho]\[FormalMu]"], If[index1 === False && index2 === False &&
index3 === True && index4 === True, Subsuperscript["\[FormalCapitalC]", "\[FormalMu]\[FormalNu]", "\[FormalRho]\[FormalSigma]"], If[index1 === True &&
index2 === True && index3 === True && index4 === False, Subsuperscript["\[FormalCapitalC]", "\[FormalRho]\[FormalSigma]\[FormalMu]", "\[FormalNu]"],
If[index1 === True && index2 === True && index3 === False && index4 === True, Subsuperscript["\[FormalCapitalC]",
"\[FormalRho]\[FormalSigma]\[FormalNu]", "\[FormalMu]"], If[index1 === True && index2 === False && index3 === True && index4 === True,
Subsuperscript["\[FormalCapitalC]", "\[FormalRho]\[FormalMu]\[FormalNu]", "\[FormalSigma]"], If[index1 === False && index2 === True && index3 === True &&
index4 === True, Subsuperscript["\[FormalCapitalC]", "\[FormalSigma]\[FormalMu]\[FormalNu]", "\[FormalRho]"], Indeterminate]]]]]]]]]]]]]]]] /;
SymbolName[metricTensor] === "MetricTensor" && Length[Dimensions[matrixRepresentation]] == 2 &&
Length[coordinates] == Length[matrixRepresentation] && BooleanQ[metricIndex1] && BooleanQ[metricIndex2] &&
BooleanQ[index1] && BooleanQ[index2] && BooleanQ[index3] && BooleanQ[index4]
WeylTensor /: MakeBoxes[weylTensor:WeylTensor[(metricTensor_)[matrixRepresentation_List, coordinates_List, metricIndex1_,
metricIndex2_], index1_, index2_, index3_, index4_], format_] :=
Module[{newMatrixRepresentation, newCoordinates, christoffelSymbols, riemannTensor, covariantRiemannTensor,
ricciTensor, ricciScalar, tensorRepresentation, newTensorRepresentation, type, symbol, dimensions, eigenvalues,
positiveEigenvalues, negativeEigenvalues, signature, icon},
newMatrixRepresentation = matrixRepresentation /. (#1 -> ToExpression[#1] & ) /@ Select[coordinates, StringQ];
newCoordinates = coordinates /. (#1 -> ToExpression[#1] & ) /@ Select[coordinates, StringQ];
christoffelSymbols = Normal[SparseArray[
(Module[{index = #1}, index -> Total[((1/2)*Inverse[newMatrixRepresentation][[index[[1]],#1]]*
(D[newMatrixRepresentation[[#1,index[[3]]]], newCoordinates[[index[[2]]]]] + D[newMatrixRepresentation[[
index[[2]],#1]], newCoordinates[[index[[3]]]]] - D[newMatrixRepresentation[[index[[2]],index[[3]]]],
newCoordinates[[#1]]]) & ) /@ Range[Length[newMatrixRepresentation]]]] & ) /@
Tuples[Range[Length[newMatrixRepresentation]], 3]]]; riemannTensor =
Normal[SparseArray[(Module[{index = #1}, index -> D[christoffelSymbols[[index[[1]],index[[2]],index[[4]]]],
newCoordinates[[index[[3]]]]] - D[christoffelSymbols[[index[[1]],index[[2]],index[[3]]]],
newCoordinates[[index[[4]]]]] + Total[(christoffelSymbols[[index[[1]],#1,index[[3]]]]*christoffelSymbols[[
#1,index[[2]],index[[4]]]] & ) /@ Range[Length[newMatrixRepresentation]]] - Total[
(christoffelSymbols[[index[[1]],#1,index[[4]]]]*christoffelSymbols[[#1,index[[2]],index[[3]]]] & ) /@
Range[Length[newMatrixRepresentation]]]] & ) /@ Tuples[Range[Length[newMatrixRepresentation]], 4]]] /.
(ToExpression[#1] -> #1 & ) /@ Select[coordinates, StringQ]; covariantRiemannTensor =
Normal[SparseArray[(Module[{index = #1}, index -> Total[(matrixRepresentation[[index[[1]],#1]]*riemannTensor[[#1,
index[[2]],index[[3]],index[[4]]]] & ) /@ Range[Length[matrixRepresentation]]]] & ) /@
Tuples[Range[Length[matrixRepresentation]], 4]]]; ricciTensor =
Normal[SparseArray[(Module[{index = #1}, index -> Total[(riemannTensor[[#1,First[index],#1,Last[index]]] & ) /@
Range[Length[matrixRepresentation]]]] & ) /@ Tuples[Range[Length[matrixRepresentation]], 2]]];
ricciScalar = Total[(Inverse[matrixRepresentation][[First[#1],Last[#1]]]*ricciTensor[[First[#1],Last[#1]]] & ) /@
Tuples[Range[Length[matrixRepresentation]], 2]]; tensorRepresentation =
Normal[SparseArray[(Module[{index = #1}, index -> covariantRiemannTensor[[index[[1]],index[[2]],index[[3]],index[[
4]]]] + (1/(Length[matrixRepresentation] - 2))*(ricciTensor[[index[[1]],index[[4]]]]*
matrixRepresentation[[index[[2]],index[[3]]]] - ricciTensor[[index[[1]],index[[3]]]]*
matrixRepresentation[[index[[2]],index[[4]]]] + ricciTensor[[index[[2]],index[[3]]]]*
matrixRepresentation[[index[[1]],index[[4]]]] - ricciTensor[[index[[2]],index[[4]]]]*
matrixRepresentation[[index[[1]],index[[3]]]]) + (1/((Length[matrixRepresentation] - 1)*
(Length[matrixRepresentation] - 2)))*(ricciScalar*(matrixRepresentation[[index[[1]],index[[3]]]]*
matrixRepresentation[[index[[2]],index[[4]]]] - matrixRepresentation[[index[[1]],index[[4]]]]*
matrixRepresentation[[index[[2]],index[[3]]]]))] & ) /@ Tuples[Range[Length[matrixRepresentation]],
4]]]; If[index1 === True && index2 === True && index3 === True && index4 === True,
newTensorRepresentation = tensorRepresentation; type = "Covariant"; symbol = Subscript["\[FormalCapitalC]", "\[FormalRho]\[FormalSigma]\[FormalMu]\[FormalNu]"],
If[index1 === False && index2 === False && index3 === False && index4 === False,
newTensorRepresentation = Normal[SparseArray[(Module[{index = #1}, index ->
Total[(Inverse[matrixRepresentation][[index[[1]],#1[[1]]]]*Inverse[matrixRepresentation][[index[[2]],
#1[[2]]]]*Inverse[matrixRepresentation][[#1[[3]],index[[3]]]]*Inverse[matrixRepresentation][[#1[[4]],
index[[4]]]]*tensorRepresentation[[#1[[1]],#1[[2]],#1[[3]],#1[[4]]]] & ) /@
Tuples[Range[Length[matrixRepresentation]], 4]]] & ) /@ Tuples[Range[Length[matrixRepresentation]],
4]]]; type = "Contravariant"; symbol = Superscript["\[FormalCapitalC]", "\[FormalRho]\[FormalSigma]\[FormalMu]\[FormalNu]"],
If[index1 === True && index2 === False && index3 === False && index4 === False,
newTensorRepresentation = Normal[SparseArray[(Module[{index = #1}, index -> Total[
(Inverse[matrixRepresentation][[index[[2]],#1[[1]]]]*Inverse[matrixRepresentation][[#1[[2]],
index[[3]]]]*Inverse[matrixRepresentation][[#1[[3]],index[[4]]]]*tensorRepresentation[[index[[1]],
#1[[1]],#1[[2]],#1[[3]]]] & ) /@ Tuples[Range[Length[matrixRepresentation]], 3]]] & ) /@
Tuples[Range[Length[matrixRepresentation]], 4]]]; type = "Mixed";
symbol = Subsuperscript["\[FormalCapitalC]", "\[FormalRho]", "\[FormalSigma]\[FormalMu]\[FormalNu]"], If[index1 === False && index2 === True &&
index3 === False && index4 === False, newTensorRepresentation =
Normal[SparseArray[(Module[{index = #1}, index -> Total[(Inverse[matrixRepresentation][[index[[1]],#1[[1]]]]*
Inverse[matrixRepresentation][[#1[[2]],index[[3]]]]*Inverse[matrixRepresentation][[#1[[3]],
index[[4]]]]*tensorRepresentation[[#1[[1]],index[[2]],#1[[2]],#1[[3]]]] & ) /@
Tuples[Range[Length[matrixRepresentation]], 3]]] & ) /@ Tuples[Range[Length[matrixRepresentation]],
4]]]; type = "Mixed"; symbol = Subsuperscript["\[FormalCapitalC]", "\[FormalSigma]", "\[FormalRho]\[FormalMu]\[FormalNu]"],
If[index1 === False && index2 === False && index3 === True && index4 === False,
newTensorRepresentation = Normal[SparseArray[(Module[{index = #1}, index -> Total[
(Inverse[matrixRepresentation][[index[[1]],#1[[1]]]]*Inverse[matrixRepresentation][[index[[2]],
#1[[2]]]]*Inverse[matrixRepresentation][[#1[[3]],index[[4]]]]*tensorRepresentation[[#1[[1]],
#1[[2]],index[[3]],#1[[3]]]] & ) /@ Tuples[Range[Length[matrixRepresentation]], 3]]] & ) /@
Tuples[Range[Length[matrixRepresentation]], 4]]]; type = "Mixed";
symbol = Subsuperscript["\[FormalCapitalC]", "\[FormalMu]", "\[FormalRho]\[FormalSigma]\[FormalNu]"], If[index1 === False && index2 === False &&
index3 === False && index4 === True, newTensorRepresentation = Normal[SparseArray[
(Module[{index = #1}, index -> Total[(Inverse[matrixRepresentation][[index[[1]],#1[[1]]]]*
Inverse[matrixRepresentation][[index[[2]],#1[[2]]]]*Inverse[matrixRepresentation][[#1[[3]],
index[[3]]]]*tensorRepresentation[[#1[[1]],#1[[2]],#1[[3]],index[[4]]]] & ) /@
Tuples[Range[Length[matrixRepresentation]], 3]]] & ) /@ Tuples[Range[Length[matrixRepresentation]],
4]]]; type = "Mixed"; symbol = Subsuperscript["\[FormalCapitalC]", "\[FormalNu]", "\[FormalRho]\[FormalSigma]\[FormalMu]"],
If[index1 === True && index2 === True && index3 === False && index4 === False,
newTensorRepresentation = Normal[SparseArray[(Module[{index = #1}, index -> Total[
(Inverse[matrixRepresentation][[#1[[1]],index[[3]]]]*Inverse[matrixRepresentation][[#1[[2]],
index[[4]]]]*tensorRepresentation[[index[[1]],index[[2]],#1[[1]],#1[[2]]]] & ) /@
Tuples[Range[Length[matrixRepresentation]], 2]]] & ) /@ Tuples[Range[Length[
matrixRepresentation]], 4]]]; type = "Mixed"; symbol = Subsuperscript["\[FormalCapitalC]", "\[FormalRho]\[FormalSigma]", "\[FormalMu]\[FormalNu]"],
If[index1 === True && index2 === False && index3 === True && index4 === False, newTensorRepresentation =
Normal[SparseArray[(Module[{index = #1}, index -> Total[(Inverse[matrixRepresentation][[index[[2]],
#1[[1]]]]*Inverse[matrixRepresentation][[#1[[2]],index[[4]]]]*tensorRepresentation[[index[[1]],
#1[[1]],index[[3]],#1[[2]]]] & ) /@ Tuples[Range[Length[matrixRepresentation]], 2]]] & ) /@
Tuples[Range[Length[matrixRepresentation]], 4]]]; type = "Mixed"; symbol = Subsuperscript["\[FormalCapitalC]",
"\[FormalRho]\[FormalMu]", "\[FormalSigma]\[FormalNu]"], If[index1 === True && index2 === False && index3 === False && index4 === True,
newTensorRepresentation = Normal[SparseArray[(Module[{index = #1}, index -> Total[
(Inverse[matrixRepresentation][[index[[2]],#1[[1]]]]*Inverse[matrixRepresentation][[#1[[2]],
index[[3]]]]*tensorRepresentation[[index[[1]],#1[[1]],#1[[2]],index[[4]]]] & ) /@
Tuples[Range[Length[matrixRepresentation]], 2]]] & ) /@ Tuples[Range[Length[
matrixRepresentation]], 4]]]; type = "Mixed"; symbol = Subsuperscript["\[FormalCapitalC]", "\[FormalRho]\[FormalNu]", "\[FormalSigma]\[FormalMu]"],
If[index1 === False && index2 === True && index3 === True && index4 === False,
newTensorRepresentation = Normal[SparseArray[(Module[{index = #1}, index -> Total[
(Inverse[matrixRepresentation][[index[[1]],#1[[1]]]]*Inverse[matrixRepresentation][[#1[[2]],
index[[4]]]]*tensorRepresentation[[#1[[1]],index[[2]],index[[3]],#1[[2]]]] & ) /@
Tuples[Range[Length[matrixRepresentation]], 2]]] & ) /@ Tuples[Range[Length[
matrixRepresentation]], 4]]]; type = "Mixed"; symbol = Subsuperscript["\[FormalCapitalC]", "\[FormalSigma]\[FormalMu]", "\[FormalRho]\[FormalNu]"],
If[index1 === False && index2 === True && index3 === False && index4 === True, newTensorRepresentation =
Normal[SparseArray[(Module[{index = #1}, index -> Total[(Inverse[matrixRepresentation][[index[[1]],
#1[[1]]]]*Inverse[matrixRepresentation][[#1[[2]],index[[3]]]]*tensorRepresentation[[#1[[1]],
index[[2]],#1[[2]],index[[4]]]] & ) /@ Tuples[Range[Length[matrixRepresentation]],
2]]] & ) /@ Tuples[Range[Length[matrixRepresentation]], 4]]]; type = "Mixed";
symbol = Subsuperscript["\[FormalCapitalC]", "\[FormalSigma]\[FormalNu]", "\[FormalRho]\[FormalMu]"], If[index1 === False && index2 === False &&
index3 === True && index4 === True, newTensorRepresentation = Normal[SparseArray[
(Module[{index = #1}, index -> Total[(Inverse[matrixRepresentation][[index[[1]],#1[[1]]]]*
Inverse[matrixRepresentation][[index[[2]],#1[[2]]]]*tensorRepresentation[[#1[[1]],#1[[2]],
index[[3]],index[[4]]]] & ) /@ Tuples[Range[Length[matrixRepresentation]], 2]]] & ) /@
Tuples[Range[Length[matrixRepresentation]], 4]]]; type = "Mixed"; symbol = Subsuperscript["\[FormalCapitalC]",
"\[FormalMu]\[FormalNu]", "\[FormalRho]\[FormalSigma]"], If[index1 === True && index2 === True && index3 === True && index4 === False,
newTensorRepresentation = Normal[SparseArray[(Module[{index = #1}, index -> Total[
(Inverse[matrixRepresentation][[#1,index[[4]]]]*tensorRepresentation[[index[[1]],index[[2]],
index[[3]],#1]] & )/Range[Length[matrixRepresentation]]]] & ) /@ Tuples[Range[
Length[matrixRepresentation]], 4]]]; type = "Mixed"; symbol = Subsuperscript["\[FormalCapitalC]",
"\[FormalRho]\[FormalSigma]\[FormalMu]", "\[FormalNu]"], If[index1 === True && index2 === True && index3 === False && index4 === True,
newTensorRepresentation = Normal[SparseArray[(Module[{index = #1}, index -> Total[
(Inverse[matrixRepresentation][[#1,index[[3]]]]*tensorRepresentation[[index[[1]],index[[2]],
#1,index[[4]]]] & ) /@ Range[Length[matrixRepresentation]]]] & ) /@ Tuples[
Range[Length[matrixRepresentation]], 4]]]; type = "Mixed"; symbol = Subsuperscript["\[FormalCapitalC]",
"\[FormalRho]\[FormalSigma]\[FormalNu]", "\[FormalMu]"], If[index1 === True && index2 === False && index3 === True && index4 === True,
newTensorRepresentation = Normal[SparseArray[(Module[{index = #1}, index -> Total[
(Inverse[matrixRepresentation][[index[[2]],#1]]*tensorRepresentation[[index[[1]],#1,
index[[3]],index[[4]]]] & ) /@ Range[Length[matrixRepresentation]]]] & ) /@
Tuples[Range[Length[matrixRepresentation]], 4]]]; type = "Mixed"; symbol = Subsuperscript[
"\[FormalCapitalC]", "\[FormalRho]\[FormalMu]\[FormalNu]", "\[FormalSigma]"], If[index1 === False && index2 === True && index3 === True &&
index4 === True, newTensorRepresentation = Normal[SparseArray[(Module[{index = #1}, index ->
Total[(Inverse[matrixRepresentation][[index[[1]],#1]]*tensorRepresentation[[#1,index[[2]],
index[[3]],index[[4]]]] & ) /@ Range[Length[matrixRepresentation]]]] & ) /@
Tuples[Range[Length[matrixRepresentation]], 4]]]; type = "Mixed"; symbol = Subsuperscript[
"\[FormalCapitalC]", "\[FormalSigma]\[FormalMu]\[FormalNu]", "\[FormalRho]"], newTensorRepresentation = ConstantArray[Indeterminate,
{Length[matrixRepresentation], Length[matrixRepresentation], Length[matrixRepresentation],
Length[matrixRepresentation]}]; type = Indeterminate; symbol = Indeterminate]]]]]]]]]]]]]]]];
dimensions = Length[matrixRepresentation]; eigenvalues = Eigenvalues[matrixRepresentation];
positiveEigenvalues = Select[eigenvalues, #1 > 0 & ]; negativeEigenvalues = Select[eigenvalues, #1 < 0 & ];
If[Length[positiveEigenvalues] + Length[negativeEigenvalues] == Length[matrixRepresentation],
If[Length[positiveEigenvalues] == Length[matrixRepresentation] || Length[negativeEigenvalues] ==
Length[matrixRepresentation], signature = "Riemannian", If[Length[positiveEigenvalues] == 1 ||
Length[negativeEigenvalues] == 1, signature = "Lorentzian", signature = "Pseudo-Riemannian"]],
signature = Indeterminate]; icon = MatrixPlot[Total[Total[newTensorRepresentation]],
ImageSize -> Dynamic[{Automatic, 3.5*(CurrentValue["FontCapHeight"]/AbsoluteCurrentValue[Magnification])}],
Frame -> False, FrameTicks -> None]; BoxForm`ArrangeSummaryBox["WeylTensor", weylTensor, icon,
{{BoxForm`SummaryItem[{"Type: ", type}], BoxForm`SummaryItem[{"Symbol: ", symbol}]},
{BoxForm`SummaryItem[{"Dimensions: ", dimensions}], BoxForm`SummaryItem[{"Signature: ", signature}]}},
{{BoxForm`SummaryItem[{"Coordinates: ", coordinates}]}}, format, "Interpretable" -> Automatic]] /;
SymbolName[metricTensor] === "MetricTensor" && Length[Dimensions[matrixRepresentation]] == 2 &&
Length[coordinates] == Length[matrixRepresentation] && BooleanQ[metricIndex1] && BooleanQ[metricIndex2] &&
BooleanQ[index1] && BooleanQ[index2] && BooleanQ[index3] && BooleanQ[index4]