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SchoutenTensor.wl
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(* ::Package:: *)
SchoutenTensor[(metricTensor_)[matrixRepresentation_List, coordinates_List, index1_, index2_]] :=
SchoutenTensor[ResourceFunction["MetricTensor"][matrixRepresentation, coordinates, index1, index2], True, True] /;
SymbolName[metricTensor] === "MetricTensor" && Length[Dimensions[matrixRepresentation]] == 2 &&
Length[coordinates] == Length[matrixRepresentation] && BooleanQ[index1] && BooleanQ[index2]
SchoutenTensor[(metricTensor_)[matrixRepresentation_List, coordinates_List, index1_, index2_], newCoordinates_List] :=
SchoutenTensor[ResourceFunction["MetricTensor"][matrixRepresentation /. Thread[coordinates -> newCoordinates],
newCoordinates, index1, index2], True, True] /; SymbolName[metricTensor] === "MetricTensor" &&
Length[Dimensions[matrixRepresentation]] == 2 && Length[coordinates] == Length[matrixRepresentation] &&
Length[newCoordinates] == Length[matrixRepresentation] && BooleanQ[index1] && BooleanQ[index2]
SchoutenTensor[(metricTensor_)[matrixRepresentation_List, coordinates_List, metricIndex1_, metricIndex2_],
newCoordinates_List, index1_, index2_] :=
SchoutenTensor[ResourceFunction["MetricTensor"][matrixRepresentation /. Thread[coordinates -> newCoordinates],
newCoordinates, metricIndex1, metricIndex2], index1, index2] /; SymbolName[metricTensor] === "MetricTensor" &&
Length[Dimensions[matrixRepresentation]] == 2 && Length[coordinates] == Length[matrixRepresentation] &&
Length[newCoordinates] == Length[matrixRepresentation] && BooleanQ[metricIndex1] && BooleanQ[metricIndex2] &&
BooleanQ[index1] && BooleanQ[index2]
SchoutenTensor[(metricTensor_)[matrixRepresentation_List, coordinates_List, metricIndex1_, metricIndex2_], index1_,
index2_]["MatrixRepresentation"] := Module[{newMatrixRepresentation, newCoordinates, christoffelSymbols,
riemannTensor, ricciTensor, ricciScalar, schoutenTensor},
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];
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]]; schoutenTensor = (1/(Length[matrixRepresentation] - 2))*
(ricciTensor - (ricciScalar/(2*(Length[matrixRepresentation] - 1)))*matrixRepresentation);
If[index1 === True && index2 === True, schoutenTensor, If[index1 === False && index2 === False,
Normal[SparseArray[(Module[{index = #1}, index -> Total[(Inverse[matrixRepresentation][[First[index],First[#1]]]*
Inverse[matrixRepresentation][[Last[#1],Last[index]]]*schoutenTensor[[First[#1],Last[#1]]] & ) /@ Tuples[
Range[Length[matrixRepresentation]], 2]]] & ) /@ Tuples[Range[Length[matrixRepresentation]], 2]]],
If[index1 === True && index2 === False, Normal[SparseArray[
(Module[{index = #1}, index -> Total[(Inverse[matrixRepresentation][[#1,Last[index]]]*schoutenTensor[[
First[index],#1]] & ) /@ Range[Length[matrixRepresentation]]]] & ) /@
Tuples[Range[Length[matrixRepresentation]], 2]]], If[index1 === False && index2 === True,
Normal[SparseArray[(Module[{index = #1}, index -> Total[(Inverse[matrixRepresentation][[First[index],#1]]*
schoutenTensor[[#1,Last[index]]] & ) /@ Range[Length[matrixRepresentation]]]] & ) /@
Tuples[Range[Length[matrixRepresentation]], 2]]], Indeterminate]]]]] /;
SymbolName[metricTensor] === "MetricTensor" && Length[Dimensions[matrixRepresentation]] == 2 &&
Length[coordinates] == Length[matrixRepresentation] && BooleanQ[metricIndex1] && BooleanQ[metricIndex2] &&
BooleanQ[index1] && BooleanQ[index2]
SchoutenTensor[(metricTensor_)[matrixRepresentation_List, coordinates_List, metricIndex1_, metricIndex2_], index1_,
index2_]["ReducedMatrixRepresentation"] :=
Module[{newMatrixRepresentation, newCoordinates, christoffelSymbols, riemannTensor, ricciTensor, ricciScalar,
schoutenTensor}, 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];
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]]; schoutenTensor = (1/(Length[matrixRepresentation] - 2))*
(ricciTensor - (ricciScalar/(2*(Length[matrixRepresentation] - 1)))*matrixRepresentation);
If[index1 === True && index2 === True, FullSimplify[schoutenTensor], If[index1 === False && index2 === False,
FullSimplify[Normal[SparseArray[(Module[{index = #1}, index -> Total[(Inverse[matrixRepresentation][[First[index],
First[#1]]]*Inverse[matrixRepresentation][[Last[#1],Last[index]]]*schoutenTensor[[First[#1],
Last[#1]]] & ) /@ Tuples[Range[Length[matrixRepresentation]], 2]]] & ) /@
Tuples[Range[Length[matrixRepresentation]], 2]]]], If[index1 === True && index2 === False,
FullSimplify[Normal[SparseArray[(Module[{index = #1}, index -> Total[(Inverse[matrixRepresentation][[#1,
Last[index]]]*schoutenTensor[[First[index],#1]] & ) /@ Range[Length[matrixRepresentation]]]] & ) /@
Tuples[Range[Length[matrixRepresentation]], 2]]]], If[index1 === False && index2 === True,
FullSimplify[Normal[SparseArray[(Module[{index = #1}, index -> Total[(Inverse[matrixRepresentation][[
First[index],#1]]*schoutenTensor[[#1,Last[index]]] & ) /@ Range[Length[
matrixRepresentation]]]] & ) /@ Tuples[Range[Length[matrixRepresentation]], 2]]]],
Indeterminate]]]]] /; SymbolName[metricTensor] === "MetricTensor" && Length[Dimensions[matrixRepresentation]] ==
2 && Length[coordinates] == Length[matrixRepresentation] && BooleanQ[metricIndex1] && BooleanQ[metricIndex2] &&
BooleanQ[index1] && BooleanQ[index2]
SchoutenTensor[(metricTensor_)[matrixRepresentation_List, coordinates_List, metricIndex1_, metricIndex2_], index1_,
index2_]["SymbolicMatrixRepresentation"] :=
Module[{newMatrixRepresentation, newCoordinates, christoffelSymbols, riemannTensor, ricciTensor, ricciScalar,
schoutenTensor}, 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]; 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]]; schoutenTensor = (1/(Length[matrixRepresentation] - 2))*
(ricciTensor - (ricciScalar/(2*(Length[matrixRepresentation] - 1)))*matrixRepresentation);
If[index1 === True && index2 === True, schoutenTensor, If[index1 === False && index2 === False,
Normal[SparseArray[(Module[{index = #1}, index -> Total[(Inverse[matrixRepresentation][[First[index],First[#1]]]*
Inverse[matrixRepresentation][[Last[#1],Last[index]]]*schoutenTensor[[First[#1],Last[#1]]] & ) /@ Tuples[
Range[Length[matrixRepresentation]], 2]]] & ) /@ Tuples[Range[Length[matrixRepresentation]], 2]]],
If[index1 === True && index2 === False, Normal[SparseArray[
(Module[{index = #1}, index -> Total[(Inverse[matrixRepresentation][[#1,Last[index]]]*schoutenTensor[[
First[index],#1]] & ) /@ Range[Length[matrixRepresentation]]]] & ) /@
Tuples[Range[Length[matrixRepresentation]], 2]]], If[index1 === False && index2 === True,
Normal[SparseArray[(Module[{index = #1}, index -> Total[(Inverse[matrixRepresentation][[First[index],#1]]*
Inverse[matrixRepresentation][[#1,Last[index]]] & ) /@ Range[Length[matrixRepresentation]]]] & ) /@
Tuples[Range[Length[matrixRepresentation]], 2]]], Indeterminate]]]]] /;
SymbolName[metricTensor] === "MetricTensor" && Length[Dimensions[matrixRepresentation]] == 2 &&
Length[coordinates] == Length[matrixRepresentation] && BooleanQ[metricIndex1] && BooleanQ[metricIndex2] &&
BooleanQ[index1] && BooleanQ[index2]
SchoutenTensor[(metricTensor_)[matrixRepresentation_List, coordinates_List, metricIndex1_, metricIndex2_], index1_,
index2_]["Trace"] := Module[{newMatrixRepresentation, newCoordinates, christoffelSymbols, riemannTensor, ricciTensor,
ricciScalar, schoutenTensor}, 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];
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]]; schoutenTensor = (1/(Length[matrixRepresentation] - 2))*
(ricciTensor - (ricciScalar/(2*(Length[matrixRepresentation] - 1)))*matrixRepresentation);
Total[(Inverse[matrixRepresentation][[First[#1],Last[#1]]]*schoutenTensor[[First[#1],Last[#1]]] & ) /@
Tuples[Range[Length[matrixRepresentation]], 2]]] /; SymbolName[metricTensor] === "MetricTensor" &&
Length[Dimensions[matrixRepresentation]] == 2 && Length[coordinates] == Length[matrixRepresentation] &&
BooleanQ[metricIndex1] && BooleanQ[metricIndex2] && BooleanQ[index1] && BooleanQ[index2]
SchoutenTensor[(metricTensor_)[matrixRepresentation_List, coordinates_List, metricIndex1_, metricIndex2_], index1_,
index2_]["ReducedTrace"] := Module[{newMatrixRepresentation, newCoordinates, christoffelSymbols, riemannTensor,
ricciTensor, ricciScalar, schoutenTensor},
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];
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]]; schoutenTensor = (1/(Length[matrixRepresentation] - 2))*
(ricciTensor - (ricciScalar/(2*(Length[matrixRepresentation] - 1)))*matrixRepresentation);
FullSimplify[Total[(Inverse[matrixRepresentation][[First[#1],Last[#1]]]*schoutenTensor[[First[#1],Last[#1]]] & ) /@
Tuples[Range[Length[matrixRepresentation]], 2]]]] /; SymbolName[metricTensor] === "MetricTensor" &&
Length[Dimensions[matrixRepresentation]] == 2 && Length[coordinates] == Length[matrixRepresentation] &&
BooleanQ[metricIndex1] && BooleanQ[metricIndex2] && BooleanQ[index1] && BooleanQ[index2]
SchoutenTensor[(metricTensor_)[matrixRepresentation_List, coordinates_List, metricIndex1_, metricIndex2_], index1_,
index2_]["SymbolicTrace"] := Module[{newMatrixRepresentation, newCoordinates, christoffelSymbols, riemannTensor,
ricciTensor, ricciScalar, schoutenTensor},
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]; 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]]; schoutenTensor = (1/(Length[matrixRepresentation] - 2))*
(ricciTensor - (ricciScalar/(2*(Length[matrixRepresentation] - 1)))*matrixRepresentation);
Total[(Inverse[matrixRepresentation][[First[#1],Last[#1]]]*schoutenTensor[[First[#1],Last[#1]]] & ) /@
Tuples[Range[Length[matrixRepresentation]], 2]]] /; SymbolName[metricTensor] === "MetricTensor" &&
Length[Dimensions[matrixRepresentation]] == 2 && Length[coordinates] == Length[matrixRepresentation] &&
BooleanQ[metricIndex1] && BooleanQ[metricIndex2] && BooleanQ[index1] && BooleanQ[index2]
SchoutenTensor[(metricTensor_)[matrixRepresentation_List, coordinates_List, metricIndex1_, metricIndex2_], index1_,
index2_]["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]
SchoutenTensor[(metricTensor_)[matrixRepresentation_List, coordinates_List, metricIndex1_, metricIndex2_], index1_,
index2_]["Coordinates"] := coordinates /; SymbolName[metricTensor] === "MetricTensor" &&
Length[Dimensions[matrixRepresentation]] == 2 && Length[coordinates] == Length[matrixRepresentation] &&
BooleanQ[metricIndex1] && BooleanQ[metricIndex2] && BooleanQ[index1] && BooleanQ[index2]
SchoutenTensor[(metricTensor_)[matrixRepresentation_List, coordinates_List, metricIndex1_, metricIndex2_], index1_,
index2_]["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]
SchoutenTensor[(metricTensor_)[matrixRepresentation_List, coordinates_List, metricIndex1_, metricIndex2_], index1_,
index2_]["Indices"] := {index1, index2} /; SymbolName[metricTensor] === "MetricTensor" &&
Length[Dimensions[matrixRepresentation]] == 2 && Length[coordinates] == Length[matrixRepresentation] &&
BooleanQ[metricIndex1] && BooleanQ[metricIndex2] && BooleanQ[index1] && BooleanQ[index2]
SchoutenTensor[(metricTensor_)[matrixRepresentation_List, coordinates_List, metricIndex1_, metricIndex2_], index1_,
index2_]["CovariantQ"] := If[index1 === True && index2 === True, True, False] /;
SymbolName[metricTensor] === "MetricTensor" && Length[Dimensions[matrixRepresentation]] == 2 &&
Length[coordinates] == Length[matrixRepresentation] && BooleanQ[metricIndex2] && BooleanQ[metricIndex2] &&
BooleanQ[index1] && BooleanQ[index2]
SchoutenTensor[(metricTensor_)[matrixRepresentation_List, coordinates_List, metricIndex1_, metricIndex2_], index1_,
index2_]["ContravariantQ"] := If[index1 === False && index2 === False, True, False] /;
SymbolName[metricTensor] === "MetricTensor" && Length[Dimensions[matrixRepresentation]] == 2 &&
Length[coordinates] == Length[matrixRepresentation] && BooleanQ[metricIndex1] && BooleanQ[metricIndex2] &&
BooleanQ[index1] && BooleanQ[index2]
SchoutenTensor[(metricTensor_)[matrixRepresentation_List, coordinates_List, metricIndex1_, metricIndex2_], index1_,
index2_]["MixedQ"] := If[(index1 === True && index2 === False) || (index1 === False && index2 === True), True,
False] /; SymbolName[metricTensor] === "MetricTensor" && Length[Dimensions[matrixRepresentation]] == 2 &&
Length[coordinates] == Length[matrixRepresentation] && BooleanQ[metricIndex1] && BooleanQ[metricIndex2] &&
BooleanQ[index1] && BooleanQ[index2]
SchoutenTensor[(metricTensor_)[matrixRepresentation_List, coordinates_List, metricIndex1_, metricIndex2_], index1_,
index2_]["Symbol"] := If[index1 === True && index2 === True, Subscript["\[FormalCapitalP]", "\[FormalMu]\[FormalNu]"],
If[index1 === False && index2 === False, Superscript["\[FormalCapitalP]", "\[FormalMu]\[FormalNu]"], If[index1 === True && index2 === False,
Subsuperscript["\[FormalCapitalP]", "\[FormalMu]", "\[FormalNu]"], If[index1 === False && index2 === True, Subsuperscript["\[FormalCapitalP]", "\[FormalNu]", "\[FormalMu]"],
Indeterminate]]]] /; SymbolName[metricTensor] === "MetricTensor" && Length[Dimensions[matrixRepresentation]] ==
2 && Length[coordinates] == Length[matrixRepresentation] && BooleanQ[metricIndex1] && BooleanQ[metricIndex2] &&
BooleanQ[index1] && BooleanQ[index2]
SchoutenTensor /: MakeBoxes[schoutenTensor:SchoutenTensor[(metricTensor_)[matrixRepresentation_List, coordinates_List,
metricIndex1_, metricIndex2_], index1_, index2_], format_] :=
Module[{newMatrixRepresentation, newCoordinates, christoffelSymbols, riemannTensor, ricciTensor, ricciScalar,
tensorRepresentation, matrixForm, 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];
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 = (1/(Length[matrixRepresentation] - 2))*
(ricciTensor - (ricciScalar/(2*(Length[matrixRepresentation] - 1)))*matrixRepresentation);
If[index1 === True && index2 === True, matrixForm = tensorRepresentation; type = "Covariant";
symbol = Subscript["\[FormalCapitalP]", "\[FormalMu]\[FormalNu]"], If[index1 === False && index2 === False,
matrixForm = Normal[SparseArray[(Module[{index = #1}, index -> Total[(Inverse[matrixRepresentation][[First[index],
First[#1]]]*Inverse[matrixRepresentation][[Last[#1],Last[index]]]*tensorRepresentation[[First[#1],
Last[#1]]] & ) /@ Tuples[Range[Length[matrixRepresentation]], 2]]] & ) /@
Tuples[Range[Length[matrixRepresentation]], 2]]]; type = "Contravariant";
symbol = Superscript["\[FormalCapitalP]", "\[FormalMu]\[FormalNu]"], If[index1 === True && index2 === False,
matrixForm = Normal[SparseArray[(Module[{index = #1}, index -> Total[(Inverse[matrixRepresentation][[#1,
Last[index]]]*tensorRepresentation[[First[index],#1]] & ) /@ Range[Length[
matrixRepresentation]]]] & ) /@ Tuples[Range[Length[matrixRepresentation]], 2]]]; type = "Mixed";
symbol = Subsuperscript["\[FormalCapitalP]", "\[FormalMu]", "\[FormalNu]"], If[index1 === False && index2 === True,
matrixForm = Normal[SparseArray[(Module[{index = #1}, index -> Total[(Inverse[matrixRepresentation][[
First[index],#1]]*tensorRepresentation[[#1,Last[index]]] & ) /@ Range[Length[
matrixRepresentation]]]] & ) /@ Tuples[Range[Length[matrixRepresentation]], 2]]]; type = "Mixed";
symbol = Subsuperscript["\[FormalCapitalP]", "\[FormalNu]", "\[FormalMu]"], matrixForm = ConstantArray[Indeterminate,
{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[matrixForm, ImageSize ->
Dynamic[{Automatic, 3.5*(CurrentValue["FontCapHeight"]/AbsoluteCurrentValue[Magnification])}], Frame -> False,
FrameTicks -> None]; BoxForm`ArrangeSummaryBox["SchoutenTensor", schoutenTensor, 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]