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DiscreteHypersurfaceDecomposition.wl
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(* ::Package:: *)
DiscreteHypersurfaceDecomposition[(metricTensor_)[matrixRepresentation_List, coordinates_List, index1_, index2_]] :=
DiscreteHypersurfaceDecomposition[MetricTensor[matrixRepresentation, coordinates, index1, index2],
{coordinates[[1]], 0, 1}, {coordinates[[2]], -2, 2}, {coordinates[[3]], -2, 2}, 100, 1] /;
SymbolName[metricTensor] === "MetricTensor" && Length[Dimensions[matrixRepresentation]] == 2 &&
Length[coordinates] == Length[matrixRepresentation] && BooleanQ[index1] && BooleanQ[index2]
DiscreteHypersurfaceDecomposition[(metricTensor_)[matrixRepresentation_List, coordinates_List, index1_, index2_],
vertexCount_Integer] := DiscreteHypersurfaceDecomposition[MetricTensor[matrixRepresentation, coordinates, index1,
index2], {coordinates[[1]], 0, 1}, {coordinates[[2]], -2, 2}, {coordinates[[3]], -2, 2}, vertexCount, 1] /;
SymbolName[metricTensor] === "MetricTensor" && Length[Dimensions[matrixRepresentation]] == 2 &&
Length[coordinates] == Length[matrixRepresentation] && BooleanQ[index1] && BooleanQ[index2]
DiscreteHypersurfaceDecomposition[(metricTensor_)[matrixRepresentation_List, coordinates_List, index1_, index2_],
vertexCount_Integer, discretizationScale_] :=
DiscreteHypersurfaceDecomposition[MetricTensor[matrixRepresentation, coordinates, index1, index2],
{coordinates[[1]], 0, 1}, {coordinates[[2]], -2, 2}, {coordinates[[3]], -2, 2}, vertexCount, discretizationScale] /;
SymbolName[metricTensor] === "MetricTensor" && Length[Dimensions[matrixRepresentation]] == 2 &&
Length[coordinates] == Length[matrixRepresentation] && BooleanQ[index1] && BooleanQ[index2]
DiscreteHypersurfaceDecomposition[(metricTensor_)[matrixRepresentation_List, coordinates_List, index1_, index2_],
{timeCoordinate_, initialTime_, finalTime_}] :=
DiscreteHypersurfaceDecomposition[MetricTensor[matrixRepresentation, coordinates, index1, index2],
{timeCoordinate, initialTime, finalTime}, {coordinates[[2]], -2, 2}, {coordinates[[3]], -2, 2}, 100, 1] /;
SymbolName[metricTensor] === "MetricTensor" && Length[Dimensions[matrixRepresentation]] == 2 &&
Length[coordinates] == Length[matrixRepresentation] && BooleanQ[index1] && BooleanQ[index2]
DiscreteHypersurfaceDecomposition[(metricTensor_)[matrixRepresentation_List, coordinates_List, index1_, index2_],
{timeCoordinate_, initialTime_, finalTime_}, vertexCount_Integer] :=
DiscreteHypersurfaceDecomposition[MetricTensor[matrixRepresentation, coordinates, index1, index2],
{timeCoordinate, initialTime, finalTime}, {coordinates[[2]], -2, 2}, {coordinates[[3]], -2, 2}, vertexCount, 1] /;
SymbolName[metricTensor] === "MetricTensor" && Length[Dimensions[matrixRepresentation]] == 2 &&
Length[coordinates] == Length[matrixRepresentation] && BooleanQ[index1] && BooleanQ[index2]
DiscreteHypersurfaceDecomposition[(metricTensor_)[matrixRepresentation_List, coordinates_List, index1_, index2_],
{timeCoordinate_, initialTime_, finalTime_}, vertexCount_Integer, discretizationScale_] :=
DiscreteHypersurfaceDecomposition[MetricTensor[matrixRepresentation, coordinates, index1, index2],
{timeCoordinate, initialTime, finalTime}, {coordinates[[2]], -2, 2}, {coordinates[[3]], -2, 2}, vertexCount,
discretizationScale] /; SymbolName[metricTensor] === "MetricTensor" && Length[Dimensions[matrixRepresentation]] ==
2 && Length[coordinates] == Length[matrixRepresentation] && BooleanQ[index1] && BooleanQ[index2]
DiscreteHypersurfaceDecomposition[(metricTensor_)[matrixRepresentation_List, coordinates_List, index1_, index2_],
{timeCoordinate_, initialTime_, finalTime_}, {coordinate1_, initialCoordinate1_, finalCoordinate1_},
{coordinate2_, initialCoordinate2_, finalCoordinate2_}] :=
DiscreteHypersurfaceDecomposition[MetricTensor[matrixRepresentation, coordinates, index1, index2],
{timeCoordinate, initialTime, finalTime}, {coordinate1, initialCoordinate1, finalCoordinate1},
{coordinate2, initialCoordinate2, finalCoordinate2}, 100, 1] /; SymbolName[metricTensor] === "MetricTensor" &&
Length[Dimensions[matrixRepresentation]] == 2 && Length[coordinates] == Length[matrixRepresentation] &&
BooleanQ[index1] && BooleanQ[index2]
DiscreteHypersurfaceDecomposition[(metricTensor_)[matrixRepresentation_List, coordinates_List, index1_, index2_],
{timeCoordinate_, initialTime_, finalTime_}, {coordinate1_, initialCoordinate1_, finalCoordinate1_},
{coordinate2_, initialCoordinate2_, finalCoordinate2_}, vertexCount_] :=
DiscreteHypersurfaceDecomposition[MetricTensor[matrixRepresentation, coordinates, index1, index2],
{timeCoordinate, initialTime, finalTime}, {coordinate1, initialCoordinate1, finalCoordinate1},
{coordinate2, initialCoordinate2, finalCoordinate2}, vertexCount, 1] /;
SymbolName[metricTensor] === "MetricTensor" && Length[Dimensions[matrixRepresentation]] == 2 &&
Length[coordinates] == Length[matrixRepresentation] && BooleanQ[index1] && BooleanQ[index2]
DiscreteHypersurfaceDecomposition[(metricTensor_)[matrixRepresentation_List, coordinates_List, index1_, index2_],
{timeCoordinate_, initialTime_, finalTime_}, {coordinate1_, initialCoordinate1_, finalCoordinate1_},
{coordinate2_, initialCoordinate2_, finalCoordinate2_}, vertexCount_Integer, discretizationScale_][
"CoordinatizedPolarGraphColored"] := Module[{newMatrixRepresentation, newCoordinates, newTimeCoordinate,
newCoordinate1, newCoordinate2, flatteningRules, christoffelSymbols, riemannTensor, covariantRiemannTensor,
contravariantRiemannTensor, kretschmannScalar, region, points, curvatureRange, edgeList},
newMatrixRepresentation = matrixRepresentation /. (#1 -> ToExpression[#1] & ) /@ Select[coordinates, StringQ];
newCoordinates = coordinates /. (#1 -> ToExpression[#1] & ) /@ Select[coordinates, StringQ];
newTimeCoordinate = timeCoordinate /. (#1 -> ToExpression[#1] & ) /@ Select[coordinates, StringQ];
newCoordinate1 = coordinate1 /. (#1 -> ToExpression[#1] & ) /@ Select[coordinates, StringQ];
newCoordinate2 = coordinate2 /. (#1 -> ToExpression[#1] & ) /@ Select[coordinates, StringQ];
flatteningRules = Thread[Complement[newCoordinates, {newTimeCoordinate, newCoordinate1, newCoordinate2}] -> 0];
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]]];
covariantRiemannTensor = Normal[SparseArray[
(Module[{index = #1}, index -> Total[(newMatrixRepresentation[[index[[1]],#1]]*riemannTensor[[#1,index[[2]],
index[[3]],index[[4]]]] & ) /@ Range[Length[newMatrixRepresentation]]]] & ) /@
Tuples[Range[Length[newMatrixRepresentation]], 4]]]; contravariantRiemannTensor =
Normal[SparseArray[(Module[{index = #1}, index -> Total[(Inverse[newMatrixRepresentation][[index[[2]],#1[[1]]]]*
Inverse[newMatrixRepresentation][[#1[[2]],index[[3]]]]*Inverse[newMatrixRepresentation][[#1[[3]],
index[[4]]]]*riemannTensor[[index[[1]],#1[[1]],#1[[2]],#1[[3]]]] & ) /@ Tuples[Range[
Length[newMatrixRepresentation]], 3]]] & ) /@ Tuples[Range[Length[newMatrixRepresentation]], 4]]];
kretschmannScalar = FullSimplify[Total[(covariantRiemannTensor[[#1[[1]],#1[[2]],#1[[3]],#1[[4]]]]*
contravariantRiemannTensor[[#1[[1]],#1[[2]],#1[[3]],#1[[4]]]] & ) /@
Tuples[Range[Length[newMatrixRepresentation]], 4]] /. flatteningRules];
region = DiscretizeRegion[ImplicitRegion[-(kretschmannScalar /. {newCoordinate1 -> Sqrt[\[FormalX]^2 + \[FormalY]^2],
newCoordinate2 -> ArcTan[\[FormalY]/\[FormalX]]} /. newTimeCoordinate -> finalTime) == \[FormalL], {\[FormalX], \[FormalY], \[FormalL]}],
{{-Max[Abs[initialCoordinate1], Abs[finalCoordinate1]], Max[Abs[initialCoordinate1], Abs[finalCoordinate1]]},
{-Max[Abs[initialCoordinate1], Abs[finalCoordinate1]], Max[Abs[initialCoordinate1], Abs[finalCoordinate1]]},
{-Max[Abs[initialCoordinate1], Abs[finalCoordinate1]], Max[Abs[initialCoordinate1], Abs[finalCoordinate1]]}}];
points = RandomPoint[region, vertexCount]; curvatureRange = Max[Last /@ points] - Min[Last /@ points];
edgeList = Apply[UndirectedEdge, Select[Tuples[points, 2], Norm[First[#1] - Last[#1]] < discretizationScale &&
First[#1] =!= Last[#1] & ], {1}]; SimpleGraph[Graph[points, edgeList,
VertexStyle -> (#1 -> ColorData["LightTemperatureMap", -(Last[#1]/curvatureRange)] & ) /@ points,
EdgeStyle -> (#1 -> ColorData["LightTemperatureMap", -(Last[Last[#1]]/curvatureRange)] & ) /@ edgeList,
VertexCoordinates -> (#1 -> #1 & ) /@ points]]] /; SymbolName[metricTensor] === "MetricTensor" &&
Length[Dimensions[matrixRepresentation]] == 2 && Length[coordinates] == Length[matrixRepresentation] &&
BooleanQ[index1] && BooleanQ[index2]
DiscreteHypersurfaceDecomposition[(metricTensor_)[matrixRepresentation_List, coordinates_List, index1_, index2_],
{timeCoordinate_, initialTime_, finalTime_}, {coordinate1_, initialCoordinate1_, finalCoordinate1_},
{coordinate2_, initialCoordinate2_, finalCoordinate2_}, vertexCount_Integer, discretizationScale_][
"PolarGraphColored"] := Module[{newMatrixRepresentation, newCoordinates, newTimeCoordinate, newCoordinate1,
newCoordinate2, flatteningRules, christoffelSymbols, riemannTensor, covariantRiemannTensor,
contravariantRiemannTensor, kretschmannScalar, region, points, curvatureRange, edgeList},
newMatrixRepresentation = matrixRepresentation /. (#1 -> ToExpression[#1] & ) /@ Select[coordinates, StringQ];
newCoordinates = coordinates /. (#1 -> ToExpression[#1] & ) /@ Select[coordinates, StringQ];
newTimeCoordinate = timeCoordinate /. (#1 -> ToExpression[#1] & ) /@ Select[coordinates, StringQ];
newCoordinate1 = coordinate1 /. (#1 -> ToExpression[#1] & ) /@ Select[coordinates, StringQ];
newCoordinate2 = coordinate2 /. (#1 -> ToExpression[#1] & ) /@ Select[coordinates, StringQ];
flatteningRules = Thread[Complement[newCoordinates, {newTimeCoordinate, newCoordinate1, newCoordinate2}] -> 0];
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]]];
covariantRiemannTensor = Normal[SparseArray[
(Module[{index = #1}, index -> Total[(newMatrixRepresentation[[index[[1]],#1]]*riemannTensor[[#1,index[[2]],
index[[3]],index[[4]]]] & ) /@ Range[Length[newMatrixRepresentation]]]] & ) /@
Tuples[Range[Length[newMatrixRepresentation]], 4]]]; contravariantRiemannTensor =
Normal[SparseArray[(Module[{index = #1}, index -> Total[(Inverse[newMatrixRepresentation][[index[[2]],#1[[1]]]]*
Inverse[newMatrixRepresentation][[#1[[2]],index[[3]]]]*Inverse[newMatrixRepresentation][[#1[[3]],
index[[4]]]]*riemannTensor[[index[[1]],#1[[1]],#1[[2]],#1[[3]]]] & ) /@ Tuples[Range[
Length[newMatrixRepresentation]], 3]]] & ) /@ Tuples[Range[Length[newMatrixRepresentation]], 4]]];
kretschmannScalar = FullSimplify[Total[(covariantRiemannTensor[[#1[[1]],#1[[2]],#1[[3]],#1[[4]]]]*
contravariantRiemannTensor[[#1[[1]],#1[[2]],#1[[3]],#1[[4]]]] & ) /@
Tuples[Range[Length[newMatrixRepresentation]], 4]] /. flatteningRules];
region = DiscretizeRegion[ImplicitRegion[-(kretschmannScalar /. {newCoordinate1 -> Sqrt[\[FormalX]^2 + \[FormalY]^2],
newCoordinate2 -> ArcTan[\[FormalY]/\[FormalX]]} /. newTimeCoordinate -> finalTime) == \[FormalL], {\[FormalX], \[FormalY], \[FormalL]}],
{{-Max[Abs[initialCoordinate1], Abs[finalCoordinate1]], Max[Abs[initialCoordinate1], Abs[finalCoordinate1]]},
{-Max[Abs[initialCoordinate1], Abs[finalCoordinate1]], Max[Abs[initialCoordinate1], Abs[finalCoordinate1]]},
{-Max[Abs[initialCoordinate1], Abs[finalCoordinate1]], Max[Abs[initialCoordinate1], Abs[finalCoordinate1]]}}];
points = RandomPoint[region, vertexCount]; curvatureRange = Max[Last /@ points] - Min[Last /@ points];
edgeList = Apply[UndirectedEdge, Select[Tuples[points, 2], Norm[First[#1] - Last[#1]] < discretizationScale &&
First[#1] =!= Last[#1] & ], {1}]; SimpleGraph[Graph[points, edgeList,
VertexStyle -> (#1 -> ColorData["LightTemperatureMap", -(Last[#1]/curvatureRange)] & ) /@ points,
EdgeStyle -> (#1 -> ColorData["LightTemperatureMap", -(Last[Last[#1]]/curvatureRange)] & ) /@ edgeList]]] /;
SymbolName[metricTensor] === "MetricTensor" && Length[Dimensions[matrixRepresentation]] == 2 &&
Length[coordinates] == Length[matrixRepresentation] && BooleanQ[index1] && BooleanQ[index2]
DiscreteHypersurfaceDecomposition[(metricTensor_)[matrixRepresentation_List, coordinates_List, index1_, index2_],
{timeCoordinate_, initialTime_, finalTime_}, {coordinate1_, initialCoordinate1_, finalCoordinate1_},
{coordinate2_, initialCoordinate2_, finalCoordinate2_}, vertexCount_Integer, discretizationScale_][
"CoordinatizedPolarGraph"] := Module[{newMatrixRepresentation, newCoordinates, newTimeCoordinate, newCoordinate1,
newCoordinate2, flatteningRules, christoffelSymbols, riemannTensor, covariantRiemannTensor,
contravariantRiemannTensor, kretschmannScalar, region, points, edgeList},
newMatrixRepresentation = matrixRepresentation /. (#1 -> ToExpression[#1] & ) /@ Select[coordinates, StringQ];
newCoordinates = coordinates /. (#1 -> ToExpression[#1] & ) /@ Select[coordinates, StringQ];
newTimeCoordinate = timeCoordinate /. (#1 -> ToExpression[#1] & ) /@ Select[coordinates, StringQ];
newCoordinate1 = coordinate1 /. (#1 -> ToExpression[#1] & ) /@ Select[coordinates, StringQ];
newCoordinate2 = coordinate2 /. (#1 -> ToExpression[#1] & ) /@ Select[coordinates, StringQ];
flatteningRules = Thread[Complement[newCoordinates, {newTimeCoordinate, newCoordinate1, newCoordinate2}] -> 0];
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]]];
covariantRiemannTensor = Normal[SparseArray[
(Module[{index = #1}, index -> Total[(newMatrixRepresentation[[index[[1]],#1]]*riemannTensor[[#1,index[[2]],
index[[3]],index[[4]]]] & ) /@ Range[Length[newMatrixRepresentation]]]] & ) /@
Tuples[Range[Length[newMatrixRepresentation]], 4]]]; contravariantRiemannTensor =
Normal[SparseArray[(Module[{index = #1}, index -> Total[(Inverse[newMatrixRepresentation][[index[[2]],#1[[1]]]]*
Inverse[newMatrixRepresentation][[#1[[2]],index[[3]]]]*Inverse[newMatrixRepresentation][[#1[[3]],
index[[4]]]]*riemannTensor[[index[[1]],#1[[1]],#1[[2]],#1[[3]]]] & ) /@ Tuples[Range[
Length[newMatrixRepresentation]], 3]]] & ) /@ Tuples[Range[Length[newMatrixRepresentation]], 4]]];
kretschmannScalar = FullSimplify[Total[(covariantRiemannTensor[[#1[[1]],#1[[2]],#1[[3]],#1[[4]]]]*
contravariantRiemannTensor[[#1[[1]],#1[[2]],#1[[3]],#1[[4]]]] & ) /@
Tuples[Range[Length[newMatrixRepresentation]], 4]] /. flatteningRules];
region = DiscretizeRegion[ImplicitRegion[-(kretschmannScalar /. {newCoordinate1 -> Sqrt[\[FormalX]^2 + \[FormalY]^2],
newCoordinate2 -> ArcTan[\[FormalY]/\[FormalX]]} /. newTimeCoordinate -> finalTime) == \[FormalL], {\[FormalX], \[FormalY], \[FormalL]}],
{{-Max[Abs[initialCoordinate1], Abs[finalCoordinate1]], Max[Abs[initialCoordinate1], Abs[finalCoordinate1]]},
{-Max[Abs[initialCoordinate1], Abs[finalCoordinate1]], Max[Abs[initialCoordinate1], Abs[finalCoordinate1]]},
{-Max[Abs[initialCoordinate1], Abs[finalCoordinate1]], Max[Abs[initialCoordinate1], Abs[finalCoordinate1]]}}];
points = RandomPoint[region, vertexCount]; edgeList = Apply[UndirectedEdge, Select[Tuples[points, 2],
Norm[First[#1] - Last[#1]] < discretizationScale && First[#1] =!= Last[#1] & ], {1}];
SimpleGraph[Graph[points, edgeList, VertexCoordinates -> (#1 -> #1 & ) /@ points]]] /;
SymbolName[metricTensor] === "MetricTensor" && Length[Dimensions[matrixRepresentation]] == 2 &&
Length[coordinates] == Length[matrixRepresentation] && BooleanQ[index1] && BooleanQ[index2]
DiscreteHypersurfaceDecomposition[(metricTensor_)[matrixRepresentation_List, coordinates_List, index1_, index2_],
{timeCoordinate_, initialTime_, finalTime_}, {coordinate1_, initialCoordinate1_, finalCoordinate1_},
{coordinate2_, initialCoordinate2_, finalCoordinate2_}, vertexCount_Integer, discretizationScale_]["PolarGraph"] :=
Module[{newMatrixRepresentation, newCoordinates, newTimeCoordinate, newCoordinate1, newCoordinate2, flatteningRules,
christoffelSymbols, riemannTensor, covariantRiemannTensor, contravariantRiemannTensor, kretschmannScalar, region,
points, edgeList}, newMatrixRepresentation = matrixRepresentation /. (#1 -> ToExpression[#1] & ) /@
Select[coordinates, StringQ]; newCoordinates = coordinates /. (#1 -> ToExpression[#1] & ) /@
Select[coordinates, StringQ]; newTimeCoordinate = timeCoordinate /. (#1 -> ToExpression[#1] & ) /@
Select[coordinates, StringQ]; newCoordinate1 = coordinate1 /. (#1 -> ToExpression[#1] & ) /@
Select[coordinates, StringQ]; newCoordinate2 = coordinate2 /. (#1 -> ToExpression[#1] & ) /@
Select[coordinates, StringQ]; flatteningRules =
Thread[Complement[newCoordinates, {newTimeCoordinate, newCoordinate1, newCoordinate2}] -> 0];
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]]];
covariantRiemannTensor = Normal[SparseArray[
(Module[{index = #1}, index -> Total[(newMatrixRepresentation[[index[[1]],#1]]*riemannTensor[[#1,index[[2]],
index[[3]],index[[4]]]] & ) /@ Range[Length[newMatrixRepresentation]]]] & ) /@
Tuples[Range[Length[newMatrixRepresentation]], 4]]]; contravariantRiemannTensor =
Normal[SparseArray[(Module[{index = #1}, index -> Total[(Inverse[newMatrixRepresentation][[index[[2]],#1[[1]]]]*
Inverse[newMatrixRepresentation][[#1[[2]],index[[3]]]]*Inverse[newMatrixRepresentation][[#1[[3]],
index[[4]]]]*riemannTensor[[index[[1]],#1[[1]],#1[[2]],#1[[3]]]] & ) /@ Tuples[Range[
Length[newMatrixRepresentation]], 3]]] & ) /@ Tuples[Range[Length[newMatrixRepresentation]], 4]]];
kretschmannScalar = FullSimplify[Total[(covariantRiemannTensor[[#1[[1]],#1[[2]],#1[[3]],#1[[4]]]]*
contravariantRiemannTensor[[#1[[1]],#1[[2]],#1[[3]],#1[[4]]]] & ) /@
Tuples[Range[Length[newMatrixRepresentation]], 4]] /. flatteningRules];
region = DiscretizeRegion[ImplicitRegion[-(kretschmannScalar /. {newCoordinate1 -> Sqrt[\[FormalX]^2 + \[FormalY]^2],
newCoordinate2 -> ArcTan[\[FormalY]/\[FormalX]]} /. newTimeCoordinate -> finalTime) == \[FormalL], {\[FormalX], \[FormalY], \[FormalL]}],
{{-Max[Abs[initialCoordinate1], Abs[finalCoordinate1]], Max[Abs[initialCoordinate1], Abs[finalCoordinate1]]},
{-Max[Abs[initialCoordinate1], Abs[finalCoordinate1]], Max[Abs[initialCoordinate1], Abs[finalCoordinate1]]},
{-Max[Abs[initialCoordinate1], Abs[finalCoordinate1]], Max[Abs[initialCoordinate1], Abs[finalCoordinate1]]}}];
points = RandomPoint[region, vertexCount]; edgeList = Apply[UndirectedEdge, Select[Tuples[points, 2],
Norm[First[#1] - Last[#1]] < discretizationScale && First[#1] =!= Last[#1] & ], {1}];
SimpleGraph[Graph[points, edgeList]]] /; SymbolName[metricTensor] === "MetricTensor" &&
Length[Dimensions[matrixRepresentation]] == 2 && Length[coordinates] == Length[matrixRepresentation] &&
BooleanQ[index1] && BooleanQ[index2]
DiscreteHypersurfaceDecomposition[(metricTensor_)[matrixRepresentation_List, coordinates_List, index1_, index2_],
{timeCoordinate_, initialTime_, finalTime_}, {coordinate1_, initialCoordinate1_, finalCoordinate1_},
{coordinate2_, initialCoordinate2_, finalCoordinate2_}, vertexCount_Integer, discretizationScale_][
"CoordinatizedPolarGraphEvolutionColored"] :=
Module[{newMatrixRepresentation, newCoordinates, newTimeCoordinate, newCoordinate1, newCoordinate2, flatteningRules,
christoffelSymbols, riemannTensor, covariantRiemannTensor, contravariantRiemannTensor, kretschmannScalar, regions,
points, curvatureRanges, edgeLists}, newMatrixRepresentation = matrixRepresentation /.
(#1 -> ToExpression[#1] & ) /@ Select[coordinates, StringQ];
newCoordinates = coordinates /. (#1 -> ToExpression[#1] & ) /@ Select[coordinates, StringQ];
newTimeCoordinate = timeCoordinate /. (#1 -> ToExpression[#1] & ) /@ Select[coordinates, StringQ];
newCoordinate1 = coordinate1 /. (#1 -> ToExpression[#1] & ) /@ Select[coordinates, StringQ];
newCoordinate2 = coordinate2 /. (#1 -> ToExpression[#1] & ) /@ Select[coordinates, StringQ];
flatteningRules = Thread[Complement[newCoordinates, {newTimeCoordinate, newCoordinate1, newCoordinate2}] -> 0];
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]]];
covariantRiemannTensor = Normal[SparseArray[
(Module[{index = #1}, index -> Total[(newMatrixRepresentation[[index[[1]],#1]]*riemannTensor[[#1,index[[2]],
index[[3]],index[[4]]]] & ) /@ Range[Length[newMatrixRepresentation]]]] & ) /@
Tuples[Range[Length[newMatrixRepresentation]], 4]]]; contravariantRiemannTensor =
Normal[SparseArray[(Module[{index = #1}, index -> Total[(Inverse[newMatrixRepresentation][[index[[2]],#1[[1]]]]*
Inverse[newMatrixRepresentation][[#1[[2]],index[[3]]]]*Inverse[newMatrixRepresentation][[#1[[3]],
index[[4]]]]*riemannTensor[[index[[1]],#1[[1]],#1[[2]],#1[[3]]]] & ) /@ Tuples[Range[
Length[newMatrixRepresentation]], 3]]] & ) /@ Tuples[Range[Length[newMatrixRepresentation]], 4]]];
kretschmannScalar = FullSimplify[Total[(covariantRiemannTensor[[#1[[1]],#1[[2]],#1[[3]],#1[[4]]]]*
contravariantRiemannTensor[[#1[[1]],#1[[2]],#1[[3]],#1[[4]]]] & ) /@
Tuples[Range[Length[newMatrixRepresentation]], 4]] /. flatteningRules];
regions = (DiscretizeRegion[ImplicitRegion[-(kretschmannScalar /. {newCoordinate1 -> Sqrt[\[FormalX]^2 + \[FormalY]^2],
newCoordinate2 -> ArcTan[\[FormalY]/\[FormalX]]} /. newTimeCoordinate -> #1) == \[FormalL], {\[FormalX], \[FormalY], \[FormalL]}],
{{-Max[Abs[initialCoordinate1], Abs[finalCoordinate1]], Max[Abs[initialCoordinate1], Abs[finalCoordinate1]]},
{-Max[Abs[initialCoordinate1], Abs[finalCoordinate1]], Max[Abs[initialCoordinate1], Abs[finalCoordinate1]]},
{-Max[Abs[initialCoordinate1], Abs[finalCoordinate1]], Max[Abs[initialCoordinate1],
Abs[finalCoordinate1]]}}] & ) /@ Range[initialTime, finalTime, (finalTime - initialTime)/10];
points = (RandomPoint[#1, vertexCount] & ) /@ regions; curvatureRanges = (Max[Last /@ #1] - Min[Last /@ #1] & ) /@
points; edgeLists = (Module[{point = #1}, Apply[UndirectedEdge, Select[Tuples[point, 2],
Norm[First[#1] - Last[#1]] < discretizationScale && First[#1] =!= Last[#1] & ], {1}]] & ) /@ points;
(Module[{index = #1}, SimpleGraph[Graph[points[[index]], edgeLists[[index]],
VertexStyle -> (#1 -> ColorData["LightTemperatureMap", -(Last[#1]/curvatureRanges[[index]])] & ) /@
points[[index]], EdgeStyle -> (#1 -> ColorData["LightTemperatureMap", -(Last[Last[#1]]/curvatureRanges[[
index]])] & ) /@ edgeLists[[index]], VertexCoordinates -> (#1 -> #1 & ) /@ points[[index]]]]] & ) /@
Range[Length[points]]] /; SymbolName[metricTensor] === "MetricTensor" &&
Length[Dimensions[matrixRepresentation]] == 2 && Length[coordinates] == Length[matrixRepresentation] &&
BooleanQ[index1] && BooleanQ[index2]
DiscreteHypersurfaceDecomposition[(metricTensor_)[matrixRepresentation_List, coordinates_List, index1_, index2_],
{timeCoordinate_, initialTime_, finalTime_}, {coordinate1_, initialCoordinate1_, finalCoordinate1_},
{coordinate2_, initialCoordinate2_, finalCoordinate2_}, vertexCount_Integer, discretizationScale_][
{"CoordinatizedPolarGraphEvolutionColored", stepCount_Integer}] :=
Module[{newMatrixRepresentation, newCoordinates, newTimeCoordinate, newCoordinate1, newCoordinate2, flatteningRules,
christoffelSymbols, riemannTensor, covariantRiemannTensor, contravariantRiemannTensor, kretschmannScalar, regions,
points, curvatureRanges, edgeLists}, newMatrixRepresentation = matrixRepresentation /.
(#1 -> ToExpression[#1] & ) /@ Select[coordinates, StringQ];
newCoordinates = coordinates /. (#1 -> ToExpression[#1] & ) /@ Select[coordinates, StringQ];
newTimeCoordinate = timeCoordinate /. (#1 -> ToExpression[#1] & ) /@ Select[coordinates, StringQ];
newCoordinate1 = coordinate1 /. (#1 -> ToExpression[#1] & ) /@ Select[coordinates, StringQ];
newCoordinate2 = coordinate2 /. (#1 -> ToExpression[#1] & ) /@ Select[coordinates, StringQ];
flatteningRules = Thread[Complement[newCoordinates, {newTimeCoordinate, newCoordinate1, newCoordinate2}] -> 0];
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]]];
covariantRiemannTensor = Normal[SparseArray[
(Module[{index = #1}, index -> Total[(newMatrixRepresentation[[index[[1]],#1]]*riemannTensor[[#1,index[[2]],
index[[3]],index[[4]]]] & ) /@ Range[Length[newMatrixRepresentation]]]] & ) /@
Tuples[Range[Length[newMatrixRepresentation]], 4]]]; contravariantRiemannTensor =
Normal[SparseArray[(Module[{index = #1}, index -> Total[(Inverse[newMatrixRepresentation][[index[[2]],#1[[1]]]]*
Inverse[newMatrixRepresentation][[#1[[2]],index[[3]]]]*Inverse[newMatrixRepresentation][[#1[[3]],
index[[4]]]]*riemannTensor[[index[[1]],#1[[1]],#1[[2]],#1[[3]]]] & ) /@ Tuples[Range[
Length[newMatrixRepresentation]], 3]]] & ) /@ Tuples[Range[Length[newMatrixRepresentation]], 4]]];
kretschmannScalar = FullSimplify[Total[(covariantRiemannTensor[[#1[[1]],#1[[2]],#1[[3]],#1[[4]]]]*
contravariantRiemannTensor[[#1[[1]],#1[[2]],#1[[3]],#1[[4]]]] & ) /@
Tuples[Range[Length[newMatrixRepresentation]], 4]] /. flatteningRules];
regions = (DiscretizeRegion[ImplicitRegion[-(kretschmannScalar /. {newCoordinate1 -> Sqrt[\[FormalX]^2 + \[FormalY]^2],
newCoordinate2 -> ArcTan[\[FormalY]/\[FormalX]]} /. newTimeCoordinate -> #1) == \[FormalL], {\[FormalX], \[FormalY], \[FormalL]}],
{{-Max[Abs[initialCoordinate1], Abs[finalCoordinate1]], Max[Abs[initialCoordinate1], Abs[finalCoordinate1]]},
{-Max[Abs[initialCoordinate1], Abs[finalCoordinate1]], Max[Abs[initialCoordinate1], Abs[finalCoordinate1]]},
{-Max[Abs[initialCoordinate1], Abs[finalCoordinate1]], Max[Abs[initialCoordinate1],
Abs[finalCoordinate1]]}}] & ) /@ Range[initialTime, finalTime, (finalTime - initialTime)/stepCount];
points = (RandomPoint[#1, vertexCount] & ) /@ regions; curvatureRanges = (Max[Last /@ #1] - Min[Last /@ #1] & ) /@
points; edgeLists = (Module[{point = #1}, Apply[UndirectedEdge, Select[Tuples[point, 2],
Norm[First[#1] - Last[#1]] < discretizationScale && First[#1] =!= Last[#1] & ], {1}]] & ) /@ points;
(Module[{index = #1}, SimpleGraph[Graph[points[[index]], edgeLists[[index]],
VertexStyle -> (#1 -> ColorData["LightTemperatureMap", -(Last[#1]/curvatureRanges[[index]])] & ) /@
points[[index]], EdgeStyle -> (#1 -> ColorData["LightTemperatureMap", -(Last[Last[#1]]/curvatureRanges[[
index]])] & ) /@ edgeLists[[index]], VertexCoordinates -> (#1 -> #1 & ) /@ points[[index]]]]] & ) /@
Range[Length[points]]] /; SymbolName[metricTensor] === "MetricTensor" &&
Length[Dimensions[matrixRepresentation]] == 2 && Length[coordinates] == Length[matrixRepresentation] &&
BooleanQ[index1] && BooleanQ[index2]
DiscreteHypersurfaceDecomposition[(metricTensor_)[matrixRepresentation_List, coordinates_List, index1_, index2_],
{timeCoordinate_, initialTime_, finalTime_}, {coordinate1_, initialCoordinate1_, finalCoordinate1_},
{coordinate2_, initialCoordinate2_, finalCoordinate2_}, vertexCount_Integer, discretizationScale_][
"PolarGraphEvolutionColored"] := Module[{newMatrixRepresentation, newCoordinates, newTimeCoordinate, newCoordinate1,
newCoordinate2, flatteningRules, christoffelSymbols, riemannTensor, covariantRiemannTensor,
contravariantRiemannTensor, kretschmannScalar, regions, points, curvatureRanges, edgeLists},
newMatrixRepresentation = matrixRepresentation /. (#1 -> ToExpression[#1] & ) /@ Select[coordinates, StringQ];
newCoordinates = coordinates /. (#1 -> ToExpression[#1] & ) /@ Select[coordinates, StringQ];
newTimeCoordinate = timeCoordinate /. (#1 -> ToExpression[#1] & ) /@ Select[coordinates, StringQ];
newCoordinate1 = coordinate1 /. (#1 -> ToExpression[#1] & ) /@ Select[coordinates, StringQ];
newCoordinate2 = coordinate2 /. (#1 -> ToExpression[#1] & ) /@ Select[coordinates, StringQ];
flatteningRules = Thread[Complement[newCoordinates, {newTimeCoordinate, newCoordinate1, newCoordinate2}] -> 0];
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]]];
covariantRiemannTensor = Normal[SparseArray[
(Module[{index = #1}, index -> Total[(newMatrixRepresentation[[index[[1]],#1]]*riemannTensor[[#1,index[[2]],
index[[3]],index[[4]]]] & ) /@ Range[Length[newMatrixRepresentation]]]] & ) /@
Tuples[Range[Length[newMatrixRepresentation]], 4]]]; contravariantRiemannTensor =
Normal[SparseArray[(Module[{index = #1}, index -> Total[(Inverse[newMatrixRepresentation][[index[[2]],#1[[1]]]]*
Inverse[newMatrixRepresentation][[#1[[2]],index[[3]]]]*Inverse[newMatrixRepresentation][[#1[[3]],
index[[4]]]]*riemannTensor[[index[[1]],#1[[1]],#1[[2]],#1[[3]]]] & ) /@ Tuples[Range[
Length[newMatrixRepresentation]], 3]]] & ) /@ Tuples[Range[Length[newMatrixRepresentation]], 4]]];
kretschmannScalar = FullSimplify[Total[(covariantRiemannTensor[[#1[[1]],#1[[2]],#1[[3]],#1[[4]]]]*
contravariantRiemannTensor[[#1[[1]],#1[[2]],#1[[3]],#1[[4]]]] & ) /@
Tuples[Range[Length[newMatrixRepresentation]], 4]] /. flatteningRules];
regions = (DiscretizeRegion[ImplicitRegion[-(kretschmannScalar /. {newCoordinate1 -> Sqrt[\[FormalX]^2 + \[FormalY]^2],
newCoordinate2 -> ArcTan[\[FormalY]/\[FormalX]]} /. newTimeCoordinate -> #1) == \[FormalL], {\[FormalX], \[FormalY], \[FormalL]}],
{{-Max[Abs[initialCoordinate1], Abs[finalCoordinate1]], Max[Abs[initialCoordinate1], Abs[finalCoordinate1]]},
{-Max[Abs[initialCoordinate1], Abs[finalCoordinate1]], Max[Abs[initialCoordinate1], Abs[finalCoordinate1]]},
{-Max[Abs[initialCoordinate1], Abs[finalCoordinate1]], Max[Abs[initialCoordinate1],
Abs[finalCoordinate1]]}}] & ) /@ Range[initialTime, finalTime, (finalTime - initialTime)/10];
points = (RandomPoint[#1, vertexCount] & ) /@ regions; curvatureRanges = (Max[Last /@ #1] - Min[Last /@ #1] & ) /@
points; edgeLists = (Module[{point = #1}, Apply[UndirectedEdge, Select[Tuples[point, 2],
Norm[First[#1] - Last[#1]] < discretizationScale && First[#1] =!= Last[#1] & ], {1}]] & ) /@ points;
(Module[{index = #1}, SimpleGraph[Graph[points[[index]], edgeLists[[index]],
VertexStyle -> (#1 -> ColorData["LightTemperatureMap", -(Last[#1]/curvatureRanges[[index]])] & ) /@
points[[index]], EdgeStyle -> (#1 -> ColorData["LightTemperatureMap", -(Last[Last[#1]]/curvatureRanges[[
index]])] & ) /@ edgeLists[[index]]]]] & ) /@ Range[Length[points]]] /;
SymbolName[metricTensor] === "MetricTensor" && Length[Dimensions[matrixRepresentation]] == 2 &&
Length[coordinates] == Length[matrixRepresentation] && BooleanQ[index1] && BooleanQ[index2]
DiscreteHypersurfaceDecomposition[(metricTensor_)[matrixRepresentation_List, coordinates_List, index1_, index2_],
{timeCoordinate_, initialTime_, finalTime_}, {coordinate1_, initialCoordinate1_, finalCoordinate1_},
{coordinate2_, initialCoordinate2_, finalCoordinate2_}, vertexCount_Integer, discretizationScale_][
{"PolarGraphEvolutionColored", stepCount_Integer}] :=
Module[{newMatrixRepresentation, newCoordinates, newTimeCoordinate, newCoordinate1, newCoordinate2, flatteningRules,
christoffelSymbols, riemannTensor, covariantRiemannTensor, contravariantRiemannTensor, kretschmannScalar, regions,
points, curvatureRanges, edgeLists}, newMatrixRepresentation = matrixRepresentation /.
(#1 -> ToExpression[#1] & ) /@ Select[coordinates, StringQ];
newCoordinates = coordinates /. (#1 -> ToExpression[#1] & ) /@ Select[coordinates, StringQ];
newTimeCoordinate = timeCoordinate /. (#1 -> ToExpression[#1] & ) /@ Select[coordinates, StringQ];
newCoordinate1 = coordinate1 /. (#1 -> ToExpression[#1] & ) /@ Select[coordinates, StringQ];
newCoordinate2 = coordinate2 /. (#1 -> ToExpression[#1] & ) /@ Select[coordinates, StringQ];
flatteningRules = Thread[Complement[newCoordinates, {newTimeCoordinate, newCoordinate1, newCoordinate2}] -> 0];
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]]];
covariantRiemannTensor = Normal[SparseArray[
(Module[{index = #1}, index -> Total[(newMatrixRepresentation[[index[[1]],#1]]*riemannTensor[[#1,index[[2]],
index[[3]],index[[4]]]] & ) /@ Range[Length[newMatrixRepresentation]]]] & ) /@
Tuples[Range[Length[newMatrixRepresentation]], 4]]]; contravariantRiemannTensor =
Normal[SparseArray[(Module[{index = #1}, index -> Total[(Inverse[newMatrixRepresentation][[index[[2]],#1[[1]]]]*
Inverse[newMatrixRepresentation][[#1[[2]],index[[3]]]]*Inverse[newMatrixRepresentation][[#1[[3]],
index[[4]]]]*riemannTensor[[index[[1]],#1[[1]],#1[[2]],#1[[3]]]] & ) /@ Tuples[Range[
Length[newMatrixRepresentation]], 3]]] & ) /@ Tuples[Range[Length[newMatrixRepresentation]], 4]]];
kretschmannScalar = FullSimplify[Total[(covariantRiemannTensor[[#1[[1]],#1[[2]],#1[[3]],#1[[4]]]]*
contravariantRiemannTensor[[#1[[1]],#1[[2]],#1[[3]],#1[[4]]]] & ) /@
Tuples[Range[Length[newMatrixRepresentation]], 4]] /. flatteningRules];
regions = (DiscretizeRegion[ImplicitRegion[-(kretschmannScalar /. {newCoordinate1 -> Sqrt[\[FormalX]^2 + \[FormalY]^2],
newCoordinate2 -> ArcTan[\[FormalY]/\[FormalX]]} /. newTimeCoordinate -> #1) == \[FormalL], {\[FormalX], \[FormalY], \[FormalL]}],
{{-Max[Abs[initialCoordinate1], Abs[finalCoordinate1]], Max[Abs[initialCoordinate1], Abs[finalCoordinate1]]},
{-Max[Abs[initialCoordinate1], Abs[finalCoordinate1]], Max[Abs[initialCoordinate1], Abs[finalCoordinate1]]},
{-Max[Abs[initialCoordinate1], Abs[finalCoordinate1]], Max[Abs[initialCoordinate1],
Abs[finalCoordinate1]]}}] & ) /@ Range[initialTime, finalTime, (finalTime - initialTime)/stepCount];
points = (RandomPoint[#1, vertexCount] & ) /@ regions; curvatureRanges = (Max[Last /@ #1] - Min[Last /@ #1] & ) /@
points; edgeLists = (Module[{point = #1}, Apply[UndirectedEdge, Select[Tuples[point, 2],
Norm[First[#1] - Last[#1]] < discretizationScale && First[#1] =!= Last[#1] & ], {1}]] & ) /@ points;
(Module[{index = #1}, SimpleGraph[Graph[points[[index]], edgeLists[[index]],
VertexStyle -> (#1 -> ColorData["LightTemperatureMap", -(Last[#1]/curvatureRanges[[index]])] & ) /@
points[[index]], EdgeStyle -> (#1 -> ColorData["LightTemperatureMap", -(Last[Last[#1]]/curvatureRanges[[
index]])] & ) /@ edgeLists[[index]]]]] & ) /@ Range[Length[points]]] /;
SymbolName[metricTensor] === "MetricTensor" && Length[Dimensions[matrixRepresentation]] == 2 &&
Length[coordinates] == Length[matrixRepresentation] && BooleanQ[index1] && BooleanQ[index2]
DiscreteHypersurfaceDecomposition[(metricTensor_)[matrixRepresentation_List, coordinates_List, index1_, index2_],
{timeCoordinate_, initialTime_, finalTime_}, {coordinate1_, initialCoordinate1_, finalCoordinate1_},
{coordinate2_, initialCoordinate2_, finalCoordinate2_}, vertexCount_Integer, discretizationScale_][
"CoordinatizedPolarGraphEvolution"] := Module[{newMatrixRepresentation, newCoordinates, newTimeCoordinate,
newCoordinate1, newCoordinate2, flatteningRules, christoffelSymbols, riemannTensor, covariantRiemannTensor,
contravariantRiemannTensor, kretschmannScalar, regions, points, edgeLists},
newMatrixRepresentation = matrixRepresentation /. (#1 -> ToExpression[#1] & ) /@ Select[coordinates, StringQ];
newCoordinates = coordinates /. (#1 -> ToExpression[#1] & ) /@ Select[coordinates, StringQ];
newTimeCoordinate = timeCoordinate /. (#1 -> ToExpression[#1] & ) /@ Select[coordinates, StringQ];
newCoordinate1 = coordinate1 /. (#1 -> ToExpression[#1] & ) /@ Select[coordinates, StringQ];
newCoordinate2 = coordinate2 /. (#1 -> ToExpression[#1] & ) /@ Select[coordinates, StringQ];
flatteningRules = Thread[Complement[newCoordinates, {newTimeCoordinate, newCoordinate1, newCoordinate2}] -> 0];
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]]];
covariantRiemannTensor = Normal[SparseArray[
(Module[{index = #1}, index -> Total[(newMatrixRepresentation[[index[[1]],#1]]*riemannTensor[[#1,index[[2]],
index[[3]],index[[4]]]] & ) /@ Range[Length[newMatrixRepresentation]]]] & ) /@
Tuples[Range[Length[newMatrixRepresentation]], 4]]]; contravariantRiemannTensor =
Normal[SparseArray[(Module[{index = #1}, index -> Total[(Inverse[newMatrixRepresentation][[index[[2]],#1[[1]]]]*
Inverse[newMatrixRepresentation][[#1[[2]],index[[3]]]]*Inverse[newMatrixRepresentation][[#1[[3]],
index[[4]]]]*riemannTensor[[index[[1]],#1[[1]],#1[[2]],#1[[3]]]] & ) /@ Tuples[Range[
Length[newMatrixRepresentation]], 3]]] & ) /@ Tuples[Range[Length[newMatrixRepresentation]], 4]]];
kretschmannScalar = FullSimplify[Total[(covariantRiemannTensor[[#1[[1]],#1[[2]],#1[[3]],#1[[4]]]]*
contravariantRiemannTensor[[#1[[1]],#1[[2]],#1[[3]],#1[[4]]]] & ) /@
Tuples[Range[Length[newMatrixRepresentation]], 4]] /. flatteningRules];
regions = (DiscretizeRegion[ImplicitRegion[-(kretschmannScalar /. {newCoordinate1 -> Sqrt[\[FormalX]^2 + \[FormalY]^2],
newCoordinate2 -> ArcTan[\[FormalY]/\[FormalX]]} /. newTimeCoordinate -> #1) == \[FormalL], {\[FormalX], \[FormalY], \[FormalL]}],
{{-Max[Abs[initialCoordinate1], Abs[finalCoordinate1]], Max[Abs[initialCoordinate1], Abs[finalCoordinate1]]},
{-Max[Abs[initialCoordinate1], Abs[finalCoordinate1]], Max[Abs[initialCoordinate1], Abs[finalCoordinate1]]},
{-Max[Abs[initialCoordinate1], Abs[finalCoordinate1]], Max[Abs[initialCoordinate1],
Abs[finalCoordinate1]]}}] & ) /@ Range[initialTime, finalTime, (finalTime - initialTime)/10];
points = (RandomPoint[#1, vertexCount] & ) /@ regions;
edgeLists = (Module[{point = #1}, Apply[UndirectedEdge, Select[Tuples[point, 2],
Norm[First[#1] - Last[#1]] < discretizationScale && First[#1] =!= Last[#1] & ], {1}]] & ) /@ points;
(Module[{index = #1}, SimpleGraph[Graph[points[[index]], edgeLists[[index]], VertexCoordinates ->
(#1 -> #1 & ) /@ points[[index]]]]] & ) /@ Range[Length[points]]] /;
SymbolName[metricTensor] === "MetricTensor" && Length[Dimensions[matrixRepresentation]] == 2 &&
Length[coordinates] == Length[matrixRepresentation] && BooleanQ[index1] && BooleanQ[index2]
DiscreteHypersurfaceDecomposition[(metricTensor_)[matrixRepresentation_List, coordinates_List, index1_, index2_],
{timeCoordinate_, initialTime_, finalTime_}, {coordinate1_, initialCoordinate1_, finalCoordinate1_},
{coordinate2_, initialCoordinate2_, finalCoordinate2_}, vertexCount_Integer, discretizationScale_][
{"CoordinatizedPolarGraphEvolution", stepCount_Integer}] :=
Module[{newMatrixRepresentation, newCoordinates, newTimeCoordinate, newCoordinate1, newCoordinate2, flatteningRules,
christoffelSymbols, riemannTensor, covariantRiemannTensor, contravariantRiemannTensor, kretschmannScalar, regions,
points, edgeLists}, newMatrixRepresentation = matrixRepresentation /. (#1 -> ToExpression[#1] & ) /@
Select[coordinates, StringQ]; newCoordinates = coordinates /. (#1 -> ToExpression[#1] & ) /@
Select[coordinates, StringQ]; newTimeCoordinate = timeCoordinate /. (#1 -> ToExpression[#1] & ) /@
Select[coordinates, StringQ]; newCoordinate1 = coordinate1 /. (#1 -> ToExpression[#1] & ) /@
Select[coordinates, StringQ]; newCoordinate2 = coordinate2 /. (#1 -> ToExpression[#1] & ) /@
Select[coordinates, StringQ]; flatteningRules =
Thread[Complement[newCoordinates, {newTimeCoordinate, newCoordinate1, newCoordinate2}] -> 0];
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]]];
covariantRiemannTensor = Normal[SparseArray[
(Module[{index = #1}, index -> Total[(newMatrixRepresentation[[index[[1]],#1]]*riemannTensor[[#1,index[[2]],
index[[3]],index[[4]]]] & ) /@ Range[Length[newMatrixRepresentation]]]] & ) /@
Tuples[Range[Length[newMatrixRepresentation]], 4]]]; contravariantRiemannTensor =
Normal[SparseArray[(Module[{index = #1}, index -> Total[(Inverse[newMatrixRepresentation][[index[[2]],#1[[1]]]]*
Inverse[newMatrixRepresentation][[#1[[2]],index[[3]]]]*Inverse[newMatrixRepresentation][[#1[[3]],
index[[4]]]]*riemannTensor[[index[[1]],#1[[1]],#1[[2]],#1[[3]]]] & ) /@ Tuples[Range[
Length[newMatrixRepresentation]], 3]]] & ) /@ Tuples[Range[Length[newMatrixRepresentation]], 4]]];
kretschmannScalar = FullSimplify[Total[(covariantRiemannTensor[[#1[[1]],#1[[2]],#1[[3]],#1[[4]]]]*
contravariantRiemannTensor[[#1[[1]],#1[[2]],#1[[3]],#1[[4]]]] & ) /@
Tuples[Range[Length[newMatrixRepresentation]], 4]] /. flatteningRules];
regions = (DiscretizeRegion[ImplicitRegion[-(kretschmannScalar /. {newCoordinate1 -> Sqrt[\[FormalX]^2 + \[FormalY]^2],
newCoordinate2 -> ArcTan[\[FormalY]/\[FormalX]]} /. newTimeCoordinate -> #1) == \[FormalL], {\[FormalX], \[FormalY], \[FormalL]}],
{{-Max[Abs[initialCoordinate1], Abs[finalCoordinate1]], Max[Abs[initialCoordinate1], Abs[finalCoordinate1]]},
{-Max[Abs[initialCoordinate1], Abs[finalCoordinate1]], Max[Abs[initialCoordinate1], Abs[finalCoordinate1]]},
{-Max[Abs[initialCoordinate1], Abs[finalCoordinate1]], Max[Abs[initialCoordinate1],
Abs[finalCoordinate1]]}}] & ) /@ Range[initialTime, finalTime, (finalTime - initialTime)/stepCount];
points = (RandomPoint[#1, vertexCount] & ) /@ regions;
edgeLists = (Module[{point = #1}, Apply[UndirectedEdge, Select[Tuples[point, 2],
Norm[First[#1] - Last[#1]] < discretizationScale && First[#1] =!= Last[#1] & ], {1}]] & ) /@ points;
(Module[{index = #1}, SimpleGraph[Graph[points[[index]], edgeLists[[index]], VertexCoordinates ->
(#1 -> #1 & ) /@ points[[index]]]]] & ) /@ Range[Length[points]]] /;
SymbolName[metricTensor] === "MetricTensor" && Length[Dimensions[matrixRepresentation]] == 2 &&
Length[coordinates] == Length[matrixRepresentation] && BooleanQ[index1] && BooleanQ[index2]
DiscreteHypersurfaceDecomposition[(metricTensor_)[matrixRepresentation_List, coordinates_List, index1_, index2_],
{timeCoordinate_, initialTime_, finalTime_}, {coordinate1_, initialCoordinate1_, finalCoordinate1_},
{coordinate2_, initialCoordinate2_, finalCoordinate2_}, vertexCount_Integer, discretizationScale_][
"PolarGraphEvolution"] := Module[{newMatrixRepresentation, newCoordinates, newTimeCoordinate, newCoordinate1,
newCoordinate2, flatteningRules, christoffelSymbols, riemannTensor, covariantRiemannTensor,
contravariantRiemannTensor, kretschmannScalar, regions, points, edgeLists},
newMatrixRepresentation = matrixRepresentation /. (#1 -> ToExpression[#1] & ) /@ Select[coordinates, StringQ];
newCoordinates = coordinates /. (#1 -> ToExpression[#1] & ) /@ Select[coordinates, StringQ];
newTimeCoordinate = timeCoordinate /. (#1 -> ToExpression[#1] & ) /@ Select[coordinates, StringQ];
newCoordinate1 = coordinate1 /. (#1 -> ToExpression[#1] & ) /@ Select[coordinates, StringQ];
newCoordinate2 = coordinate2 /. (#1 -> ToExpression[#1] & ) /@ Select[coordinates, StringQ];
flatteningRules = Thread[Complement[newCoordinates, {newTimeCoordinate, newCoordinate1, newCoordinate2}] -> 0];
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]]];
covariantRiemannTensor = Normal[SparseArray[
(Module[{index = #1}, index -> Total[(newMatrixRepresentation[[index[[1]],#1]]*riemannTensor[[#1,index[[2]],
index[[3]],index[[4]]]] & ) /@ Range[Length[newMatrixRepresentation]]]] & ) /@
Tuples[Range[Length[newMatrixRepresentation]], 4]]]; contravariantRiemannTensor =
Normal[SparseArray[(Module[{index = #1}, index -> Total[(Inverse[newMatrixRepresentation][[index[[2]],#1[[1]]]]*
Inverse[newMatrixRepresentation][[#1[[2]],index[[3]]]]*Inverse[newMatrixRepresentation][[#1[[3]],
index[[4]]]]*riemannTensor[[index[[1]],#1[[1]],#1[[2]],#1[[3]]]] & ) /@ Tuples[Range[
Length[newMatrixRepresentation]], 3]]] & ) /@ Tuples[Range[Length[newMatrixRepresentation]], 4]]];
kretschmannScalar = FullSimplify[Total[(covariantRiemannTensor[[#1[[1]],#1[[2]],#1[[3]],#1[[4]]]]*
contravariantRiemannTensor[[#1[[1]],#1[[2]],#1[[3]],#1[[4]]]] & ) /@
Tuples[Range[Length[newMatrixRepresentation]], 4]] /. flatteningRules];
regions = (DiscretizeRegion[ImplicitRegion[-(kretschmannScalar /. {newCoordinate1 -> Sqrt[\[FormalX]^2 + \[FormalY]^2],
newCoordinate2 -> ArcTan[\[FormalY]/\[FormalX]]} /. newTimeCoordinate -> #1) == \[FormalL], {\[FormalX], \[FormalY], \[FormalL]}],
{{-Max[Abs[initialCoordinate1], Abs[finalCoordinate1]], Max[Abs[initialCoordinate1], Abs[finalCoordinate1]]},
{-Max[Abs[initialCoordinate1], Abs[finalCoordinate1]], Max[Abs[initialCoordinate1], Abs[finalCoordinate1]]},
{-Max[Abs[initialCoordinate1], Abs[finalCoordinate1]], Max[Abs[initialCoordinate1],
Abs[finalCoordinate1]]}}] & ) /@ Range[initialTime, finalTime, (finalTime - initialTime)/10];
points = (RandomPoint[#1, vertexCount] & ) /@ regions;
edgeLists = (Module[{point = #1}, Apply[UndirectedEdge, Select[Tuples[point, 2],
Norm[First[#1] - Last[#1]] < discretizationScale && First[#1] =!= Last[#1] & ], {1}]] & ) /@ points;
(Module[{index = #1}, SimpleGraph[Graph[points[[index]], edgeLists[[index]]]]] & ) /@ Range[Length[points]]] /;
SymbolName[metricTensor] === "MetricTensor" && Length[Dimensions[matrixRepresentation]] == 2 &&
Length[coordinates] == Length[matrixRepresentation] && BooleanQ[index1] && BooleanQ[index2]
DiscreteHypersurfaceDecomposition[(metricTensor_)[matrixRepresentation_List, coordinates_List, index1_, index2_],
{timeCoordinate_, initialTime_, finalTime_}, {coordinate1_, initialCoordinate1_, finalCoordinate1_},
{coordinate2_, initialCoordinate2_, finalCoordinate2_}, vertexCount_Integer, discretizationScale_][
{"PolarGraphEvolution", stepCount_Integer}] :=
Module[{newMatrixRepresentation, newCoordinates, newTimeCoordinate, newCoordinate1, newCoordinate2, flatteningRules,
christoffelSymbols, riemannTensor, covariantRiemannTensor, contravariantRiemannTensor, kretschmannScalar, regions,
points, edgeLists}, newMatrixRepresentation = matrixRepresentation /. (#1 -> ToExpression[#1] & ) /@
Select[coordinates, StringQ]; newCoordinates = coordinates /. (#1 -> ToExpression[#1] & ) /@
Select[coordinates, StringQ]; newTimeCoordinate = timeCoordinate /. (#1 -> ToExpression[#1] & ) /@
Select[coordinates, StringQ]; newCoordinate1 = coordinate1 /. (#1 -> ToExpression[#1] & ) /@
Select[coordinates, StringQ]; newCoordinate2 = coordinate2 /. (#1 -> ToExpression[#1] & ) /@
Select[coordinates, StringQ]; flatteningRules =
Thread[Complement[newCoordinates, {newTimeCoordinate, newCoordinate1, newCoordinate2}] -> 0];
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]]];
covariantRiemannTensor = Normal[SparseArray[
(Module[{index = #1}, index -> Total[(newMatrixRepresentation[[index[[1]],#1]]*riemannTensor[[#1,index[[2]],
index[[3]],index[[4]]]] & ) /@ Range[Length[newMatrixRepresentation]]]] & ) /@
Tuples[Range[Length[newMatrixRepresentation]], 4]]]; contravariantRiemannTensor =
Normal[SparseArray[(Module[{index = #1}, index -> Total[(Inverse[newMatrixRepresentation][[index[[2]],#1[[1]]]]*
Inverse[newMatrixRepresentation][[#1[[2]],index[[3]]]]*Inverse[newMatrixRepresentation][[#1[[3]],
index[[4]]]]*riemannTensor[[index[[1]],#1[[1]],#1[[2]],#1[[3]]]] & ) /@ Tuples[Range[
Length[newMatrixRepresentation]], 3]]] & ) /@ Tuples[Range[Length[newMatrixRepresentation]], 4]]];
kretschmannScalar = FullSimplify[Total[(covariantRiemannTensor[[#1[[1]],#1[[2]],#1[[3]],#1[[4]]]]*
contravariantRiemannTensor[[#1[[1]],#1[[2]],#1[[3]],#1[[4]]]] & ) /@
Tuples[Range[Length[newMatrixRepresentation]], 4]] /. flatteningRules];
regions = (DiscretizeRegion[ImplicitRegion[-(kretschmannScalar /. {newCoordinate1 -> Sqrt[\[FormalX]^2 + \[FormalY]^2],
newCoordinate2 -> ArcTan[\[FormalY]/\[FormalX]]} /. newTimeCoordinate -> #1) == \[FormalL], {\[FormalX], \[FormalY], \[FormalL]}],
{{-Max[Abs[initialCoordinate1], Abs[finalCoordinate1]], Max[Abs[initialCoordinate1], Abs[finalCoordinate1]]},
{-Max[Abs[initialCoordinate1], Abs[finalCoordinate1]], Max[Abs[initialCoordinate1], Abs[finalCoordinate1]]},
{-Max[Abs[initialCoordinate1], Abs[finalCoordinate1]], Max[Abs[initialCoordinate1],
Abs[finalCoordinate1]]}}] & ) /@ Range[initialTime, finalTime, (finalTime - initialTime)/stepCount];
points = (RandomPoint[#1, vertexCount] & ) /@ regions;
edgeLists = (Module[{point = #1}, Apply[UndirectedEdge, Select[Tuples[point, 2],
Norm[First[#1] - Last[#1]] < discretizationScale && First[#1] =!= Last[#1] & ], {1}]] & ) /@ points;
(Module[{index = #1}, SimpleGraph[Graph[points[[index]], edgeLists[[index]]]]] & ) /@ Range[Length[points]]] /;
SymbolName[metricTensor] === "MetricTensor" && Length[Dimensions[matrixRepresentation]] == 2 &&
Length[coordinates] == Length[matrixRepresentation] && BooleanQ[index1] && BooleanQ[index2]
DiscreteHypersurfaceDecomposition[(metricTensor_)[matrixRepresentation_List, coordinates_List, index1_, index2_],
{timeCoordinate_, initialTime_, finalTime_}, {coordinate1_, initialCoordinate1_, finalCoordinate1_},
{coordinate2_, initialCoordinate2_, finalCoordinate2_}, vertexCount_Integer, discretizationScale_][
"CoordinatizedCartesianGraphColored"] :=
Module[{newMatrixRepresentation, newCoordinates, newTimeCoordinate, newCoordinate1, newCoordinate2, flatteningRules,
christoffelSymbols, riemannTensor, covariantRiemannTensor, contravariantRiemannTensor, kretschmannScalar, region,
points, curvatureRange, edgeList}, newMatrixRepresentation = matrixRepresentation /.
(#1 -> ToExpression[#1] & ) /@ Select[coordinates, StringQ];
newCoordinates = coordinates /. (#1 -> ToExpression[#1] & ) /@ Select[coordinates, StringQ];
newTimeCoordinate = timeCoordinate /. (#1 -> ToExpression[#1] & ) /@ Select[coordinates, StringQ];
newCoordinate1 = coordinate1 /. (#1 -> ToExpression[#1] & ) /@ Select[coordinates, StringQ];
newCoordinate2 = coordinate2 /. (#1 -> ToExpression[#1] & ) /@ Select[coordinates, StringQ];
flatteningRules = Thread[Complement[newCoordinates, {newTimeCoordinate, newCoordinate1, newCoordinate2}] -> 0];
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]]];
covariantRiemannTensor = Normal[SparseArray[
(Module[{index = #1}, index -> Total[(newMatrixRepresentation[[index[[1]],#1]]*riemannTensor[[#1,index[[2]],
index[[3]],index[[4]]]] & ) /@ Range[Length[newMatrixRepresentation]]]] & ) /@
Tuples[Range[Length[newMatrixRepresentation]], 4]]]; contravariantRiemannTensor =
Normal[SparseArray[(Module[{index = #1}, index -> Total[(Inverse[newMatrixRepresentation][[index[[2]],#1[[1]]]]*
Inverse[newMatrixRepresentation][[#1[[2]],index[[3]]]]*Inverse[newMatrixRepresentation][[#1[[3]],
index[[4]]]]*riemannTensor[[index[[1]],#1[[1]],#1[[2]],#1[[3]]]] & ) /@ Tuples[Range[
Length[newMatrixRepresentation]], 3]]] & ) /@ Tuples[Range[Length[newMatrixRepresentation]], 4]]];
kretschmannScalar = FullSimplify[Total[(covariantRiemannTensor[[#1[[1]],#1[[2]],#1[[3]],#1[[4]]]]*
contravariantRiemannTensor[[#1[[1]],#1[[2]],#1[[3]],#1[[4]]]] & ) /@
Tuples[Range[Length[newMatrixRepresentation]], 4]] /. flatteningRules];
region = DiscretizeRegion[ImplicitRegion[-(kretschmannScalar /. {newCoordinate1 -> \[FormalX], newCoordinate2 -> \[FormalY]} /.
newTimeCoordinate -> finalTime) == \[FormalL], {\[FormalX], \[FormalY], \[FormalL]}], {{initialCoordinate1, finalCoordinate1},
{initialCoordinate2, finalCoordinate2}, {-Max[Abs[initialCoordinate1], Abs[finalCoordinate1],
Abs[initialCoordinate2], Abs[finalCoordinate2]], Max[Abs[initialCoordinate1], Abs[finalCoordinate1],
Abs[initialCoordinate2], Abs[finalCoordinate2]]}}]; points = RandomPoint[region, vertexCount];
curvatureRange = Max[Last /@ points] - Min[Last /@ points];
edgeList = Apply[UndirectedEdge, Select[Tuples[points, 2], Norm[First[#1] - Last[#1]] < discretizationScale &&
First[#1] =!= Last[#1] & ], {1}]; SimpleGraph[Graph[points, edgeList,
VertexStyle -> (#1 -> ColorData["LightTemperatureMap", -(Last[#1]/curvatureRange)] & ) /@ points,
EdgeStyle -> (#1 -> ColorData["LightTemperatureMap", -(Last[Last[#1]]/curvatureRange)] & ) /@ edgeList,
VertexCoordinates -> (#1 -> #1 & ) /@ points]]] /; SymbolName[metricTensor] === "MetricTensor" &&
Length[Dimensions[matrixRepresentation]] == 2 && Length[coordinates] == Length[matrixRepresentation] &&
BooleanQ[index1] && BooleanQ[index2]
DiscreteHypersurfaceDecomposition[(metricTensor_)[matrixRepresentation_List, coordinates_List, index1_, index2_],
{timeCoordinate_, initialTime_, finalTime_}, {coordinate1_, initialCoordinate1_, finalCoordinate1_},
{coordinate2_, initialCoordinate2_, finalCoordinate2_}, vertexCount_Integer, discretizationScale_][
"CartesianGraphColored"] := Module[{newMatrixRepresentation, newCoordinates, newTimeCoordinate, newCoordinate1,
newCoordinate2, flatteningRules, christoffelSymbols, riemannTensor, covariantRiemannTensor,
contravariantRiemannTensor, kretschmannScalar, region, points, curvatureRange, edgeList},
newMatrixRepresentation = matrixRepresentation /. (#1 -> ToExpression[#1] & ) /@ Select[coordinates, StringQ];
newCoordinates = coordinates /. (#1 -> ToExpression[#1] & ) /@ Select[coordinates, StringQ];
newTimeCoordinate = timeCoordinate /. (#1 -> ToExpression[#1] & ) /@ Select[coordinates, StringQ];
newCoordinate1 = coordinate1 /. (#1 -> ToExpression[#1] & ) /@ Select[coordinates, StringQ];
newCoordinate2 = coordinate2 /. (#1 -> ToExpression[#1] & ) /@ Select[coordinates, StringQ];
flatteningRules = Thread[Complement[newCoordinates, {newTimeCoordinate, newCoordinate1, newCoordinate2}] -> 0];
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]]];
covariantRiemannTensor = Normal[SparseArray[
(Module[{index = #1}, index -> Total[(newMatrixRepresentation[[index[[1]],#1]]*riemannTensor[[#1,index[[2]],
index[[3]],index[[4]]]] & ) /@ Range[Length[newMatrixRepresentation]]]] & ) /@
Tuples[Range[Length[newMatrixRepresentation]], 4]]]; contravariantRiemannTensor =
Normal[SparseArray[(Module[{index = #1}, index -> Total[(Inverse[newMatrixRepresentation][[index[[2]],#1[[1]]]]*
Inverse[newMatrixRepresentation][[#1[[2]],index[[3]]]]*Inverse[newMatrixRepresentation][[#1[[3]],
index[[4]]]]*riemannTensor[[index[[1]],#1[[1]],#1[[2]],#1[[3]]]] & ) /@ Tuples[Range[
Length[newMatrixRepresentation]], 3]]] & ) /@ Tuples[Range[Length[newMatrixRepresentation]], 4]]];
kretschmannScalar = FullSimplify[Total[(covariantRiemannTensor[[#1[[1]],#1[[2]],#1[[3]],#1[[4]]]]*
contravariantRiemannTensor[[#1[[1]],#1[[2]],#1[[3]],#1[[4]]]] & ) /@
Tuples[Range[Length[newMatrixRepresentation]], 4]] /. flatteningRules];
region = DiscretizeRegion[ImplicitRegion[-(kretschmannScalar /. {newCoordinate1 -> \[FormalX], newCoordinate2 -> \[FormalY]} /.
newTimeCoordinate -> finalTime) == \[FormalL], {\[FormalX], \[FormalY], \[FormalL]}], {{initialCoordinate1, finalCoordinate1},
{initialCoordinate2, finalCoordinate2}, {-Max[Abs[initialCoordinate1], Abs[finalCoordinate1],
Abs[initialCoordinate2], Abs[finalCoordinate2]], Max[Abs[initialCoordinate1], Abs[finalCoordinate1],
Abs[initialCoordinate2], Abs[finalCoordinate2]]}}]; points = RandomPoint[region, vertexCount];
curvatureRange = Max[Last /@ points] - Min[Last /@ points];
edgeList = Apply[UndirectedEdge, Select[Tuples[points, 2], Norm[First[#1] - Last[#1]] < discretizationScale &&
First[#1] =!= Last[#1] & ], {1}]; SimpleGraph[Graph[points, edgeList,
VertexStyle -> (#1 -> ColorData["LightTemperatureMap", -(Last[#1]/curvatureRange)] & ) /@ points,
EdgeStyle -> (#1 -> ColorData["LightTemperatureMap", -(Last[Last[#1]]/curvatureRange)] & ) /@ edgeList]]] /;
SymbolName[metricTensor] === "MetricTensor" && Length[Dimensions[matrixRepresentation]] == 2 &&
Length[coordinates] == Length[matrixRepresentation] && BooleanQ[index1] && BooleanQ[index2]
DiscreteHypersurfaceDecomposition[(metricTensor_)[matrixRepresentation_List, coordinates_List, index1_, index2_],
{timeCoordinate_, initialTime_, finalTime_}, {coordinate1_, initialCoordinate1_, finalCoordinate1_},
{coordinate2_, initialCoordinate2_, finalCoordinate2_}, vertexCount_Integer, discretizationScale_][
"CoordinatizedCartesianGraph"] := Module[{newMatrixRepresentation, newCoordinates, newTimeCoordinate, newCoordinate1,
newCoordinate2, flatteningRules, christoffelSymbols, riemannTensor, covariantRiemannTensor,
contravariantRiemannTensor, kretschmannScalar, region, points, edgeList},
newMatrixRepresentation = matrixRepresentation /. (#1 -> ToExpression[#1] & ) /@ Select[coordinates, StringQ];
newCoordinates = coordinates /. (#1 -> ToExpression[#1] & ) /@ Select[coordinates, StringQ];
newTimeCoordinate = timeCoordinate /. (#1 -> ToExpression[#1] & ) /@ Select[coordinates, StringQ];
newCoordinate1 = coordinate1 /. (#1 -> ToExpression[#1] & ) /@ Select[coordinates, StringQ];
newCoordinate2 = coordinate2 /. (#1 -> ToExpression[#1] & ) /@ Select[coordinates, StringQ];
flatteningRules = Thread[Complement[newCoordinates, {newTimeCoordinate, newCoordinate1, newCoordinate2}] -> 0];
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]]];
covariantRiemannTensor = Normal[SparseArray[
(Module[{index = #1}, index -> Total[(newMatrixRepresentation[[index[[1]],#1]]*riemannTensor[[#1,index[[2]],
index[[3]],index[[4]]]] & ) /@ Range[Length[newMatrixRepresentation]]]] & ) /@
Tuples[Range[Length[newMatrixRepresentation]], 4]]]; contravariantRiemannTensor =
Normal[SparseArray[(Module[{index = #1}, index -> Total[(Inverse[newMatrixRepresentation][[index[[2]],#1[[1]]]]*
Inverse[newMatrixRepresentation][[#1[[2]],index[[3]]]]*Inverse[newMatrixRepresentation][[#1[[3]],
index[[4]]]]*riemannTensor[[index[[1]],#1[[1]],#1[[2]],#1[[3]]]] & ) /@ Tuples[Range[
Length[newMatrixRepresentation]], 3]]] & ) /@ Tuples[Range[Length[newMatrixRepresentation]], 4]]];
kretschmannScalar = FullSimplify[Total[(covariantRiemannTensor[[#1[[1]],#1[[2]],#1[[3]],#1[[4]]]]*
contravariantRiemannTensor[[#1[[1]],#1[[2]],#1[[3]],#1[[4]]]] & ) /@
Tuples[Range[Length[newMatrixRepresentation]], 4]] /. flatteningRules];
region = DiscretizeRegion[ImplicitRegion[-(kretschmannScalar /. {newCoordinate1 -> \[FormalX], newCoordinate2 -> \[FormalY]} /.
newTimeCoordinate -> finalTime) == \[FormalL], {\[FormalX], \[FormalY], \[FormalL]}], {{initialCoordinate1, finalCoordinate1},
{initialCoordinate2, finalCoordinate2}, {-Max[Abs[initialCoordinate1], Abs[finalCoordinate1],
Abs[initialCoordinate2], Abs[finalCoordinate2]], Max[Abs[initialCoordinate1], Abs[finalCoordinate1],
Abs[initialCoordinate2], Abs[finalCoordinate2]]}}]; points = RandomPoint[region, vertexCount];
edgeList = Apply[UndirectedEdge, Select[Tuples[points, 2], Norm[First[#1] - Last[#1]] < discretizationScale &&
First[#1] =!= Last[#1] & ], {1}]; SimpleGraph[Graph[points, edgeList,
VertexCoordinates -> (#1 -> #1 & ) /@ points]]] /; SymbolName[metricTensor] === "MetricTensor" &&
Length[Dimensions[matrixRepresentation]] == 2 && Length[coordinates] == Length[matrixRepresentation] &&
BooleanQ[index1] && BooleanQ[index2]
DiscreteHypersurfaceDecomposition[(metricTensor_)[matrixRepresentation_List, coordinates_List, index1_, index2_],
{timeCoordinate_, initialTime_, finalTime_}, {coordinate1_, initialCoordinate1_, finalCoordinate1_},
{coordinate2_, initialCoordinate2_, finalCoordinate2_}, vertexCount_Integer, discretizationScale_][
"CartesianGraph"] := Module[{newMatrixRepresentation, newCoordinates, newTimeCoordinate, newCoordinate1,
newCoordinate2, flatteningRules, christoffelSymbols, riemannTensor, covariantRiemannTensor,
contravariantRiemannTensor, kretschmannScalar, region, points, edgeList},
newMatrixRepresentation = matrixRepresentation /. (#1 -> ToExpression[#1] & ) /@ Select[coordinates, StringQ];
newCoordinates = coordinates /. (#1 -> ToExpression[#1] & ) /@ Select[coordinates, StringQ];
newTimeCoordinate = timeCoordinate /. (#1 -> ToExpression[#1] & ) /@ Select[coordinates, StringQ];
newCoordinate1 = coordinate1 /. (#1 -> ToExpression[#1] & ) /@ Select[coordinates, StringQ];
newCoordinate2 = coordinate2 /. (#1 -> ToExpression[#1] & ) /@ Select[coordinates, StringQ];
flatteningRules = Thread[Complement[newCoordinates, {newTimeCoordinate, newCoordinate1, newCoordinate2}] -> 0];
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]]];
covariantRiemannTensor = Normal[SparseArray[
(Module[{index = #1}, index -> Total[(newMatrixRepresentation[[index[[1]],#1]]*riemannTensor[[#1,index[[2]],
index[[3]],index[[4]]]] & ) /@ Range[Length[newMatrixRepresentation]]]] & ) /@
Tuples[Range[Length[newMatrixRepresentation]], 4]]]; contravariantRiemannTensor =
Normal[SparseArray[(Module[{index = #1}, index -> Total[(Inverse[newMatrixRepresentation][[index[[2]],#1[[1]]]]*
Inverse[newMatrixRepresentation][[#1[[2]],index[[3]]]]*Inverse[newMatrixRepresentation][[#1[[3]],
index[[4]]]]*riemannTensor[[index[[1]],#1[[1]],#1[[2]],#1[[3]]]] & ) /@ Tuples[Range[
Length[newMatrixRepresentation]], 3]]] & ) /@ Tuples[Range[Length[newMatrixRepresentation]], 4]]];
kretschmannScalar = FullSimplify[Total[(covariantRiemannTensor[[#1[[1]],#1[[2]],#1[[3]],#1[[4]]]]*
contravariantRiemannTensor[[#1[[1]],#1[[2]],#1[[3]],#1[[4]]]] & ) /@
Tuples[Range[Length[newMatrixRepresentation]], 4]] /. flatteningRules];
region = DiscretizeRegion[ImplicitRegion[-(kretschmannScalar /. {newCoordinate1 -> \[FormalX], newCoordinate2 -> \[FormalY]} /.
newTimeCoordinate -> finalTime) == \[FormalL], {\[FormalX], \[FormalY], \[FormalL]}], {{initialCoordinate1, finalCoordinate1},
{initialCoordinate2, finalCoordinate2}, {-Max[Abs[initialCoordinate1], Abs[finalCoordinate1],
Abs[initialCoordinate2], Abs[finalCoordinate2]], Max[Abs[initialCoordinate1], Abs[finalCoordinate1],
Abs[initialCoordinate2], Abs[finalCoordinate2]]}}]; points = RandomPoint[region, vertexCount];
edgeList = Apply[UndirectedEdge, Select[Tuples[points, 2], Norm[First[#1] - Last[#1]] < discretizationScale &&
First[#1] =!= Last[#1] & ], {1}]; SimpleGraph[Graph[points, edgeList]]] /;
SymbolName[metricTensor] === "MetricTensor" && Length[Dimensions[matrixRepresentation]] == 2 &&
Length[coordinates] == Length[matrixRepresentation] && BooleanQ[index1] && BooleanQ[index2]
DiscreteHypersurfaceDecomposition[(metricTensor_)[matrixRepresentation_List, coordinates_List, index1_, index2_],
{timeCoordinate_, initialTime_, finalTime_}, {coordinate1_, initialCoordinate1_, finalCoordinate1_},
{coordinate2_, initialCoordinate2_, finalCoordinate2_}, vertexCount_Integer, discretizationScale_][
"CoordinatizedCartesianGraphEvolutionColored"] :=
Module[{newMatrixRepresentation, newCoordinates, newTimeCoordinate, newCoordinate1, newCoordinate2, flatteningRules,
christoffelSymbols, riemannTensor, covariantRiemannTensor, contravariantRiemannTensor, kretschmannScalar, regions,
points, curvatureRanges, edgeLists}, newMatrixRepresentation = matrixRepresentation /.
(#1 -> ToExpression[#1] & ) /@ Select[coordinates, StringQ];
newCoordinates = coordinates /. (#1 -> ToExpression[#1] & ) /@ Select[coordinates, StringQ];
newTimeCoordinate = timeCoordinate /. (#1 -> ToExpression[#1] & ) /@ Select[coordinates, StringQ];
newCoordinate1 = coordinate1 /. (#1 -> ToExpression[#1] & ) /@ Select[coordinates, StringQ];
newCoordinate2 = coordinate2 /. (#1 -> ToExpression[#1] & ) /@ Select[coordinates, StringQ];
flatteningRules = Thread[Complement[newCoordinates, {newTimeCoordinate, newCoordinate1, newCoordinate2}] -> 0];
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]]];
covariantRiemannTensor = Normal[SparseArray[
(Module[{index = #1}, index -> Total[(newMatrixRepresentation[[index[[1]],#1]]*riemannTensor[[#1,index[[2]],
index[[3]],index[[4]]]] & ) /@ Range[Length[newMatrixRepresentation]]]] & ) /@
Tuples[Range[Length[newMatrixRepresentation]], 4]]]; contravariantRiemannTensor =
Normal[SparseArray[(Module[{index = #1}, index -> Total[(Inverse[newMatrixRepresentation][[index[[2]],#1[[1]]]]*
Inverse[newMatrixRepresentation][[#1[[2]],index[[3]]]]*Inverse[newMatrixRepresentation][[#1[[3]],
index[[4]]]]*riemannTensor[[index[[1]],#1[[1]],#1[[2]],#1[[3]]]] & ) /@ Tuples[Range[
Length[newMatrixRepresentation]], 3]]] & ) /@ Tuples[Range[Length[newMatrixRepresentation]], 4]]];
kretschmannScalar = FullSimplify[Total[(covariantRiemannTensor[[#1[[1]],#1[[2]],#1[[3]],#1[[4]]]]*
contravariantRiemannTensor[[#1[[1]],#1[[2]],#1[[3]],#1[[4]]]] & ) /@
Tuples[Range[Length[newMatrixRepresentation]], 4]] /. flatteningRules];
regions = (DiscretizeRegion[ImplicitRegion[-(kretschmannScalar /. {newCoordinate1 -> \[FormalX], newCoordinate2 -> \[FormalY]} /.
newTimeCoordinate -> #1) == \[FormalL], {\[FormalX], \[FormalY], \[FormalL]}], {{initialCoordinate1, finalCoordinate1},
{initialCoordinate2, finalCoordinate2}, {-Max[Abs[initialCoordinate1], Abs[finalCoordinate1],
Abs[initialCoordinate2], Abs[finalCoordinate2]], Max[Abs[initialCoordinate1], Abs[finalCoordinate1],
Abs[initialCoordinate2], Abs[finalCoordinate2]]}}] & ) /@ Range[initialTime, finalTime,
(finalTime - initialTime)/10]; points = (RandomPoint[#1, vertexCount] & ) /@ regions;
curvatureRanges = (Max[Last /@ #1] - Min[Last /@ #1] & ) /@ points;
edgeLists = (Module[{point = #1}, Apply[UndirectedEdge, Select[Tuples[point, 2],
Norm[First[#1] - Last[#1]] < discretizationScale && First[#1] =!= Last[#1] & ], {1}]] & ) /@ points;
(Module[{index = #1}, SimpleGraph[Graph[points[[index]], edgeLists[[index]],
VertexStyle -> (#1 -> ColorData["LightTemperatureMap", -(Last[#1]/curvatureRanges[[index]])] & ) /@
points[[index]], EdgeStyle -> (#1 -> ColorData["LightTemperatureMap", -(Last[Last[#1]]/curvatureRanges[[
index]])] & ) /@ edgeLists[[index]], VertexCoordinates -> (#1 -> #1 & ) /@ points[[index]]]]] & ) /@
Range[Length[points]]] /; SymbolName[metricTensor] === "MetricTensor" &&
Length[Dimensions[matrixRepresentation]] == 2 && Length[coordinates] == Length[matrixRepresentation] &&
BooleanQ[index1] && BooleanQ[index2]
DiscreteHypersurfaceDecomposition[(metricTensor_)[matrixRepresentation_List, coordinates_List, index1_, index2_],
{timeCoordinate_, initialTime_, finalTime_}, {coordinate1_, initialCoordinate1_, finalCoordinate1_},
{coordinate2_, initialCoordinate2_, finalCoordinate2_}, vertexCount_Integer, discretizationScale_][
{"CoordinatizedCartesianGraphEvolutionColored", stepCount_Integer}] :=
Module[{newMatrixRepresentation, newCoordinates, newTimeCoordinate, newCoordinate1, newCoordinate2, flatteningRules,
christoffelSymbols, riemannTensor, covariantRiemannTensor, contravariantRiemannTensor, kretschmannScalar, regions,
points, curvatureRanges, edgeLists}, newMatrixRepresentation = matrixRepresentation /.
(#1 -> ToExpression[#1] & ) /@ Select[coordinates, StringQ];
newCoordinates = coordinates /. (#1 -> ToExpression[#1] & ) /@ Select[coordinates, StringQ];
newTimeCoordinate = timeCoordinate /. (#1 -> ToExpression[#1] & ) /@ Select[coordinates, StringQ];
newCoordinate1 = coordinate1 /. (#1 -> ToExpression[#1] & ) /@ Select[coordinates, StringQ];
newCoordinate2 = coordinate2 /. (#1 -> ToExpression[#1] & ) /@ Select[coordinates, StringQ];
flatteningRules = Thread[Complement[newCoordinates, {newTimeCoordinate, newCoordinate1, newCoordinate2}] -> 0];
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]]];
covariantRiemannTensor = Normal[SparseArray[
(Module[{index = #1}, index -> Total[(newMatrixRepresentation[[index[[1]],#1]]*riemannTensor[[#1,index[[2]],
index[[3]],index[[4]]]] & ) /@ Range[Length[newMatrixRepresentation]]]] & ) /@
Tuples[Range[Length[newMatrixRepresentation]], 4]]]; contravariantRiemannTensor =
Normal[SparseArray[(Module[{index = #1}, index -> Total[(Inverse[newMatrixRepresentation][[index[[2]],#1[[1]]]]*
Inverse[newMatrixRepresentation][[#1[[2]],index[[3]]]]*Inverse[newMatrixRepresentation][[#1[[3]],
index[[4]]]]*riemannTensor[[index[[1]],#1[[1]],#1[[2]],#1[[3]]]] & ) /@ Tuples[Range[
Length[newMatrixRepresentation]], 3]]] & ) /@ Tuples[Range[Length[newMatrixRepresentation]], 4]]];
kretschmannScalar = FullSimplify[Total[(covariantRiemannTensor[[#1[[1]],#1[[2]],#1[[3]],#1[[4]]]]*
contravariantRiemannTensor[[#1[[1]],#1[[2]],#1[[3]],#1[[4]]]] & ) /@
Tuples[Range[Length[newMatrixRepresentation]], 4]] /. flatteningRules];
regions = (DiscretizeRegion[ImplicitRegion[-(kretschmannScalar /. {newCoordinate1 -> \[FormalX], newCoordinate2 -> \[FormalY]} /.
newTimeCoordinate -> #1) == \[FormalL], {\[FormalX], \[FormalY], \[FormalL]}], {{initialCoordinate1, finalCoordinate1},
{initialCoordinate2, finalCoordinate2}, {-Max[Abs[initialCoordinate1], Abs[finalCoordinate1],
Abs[initialCoordinate2], Abs[finalCoordinate2]], Max[Abs[initialCoordinate1], Abs[finalCoordinate1],
Abs[initialCoordinate2], Abs[finalCoordinate2]]}}] & ) /@ Range[initialTime, finalTime,
(finalTime - initialTime)/stepCount]; points = (RandomPoint[#1, vertexCount] & ) /@ regions;
curvatureRanges = (Max[Last /@ #1] - Min[Last /@ #1] & ) /@ points;
edgeLists = (Module[{point = #1}, Apply[UndirectedEdge, Select[Tuples[point, 2],
Norm[First[#1] - Last[#1]] < discretizationScale && First[#1] =!= Last[#1] & ], {1}]] & ) /@ points;
(Module[{index = #1}, SimpleGraph[Graph[points[[index]], edgeLists[[index]],
VertexStyle -> (#1 -> ColorData["LightTemperatureMap", -(Last[#1]/curvatureRanges[[index]])] & ) /@
points[[index]], EdgeStyle -> (#1 -> ColorData["LightTemperatureMap", -(Last[Last[#1]]/curvatureRanges[[
index]])] & ) /@ edgeLists[[index]], VertexCoordinates -> (#1 -> #1 & ) /@ points[[index]]]]] & ) /@
Range[Length[points]]] /; SymbolName[metricTensor] === "MetricTensor" &&
Length[Dimensions[matrixRepresentation]] == 2 && Length[coordinates] == Length[matrixRepresentation] &&
BooleanQ[index1] && BooleanQ[index2]
DiscreteHypersurfaceDecomposition[(metricTensor_)[matrixRepresentation_List, coordinates_List, index1_, index2_],
{timeCoordinate_, initialTime_, finalTime_}, {coordinate1_, initialCoordinate1_, finalCoordinate1_},
{coordinate2_, initialCoordinate2_, finalCoordinate2_}, vertexCount_Integer, discretizationScale_][
"CartesianGraphEvolutionColored"] := Module[{newMatrixRepresentation, newCoordinates, newTimeCoordinate,
newCoordinate1, newCoordinate2, flatteningRules, christoffelSymbols, riemannTensor, covariantRiemannTensor,
contravariantRiemannTensor, kretschmannScalar, regions, points, curvatureRanges, edgeLists},
newMatrixRepresentation = matrixRepresentation /. (#1 -> ToExpression[#1] & ) /@ Select[coordinates, StringQ];
newCoordinates = coordinates /. (#1 -> ToExpression[#1] & ) /@ Select[coordinates, StringQ];
newTimeCoordinate = timeCoordinate /. (#1 -> ToExpression[#1] & ) /@ Select[coordinates, StringQ];
newCoordinate1 = coordinate1 /. (#1 -> ToExpression[#1] & ) /@ Select[coordinates, StringQ];
newCoordinate2 = coordinate2 /. (#1 -> ToExpression[#1] & ) /@ Select[coordinates, StringQ];
flatteningRules = Thread[Complement[newCoordinates, {newTimeCoordinate, newCoordinate1, newCoordinate2}] -> 0];
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]]];
covariantRiemannTensor = Normal[SparseArray[
(Module[{index = #1}, index -> Total[(newMatrixRepresentation[[index[[1]],#1]]*riemannTensor[[#1,index[[2]],
index[[3]],index[[4]]]] & ) /@ Range[Length[newMatrixRepresentation]]]] & ) /@
Tuples[Range[Length[newMatrixRepresentation]], 4]]]; contravariantRiemannTensor =
Normal[SparseArray[(Module[{index = #1}, index -> Total[(Inverse[newMatrixRepresentation][[index[[2]],#1[[1]]]]*
Inverse[newMatrixRepresentation][[#1[[2]],index[[3]]]]*Inverse[newMatrixRepresentation][[#1[[3]],
index[[4]]]]*riemannTensor[[index[[1]],#1[[1]],#1[[2]],#1[[3]]]] & ) /@ Tuples[Range[
Length[newMatrixRepresentation]], 3]]] & ) /@ Tuples[Range[Length[newMatrixRepresentation]], 4]]];
kretschmannScalar = FullSimplify[Total[(covariantRiemannTensor[[#1[[1]],#1[[2]],#1[[3]],#1[[4]]]]*
contravariantRiemannTensor[[#1[[1]],#1[[2]],#1[[3]],#1[[4]]]] & ) /@
Tuples[Range[Length[newMatrixRepresentation]], 4]] /. flatteningRules];
regions = (DiscretizeRegion[ImplicitRegion[-(kretschmannScalar /. {newCoordinate1 -> \[FormalX], newCoordinate2 -> \[FormalY]} /.
newTimeCoordinate -> #1) == \[FormalL], {\[FormalX], \[FormalY], \[FormalL]}], {{initialCoordinate1, finalCoordinate1},
{initialCoordinate2, finalCoordinate2}, {-Max[Abs[initialCoordinate1], Abs[finalCoordinate1],
Abs[initialCoordinate2], Abs[finalCoordinate2]], Max[Abs[initialCoordinate1], Abs[finalCoordinate1],
Abs[initialCoordinate2], Abs[finalCoordinate2]]}}] & ) /@ Range[initialTime, finalTime,
(finalTime - initialTime)/10]; points = (RandomPoint[#1, vertexCount] & ) /@ regions;
curvatureRanges = (Max[Last /@ #1] - Min[Last /@ #1] & ) /@ points;
edgeLists = (Module[{point = #1}, Apply[UndirectedEdge, Select[Tuples[point, 2],
Norm[First[#1] - Last[#1]] < discretizationScale && First[#1] =!= Last[#1] & ], {1}]] & ) /@ points;
(Module[{index = #1}, SimpleGraph[Graph[points[[index]], edgeLists[[index]],
VertexStyle -> (#1 -> ColorData["LightTemperatureMap", -(Last[#1]/curvatureRanges[[index]])] & ) /@
points[[index]], EdgeStyle -> (#1 -> ColorData["LightTemperatureMap", -(Last[Last[#1]]/curvatureRanges[[
index]])] & ) /@ edgeLists[[index]]]]] & ) /@ Range[Length[points]]] /;
SymbolName[metricTensor] === "MetricTensor" && Length[Dimensions[matrixRepresentation]] == 2 &&
Length[coordinates] == Length[matrixRepresentation] && BooleanQ[index1] && BooleanQ[index2]
DiscreteHypersurfaceDecomposition[(metricTensor_)[matrixRepresentation_List, coordinates_List, index1_, index2_],
{timeCoordinate_, initialTime_, finalTime_}, {coordinate1_, initialCoordinate1_, finalCoordinate1_},
{coordinate2_, initialCoordinate2_, finalCoordinate2_}, vertexCount_Integer, discretizationScale_][
{"CartesianGraphEvolutionColored", stepCount_Integer}] :=
Module[{newMatrixRepresentation, newCoordinates, newTimeCoordinate, newCoordinate1, newCoordinate2, flatteningRules,
christoffelSymbols, riemannTensor, covariantRiemannTensor, contravariantRiemannTensor, kretschmannScalar, regions,
points, curvatureRanges, edgeLists}, newMatrixRepresentation = matrixRepresentation /.
(#1 -> ToExpression[#1] & ) /@ Select[coordinates, StringQ];
newCoordinates = coordinates /. (#1 -> ToExpression[#1] & ) /@ Select[coordinates, StringQ];
newTimeCoordinate = timeCoordinate /. (#1 -> ToExpression[#1] & ) /@ Select[coordinates, StringQ];
newCoordinate1 = coordinate1 /. (#1 -> ToExpression[#1] & ) /@ Select[coordinates, StringQ];
newCoordinate2 = coordinate2 /. (#1 -> ToExpression[#1] & ) /@ Select[coordinates, StringQ];
flatteningRules = Thread[Complement[newCoordinates, {newTimeCoordinate, newCoordinate1, newCoordinate2}] -> 0];
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]]];
covariantRiemannTensor = Normal[SparseArray[
(Module[{index = #1}, index -> Total[(newMatrixRepresentation[[index[[1]],#1]]*riemannTensor[[#1,index[[2]],