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MetricTensor.wl
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(* ::Package:: *)
MetricTensor[] := {"Symmetric", "SymmetricField", "Asymmetric", "AsymmetricField", "Euclidean", "Minkowski",
"Schwarzschild", "IsotropicSchwarzschild", "EddingtonFinkelstein", "IngoingEddingtonFinkelstein",
"OutgoingEddingtonFinkelstein", "GullstrandPainleve", "IngoingGullstrandPainleve", "OutgoingGullstrandPainleve",
"KruskalSzekeres", "Kerr", "ReissnerNordstrom", "KerrNewman", "Godel", "FLRW"}
MetricTensor[dimensionCount_Integer] :=
MetricTensor[Normal[SparseArray[(#1 -> Subscript["\[FormalG]", Sort[#1]] & ) /@ Tuples[Range[dimensionCount], 2]]],
(Superscript["\[FormalX]", ToString[#1]] & ) /@ Range[dimensionCount], True, True]
MetricTensor[dimensionCount_Integer, coordinates_List] :=
MetricTensor[Normal[SparseArray[(#1 -> Subscript["\[FormalG]", Sort[#1]] & ) /@ Tuples[Range[dimensionCount], 2]]],
coordinates, True, True] /; Length[coordinates] == dimensionCount
MetricTensor[dimensionCount_Integer, index1_, index2_] :=
MetricTensor[Normal[SparseArray[(#1 -> Subscript["\[FormalG]", Sort[#1]] & ) /@ Tuples[Range[dimensionCount], 2]]],
(Superscript["\[FormalX]", ToString[#1]] & ) /@ Range[dimensionCount], index1, index2] /;
BooleanQ[index1] && BooleanQ[index2]
MetricTensor[dimensionCount_Integer, coordinates_List, index1_, index2_] :=
MetricTensor[Normal[SparseArray[(#1 -> Subscript["\[FormalG]", Sort[#1]] & ) /@ Tuples[Range[dimensionCount], 2]]],
coordinates, index1, index2] /; Length[coordinates] == dimensionCount && BooleanQ[index1] && BooleanQ[index2]
MetricTensor["Symmetric"] := MetricTensor[Normal[SparseArray[(#1 -> Subscript["\[FormalG]", Sort[#1]] & ) /@
Tuples[Range[4], 2]]], (Superscript["\[FormalX]", ToString[#1]] & ) /@ Range[4], True, True]
MetricTensor["Symmetric", coordinates_List] :=
MetricTensor[Normal[SparseArray[(#1 -> Subscript["\[FormalG]", Sort[#1]] & ) /@ Tuples[Range[4], 2]]], coordinates, True,
True] /; Length[coordinates] == 4
MetricTensor["Symmetric", index1_, index2_] :=
MetricTensor[Normal[SparseArray[(#1 -> Subscript["\[FormalG]", Sort[#1]] & ) /@ Tuples[Range[4], 2]]],
(Superscript["\[FormalX]", ToString[#1]] & ) /@ Range[4], index1, index2] /; BooleanQ[index1] && BooleanQ[index2]
MetricTensor["Symmetric", coordinates_List, index1_, index2_] :=
MetricTensor[Normal[SparseArray[(#1 -> Subscript["\[FormalG]", Sort[#1]] & ) /@ Tuples[Range[4], 2]]], coordinates, index1,
index2] /; Length[coordinates] == 4 && BooleanQ[index1] && BooleanQ[index2]
MetricTensor[{"Symmetric", dimensionCount_Integer}] :=
MetricTensor[Normal[SparseArray[(#1 -> Subscript["\[FormalG]", Sort[#1]] & ) /@ Tuples[Range[dimensionCount], 2]]],
(Superscript["\[FormalX]", ToString[#1]] & ) /@ Range[dimensionCount], True, True]
MetricTensor[{"Symmetric", dimensionCount_Integer}, coordinates_List] :=
MetricTensor[Normal[SparseArray[(#1 -> Subscript["\[FormalG]", Sort[#1]] & ) /@ Tuples[Range[dimensionCount], 2]]],
coordinates, True, True] /; Length[coordinates] == dimensionCount
MetricTensor[{"Symmetric", dimensionCount_Integer}, index1_, index2_] :=
MetricTensor[Normal[SparseArray[(#1 -> Subscript["\[FormalG]", Sort[#1]] & ) /@ Tuples[Range[dimensionCount], 2]]],
(Superscript["\[FormalX]", ToString[#1]] & ) /@ Range[dimensionCount], index1, index2] /;
BooleanQ[index1] && BooleanQ[index2]
MetricTensor[{"Symmetric", dimensionCount_Integer}, coordinates_List, index1_, index2_] :=
MetricTensor[Normal[SparseArray[(#1 -> Subscript["\[FormalG]", Sort[#1]] & ) /@ Tuples[Range[dimensionCount], 2]]],
coordinates, index1, index2] /; Length[coordinates] == dimensionCount && BooleanQ[index1] && BooleanQ[index2]
MetricTensor["SymmetricField"] :=
MetricTensor[
Normal[SparseArray[(Module[{index = #1}, index -> Subscript["\[FormalG]", Sort[index]] @@
(Superscript["\[FormalX]", ToString[#1]] & ) /@ Range[4]] & ) /@ Tuples[Range[4], 2]]],
(Superscript["\[FormalX]", ToString[#1]] & ) /@ Range[4], True, True]
MetricTensor["SymmetricField", coordinates_List] :=
MetricTensor[Normal[SparseArray[(Module[{index = #1}, index -> Subscript["\[FormalG]", Sort[index]] @@ coordinates] & ) /@
Tuples[Range[4], 2]]], coordinates, True, True] /; Length[coordinates] == 4
MetricTensor["SymmetricField", index1_, index2_] :=
MetricTensor[Normal[SparseArray[(Module[{index = #1}, index -> Subscript["\[FormalG]", Sort[index]] @@
(Superscript["\[FormalX]", ToString[#1]] & ) /@ Range[4]] & ) /@ Tuples[Range[4], 2]]],
(Superscript["\[FormalX]", ToString[#1]] & ) /@ Range[4], index1, index2] /; BooleanQ[index1] && BooleanQ[index2]
MetricTensor["SymmetricField", coordinates_List, index1_, index2_] :=
MetricTensor[Normal[SparseArray[(Module[{index = #1}, index -> Subscript["\[FormalG]", Sort[index]] @@ coordinates] & ) /@
Tuples[Range[4], 2]]], coordinates, index1, index2] /; Length[coordinates] == 4 && BooleanQ[index1] &&
BooleanQ[index2]
MetricTensor[{"SymmetricField", dimensionCount_Integer}] :=
MetricTensor[
Normal[SparseArray[(Module[{index = #1}, index -> Subscript["\[FormalG]", Sort[index]] @@
(Superscript["\[FormalX]", ToString[#1]] & ) /@ Range[dimensionCount]] & ) /@ Tuples[Range[dimensionCount], 2]]],
(Superscript["\[FormalX]", ToString[#1]] & ) /@ Range[dimensionCount], True, True]
MetricTensor[{"SymmetricField", dimensionCount_Integer}, coordinates_List] :=
MetricTensor[Normal[SparseArray[(Module[{index = #1}, index -> Subscript["\[FormalG]", Sort[index]] @@ coordinates] & ) /@
Tuples[Range[dimensionCount], 2]]], coordinates, True, True] /; Length[coordinates] == dimensionCount
MetricTensor[{"SymmetricField", dimensionCount_Integer}, index1_, index2_] :=
MetricTensor[Normal[SparseArray[(Module[{index = #1}, index -> Subscript["\[FormalG]", Sort[index]] @@
(Superscript["\[FormalX]", ToString[#1]] & ) /@ Range[dimensionCount]] & ) /@ Tuples[Range[dimensionCount], 2]]],
(Superscript["\[FormalX]", ToString[#1]] & ) /@ Range[dimensionCount], index1, index2] /;
BooleanQ[index1] && BooleanQ[index2]
MetricTensor[{"SymmetricField", dimensionCount_Integer}, coordinates_List, index1_, index2_] :=
MetricTensor[Normal[SparseArray[(Module[{index = #1}, index -> Subscript["\[FormalG]", Sort[index]] @@ coordinates] & ) /@
Tuples[Range[dimensionCount], 2]]], coordinates, index1, index2] /; Length[coordinates] == dimensionCount &&
BooleanQ[index1] && BooleanQ[index2]
MetricTensor["Asymmetric"] := MetricTensor[Normal[SparseArray[(#1 -> Subscript["g", #1] & ) /@ Tuples[Range[4], 2]]],
(Superscript["\[FormalX]", ToString[#1]] & ) /@ Range[4], True, True]
MetricTensor["Asymmetric", coordinates_List] :=
MetricTensor[Normal[SparseArray[(#1 -> Subscript["\[FormalG]", #1] & ) /@ Tuples[Range[4], 2]]], coordinates, True, True] /;
Length[coordinates] == 4
MetricTensor["Asymmetric", index1_, index2_] :=
MetricTensor[Normal[SparseArray[(#1 -> Subscript["\[FormalG]", #1] & ) /@ Tuples[Range[4], 2]]],
(Superscript["\[FormalX]", ToString[#1]] & ) /@ Range[4], index1, index2] /; BooleanQ[index1] && BooleanQ[index2]
MetricTensor["Asymmetric", coordinates_List, index1_, index2_] :=
MetricTensor[Normal[SparseArray[(#1 -> Subscript["\[FormalG]", #1] & ) /@ Tuples[Range[4], 2]]], coordinates, index1,
index2] /; Length[coordinates] == 4 && BooleanQ[index1] && BooleanQ[index2]
MetricTensor[{"Asymmetric", dimensionCount_Integer}] :=
MetricTensor[Normal[SparseArray[(#1 -> Subscript["\[FormalG]", #1] & ) /@ Tuples[Range[dimensionCount], 2]]],
(Superscript["\[FormalX]", ToString[#1]] & ) /@ Range[dimensionCount], True, True]
MetricTensor[{"Asymmetric", dimensionCount_Integer}, coordinates_List] :=
MetricTensor[Normal[SparseArray[(#1 -> Subscript["\[FormalG]", #1] & ) /@ Tuples[Range[dimensionCount], 2]]], coordinates,
True, True] /; Length[coordinates] == dimensionCount
MetricTensor[{"Asymmetric", dimensionCount_Integer}, index1_, index2_] :=
MetricTensor[Normal[SparseArray[(#1 -> Subscript["\[FormalG]", #1] & ) /@ Tuples[Range[dimensionCount], 2]]],
(Superscript["\[FormalX]", ToString[#1]] & ) /@ Range[dimensionCount], index1, index2] /;
BooleanQ[index1] && BooleanQ[index2]
MetricTensor[{"Asymmetric", dimensionCount_Integer}, coordinates_List, index1_, index2_] :=
MetricTensor[Normal[SparseArray[(#1 -> Subscript["\[FormalG]", #1] & ) /@ Tuples[Range[dimensionCount], 2]]], coordinates,
index1, index2] /; Length[coordinates] == dimensionCount && BooleanQ[index1] && BooleanQ[index2]
MetricTensor["AsymmetricField"] :=
MetricTensor[
Normal[SparseArray[(Module[{index = #1}, index -> Subscript["\[FormalG]", index] @@ (Superscript["\[FormalX]", ToString[#1]] & ) /@
Range[4]] & ) /@ Tuples[Range[4], 2]]], (Superscript["\[FormalX]", ToString[#1]] & ) /@ Range[4], True, True]
MetricTensor["AsymmetricField", coordinates_List] :=
MetricTensor[Normal[SparseArray[(Module[{index = #1}, index -> Subscript["\[FormalG]", index] @@ coordinates] & ) /@
Tuples[Range[4], 2]]], coordinates, True, True] /; Length[coordinates] == 4
MetricTensor["AsymmetricField", index1_, index2_] :=
MetricTensor[Normal[SparseArray[(Module[{index = #1}, index -> Subscript["\[FormalG]", index] @@
(Superscript["\[FormalX]", ToString[#1]] & ) /@ Range[4]] & ) /@ Tuples[Range[4], 2]]],
(Superscript["\[FormalX]", ToString[#1]] & ) /@ Range[4], index1, index2] /; BooleanQ[index1] && BooleanQ[index2]
MetricTensor["AsymmetricField", coordinates_List, index1_, index2_] :=
MetricTensor[Normal[SparseArray[(Module[{index = #1}, index -> Subscript["\[FormalG]", index] @@ coordinates] & ) /@
Tuples[Range[4], 2]]], coordinates, index1, index2] /; Length[coordinates] == 4 && BooleanQ[index1] &&
BooleanQ[index2]
MetricTensor[{"AsymmetricField", dimensionCount_Integer}] :=
MetricTensor[
Normal[SparseArray[(Module[{index = #1}, index -> Subscript["\[FormalG]", index] @@ (Superscript["\[FormalX]", ToString[#1]] & ) /@
Range[dimensionCount]] & ) /@ Tuples[Range[dimensionCount], 2]]],
(Superscript["\[FormalX]", ToString[#1]] & ) /@ Range[dimensionCount], True, True]
MetricTensor[{"AsymmetricField", dimensionCount_Integer}, coordinates_List] :=
MetricTensor[Normal[SparseArray[(Module[{index = #1}, index -> Subscript["\[FormalG]", index] @@ coordinates] & ) /@
Tuples[Range[dimensionCount], 2]]], coordinates, True, True] /; Length[coordinates] == dimensionCount
MetricTensor[{"AsymmetricField", dimensionCount_Integer}, index1_, index2_] :=
MetricTensor[Normal[SparseArray[(Module[{index = #1}, index -> Subscript["\[FormalG]", index] @@
(Superscript["\[FormalX]", ToString[#1]] & ) /@ Range[dimensionCount]] & ) /@ Tuples[Range[dimensionCount], 2]]],
(Superscript["\[FormalX]", ToString[#1]] & ) /@ Range[dimensionCount], index1, index2] /;
BooleanQ[index1] && BooleanQ[index2]
MetricTensor[{"AsymmetricField", dimensionCount_Integer}, coordinates_List, index1_, index2_] :=
MetricTensor[Normal[SparseArray[(Module[{index = #1}, index -> Subscript["\[FormalG]", index] @@ coordinates] & ) /@
Tuples[Range[dimensionCount], 2]]], coordinates, index1, index2] /; Length[coordinates] == dimensionCount &&
BooleanQ[index1] && BooleanQ[index2]
MetricTensor["Euclidean"] := MetricTensor[DiagonalMatrix[ConstantArray[1, 3]],
(Superscript["\[FormalX]", ToString[#1]] & ) /@ Range[3], True, True]
MetricTensor["Euclidean", coordinates_List] := MetricTensor[DiagonalMatrix[ConstantArray[1, 3]], coordinates, True,
True] /; Length[coordinates] == 3
MetricTensor["Euclidean", index1_, index2_] := MetricTensor[DiagonalMatrix[ConstantArray[1, 3]],
(Superscript["\[FormalX]", ToString[#1]] & ) /@ Range[3], index1, index2] /; BooleanQ[index1] && BooleanQ[index2]
MetricTensor["Euclidean", coordinates_List, index1_, index2_] :=
MetricTensor[DiagonalMatrix[ConstantArray[1, 3]], coordinates, index1, index2] /;
Length[coordinates] == 3 && BooleanQ[index1] && BooleanQ[index2]
MetricTensor[{"Euclidean", dimensionCount_Integer}] := MetricTensor[DiagonalMatrix[ConstantArray[1, dimensionCount]],
(Superscript["\[FormalX]", ToString[#1]] & ) /@ Range[dimensionCount], True, True]
MetricTensor[{"Euclidean", dimensionCount_Integer}, coordinates_List] :=
MetricTensor[DiagonalMatrix[ConstantArray[1, dimensionCount]], coordinates, True, True] /;
Length[coordinates] == dimensionCount
MetricTensor[{"Euclidean", dimensionCount_Integer}, index1_, index2_] :=
MetricTensor[DiagonalMatrix[ConstantArray[1, dimensionCount]], (Superscript["\[FormalX]", ToString[#1]] & ) /@
Range[dimensionCount], index1, index2] /; BooleanQ[index1] && BooleanQ[index2]
MetricTensor[{"Euclidean", dimensionCount_Integer}, coordinates_List, index1_, index2_] :=
MetricTensor[DiagonalMatrix[ConstantArray[1, dimensionCount]], coordinates, index1, index2] /;
Length[coordinates] == dimensionCount && BooleanQ[index1] && BooleanQ[index2]
MetricTensor["Minkowski"] := MetricTensor[DiagonalMatrix[Join[{-1}, ConstantArray[1, 3]]],
Join[{"\[FormalT]"}, (Superscript["\[FormalX]", ToString[#1]] & ) /@ Range[3]], True, True]
MetricTensor["Minkowski", coordinates_List] := MetricTensor[DiagonalMatrix[Join[{-1}, ConstantArray[1, 3]]], coordinates,
True, True] /; Length[coordinates] == 4
MetricTensor["Minkowski", index1_, index2_] := MetricTensor[DiagonalMatrix[Join[{-1}, ConstantArray[1, 3]]],
Join[{"\[FormalT]"}, (Superscript["\[FormalX]", ToString[#1]] & ) /@ Range[3]], index1, index2] /;
BooleanQ[index1] && BooleanQ[index2]
MetricTensor["Minkowski", coordinates_List, index1_, index2_] :=
MetricTensor[DiagonalMatrix[Join[{-1}, ConstantArray[1, 3]]], coordinates, index1, index2] /;
Length[coordinates] == 4 && BooleanQ[index1] && BooleanQ[index2]
MetricTensor[{"Minkowski", dimensionCount_Integer}] :=
MetricTensor[DiagonalMatrix[Join[{-1}, ConstantArray[1, dimensionCount - 1]]],
Join[{"\[FormalT]"}, (Superscript["\[FormalX]", ToString[#1]] & ) /@ Range[dimensionCount - 1]], True, True]
MetricTensor[{"Minkowski", dimensionCount_Integer}, coordinates_List] :=
MetricTensor[DiagonalMatrix[Join[{-1}, ConstantArray[1, dimensionCount - 1]]], coordinates, True, True] /;
Length[coordinates] == dimensionCount
MetricTensor[{"Minkowski", dimensionCount_Integer}, index1_, index2_] :=
MetricTensor[DiagonalMatrix[Join[{-1}, ConstantArray[1, dimensionCount - 1]]],
Join[{"\[FormalT]"}, (Superscript["\[FormalX]", ToString[#1]] & ) /@ Range[dimensionCount - 1]], index1, index2] /;
BooleanQ[index1] && BooleanQ[index2]
MetricTensor[{"Minkowski", dimensionCount_Integer}, coordinates_List, index1_, index2_] :=
MetricTensor[DiagonalMatrix[Join[{-1}, ConstantArray[1, dimensionCount - 1]]], coordinates, index1, index2] /;
Length[coordinates] == dimensionCount && BooleanQ[index1] && BooleanQ[index2]
MetricTensor["Schwarzschild"] := MetricTensor[DiagonalMatrix[{-(1 - (2*"\[FormalCapitalM]")/"\[FormalR]"), 1/(1 - (2*"\[FormalCapitalM]")/"\[FormalR]"), "\[FormalR]"^2,
"\[FormalR]"^2*Sin["\[FormalTheta]"]^2}], {"\[FormalT]", "\[FormalR]", "\[FormalTheta]", "\[FormalPhi]"}, True, True]
MetricTensor["Schwarzschild", coordinates_List] :=
MetricTensor[DiagonalMatrix[{-(1 - (2*"\[FormalCapitalM]")/coordinates[[2]]), 1/(1 - (2*"\[FormalCapitalM]")/coordinates[[2]]), coordinates[[2]]^2,
coordinates[[2]]^2*Sin[coordinates[[3]]]^2}], coordinates, True, True] /; Length[coordinates] == 4
MetricTensor["Schwarzschild", index1_, index2_] :=
MetricTensor[DiagonalMatrix[{-(1 - (2*"\[FormalCapitalM]")/"\[FormalR]"), 1/(1 - (2*"\[FormalCapitalM]")/"\[FormalR]"), "\[FormalR]"^2, "\[FormalR]"^2*Sin["\[Theta]"]^2}],
{"\[FormalT]", "\[FormalR]", "\[FormalTheta]", "\[FormalPhi]"}, index1, index2] /; BooleanQ[index1] && BooleanQ[index2]
MetricTensor["Schwarzschild", coordinates_List, index1_, index2_] :=
MetricTensor[DiagonalMatrix[{-(1 - (2*"\[FormalCapitalM]")/coordinates[[2]]), 1/(1 - (2*"\[FormalCapitalM]")/coordinates[[2]]), coordinates[[2]]^2,
coordinates[[2]]^2*Sin[coordinates[[3]]]^2}], coordinates, index1, index2] /;
Length[coordinates] == 4 && BooleanQ[index1] && BooleanQ[index2]
MetricTensor[{"Schwarzschild", mass_}] := MetricTensor[DiagonalMatrix[{-(1 - (2*mass)/"\[FormalR]"), 1/(1 - (2*mass)/"\[FormalR]"),
"\[FormalR]"^2, "\[FormalR]"^2*Sin["\[FormalTheta]"]^2}], {"\[FormalT]", "\[FormalR]", "\[FormalTheta]", "\[FormalPhi]"}, True, True]
MetricTensor[{"Schwarzschild", mass_}, coordinates_List] :=
MetricTensor[DiagonalMatrix[{-(1 - (2*mass)/coordinates[[2]]), 1/(1 - (2*mass)/coordinates[[2]]), coordinates[[2]]^2,
coordinates[[2]]^2*Sin[coordinates[[3]]]^2}], coordinates, True, True] /; Length[coordinates] == 4
MetricTensor[{"Schwarzschild", mass_}, index1_, index2_] :=
MetricTensor[DiagonalMatrix[{-(1 - (2*mass)/"\[FormalR]"), 1/(1 - (2*mass)/"\[FormalR]"), "\[FormalR]"^2, "\[FormalR]"^2*Sin["\[FormalTheta]"]^2}],
{"\[FormalT]", "\[FormalR]", "\[FormalTheta]", "\[FormalPhi]"}, index1, index2] /; BooleanQ[index1] && BooleanQ[index2]
MetricTensor[{"Schwarzschild", mass_}, coordinates_List, index1_, index2_] :=
MetricTensor[DiagonalMatrix[{-(1 - (2*mass)/coordinates[[2]]), 1/(1 - (2*mass)/coordinates[[2]]), coordinates[[2]]^2,
coordinates[[2]]^2*Sin[coordinates[[3]]]^2}], coordinates, index2, index2] /;
Length[coordinates] == 4 && BooleanQ[index1] && BooleanQ[index2]
MetricTensor["IsotropicSchwarzschild"] :=
MetricTensor[DiagonalMatrix[Join[{-((1 - "\[FormalCapitalM]"/(2*Sqrt["\[FormalX]"^2 + "\[FormalY]"^2 + "\[FormalZ]"^2]))^2/
(1 + "\[FormalCapitalM]"/(2*Sqrt["\[FormalX]"^2 + "\[FormalY]"^2 + "\[FormalZ]"^2]))^2)},
ConstantArray[(1 + "\[FormalCapitalM]"/(2*Sqrt["\[FormalX]"^2 + "\[FormalY]"^2 + "\[FormalZ]"^2]))^4, 3]]], {"\[FormalT]", "\[FormalX]", "\[FormalY]", "\[FormalZ]"}, True, True]
MetricTensor["IsotropicSchwarzschild", coordinates_List] :=
MetricTensor[DiagonalMatrix[Join[{-((1 - "\[FormalCapitalM]"/(2*Sqrt[coordinates[[2]]^2 + coordinates[[3]]^2 + coordinates[[4]]^2]))^
2/(1 + "\[FormalCapitalM]"/(2*Sqrt[coordinates[[2]]^2 + coordinates[[3]]^2 + coordinates[[4]]^2]))^2)},
ConstantArray[(1 + "\[FormalCapitalM]"/(2*Sqrt[coordinates[[2]]^2 + coordinates[[3]]^2 + coordinates[[4]]^2]))^4, 3]]],
coordinates, True, True] /; Length[coordinates] == 4
MetricTensor["IsotropicSchwarzschild", index1_, index2_] :=
MetricTensor[DiagonalMatrix[Join[{-((1 - "\[FormalCapitalM]"/(2*Sqrt["\[FormalX]"^2 + "\[FormalY]"^2 + "\[FormalZ]"^2]))^2/
(1 + "\[FormalCapitalM]"/(2*Sqrt["\[FormalX]"^2 + "\[FormalY]"^2 + "\[FormalZ]"^2]))^2)},
ConstantArray[(1 + "\[FormalCapitalM]"/(2*Sqrt["\[FormalX]"^2 + "\[FormalY]"^2 + "\[FormalZ]"^2]))^4, 3]]], {"\[FormalT]", "\[FormalX]", "\[FormalY]", "\[FormalZ]"}, index1,
index2] /; BooleanQ[index1] && BooleanQ[index2]
MetricTensor["IsotropicSchwarzschild", coordinates_List, index1_, index2_] :=
MetricTensor[DiagonalMatrix[Join[{-((1 - "\[FormalCapitalM]"/(2*Sqrt[coordinates[[2]]^2 + coordinates[[3]]^2 + coordinates[[4]]^2]))^
2/(1 + "\[FormalCapitalM]"/(2*Sqrt[coordinates[[2]]^2 + coordinates[[3]]^2 + coordinates[[4]]^2]))^2)},
ConstantArray[(1 + "\[FormalCapitalM]"/(2*Sqrt[coordinates[[2]]^2 + coordinates[[3]]^2 + coordinates[[4]]^2]))^4, 3]]],
coordinates, index1, index2] /; Length[coordinates] == 4 && BooleanQ[index1] && BooleanQ[index2]
MetricTensor[{"IsotropicSchwarzschild", mass_}] :=
MetricTensor[DiagonalMatrix[Join[{-((1 - mass/(2*Sqrt["\[FormalX]"^2 + "\[FormalY]"^2 + "\[FormalZ]"^2]))^2/
(1 + mass/(2*Sqrt["\[FormalX]"^2 + "\[FormalY]"^2 + "\[FormalZ]"^2]))^2)},
ConstantArray[(1 + mass/(2*Sqrt["\[FormalX]"^2 + "\[FormalY]"^2 + "\[FormalZ]"^2]))^4, 3]]], {"\[FormalT]", "\[FormalX]", "\[FormalY]", "\[FormalZ]"}, True, True]
MetricTensor[{"IsotropicSchwarzschild", mass_}, coordinates_List] :=
MetricTensor[DiagonalMatrix[Join[{-((1 - mass/(2*Sqrt[coordinates[[2]]^2 + coordinates[[3]]^2 + coordinates[[4]]^2]))^2/
(1 + mass/(2*Sqrt[coordinates[[2]]^2 + coordinates[[3]]^2 + coordinates[[4]]^2]))^2)},
ConstantArray[(1 + mass/(2*Sqrt[coordinates[[2]]^2 + coordinates[[3]]^2 + coordinates[[4]]^2]))^4, 3]]],
coordinates, True, True] /; Length[coordinates] == 4
MetricTensor[{"IsotropicSchwarzschild", mass_}, index1_, index2_] :=
MetricTensor[DiagonalMatrix[Join[{-((1 - mass/(2*Sqrt["\[FormalX]"^2 + "\[FormalY]"^2 + "\[FormalZ]"^2]))^2/
(1 + mass/(2*Sqrt["\[FormalX]"^2 + "\[FormalY]"^2 + "\[FormalZ]"^2]))^2)},
ConstantArray[(1 + mass/(2*Sqrt["\[FormalX]"^2 + "\[FormalY]"^2 + "\[FormalZ]"^2]))^4, 3]]], {"\[FormalT]", "\[FormalX]", "\[FormalY]", "\[FormalZ]"}, index1,
index2] /; BooleanQ[index1] && BooleanQ[index2]
MetricTensor[{"IsotropicSchwarzschild", mass_}, coordinates_List, index1_, index2_] :=
MetricTensor[DiagonalMatrix[Join[{-((1 - mass/(2*Sqrt[coordinates[[2]]^2 + coordinates[[3]]^2 + coordinates[[4]]^2]))^2/
(1 + mass/(2*Sqrt[coordinates[[2]]^2 + coordinates[[3]]^2 + coordinates[[4]]^2]))^2)},
ConstantArray[(1 + mass/(2*Sqrt[coordinates[[2]]^2 + coordinates[[3]]^2 + coordinates[[4]]^2]))^4, 3]]],
coordinates, index1, index2] /; Length[coordinates] == 4 && BooleanQ[index1] && BooleanQ[index2]
MetricTensor["EddingtonFinkelstein"] :=
MetricTensor[Normal[SparseArray[{{1, 1} -> -(1 - (2*"\[FormalCapitalM]")/"\[FormalR]"), {1, 2} -> \[PlusMinus]1, {2, 1} -> \[PlusMinus]1, {3, 3} -> "\[FormalR]"^2,
{4, 4} -> "\[FormalR]"^2*Sin["\[FormalTheta]"]^2}]], {"\[FormalCapitalT]", "\[FormalR]", "\[FormalTheta]", "\[FormalPhi]"}, True, True]
MetricTensor["EddingtonFinkelstein", coordinates_List] :=
MetricTensor[Normal[SparseArray[{{1, 1} -> -(1 - (2*"\[FormalCapitalM]")/coordinates[[2]]), {1, 2} -> \[PlusMinus]1, {2, 1} -> \[PlusMinus]1,
{3, 3} -> coordinates[[2]]^2, {4, 4} -> coordinates[[2]]^2*Sin[coordinates[[3]]]^2}]], coordinates, True, True] /;
Length[coordinates] == 4
MetricTensor["EddingtonFinkelstein", index1_, index2_] :=
MetricTensor[Normal[SparseArray[{{1, 1} -> -(1 - (2*"\[FormalCapitalM]")/"\[FormalR]"), {1, 2} -> \[PlusMinus]1, {2, 1} -> \[PlusMinus]1, {3, 3} -> "\[FormalR]"^2,
{4, 4} -> "\[FormalR]"^2*Sin["\[FormalTheta]"]^2}]], {"\[FormalCapitalT]", "\[FormalR]", "\[FormalTheta]", "\[FormalPhi]"}, index1, index2] /;
BooleanQ[index1] && BooleanQ[index2]
MetricTensor["EddingtonFinkelstein", coordinates_List, index1_, index2_] :=
MetricTensor[Normal[SparseArray[{{1, 1} -> -(1 - (2*"\[FormalCapitalM]")/coordinates[[2]]), {1, 2} -> \[PlusMinus]1, {2, 1} -> \[PlusMinus]1,
{3, 3} -> coordinates[[2]]^2, {4, 4} -> coordinates[[2]]^2*Sin[coordinates[[3]]]^2}]], coordinates, index1,
index2] /; Length[coordinates] == 4 && BooleanQ[index1] && BooleanQ[index2]
MetricTensor[{"EddingtonFinkelstein", mass_}] :=
MetricTensor[Normal[SparseArray[{{1, 1} -> -(1 - (2*mass)/"\[FormalR]"), {1, 2} -> \[PlusMinus]1, {2, 1} -> \[PlusMinus]1, {3, 3} -> "\[FormalR]"^2,
{4, 4} -> "\[FormalR]"^2*Sin["\[FormalTheta]"]^2}]], {"\[FormalCapitalT]", "\[FormalR]", "\[FormalTheta]", "\[FormalPhi]"}, True, True]
MetricTensor[{"EddingtonFinkelstein", mass_}, coordinates_List] :=
MetricTensor[Normal[SparseArray[{{1, 1} -> -(1 - (2*mass)/coordinates[[2]]), {1, 2} -> \[PlusMinus]1, {2, 1} -> \[PlusMinus]1,
{3, 3} -> coordinates[[2]]^2, {4, 4} -> coordinates[[2]]^2*Sin[coordinates[[3]]]^2}]], coordinates, True, True] /;
Length[coordinates] == 4
MetricTensor[{"EddingtonFinkelstein", mass_}, index1_, index2_] :=
MetricTensor[Normal[SparseArray[{{1, 1} -> -(1 - (2*mass)/"\[FormalR]"), {1, 2} -> \[PlusMinus]1, {2, 1} -> \[PlusMinus]1, {3, 3} -> "\[FormalR]"^2,
{4, 4} -> "\[FormalR]"^2*Sin["\[FormalTheta]"]^2}]], {"\[FormalCapitalT]", "\[FormalR]", "\[FormalTheta]", "\[FormalPhi]"}, index1, index2] /;
BooleanQ[index1] && BooleanQ[index2]
MetricTensor[{"EddingtonFinkelstein", mass_}, coordinates_List, index1_, index2_] :=
MetricTensor[Normal[SparseArray[{{1, 1} -> -(1 - (2*mass)/coordinates[[2]]), {1, 2} -> \[PlusMinus]1, {2, 1} -> \[PlusMinus]1,
{3, 3} -> coordinates[[2]]^2, {4, 4} -> coordinates[[2]]^2*Sin[coordinates[[3]]]^2}]], coordinates, index1,
index2] /; Length[coordinates] == 4 && BooleanQ[index1] && BooleanQ[index2]
MetricTensor["IngoingEddingtonFinkelstein"] :=
MetricTensor[Normal[SparseArray[{{1, 1} -> -(1 - (2*"\[FormalCapitalM]")/"\[FormalR]"), {1, 2} -> 1, {2, 1} -> 1, {3, 3} -> "\[FormalR]"^2,
{4, 4} -> "\[FormalR]"^2*Sin["\[FormalTheta]"]^2}]], {"\[FormalV]", "\[FormalR]", "\[FormalTheta]", "\[FormalPhi]"}, True, True]
MetricTensor["IngoingEddingtonFinkelstein", coordinates_List] :=
MetricTensor[Normal[SparseArray[{{1, 1} -> -(1 - (2*"\[FormalCapitalM]")/coordinates[[2]]), {1, 2} -> 1, {2, 1} -> 1,
{3, 3} -> coordinates[[2]]^2, {4, 4} -> coordinates[[2]]^2*Sin[coordinates[[3]]]^2}]], coordinates, True, True] /;
Length[coordinates] == 4
MetricTensor["IngoingEddingtonFinkelstein", index1_, index2_] :=
MetricTensor[Normal[SparseArray[{{1, 1} -> -(1 - (2*"\[FormalCapitalM]")/"\[FormalR]"), {1, 2} -> 1, {2, 1} -> 1, {3, 3} -> "\[FormalR]"^2,
{4, 4} -> "\[FormalR]"^2*Sin["\[FormalTheta]"]^2}]], {"\[FormalV]", "\[FormalR]", "\[FormalTheta]", "\[FormalPhi]"}, index1, index2] /;
BooleanQ[index1] && BooleanQ[index2]
MetricTensor["IngoingEddingtonFinkelstein", coordinates_List, index1_, index2_] :=
MetricTensor[Normal[SparseArray[{{1, 1} -> -(1 - (2*"\[FormalCapitalM]")/coordinates[[2]]), {1, 2} -> 1, {2, 1} -> 1,
{3, 3} -> coordinates[[2]]^2, {4, 4} -> coordinates[[2]]^2*Sin[coordinates[[3]]]^2}]], coordinates, index1,
index2] /; Length[coordinates] == 4 && BooleanQ[index1] && BooleanQ[index2]
MetricTensor[{"IngoingEddingtonFinkelstein", mass_}] :=
MetricTensor[Normal[SparseArray[{{1, 1} -> -(1 - (2*mass)/"\[FormalR]"), {1, 2} -> 1, {2, 1} -> 1, {3, 3} -> "\[FormalR]"^2,
{4, 4} -> "\[FormalR]"^2*Sin["\[FormalTheta]"]^2}]], {"\[FormalV]", "\[FormalR]", "\[FormalTheta]", "\[FormalPhi]"}, True, True]
MetricTensor[{"IngoingEddingtonFinkelstein", mass_}, coordinates_List] :=
MetricTensor[Normal[SparseArray[{{1, 1} -> -(1 - (2*mass)/coordinates[[2]]), {1, 2} -> 1, {2, 1} -> 1,
{3, 3} -> coordinates[[2]]^2, {4, 4} -> coordinates[[2]]^2*Sin[coordinates[[3]]]^2}]], coordinates, True, True] /;
Length[coordinates] == 4
MetricTensor[{"IngoingEddingtonFinkelstein", mass_}, index1_, index2_] :=
MetricTensor[Normal[SparseArray[{{1, 1} -> -(1 - (2*mass)/"\[FormalR]"), {1, 2} -> 1, {2, 1} -> 1, {3, 3} -> "\[FormalR]"^2,
{4, 4} -> "\[FormalR]"^2*Sin["\[FormalTheta]"]^2}]], {"\[FormalV]", "\[FormalR]", "\[FormalTheta]", "\[FormalPhi]"}, index1, index2] /;
BooleanQ[index1] && BooleanQ[index2]
MetricTensor[{"IngoingEddingtonFinkelstein", mass_}, coordinates_List, index1_, index2_] :=
MetricTensor[Normal[SparseArray[{{1, 1} -> -(1 - (2*mass)/coordinates[[2]]), {1, 2} -> 1, {2, 1} -> 1,
{3, 3} -> coordinates[[2]]^2, {4, 4} -> coordinates[[2]]^2*Sin[coordinates[[3]]]^2}]], coordinates, index1,
index2] /; Length[coordinates] == 4 && BooleanQ[index1] && BooleanQ[index2]
MetricTensor["OutgoingEddingtonFinkelstein"] :=
MetricTensor[Normal[SparseArray[{{1, 1} -> -(1 - (2*"\[FormalCapitalM]")/"\[FormalR]"), {1, 2} -> -1, {2, 1} -> -1, {3, 3} -> "\[FormalR]"^2,
{4, 4} -> "\[FormalR]"^2*Sin["\[FormalTheta]"]^2}]], {"\[FormalU]", "\[FormalR]", "\[FormalTheta]", "\[FormalPhi]"}, True, True]
MetricTensor["OutgoingEddingtonFinkelstein", coordinates_List] :=
MetricTensor[Normal[SparseArray[{{1, 1} -> -(1 - (2*"\[FormalCapitalM]")/coordinates[[2]]), {1, 2} -> -1, {2, 1} -> -1,
{3, 3} -> coordinates[[2]]^2, {4, 4} -> coordinates[[2]]^2*Sin[coordinates[[3]]]^2}]], coordinates, True, True] /;
Length[coordinates] == 4
MetricTensor["OutgoingEddingtonFinkelstein", index1_, index2_] :=
MetricTensor[Normal[SparseArray[{{1, 1} -> -(1 - (2*"\[FormalCapitalM]")/"\[FormalR]"), {1, 2} -> -1, {2, 1} -> -1, {3, 3} -> "\[FormalR]"^2,
{4, 4} -> "\[FormalR]"^2*Sin["\[FormalTheta]"]^2}]], {"\[FormalU]", "\[FormalR]", "\[FormalTheta]", "\[FormalPhi]"}, index1, index2] /;
BooleanQ[index1] && BooleanQ[index2]
MetricTensor["OutgoingEddingtonFinkelstein", coordinates_List, index1_, index2_] :=
MetricTensor[Normal[SparseArray[{{1, 1} -> -(1 - (2*"\[FormalCapitalM]")/coordinates[[2]]), {1, 2} -> -1, {2, 1} -> -1,
{3, 3} -> coordinates[[2]]^2, {4, 4} -> coordinates[[2]]^2*Sin[coordinates[[3]]]^2}]], coordinates, index1,
index2] /; Length[coordinates] == 4 && BooleanQ[index1] && BooleanQ[index2]
MetricTensor[{"OutgoingEddingtonFinkelstein", mass_}] :=
MetricTensor[Normal[SparseArray[{{1, 1} -> -(1 - (2*mass)/"\[FormalR]"), {1, 2} -> -1, {2, 1} -> -1, {3, 3} -> "\[FormalR]"^2,
{4, 4} -> "\[FormalR]"^2*Sin["\[FormalTheta]"]^2}]], {"\[FormalU]", "\[FormalR]", "\[FormalTheta]", "\[FormalPhi]"}, True, True]
MetricTensor[{"OutgoingEddingtonFinkelstein", mass_}, coordinates_List] :=
MetricTensor[Normal[SparseArray[{{1, 1} -> -(1 - (2*mass)/coordinates[[2]]), {1, 2} -> -1, {2, 1} -> -1,
{3, 3} -> coordinates[[2]]^2, {4, 4} -> coordinates[[2]]^2*Sin[coordinates[[3]]]^2}]], coordinates, True, True] /;
Length[coordinates] == 4
MetricTensor[{"OutgoingEddingtonFinkelstein", mass_}, index1_, index2_] :=
MetricTensor[Normal[SparseArray[{{1, 1} -> -(1 - (2*mass)/"\[FormalR]"), {1, 2} -> -1, {2, 1} -> -1, {3, 3} -> "\[FormalR]"^2,
{4, 4} -> "\[FormalR]"^2*Sin["\[FormalTheta]"]^2}]], {"\[FormalU]", "\[FormalR]", "\[FormalTheta]", "\[FormalPhi]"}, index1, index2] /;
BooleanQ[index1] && BooleanQ[index2]
MetricTensor[{"OutgoingEddingtonFinkelstein", mass_}, coordinates_List, index1_, index2_] :=
MetricTensor[Normal[SparseArray[{{1, 1} -> -(1 - (2*mass)/coordinates[[2]]), {1, 2} -> -1, {2, 1} -> -1,
{3, 3} -> coordinates[[2]]^2, {4, 4} -> coordinates[[2]]^2*Sin[coordinates[[3]]]^2}]], coordinates, index1,
index2] /; Length[coordinates] == 4 && BooleanQ[index1] && BooleanQ[index2]
MetricTensor["GullstrandPainleve"] :=
MetricTensor[Normal[SparseArray[{{1, 1} -> -(1 - (2*"\[FormalCapitalM]")/"\[FormalR]"), {2, 2} -> 1, {1, 2} -> \[PlusMinus]Sqrt[(2*"\[FormalCapitalM]")/"\[FormalR]"],
{2, 1} -> \[PlusMinus]Sqrt[(2*"\[FormalCapitalM]")/"\[FormalR]"], {3, 3} -> "\[FormalR]"^2, {4, 4} -> "\[FormalR]"^2*Sin["\[FormalTheta]"]^2}]],
{"\[FormalCapitalT]", "\[FormalR]", "\[FormalTheta]", "\[FormalPhi]"}, True, True]
MetricTensor["GullstrandPainleve", coordinates_List] :=
MetricTensor[Normal[SparseArray[{{1, 1} -> -(1 - (2*"\[FormalCapitalM]")/coordinates[[2]]), {2, 2} -> 1,
{1, 2} -> \[PlusMinus]Sqrt[(2*"\[FormalCapitalM]")/coordinates[[2]]], {2, 1} -> \[PlusMinus]Sqrt[(2*"\[FormalCapitalM]")/coordinates[[2]]],
{3, 3} -> coordinates[[2]]^2, {4, 4} -> coordinates[[2]]^2*Sin[coordinates[[3]]]^2}]], coordinates, True, True] /;
Length[coordinates] == 4
MetricTensor["GullstrandPainleve", index1_, index2_] :=
MetricTensor[Normal[SparseArray[{{1, 1} -> -(1 - (2*"\[FormalCapitalM]")/"\[FormalR]"), {2, 2} -> 1, {1, 2} -> \[PlusMinus]Sqrt[(2*"\[FormalCapitalM]")/"\[FormalR]"],
{2, 1} -> \[PlusMinus]Sqrt[(2*"\[FormalCapitalM]")/"\[FormalR]"], {3, 3} -> "\[FormalR]"^2, {4, 4} -> "\[FormalR]"^2*Sin["\[FormalTheta]"]^2}]],
{"\[FormalCapitalT]", "\[FormalR]", "\[FormalTheta]", "\[FormalPhi]"}, index1, index2] /; BooleanQ[index1] && BooleanQ[index2]
MetricTensor["GullstrandPainleve", coordinates_List, index1_, index2_] :=
MetricTensor[Normal[SparseArray[{{1, 1} -> -(1 - (2*"\[FormalCapitalM]")/coordinates[[2]]), {2, 2} -> 1,
{1, 2} -> \[PlusMinus]Sqrt[(2*"\[FormalCapitalM]")/coordinates[[2]]], {2, 1} -> \[PlusMinus]Sqrt[(2*"\[FormalCapitalM]")/coordinates[[2]]],
{3, 3} -> coordinates[[2]]^2, {4, 4} -> coordinates[[2]]^2*Sin[coordinates[[3]]]^2}]], coordinates, index1,
index2] /; Length[coordinates] == 4 && BooleanQ[index1] && BooleanQ[index2]
MetricTensor[{"GullstrandPainleve", mass_}] :=
MetricTensor[Normal[SparseArray[{{1, 1} -> -(1 - (2*mass)/"\[FormalR]"), {2, 2} -> 1, {1, 2} -> \[PlusMinus]Sqrt[(2*mass)/"\[FormalR]"],
{2, 1} -> \[PlusMinus]Sqrt[(2*mass)/"\[FormalR]"], {3, 3} -> "\[FormalR]"^2, {4, 4} -> "\[FormalR]"^2*Sin["\[FormalTheta]"]^2}]],
{"\[FormalCapitalT]", "\[FormalR]", "\[FormalTheta]", "\[FormalPhi]"}, True, True]
MetricTensor[{"GullstrandPainleve", mass_}, coordinates_List] :=
MetricTensor[Normal[SparseArray[{{1, 1} -> -(1 - (2*mass)/coordinates[[2]]), {2, 2} -> 1,
{1, 2} -> \[PlusMinus]Sqrt[(2*mass)/coordinates[[2]]], {2, 1} -> \[PlusMinus]Sqrt[(2*mass)/coordinates[[2]]],
{3, 3} -> coordinates[[2]]^2, {4, 4} -> coordinates[[2]]^2*Sin[coordinates[[3]]]^2}]], coordinates, True, True] /;
Length[coordinates] == 4
MetricTensor[{"GullstrandPainleve", mass_}, index1_, index2_] :=
MetricTensor[Normal[SparseArray[{{1, 1} -> -(1 - (2*mass)/"\[FormalR]"), {2, 2} -> 1, {1, 2} -> \[PlusMinus]Sqrt[(2*mass)/"\[FormalR]"],
{2, 1} -> \[PlusMinus]Sqrt[(2*mass)/"\[FormalR]"], {3, 3} -> "\[FormalR]"^2, {4, 4} -> "\[FormalR]"^2*Sin["\[FormalTheta]"]^2}]],
{"\[FormalCapitalT]", "\[FormalR]", "\[FormalTheta]", "\[FormalPhi]"}, index1, index2] /; BooleanQ[index1] && BooleanQ[index2]
MetricTensor[{"GullstrandPainleve", mass_}, coordinates_List, index1_, index2_] :=
MetricTensor[Normal[SparseArray[{{1, 1} -> -(1 - (2*mass)/coordinates[[2]]), {2, 2} -> 1,
{1, 2} -> \[PlusMinus]Sqrt[(2*mass)/coordinates[[2]]], {2, 1} -> \[PlusMinus]Sqrt[(2*mass)/coordinates[[2]]],
{3, 3} -> coordinates[[2]]^2, {4, 4} -> coordinates[[2]]^2*Sin[coordinates[[3]]]^2}]], coordinates, index1,
index2] /; Length[coordinates] == 4 && BooleanQ[index1] && BooleanQ[index2]
MetricTensor["IngoingGullstrandPainleve"] :=
MetricTensor[Normal[SparseArray[{{1, 1} -> -(1 - (2*"\[FormalCapitalM]")/"\[FormalR]"), {2, 2} -> 1, {1, 2} -> Sqrt[(2*"\[FormalCapitalM]")/"\[FormalR]"],
{2, 1} -> Sqrt[(2*"\[FormalCapitalM]")/"\[FormalR]"], {3, 3} -> "\[FormalR]"^2, {4, 4} -> "\[FormalR]"^2*Sin["\[FormalTheta]"]^2}]],
{"\[FormalCapitalT]", "\[FormalR]", "\[FormalTheta]", "\[FormalPhi]"}, True, True]
MetricTensor["IngoingGullstrandPainleve", coordinates_List] :=
MetricTensor[Normal[SparseArray[{{1, 1} -> -(1 - (2*"\[FormalCapitalM]")/coordinates[[2]]), {2, 2} -> 1,
{1, 2} -> Sqrt[(2*"\[FormalCapitalM]")/coordinates[[2]]], {2, 1} -> Sqrt[(2*"\[FormalCapitalM]")/coordinates[[2]]],
{3, 3} -> coordinates[[2]]^2, {4, 4} -> coordinates[[2]]^2*Sin[coordinates[[3]]]^2}]], coordinates, True, True] /;
Length[coordinates] == 4
MetricTensor["IngoingGullstrandPainleve", index1_, index2_] :=
MetricTensor[Normal[SparseArray[{{1, 1} -> -(1 - (2*"\[FormalCapitalM]")/"\[FormalR]"), {2, 2} -> 1, {1, 2} -> Sqrt[(2*"\[FormalCapitalM]")/"\[FormalR]"],
{2, 1} -> Sqrt[(2*"\[FormalCapitalM]")/"\[FormalR]"], {3, 3} -> "\[FormalR]"^2, {4, 4} -> "\[FormalR]"^2*Sin["\[FormalTheta]"]^2}]],
{"\[FormalCapitalT]", "\[FormalR]", "\[FormalTheta]", "\[FormalPhi]"}, index1, index2] /; BooleanQ[index1] && BooleanQ[index2]
MetricTensor["IngoingGullstrandPainleve", coordinates_List, index1_, index2_] :=
MetricTensor[Normal[SparseArray[{{1, 1} -> -(1 - (2*"\[FormalCapitalM]")/coordinates[[2]]), {2, 2} -> 1,
{1, 2} -> Sqrt[(2*"\[FormalCapitalM]")/coordinates[[2]]], {2, 1} -> Sqrt[(2*"\[FormalCapitalM]")/coordinates[[2]]],
{3, 3} -> coordinates[[2]]^2, {4, 4} -> coordinates[[2]]^2*Sin[coordinates[[3]]]^2}]], coordinates, index1,
index2] /; Length[coordinates] == 4 && BooleanQ[index1] && BooleanQ[index2]
MetricTensor[{"IngoingGullstrandPainleve", mass_}] :=
MetricTensor[Normal[SparseArray[{{1, 1} -> -(1 - (2*mass)/"\[FormalR]"), {2, 2} -> 1, {1, 2} -> Sqrt[(2*mass)/"\[FormalR]"],
{2, 1} -> Sqrt[(2*mass)/"\[FormalR]"], {3, 3} -> "\[FormalR]"^2, {4, 4} -> "\[FormalR]"^2*Sin["\[FormalTheta]"]^2}]], {"\[FormalCapitalT]", "\[FormalR]", "\[FormalTheta]", "\[FormalPhi]"},
True, True]
MetricTensor[{"IngoingGullstrandPainleve", mass_}, coordinates_List] :=
MetricTensor[Normal[SparseArray[{{1, 1} -> -(1 - (2*mass)/coordinates[[2]]), {2, 2} -> 1,
{1, 2} -> Sqrt[(2*mass)/coordinates[[2]]], {2, 1} -> Sqrt[(2*mass)/coordinates[[2]]],
{3, 3} -> coordinates[[2]]^2, {4, 4} -> coordinates[[2]]^2*Sin[coordinates[[3]]]^2}]], coordinates, True, True] /;
Length[coordinates] == 4
MetricTensor[{"IngoingGullstrandPainleve", mass_}, index1_, index2_] :=
MetricTensor[Normal[SparseArray[{{1, 1} -> -(1 - (2*mass)/"\[FormalR]"), {2, 2} -> 1, {1, 2} -> Sqrt[(2*mass)/"\[FormalR]"],
{2, 1} -> Sqrt[(2*mass)/"\[FormalR]"], {3, 3} -> "\[FormalR]"^2, {4, 4} -> "\[FormalR]"^2*Sin["\[FormalTheta]"]^2}]],
{"\[FormalCapitalT]", "\[FormalR]", "\[FormalTheta]", "\[FormalPhi]"}, index1, index2] /; BooleanQ[index1] && BooleanQ[index2]
MetricTensor[{"IngoingGullstrandPainleve", mass_}, coordinates_List, index1_, index2_] :=
MetricTensor[Normal[SparseArray[{{1, 1} -> -(1 - (2*mass)/coordinates[[2]]), {2, 2} -> 1,
{1, 2} -> Sqrt[(2*mass)/coordinates[[2]]], {2, 1} -> Sqrt[(2*mass)/coordinates[[2]]],
{3, 3} -> coordinates[[2]]^2, {4, 4} -> coordinates[[2]]^2*Sin[coordinates[[3]]]^2}]], coordinates, index1,
index2] /; Length[coordinates] == 4 && BooleanQ[index1] && BooleanQ[index2]
MetricTensor["OutgoingGullstrandPainleve"] :=
MetricTensor[Normal[SparseArray[{{1, 1} -> -(1 - (2*"\[FormalCapitalM]")/"\[FormalR]"), {2, 2} -> 1, {1, 2} -> -Sqrt[(2*"\[FormalCapitalM]")/"\[FormalR]"],
{2, 1} -> -Sqrt[(2*"\[FormalCapitalM]")/"\[FormalR]"], {3, 3} -> "\[FormalR]"^2, {4, 4} -> "\[FormalR]"^2*Sin["\[FormalTheta]"]^2}]],
{"\[FormalCapitalT]", "\[FormalR]", "\[FormalTheta]", "\[FormalPhi]"}, True, True]
MetricTensor["OutgoingGullstrandPainleve", coordinates_List] :=
MetricTensor[Normal[SparseArray[{{1, 1} -> -(1 - (2*"\[FormalCapitalM]")/coordinates[[2]]), {2, 2} -> 1,
{1, 2} -> -Sqrt[(2*"\[FormalCapitalM]")/coordinates[[2]]], {2, 1} -> -Sqrt[(2*"\[FormalCapitalM]")/coordinates[[2]]],
{3, 3} -> coordinates[[2]]^2, {4, 4} -> coordinates[[2]]^2*Sin[coordinates[[3]]]^2}]], coordinates, True, True] /;
Length[coordinates] == 4
MetricTensor["OutgoingGullstrandPainleve", index1_, index2_] :=
MetricTensor[Normal[SparseArray[{{1, 1} -> -(1 - (2*"\[FormalCapitalM]")/"\[FormalR]"), {2, 2} -> 1, {1, 2} -> -Sqrt[(2*"\[FormalCapitalM]")/"\[FormalR]"],
{2, 1} -> -Sqrt[(2*"\[FormalCapitalM]")/"\[FormalR]"], {3, 3} -> "\[FormalR]"^2, {4, 4} -> "\[FormalR]"^2*Sin["\[FormalTheta]"]^2}]],
{"\[FormalCapitalT]", "\[FormalR]", "\[FormalTheta]", "\[FormalPhi]"}, index1, index2] /; BooleanQ[index1] && BooleanQ[index2]
MetricTensor["OutgoingGullstrandPainleve", coordinates_List, index1_, index2_] :=
MetricTensor[Normal[SparseArray[{{1, 1} -> -(1 - (2*"\[FormalCapitalM]")/coordinates[[2]]), {2, 2} -> 1,
{1, 2} -> -Sqrt[(2*"\[FormalCapitalM]")/coordinates[[2]]], {2, 1} -> -Sqrt[(2*"\[FormalCapitalM]")/coordinates[[2]]],
{3, 3} -> coordinates[[2]]^2, {4, 4} -> coordinates[[2]]^2*Sin[coordinates[[3]]]^2}]], coordinates, index1,
index2] /; Length[coordinates] == 4 && BooleanQ[index1] && BooleanQ[index2]
MetricTensor[{"OutgoingGullstrandPainleve", mass_}] :=
MetricTensor[Normal[SparseArray[{{1, 1} -> -(1 - (2*mass)/"\[FormalR]"), {2, 2} -> 1, {1, 2} -> -Sqrt[(2*mass)/"\[FormalR]"],
{2, 1} -> -Sqrt[(2*mass)/"\[FormalR]"], {3, 3} -> "\[FormalR]"^2, {4, 4} -> "\[FormalR]"^2*Sin["\[FormalTheta]"]^2}]],
{"\[FormalCapitalT]", "\[FormalR]", "\[FormalTheta]", "\[FormalPhi]"}, True, True]
MetricTensor[{"OutgoingGullstrandPainleve", mass_}, coordinates_List] :=
MetricTensor[Normal[SparseArray[{{1, 1} -> -(1 - (2*mass)/coordinates[[2]]), {2, 2} -> 1,
{1, 2} -> -Sqrt[(2*mass)/coordinates[[2]]], {2, 1} -> -Sqrt[(2*mass)/coordinates[[2]]],
{3, 3} -> coordinates[[2]]^2, {4, 4} -> coordinates[[2]]^2*Sin[coordinates[[3]]]^2}]], coordinates, True, True] /;
Length[coordinates] == 4
MetricTensor[{"OutgoingGullstrandPainleve", mass_}, index1_, index2_] :=
MetricTensor[Normal[SparseArray[{{1, 1} -> -(1 - (2*mass)/"\[FormalR]"), {2, 2} -> 1, {1, 2} -> -Sqrt[(2*mass)/"\[FormalR]"],
{2, 1} -> -Sqrt[(2*mass)/"\[FormalR]"], {3, 3} -> "\[FormalR]"^2, {4, 4} -> "\[FormalR]"^2*Sin["\[FormalTheta]"]^2}]],
{"\[FormalCapitalT]", "\[FormalR]", "\[FormalTheta]", "\[FormalPhi]"}, index1, index2] /; BooleanQ[index1] && BooleanQ[index2]
MetricTensor[{"OutgoingGullstrandPainleve", mass_}, coordinates_List, index1_, index2_] :=
MetricTensor[Normal[SparseArray[{{1, 1} -> -(1 - (2*mass)/coordinates[[2]]), {2, 2} -> 1,
{1, 2} -> -Sqrt[(2*mass)/coordinates[[2]]], {2, 1} -> -Sqrt[(2*mass)/coordinates[[2]]],
{3, 3} -> coordinates[[2]]^2, {4, 4} -> coordinates[[2]]^2*Sin[coordinates[[3]]]^2}]], coordinates, index1,
index2] /; Length[coordinates] == 4 && BooleanQ[index1] && BooleanQ[index2]
MetricTensor["KruskalSzekeres"] := Module[{radialCoordinate},
radialCoordinate = (2*"\[FormalCapitalM]")*(1 + ProductLog[("\[FormalCapitalX]"^2 - "\[FormalCapitalT]"^2)/E]);
MetricTensor[DiagonalMatrix[{(-((32*"\[FormalCapitalM]"^3)/radialCoordinate))*Exp[-(radialCoordinate/(2*"\[FormalCapitalM]"))],
((32*"\[FormalCapitalM]"^3)/radialCoordinate)*Exp[-(radialCoordinate/(2*"\[FormalCapitalM]"))], radialCoordinate^2,
radialCoordinate^2*Sin["\[FormalTheta]"]^2}], {"\[FormalCapitalT]", "\[FormalCapitalX]", "\[FormalTheta]", "\[FormalPhi]"}, True, True]]
MetricTensor["KruskalSzekeres", coordinates_List] :=
Module[{radialCoordinate}, radialCoordinate = (2*"\[FormalCapitalM]")*(1 + ProductLog[(coordinates[[2]]^2 - coordinates[[1]]^2)/E]);
MetricTensor[DiagonalMatrix[{(-((32*"\[FormalCapitalM]"^3)/radialCoordinate))*Exp[-(radialCoordinate/(2*"\[FormalCapitalM]"))],
((32*"\[FormalCapitalM]"^3)/radialCoordinate)*Exp[-(radialCoordinate/(2*"\[FormalCapitalM]"))], radialCoordinate^2,
radialCoordinate^2*Sin[coordinates[[3]]]^2}], coordinates, True, True]] /; Length[coordinates] == 4
MetricTensor["KruskalSzekeres", index1_, index2_] :=
Module[{radialCoordinate}, radialCoordinate = (2*"\[FormalCapitalM]")*(1 + ProductLog[("\[FormalCapitalX]"^2 - "\[FormalCapitalT]"^2)/E]);
MetricTensor[DiagonalMatrix[{(-((32*"\[FormalCapitalM]"^3)/radialCoordinate))*Exp[-(radialCoordinate/(2*"\[FormalCapitalM]"))],
((32*"\[FormalCapitalM]"^3)/radialCoordinate)*Exp[-(radialCoordinate/(2*"\[FormalCapitalM]"))], radialCoordinate^2,
radialCoordinate^2*Sin["\[FormalTheta]"]^2}], {"\[FormalCapitalT]", "\[FormalCapitalX]", "\[FormalTheta]", "\[FormalPhi]"}, index1, index2]] /;
BooleanQ[index1] && BooleanQ[index2]
MetricTensor["KruskalSzekeres", coordinates_List, index1_, index2_] :=
Module[{radialCoordinate}, radialCoordinate = (2*"\[FormalCapitalM]")*(1 + ProductLog[(coordinates[[2]]^2 - coordinates[[1]]^2)/E]);
MetricTensor[DiagonalMatrix[{(-((32*"\[FormalCapitalM]"^3)/radialCoordinate))*Exp[-(radialCoordinate/(2*"\[FormalCapitalM]"))],
((32*"\[FormalCapitalM]"^3)/radialCoordinate)*Exp[-(radialCoordinate/(2*"\[FormalCapitalM]"))], radialCoordinate^2,
radialCoordinate^2*Sin[coordinates[[3]]]^2}], coordinates, index1, index2]] /;
Length[coordinates] == 4 && BooleanQ[index1] && BooleanQ[index2]
MetricTensor[{"KruskalSzekeres", mass_}] := Module[{radialCoordinate},
radialCoordinate = (2*mass)*(1 + ProductLog[("\[FormalCapitalX]"^2 - "\[FormalCapitalT]"^2)/E]);
MetricTensor[DiagonalMatrix[{(-((32*mass^3)/radialCoordinate))*Exp[-(radialCoordinate/(2*mass))],
((32*mass^3)/radialCoordinate)*Exp[-(radialCoordinate/(2*mass))], radialCoordinate^2,
radialCoordinate^2*Sin["\[FormalTheta]"]^2}], {"\[FormalCapitalT]", "\[FormalCapitalX]", "\[FormalTheta]", "\[FormalPhi]"}, True, True]]
MetricTensor[{"KruskalSzekeres", mass_}, coordinates_List] :=
Module[{radialCoordinate}, radialCoordinate = (2*mass)*(1 + ProductLog[(coordinates[[2]]^2 - coordinates[[1]]^2)/E]);
MetricTensor[DiagonalMatrix[{(-((32*mass^3)/radialCoordinate))*Exp[-(radialCoordinate/(2*mass))],
((32*mass^3)/radialCoordinate)*Exp[-(radialCoordinate/(2*mass))], radialCoordinate^2,
radialCoordinate^2*Sin[coordinates[[3]]]^2}], coordinates, True, True]] /; Length[coordinates] == 4
MetricTensor[{"KruskalSzekeres", mass_}, index1_, index2_] :=
Module[{radialCoordinate}, radialCoordinate = (2*mass)*(1 + ProductLog[("\[FormalCapitalX]"^2 - "\[FormalCapitalT]"^2)/E]);
MetricTensor[DiagonalMatrix[{(-((32*mass^3)/radialCoordinate))*Exp[-(radialCoordinate/(2*mass))],
((32*mass^3)/radialCoordinate)*Exp[-(radialCoordinate/(2*mass))], radialCoordinate^2,
radialCoordinate^2*Sin["\[FormalTheta]"]^2}], {"\[FormalCapitalT]", "\[FormalCapitalX]", "\[FormalTheta]", "\[FormalPhi]"}, index1, index2]] /;
BooleanQ[index1] && BooleanQ[index2]
MetricTensor[{"KruskalSzekeres", mass_}, coordinates_List, index1_, index2_] :=
Module[{radialCoordinate}, radialCoordinate = (2*mass)*(1 + ProductLog[(coordinates[[2]]^2 - coordinates[[1]]^2)/E]);
MetricTensor[DiagonalMatrix[{(-((32*mass^3)/radialCoordinate))*Exp[-(radialCoordinate/(2*mass))],
((32*mass^3)/radialCoordinate)*Exp[-(radialCoordinate/(2*mass))], radialCoordinate^2,
radialCoordinate^2*Sin[coordinates[[3]]]^2}], coordinates, index1, index2]] /;
Length[coordinates] == 4 && BooleanQ[index1] && BooleanQ[index2]
MetricTensor["Kerr"] := MetricTensor[
Normal[SparseArray[{{1, 1} -> -(1 - (2*"\[FormalCapitalM]"*"\[FormalR]")/("\[FormalR]"^2 + ("\[FormalCapitalJ]"/"\[FormalCapitalM]")^2*Cos["\[FormalTheta]"]^2)),
{2, 2} -> ("\[FormalR]"^2 + ("\[FormalCapitalJ]"/"\[FormalCapitalM]")^2*Cos["\[FormalTheta]"]^2)/("\[FormalR]"^2 - 2*"\[FormalCapitalM]"*"\[FormalR]" + ("\[FormalCapitalJ]"/"\[FormalCapitalM]")^2),
{3, 3} -> "\[FormalR]"^2 + ("\[FormalCapitalJ]"/"\[FormalCapitalM]")^2*Cos["\[FormalTheta]"]^2,
{4, 4} -> ("\[FormalR]"^2 + ("\[FormalCapitalJ]"/"\[FormalCapitalM]")^2 + (2*"\[FormalR]"*("\[FormalCapitalJ]"^2/"\[FormalCapitalM]")*Sin["\[FormalTheta]"]^2)/
("\[FormalR]"^2 + ("\[FormalCapitalJ]"/"\[FormalCapitalM]")^2*Cos["\[FormalTheta]"]^2))*Sin["\[FormalTheta]"]^2,
{1, 4} -> (-2*"\[FormalR]"*"\[FormalCapitalJ]"*Sin["\[FormalTheta]"]^2)/("\[FormalR]"^2 + ("\[FormalCapitalJ]"/"\[FormalCapitalM]")^2*Cos["\[FormalTheta]"]^2),
{4, 1} -> (-2*"\[FormalR]"*"\[FormalCapitalJ]"*Sin["\[FormalTheta]"]^2)/("\[FormalR]"^2 + ("\[FormalCapitalJ]"/"\[FormalCapitalM]")^2*Cos["\[FormalTheta]"]^2)}]], {"\[FormalT]", "\[FormalR]", "\[FormalTheta]", "\[FormalPhi]"},
True, True]
MetricTensor["Kerr", coordinates_List] :=
MetricTensor[Normal[SparseArray[
{{1, 1} -> -(1 - (2*"\[FormalCapitalM]"*coordinates[[2]])/(coordinates[[2]]^2 + ("\[FormalCapitalJ]"/"\[FormalCapitalM]")^2*Cos[coordinates[[3]]]^2)),
{2, 2} -> (coordinates[[2]]^2 + ("\[FormalCapitalJ]"/"\[FormalCapitalM]")^2*Cos[coordinates[[3]]]^2)/(coordinates[[2]]^2 -
2*"\[FormalCapitalM]"*coordinates[[2]] + ("\[FormalCapitalJ]"/"\[FormalCapitalM]")^2), {3, 3} -> coordinates[[2]]^2 +
("\[FormalCapitalJ]"/"\[FormalCapitalM]")^2*Cos[coordinates[[3]]]^2, {4, 4} -> (coordinates[[2]]^2 + ("\[FormalCapitalJ]"/"\[FormalCapitalM]")^2 +
(2*coordinates[[2]]*("\[FormalCapitalJ]"^2/"\[FormalCapitalM]")*Sin[coordinates[[3]]]^2)/(coordinates[[2]]^2 +
("\[FormalCapitalJ]"/"\[FormalCapitalM]")^2*Cos[coordinates[[3]]]^2))*Sin[coordinates[[3]]]^2,
{1, 4} -> (-2*coordinates[[2]]*"\[FormalCapitalJ]"*Sin[coordinates[[3]]]^2)/(coordinates[[2]]^2 +
("\[FormalCapitalJ]"/"\[FormalCapitalM]")^2*Cos[coordinates[[3]]]^2), {4, 1} -> (-2*coordinates[[2]]*"\[FormalCapitalJ]"*Sin[coordinates[[3]]]^2)/
(coordinates[[2]]^2 + ("\[FormalCapitalJ]"/"\[FormalCapitalM]")^2*Cos[coordinates[[3]]]^2)}]], coordinates, True, True] /;
Length[coordinates] == 4
MetricTensor["Kerr", index1_, index2_] :=
MetricTensor[Normal[SparseArray[{{1, 1} -> -(1 - (2*"\[FormalCapitalM]"*"\[FormalR]")/("\[FormalR]"^2 + ("\[FormalCapitalJ]"/"\[FormalCapitalM]")^2*Cos["\[FormalTheta]"]^2)),
{2, 2} -> ("\[FormalR]"^2 + ("\[FormalCapitalJ]"/"\[FormalCapitalM]")^2*Cos["\[FormalTheta]"]^2)/("\[FormalR]"^2 - 2*"\[FormalCapitalM]"*"\[FormalR]" + ("\[FormalCapitalJ]"/"\[FormalCapitalM]")^2),
{3, 3} -> "\[FormalR]"^2 + ("\[FormalCapitalJ]"/"\[FormalCapitalM]")^2*Cos["\[FormalTheta]"]^2,
{4, 4} -> ("\[FormalR]"^2 + ("\[FormalCapitalJ]"/"\[FormalCapitalM]")^2 + (2*"\[FormalR]"*("\[FormalCapitalJ]"^2/"\[FormalCapitalM]")*Sin["\[FormalTheta]"]^2)/
("\[FormalR]"^2 + ("\[FormalCapitalJ]"/"\[FormalCapitalM]")^2*Cos["\[FormalTheta]"]^2))*Sin["\[FormalTheta]"]^2, {1, 4} -> (-2*"\[FormalR]"*"\[FormalCapitalJ]"*Sin["\[FormalTheta]"]^2)/
("\[FormalR]"^2 + ("\[FormalCapitalJ]"/"\[FormalCapitalM]")^2*Cos["\[FormalTheta]"]^2), {4, 1} -> (-2*"\[FormalR]"*"\[FormalCapitalJ]"*Sin["\[FormalTheta]"]^2)/
("\[FormalR]"^2 + ("\[FormalCapitalJ]"/"\[FormalCapitalM]")^2*Cos["\[FormalTheta]"]^2)}]], {"\[FormalT]", "\[FormalR]", "\[FormalTheta]", "\[FormalPhi]"}, index1, index2] /;
BooleanQ[index1] && BooleanQ[index2]
MetricTensor["Kerr", coordinates_List, index1_, index2_] :=
MetricTensor[Normal[SparseArray[
{{1, 1} -> -(1 - (2*"\[FormalCapitalM]"*coordinates[[2]])/(coordinates[[2]]^2 + ("\[FormalCapitalJ]"/"\[FormalCapitalM]")^2*Cos[coordinates[[3]]]^2)),
{2, 2} -> (coordinates[[2]]^2 + ("\[FormalCapitalJ]"/"\[FormalCapitalM]")^2*Cos[coordinates[[3]]]^2)/(coordinates[[2]]^2 -
2*"\[FormalCapitalM]"*coordinates[[2]] + ("\[FormalCapitalJ]"/"\[FormalCapitalM]")^2), {3, 3} -> coordinates[[2]]^2 +
("\[FormalCapitalJ]"/"\[FormalCapitalM]")^2*Cos[coordinates[[3]]]^2, {4, 4} -> (coordinates[[2]]^2 + ("\[FormalCapitalJ]"/"\[FormalCapitalM]")^2 +
(2*coordinates[[2]]*("\[FormalCapitalJ]"^2/"\[FormalCapitalM]")*Sin[coordinates[[3]]]^2)/(coordinates[[2]]^2 +
("\[FormalCapitalJ]"/"\[FormalCapitalM]")^2*Cos[coordinates[[3]]]^2))*Sin[coordinates[[3]]]^2,
{1, 4} -> (-2*coordinates[[2]]*"\[FormalCapitalJ]"*Sin[coordinates[[3]]]^2)/(coordinates[[2]]^2 +
("\[FormalCapitalJ]"/"\[FormalCapitalM]")^2*Cos[coordinates[[3]]]^2), {4, 1} -> (-2*coordinates[[2]]*"\[FormalCapitalJ]"*Sin[coordinates[[3]]]^2)/
(coordinates[[2]]^2 + ("\[FormalCapitalJ]"/"\[FormalCapitalM]")^2*Cos[coordinates[[3]]]^2)}]], coordinates, index1, index2] /;
Length[coordinates] == 4 && BooleanQ[index1] && BooleanQ[index2]
MetricTensor[{"Kerr", mass_}] :=
MetricTensor[Normal[SparseArray[{{1, 1} -> -(1 - (2*mass*"\[FormalR]")/("\[FormalR]"^2 + ("\[FormalCapitalJ]"/mass)^2*Cos["\[FormalTheta]"]^2)),
{2, 2} -> ("\[FormalR]"^2 + ("\[FormalCapitalJ]"/mass)^2*Cos["\[FormalTheta]"]^2)/("\[FormalR]"^2 - 2*mass*"\[FormalR]" + ("\[FormalCapitalJ]"/mass)^2),
{3, 3} -> "\[FormalR]"^2 + ("\[FormalCapitalJ]"/mass)^2*Cos["\[FormalTheta]"]^2,
{4, 4} -> ("\[FormalR]"^2 + ("\[FormalCapitalJ]"/mass)^2 + (2*"\[FormalR]"*("\[FormalCapitalJ]"^2/mass)*Sin["\[FormalTheta]"]^2)/
("\[FormalR]"^2 + ("\[FormalCapitalJ]"/mass)^2*Cos["\[FormalTheta]"]^2))*Sin["\[FormalTheta]"]^2, {1, 4} -> (-2*"\[FormalR]"*"\[FormalCapitalJ]"*Sin["\[FormalTheta]"]^2)/
("\[FormalR]"^2 + ("\[FormalCapitalJ]"/mass)^2*Cos["\[FormalTheta]"]^2), {4, 1} -> (-2*"\[FormalR]"*"\[FormalCapitalJ]"*Sin["\[FormalTheta]"]^2)/
("\[FormalR]"^2 + ("\[FormalCapitalJ]"/mass)^2*Cos["\[FormalTheta]"]^2)}]], {"\[FormalT]", "\[FormalR]", "\[FormalTheta]", "\[FormalPhi]"}, True, True]
MetricTensor[{"Kerr", mass_}, coordinates_] :=
MetricTensor[Normal[SparseArray[
{{1, 1} -> -(1 - (2*mass*coordinates[[2]])/(coordinates[[2]]^2 + ("\[FormalCapitalJ]"/mass)^2*Cos[coordinates[[3]]]^2)),
{2, 2} -> (coordinates[[2]]^2 + ("\[FormalCapitalJ]"/mass)^2*Cos[coordinates[[3]]]^2)/(coordinates[[2]]^2 -
2*mass*coordinates[[2]] + ("\[FormalCapitalJ]"/mass)^2), {3, 3} -> coordinates[[2]]^2 +
("\[FormalCapitalJ]"/mass)^2*Cos[coordinates[[3]]]^2, {4, 4} -> (coordinates[[2]]^2 + ("\[FormalCapitalJ]"/mass)^2 +
(2*coordinates[[2]]*("\[FormalCapitalJ]"^2/mass)*Sin[coordinates[[3]]]^2)/(coordinates[[2]]^2 +
("\[FormalCapitalJ]"/mass)^2*Cos[coordinates[[3]]]^2))*Sin[coordinates[[3]]]^2,
{1, 4} -> (-2*coordinates[[2]]*"\[FormalCapitalJ]"*Sin[coordinates[[3]]]^2)/(coordinates[[2]]^2 +
("\[FormalCapitalJ]"/mass)^2*Cos[coordinates[[3]]]^2), {4, 1} -> (-2*coordinates[[2]]*"\[FormalCapitalJ]"*Sin[coordinates[[3]]]^2)/
(coordinates[[2]]^2 + ("\[FormalCapitalJ]"/mass)^2*Cos[coordinates[[3]]]^2)}]], coordinates, True, True] /;
Length[coordinates] == 4
MetricTensor[{"Kerr", mass_}, index1_, index2_] :=
MetricTensor[Normal[SparseArray[{{1, 1} -> -(1 - (2*mass*"\[FormalR]")/("\[FormalR]"^2 + ("\[FormalCapitalJ]"/mass)^2*Cos["\[FormalTheta]"]^2)),
{2, 2} -> ("\[FormalR]"^2 + ("\[FormalCapitalJ]"/mass)^2*Cos["\[FormalTheta]"]^2)/("\[FormalR]"^2 - 2*mass*"\[FormalR]" + ("\[FormalCapitalJ]"/mass)^2),
{3, 3} -> "\[FormalR]"^2 + ("\[FormalCapitalJ]"/mass)^2*Cos["\[FormalTheta]"]^2,
{4, 4} -> ("\[FormalR]"^2 + ("\[FormalCapitalJ]"/mass)^2 + (2*"\[FormalR]"*("\[FormalCapitalJ]"^2/mass)*Sin["\[FormalTheta]"]^2)/
("\[FormalR]"^2 + ("\[FormalCapitalJ]"/mass)^2*Cos["\[FormalTheta]"]^2))*Sin["\[FormalTheta]"]^2, {1, 4} -> (-2*"\[FormalR]"*"\[FormalCapitalJ]"*Sin["\[FormalTheta]"]^2)/
("\[FormalR]"^2 + ("\[FormalCapitalJ]"/mass)^2*Cos["\[FormalTheta]"]^2), {4, 1} -> (-2*"\[FormalR]"*"\[FormalCapitalJ]"*Sin["\[FormalTheta]"]^2)/
("\[FormalR]"^2 + ("\[FormalCapitalJ]"/mass)^2*Cos["\[FormalTheta]"]^2)}]], {"\[FormalT]", "\[FormalR]", "\[FormalTheta]", "\[FormalPhi]"}, index1, index2] /;
BooleanQ[index1] && BooleanQ[index2]
MetricTensor[{"Kerr", mass_}, coordinates_List, index1_, index2_] :=
MetricTensor[Normal[SparseArray[
{{1, 1} -> -(1 - (2*mass*coordinates[[2]])/(coordinates[[2]]^2 + ("\[FormalCapitalJ]"/mass)^2*Cos[coordinates[[3]]]^2)),
{2, 2} -> (coordinates[[2]]^2 + ("\[FormalCapitalJ]"/mass)^2*Cos[coordinates[[3]]]^2)/(coordinates[[2]]^2 -
2*mass*coordinates[[2]] + ("\[FormalCapitalJ]"/mass)^2), {3, 3} -> coordinates[[2]]^2 +
("\[FormalCapitalJ]"/mass)^2*Cos[coordinates[[3]]]^2, {4, 4} -> (coordinates[[2]]^2 + ("\[FormalCapitalJ]"/mass)^2 +
(2*coordinates[[2]]*("\[FormalCapitalJ]"^2/mass)*Sin[coordinates[[3]]]^2)/(coordinates[[2]]^2 +
("\[FormalCapitalJ]"/mass)^2*Cos[coordinates[[3]]]^2))*Sin[coordinates[[3]]]^2,
{1, 4} -> (-2*coordinates[[2]]*"\[FormalCapitalJ]"*Sin[coordinates[[3]]]^2)/(coordinates[[2]]^2 +
("\[FormalCapitalJ]"/mass)^2*Cos[coordinates[[3]]]^2), {4, 1} -> (-2*coordinates[[2]]*"\[FormalCapitalJ]"*Sin[coordinates[[3]]]^2)/
(coordinates[[2]]^2 + ("\[FormalCapitalJ]"/mass)^2*Cos[coordinates[[3]]]^2)}]], coordinates, index1, index2] /;
Length[coordinates] == 4 && BooleanQ[index1] && BooleanQ[index2]
MetricTensor[{"Kerr", mass_, angularMomentum_}] :=
MetricTensor[Normal[SparseArray[{{1, 1} -> -(1 - (2*mass*"\[FormalR]")/("\[FormalR]"^2 + (angularMomentum/mass)^2*Cos["\[FormalTheta]"]^2)),
{2, 2} -> ("\[FormalR]"^2 + (angularMomentum/mass)^2*Cos["\[FormalTheta]"]^2)/("\[FormalR]"^2 - 2*mass*"\[FormalR]" + (angularMomentum/mass)^2),
{3, 3} -> "\[FormalR]"^2 + (angularMomentum/mass)^2*Cos["\[FormalTheta]"]^2,
{4, 4} -> ("\[FormalR]"^2 + (angularMomentum/mass)^2 + (2*"\[FormalR]"*(angularMomentum^2/mass)*Sin["\[FormalTheta]"]^2)/
("\[FormalR]"^2 + (angularMomentum/mass)^2*Cos["\[FormalTheta]"]^2))*Sin["\[FormalTheta]"]^2,
{1, 4} -> (-2*"\[FormalR]"*angularMomentum*Sin["\[FormalTheta]"]^2)/("\[FormalR]"^2 + (angularMomentum/mass)^2*Cos["\[FormalTheta]"]^2),
{4, 1} -> (-2*"\[FormalR]"*angularMomentum*Sin["\[FormalTheta]"]^2)/("\[FormalR]"^2 + (angularMomentum/mass)^2*Cos["\[FormalTheta]"]^2)}]],
{"\[FormalT]", "\[FormalR]", "\[FormalTheta]", "\[FormalPhi]"}, True, True]
MetricTensor[{"Kerr", mass_, angularMomentum_}, coordinates_List] :=
MetricTensor[Normal[SparseArray[
{{1, 1} -> -(1 - (2*mass*coordinates[[2]])/(coordinates[[2]]^2 + (angularMomentum/mass)^2*
Cos[coordinates[[3]]]^2)), {2, 2} -> (coordinates[[2]]^2 + (angularMomentum/mass)^2*Cos[coordinates[[3]]]^2)/
(coordinates[[2]]^2 - 2*mass*coordinates[[2]] + (angularMomentum/mass)^2),
{3, 3} -> coordinates[[2]]^2 + (angularMomentum/mass)^2*Cos[coordinates[[3]]]^2,
{4, 4} -> (coordinates[[2]]^2 + (angularMomentum/mass)^2 + (2*coordinates[[2]]*(angularMomentum^2/mass)*
Sin[coordinates[[3]]]^2)/(coordinates[[2]]^2 + (angularMomentum/mass)^2*Cos[coordinates[[3]]]^2))*
Sin[coordinates[[3]]]^2, {1, 4} -> (-2*coordinates[[2]]*angularMomentum*Sin[coordinates[[3]]]^2)/
(coordinates[[2]]^2 + (angularMomentum/mass)^2*Cos[coordinates[[3]]]^2),
{4, 1} -> (-2*coordinates[[2]]*angularMomentum*Sin[coordinates[[3]]]^2)/(coordinates[[2]]^2 +
(angularMomentum/mass)^2*Cos[coordinates[[3]]]^2)}]], coordinates, True, True] /; Length[coordinates] == 4
MetricTensor[{"Kerr", mass_, angularMomentum_}, index1_, index2_] :=
MetricTensor[Normal[SparseArray[{{1, 1} -> -(1 - (2*mass*"\[FormalR]")/("\[FormalR]"^2 + (angularMomentum/mass)^2*Cos["\[FormalTheta]"]^2)),
{2, 2} -> ("\[FormalR]"^2 + (angularMomentum/mass)^2*Cos["\[FormalTheta]"]^2)/("\[FormalR]"^2 - 2*mass*"\[FormalR]" + (angularMomentum/mass)^2),
{3, 3} -> "\[FormalR]"^2 + (angularMomentum/mass)^2*Cos["\[FormalTheta]"]^2,
{4, 4} -> ("\[FormalR]"^2 + (angularMomentum/mass)^2 + (2*"\[FormalR]"*(angularMomentum^2/mass)*Sin["\[FormalTheta]"]^2)/
("\[FormalR]"^2 + (angularMomentum/mass)^2*Cos["\[FormalTheta]"]^2))*Sin["\[FormalTheta]"]^2,
{1, 4} -> (-2*"\[FormalR]"*angularMomentum*Sin["\[FormalTheta]"]^2)/("\[FormalR]"^2 + (angularMomentum/mass)^2*Cos["\[FormalTheta]"]^2),
{4, 1} -> (-2*"\[FormalR]"*angularMomentum*Sin["\[FormalTheta]"]^2)/("\[FormalR]"^2 + (angularMomentum/mass)^2*Cos["\[FormalTheta]"]^2)}]],
{"\[FormalT]", "\[FormalR]", "\[FormalTheta]", "\[FormalPhi]"}, index1, index2] /; BooleanQ[index1] && BooleanQ[index2]
MetricTensor[{"Kerr", mass_, angularMomentum_}, coordinates_List, index1_, index2_] :=
MetricTensor[Normal[SparseArray[
{{1, 1} -> -(1 - (2*mass*coordinates[[2]])/(coordinates[[2]]^2 + (angularMomentum/mass)^2*
Cos[coordinates[[3]]]^2)), {2, 2} -> (coordinates[[2]]^2 + (angularMomentum/mass)^2*Cos[coordinates[[3]]]^2)/
(coordinates[[2]]^2 - 2*mass*coordinates[[2]] + (angularMomentum/mass)^2),
{3, 3} -> coordinates[[2]]^2 + (angularMomentum/mass)^2*Cos[coordinates[[3]]]^2,
{4, 4} -> (coordinates[[2]]^2 + (angularMomentum/mass)^2 + (2*coordinates[[2]]*(angularMomentum^2/mass)*
Sin[coordinates[[3]]]^2)/(coordinates[[2]]^2 + (angularMomentum/mass)^2*Cos[coordinates[[3]]]^2))*
Sin[coordinates[[3]]]^2, {1, 4} -> (-2*coordinates[[2]]*angularMomentum*Sin[coordinates[[3]]]^2)/
(coordinates[[2]]^2 + (angularMomentum/mass)^2*Cos[coordinates[[3]]]^2),
{4, 1} -> (-2*coordinates[[2]]*angularMomentum*Sin[coordinates[[3]]]^2)/(coordinates[[2]]^2 +
(angularMomentum/mass)^2*Cos[coordinates[[3]]]^2)}]], coordinates, index1, index2] /;
Length[coordinates] == 4 && BooleanQ[index1] && BooleanQ[index2]
MetricTensor["ReissnerNordstrom"] := MetricTensor[DiagonalMatrix[{-(1 - (2*"\[FormalCapitalM]")/"\[FormalR]" + "\[FormalCapitalQ]"^2/(4*Pi*"\[FormalR]"^2)),
1/(1 - (2*"\[FormalCapitalM]")/"\[FormalR]" + "\[FormalCapitalQ]"^2/(4*Pi*"\[FormalR]"^2)), "\[FormalR]"^2, "\[FormalR]"^2*Sin["\[FormalTheta]"]^2}], {"\[FormalT]", "\[FormalR]", "\[FormalTheta]", "\[FormalPhi]"},
True, True]
MetricTensor["ReissnerNordstrom", coordinates_List] :=
MetricTensor[DiagonalMatrix[{-(1 - (2*"\[FormalCapitalM]")/coordinates[[2]] + "\[FormalCapitalQ]"^2/(4*Pi*coordinates[[2]]^2)),
1/(1 - (2*"\[FormalCapitalM]")/coordinates[[2]] + "\[FormalCapitalQ]"^2/(4*Pi*coordinates[[2]]^2)), coordinates[[2]]^2,
coordinates[[2]]^2*Sin[coordinates[[3]]]^2}], coordinates, True, True] /; Length[coordinates] == 4
MetricTensor["ReissnerNordstrom", index1_, index2_] :=
MetricTensor[DiagonalMatrix[{-(1 - (2*"\[FormalCapitalM]")/"\[FormalR]" + "\[FormalCapitalQ]"^2/(4*Pi*"\[FormalR]"^2)),
1/(1 - (2*"\[FormalCapitalM]")/"\[FormalR]" + "\[FormalCapitalQ]"^2/(4*Pi*"\[FormalR]"^2)), "\[FormalR]"^2, "\[FormalR]"^2*Sin["\[FormalTheta]"]^2}], {"\[FormalT]", "\[FormalR]", "\[FormalTheta]", "\[FormalPhi]"},
index1, index2] /; BooleanQ[index1] && BooleanQ[index2]
MetricTensor["ReissnerNordstrom", coordinates_List, index1_, index2_] :=
MetricTensor[DiagonalMatrix[{-(1 - (2*"\[FormalCapitalM]")/coordinates[[2]] + "\[FormalCapitalQ]"^2/(4*Pi*coordinates[[2]]^2)),
1/(1 - (2*"\[FormalCapitalM]")/coordinates[[2]] + "\[FormalCapitalQ]"^2/(4*Pi*coordinates[[2]]^2)), coordinates[[2]]^2,
coordinates[[2]]^2*Sin[coordinates[[3]]]^2}], coordinates, index1, index2] /;
Length[coordinates] == 4 && BooleanQ[index1] && BooleanQ[index2]
MetricTensor[{"ReissnerNordstrom", mass_}] :=
MetricTensor[DiagonalMatrix[{-(1 - (2*mass)/"\[FormalR]" + "\[FormalCapitalQ]"^2/(4*Pi*"\[FormalR]"^2)),
1/(1 - (2*mass)/"\[FormalR]" + "\[FormalCapitalQ]"^2/(4*Pi*"\[FormalR]"^2)), "\[FormalR]"^2, "\[FormalR]"^2*Sin["\[FormalTheta]"]^2}], {"\[FormalT]", "\[FormalR]", "\[FormalTheta]", "\[FormalPhi]"},
True, True]
MetricTensor[{"ReissnerNordstrom", mass_}, coordinates_List] :=
MetricTensor[DiagonalMatrix[{-(1 - (2*mass)/coordinates[[2]] + "\[FormalCapitalQ]"^2/(4*Pi*coordinates[[2]]^2)),
1/(1 - (2*mass)/coordinates[[2]] + "\[FormalCapitalQ]"^2/(4*Pi*coordinates[[2]]^2)), coordinates[[2]]^2,
coordinates[[2]]^2*Sin[coordinates[[3]]]^2}], coordinates, True, True] /; Length[coordinates] == 4
MetricTensor[{"ReissnerNordstrom", mass_}, index1_, index2_] :=
MetricTensor[DiagonalMatrix[{-(1 - (2*mass)/"\[FormalR]" + "\[FormalCapitalQ]"^2/(4*Pi*"\[FormalR]"^2)),
1/(1 - (2*mass)/"\[FormalR]" + "\[FormalCapitalQ]"^2/(4*Pi*"\[FormalR]"^2)), "\[FormalR]"^2, "\[FormalR]"^2*Sin["\[FormalTheta]"]^2}], {"\[FormalT]", "\[FormalR]", "\[FormalTheta]", "\[FormalPhi]"},
index1, index2] /; BooleanQ[index1] && BooleanQ[index2]
MetricTensor[{"ReissnerNordstrom", mass_}, coordinates_List, index1_, index2_] :=
MetricTensor[DiagonalMatrix[{-(1 - (2*mass)/coordinates[[2]] + "\[FormalCapitalQ]"^2/(4*Pi*coordinates[[2]]^2)),
1/(1 - (2*mass)/coordinates[[2]] + "\[FormalCapitalQ]"^2/(4*Pi*coordinates[[2]]^2)), coordinates[[2]]^2,
coordinates[[2]]^2*Sin[coordinates[[3]]]^2}], coordinates, index1, index2] /;
Length[coordinates] == 4 && BooleanQ[index1] && BooleanQ[index2]
MetricTensor[{"ReissnerNordstrom", mass_, charge_}] :=
MetricTensor[DiagonalMatrix[{-(1 - (2*mass)/"\[FormalR]" + charge^2/(4*Pi*"\[FormalR]"^2)),
1/(1 - (2*mass)/"\[FormalR]" + charge^2/(4*Pi*"\[FormalR]"^2)), "\[FormalR]"^2, "\[FormalR]"^2*Sin["\[FormalTheta]"]^2}], {"\[FormalT]", "\[FormalR]", "\[FormalTheta]", "\[FormalPhi]"},
True, True]
MetricTensor[{"ReissnerNordstrom", mass_, charge_}, coordinates_List] :=
MetricTensor[DiagonalMatrix[{-(1 - (2*mass)/coordinates[[2]] + charge^2/(4*Pi*coordinates[[2]]^2)),
1/(1 - (2*mass)/coordinates[[2]] + charge^2/(4*Pi*coordinates[[2]]^2)), coordinates[[2]]^2,
coordinates[[2]]^2*Sin[coordinates[[3]]]^2}], coordinates, True, True] /; Length[coordinates] == 4
MetricTensor[{"ReissnerNordstrom", mass_, charge_}, index1_, index2_] :=
MetricTensor[DiagonalMatrix[{-(1 - (2*mass)/"\[FormalR]" + charge^2/(4*Pi*"\[FormalR]"^2)),
1/(1 - (2*mass)/"\[FormalR]" + charge^2/(4*Pi*"\[FormalR]"^2)), "\[FormalR]"^2, "\[FormalR]"^2*Sin["\[FormalTheta]"]^2}], {"\[FormalT]", "\[FormalR]", "\[FormalTheta]", "\[FormalPhi]"},
index1, index2] /; BooleanQ[index1] && BooleanQ[index2]
MetricTensor[{"ReissnerNordstrom", mass_, charge_}, coordinates_List, index1_, index2_] :=
MetricTensor[DiagonalMatrix[{-(1 - (2*mass)/coordinates[[2]] + charge^2/(4*Pi*coordinates[[2]]^2)),
1/(1 - (2*mass)/coordinates[[2]] + charge^2/(4*Pi*coordinates[[2]]^2)), coordinates[[2]]^2,
coordinates[[2]]^2*Sin[coordinates[[3]]]^2}], coordinates, index1, index2] /;
Length[coordinates] == 4 && BooleanQ[index1] && BooleanQ[index2]
MetricTensor["KerrNewman"] :=
MetricTensor[Normal[SparseArray[{{1, 1} -> (("\[FormalCapitalJ]"/"\[FormalCapitalM]")^2*Sin["\[FormalTheta]"]^2 - "\[FormalR]"^2 + 2*"\[FormalCapitalM]"*"\[FormalR]" - ("\[FormalCapitalJ]"/"\[FormalCapitalM]")^2 -
"\[FormalCapitalQ]"^2/(4*Pi))/("\[FormalR]"^2 + ("\[FormalCapitalJ]"/"\[FormalCapitalM]")^2*Cos["\[FormalTheta]"]^2), {2, 2} -> ("\[FormalR]"^2 + ("\[FormalCapitalJ]"/"\[FormalCapitalM]")^2*Cos["\[FormalTheta]"]^2)/
("\[FormalR]"^2 - 2*"\[FormalCapitalM]"*"\[FormalR]" + ("\[FormalCapitalJ]"/"\[FormalCapitalM]")^2 + "\[FormalCapitalQ]"^2/(4*Pi)), {3, 3} -> "\[FormalR]"^2 + ("\[FormalCapitalJ]"/"\[FormalCapitalM]")^2*Cos["\[FormalTheta]"]^2,
{4, 4} -> ((("\[FormalR]"^2 + ("\[FormalCapitalJ]"/"\[FormalCapitalM]")^2)^2 - ("\[FormalCapitalJ]"/"\[FormalCapitalM]")^2*("\[FormalR]"^2 - 2*"\[FormalCapitalM]"*"\[FormalR]" + ("\[FormalCapitalJ]"/"\[FormalCapitalM]")^2 +
"\[FormalCapitalQ]"^2/(4*Pi))*Sin["\[FormalTheta]"]^2)/("\[FormalR]"^2 + ("\[FormalCapitalJ]"/"\[FormalCapitalM]")^2*Cos["\[FormalTheta]"]^2))*Sin["\[FormalTheta]"]^2,
{1, 4} -> ((("\[FormalCapitalJ]"/"\[FormalCapitalM]")*("\[FormalR]"^2 - 2*"\[FormalCapitalM]"*"\[FormalR]" + ("\[FormalCapitalJ]"/"\[FormalCapitalM]")^2 + "\[FormalCapitalQ]"^2/(4*Pi)) -
("\[FormalR]"^2 + ("\[FormalCapitalJ]"/"\[FormalCapitalM]")^2)*("\[FormalCapitalJ]"/"\[FormalCapitalM]"))/("\[FormalR]"^2 + ("\[FormalCapitalJ]"/"\[FormalCapitalM]")^2*Cos["\[FormalTheta]"]^2))*Sin["\[FormalTheta]"]^2,
{4, 1} -> ((("\[FormalCapitalJ]"/"\[FormalCapitalM]")*("\[FormalR]"^2 - 2*"\[FormalCapitalM]"*"\[FormalR]" + ("\[FormalCapitalJ]"/"\[FormalCapitalM]")^2 + "\[FormalCapitalQ]"^2/(4*Pi)) -
("\[FormalR]"^2 + ("\[FormalCapitalJ]"/"\[FormalCapitalM]")^2)*("\[FormalCapitalJ]"/"\[FormalCapitalM]"))/("\[FormalR]"^2 + ("\[FormalCapitalJ]"/"\[FormalCapitalM]")^2*Cos["\[FormalTheta]"]^2))*Sin["\[FormalTheta]"]^2}]],
{"\[FormalT]", "\[FormalR]", "\[FormalTheta]", "\[FormalPhi]"}, True, True]
MetricTensor["KerrNewman", coordinates_List] :=
MetricTensor[Normal[SparseArray[{{1, 1} -> (("\[FormalCapitalJ]"/"\[FormalCapitalM]")^2*Sin[coordinates[[3]]]^2 - coordinates[[2]]^2 +
2*"\[FormalCapitalM]"*coordinates[[2]] - ("\[FormalCapitalJ]"/"\[FormalCapitalM]")^2 - "\[FormalCapitalQ]"^2/(4*Pi))/(coordinates[[2]]^2 +
("\[FormalCapitalJ]"/"\[FormalCapitalM]")^2*Cos[coordinates[[3]]]^2),
{2, 2} -> (coordinates[[2]]^2 + ("\[FormalCapitalJ]"/"\[FormalCapitalM]")^2*Cos[coordinates[[3]]]^2)/(coordinates[[2]]^2 -
2*"\[FormalCapitalM]"*coordinates[[2]] + ("\[FormalCapitalJ]"/"\[FormalCapitalM]")^2 + "\[FormalCapitalQ]"^2/(4*Pi)),
{3, 3} -> coordinates[[2]]^2 + ("\[FormalCapitalJ]"/"\[FormalCapitalM]")^2*Cos[coordinates[[3]]]^2,
{4, 4} -> (((coordinates[[2]]^2 + ("\[FormalCapitalJ]"/"\[FormalCapitalM]")^2)^2 - ("\[FormalCapitalJ]"/"\[FormalCapitalM]")^2*(coordinates[[2]]^2 -
2*"\[FormalCapitalM]"*coordinates[[2]] + ("\[FormalCapitalJ]"/"\[FormalCapitalM]")^2 + "\[FormalCapitalQ]"^2/(4*Pi))*Sin[coordinates[[3]]]^2)/
(coordinates[[2]]^2 + ("\[FormalCapitalJ]"/"\[FormalCapitalM]")^2*Cos[coordinates[[3]]]^2))*Sin[coordinates[[3]]]^2,
{1, 4} -> ((("\[FormalCapitalJ]"/"\[FormalCapitalM]")*(coordinates[[2]]^2 - 2*"\[FormalCapitalM]"*coordinates[[2]] + ("\[FormalCapitalJ]"/"\[FormalCapitalM]")^2 + "\[FormalCapitalQ]"^2/(4*Pi)) -
(coordinates[[2]]^2 + ("\[FormalCapitalJ]"/"\[FormalCapitalM]")^2)*("\[FormalCapitalJ]"/"\[FormalCapitalM]"))/(coordinates[[2]]^2 +
("\[FormalCapitalJ]"/"\[FormalCapitalM]")^2*Cos[coordinates[[3]]]^2))*Sin[coordinates[[3]]]^2,
{4, 1} -> ((("\[FormalCapitalJ]"/"\[FormalCapitalM]")*(coordinates[[2]]^2 - 2*"\[FormalCapitalM]"*coordinates[[2]] + ("\[FormalCapitalJ]"/"\[FormalCapitalM]")^2 + "\[FormalCapitalQ]"^2/(4*Pi)) -
(coordinates[[2]]^2 + ("\[FormalCapitalJ]"/"\[FormalCapitalM]")^2)*("\[FormalCapitalJ]"/"\[FormalCapitalM]"))/(coordinates[[2]]^2 +
("\[FormalCapitalJ]"/"\[FormalCapitalM]")^2*Cos[coordinates[[3]]]^2))*Sin[coordinates[[3]]]^2}]], coordinates, True, True] /;
Length[coordinates] == 4
MetricTensor["KerrNewman", index1_, index2_] :=
MetricTensor[Normal[SparseArray[{{1, 1} -> (("\[FormalCapitalJ]"/"\[FormalCapitalM]")^2*Sin["\[FormalTheta]"]^2 - "\[FormalR]"^2 + 2*"\[FormalCapitalM]"*"\[FormalR]" - ("\[FormalCapitalJ]"/"\[FormalCapitalM]")^2 -
"\[FormalCapitalQ]"^2/(4*Pi))/("\[FormalR]"^2 + ("\[FormalCapitalJ]"/"\[FormalCapitalM]")^2*Cos["\[FormalTheta]"]^2), {2, 2} -> ("\[FormalR]"^2 + ("\[FormalCapitalJ]"/"\[FormalCapitalM]")^2*Cos["\[FormalTheta]"]^2)/
("\[FormalR]"^2 - 2*"\[FormalCapitalM]"*"\[FormalR]" + ("\[FormalCapitalJ]"/"\[FormalCapitalM]")^2 + "\[FormalCapitalQ]"^2/(4*Pi)), {3, 3} -> "\[FormalR]"^2 + ("\[FormalCapitalJ]"/"\[FormalCapitalM]")^2*Cos["\[FormalTheta]"]^2,
{4, 4} -> ((("\[FormalR]"^2 + ("\[FormalCapitalJ]"/"\[FormalCapitalM]")^2)^2 - ("\[FormalCapitalJ]"/"\[FormalCapitalM]")^2*("\[FormalR]"^2 - 2*"\[FormalCapitalM]"*"\[FormalR]" + ("\[FormalCapitalJ]"/"\[FormalCapitalM]")^2 +
"\[FormalCapitalQ]"^2/(4*Pi))*Sin["\[FormalTheta]"]^2)/("\[FormalR]"^2 + ("\[FormalCapitalJ]"/"\[FormalCapitalM]")^2*Cos["\[FormalTheta]"]^2))*Sin["\[FormalTheta]"]^2,
{1, 4} -> ((("\[FormalCapitalJ]"/"\[FormalCapitalM]")*("\[FormalR]"^2 - 2*"\[FormalCapitalM]"*"\[FormalR]" + ("\[FormalCapitalJ]"/"\[FormalCapitalM]")^2 + "\[FormalCapitalQ]"^2/(4*Pi)) -
("\[FormalR]"^2 + ("\[FormalCapitalJ]"/"\[FormalCapitalM]")^2)*("\[FormalCapitalJ]"/"\[FormalCapitalM]"))/("\[FormalR]"^2 + ("\[FormalCapitalJ]"/"\[FormalCapitalM]")^2*Cos["\[FormalTheta]"]^2))*Sin["\[FormalTheta]"]^2,
{4, 1} -> ((("\[FormalCapitalJ]"/"\[FormalCapitalM]")*("\[FormalR]"^2 - 2*"\[FormalCapitalM]"*"\[FormalR]" + ("\[FormalCapitalJ]"/"\[FormalCapitalM]")^2 + "\[FormalCapitalQ]"^2/(4*Pi)) -
("\[FormalR]"^2 + ("\[FormalCapitalJ]"/"\[FormalCapitalM]")^2)*("\[FormalCapitalJ]"/"\[FormalCapitalM]"))/("\[FormalR]"^2 + ("\[FormalCapitalJ]"/"\[FormalCapitalM]")^2*Cos["\[FormalTheta]"]^2))*Sin["\[FormalTheta]"]^2}]],
{"\[FormalT]", "\[FormalR]", "\[FormalTheta]", "\[FormalPhi]"}, index1, index2] /; BooleanQ[index1] && BooleanQ[index2]
MetricTensor["KerrNewman", coordinates_List, index1_, index2_] :=
MetricTensor[Normal[SparseArray[{{1, 1} -> (("\[FormalCapitalJ]"/"\[FormalCapitalM]")^2*Sin[coordinates[[3]]]^2 - coordinates[[2]]^2 +
2*"\[FormalCapitalM]"*coordinates[[2]] - ("\[FormalCapitalJ]"/"\[FormalCapitalM]")^2 - "\[FormalCapitalQ]"^2/(4*Pi))/(coordinates[[2]]^2 +
("\[FormalCapitalJ]"/"\[FormalCapitalM]")^2*Cos[coordinates[[3]]]^2),
{2, 2} -> (coordinates[[2]]^2 + ("\[FormalCapitalJ]"/"\[FormalCapitalM]")^2*Cos[coordinates[[3]]]^2)/(coordinates[[2]]^2 -
2*"\[FormalCapitalM]"*coordinates[[2]] + ("\[FormalCapitalJ]"/"\[FormalCapitalM]")^2 + "\[FormalCapitalQ]"^2/(4*Pi)),
{3, 3} -> coordinates[[2]]^2 + ("\[FormalCapitalJ]"/"\[FormalCapitalM]")^2*Cos[coordinates[[3]]]^2,
{4, 4} -> (((coordinates[[2]]^2 + ("\[FormalCapitalJ]"/"\[FormalCapitalM]")^2)^2 - ("\[FormalCapitalJ]"/"\[FormalCapitalM]")^2*(coordinates[[2]]^2 -
2*"\[FormalCapitalM]"*coordinates[[2]] + ("\[FormalCapitalJ]"/"\[FormalCapitalM]")^2 + "\[FormalCapitalQ]"^2/(4*Pi))*Sin[coordinates[[3]]]^2)/
(coordinates[[2]]^2 + ("\[FormalCapitalJ]"/"\[FormalCapitalM]")^2*Cos[coordinates[[3]]]^2))*Sin[coordinates[[3]]]^2,
{1, 4} -> ((("\[FormalCapitalJ]"/"\[FormalCapitalM]")*(coordinates[[2]]^2 - 2*"\[FormalCapitalM]"*coordinates[[2]] + ("\[FormalCapitalJ]"/"\[FormalCapitalM]")^2 + "\[FormalCapitalQ]"^2/(4*Pi)) -
(coordinates[[2]]^2 + ("\[FormalCapitalJ]"/"\[FormalCapitalM]")^2)*("\[FormalCapitalJ]"/"\[FormalCapitalM]"))/(coordinates[[2]]^2 +
("\[FormalCapitalJ]"/"\[FormalCapitalM]")^2*Cos[coordinates[[3]]]^2))*Sin[coordinates[[3]]]^2,
{4, 1} -> ((("\[FormalCapitalJ]"/"\[FormalCapitalM]")*(coordinates[[2]]^2 - 2*"\[FormalCapitalM]"*coordinates[[2]] + ("\[FormalCapitalJ]"/"\[FormalCapitalM]")^2 + "\[FormalCapitalQ]"^2/(4*Pi)) -
(coordinates[[2]]^2 + ("\[FormalCapitalJ]"/"\[FormalCapitalM]")^2)*("\[FormalCapitalJ]"/"\[FormalCapitalM]"))/(coordinates[[2]]^2 +
("\[FormalCapitalJ]"/"\[FormalCapitalM]")^2*Cos[coordinates[[3]]]^2))*Sin[coordinates[[3]]]^2}]], coordinates, index1, index2] /;
Length[coordinates] == 4 && BooleanQ[index1] && BooleanQ[index2]
MetricTensor[{"KerrNewman", mass_}] :=
MetricTensor[Normal[SparseArray[{{1, 1} -> (("\[FormalCapitalJ]"/mass)^2*Sin["\[FormalTheta]"]^2 - "\[FormalR]"^2 + 2*mass*"\[FormalR]" - ("\[FormalCapitalJ]"/mass)^2 -
"\[FormalCapitalQ]"^2/(4*Pi))/("\[FormalR]"^2 + ("\[FormalCapitalJ]"/mass)^2*Cos["\[FormalTheta]"]^2), {2, 2} -> ("\[FormalR]"^2 + ("\[FormalCapitalJ]"/mass)^2*Cos["\[FormalTheta]"]^2)/
("\[FormalR]"^2 - 2*mass*"\[FormalR]" + ("\[FormalCapitalJ]"/mass)^2 + "\[FormalCapitalQ]"^2/(4*Pi)), {3, 3} -> "\[FormalR]"^2 + ("\[FormalCapitalJ]"/mass)^2*Cos["\[FormalTheta]"]^2,
{4, 4} -> ((("\[FormalR]"^2 + ("\[FormalCapitalJ]"/mass)^2)^2 - ("\[FormalCapitalJ]"/mass)^2*("\[FormalR]"^2 - 2*mass*"\[FormalR]" + ("\[FormalCapitalJ]"/mass)^2 +
"\[FormalCapitalQ]"^2/(4*Pi))*Sin["\[FormalTheta]"]^2)/("\[FormalR]"^2 + ("\[FormalCapitalJ]"/mass)^2*Cos["\[FormalTheta]"]^2))*Sin["\[FormalTheta]"]^2,
{1, 4} -> ((("\[FormalCapitalJ]"/mass)*("\[FormalR]"^2 - 2*mass*"\[FormalR]" + ("\[FormalCapitalJ]"/mass)^2 + "\[FormalCapitalQ]"^2/(4*Pi)) -
("\[FormalR]"^2 + ("\[FormalCapitalJ]"/mass)^2)*("\[FormalCapitalJ]"/mass))/("\[FormalR]"^2 + ("\[FormalCapitalJ]"/mass)^2*Cos["\[FormalTheta]"]^2))*Sin["\[FormalTheta]"]^2,
{4, 1} -> ((("\[FormalCapitalJ]"/mass)*("\[FormalR]"^2 - 2*mass*"\[FormalR]" + ("\[FormalCapitalJ]"/mass)^2 + "\[FormalCapitalQ]"^2/(4*Pi)) -
("\[FormalR]"^2 + ("\[FormalCapitalJ]"/mass)^2)*("\[FormalCapitalJ]"/mass))/("\[FormalR]"^2 + ("\[FormalCapitalJ]"/mass)^2*Cos["\[FormalTheta]"]^2))*Sin["\[FormalTheta]"]^2}]],
{"\[FormalT]", "\[FormalR]", "\[FormalTheta]", "\[FormalPhi]"}, True, True]
MetricTensor[{"KerrNewman", mass_}, coordinates_List] :=
MetricTensor[Normal[SparseArray[{{1, 1} -> (("\[FormalCapitalJ]"/mass)^2*Sin[coordinates[[3]]]^2 - coordinates[[2]]^2 +
2*mass*coordinates[[2]] - ("\[FormalCapitalJ]"/mass)^2 - "\[FormalCapitalQ]"^2/(4*Pi))/(coordinates[[2]]^2 +
("\[FormalCapitalJ]"/mass)^2*Cos[coordinates[[3]]]^2),
{2, 2} -> (coordinates[[2]]^2 + ("\[FormalCapitalJ]"/mass)^2*Cos[coordinates[[3]]]^2)/(coordinates[[2]]^2 -
2*mass*coordinates[[2]] + ("\[FormalCapitalJ]"/mass)^2 + "\[FormalCapitalQ]"^2/(4*Pi)),
{3, 3} -> coordinates[[2]]^2 + ("\[FormalCapitalJ]"/mass)^2*Cos[coordinates[[3]]]^2,
{4, 4} -> (((coordinates[[2]]^2 + ("\[FormalCapitalJ]"/mass)^2)^2 - ("\[FormalCapitalJ]"/mass)^2*(coordinates[[2]]^2 -
2*mass*coordinates[[2]] + ("\[FormalCapitalJ]"/mass)^2 + "\[FormalCapitalQ]"^2/(4*Pi))*Sin[coordinates[[3]]]^2)/
(coordinates[[2]]^2 + ("\[FormalCapitalJ]"/mass)^2*Cos[coordinates[[3]]]^2))*Sin[coordinates[[3]]]^2,
{1, 4} -> ((("\[FormalCapitalJ]"/mass)*(coordinates[[2]]^2 - 2*mass*coordinates[[2]] + ("\[FormalCapitalJ]"/mass)^2 + "\[FormalCapitalQ]"^2/(4*Pi)) -
(coordinates[[2]]^2 + ("\[FormalCapitalJ]"/mass)^2)*("\[FormalCapitalJ]"/mass))/(coordinates[[2]]^2 +
("\[FormalCapitalJ]"/mass)^2*Cos[coordinates[[3]]]^2))*Sin[coordinates[[3]]]^2,
{4, 1} -> ((("\[FormalCapitalJ]"/mass)*(coordinates[[2]]^2 - 2*mass*coordinates[[2]] + ("\[FormalCapitalJ]"/mass)^2 + "\[FormalCapitalQ]"^2/(4*Pi)) -
(coordinates[[2]]^2 + ("\[FormalCapitalJ]"/mass)^2)*("\[FormalCapitalJ]"/mass))/(coordinates[[2]]^2 +
("\[FormalCapitalJ]"/mass)^2*Cos[coordinates[[3]]]^2))*Sin[coordinates[[3]]]^2}]], coordinates, True, True] /;
Length[coordinates] == 4
MetricTensor[{"KerrNewman", mass_}, index1_, index2_] :=
MetricTensor[Normal[SparseArray[{{1, 1} -> (("\[FormalCapitalJ]"/mass)^2*Sin["\[FormalTheta]"]^2 - "\[FormalR]"^2 + 2*mass*"\[FormalR]" - ("\[FormalCapitalJ]"/mass)^2 -
"\[FormalCapitalQ]"^2/(4*Pi))/("\[FormalR]"^2 + ("\[FormalCapitalJ]"/mass)^2*Cos["\[FormalTheta]"]^2), {2, 2} -> ("\[FormalR]"^2 + ("\[FormalCapitalJ]"/mass)^2*Cos["\[FormalTheta]"]^2)/
("\[FormalR]"^2 - 2*mass*"\[FormalR]" + ("\[FormalCapitalJ]"/mass)^2 + "\[FormalCapitalQ]"^2/(4*Pi)), {3, 3} -> "\[FormalR]"^2 + ("\[FormalCapitalJ]"/mass)^2*Cos["\[FormalTheta]"]^2,
{4, 4} -> ((("\[FormalR]"^2 + ("\[FormalCapitalJ]"/mass)^2)^2 - ("\[FormalCapitalJ]"/mass)^2*("\[FormalR]"^2 - 2*mass*"\[FormalR]" + ("\[FormalCapitalJ]"/mass)^2 +
"\[FormalCapitalQ]"^2/(4*Pi))*Sin["\[FormalTheta]"]^2)/("\[FormalR]"^2 + ("\[FormalCapitalJ]"/mass)^2*Cos["\[FormalTheta]"]^2))*Sin["\[FormalTheta]"]^2,
{1, 4} -> ((("\[FormalCapitalJ]"/mass)*("\[FormalR]"^2 - 2*mass*"\[FormalR]" + ("\[FormalCapitalJ]"/mass)^2 + "\[FormalCapitalQ]"^2/(4*Pi)) -
("\[FormalR]"^2 + ("\[FormalCapitalJ]"/mass)^2)*("\[FormalCapitalJ]"/mass))/("\[FormalR]"^2 + ("\[FormalCapitalJ]"/mass)^2*Cos["\[FormalTheta]"]^2))*Sin["\[FormalTheta]"]^2,
{4, 1} -> ((("\[FormalCapitalJ]"/mass)*("\[FormalR]"^2 - 2*mass*"\[FormalR]" + ("\[FormalCapitalJ]"/mass)^2 + "\[FormalCapitalQ]"^2/(4*Pi)) -
("\[FormalR]"^2 + ("\[FormalCapitalJ]"/mass)^2)*("\[FormalCapitalJ]"/mass))/("\[FormalR]"^2 + ("\[FormalCapitalJ]"/mass)^2*Cos["\[FormalTheta]"]^2))*Sin["\[FormalTheta]"]^2}]],
{"\[FormalT]", "\[FormalR]", "\[FormalTheta]", "\[FormalPhi]"}, index1, index2] /; BooleanQ[index1] && BooleanQ[index2]
MetricTensor[{"KerrNewman", mass_}, coordinates_List, index1_, index2_] :=
MetricTensor[Normal[SparseArray[{{1, 1} -> (("\[FormalCapitalJ]"/mass)^2*Sin[coordinates[[3]]]^2 - coordinates[[2]]^2 +
2*mass*coordinates[[2]] - ("\[FormalCapitalJ]"/mass)^2 - "\[FormalCapitalQ]"^2/(4*Pi))/(coordinates[[2]]^2 +
("\[FormalCapitalJ]"/mass)^2*Cos[coordinates[[3]]]^2),
{2, 2} -> (coordinates[[2]]^2 + ("\[FormalCapitalJ]"/mass)^2*Cos[coordinates[[3]]]^2)/(coordinates[[2]]^2 -
2*mass*coordinates[[2]] + ("\[FormalCapitalJ]"/mass)^2 + "\[FormalCapitalQ]"^2/(4*Pi)),
{3, 3} -> coordinates[[2]]^2 + ("\[FormalCapitalJ]"/mass)^2*Cos[coordinates[[3]]]^2,
{4, 4} -> (((coordinates[[2]]^2 + ("\[FormalCapitalJ]"/mass)^2)^2 - ("\[FormalCapitalJ]"/mass)^2*(coordinates[[2]]^2 -
2*mass*coordinates[[2]] + ("\[FormalCapitalJ]"/mass)^2 + "\[FormalCapitalQ]"^2/(4*Pi))*Sin[coordinates[[3]]]^2)/
(coordinates[[2]]^2 + ("\[FormalCapitalJ]"/mass)^2*Cos[coordinates[[3]]]^2))*Sin[coordinates[[3]]]^2,
{1, 4} -> ((("\[FormalCapitalJ]"/mass)*(coordinates[[2]]^2 - 2*mass*coordinates[[2]] + ("\[FormalCapitalJ]"/mass)^2 + "\[FormalCapitalQ]"^2/(4*Pi)) -
(coordinates[[2]]^2 + ("\[FormalCapitalJ]"/mass)^2)*("\[FormalCapitalJ]"/mass))/(coordinates[[2]]^2 +
("\[FormalCapitalJ]"/mass)^2*Cos[coordinates[[3]]]^2))*Sin[coordinates[[3]]]^2,
{4, 1} -> ((("\[FormalCapitalJ]"/mass)*(coordinates[[2]]^2 - 2*mass*coordinates[[2]] + ("\[FormalCapitalJ]"/mass)^2 + "\[FormalCapitalQ]"^2/(4*Pi)) -
(coordinates[[2]]^2 + ("\[FormalCapitalJ]"/mass)^2)*("\[FormalCapitalJ]"/mass))/(coordinates[[2]]^2 +
("\[FormalCapitalJ]"/mass)^2*Cos[coordinates[[3]]]^2))*Sin[coordinates[[3]]]^2}]], coordinates, index1, index2] /;
Length[coordinates] == 4 && BooleanQ[index1] && BooleanQ[index2]
MetricTensor[{"KerrNewman", mass_, angularMomentum_}] :=
MetricTensor[Normal[SparseArray[{{1, 1} -> ((angularMomentum/mass)^2*Sin["\[FormalTheta]"]^2 - "\[FormalR]"^2 + 2*mass*"\[FormalR]" -
(angularMomentum/mass)^2 - "\[FormalCapitalQ]"^2/(4*Pi))/("\[FormalR]"^2 + (angularMomentum/mass)^2*Cos["\[FormalTheta]"]^2),
{2, 2} -> ("\[FormalR]"^2 + (angularMomentum/mass)^2*Cos["\[FormalTheta]"]^2)/("\[FormalR]"^2 - 2*mass*"\[FormalR]" + (angularMomentum/mass)^2 +
"\[FormalCapitalQ]"^2/(4*Pi)), {3, 3} -> "\[FormalR]"^2 + (angularMomentum/mass)^2*Cos["\[FormalTheta]"]^2,
{4, 4} -> ((("\[FormalR]"^2 + (angularMomentum/mass)^2)^2 - (angularMomentum/mass)^2*("\[FormalR]"^2 - 2*mass*"\[FormalR]" +
(angularMomentum/mass)^2 + "\[FormalCapitalQ]"^2/(4*Pi))*Sin["\[FormalTheta]"]^2)/("\[FormalR]"^2 + (angularMomentum/mass)^2*Cos["\[FormalTheta]"]^2))*
Sin["\[FormalTheta]"]^2, {1, 4} -> (((angularMomentum/mass)*("\[FormalR]"^2 - 2*mass*"\[FormalR]" + (angularMomentum/mass)^2 +
"\[FormalCapitalQ]"^2/(4*Pi)) - ("\[FormalR]"^2 + (angularMomentum/mass)^2)*(angularMomentum/mass))/
("\[FormalR]"^2 + (angularMomentum/mass)^2*Cos["\[FormalTheta]"]^2))*Sin["\[FormalTheta]"]^2,
{4, 1} -> (((angularMomentum/mass)*("\[FormalR]"^2 - 2*mass*"\[FormalR]" + (angularMomentum/mass)^2 + "\[FormalCapitalQ]"^2/(4*Pi)) -
("\[FormalR]"^2 + (angularMomentum/mass)^2)*(angularMomentum/mass))/("\[FormalR]"^2 + (angularMomentum/mass)^2*Cos["\[FormalTheta]"]^2))*
Sin["\[FormalTheta]"]^2}]], {"\[FormalT]", "\[FormalR]", "\[FormalTheta]", "\[FormalPhi]"}, True, True]
MetricTensor[{"KerrNewman", mass_, angularMomentum_}, coordinates_List] :=
MetricTensor[Normal[SparseArray[{{1, 1} -> ((angularMomentum/mass)^2*Sin[coordinates[[3]]]^2 - coordinates[[2]]^2 +
2*mass*coordinates[[2]] - (angularMomentum/mass)^2 - "\[FormalCapitalQ]"^2/(4*Pi))/(coordinates[[2]]^2 +
(angularMomentum/mass)^2*Cos[coordinates[[3]]]^2),
{2, 2} -> (coordinates[[2]]^2 + (angularMomentum/mass)^2*Cos[coordinates[[3]]]^2)/
(coordinates[[2]]^2 - 2*mass*coordinates[[2]] + (angularMomentum/mass)^2 + "\[FormalCapitalQ]"^2/(4*Pi)),
{3, 3} -> coordinates[[2]]^2 + (angularMomentum/mass)^2*Cos[coordinates[[3]]]^2,
{4, 4} -> (((coordinates[[2]]^2 + (angularMomentum/mass)^2)^2 - (angularMomentum/mass)^2*
(coordinates[[2]]^2 - 2*mass*coordinates[[2]] + (angularMomentum/mass)^2 + "\[FormalCapitalQ]"^2/(4*Pi))*
Sin[coordinates[[3]]]^2)/(coordinates[[2]]^2 + (angularMomentum/mass)^2*Cos[coordinates[[3]]]^2))*
Sin[coordinates[[3]]]^2, {1, 4} -> (((angularMomentum/mass)*(coordinates[[2]]^2 - 2*mass*coordinates[[2]] +
(angularMomentum/mass)^2 + "\[FormalCapitalQ]"^2/(4*Pi)) - (coordinates[[2]]^2 + (angularMomentum/mass)^2)*
(angularMomentum/mass))/(coordinates[[2]]^2 + (angularMomentum/mass)^2*Cos[coordinates[[3]]]^2))*
Sin[coordinates[[3]]]^2, {4, 1} -> (((angularMomentum/mass)*(coordinates[[2]]^2 - 2*mass*coordinates[[2]] +
(angularMomentum/mass)^2 + "\[FormalCapitalQ]"^2/(4*Pi)) - (coordinates[[2]]^2 + (angularMomentum/mass)^2)*
(angularMomentum/mass))/(coordinates[[2]]^2 + (angularMomentum/mass)^2*Cos[coordinates[[3]]]^2))*
Sin[coordinates[[3]]]^2}]], coordinates, True, True] /; Length[coordinates] == 4
MetricTensor[{"KerrNewman", mass_, angularMomentum_}, index1_, index2_] :=
MetricTensor[Normal[SparseArray[{{1, 1} -> ((angularMomentum/mass)^2*Sin["\[FormalTheta]"]^2 - "\[FormalR]"^2 + 2*mass*"\[FormalR]" -
(angularMomentum/mass)^2 - "\[FormalCapitalQ]"^2/(4*Pi))/("\[FormalR]"^2 + (angularMomentum/mass)^2*Cos["\[FormalTheta]"]^2),
{2, 2} -> ("\[FormalR]"^2 + (angularMomentum/mass)^2*Cos["\[FormalTheta]"]^2)/("\[FormalR]"^2 - 2*mass*"\[FormalR]" + (angularMomentum/mass)^2 +
"\[FormalCapitalQ]"^2/(4*Pi)), {3, 3} -> "\[FormalR]"^2 + (angularMomentum/mass)^2*Cos["\[FormalTheta]"]^2,
{4, 4} -> ((("\[FormalR]"^2 + (angularMomentum/mass)^2)^2 - (angularMomentum/mass)^2*("\[FormalR]"^2 - 2*mass*"\[FormalR]" +
(angularMomentum/mass)^2 + "\[FormalCapitalQ]"^2/(4*Pi))*Sin["\[FormalTheta]"]^2)/("\[FormalR]"^2 + (angularMomentum/mass)^2*Cos["\[FormalTheta]"]^2))*
Sin["\[FormalTheta]"]^2, {1, 4} -> (((angularMomentum/mass)*("\[FormalR]"^2 - 2*mass*"\[FormalR]" + (angularMomentum/mass)^2 +
"\[FormalCapitalQ]"^2/(4*Pi)) - ("\[FormalR]"^2 + (angularMomentum/mass)^2)*(angularMomentum/mass))/
("\[FormalR]"^2 + (angularMomentum/mass)^2*Cos["\[FormalTheta]"]^2))*Sin["\[FormalTheta]"]^2,
{4, 1} -> (((angularMomentum/mass)*("\[FormalR]"^2 - 2*mass*"\[FormalR]" + (angularMomentum/mass)^2 + "\[FormalCapitalQ]"^2/(4*Pi)) -
("\[FormalR]"^2 + (angularMomentum/mass)^2)*(angularMomentum/mass))/("\[FormalR]"^2 + (angularMomentum/mass)^2*
Cos["\[FormalTheta]"]^2))*Sin["\[FormalTheta]"]^2}]], {"\[FormalT]", "\[FormalR]", "\[FormalTheta]", "\[FormalPhi]"}, index1, index2] /;
BooleanQ[index1] && BooleanQ[index2]
MetricTensor[{"KerrNewman", mass_, angularMomentum_}, coordinates_List, index1_, index2_] :=
MetricTensor[Normal[SparseArray[{{1, 1} -> ((angularMomentum/mass)^2*Sin[coordinates[[3]]]^2 - coordinates[[2]]^2 +
2*mass*coordinates[[2]] - (angularMomentum/mass)^2 - "\[FormalCapitalQ]"^2/(4*Pi))/(coordinates[[2]]^2 +
(angularMomentum/mass)^2*Cos[coordinates[[3]]]^2),
{2, 2} -> (coordinates[[2]]^2 + (angularMomentum/mass)^2*Cos[coordinates[[3]]]^2)/
(coordinates[[2]]^2 - 2*mass*coordinates[[2]] + (angularMomentum/mass)^2 + "\[FormalCapitalQ]"^2/(4*Pi)),
{3, 3} -> coordinates[[2]]^2 + (angularMomentum/mass)^2*Cos[coordinates[[3]]]^2,
{4, 4} -> (((coordinates[[2]]^2 + (angularMomentum/mass)^2)^2 - (angularMomentum/mass)^2*
(coordinates[[2]]^2 - 2*mass*coordinates[[2]] + (angularMomentum/mass)^2 + "\[FormalCapitalQ]"^2/(4*Pi))*
Sin[coordinates[[3]]]^2)/(coordinates[[2]]^2 + (angularMomentum/mass)^2*Cos[coordinates[[3]]]^2))*
Sin[coordinates[[3]]]^2, {1, 4} -> (((angularMomentum/mass)*(coordinates[[2]]^2 - 2*mass*coordinates[[2]] +
(angularMomentum/mass)^2 + "\[FormalCapitalQ]"^2/(4*Pi)) - (coordinates[[2]]^2 + (angularMomentum/mass)^2)*
(angularMomentum/mass))/(coordinates[[2]]^2 + (angularMomentum/mass)^2*Cos[coordinates[[3]]]^2))*
Sin[coordinates[[3]]]^2, {4, 1} -> (((angularMomentum/mass)*(coordinates[[2]]^2 - 2*mass*coordinates[[2]] +
(angularMomentum/mass)^2 + "\[FormalCapitalQ]"^2/(4*Pi)) - (coordinates[[2]]^2 + (angularMomentum/mass)^2)*
(angularMomentum/mass))/(coordinates[[2]]^2 + (angularMomentum/mass)^2*Cos[coordinates[[3]]]^2))*
Sin[coordinates[[3]]]^2}]], coordinates, index1, index2] /; Length[coordinates] == 4 && BooleanQ[index1] &&
BooleanQ[index2]
MetricTensor[{"KerrNewman", mass_, angularMomentum_, charge_}] :=
MetricTensor[Normal[SparseArray[{{1, 1} -> ((angularMomentum/mass)^2*Sin["\[FormalTheta]"]^2 - "\[FormalR]"^2 + 2*mass*"\[FormalR]" -
(angularMomentum/mass)^2 - charge^2/(4*Pi))/("\[FormalR]"^2 + (angularMomentum/mass)^2*Cos["\[FormalTheta]"]^2),
{2, 2} -> ("\[FormalR]"^2 + (angularMomentum/mass)^2*Cos["\[FormalTheta]"]^2)/("\[FormalR]"^2 - 2*mass*"\[FormalR]" + (angularMomentum/mass)^2 +
charge^2/(4*Pi)), {3, 3} -> "\[FormalR]"^2 + (angularMomentum/mass)^2*Cos["\[FormalTheta]"]^2,
{4, 4} -> ((("\[FormalR]"^2 + (angularMomentum/mass)^2)^2 - (angularMomentum/mass)^2*("\[FormalR]"^2 - 2*mass*"\[FormalR]" +
(angularMomentum/mass)^2 + charge^2/(4*Pi))*Sin["\[FormalTheta]"]^2)/("\[FormalR]"^2 + (angularMomentum/mass)^2*Cos["\[FormalTheta]"]^2))*
Sin["\[FormalTheta]"]^2, {1, 4} -> (((angularMomentum/mass)*("\[FormalR]"^2 - 2*mass*"\[FormalR]" + (angularMomentum/mass)^2 +
charge^2/(4*Pi)) - ("\[FormalR]"^2 + (angularMomentum/mass)^2)*(angularMomentum/mass))/
("\[FormalR]"^2 + (angularMomentum/mass)^2*Cos["\[FormalTheta]"]^2))*Sin["\[FormalTheta]"]^2,
{4, 1} -> (((angularMomentum/mass)*("\[FormalR]"^2 - 2*mass*"\[FormalR]" + (angularMomentum/mass)^2 + charge^2/(4*Pi)) -
("\[FormalR]"^2 + (angularMomentum/mass)^2)*(angularMomentum/mass))/("\[FormalR]"^2 + (angularMomentum/mass)^2*Cos["\[FormalTheta]"]^2))*
Sin["\[FormalTheta]"]^2}]], {"\[FormalT]", "\[FormalR]", "\[FormalTheta]", "\[FormalPhi]"}, True, True]
MetricTensor[{"KerrNewman", mass_, angularMomentum_, charge_}, coordinates_List] :=
MetricTensor[Normal[SparseArray[{{1, 1} -> ((angularMomentum/mass)^2*Sin[coordinates[[3]]]^2 - coordinates[[2]]^2 +
2*mass*coordinates[[2]] - (angularMomentum/mass)^2 - charge^2/(4*Pi))/(coordinates[[2]]^2 +
(angularMomentum/mass)^2*Cos[coordinates[[3]]]^2),
{2, 2} -> (coordinates[[2]]^2 + (angularMomentum/mass)^2*Cos[coordinates[[3]]]^2)/
(coordinates[[2]]^2 - 2*mass*coordinates[[2]] + (angularMomentum/mass)^2 + charge^2/(4*Pi)),
{3, 3} -> coordinates[[2]]^2 + (angularMomentum/mass)^2*Cos[coordinates[[3]]]^2,
{4, 4} -> (((coordinates[[2]]^2 + (angularMomentum/mass)^2)^2 - (angularMomentum/mass)^2*
(coordinates[[2]]^2 - 2*mass*coordinates[[2]] + (angularMomentum/mass)^2 + charge^2/(4*Pi))*
Sin[coordinates[[3]]]^2)/(coordinates[[2]]^2 + (angularMomentum/mass)^2*Cos[coordinates[[3]]]^2))*
Sin[coordinates[[3]]]^2, {1, 4} -> (((angularMomentum/mass)*(coordinates[[2]]^2 - 2*mass*coordinates[[2]] +
(angularMomentum/mass)^2 + charge^2/(4*Pi)) - (coordinates[[2]]^2 + (angularMomentum/mass)^2)*
(angularMomentum/mass))/(coordinates[[2]]^2 + (angularMomentum/mass)^2*Cos[coordinates[[3]]]^2))*
Sin[coordinates[[3]]]^2, {4, 1} -> (((angularMomentum/mass)*(coordinates[[2]]^2 - 2*mass*coordinates[[2]] +
(angularMomentum/mass)^2 + charge^2/(4*Pi)) - (coordinates[[2]]^2 + (angularMomentum/mass)^2)*
(angularMomentum/mass))/(coordinates[[2]]^2 + (angularMomentum/mass)^2*Cos[coordinates[[3]]]^2))*
Sin[coordinates[[3]]]^2}]], coordinates, True, True] /; Length[coordinates] == 4
MetricTensor[{"KerrNewman", mass_, angularMomentum_, charge_}, index1_, index2_] :=
MetricTensor[Normal[SparseArray[{{1, 1} -> ((angularMomentum/mass)^2*Sin["\[FormalTheta]"]^2 - "\[FormalR]"^2 + 2*mass*"\[FormalR]" -
(angularMomentum/mass)^2 - charge^2/(4*Pi))/("\[FormalR]"^2 + (angularMomentum/mass)^2*Cos["\[FormalTheta]"]^2),
{2, 2} -> ("\[FormalR]"^2 + (angularMomentum/mass)^2*Cos["\[FormalTheta]"]^2)/("\[FormalR]"^2 - 2*mass*"\[FormalR]" + (angularMomentum/mass)^2 +
charge^2/(4*Pi)), {3, 3} -> "\[FormalR]"^2 + (angularMomentum/mass)^2*Cos["\[FormalTheta]"]^2,
{4, 4} -> ((("\[FormalR]"^2 + (angularMomentum/mass)^2)^2 - (angularMomentum/mass)^2*("\[FormalR]"^2 - 2*mass*"\[FormalR]" +
(angularMomentum/mass)^2 + charge^2/(4*Pi))*Sin["\[FormalTheta]"]^2)/("\[FormalR]"^2 + (angularMomentum/mass)^2*Cos["\[FormalTheta]"]^2))*
Sin["\[FormalTheta]"]^2, {1, 4} -> (((angularMomentum/mass)*("\[FormalR]"^2 - 2*mass*"\[FormalR]" + (angularMomentum/mass)^2 +
charge^2/(4*Pi)) - ("\[FormalR]"^2 + (angularMomentum/mass)^2)*(angularMomentum/mass))/
("\[FormalR]"^2 + (angularMomentum/mass)^2*Cos["\[FormalTheta]"]^2))*Sin["\[FormalTheta]"]^2,
{4, 1} -> (((angularMomentum/mass)*("\[FormalR]"^2 - 2*mass*"\[FormalR]" + (angularMomentum/mass)^2 + charge^2/(4*Pi)) -
("\[FormalR]"^2 + (angularMomentum/mass)^2)*(angularMomentum/mass))/("\[FormalR]"^2 + (angularMomentum/mass)^2*
Cos["\[FormalTheta]"]^2))*Sin["\[FormalTheta]"]^2}]], {"\[FormalT]", "\[FormalR]", "\[FormalTheta]", "\[FormalPhi]"}, index1, index2] /;
BooleanQ[index1] && BooleanQ[index2]
MetricTensor[{"KerrNewman", mass_, angularMomentum_, charge_}, coordinates_List, index1_, index2_] :=
MetricTensor[Normal[SparseArray[{{1, 1} -> ((angularMomentum/mass)^2*Sin[coordinates[[3]]]^2 - coordinates[[2]]^2 +
2*mass*coordinates[[2]] - (angularMomentum/mass)^2 - charge^2/(4*Pi))/(coordinates[[2]]^2 +
(angularMomentum/mass)^2*Cos[coordinates[[3]]]^2),
{2, 2} -> (coordinates[[2]]^2 + (angularMomentum/mass)^2*Cos[coordinates[[3]]]^2)/
(coordinates[[2]]^2 - 2*mass*coordinates[[2]] + (angularMomentum/mass)^2 + charge^2/(4*Pi)),
{3, 3} -> coordinates[[2]]^2 + (angularMomentum/mass)^2*Cos[coordinates[[3]]]^2,
{4, 4} -> (((coordinates[[2]]^2 + (angularMomentum/mass)^2)^2 - (angularMomentum/mass)^2*
(coordinates[[2]]^2 - 2*mass*coordinates[[2]] + (angularMomentum/mass)^2 + charge^2/(4*Pi))*
Sin[coordinates[[3]]]^2)/(coordinates[[2]]^2 + (angularMomentum/mass)^2*Cos[coordinates[[3]]]^2))*
Sin[coordinates[[3]]]^2, {1, 4} -> (((angularMomentum/mass)*(coordinates[[2]]^2 - 2*mass*coordinates[[2]] +
(angularMomentum/mass)^2 + charge^2/(4*Pi)) - (coordinates[[2]]^2 + (angularMomentum/mass)^2)*
(angularMomentum/mass))/(coordinates[[2]]^2 + (angularMomentum/mass)^2*Cos[coordinates[[3]]]^2))*
Sin[coordinates[[3]]]^2, {4, 1} -> (((angularMomentum/mass)*(coordinates[[2]]^2 - 2*mass*coordinates[[2]] +
(angularMomentum/mass)^2 + charge^2/(4*Pi)) - (coordinates[[2]]^2 + (angularMomentum/mass)^2)*
(angularMomentum/mass))/(coordinates[[2]]^2 + (angularMomentum/mass)^2*Cos[coordinates[[3]]]^2))*
Sin[coordinates[[3]]]^2}]], coordinates, index1, index2] /; Length[coordinates] == 4 && BooleanQ[index1] &&
BooleanQ[index2]
MetricTensor["Godel"] := MetricTensor[Normal[SparseArray[{{1, 1} -> -(1/(2*"\[FormalOmega]"^2)), {2, 2} -> 1/(2*"\[FormalOmega]"^2),
{3, 3} -> -(Exp[2*"\[FormalX]"]/(4*"\[FormalOmega]"^2)), {4, 4} -> 1/(2*"\[FormalOmega]"^2), {1, 3} -> -(Exp["\[FormalX]"]/(2*"\[FormalOmega]"^2)),
{3, 1} -> -(Exp["\[FormalX]"]/(2*"\[FormalOmega]"^2))}]], {"\[FormalT]", "\[FormalX]", "\[FormalY]", "\[FormalZ]"}, True, True]
MetricTensor["Godel", coordinates_List] :=
MetricTensor[Normal[SparseArray[{{1, 1} -> -(1/(2*"\[FormalOmega]"^2)), {2, 2} -> 1/(2*"\[FormalOmega]"^2),
{3, 3} -> -(Exp[2*coordinates[[2]]]/(4*"\[FormalOmega]"^2)), {4, 4} -> 1/(2*"\[FormalOmega]"^2),
{1, 3} -> -(Exp[coordinates[[2]]]/(2*"\[FormalOmega]"^2)), {3, 1} -> -(Exp[coordinates[[2]]]/(2*"\[FormalOmega]"^2))}]], coordinates,
True, True] /; Length[coordinates] == 4
MetricTensor["Godel", index1_, index2_] :=
MetricTensor[Normal[SparseArray[{{1, 1} -> -(1/(2*"\[FormalOmega]"^2)), {2, 2} -> 1/(2*"\[FormalOmega]"^2),
{3, 3} -> -(Exp[2*"\[FormalX]"]/(4*"\[FormalOmega]"^2)), {4, 4} -> 1/(2*"\[FormalOmega]"^2), {1, 3} -> -(Exp["\[FormalX]"]/(2*"\[FormalOmega]"^2)),
{3, 1} -> -(Exp["\[FormalX]"]/(2*"\[FormalOmega]"^2))}]], {"\[FormalT]", "\[FormalX]", "\[FormalY]", "\[FormalZ]"}, index1, index2] /;
BooleanQ[index1] && BooleanQ[index2]
MetricTensor["Godel", coordinates_List, index1_, index2_] :=
MetricTensor[Normal[SparseArray[{{1, 1} -> -(1/(2*"\[FormalOmega]"^2)), {2, 2} -> 1/(2*"\[FormalOmega]"^2),
{3, 3} -> -(Exp[2*coordinates[[2]]]/(4*"\[FormalOmega]"^2)), {4, 4} -> 1/(2*"\[FormalOmega]"^2),
{1, 3} -> -(Exp[coordinates[[2]]]/(2*"\[FormalOmega]"^2)), {3, 1} -> -(Exp[coordinates[[2]]]/(2*"\[FormalOmega]"^2))}]], coordinates,
index1, index2] /; Length[coordinates] == 4 && BooleanQ[index1] && BooleanQ[index2]
MetricTensor[{"Godel", angularVelocity_}] :=
MetricTensor[Normal[SparseArray[{{1, 1} -> -(1/(2*angularVelocity^2)), {2, 2} -> 1/(2*angularVelocity^2),
{3, 3} -> -(Exp[2*"\[FormalX]"]/(4*angularVelocity^2)), {4, 4} -> 1/(2*angularVelocity^2),
{1, 3} -> -(Exp["\[FormalX]"]/(2*angularVelocity^2)), {3, 1} -> -(Exp["\[FormalX]"]/(2*angularVelocity^2))}]],
{"\[FormalT]", "\[FormalX]", "\[FormalY]", "\[FormalZ]"}, True, True]
MetricTensor[{"Godel", angularVelocity_}, coordinates_List] :=
MetricTensor[Normal[SparseArray[{{1, 1} -> -(1/(2*angularVelocity^2)), {2, 2} -> 1/(2*angularVelocity^2),
{3, 3} -> -(Exp[2*coordinates[[2]]]/(4*angularVelocity^2)), {4, 4} -> 1/(2*angularVelocity^2),
{1, 3} -> -(Exp[coordinates[[2]]]/(2*angularVelocity^2)),
{3, 1} -> -(Exp[coordinates[[2]]]/(2*angularVelocity^2))}]], coordinates, True, True] /; Length[coordinates] == 4
MetricTensor[{"Godel", angularVelocity_}, index1_, index2_] :=
MetricTensor[Normal[SparseArray[{{1, 1} -> -(1/(2*angularVelocity^2)), {2, 2} -> 1/(2*angularVelocity^2),
{3, 3} -> -(Exp[2*"\[FormalX]"]/(4*angularVelocity^2)), {4, 4} -> 1/(2*angularVelocity^2),
{1, 3} -> -(Exp["\[FormalX]"]/(2*angularVelocity^2)), {3, 1} -> -(Exp["\[FormalX]"]/(2*angularVelocity^2))}]],
{"\[FormalT]", "\[FormalX]", "\[FormalY]", "\[FormalZ]"}, index1, index2] /; BooleanQ[index1] && BooleanQ[index2]
MetricTensor[{"Godel", angularVelocity_}, coordinates_List, index1_, index2_] :=
MetricTensor[Normal[SparseArray[{{1, 1} -> -(1/(2*angularVelocity^2)), {2, 2} -> 1/(2*angularVelocity^2),
{3, 3} -> -(Exp[2*coordinates[[2]]]/(4*angularVelocity^2)), {4, 4} -> 1/(2*angularVelocity^2),
{1, 3} -> -(Exp[coordinates[[2]]]/(2*angularVelocity^2)),
{3, 1} -> -(Exp[coordinates[[2]]]/(2*angularVelocity^2))}]], coordinates, index1, index2] /;
Length[coordinates] == 4 && BooleanQ[index1] && BooleanQ[index2]
MetricTensor["FLRW"] := MetricTensor[DiagonalMatrix[{-1, "\[FormalA]"["\[FormalT]"]^2/(1 - "\[FormalK]"*"\[FormalR]"^2), "\[FormalA]"["\[FormalT]"]^2*"\[FormalR]"^2,
"\[FormalA]"["\[FormalT]"]^2*"\[FormalR]"^2*Sin["\[FormalTheta]"]^2}], {"\[FormalT]", "\[FormalR]", "\[FormalTheta]", "\[FormalPhi]"}, True, True]
MetricTensor["FLRW", coordinates_List] :=
MetricTensor[DiagonalMatrix[{-1, "\[FormalA]"[coordinates[[1]]]^2/(1 - "\[FormalK]"*coordinates[[2]]^2),
"\[FormalA]"[coordinates[[1]]]^2*coordinates[[2]]^2, "\[FormalA]"[coordinates[[1]]]^2*coordinates[[2]]^2*
Sin[coordinates[[3]]]^2}], coordinates, True, True] /; Length[coordinates] == 4
MetricTensor["FLRW", index1_, index2_] :=
MetricTensor[DiagonalMatrix[{-1, "\[FormalA]"["\[FormalT]"]^2/(1 - "\[FormalK]"*"\[FormalR]"^2), "\[FormalA]"["\[FormalT]"]^2*"\[FormalR]"^2,
"\[FormalA]"["\[FormalT]"]^2*"\[FormalR]"^2*Sin["\[FormalTheta]"]^2}], {"\[FormalT]", "\[FormalR]", "\[FormalTheta]", "\[FormalPhi]"}, index1, index2] /;
BooleanQ[index1] && BooleanQ[index2]
MetricTensor["FLRW", coordinates_List, index1_, index2_] :=
MetricTensor[DiagonalMatrix[{-1, "\[FormalA]"[coordinates[[1]]]^2/(1 - "\[FormalK]"*coordinates[[2]]^2),
"\[FormalA]"[coordinates[[1]]]^2*coordinates[[2]]^2, "\[FormalA]"[coordinates[[1]]]^2*coordinates[[2]]^2*
Sin[coordinates[[3]]]^2}], coordinates, index1, index2] /; Length[coordinates] == 4 && BooleanQ[index1] &&
BooleanQ[index2]
MetricTensor[{"FLRW", curvature_}] := MetricTensor[DiagonalMatrix[{-1, "\[FormalA]"["\[FormalT]"]^2/(1 - curvature*"\[FormalR]"^2),
"\[FormalA]"["\[FormalT]"]^2*"\[FormalR]"^2, "\[FormalA]"["\[FormalT]"]^2*"\[FormalR]"^2*Sin["\[FormalTheta]"]^2}], {"\[FormalT]", "\[FormalR]", "\[FormalTheta]", "\[FormalPhi]"}, True, True]
MetricTensor[{"FLRW", curvature_}, coordinates_List] :=
MetricTensor[DiagonalMatrix[{-1, "\[FormalA]"[coordinates[[1]]]^2/(1 - curvature*coordinates[[2]]^2),
"\[FormalA]"[coordinates[[1]]]^2*coordinates[[2]]^2, "\[FormalA]"[coordinates[[1]]]^2*coordinates[[2]]^2*
Sin[coordinates[[3]]]^2}], coordinates, True, True] /; Length[coordinates] == 4
MetricTensor[{"FLRW", curvature_}, index1_, index2_] :=
MetricTensor[DiagonalMatrix[{-1, "\[FormalA]"["\[FormalT]"]^2/(1 - curvature*"\[FormalR]"^2), "\[FormalA]"["\[FormalT]"]^2*"\[FormalR]"^2,
"\[FormalA]"["\[FormalT]"]^2*"\[FormalR]"^2*Sin["\[FormalTheta]"]^2}], {"\[FormalT]", "\[FormalR]", "\[FormalTheta]", "\[FormalPhi]"}, index1, index2] /;
BooleanQ[index1] && BooleanQ[index2]
MetricTensor[{"FLRW", curvature_}, coordinates_List, index1_, index2_] :=
MetricTensor[DiagonalMatrix[{-1, "\[FormalA]"[coordinates[[1]]]^2/(1 - curvature*coordinates[[2]]^2),
"\[FormalA]"[coordinates[[1]]]^2*coordinates[[2]]^2, "\[FormalA]"[coordinates[[1]]]^2*coordinates[[2]]^2*
Sin[coordinates[[3]]]^2}], coordinates, index1, index2] /; Length[coordinates] == 4 && BooleanQ[index1] &&
BooleanQ[index2]
MetricTensor[{"FLRW", curvature_, scaleFactor_}] :=
MetricTensor[DiagonalMatrix[{-1, scaleFactor["\[FormalT]"]^2/(1 - curvature*"\[FormalR]"^2), scaleFactor["\[FormalT]"]^2*"\[FormalR]"^2,
scaleFactor["\[FormalT]"]^2*"\[FormalR]"^2*Sin["\[FormalTheta]"]^2}], {"\[FormalT]", "\[FormalR]", "\[FormalTheta]", "\[FormalPhi]"}, True, True]
MetricTensor[{"FLRW", curvature_, scaleFactor_}, coordinates_List] :=
MetricTensor[DiagonalMatrix[{-1, scaleFactor[coordinates[[1]]]^2/(1 - curvature*coordinates[[2]]^2),
scaleFactor[coordinates[[1]]]^2*coordinates[[2]]^2, scaleFactor[coordinates[[1]]]^2*coordinates[[2]]^2*
Sin[coordinates[[3]]]^2}], coordinates, True, True] /; Length[coordinates] == 4
MetricTensor[{"FLRW", curvature_, scaleFactor_}, index1_, index2_] :=
MetricTensor[DiagonalMatrix[{-1, scaleFactor["\[FormalT]"]^2/(1 - curvature*"\[FormalR]"^2), scaleFactor["\[FormalT]"]^2*"\[FormalR]"^2,
scaleFactor["\[FormalT]"]^2*"\[FormalR]"^2*Sin["\[FormalTheta]"]^2}], {"\[FormalT]", "\[FormalR]", "\[FormalTheta]", "\[FormalPhi]"}, index1, index2] /;
BooleanQ[index1] && BooleanQ[index2]
MetricTensor[{"FLRW", curvature_, scaleFactor_}, coordinates_List, index1_, index2_] :=
MetricTensor[DiagonalMatrix[{-1, scaleFactor[coordinates[[1]]]^2/(1 - curvature*coordinates[[2]]^2),
scaleFactor[coordinates[[1]]]^2*coordinates[[2]]^2, scaleFactor[coordinates[[1]]]^2*coordinates[[2]]^2*
Sin[coordinates[[3]]]^2}], coordinates, index1, index2] /; Length[coordinates] == 4 && BooleanQ[index1] &&
BooleanQ[index2]
MetricTensor[matrixRepresentation_List] := MetricTensor[matrixRepresentation, (Superscript["\[FormalX]", ToString[#1]] & ) /@
Range[Length[matrixRepresentation]], True, True] /; Length[Dimensions[matrixRepresentation]] == 2
MetricTensor[matrixRepresentation_List, coordinates_List] :=
MetricTensor[matrixRepresentation, coordinates, True, True] /; Length[Dimensions[matrixRepresentation]] == 2 &&
Length[coordinates] == Length[matrixRepresentation]
MetricTensor[matrixRepresentation_List, index1_, index2_] :=
MetricTensor[matrixRepresentation, (Superscript["\[FormalX]", ToString[#1]] & ) /@ Range[Length[matrixRepresentation]],
index1, index2] /; Length[Dimensions[matrixRepresentation]] == 2 && BooleanQ[index1] && BooleanQ[index2]
MetricTensor[matrixRepresentation_List, coordinates_List, index1_, index2_]["MatrixRepresentation"] :=
If[index1 === True && index2 === True, matrixRepresentation, If[index1 === False && index2 === False,
Inverse[matrixRepresentation], If[index1 === True && index2 === False,
Normal[SparseArray[(Module[{index = #1}, index -> Total[(matrixRepresentation[[First[index],#1]]*
Inverse[matrixRepresentation][[#1,Last[index]]] & ) /@ Range[Length[matrixRepresentation]]]] & ) /@
Tuples[Range[Length[matrixRepresentation]], 2]]], If[index1 === False && index2 === True,
Normal[SparseArray[(Module[{index = #1}, index -> Total[(Inverse[matrixRepresentation][[First[index],#1]]*
matrixRepresentation[[#1,Last[index]]] & ) /@ Range[Length[matrixRepresentation]]]] & ) /@
Tuples[Range[Length[matrixRepresentation]], 2]]], Indeterminate]]]] /;
Length[Dimensions[matrixRepresentation]] == 2 && Length[coordinates] == Length[matrixRepresentation] &&
BooleanQ[index1] && BooleanQ[index2]
MetricTensor[matrixRepresentation_List, coordinates_List, index1_, index2_]["ReducedMatrixRepresentation"] :=
If[index1 === True && index2 === True, FullSimplify[matrixRepresentation], If[index1 === False && index2 === False,
FullSimplify[Inverse[matrixRepresentation]], If[index1 === True && index2 === False,
FullSimplify[Normal[SparseArray[(Module[{index = #1}, index -> Total[(matrixRepresentation[[First[index],#1]]*
Inverse[matrixRepresentation][[#1,Last[index]]] & ) /@ Range[Length[matrixRepresentation]]]] & ) /@