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ADMStressEnergyDecomposition.wl
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(* ::Package:: *)
ADMStressEnergyDecomposition["PerfectFluid"] := Module[{lapseFunction, shiftVector, matrixRepresentation, shiftCovector,
spacetimeMetricTensor}, lapseFunction = "\[FormalAlpha]" @@ Join[{"\[FormalT]"}, (Superscript["\[FormalX]", ToString[#1]] & ) /@ Range[3]];
shiftVector = (Module[{index = #1}, Superscript["\[FormalBeta]", index] @@ Join[{"\[FormalT]"},
(Superscript["\[FormalX]", ToString[#1]] & ) /@ Range[3]]] & ) /@ Range[3];
matrixRepresentation = DiagonalMatrix[ConstantArray[1, 3]];
shiftCovector = Normal[SparseArray[
(Module[{index = #1}, index -> Total[(matrixRepresentation[[index,#1]]*shiftVector[[#1]] & ) /@
Range[Length[matrixRepresentation]]]] & ) /@ Range[Length[matrixRepresentation]]]];
spacetimeMetricTensor = Normal[SparseArray[
Join[{{1, 1} -> Total[(shiftVector[[#1]]*shiftCovector[[#1]] & ) /@ Range[Length[matrixRepresentation]]] -
lapseFunction^2}, (Module[{index = #1}, {1, index + 1} -> Total[(matrixRepresentation[[index,#1]]*
shiftVector[[#1]] & ) /@ Range[Length[matrixRepresentation]]]] & ) /@
Range[Length[matrixRepresentation]], (Module[{index = #1}, {index + 1, 1} ->
Total[(matrixRepresentation[[#1,index]]*shiftVector[[#1]] & ) /@ Range[Length[matrixRepresentation]]]] & ) /@
Range[Length[matrixRepresentation]], ({First[#1] + 1, Last[#1] + 1} -> matrixRepresentation[[First[#1],
Last[#1]]] & ) /@ Tuples[Range[Length[matrixRepresentation]], 2]]]];
ADMStressEnergyDecomposition[ResourceFunction["ADMDecomposition"][ResourceFunction["MetricTensor"][
matrixRepresentation, (Superscript["\[FormalX]", ToString[#1]] & ) /@ Range[3], True, True], "\[FormalT]", lapseFunction,
shiftVector], Normal[SparseArray[(#1 -> ("\[FormalRho]" + "\[FormalCapitalP]")*Superscript["\[FormalU]", ToString[First[#1]]]*
Superscript["\[FormalU]", ToString[Last[#1]]] + "\[FormalCapitalP]"*Inverse[spacetimeMetricTensor][[First[#1],Last[#1]]] & ) /@
Tuples[Range[4], 2]]]]]
ADMStressEnergyDecomposition["PerfectFluid", (admDecomposition_)[(metricTensor_)[matrixRepresentation_List,
coordinates_List, index1_, index2_], timeCoordinate_, lapseFunction_, shiftVector_List]] :=
Module[{shiftCovector, spacetimeMetricTensor},
shiftCovector = Normal[SparseArray[(Module[{index = #1}, index -> Total[(matrixRepresentation[[index,#1]]*
shiftVector[[#1]] & ) /@ Range[Length[matrixRepresentation]]]] & ) /@
Range[Length[matrixRepresentation]]]]; spacetimeMetricTensor =
Normal[SparseArray[Join[{{1, 1} -> Total[(shiftVector[[#1]]*shiftCovector[[#1]] & ) /@
Range[Length[matrixRepresentation]]] - lapseFunction^2},
(Module[{index = #1}, {1, index + 1} -> Total[(matrixRepresentation[[index,#1]]*shiftVector[[#1]] & ) /@ Range[
Length[matrixRepresentation]]]] & ) /@ Range[Length[matrixRepresentation]],
(Module[{index = #1}, {index + 1, 1} -> Total[(matrixRepresentation[[#1,index]]*shiftVector[[#1]] & ) /@ Range[
Length[matrixRepresentation]]]] & ) /@ Range[Length[matrixRepresentation]],
({First[#1] + 1, Last[#1] + 1} -> matrixRepresentation[[First[#1],Last[#1]]] & ) /@
Tuples[Range[Length[matrixRepresentation]], 2]]]]; ADMStressEnergyDecomposition[
ResourceFunction["ADMDecomposition"][ResourceFunction["MetricTensor"][matrixRepresentation, coordinates, index1,
index2], timeCoordinate, lapseFunction, shiftVector],
Normal[SparseArray[(#1 -> ("\[FormalRho]" + "\[FormalCapitalP]")*Superscript["\[FormalU]", ToString[First[#1]]]*Superscript["\[FormalU]",
ToString[Last[#1]]] + "\[FormalCapitalP]"*Inverse[spacetimeMetricTensor][[First[#1],Last[#1]]] & ) /@
Tuples[Range[Length[spacetimeMetricTensor]], 2]]]]] /; SymbolName[admDecomposition] === "ADMDecomposition" &&
SymbolName[metricTensor] === "MetricTensor" && Length[Dimensions[matrixRepresentation]] == 2 &&
Length[coordinates] == Length[matrixRepresentation] && BooleanQ[index1] && BooleanQ[index2] &&
Length[shiftVector] == Length[matrixRepresentation]
ADMStressEnergyDecomposition[{"PerfectFluid", fourVelocity_List}] :=
Module[{lapseFunction, shiftVector, matrixRepresentation, shiftCovector, spacetimeMetricTensor},
lapseFunction = "\[FormalAlpha]" @@ Join[{"\[FormalT]"}, (Superscript["\[FormalX]", ToString[#1]] & ) /@ Range[Length[fourVelocity] - 1]];
shiftVector = (Module[{index = #1}, Superscript["\[FormalBeta]", index] @@ Join[{"\[FormalT]"},
(Superscript["\[FormalX]", ToString[#1]] & ) /@ Range[Length[fourVelocity] - 1]]] & ) /@
Range[Length[fourVelocity] - 1]; matrixRepresentation = DiagonalMatrix[ConstantArray[1, Length[fourVelocity] - 1]];
shiftCovector = Normal[SparseArray[
(Module[{index = #1}, index -> Total[(matrixRepresentation[[index,#1]]*shiftVector[[#1]] & ) /@
Range[Length[matrixRepresentation]]]] & ) /@ Range[Length[matrixRepresentation]]]];
spacetimeMetricTensor = Normal[SparseArray[
Join[{{1, 1} -> Total[(shiftVector[[#1]]*shiftCovector[[#1]] & ) /@ Range[Length[matrixRepresentation]]] -
lapseFunction^2}, (Module[{index = #1}, {1, index + 1} -> Total[(matrixRepresentation[[index,#1]]*
shiftVector[[#1]] & ) /@ Range[Length[matrixRepresentation]]]] & ) /@
Range[Length[matrixRepresentation]], (Module[{index = #1}, {index + 1, 1} ->
Total[(matrixRepresentation[[#1,index]]*shiftVector[[#1]] & ) /@ Range[Length[matrixRepresentation]]]] & ) /@
Range[Length[matrixRepresentation]], ({First[#1] + 1, Last[#1] + 1} -> matrixRepresentation[[First[#1],
Last[#1]]] & ) /@ Tuples[Range[Length[matrixRepresentation]], 2]]]];
ADMStressEnergyDecomposition[ResourceFunction["ADMDecomposition"][ResourceFunction["MetricTensor"][
matrixRepresentation, (Superscript["\[FormalX]", ToString[#1]] & ) /@ Range[Length[fourVelocity] - 1], True, True],
"\[FormalT]", lapseFunction, shiftVector],
Normal[SparseArray[(#1 -> ("\[FormalRho]" + "\[FormalCapitalP]")*fourVelocity[[First[#1]]]*fourVelocity[[Last[#1]]] +
"\[FormalCapitalP]"*Inverse[spacetimeMetricTensor][[First[#1],Last[#1]]] & ) /@ Tuples[Range[Length[spacetimeMetricTensor]],
2]]]]]
ADMStressEnergyDecomposition[{"PerfectFluid", fourVelocity_List},
(admDecomposition_)[(metricTensor_)[matrixRepresentation_List, coordinates_List, index1_, index2_], timeCoordinate_,
lapseFunction_, shiftVector_List]] := Module[{shiftCovector, spacetimeMetricTensor},
shiftCovector = Normal[SparseArray[(Module[{index = #1}, index -> Total[(matrixRepresentation[[index,#1]]*
shiftVector[[#1]] & ) /@ Range[Length[matrixRepresentation]]]] & ) /@
Range[Length[matrixRepresentation]]]]; spacetimeMetricTensor =
Normal[SparseArray[Join[{{1, 1} -> Total[(shiftVector[[#1]]*shiftCovector[[#1]] & ) /@
Range[Length[matrixRepresentation]]] - lapseFunction^2},
(Module[{index = #1}, {1, index + 1} -> Total[(matrixRepresentation[[index,#1]]*shiftVector[[#1]] & ) /@ Range[
Length[matrixRepresentation]]]] & ) /@ Range[Length[matrixRepresentation]],
(Module[{index = #1}, {index + 1, 1} -> Total[(matrixRepresentation[[#1,index]]*shiftVector[[#1]] & ) /@ Range[
Length[matrixRepresentation]]]] & ) /@ Range[Length[matrixRepresentation]],
({First[#1] + 1, Last[#1] + 1} -> matrixRepresentation[[First[#1],Last[#1]]] & ) /@
Tuples[Range[Length[matrixRepresentation]], 2]]]]; ADMStressEnergyDecomposition[
ResourceFunction["ADMDecomposition"][ResourceFunction["MetricTensor"][matrixRepresentation, coordinates, index1,
index2], timeCoordinate, lapseFunction, shiftVector],
Normal[SparseArray[(#1 -> ("\[FormalRho]" + "\[FormalCapitalP]")*fourVelocity[[First[#1]]]*fourVelocity[[Last[#1]]] +
"\[FormalCapitalP]"*Inverse[spacetimeMetricTensor][[First[#1],Last[#1]]] & ) /@
Tuples[Range[Length[spacetimeMetricTensor]], 2]]]]] /; SymbolName[admDecomposition] === "ADMDecomposition" &&
SymbolName[metricTensor] === "MetricTensor" && Length[Dimensions[matrixRepresentation]] == 2 &&
Length[coordinates] == Length[matrixRepresentation] && BooleanQ[index1] && BooleanQ[index2] &&
Length[shiftVector] == Length[matrixRepresentation] && Length[fourVelocity] == Length[matrixRepresentation] + 1
ADMStressEnergyDecomposition[{"PerfectFluid", fourVelocity_List, density_}] :=
Module[{lapseFunction, shiftVector, matrixRepresentation, shiftCovector, spacetimeMetricTensor},
lapseFunction = "\[FormalAlpha]" @@ Join[{"\[FormalT]"}, (Superscript["\[FormalX]", ToString[#1]] & ) /@ Range[Length[fourVelocity] - 1]];
shiftVector = (Module[{index = #1}, Superscript["\[FormalBeta]", index] @@ Join[{"\[FormalT]"},
(Superscript["\[FormalX]", ToString[#1]] & ) /@ Range[Length[fourVelocity] - 1]]] & ) /@
Range[Length[fourVelocity] - 1]; matrixRepresentation = DiagonalMatrix[ConstantArray[1, Length[fourVelocity] - 1]];
shiftCovector = Normal[SparseArray[
(Module[{index = #1}, index -> Total[(matrixRepresentation[[index,#1]]*shiftVector[[#1]] & ) /@
Range[Length[matrixRepresentation]]]] & ) /@ Range[Length[matrixRepresentation]]]];
spacetimeMetricTensor = Normal[SparseArray[
Join[{{1, 1} -> Total[(shiftVector[[#1]]*shiftCovector[[#1]] & ) /@ Range[Length[matrixRepresentation]]] -
lapseFunction^2}, (Module[{index = #1}, {1, index + 1} -> Total[(matrixRepresentation[[index,#1]]*
shiftVector[[#1]] & ) /@ Range[Length[matrixRepresentation]]]] & ) /@
Range[Length[matrixRepresentation]], (Module[{index = #1}, {index + 1, 1} ->
Total[(matrixRepresentation[[#1,index]]*shiftVector[[#1]] & ) /@ Range[Length[matrixRepresentation]]]] & ) /@
Range[Length[matrixRepresentation]], ({First[#1] + 1, Last[#1] + 1} -> matrixRepresentation[[First[#1],
Last[#1]]] & ) /@ Tuples[Range[Length[matrixRepresentation]], 2]]]];
ADMStressEnergyDecomposition[ResourceFunction["ADMDecomposition"][ResourceFunction["MetricTensor"][
matrixRepresentation, (Superscript["\[FormalX]", ToString[#1]] & ) /@ Range[Length[fourVelocity] - 1], True, True],
"\[FormalT]", lapseFunction, shiftVector],
Normal[SparseArray[(#1 -> (density + "\[FormalCapitalP]")*fourVelocity[[First[#1]]]*fourVelocity[[Last[#1]]] +
"\[FormalCapitalP]"*Inverse[spacetimeMetricTensor][[First[#1],Last[#1]]] & ) /@ Tuples[Range[Length[spacetimeMetricTensor]],
2]]]]]
ADMStressEnergyDecomposition[{"PerfectFluid", fourVelocity_List, density_},
(admDecomposition_)[(metricTensor_)[matrixRepresentation_List, coordinates_List, index1_, index2_], timeCoordinate_,
lapseFunction_, shiftVector_List]] := Module[{shiftCovector, spacetimeMetricTensor},
shiftCovector = Normal[SparseArray[(Module[{index = #1}, index -> Total[(matrixRepresentation[[index,#1]]*
shiftVector[[#1]] & ) /@ Range[Length[matrixRepresentation]]]] & ) /@
Range[Length[matrixRepresentation]]]]; spacetimeMetricTensor =
Normal[SparseArray[Join[{{1, 1} -> Total[(shiftVector[[#1]]*shiftCovector[[#1]] & ) /@
Range[Length[matrixRepresentation]]] - lapseFunction^2},
(Module[{index = #1}, {1, index + 1} -> Total[(matrixRepresentation[[index,#1]]*shiftVector[[#1]] & ) /@ Range[
Length[matrixRepresentation]]]] & ) /@ Range[Length[matrixRepresentation]],
(Module[{index = #1}, {index + 1, 1} -> Total[(matrixRepresentation[[#1,index]]*shiftVector[[#1]] & ) /@ Range[
Length[matrixRepresentation]]]] & ) /@ Range[Length[matrixRepresentation]],
({First[#1] + 1, Last[#1] + 1} -> matrixRepresentation[[First[#1],Last[#1]]] & ) /@
Tuples[Range[Length[matrixRepresentation]], 2]]]]; ADMStressEnergyDecomposition[
ResourceFunction["ADMDecomposition"][ResourceFunction["MetricTensor"][matrixRepresentation, coordinates, index1,
index2], timeCoordinate, lapseFunction, shiftVector],
Normal[SparseArray[(#1 -> (density + "\[FormalCapitalP]")*fourVelocity[[First[#1]]]*fourVelocity[[Last[#1]]] +
"\[FormalCapitalP]"*Inverse[spacetimeMetricTensor][[First[#1],Last[#1]]] & ) /@
Tuples[Range[Length[spacetimeMetricTensor]], 2]]]]] /; SymbolName[admDecomposition] === "ADMDecomposition" &&
SymbolName[metricTensor] === "MetricTensor" && Length[Dimensions[matrixRepresentation]] == 2 &&
Length[coordinates] == Length[matrixRepresentation] && BooleanQ[index1] && BooleanQ[index2] &&
Length[shiftVector] == Length[matrixRepresentation] && Length[fourVelocity] == Length[matrixRepresentation] + 1
ADMStressEnergyDecomposition[{"PerfectFluid", fourVelocity_List, density_, pressure_}] :=
Module[{lapseFunction, shiftVector, matrixRepresentation, shiftCovector, spacetimeMetricTensor},
lapseFunction = "\[FormalAlpha]" @@ Join[{"\[FormalT]"}, (Superscript["\[FormalX]", ToString[#1]] & ) /@ Range[Length[fourVelocity] - 1]];
shiftVector = (Module[{index = #1}, Superscript["\[FormalBeta]", index] @@ Join[{"\[FormalT]"},
(Superscript["\[FormalX]", ToString[#1]] & ) /@ Range[Length[fourVelocity] - 1]]] & ) /@
Range[Length[fourVelocity] - 1]; matrixRepresentation = DiagonalMatrix[ConstantArray[1, Length[fourVelocity] - 1]];
shiftCovector = Normal[SparseArray[
(Module[{index = #1}, index -> Total[(matrixRepresentation[[index,#1]]*shiftVector[[#1]] & ) /@
Range[Length[matrixRepresentation]]]] & ) /@ Range[Length[matrixRepresentation]]]];
spacetimeMetricTensor = Normal[SparseArray[
Join[{{1, 1} -> Total[(shiftVector[[#1]]*shiftCovector[[#1]] & ) /@ Range[Length[matrixRepresentation]]] -
lapseFunction^2}, (Module[{index = #1}, {1, index + 1} -> Total[(matrixRepresentation[[index,#1]]*
shiftVector[[#1]] & ) /@ Range[Length[matrixRepresentation]]]] & ) /@
Range[Length[matrixRepresentation]], (Module[{index = #1}, {index + 1, 1} ->
Total[(matrixRepresentation[[#1,index]]*shiftVector[[#1]] & ) /@ Range[Length[matrixRepresentation]]]] & ) /@
Range[Length[matrixRepresentation]], ({First[#1] + 1, Last[#1] + 1} -> matrixRepresentation[[First[#1],
Last[#1]]] & ) /@ Tuples[Range[Length[matrixRepresentation]], 2]]]];
ADMStressEnergyDecomposition[ResourceFunction["ADMDecomposition"][ResourceFunction["MetricTensor"][
matrixRepresentation, (Superscript["\[FormalX]", ToString[#1]] & ) /@ Range[Length[fourVelocity] - 1], True, True],
"\[FormalT]", lapseFunction, shiftVector],
Normal[SparseArray[(#1 -> (density + pressure)*fourVelocity[[First[#1]]]*fourVelocity[[Last[#1]]] +
pressure*Inverse[spacetimeMetricTensor][[First[#1],Last[#1]]] & ) /@
Tuples[Range[Length[spacetimeMetricTensor]], 2]]]]]
ADMStressEnergyDecomposition[{"PerfectFluid", fourVelocity_List, density_, pressure_},
(admDecomposition_)[(metricTensor_)[matrixRepresentation_List, coordinates_List, index1_, index2_], timeCoordinate_,
lapseFunction_, shiftVector_List]] := Module[{shiftCovector, spacetimeMetricTensor},
shiftCovector = Normal[SparseArray[(Module[{index = #1}, index -> Total[(matrixRepresentation[[index,#1]]*
shiftVector[[#1]] & ) /@ Range[Length[matrixRepresentation]]]] & ) /@
Range[Length[matrixRepresentation]]]]; spacetimeMetricTensor =
Normal[SparseArray[Join[{{1, 1} -> Total[(shiftVector[[#1]]*shiftCovector[[#1]] & ) /@
Range[Length[matrixRepresentation]]] - lapseFunction^2},
(Module[{index = #1}, {1, index + 1} -> Total[(matrixRepresentation[[index,#1]]*shiftVector[[#1]] & ) /@ Range[
Length[matrixRepresentation]]]] & ) /@ Range[Length[matrixRepresentation]],
(Module[{index = #1}, {index + 1, 1} -> Total[(matrixRepresentation[[#1,index]]*shiftVector[[#1]] & ) /@ Range[
Length[matrixRepresentation]]]] & ) /@ Range[Length[matrixRepresentation]],
({First[#1] + 1, Last[#1] + 1} -> matrixRepresentation[[First[#1],Last[#1]]] & ) /@
Tuples[Range[Length[matrixRepresentation]], 2]]]]; ADMStressEnergyDecomposition[
ResourceFunction["ADMDecomposition"][ResourceFunction["MetricTensor"][matrixRepresentation, coordinates, index1,
index2], timeCoordinate, lapseFunction, shiftVector],
Normal[SparseArray[(#1 -> (density + pressure)*fourVelocity[[First[#1]]]*fourVelocity[[Last[#1]]] +
pressure*Inverse[spacetimeMetricTensor][[First[#1],Last[#1]]] & ) /@
Tuples[Range[Length[spacetimeMetricTensor]], 2]]]]] /; SymbolName[admDecomposition] === "ADMDecomposition" &&
SymbolName[metricTensor] === "MetricTensor" && Length[Dimensions[matrixRepresentation]] == 2 &&
Length[coordinates] == Length[matrixRepresentation] && BooleanQ[index1] && BooleanQ[index2] &&
Length[shiftVector] == Length[matrixRepresentation] && Length[fourVelocity] == Length[matrixRepresentation] + 1
ADMStressEnergyDecomposition["PerfectFluidField"] := Module[{lapseFunction, shiftVector, matrixRepresentation,
shiftCovector, spacetimeMetricTensor},
lapseFunction = "\[FormalAlpha]" @@ Join[{"\[FormalT]"}, (Superscript["\[FormalX]", ToString[#1]] & ) /@ Range[3]];
shiftVector = (Module[{index = #1}, Superscript["\[FormalBeta]", index] @@ Join[{"\[FormalT]"},
(Superscript["\[FormalX]", ToString[#1]] & ) /@ Range[3]]] & ) /@ Range[3];
matrixRepresentation = DiagonalMatrix[ConstantArray[1, 3]];
shiftCovector = Normal[SparseArray[
(Module[{index = #1}, index -> Total[(matrixRepresentation[[index,#1]]*shiftVector[[#1]] & ) /@
Range[Length[matrixRepresentation]]]] & ) /@ Range[Length[matrixRepresentation]]]];
spacetimeMetricTensor = Normal[SparseArray[
Join[{{1, 1} -> Total[(shiftVector[[#1]]*shiftCovector[[#1]] & ) /@ Range[Length[matrixRepresentation]]] -
lapseFunction^2}, (Module[{index = #1}, {1, index + 1} -> Total[(matrixRepresentation[[index,#1]]*
shiftVector[[#1]] & ) /@ Range[Length[matrixRepresentation]]]] & ) /@
Range[Length[matrixRepresentation]], (Module[{index = #1}, {index + 1, 1} ->
Total[(matrixRepresentation[[#1,index]]*shiftVector[[#1]] & ) /@ Range[Length[matrixRepresentation]]]] & ) /@
Range[Length[matrixRepresentation]], ({First[#1] + 1, Last[#1] + 1} -> matrixRepresentation[[First[#1],
Last[#1]]] & ) /@ Tuples[Range[Length[matrixRepresentation]], 2]]]];
ADMStressEnergyDecomposition[ResourceFunction["ADMDecomposition"][ResourceFunction["MetricTensor"][
matrixRepresentation, (Superscript["\[FormalX]", ToString[#1]] & ) /@ Range[3], True, True], "\[FormalT]", lapseFunction,
shiftVector], Normal[SparseArray[
(Module[{index = #1}, index -> ("\[FormalRho]" @@ Join[{"\[FormalT]"}, (Superscript["\[FormalX]", ToString[#1]] & ) /@ Range[3]] +
"\[FormalCapitalP]" @@ Join[{"\[FormalT]"}, (Superscript["\[FormalX]", ToString[#1]] & ) /@ Range[3]])*
Superscript["\[FormalU]", ToString[First[index]]] @@ Join[{"\[FormalT]"}, (Superscript["\[FormalX]", ToString[#1]] & ) /@
Range[3]]*Superscript["\[FormalU]", ToString[Last[index]]] @@ Join[{"\[FormalT]"}, (Superscript["\[FormalX]",
ToString[#1]] & ) /@ Range[3]] + "\[FormalCapitalP]" @@ Join[{"\[FormalT]"}, (Superscript["\[FormalX]", ToString[#1]] & ) /@
Range[3]]*Inverse[spacetimeMetricTensor][[First[#1],Last[#1]]]] & ) /@ Tuples[Range[4], 2]]]]]
ADMStressEnergyDecomposition["PerfectFluidField", (admDecomposition_)[(metricTensor_)[matrixRepresentation_List,
coordinates_List, index1_, index2_], timeCoordinate_, lapseFunction_, shiftVector_List]] :=
Module[{shiftCovector, spacetimeMetricTensor},
shiftCovector = Normal[SparseArray[(Module[{index = #1}, index -> Total[(matrixRepresentation[[index,#1]]*
shiftVector[[#1]] & ) /@ Range[Length[matrixRepresentation]]]] & ) /@
Range[Length[matrixRepresentation]]]]; spacetimeMetricTensor =
Normal[SparseArray[Join[{{1, 1} -> Total[(shiftVector[[#1]]*shiftCovector[[#1]] & ) /@
Range[Length[matrixRepresentation]]] - lapseFunction^2},
(Module[{index = #1}, {1, index + 1} -> Total[(matrixRepresentation[[index,#1]]*shiftVector[[#1]] & ) /@ Range[
Length[matrixRepresentation]]]] & ) /@ Range[Length[matrixRepresentation]],
(Module[{index = #1}, {index + 1, 1} -> Total[(matrixRepresentation[[#1,index]]*shiftVector[[#1]] & ) /@ Range[
Length[matrixRepresentation]]]] & ) /@ Range[Length[matrixRepresentation]],
({First[#1] + 1, Last[#1] + 1} -> matrixRepresentation[[First[#1],Last[#1]]] & ) /@
Tuples[Range[Length[matrixRepresentation]], 2]]]]; ADMStressEnergyDecomposition[
ResourceFunction["ADMDecomposition"][ResourceFunction["MetricTensor"][matrixRepresentation, coordinates, index1,
index2], timeCoordinate, lapseFunction, shiftVector],
Normal[SparseArray[(Module[{index = #1}, index -> ("\[FormalRho]" @@ Join[{timeCoordinate}, coordinates] + "\[FormalCapitalP]" @@
Join[{timeCoordinate}, coordinates])*Superscript["\[FormalU]", ToString[First[index]]] @@ Join[{timeCoordinate},
coordinates]*Superscript["\[FormalU]", ToString[Last[index]]] @@ Join[{timeCoordinate}, coordinates] +
"\[FormalCapitalP]" @@ Join[{timeCoordinate}, coordinates]*Inverse[spacetimeMetricTensor][[First[index],Last[
index]]]] & ) /@ Tuples[Range[Length[spacetimeMetricTensor]], 2]]]]] /;
SymbolName[admDecomposition] === "ADMDecomposition" && SymbolName[metricTensor] === "MetricTensor" &&
Length[Dimensions[matrixRepresentation]] == 2 && Length[coordinates] == Length[matrixRepresentation] &&
BooleanQ[index1] && BooleanQ[index2] && Length[shiftVector] == Length[matrixRepresentation]
ADMStressEnergyDecomposition[{"PerfectFluidField", fourVelocity_List}] :=
Module[{lapseFunction, shiftVector, matrixRepresentation, shiftCovector, spacetimeMetricTensor},
lapseFunction = "\[FormalAlpha]" @@ Join[{"\[FormalT]"}, (Superscript["\[FormalX]", ToString[#1]] & ) /@ Range[Length[fourVelocity] - 1]];
shiftVector = (Module[{index = #1}, Superscript["\[FormalBeta]", index] @@ Join[{"\[FormalT]"},
(Superscript["\[FormalX]", ToString[#1]] & ) /@ Range[Length[fourVelocity] - 1]]] & ) /@
Range[Length[fourVelocity] - 1]; matrixRepresentation = DiagonalMatrix[ConstantArray[1, Length[fourVelocity] - 1]];
shiftCovector = Normal[SparseArray[
(Module[{index = #1}, index -> Total[(matrixRepresentation[[index,#1]]*shiftVector[[#1]] & ) /@
Range[Length[matrixRepresentation]]]] & ) /@ Range[Length[matrixRepresentation]]]];
spacetimeMetricTensor = Normal[SparseArray[
Join[{{1, 1} -> Total[(shiftVector[[#1]]*shiftCovector[[#1]] & ) /@ Range[Length[matrixRepresentation]]] -
lapseFunction^2}, (Module[{index = #1}, {1, index + 1} -> Total[(matrixRepresentation[[index,#1]]*
shiftVector[[#1]] & ) /@ Range[Length[matrixRepresentation]]]] & ) /@
Range[Length[matrixRepresentation]], (Module[{index = #1}, {index + 1, 1} ->
Total[(matrixRepresentation[[#1,index]]*shiftVector[[#1]] & ) /@ Range[Length[matrixRepresentation]]]] & ) /@
Range[Length[matrixRepresentation]], ({First[#1] + 1, Last[#1] + 1} -> matrixRepresentation[[First[#1],
Last[#1]]] & ) /@ Tuples[Range[Length[matrixRepresentation]], 2]]]];
ADMStressEnergyDecomposition[ResourceFunction["ADMDecomposition"][ResourceFunction["MetricTensor"][
matrixRepresentation, (Superscript["\[FormalX]", ToString[#1]] & ) /@ Range[Length[fourVelocity] - 1], True, True],
"\[FormalT]", lapseFunction, shiftVector],
Normal[SparseArray[(Module[{index = #1}, index -> ("\[FormalRho]" @@ Join[{"\[FormalT]"}, (Superscript["\[FormalX]", ToString[#1]] & ) /@
Range[Length[fourVelocity] - 1]] + "\[FormalCapitalP]" @@ Join[{"\[FormalT]"}, (Superscript["\[FormalX]", ToString[#1]] & ) /@
Range[Length[fourVelocity] - 1]])*fourVelocity[[First[index]]] @@ Join[{"\[FormalT]"},
(Superscript["\[FormalX]", ToString[#1]] & ) /@ Range[Length[fourVelocity] - 1]]*fourVelocity[[Last[index]]] @@
Join[{"\[FormalT]"}, (Superscript["\[FormalX]", ToString[#1]] & ) /@ Range[Length[fourVelocity] - 1]] +
"\[FormalCapitalP]" @@ Join[{"\[FormalT]"}, (Superscript["\[FormalX]", ToString[#1]] & ) /@ Range[Length[fourVelocity] - 1]]*
Inverse[spacetimeMetricTensor][[First[index],Last[index]]]] & ) /@ Tuples[Range[Length[fourVelocity]], 2]]]]]
ADMStressEnergyDecomposition[{"PerfectFluidField", fourVelocity_List},
(admDecomposition_)[(metricTensor_)[matrixRepresentation_List, coordinates_List, index1_, index2_], timeCoordinate_,
lapseFunction_, shiftVector_List]] := Module[{shiftCovector, spacetimeMetricTensor},
shiftCovector = Normal[SparseArray[(Module[{index = #1}, index -> Total[(matrixRepresentation[[index,#1]]*
shiftVector[[#1]] & ) /@ Range[Length[matrixRepresentation]]]] & ) /@
Range[Length[matrixRepresentation]]]]; spacetimeMetricTensor =
Normal[SparseArray[Join[{{1, 1} -> Total[(shiftVector[[#1]]*shiftCovector[[#1]] & ) /@
Range[Length[matrixRepresentation]]] - lapseFunction^2},
(Module[{index = #1}, {1, index + 1} -> Total[(matrixRepresentation[[index,#1]]*shiftVector[[#1]] & ) /@ Range[
Length[matrixRepresentation]]]] & ) /@ Range[Length[matrixRepresentation]],
(Module[{index = #1}, {index + 1, 1} -> Total[(matrixRepresentation[[#1,index]]*shiftVector[[#1]] & ) /@ Range[
Length[matrixRepresentation]]]] & ) /@ Range[Length[matrixRepresentation]],
({First[#1] + 1, Last[#1] + 1} -> matrixRepresentation[[First[#1],Last[#1]]] & ) /@
Tuples[Range[Length[matrixRepresentation]], 2]]]]; ADMStressEnergyDecomposition[
ResourceFunction["ADMDecomposition"][ResourceFunction["MetricTensor"][matrixRepresentation, coordinates, index1,
index2], timeCoordinate, lapseFunction, shiftVector],
Normal[SparseArray[(Module[{index = #1}, index -> ("\[FormalRho]" @@ Join[{timeCoordinate}, coordinates] + "\[FormalCapitalP]" @@
Join[{timeCoordinate}, coordinates])*fourVelocity[[First[index]]] @@ Join[{timeCoordinate}, coordinates]*
fourVelocity[[Last[index]]] @@ Join[{timeCoordinate}, coordinates] + "\[FormalCapitalP]" @@ Join[{timeCoordinate},
coordinates]*Inverse[spacetimeMetricTensor][[First[index],Last[index]]]] & ) /@
Tuples[Range[Length[spacetimeMetricTensor]], 2]]]]] /; SymbolName[admDecomposition] === "ADMDecomposition" &&
SymbolName[metricTensor] === "MetricTensor" && Length[Dimensions[matrixRepresentation]] == 2 &&
Length[coordinates] == Length[matrixRepresentation] && BooleanQ[index1] && BooleanQ[index2] &&
Length[shiftVector] == Length[matrixRepresentation] && Length[fourVelocity] == Length[matrixRepresentation] + 1
ADMStressEnergyDecomposition[{"PerfectFluidField", fourVelocity_List, density_}] :=
Module[{lapseFunction, shiftVector, matrixRepresentation, shiftCovector, spacetimeMetricTensor},
lapseFunction = "\[FormalAlpha]" @@ Join[{"\[FormalT]"}, (Superscript["\[FormalX]", ToString[#1]] & ) /@ Range[Length[fourVelocity] - 1]];
shiftVector = (Module[{index = #1}, Superscript["\[FormalBeta]", index] @@ Join[{"\[FormalT]"},
(Superscript["\[FormalX]", ToString[#1]] & ) /@ Range[Length[fourVelocity] - 1]]] & ) /@
Range[Length[fourVelocity] - 1]; matrixRepresentation = DiagonalMatrix[ConstantArray[1, Length[fourVelocity] - 1]];
shiftCovector = Normal[SparseArray[
(Module[{index = #1}, index -> Total[(matrixRepresentation[[index,#1]]*shiftVector[[#1]] & ) /@
Range[Length[matrixRepresentation]]]] & ) /@ Range[Length[matrixRepresentation]]]];
spacetimeMetricTensor = Normal[SparseArray[
Join[{{1, 1} -> Total[(shiftVector[[#1]]*shiftCovector[[#1]] & ) /@ Range[Length[matrixRepresentation]]] -
lapseFunction^2}, (Module[{index = #1}, {1, index + 1} -> Total[(matrixRepresentation[[index,#1]]*
shiftVector[[#1]] & ) /@ Range[Length[matrixRepresentation]]]] & ) /@
Range[Length[matrixRepresentation]], (Module[{index = #1}, {index + 1, 1} ->
Total[(matrixRepresentation[[#1,index]]*shiftVector[[#1]] & ) /@ Range[Length[matrixRepresentation]]]] & ) /@
Range[Length[matrixRepresentation]], ({First[#1] + 1, Last[#1] + 1} -> matrixRepresentation[[First[#1],
Last[#1]]] & ) /@ Tuples[Range[Length[matrixRepresentation]], 2]]]];
ADMStressEnergyDecomposition[ResourceFunction["ADMDecomposition"][ResourceFunction["MetricTensor"][
matrixRepresentation, (Superscript["\[FormalX]", ToString[#1]] & ) /@ Range[Length[fourVelocity] - 1], True, True],
"\[FormalT]", lapseFunction, shiftVector],
Normal[SparseArray[(Module[{index = #1}, index -> (density @@ Join[{"\[FormalT]"}, (Superscript["\[FormalX]", ToString[#1]] & ) /@
Range[Length[fourVelocity] - 1]] + "\[FormalCapitalP]" @@ Join[{"\[FormalT]"}, (Superscript["\[FormalX]", ToString[#1]] & ) /@
Range[Length[fourVelocity] - 1]])*fourVelocity[[First[index]]] @@ Join[{"\[FormalT]"},
(Superscript["\[FormalX]", ToString[#1]] & ) /@ Range[Length[fourVelocity] - 1]]*fourVelocity[[Last[index]]] @@
Join[{"\[FormalT]"}, (Superscript["\[FormalX]", ToString[#1]] & ) /@ Range[Length[fourVelocity] - 1]] +
"\[FormalCapitalP]" @@ Join[{"\[FormalT]"}, (Superscript["\[FormalX]", ToString[#1]] & ) /@ Range[Length[fourVelocity] - 1]]*
Inverse[spacetimeMetricTensor][[First[index],Last[index]]]] & ) /@ Tuples[Range[Length[fourVelocity]], 2]]]]]
ADMStressEnergyDecomposition[{"PerfectFluidField", fourVelocity_List, density_},
(admDecomposition_)[(metricTensor_)[matrixRepresentation_List, coordinates_List, index1_, index2_], timeCoordinate_,
lapseFunction_, shiftVector_List]] := Module[{shiftCovector, spacetimeMetricTensor},
shiftCovector = Normal[SparseArray[(Module[{index = #1}, index -> Total[(matrixRepresentation[[index,#1]]*
shiftVector[[#1]] & ) /@ Range[Length[matrixRepresentation]]]] & ) /@
Range[Length[matrixRepresentation]]]]; spacetimeMetricTensor =
Normal[SparseArray[Join[{{1, 1} -> Total[(shiftVector[[#1]]*shiftCovector[[#1]] & ) /@
Range[Length[matrixRepresentation]]] - lapseFunction^2},
(Module[{index = #1}, {1, index + 1} -> Total[(matrixRepresentation[[index,#1]]*shiftVector[[#1]] & ) /@ Range[
Length[matrixRepresentation]]]] & ) /@ Range[Length[matrixRepresentation]],
(Module[{index = #1}, {index + 1, 1} -> Total[(matrixRepresentation[[#1,index]]*shiftVector[[#1]] & ) /@ Range[
Length[matrixRepresentation]]]] & ) /@ Range[Length[matrixRepresentation]],
({First[#1] + 1, Last[#1] + 1} -> matrixRepresentation[[First[#1],Last[#1]]] & ) /@
Tuples[Range[Length[matrixRepresentation]], 2]]]]; ADMStressEnergyDecomposition[
ResourceFunction["ADMDecomposition"][ResourceFunction["MetricTensor"][matrixRepresentation, coordinates, index1,
index2], timeCoordinate, lapseFunction, shiftVector],
Normal[SparseArray[(Module[{index = #1}, index -> (density @@ Join[{timeCoordinate}, coordinates] + "\[FormalCapitalP]" @@
Join[{timeCoordinate}, coordinates])*fourVelocity[[First[index]]] @@ Join[{timeCoordinate}, coordinates]*
fourVelocity[[Last[index]]] @@ Join[{timeCoordinate}, coordinates] + "\[FormalCapitalP]" @@ Join[{timeCoordinate},
coordinates]*Inverse[spacetimeMetricTensor][[First[index],Last[index]]]] & ) /@
Tuples[Range[Length[spacetimeMetricTensor]], 2]]]]] /; SymbolName[admDecomposition] === "ADMDecomposition" &&
SymbolName[metricTensor] === "MetricTensor" && Length[Dimensions[matrixRepresentation]] == 2 &&
Length[coordinates] == Length[matrixRepresentation] && BooleanQ[index1] && BooleanQ[index2] &&
Length[shiftVector] == Length[matrixRepresentation] && Length[fourVelocity] == Length[matrixRepresentation] + 1
ADMStressEnergyDecomposition[{"PerfectFluidField", fourVelocity_List, density_, pressure_}] :=
Module[{lapseFunction, shiftVector, matrixRepresentation, shiftCovector, spacetimeMetricTensor},
lapseFunction = "\[FormalAlpha]" @@ Join[{"\[FormalT]"}, (Superscript["\[FormalX]", ToString[#1]] & ) /@ Range[Length[fourVelocity] - 1]];
shiftVector = (Module[{index = #1}, Superscript["\[FormalBeta]", index] @@ Join[{"\[FormalT]"},
(Superscript["\[FormalX]", ToString[#1]] & ) /@ Range[Length[fourVelocity] - 1]]] & ) /@
Range[Length[fourVelocity] - 1]; matrixRepresentation = DiagonalMatrix[ConstantArray[1, Length[fourVelocity] - 1]];
shiftCovector = Normal[SparseArray[
(Module[{index = #1}, index -> Total[(matrixRepresentation[[index,#1]]*shiftVector[[#1]] & ) /@
Range[Length[matrixRepresentation]]]] & ) /@ Range[Length[matrixRepresentation]]]];
spacetimeMetricTensor = Normal[SparseArray[
Join[{{1, 1} -> Total[(shiftVector[[#1]]*shiftCovector[[#1]] & ) /@ Range[Length[matrixRepresentation]]] -
lapseFunction^2}, (Module[{index = #1}, {1, index + 1} -> Total[(matrixRepresentation[[index,#1]]*
shiftVector[[#1]] & ) /@ Range[Length[matrixRepresentation]]]] & ) /@
Range[Length[matrixRepresentation]], (Module[{index = #1}, {index + 1, 1} ->
Total[(matrixRepresentation[[#1,index]]*shiftVector[[#1]] & ) /@ Range[Length[matrixRepresentation]]]] & ) /@
Range[Length[matrixRepresentation]], ({First[#1] + 1, Last[#1] + 1} -> matrixRepresentation[[First[#1],
Last[#1]]] & ) /@ Tuples[Range[Length[matrixRepresentation]], 2]]]];
ADMStressEnergyDecomposition[ResourceFunction["ADMDecomposition"][ResourceFunction["MetricTensor"][
matrixRepresentation, (Superscript["\[FormalX]", ToString[#1]] & ) /@ Range[Length[fourVelocity] - 1], True, True],
"\[FormalT]", lapseFunction, shiftVector],
Normal[SparseArray[(Module[{index = #1}, index -> (density @@ Join[{"\[FormalT]"}, (Superscript["\[FormalX]", ToString[#1]] & ) /@
Range[Length[fourVelocity] - 1]] + pressure @@ Join[{"\[FormalT]"}, (Superscript["\[FormalX]", ToString[#1]] & ) /@
Range[Length[fourVelocity] - 1]])*fourVelocity[[First[index]]] @@ Join[{"\[FormalT]"},
(Superscript["\[FormalX]", ToString[#1]] & ) /@ Range[Length[fourVelocity] - 1]]*fourVelocity[[Last[index]]] @@
Join[{"\[FormalT]"}, (Superscript["\[FormalX]", ToString[#1]] & ) /@ Range[Length[fourVelocity] - 1]] +
pressure @@ Join[{"\[FormalT]"}, (Superscript["\[FormalX]", ToString[#1]] & ) /@ Range[Length[fourVelocity] - 1]]*
Inverse[spacetimeMetricTensor][[First[index],Last[index]]]] & ) /@ Tuples[Range[Length[fourVelocity]], 2]]]]]
ADMStressEnergyDecomposition[{"PerfectFluidField", fourVelocity_List, density_, pressure_},
(admDecomposition_)[(metricTensor_)[matrixRepresentation_List, coordinates_List, index1_, index2_], timeCoordinate_,
lapseFunction_, shiftVector_List]] := Module[{shiftCovector, spacetimeMetricTensor},
shiftCovector = Normal[SparseArray[(Module[{index = #1}, index -> Total[(matrixRepresentation[[index,#1]]*
shiftVector[[#1]] & ) /@ Range[Length[matrixRepresentation]]]] & ) /@
Range[Length[matrixRepresentation]]]]; spacetimeMetricTensor =
Normal[SparseArray[Join[{{1, 1} -> Total[(shiftVector[[#1]]*shiftCovector[[#1]] & ) /@
Range[Length[matrixRepresentation]]] - lapseFunction^2},
(Module[{index = #1}, {1, index + 1} -> Total[(matrixRepresentation[[index,#1]]*shiftVector[[#1]] & ) /@ Range[
Length[matrixRepresentation]]]] & ) /@ Range[Length[matrixRepresentation]],
(Module[{index = #1}, {index + 1, 1} -> Total[(matrixRepresentation[[#1,index]]*shiftVector[[#1]] & ) /@ Range[
Length[matrixRepresentation]]]] & ) /@ Range[Length[matrixRepresentation]],
({First[#1] + 1, Last[#1] + 1} -> matrixRepresentation[[First[#1],Last[#1]]] & ) /@
Tuples[Range[Length[matrixRepresentation]], 2]]]]; ADMStressEnergyDecomposition[
ResourceFunction["ADMDecomposition"][ResourceFunction["MetricTensor"][matrixRepresentation, coordinates, index1,
index2], timeCoordinate, lapseFunction, shiftVector],
Normal[SparseArray[(Module[{index = #1}, index -> (density @@ Join[{timeCoordinate}, coordinates] + pressure @@
Join[{timeCoordinate}, coordinates])*fourVelocity[[First[index]]] @@ Join[{timeCoordinate}, coordinates]*
fourVelocity[[Last[index]]] @@ Join[{timeCoordinate}, coordinates] + pressure @@ Join[{timeCoordinate},
coordinates]*Inverse[spacetimeMetricTensor][[First[index],Last[index]]]] & ) /@
Tuples[Range[Length[spacetimeMetricTensor]], 2]]]]] /; SymbolName[admDecomposition] === "ADMDecomposition" &&
SymbolName[metricTensor] === "MetricTensor" && Length[Dimensions[matrixRepresentation]] == 2 &&
Length[coordinates] == Length[matrixRepresentation] && BooleanQ[index1] && BooleanQ[index2] &&
Length[shiftVector] == Length[matrixRepresentation] && Length[fourVelocity] == Length[matrixRepresentation] + 1
ADMStressEnergyDecomposition["Dust"] := Module[{lapseFunction, shiftVector, matrixRepresentation},
lapseFunction = "\[FormalAlpha]" @@ Join[{"\[FormalT]"}, (Superscript["\[FormalX]", ToString[#1]] & ) /@ Range[3]];
shiftVector = (Module[{index = #1}, Superscript["\[FormalBeta]", index] @@ Join[{"\[FormalT]"},
(Superscript["\[FormalX]", ToString[#1]] & ) /@ Range[3]]] & ) /@ Range[3];
matrixRepresentation = DiagonalMatrix[ConstantArray[1, 3]]; ADMStressEnergyDecomposition[
ResourceFunction["ADMDecomposition"][ResourceFunction["MetricTensor"][matrixRepresentation,
(Superscript["\[FormalX]", ToString[#1]] & ) /@ Range[3], True, True], "\[FormalT]", lapseFunction, shiftVector],
Normal[SparseArray[(#1 -> "\[FormalRho]"*Superscript["\[FormalU]", ToString[First[#1]]]*Superscript["\[FormalU]",
ToString[Last[#1]]] & ) /@ Tuples[Range[4], 2]]]]]
ADMStressEnergyDecomposition["Dust", (admDecomposition_)[(metricTensor_)[matrixRepresentation_List, coordinates_List,
index1_, index2_], timeCoordinate_, lapseFunction_, shiftVector_List]] :=
ADMStressEnergyDecomposition[ResourceFunction["ADMDecomposition"][ResourceFunction["MetricTensor"][
matrixRepresentation, coordinates, index1, index2], timeCoordinate, lapseFunction, shiftVector],
Normal[SparseArray[(#1 -> "\[FormalRho]"*Superscript["\[FormalU]", ToString[First[#1]]]*Superscript["\[FormalU]", ToString[Last[#1]]] & ) /@
Tuples[Range[Length[matrixRepresentation] + 1], 2]]]] /; SymbolName[admDecomposition] === "ADMDecomposition" &&
SymbolName[metricTensor] === "MetricTensor" && Length[Dimensions[matrixRepresentation]] == 2 &&
Length[coordinates] == Length[matrixRepresentation] && BooleanQ[index1] && BooleanQ[index2] &&
Length[shiftVector] == Length[matrixRepresentation]
ADMStressEnergyDecomposition[{"Dust", fourVelocity_List}] := Module[{lapseFunction, shiftVector, matrixRepresentation},
lapseFunction = "\[FormalAlpha]" @@ Join[{"\[FormalT]"}, (Superscript["\[FormalX]", ToString[#1]] & ) /@ Range[Length[fourVelocity] - 1]];
shiftVector = (Module[{index = #1}, Superscript["\[FormalBeta]", index] @@ Join[{"\[FormalT]"},
(Superscript["\[FormalX]", ToString[#1]] & ) /@ Range[Length[fourVelocity] - 1]]] & ) /@
Range[Length[fourVelocity] - 1]; matrixRepresentation = DiagonalMatrix[ConstantArray[1, Length[fourVelocity] - 1]];
ADMStressEnergyDecomposition[ResourceFunction["ADMDecomposition"][ResourceFunction["MetricTensor"][
matrixRepresentation, (Superscript["\[FormalX]", ToString[#1]] & ) /@ Range[Length[fourVelocity] - 1], True, True],
"\[FormalT]", lapseFunction, shiftVector],
Normal[SparseArray[(#1 -> "\[FormalRho]"*fourVelocity[[First[#1]]]*fourVelocity[[Last[#1]]] & ) /@
Tuples[Range[Length[fourVelocity]], 2]]]]]
ADMStressEnergyDecomposition[{"Dust", fourVelocity_List}, (admDecomposition_)[(metricTensor_)[matrixRepresentation_List,
coordinates_List, index1_, index2_], timeCoordinate_, lapseFunction_, shiftVector_List]] :=
ADMStressEnergyDecomposition[ResourceFunction["ADMDecomposition"][ResourceFunction["MetricTensor"][
matrixRepresentation, coordinates, index1, index2], timeCoordinate, lapseFunction, shiftVector],
Normal[SparseArray[(#1 -> "\[FormalRho]"*fourVelocity[[First[#1]]]*fourVelocity[[Last[#1]]] & ) /@
Tuples[Range[Length[matrixRepresentation] + 1], 2]]]] /; SymbolName[admDecomposition] === "ADMDecomposition" &&
SymbolName[metricTensor] === "MetricTensor" && Length[Dimensions[matrixRepresentation]] == 2 &&
Length[coordinates] == Length[matrixRepresentation] && BooleanQ[index1] && BooleanQ[index2] &&
Length[shiftVector] == Length[matrixRepresentation] && Length[fourVelocity] == Length[matrixRepresentation] + 1
ADMStressEnergyDecomposition[{"Dust", fourVelocity_List, density_}] :=
Module[{lapseFunction, shiftVector, matrixRepresentation},
lapseFunction = "\[FormalAlpha]" @@ Join[{"\[FormalT]"}, (Superscript["\[FormalX]", ToString[#1]] & ) /@ Range[Length[fourVelocity] - 1]];
shiftVector = (Module[{index = #1}, Superscript["\[FormalBeta]", index] @@ Join[{"\[FormalT]"},
(Superscript["\[FormalX]", ToString[#1]] & ) /@ Range[Length[fourVelocity] - 1]]] & ) /@
Range[Length[fourVelocity] - 1]; matrixRepresentation = DiagonalMatrix[ConstantArray[1, Length[fourVelocity] - 1]];
ADMStressEnergyDecomposition[ResourceFunction["ADMDecomposition"][ResourceFunction["MetricTensor"][
matrixRepresentation, (Superscript["\[FormalX]", ToString[#1]] & ) /@ Range[Length[fourVelocity] - 1], True, True],
"\[FormalT]", lapseFunction, shiftVector],
Normal[SparseArray[(#1 -> density*fourVelocity[[First[#1]]]*fourVelocity[[Last[#1]]] & ) /@
Tuples[Range[Length[fourVelocity]], 2]]]]]
ADMStressEnergyDecomposition[{"Dust", fourVelocity_List, density_},
(admDecomposition_)[(metricTensor_)[matrixRepresentation_List, coordinates_List, index1_, index2_], timeCoordinate_,
lapseFunction_, shiftVector_List]] := ADMStressEnergyDecomposition[ResourceFunction["ADMDecomposition"][
ResourceFunction["MetricTensor"][matrixRepresentation, coordinates, index1, index2], timeCoordinate, lapseFunction,
shiftVector], Normal[SparseArray[(#1 -> density*fourVelocity[[First[#1]]]*fourVelocity[[Last[#1]]] & ) /@
Tuples[Range[Length[matrixRepresentation] + 1], 2]]]] /; SymbolName[admDecomposition] === "ADMDecomposition" &&
SymbolName[metricTensor] === "MetricTensor" && Length[Dimensions[matrixRepresentation]] == 2 &&
Length[coordinates] == Length[matrixRepresentation] && BooleanQ[index1] && BooleanQ[index2] &&
Length[shiftVector] == Length[matrixRepresentation] && Length[fourVelocity] == Length[matrixRepresentation] + 1
ADMStressEnergyDecomposition["DustField"] := Module[{lapseFunction, shiftVector, matrixRepresentation},
lapseFunction = "\[FormalAlpha]" @@ Join[{"\[FormalT]"}, (Superscript["\[FormalX]", ToString[#1]] & ) /@ Range[3]];
shiftVector = (Module[{index = #1}, Superscript["\[FormalBeta]", index] @@ Join[{"\[FormalT]"},
(Superscript["\[FormalX]", ToString[#1]] & ) /@ Range[3]]] & ) /@ Range[3];
matrixRepresentation = DiagonalMatrix[ConstantArray[1, 3]]; ADMStressEnergyDecomposition[
ResourceFunction["ADMDecomposition"][ResourceFunction["MetricTensor"][matrixRepresentation,
(Superscript["\[FormalX]", ToString[#1]] & ) /@ Range[3], True, True], "\[FormalT]", lapseFunction, shiftVector],
Normal[SparseArray[(Module[{index = #1}, index -> "\[FormalRho]" @@ Join[{"\[FormalT]"}, (Superscript["\[FormalX]", ToString[#1]] & ) /@
Range[3]]*Superscript["\[FormalU]", ToString[First[index]]] @@ Join[{"\[FormalT]"}, (Superscript["\[FormalX]",
ToString[#1]] & ) /@ Range[3]]*Superscript["\[FormalU]", ToString[Last[index]]] @@
Join[{"\[FormalT]"}, (Superscript["\[FormalX]", ToString[#1]] & ) /@ Range[3]]] & ) /@ Tuples[Range[4], 2]]]]]
ADMStressEnergyDecomposition["DustField", (admDecomposition_)[(metricTensor_)[matrixRepresentation_List,
coordinates_List, index1_, index2_], timeCoordinate_, lapseFunction_, shiftVector_List]] :=
ADMStressEnergyDecomposition[ResourceFunction["ADMDecomposition"][ResourceFunction["MetricTensor"][
matrixRepresentation, coordinates, index1, index2], timeCoordinate, lapseFunction, shiftVector],
Normal[SparseArray[(Module[{index = #1}, index -> "\[FormalRho]" @@ Join[{timeCoordinate}, coordinates]*
Superscript["\[FormalU]", ToString[First[index]]] @@ Join[{timeCoordinate}, coordinates]*
Superscript["\[FormalU]", ToString[Last[index]]] @@ Join[{timeCoordinate}, coordinates]] & ) /@
Tuples[Range[Length[matrixRepresentation] + 1], 2]]]] /; SymbolName[admDecomposition] === "ADMDecomposition" &&
SymbolName[metricTensor] === "MetricTensor" && Length[Dimensions[matrixRepresentation]] == 2 &&
Length[coordinates] == Length[matrixRepresentation] && BooleanQ[index1] && BooleanQ[index2] &&
Length[shiftVector] == Length[matrixRepresentation]
ADMStressEnergyDecomposition[{"DustField", fourVelocity_List}] :=
Module[{lapseFunction, shiftVector, matrixRepresentation},
lapseFunction = "\[FormalAlpha]" @@ Join[{"\[FormalT]"}, (Superscript["\[FormalX]", ToString[#1]] & ) /@ Range[Length[fourVelocity] - 1]];
shiftVector = (Module[{index = #1}, Superscript["\[FormalBeta]", index] @@ Join[{"\[FormalT]"},
(Superscript["\[FormalX]", ToString[#1]] & ) /@ Range[Length[fourVelocity] - 1]]] & ) /@
Range[Length[fourVelocity] - 1]; matrixRepresentation = DiagonalMatrix[ConstantArray[1, Length[fourVelocity] - 1]];
ADMStressEnergyDecomposition[ResourceFunction["ADMDecomposition"][ResourceFunction["MetricTensor"][
matrixRepresentation, (Superscript["\[FormalX]", ToString[#1]] & ) /@ Range[Length[fourVelocity] - 1], True, True],
"\[FormalT]", lapseFunction, shiftVector],
Normal[SparseArray[(Module[{index = #1}, index -> "\[FormalRho]" @@ Join[{"\[FormalT]"}, (Superscript["\[FormalX]", ToString[#1]] & ) /@
Range[Length[fourVelocity] - 1]]*fourVelocity[[First[index]]] @@ Join[{"\[FormalT]"},
(Superscript["\[FormalX]", ToString[#1]] & ) /@ Range[Length[fourVelocity] - 1]]*fourVelocity[[Last[index]]] @@
Join[{"\[FormalT]"}, (Superscript["\[FormalX]", ToString[#1]] & ) /@ Range[Length[fourVelocity] - 1]]] & ) /@
Tuples[Range[Length[fourVelocity]], 2]]]]]
ADMStressEnergyDecomposition[{"DustField", fourVelocity_List},
(admDecomposition_)[(metricTensor_)[matrixRepresentation_List, coordinates_List, index1_, index2_], timeCoordinate_,
lapseFunction_, shiftVector_List]] := ADMStressEnergyDecomposition[ResourceFunction["ADMDecomposition"][
ResourceFunction["MetricTensor"][matrixRepresentation, coordinates, index1, index2], timeCoordinate, lapseFunction,
shiftVector], Normal[SparseArray[(Module[{index = #1}, index -> "\[FormalRho]" @@ Join[{timeCoordinate}, coordinates]*
fourVelocity[[First[index]]] @@ Join[{timeCoordinate}, coordinates]*fourVelocity[[Last[index]]] @@
Join[{timeCoordinate}, coordinates]] & ) /@ Tuples[Range[Length[matrixRepresentation] + 1], 2]]]] /;
SymbolName[admDecomposition] === "ADMDecomposition" && SymbolName[metricTensor] === "MetricTensor" &&
Length[Dimensions[matrixRepresentation]] == 2 && Length[coordinates] == Length[matrixRepresentation] &&
BooleanQ[index1] && BooleanQ[index2] && Length[shiftVector] == Length[matrixRepresentation] &&
Length[fourVelocity] == Length[matrixRepresentation] + 1
ADMStressEnergyDecomposition[{"DustField", fourVelocity_List, density_}] :=
Module[{lapseFunction, shiftVector, matrixRepresentation},
lapseFunction = "\[FormalAlpha]" @@ Join[{"\[FormalT]"}, (Superscript["\[FormalX]", ToString[#1]] & ) /@ Range[Length[fourVelocity] - 1]];
shiftVector = (Module[{index = #1}, Superscript["\[FormalBeta]", index] @@ Join[{"\[FormalT]"},
(Superscript["\[FormalX]", ToString[#1]] & ) /@ Range[Length[fourVelocity] - 1]]] & ) /@
Range[Length[fourVelocity] - 1]; matrixRepresentation = DiagonalMatrix[ConstantArray[1, Length[fourVelocity] - 1]];
ADMStressEnergyDecomposition[ResourceFunction["ADMDecomposition"][ResourceFunction["MetricTensor"][
matrixRepresentation, (Superscript["\[FormalX]", ToString[#1]] & ) /@ Range[Length[fourVelocity] - 1], True, True],
"\[FormalT]", lapseFunction, shiftVector],
Normal[SparseArray[(Module[{index = #1}, index -> density @@ Join[{"\[FormalT]"}, (Superscript["\[FormalX]", ToString[#1]] & ) /@
Range[Length[fourVelocity] - 1]]*fourVelocity[[First[index]]] @@ Join[{"\[FormalT]"},
(Superscript["\[FormalX]", ToString[#1]] & ) /@ Range[Length[fourVelocity] - 1]]*fourVelocity[[Last[index]]] @@
Join[{"\[FormalT]"}, (Superscript["\[FormalX]", ToString[#1]] & ) /@ Range[Length[fourVelocity] - 1]]] & ) /@
Tuples[Range[Length[fourVelocity]], 2]]]]]
ADMStressEnergyDecomposition[{"DustField", fourVelocity_List, density_},
(admDecomposition_)[(metricTensor_)[matrixRepresentation_List, coordinates_List, index1_, index2_], timeCoordinate_,
lapseFunction_, shiftVector_List]] := ADMStressEnergyDecomposition[ResourceFunction["ADMDecomposition"][
ResourceFunction["MetricTensor"][matrixRepresentation, coordinates, index1, index2], timeCoordinate, lapseFunction,
shiftVector], Normal[SparseArray[(Module[{index = #1}, index -> density @@ Join[{timeCoordinate}, coordinates]*
fourVelocity[[First[index]]] @@ Join[{timeCoordinate}, coordinates]*fourVelocity[[Last[index]]] @@
Join[{timeCoordinate}, coordinates]] & ) /@ Tuples[Range[Length[matrixRepresentation] + 1], 2]]]] /;
SymbolName[admDecomposition] === "ADMDecomposition" && SymbolName[metricTensor] === "MetricTensor" &&
Length[Dimensions[matrixRepresentation]] == 2 && Length[coordinates] == Length[matrixRepresentation] &&
BooleanQ[index1] && BooleanQ[index2] && Length[shiftVector] == Length[matrixRepresentation] &&
Length[fourVelocity] == Length[matrixRepresentation] + 1
ADMStressEnergyDecomposition["Radiation"] := Module[{lapseFunction, shiftVector, matrixRepresentation, shiftCovector,
spacetimeMetricTensor}, lapseFunction = "\[FormalAlpha]" @@ Join[{"\[FormalT]"}, (Superscript["\[FormalX]", ToString[#1]] & ) /@ Range[3]];
shiftVector = (Module[{index = #1}, Superscript["\[FormalBeta]", index] @@ Join[{"\[FormalT]"},
(Superscript["\[FormalX]", ToString[#1]] & ) /@ Range[3]]] & ) /@ Range[3];
matrixRepresentation = DiagonalMatrix[ConstantArray[1, 3]];
shiftCovector = Normal[SparseArray[
(Module[{index = #1}, index -> Total[(matrixRepresentation[[index,#1]]*shiftVector[[#1]] & ) /@
Range[Length[matrixRepresentation]]]] & ) /@ Range[Length[matrixRepresentation]]]];
spacetimeMetricTensor = Normal[SparseArray[
Join[{{1, 1} -> Total[(shiftVector[[#1]]*shiftCovector[[#1]] & ) /@ Range[Length[matrixRepresentation]]] -
lapseFunction^2}, (Module[{index = #1}, {1, index + 1} -> Total[(matrixRepresentation[[index,#1]]*
shiftVector[[#1]] & ) /@ Range[Length[matrixRepresentation]]]] & ) /@
Range[Length[matrixRepresentation]], (Module[{index = #1}, {index + 1, 1} ->
Total[(matrixRepresentation[[#1,index]]*shiftVector[[#1]] & ) /@ Range[Length[matrixRepresentation]]]] & ) /@
Range[Length[matrixRepresentation]], ({First[#1] + 1, Last[#1] + 1} -> matrixRepresentation[[First[#1],
Last[#1]]] & ) /@ Tuples[Range[Length[matrixRepresentation]], 2]]]];
ADMStressEnergyDecomposition[ResourceFunction["ADMDecomposition"][ResourceFunction["MetricTensor"][
matrixRepresentation, (Superscript["\[FormalX]", ToString[#1]] & ) /@ Range[3], True, True], "\[FormalT]", lapseFunction,
shiftVector], Normal[SparseArray[(#1 -> (4*"\[FormalCapitalP]")*Superscript["\[FormalU]", ToString[First[#1]]]*
Superscript["\[FormalU]", ToString[Last[#1]]] + "\[FormalCapitalP]"*Inverse[spacetimeMetricTensor][[First[#1],Last[#1]]] & ) /@
Tuples[Range[4], 2]]]]]
ADMStressEnergyDecomposition["Radiation", (admDecomposition_)[(metricTensor_)[matrixRepresentation_List,
coordinates_List, index1_, index2_], timeCoordinate_, lapseFunction_, shiftVector_List]] :=
Module[{shiftCovector, spacetimeMetricTensor},
shiftCovector = Normal[SparseArray[(Module[{index = #1}, index -> Total[(matrixRepresentation[[index,#1]]*
shiftVector[[#1]] & ) /@ Range[Length[matrixRepresentation]]]] & ) /@
Range[Length[matrixRepresentation]]]]; spacetimeMetricTensor =
Normal[SparseArray[Join[{{1, 1} -> Total[(shiftVector[[#1]]*shiftCovector[[#1]] & ) /@
Range[Length[matrixRepresentation]]] - lapseFunction^2},
(Module[{index = #1}, {1, index + 1} -> Total[(matrixRepresentation[[index,#1]]*shiftVector[[#1]] & ) /@ Range[
Length[matrixRepresentation]]]] & ) /@ Range[Length[matrixRepresentation]],
(Module[{index = #1}, {index + 1, 1} -> Total[(matrixRepresentation[[#1,index]]*shiftVector[[#1]] & ) /@ Range[
Length[matrixRepresentation]]]] & ) /@ Range[Length[matrixRepresentation]],
({First[#1] + 1, Last[#1] + 1} -> matrixRepresentation[[First[#1],Last[#1]]] & ) /@
Tuples[Range[Length[matrixRepresentation]], 2]]]]; ADMStressEnergyDecomposition[
ResourceFunction["ADMDecomposition"][ResourceFunction["MetricTensor"][matrixRepresentation, coordinates, index1,
index2], timeCoordinate, lapseFunction, shiftVector],
Normal[SparseArray[(#1 -> (Length[spacetimeMetricTensor]*"\[FormalCapitalP]")*Superscript["\[FormalU]", ToString[First[#1]]]*
Superscript["\[FormalU]", ToString[Last[#1]]] + "\[FormalCapitalP]"*Inverse[spacetimeMetricTensor][[First[#1],Last[#1]]] & ) /@
Tuples[Range[Length[spacetimeMetricTensor]], 2]]]]] /; SymbolName[admDecomposition] === "ADMDecomposition" &&
SymbolName[metricTensor] === "MetricTensor" && Length[Dimensions[matrixRepresentation]] == 2 &&
Length[coordinates] == Length[matrixRepresentation] && BooleanQ[index1] && BooleanQ[index2] &&
Length[shiftVector] == Length[matrixRepresentation]
ADMStressEnergyDecomposition[{"Radiation", fourVelocity_List}] :=
Module[{lapseFunction, shiftVector, matrixRepresentation, shiftCovector, spacetimeMetricTensor},
lapseFunction = "\[FormalAlpha]" @@ Join[{"\[FormalT]"}, (Superscript["\[FormalX]", ToString[#1]] & ) /@ Range[Length[fourVelocity] - 1]];
shiftVector = (Module[{index = #1}, Superscript["\[FormalBeta]", index] @@ Join[{"\[FormalT]"},
(Superscript["\[FormalX]", ToString[#1]] & ) /@ Range[Length[fourVelocity] - 1]]] & ) /@
Range[Length[fourVelocity] - 1]; matrixRepresentation = DiagonalMatrix[ConstantArray[1, Length[fourVelocity] - 1]];
shiftCovector = Normal[SparseArray[
(Module[{index = #1}, index -> Total[(matrixRepresentation[[index,#1]]*shiftVector[[#1]] & ) /@
Range[Length[matrixRepresentation]]]] & ) /@ Range[Length[matrixRepresentation]]]];
spacetimeMetricTensor = Normal[SparseArray[
Join[{{1, 1} -> Total[(shiftVector[[#1]]*shiftCovector[[#1]] & ) /@ Range[Length[matrixRepresentation]]] -
lapseFunction^2}, (Module[{index = #1}, {1, index + 1} -> Total[(matrixRepresentation[[index,#1]]*
shiftVector[[#1]] & ) /@ Range[Length[matrixRepresentation]]]] & ) /@
Range[Length[matrixRepresentation]], (Module[{index = #1}, {index + 1, 1} ->
Total[(matrixRepresentation[[#1,index]]*shiftVector[[#1]] & ) /@ Range[Length[matrixRepresentation]]]] & ) /@
Range[Length[matrixRepresentation]], ({First[#1] + 1, Last[#1] + 1} -> matrixRepresentation[[First[#1],
Last[#1]]] & ) /@ Tuples[Range[Length[matrixRepresentation]], 2]]]];
ADMStressEnergyDecomposition[ResourceFunction["ADMDecomposition"][ResourceFunction["MetricTensor"][
matrixRepresentation, (Superscript["\[FormalX]", ToString[#1]] & ) /@ Range[Length[fourVelocity] - 1], True, True],
"\[FormalT]", lapseFunction, shiftVector],
Normal[SparseArray[(#1 -> (Length[fourVelocity]*"\[FormalCapitalP]")*fourVelocity[[First[#1]]]*fourVelocity[[Last[#1]]] +
"\[FormalCapitalP]"*Inverse[spacetimeMetricTensor][[First[#1],Last[#1]]] & ) /@ Tuples[Range[Length[spacetimeMetricTensor]],
2]]]]]
ADMStressEnergyDecomposition[{"Radiation", fourVelocity_List},
(admDecomposition_)[(metricTensor_)[matrixRepresentation_List, coordinates_List, index1_, index2_], timeCoordinate_,
lapseFunction_, shiftVector_List]] := Module[{shiftCovector, spacetimeMetricTensor},
shiftCovector = Normal[SparseArray[(Module[{index = #1}, index -> Total[(matrixRepresentation[[index,#1]]*
shiftVector[[#1]] & ) /@ Range[Length[matrixRepresentation]]]] & ) /@
Range[Length[matrixRepresentation]]]]; spacetimeMetricTensor =
Normal[SparseArray[Join[{{1, 1} -> Total[(shiftVector[[#1]]*shiftCovector[[#1]] & ) /@
Range[Length[matrixRepresentation]]] - lapseFunction^2},
(Module[{index = #1}, {1, index + 1} -> Total[(matrixRepresentation[[index,#1]]*shiftVector[[#1]] & ) /@ Range[
Length[matrixRepresentation]]]] & ) /@ Range[Length[matrixRepresentation]],
(Module[{index = #1}, {index + 1, 1} -> Total[(matrixRepresentation[[#1,index]]*shiftVector[[#1]] & ) /@ Range[
Length[matrixRepresentation]]]] & ) /@ Range[Length[matrixRepresentation]],
({First[#1] + 1, Last[#1] + 1} -> matrixRepresentation[[First[#1],Last[#1]]] & ) /@
Tuples[Range[Length[matrixRepresentation]], 2]]]]; ADMStressEnergyDecomposition[
ResourceFunction["ADMDecomposition"][ResourceFunction["MetricTensor"][matrixRepresentation, coordinates, index1,
index2], timeCoordinate, lapseFunction, shiftVector],
Normal[SparseArray[(#1 -> (Length[fourVelocity]*"\[FormalCapitalP]")*fourVelocity[[First[#1]]]*fourVelocity[[Last[#1]]] +
"\[FormalCapitalP]"*Inverse[spacetimeMetricTensor][[First[#1],Last[#1]]] & ) /@
Tuples[Range[Length[spacetimeMetricTensor]], 2]]]]] /; SymbolName[admDecomposition] === "ADMDecomposition" &&
SymbolName[metricTensor] === "MetricTensor" && Length[Dimensions[matrixRepresentation]] == 2 &&
Length[coordinates] == Length[matrixRepresentation] && BooleanQ[index1] && BooleanQ[index2] &&
Length[shiftVector] == Length[matrixRepresentation] && Length[fourVelocity] == Length[matrixRepresentation] + 1
ADMStressEnergyDecomposition[{"Radiation", fourVelocity_List, pressure_}] :=
Module[{lapseFunction, shiftVector, matrixRepresentation, shiftCovector, spacetimeMetricTensor},
lapseFunction = "\[FormalAlpha]" @@ Join[{"\[FormalT]"}, (Superscript["\[FormalX]", ToString[#1]] & ) /@ Range[Length[fourVelocity] - 1]];
shiftVector = (Module[{index = #1}, Superscript["\[FormalBeta]", index] @@ Join[{"\[FormalT]"},
(Superscript["\[FormalX]", ToString[#1]] & ) /@ Range[Length[fourVelocity] - 1]]] & ) /@
Range[Length[fourVelocity] - 1]; matrixRepresentation = DiagonalMatrix[ConstantArray[1, Length[fourVelocity] - 1]];
shiftCovector = Normal[SparseArray[
(Module[{index = #1}, index -> Total[(matrixRepresentation[[index,#1]]*shiftVector[[#1]] & ) /@
Range[Length[matrixRepresentation]]]] & ) /@ Range[Length[matrixRepresentation]]]];
spacetimeMetricTensor = Normal[SparseArray[
Join[{{1, 1} -> Total[(shiftVector[[#1]]*shiftCovector[[#1]] & ) /@ Range[Length[matrixRepresentation]]] -
lapseFunction^2}, (Module[{index = #1}, {1, index + 1} -> Total[(matrixRepresentation[[index,#1]]*
shiftVector[[#1]] & ) /@ Range[Length[matrixRepresentation]]]] & ) /@
Range[Length[matrixRepresentation]], (Module[{index = #1}, {index + 1, 1} ->
Total[(matrixRepresentation[[#1,index]]*shiftVector[[#1]] & ) /@ Range[Length[matrixRepresentation]]]] & ) /@
Range[Length[matrixRepresentation]], ({First[#1] + 1, Last[#1] + 1} -> matrixRepresentation[[First[#1],
Last[#1]]] & ) /@ Tuples[Range[Length[matrixRepresentation]], 2]]]];
ADMStressEnergyDecomposition[ResourceFunction["ADMDecomposition"][ResourceFunction["MetricTensor"][
matrixRepresentation, (Superscript["\[FormalX]", ToString[#1]] & ) /@ Range[Length[fourVelocity] - 1], True, True],
"\[FormalT]", lapseFunction, shiftVector],
Normal[SparseArray[(#1 -> (Length[fourVelocity]*pressure)*fourVelocity[[First[#1]]]*fourVelocity[[Last[#1]]] +
pressure*Inverse[spacetimeMetricTensor][[First[#1],Last[#1]]] & ) /@
Tuples[Range[Length[spacetimeMetricTensor]], 2]]]]]
ADMStressEnergyDecomposition[{"Radiation", fourVelocity_List, pressure_},
(admDecomposition_)[(metricTensor_)[matrixRepresentation_List, coordinates_List, index1_, index2_], timeCoordinate_,
lapseFunction_, shiftVector_List]] := Module[{shiftCovector, spacetimeMetricTensor},
shiftCovector = Normal[SparseArray[(Module[{index = #1}, index -> Total[(matrixRepresentation[[index,#1]]*
shiftVector[[#1]] & ) /@ Range[Length[matrixRepresentation]]]] & ) /@
Range[Length[matrixRepresentation]]]]; spacetimeMetricTensor =
Normal[SparseArray[Join[{{1, 1} -> Total[(shiftVector[[#1]]*shiftCovector[[#1]] & ) /@
Range[Length[matrixRepresentation]]] - lapseFunction^2},
(Module[{index = #1}, {1, index + 1} -> Total[(matrixRepresentation[[index,#1]]*shiftVector[[#1]] & ) /@ Range[
Length[matrixRepresentation]]]] & ) /@ Range[Length[matrixRepresentation]],
(Module[{index = #1}, {index + 1, 1} -> Total[(matrixRepresentation[[#1,index]]*shiftVector[[#1]] & ) /@ Range[
Length[matrixRepresentation]]]] & ) /@ Range[Length[matrixRepresentation]],
({First[#1] + 1, Last[#1] + 1} -> matrixRepresentation[[First[#1],Last[#1]]] & ) /@
Tuples[Range[Length[matrixRepresentation]], 2]]]]; ADMStressEnergyDecomposition[
ResourceFunction["ADMDecomposition"][ResourceFunction["MetricTensor"][matrixRepresentation, coordinates, index1,
index2], timeCoordinate, lapseFunction, shiftVector],
Normal[SparseArray[(#1 -> (Length[fourVelocity]*pressure)*fourVelocity[[First[#1]]]*fourVelocity[[Last[#1]]] +
pressure*Inverse[spacetimeMetricTensor][[First[#1],Last[#1]]] & ) /@
Tuples[Range[Length[spacetimeMetricTensor]], 2]]]]] /; SymbolName[admDecomposition] === "ADMDecomposition" &&
SymbolName[metricTensor] === "MetricTensor" && Length[Dimensions[matrixRepresentation]] == 2 &&
Length[coordinates] == Length[matrixRepresentation] && BooleanQ[index1] && BooleanQ[index2] &&
Length[shiftVector] == Length[matrixRepresentation] && Length[fourVelocity] == Length[matrixRepresentation] + 1
ADMStressEnergyDecomposition["RadiationField"] := Module[{lapseFunction, shiftVector, matrixRepresentation,
shiftCovector, spacetimeMetricTensor},
lapseFunction = "\[FormalAlpha]" @@ Join[{"\[FormalT]"}, (Superscript["\[FormalX]", ToString[#1]] & ) /@ Range[3]];
shiftVector = (Module[{index = #1}, Superscript["\[FormalBeta]", index] @@ Join[{"\[FormalT]"},
(Superscript["\[FormalX]", ToString[#1]] & ) /@ Range[3]]] & ) /@ Range[3];
matrixRepresentation = DiagonalMatrix[ConstantArray[1, 3]];
shiftCovector = Normal[SparseArray[
(Module[{index = #1}, index -> Total[(matrixRepresentation[[index,#1]]*shiftVector[[#1]] & ) /@
Range[Length[matrixRepresentation]]]] & ) /@ Range[Length[matrixRepresentation]]]];
spacetimeMetricTensor = Normal[SparseArray[
Join[{{1, 1} -> Total[(shiftVector[[#1]]*shiftCovector[[#1]] & ) /@ Range[Length[matrixRepresentation]]] -
lapseFunction^2}, (Module[{index = #1}, {1, index + 1} -> Total[(matrixRepresentation[[index,#1]]*
shiftVector[[#1]] & ) /@ Range[Length[matrixRepresentation]]]] & ) /@
Range[Length[matrixRepresentation]], (Module[{index = #1}, {index + 1, 1} ->
Total[(matrixRepresentation[[#1,index]]*shiftVector[[#1]] & ) /@ Range[Length[matrixRepresentation]]]] & ) /@
Range[Length[matrixRepresentation]], ({First[#1] + 1, Last[#1] + 1} -> matrixRepresentation[[First[#1],
Last[#1]]] & ) /@ Tuples[Range[Length[matrixRepresentation]], 2]]]];
ADMStressEnergyDecomposition[ResourceFunction["ADMDecomposition"][ResourceFunction["MetricTensor"][
matrixRepresentation, (Superscript["\[FormalX]", ToString[#1]] & ) /@ Range[3], True, True], "\[FormalT]", lapseFunction,
shiftVector], Normal[SparseArray[
(Module[{index = #1}, index -> (4*"\[FormalCapitalP]" @@ Join[{"\[FormalT]"}, (Superscript["\[FormalX]", ToString[#1]] & ) /@ Range[3]])*
Superscript["\[FormalU]", ToString[First[index]]] @@ Join[{"\[FormalT]"}, (Superscript["\[FormalX]", ToString[#1]] & ) /@
Range[3]]*Superscript["\[FormalU]", ToString[Last[index]]] @@ Join[{"\[FormalT]"}, (Superscript["\[FormalX]",
ToString[#1]] & ) /@ Range[3]] + "\[FormalCapitalP]" @@ Join[{"\[FormalT]"}, (Superscript["\[FormalX]", ToString[#1]] & ) /@
Range[3]]*Inverse[spacetimeMetricTensor][[First[#1],Last[#1]]]] & ) /@ Tuples[Range[4], 2]]]]]
ADMStressEnergyDecomposition["RadiationField", (admDecomposition_)[(metricTensor_)[matrixRepresentation_List,
coordinates_List, index1_, index2_], timeCoordinate_, lapseFunction_, shiftVector_List]] :=
Module[{shiftCovector, spacetimeMetricTensor},
shiftCovector = Normal[SparseArray[(Module[{index = #1}, index -> Total[(matrixRepresentation[[index,#1]]*
shiftVector[[#1]] & ) /@ Range[Length[matrixRepresentation]]]] & ) /@
Range[Length[matrixRepresentation]]]]; spacetimeMetricTensor =
Normal[SparseArray[Join[{{1, 1} -> Total[(shiftVector[[#1]]*shiftCovector[[#1]] & ) /@
Range[Length[matrixRepresentation]]] - lapseFunction^2},
(Module[{index = #1}, {1, index + 1} -> Total[(matrixRepresentation[[index,#1]]*shiftVector[[#1]] & ) /@ Range[
Length[matrixRepresentation]]]] & ) /@ Range[Length[matrixRepresentation]],
(Module[{index = #1}, {index + 1, 1} -> Total[(matrixRepresentation[[#1,index]]*shiftVector[[#1]] & ) /@ Range[
Length[matrixRepresentation]]]] & ) /@ Range[Length[matrixRepresentation]],
({First[#1] + 1, Last[#1] + 1} -> matrixRepresentation[[First[#1],Last[#1]]] & ) /@
Tuples[Range[Length[matrixRepresentation]], 2]]]]; ADMStressEnergyDecomposition[
ResourceFunction["ADMDecomposition"][ResourceFunction["MetricTensor"][matrixRepresentation, coordinates, index1,
index2], timeCoordinate, lapseFunction, shiftVector],
Normal[SparseArray[(Module[{index = #1}, index -> (Length[spacetimeMetricTensor]*"\[FormalCapitalP]" @@ Join[{timeCoordinate},
coordinates])*Superscript["\[FormalU]", ToString[First[index]]] @@ Join[{timeCoordinate}, coordinates]*
Superscript["\[FormalU]", ToString[Last[index]]] @@ Join[{timeCoordinate}, coordinates] +
"\[FormalCapitalP]" @@ Join[{timeCoordinate}, coordinates]*Inverse[spacetimeMetricTensor][[First[#1],Last[#1]]]] & ) /@
Tuples[Range[Length[spacetimeMetricTensor]], 2]]]]] /; SymbolName[admDecomposition] === "ADMDecomposition" &&
SymbolName[metricTensor] === "MetricTensor" && Length[Dimensions[matrixRepresentation]] == 2 &&
Length[coordinates] == Length[matrixRepresentation] && BooleanQ[index1] && BooleanQ[index2] &&
Length[shiftVector] == Length[matrixRepresentation]
ADMStressEnergyDecomposition[{"RadiationField", fourVelocity_List}] :=
Module[{lapseFunction, shiftVector, matrixRepresentation, shiftCovector, spacetimeMetricTensor},
lapseFunction = "\[FormalAlpha]" @@ Join[{"\[FormalT]"}, (Superscript["\[FormalX]", ToString[#1]] & ) /@ Range[Length[fourVelocity] - 1]];
shiftVector = (Module[{index = #1}, Superscript["\[FormalBeta]", index] @@ Join[{"\[FormalT]"},
(Superscript["\[FormalX]", ToString[#1]] & ) /@ Range[Length[fourVelocity] - 1]]] & ) /@
Range[Length[fourVelocity] - 1]; matrixRepresentation = DiagonalMatrix[ConstantArray[1, Length[fourVelocity] - 1]];
shiftCovector = Normal[SparseArray[
(Module[{index = #1}, index -> Total[(matrixRepresentation[[index,#1]]*shiftVector[[#1]] & ) /@
Range[Length[matrixRepresentation]]]] & ) /@ Range[Length[matrixRepresentation]]]];
spacetimeMetricTensor = Normal[SparseArray[
Join[{{1, 1} -> Total[(shiftVector[[#1]]*shiftCovector[[#1]] & ) /@ Range[Length[matrixRepresentation]]] -
lapseFunction^2}, (Module[{index = #1}, {1, index + 1} -> Total[(matrixRepresentation[[index,#1]]*
shiftVector[[#1]] & ) /@ Range[Length[matrixRepresentation]]]] & ) /@
Range[Length[matrixRepresentation]], (Module[{index = #1}, {index + 1, 1} ->
Total[(matrixRepresentation[[#1,index]]*shiftVector[[#1]] & ) /@ Range[Length[matrixRepresentation]]]] & ) /@
Range[Length[matrixRepresentation]], ({First[#1] + 1, Last[#1] + 1} -> matrixRepresentation[[First[#1],
Last[#1]]] & ) /@ Tuples[Range[Length[matrixRepresentation]], 2]]]];
ADMStressEnergyDecomposition[ResourceFunction["ADMDecomposition"][ResourceFunction["MetricTensor"][
matrixRepresentation, (Superscript["\[FormalX]", ToString[#1]] & ) /@ Range[Length[fourVelocity] - 1], True, True],
"\[FormalT]", lapseFunction, shiftVector],
Normal[SparseArray[(Module[{index = #1}, index -> (Length[fourVelocity]*"\[FormalCapitalP]" @@ Join[{"\[FormalT]"},
(Superscript["\[FormalX]", ToString[#1]] & ) /@ Range[Length[fourVelocity] - 1]])*
fourVelocity[[First[index]]] @@ Join[{"\[FormalT]"}, (Superscript["\[FormalX]", ToString[#1]] & ) /@
Range[Length[fourVelocity] - 1]]*fourVelocity[[Last[index]]] @@ Join[{"\[FormalT]"},
(Superscript["\[FormalX]", ToString[#1]] & ) /@ Range[Length[fourVelocity] - 1]] +
"\[FormalCapitalP]" @@ Join[{"\[FormalT]"}, (Superscript["\[FormalX]", ToString[#1]] & ) /@ Range[Length[fourVelocity] - 1]]*
Inverse[spacetimeMetricTensor][[First[#1],Last[#1]]]] & ) /@ Tuples[Range[Length[fourVelocity]], 2]]]]]
ADMStressEnergyDecomposition[{"RadiationField", fourVelocity_List},
(admDecomposition_)[(metricTensor_)[matrixRepresentation_List, coordinates_List, index1_, index2_], timeCoordinate_,
lapseFunction_, shiftVector_List]] := Module[{shiftCovector, spacetimeMetricTensor},
shiftCovector = Normal[SparseArray[(Module[{index = #1}, index -> Total[(matrixRepresentation[[index,#1]]*
shiftVector[[#1]] & ) /@ Range[Length[matrixRepresentation]]]] & ) /@
Range[Length[matrixRepresentation]]]]; spacetimeMetricTensor =
Normal[SparseArray[Join[{{1, 1} -> Total[(shiftVector[[#1]]*shiftCovector[[#1]] & ) /@
Range[Length[matrixRepresentation]]] - lapseFunction^2},
(Module[{index = #1}, {1, index + 1} -> Total[(matrixRepresentation[[index,#1]]*shiftVector[[#1]] & ) /@ Range[
Length[matrixRepresentation]]]] & ) /@ Range[Length[matrixRepresentation]],
(Module[{index = #1}, {index + 1, 1} -> Total[(matrixRepresentation[[#1,index]]*shiftVector[[#1]] & ) /@ Range[
Length[matrixRepresentation]]]] & ) /@ Range[Length[matrixRepresentation]],
({First[#1] + 1, Last[#1] + 1} -> matrixRepresentation[[First[#1],Last[#1]]] & ) /@
Tuples[Range[Length[matrixRepresentation]], 2]]]]; ADMStressEnergyDecomposition[
ResourceFunction["ADMDecomposition"][ResourceFunction["MetricTensor"][matrixRepresentation, coordinates, index1,
index2], timeCoordinate, lapseFunction, shiftVector],
Normal[SparseArray[(Module[{index = #1}, index -> (Length[fourVelocity]*"\[FormalCapitalP]" @@ Join[{timeCoordinate},
coordinates])*fourVelocity[[First[index]]] @@ Join[{timeCoordinate}, coordinates]*
fourVelocity[[Last[index]]] @@ Join[{timeCoordinate}, coordinates] + "\[FormalCapitalP]" @@ Join[{timeCoordinate},
coordinates]*Inverse[spacetimeMetricTensor][[First[#1],Last[#1]]]] & ) /@
Tuples[Range[Length[spacetimeMetricTensor]], 2]]]]] /; SymbolName[admDecomposition] === "ADMDecomposition" &&
SymbolName[metricTensor] === "MetricTensor" && Length[Dimensions[matrixRepresentation]] == 2 &&
Length[coordinates] == Length[matrixRepresentation] && BooleanQ[index1] && BooleanQ[index2] &&
Length[shiftVector] == Length[matrixRepresentation] && Length[fourVelocity] == Length[matrixRepresentation] + 1
ADMStressEnergyDecomposition[{"RadiationField", fourVelocity_List, pressure_}] :=
Module[{lapseFunction, shiftVector, matrixRepresentation, shiftCovector, spacetimeMetricTensor},
lapseFunction = "\[FormalAlpha]" @@ Join[{"\[FormalT]"}, (Superscript["\[FormalX]", ToString[#1]] & ) /@ Range[Length[fourVelocity] - 1]];
shiftVector = (Module[{index = #1}, Superscript["\[FormalBeta]", index] @@ Join[{"\[FormalT]"},
(Superscript["\[FormalX]", ToString[#1]] & ) /@ Range[Length[fourVelocity] - 1]]] & ) /@
Range[Length[fourVelocity] - 1]; matrixRepresentation = DiagonalMatrix[ConstantArray[1, Length[fourVelocity] - 1]];
shiftCovector = Normal[SparseArray[
(Module[{index = #1}, index -> Total[(matrixRepresentation[[index,#1]]*shiftVector[[#1]] & ) /@
Range[Length[matrixRepresentation]]]] & ) /@ Range[Length[matrixRepresentation]]]];
spacetimeMetricTensor = Normal[SparseArray[
Join[{{1, 1} -> Total[(shiftVector[[#1]]*shiftCovector[[#1]] & ) /@ Range[Length[matrixRepresentation]]] -
lapseFunction^2}, (Module[{index = #1}, {1, index + 1} -> Total[(matrixRepresentation[[index,#1]]*
shiftVector[[#1]] & ) /@ Range[Length[matrixRepresentation]]]] & ) /@
Range[Length[matrixRepresentation]], (Module[{index = #1}, {index + 1, 1} ->
Total[(matrixRepresentation[[#1,index]]*shiftVector[[#1]] & ) /@ Range[Length[matrixRepresentation]]]] & ) /@
Range[Length[matrixRepresentation]], ({First[#1] + 1, Last[#1] + 1} -> matrixRepresentation[[First[#1],
Last[#1]]] & ) /@ Tuples[Range[Length[matrixRepresentation]], 2]]]];
ADMStressEnergyDecomposition[ResourceFunction["ADMDecomposition"][ResourceFunction["MetricTensor"][
matrixRepresentation, (Superscript["\[FormalX]", ToString[#1]] & ) /@ Range[Length[fourVelocity] - 1], True, True],
"\[FormalT]", lapseFunction, shiftVector],
Normal[SparseArray[(Module[{index = #1}, index -> (Length[fourVelocity]*pressure @@ Join[{"\[FormalT]"},
(Superscript["\[FormalX]", ToString[#1]] & ) /@ Range[Length[fourVelocity] - 1]])*
fourVelocity[[First[index]]] @@ Join[{"\[FormalT]"}, (Superscript["\[FormalX]", ToString[#1]] & ) /@
Range[Length[fourVelocity] - 1]]*fourVelocity[[Last[index]]] @@ Join[{"\[FormalT]"},
(Superscript["\[FormalX]", ToString[#1]] & ) /@ Range[Length[fourVelocity] - 1]] +
pressure @@ Join[{"\[FormalT]"}, (Superscript["\[FormalX]", ToString[#1]] & ) /@ Range[Length[fourVelocity] - 1]]*
Inverse[spacetimeMetricTensor][[First[#1],Last[#1]]]] & ) /@ Tuples[Range[Length[fourVelocity]], 2]]]]]
ADMStressEnergyDecomposition[{"RadiationField", fourVelocity_List, pressure_},
(admDecomposition_)[(metricTensor_)[matrixRepresentation_List, coordinates_List, index1_, index2_], timeCoordinate_,
lapseFunction_, shiftVector_List]] := Module[{shiftCovector, spacetimeMetricTensor},
shiftCovector = Normal[SparseArray[(Module[{index = #1}, index -> Total[(matrixRepresentation[[index,#1]]*
shiftVector[[#1]] & ) /@ Range[Length[matrixRepresentation]]]] & ) /@
Range[Length[matrixRepresentation]]]]; spacetimeMetricTensor =
Normal[SparseArray[Join[{{1, 1} -> Total[(shiftVector[[#1]]*shiftCovector[[#1]] & ) /@
Range[Length[matrixRepresentation]]] - lapseFunction^2},
(Module[{index = #1}, {1, index + 1} -> Total[(matrixRepresentation[[index,#1]]*shiftVector[[#1]] & ) /@ Range[
Length[matrixRepresentation]]]] & ) /@ Range[Length[matrixRepresentation]],
(Module[{index = #1}, {index + 1, 1} -> Total[(matrixRepresentation[[#1,index]]*shiftVector[[#1]] & ) /@ Range[
Length[matrixRepresentation]]]] & ) /@ Range[Length[matrixRepresentation]],
({First[#1] + 1, Last[#1] + 1} -> matrixRepresentation[[First[#1],Last[#1]]] & ) /@
Tuples[Range[Length[matrixRepresentation]], 2]]]]; ADMStressEnergyDecomposition[
ResourceFunction["ADMDecomposition"][ResourceFunction["MetricTensor"][matrixRepresentation, coordinates, index1,
index2], timeCoordinate, lapseFunction, shiftVector],
Normal[SparseArray[(Module[{index = #1}, index -> (Length[fourVelocity]*pressure @@ Join[{timeCoordinate},
coordinates])*fourVelocity[[First[index]]] @@ Join[{timeCoordinate}, coordinates]*
fourVelocity[[Last[index]]] @@ Join[{timeCoordinate}, coordinates] + pressure @@ Join[{timeCoordinate},
coordinates]*Inverse[spacetimeMetricTensor][[First[#1],Last[#1]]]] & ) /@
Tuples[Range[Length[spacetimeMetricTensor]], 2]]]]] /; SymbolName[admDecomposition] === "ADMDecomposition" &&
SymbolName[metricTensor] === "MetricTensor" && Length[Dimensions[matrixRepresentation]] == 2 &&
Length[coordinates] == Length[matrixRepresentation] && BooleanQ[index1] && BooleanQ[index2] &&
Length[shiftVector] == Length[matrixRepresentation] && Length[fourVelocity] == Length[matrixRepresentation] + 1
ADMStressEnergyDecomposition[(admDecomposition_)[(metricTensor_)[metricMatrixRepresentation_List, coordinates_List,
index1_, index2_], timeCoordinate_, lapseFunction_, shiftVector_List], matrixRepresentation_List][
"ADMDecomposition"] := ResourceFunction["ADMDecomposition"][ResourceFunction["MetricTensor"][
metricMatrixRepresentation, coordinates, index1, index2], timeCoordinate, lapseFunction, shiftVector] /;
SymbolName[admDecomposition] === "ADMDecomposition" && SymbolName[metricTensor] === "MetricTensor" &&
Length[Dimensions[metricMatrixRepresentation]] == 2 && Length[coordinates] == Length[metricMatrixRepresentation] &&
BooleanQ[index1] && BooleanQ[index2] && Length[shiftVector] == Length[metricMatrixRepresentation] &&
Length[Dimensions[matrixRepresentation]] == 2 && Length[coordinates] + 1 == Length[matrixRepresentation]
ADMStressEnergyDecomposition[(admDecomposition_)[(metricTensor_)[metricMatrixRepresentation_List, coordinates_List,
index1_, index2_], timeCoordinate_, lapseFunction_, shiftVector_List], matrixRepresentation_List][
"SpatialMetricTensor"] := ResourceFunction["MetricTensor"][metricMatrixRepresentation, coordinates, index1, index2] /;
SymbolName[admDecomposition] === "ADMDecomposition" && SymbolName[metricTensor] === "MetricTensor" &&
Length[Dimensions[metricMatrixRepresentation]] == 2 && Length[coordinates] == Length[metricMatrixRepresentation] &&
BooleanQ[index1] && BooleanQ[index2] && Length[shiftVector] == Length[metricMatrixRepresentation] &&
Length[Dimensions[matrixRepresentation]] == 2 && Length[coordinates] + 1 == Length[matrixRepresentation]
ADMStressEnergyDecomposition[(admDecomposition_)[(metricTensor_)[metricMatrixRepresentation_List, coordinates_List,
index1_, index2_], timeCoordinate_, lapseFunction_, shiftVector_List], matrixRepresentation_List][
"SpacetimeMetricTensor"] :=
Module[{shiftCovector},
shiftCovector = Normal[SparseArray[(Module[{index = #1}, index -> Total[(metricMatrixRepresentation[[index,#1]]*
shiftVector[[#1]] & ) /@ Range[Length[metricMatrixRepresentation]]]] & ) /@
Range[Length[metricMatrixRepresentation]]]]; ResourceFunction["MetricTensor"][
Normal[SparseArray[Join[{{1, 1} -> Total[(shiftVector[[#1]]*shiftCovector[[#1]] & ) /@
Range[Length[metricMatrixRepresentation]]] - lapseFunction^2},
(Module[{index = #1}, {1, index + 1} -> Total[(metricMatrixRepresentation[[index,#1]]*shiftVector[[#1]] & ) /@
Range[Length[metricMatrixRepresentation]]]] & ) /@ Range[Length[metricMatrixRepresentation]],
(Module[{index = #1}, {index + 1, 1} -> Total[(metricMatrixRepresentation[[#1,index]]*shiftVector[[#1]] & ) /@
Range[Length[metricMatrixRepresentation]]]] & ) /@ Range[Length[metricMatrixRepresentation]],
({First[#1] + 1, Last[#1] + 1} -> metricMatrixRepresentation[[First[#1],Last[#1]]] & ) /@
Tuples[Range[Length[metricMatrixRepresentation]], 2]]]], Join[{timeCoordinate}, coordinates], True, True]] /;
SymbolName[admDecomposition] === "ADMDecomposition" && SymbolName[metricTensor] === "MetricTensor" &&
Length[Dimensions[metricMatrixRepresentation]] == 2 && Length[coordinates] == Length[metricMatrixRepresentation] &&
BooleanQ[index1] && BooleanQ[index2] && Length[shiftVector] == Length[metricMatrixRepresentation] &&
Length[Dimensions[matrixRepresentation]] == 2 && Length[coordinates] + 1 == Length[matrixRepresentation]
ADMStressEnergyDecomposition[(admDecomposition_)[(metricTensor_)[metricMatrixRepresentation_List, coordinates_List,
index1_, index2_], timeCoordinate_, lapseFunction_, shiftVector_List], matrixRepresentation_List][
"StressEnergyTensor"] :=
Module[{shiftCovector},
shiftCovector = Normal[SparseArray[(Module[{index = #1}, index -> Total[(metricMatrixRepresentation[[index,#1]]*
shiftVector[[#1]] & ) /@ Range[Length[metricMatrixRepresentation]]]] & ) /@
Range[Length[metricMatrixRepresentation]]]]; ResourceFunction["StressEnergyTensor"][matrixRepresentation,
ResourceFunction["MetricTensor"][Normal[SparseArray[
Join[{{1, 1} -> Total[(shiftVector[[#1]]*shiftCovector[[#1]] & ) /@ Range[Length[metricMatrixRepresentation]]] -
lapseFunction^2}, (Module[{index = #1}, {1, index + 1} -> Total[(metricMatrixRepresentation[[index,#1]]*
shiftVector[[#1]] & )[Range[Length[metricMatrixRepresentation]]]]] & ) /@
Range[Length[metricMatrixRepresentation]], (Module[{index = #1}, {index + 1, 1} ->
Total[(metricMatrixRepresentation[[#1,index]]*shiftVector[[#1]] & ) /@
Range[Length[metricMatrixRepresentation]]]] & ) /@ Range[Length[metricMatrixRepresentation]],
({First[#1] + 1, Last[#1] + 1} -> metricMatrixRepresentation[[First[#1],Last[#1]]] & ) /@
Tuples[Range[Length[metricMatrixRepresentation]], 2]]]], Join[{timeCoordinate}, coordinates], True, True],
False, False]] /; SymbolName[admDecomposition] === "ADMDecomposition" &&
SymbolName[metricTensor] === "MetricTensor" && Length[Dimensions[metricMatrixRepresentation]] == 2 &&
Length[coordinates] == Length[metricMatrixRepresentation] && BooleanQ[index1] && BooleanQ[index2] &&
Length[shiftVector] == Length[metricMatrixRepresentation] && Length[Dimensions[matrixRepresentation]] == 2 &&
Length[coordinates] + 1 == Length[matrixRepresentation]
ADMStressEnergyDecomposition[(admDecomposition_)[(metricTensor_)[metricMatrixRepresentation_List, coordinates_List,
index1_, index2_], timeCoordinate_, lapseFunction_, shiftVector_List], matrixRepresentation_List][
"EnergyDensity"] := Module[{newMatrixRepresentation, newMetricMatrixRepresentation, newCoordinates, newTimeCoordinate,
newLapseFunction, newShiftVector, shiftCovector, spacetimeMetricTensor, covariantStressEnergyTensor, normalVector},
newMatrixRepresentation = matrixRepresentation /. (#1 -> ToExpression[#1] & ) /@
Select[Join[coordinates, {timeCoordinate}], StringQ]; newMetricMatrixRepresentation =
metricMatrixRepresentation /. (#1 -> ToExpression[#1] & ) /@ Select[Join[coordinates, {timeCoordinate}], StringQ];
newCoordinates = coordinates /. (#1 -> ToExpression[#1] & ) /@ Select[Join[coordinates, {timeCoordinate}], StringQ];
newTimeCoordinate = timeCoordinate /. (#1 -> ToExpression[#1] & ) /@ Select[Join[coordinates, {timeCoordinate}],
StringQ]; newLapseFunction = lapseFunction /. (#1 -> ToExpression[#1] & ) /@
Select[Join[coordinates, {timeCoordinate}], StringQ]; newShiftVector =
shiftVector /. (#1 -> ToExpression[#1] & ) /@ Select[Join[coordinates, {timeCoordinate}], StringQ];
shiftCovector = Normal[SparseArray[(Module[{index = #1}, index -> Total[(newMetricMatrixRepresentation[[index,#1]]*
newShiftVector[[#1]] & ) /@ Range[Length[newMetricMatrixRepresentation]]]] & ) /@
Range[Length[newMetricMatrixRepresentation]]]]; spacetimeMetricTensor =
Normal[SparseArray[Join[{{1, 1} -> Total[(newShiftVector[[#1]]*shiftCovector[[#1]] & ) /@
Range[Length[newMetricMatrixRepresentation]]] - newLapseFunction^2},
(Module[{index = #1}, {1, index + 1} -> Total[(newMetricMatrixRepresentation[[index,#1]]*newShiftVector[[
#1]] & ) /@ Range[Length[newMetricMatrixRepresentation]]]] & ) /@
Range[Length[newMetricMatrixRepresentation]],
(Module[{index = #1}, {index + 1, 1} -> Total[(newMetricMatrixRepresentation[[#1,index]]*newShiftVector[[
#1]] & ) /@ Range[Length[newMetricMatrixRepresentation]]]] & ) /@
Range[Length[newMetricMatrixRepresentation]], ({First[#1] + 1, Last[#1] + 1} -> newMetricMatrixRepresentation[[
First[#1],Last[#1]]] & ) /@ Tuples[Range[Length[newMetricMatrixRepresentation]], 2]]]];
covariantStressEnergyTensor = Normal[SparseArray[
(Module[{index = #1}, index -> Total[(spacetimeMetricTensor[[First[index],First[#1]]]*spacetimeMetricTensor[[
Last[#1],Last[index]]]*newMatrixRepresentation[[First[#1],Last[#1]]] & ) /@
Tuples[Range[Length[spacetimeMetricTensor]], 2]]] & ) /@ Tuples[Range[Length[spacetimeMetricTensor]], 2]]];
normalVector = Normal[SparseArray[
(Module[{index = #1}, index -> -Total[(newLapseFunction*Inverse[spacetimeMetricTensor][[index,#1]]*
D[newTimeCoordinate, Join[{newTimeCoordinate}, newCoordinates][[#1]]] & ) /@ Range[
Length[spacetimeMetricTensor]]]] & ) /@ Range[Length[spacetimeMetricTensor]]]];
Total[(covariantStressEnergyTensor[[First[#1],Last[#1]]]*normalVector[[First[#1]]]*normalVector[[Last[#1]]] & ) /@
Tuples[Range[Length[spacetimeMetricTensor]], 2]] /. (ToExpression[#1] -> #1 & ) /@
Select[Join[coordinates, {timeCoordinate}], StringQ]] /; SymbolName[admDecomposition] === "ADMDecomposition" &&
SymbolName[metricTensor] === "MetricTensor" && Length[Dimensions[metricMatrixRepresentation]] == 2 &&
Length[coordinates] == Length[metricMatrixRepresentation] && BooleanQ[index1] && BooleanQ[index2] &&
Length[shiftVector] == Length[metricMatrixRepresentation] && Length[Dimensions[matrixRepresentation]] == 2 &&
Length[coordinates] + 1 == Length[matrixRepresentation]
ADMStressEnergyDecomposition[(admDecomposition_)[(metricTensor_)[metricMatrixRepresentation_List, coordinates_List,
index1_, index2_], timeCoordinate_, lapseFunction_, shiftVector_List], matrixRepresentation_List][
"ReducedEnergyDensity"] := Module[{newMatrixRepresentation, newMetricMatrixRepresentation, newCoordinates,
newTimeCoordinate, newLapseFunction, newShiftVector, shiftCovector, spacetimeMetricTensor,
covariantStressEnergyTensor, normalVector},
newMatrixRepresentation = matrixRepresentation /. (#1 -> ToExpression[#1] & ) /@
Select[Join[coordinates, {timeCoordinate}], StringQ]; newMetricMatrixRepresentation =
metricMatrixRepresentation /. (#1 -> ToExpression[#1] & ) /@ Select[Join[coordinates, {timeCoordinate}], StringQ];
newCoordinates = coordinates /. (#1 -> ToExpression[#1] & ) /@ Select[Join[coordinates, {timeCoordinate}], StringQ];
newTimeCoordinate = timeCoordinate /. (#1 -> ToExpression[#1] & ) /@ Select[Join[coordinates, {timeCoordinate}],
StringQ]; newLapseFunction = lapseFunction /. (#1 -> ToExpression[#1] & ) /@
Select[Join[coordinates, {timeCoordinate}], StringQ]; newShiftVector =
shiftVector /. (#1 -> ToExpression[#1] & ) /@ Select[Join[coordinates, {timeCoordinate}], StringQ];
shiftCovector = Normal[SparseArray[(Module[{index = #1}, index -> Total[(newMetricMatrixRepresentation[[index,#1]]*
newShiftVector[[#1]] & ) /@ Range[Length[newMetricMatrixRepresentation]]]] & ) /@
Range[Length[newMetricMatrixRepresentation]]]]; spacetimeMetricTensor =
Normal[SparseArray[Join[{{1, 1} -> Total[(newShiftVector[[#1]]*shiftCovector[[#1]] & ) /@
Range[Length[newMetricMatrixRepresentation]]] - newLapseFunction^2},
(Module[{index = #1}, {1, index + 1} -> Total[(newMetricMatrixRepresentation[[index,#1]]*newShiftVector[[
#1]] & ) /@ Range[Length[newMetricMatrixRepresentation]]]] & ) /@
Range[Length[newMetricMatrixRepresentation]],
(Module[{index = #1}, {index + 1, 1} -> Total[(newMetricMatrixRepresentation[[#1,index]]*newShiftVector[[
#1]] & ) /@ Range[Length[newMetricMatrixRepresentation]]]] & ) /@
Range[Length[newMetricMatrixRepresentation]], ({First[#1] + 1, Last[#1] + 1} -> newMetricMatrixRepresentation[[
First[#1],Last[#1]]] & ) /@ Tuples[Range[Length[newMetricMatrixRepresentation]], 2]]]];
covariantStressEnergyTensor = Normal[SparseArray[
(Module[{index = #1}, index -> Total[(spacetimeMetricTensor[[First[index],First[#1]]]*spacetimeMetricTensor[[
Last[#1],Last[index]]]*newMatrixRepresentation[[First[#1],Last[#1]]] & ) /@
Tuples[Range[Length[spacetimeMetricTensor]], 2]]] & ) /@ Tuples[Range[Length[spacetimeMetricTensor]], 2]]];
normalVector = Normal[SparseArray[
(Module[{index = #1}, index -> -Total[(newLapseFunction*Inverse[spacetimeMetricTensor][[index,#1]]*
D[newTimeCoordinate, Join[{newTimeCoordinate}, newCoordinates][[#1]]] & ) /@ Range[
Length[spacetimeMetricTensor]]]] & ) /@ Range[Length[spacetimeMetricTensor]]]];
FullSimplify[Total[(covariantStressEnergyTensor[[First[#1],Last[#1]]]*normalVector[[First[#1]]]*
normalVector[[Last[#1]]] & ) /@ Tuples[Range[Length[spacetimeMetricTensor]], 2]] /.
(ToExpression[#1] -> #1 & ) /@ Select[Join[coordinates, {timeCoordinate}], StringQ]]] /;
SymbolName[admDecomposition] === "ADMDecomposition" && SymbolName[metricTensor] === "MetricTensor" &&
Length[Dimensions[metricMatrixRepresentation]] == 2 && Length[coordinates] == Length[metricMatrixRepresentation] &&
BooleanQ[index1] && BooleanQ[index2] && Length[shiftVector] == Length[metricMatrixRepresentation] &&
Length[Dimensions[matrixRepresentation]] == 2 && Length[coordinates] + 1 == Length[matrixRepresentation]
ADMStressEnergyDecomposition[(admDecomposition_)[(metricTensor_)[metricMatrixRepresentation_List, coordinates_List,
index1_, index2_], timeCoordinate_, lapseFunction_, shiftVector_List], matrixRepresentation_List][
"SymbolicEnergyDensity"] := Module[{newMatrixRepresentation, newMetricMatrixRepresentation, newCoordinates,
newTimeCoordinate, newLapseFunction, newShiftVector, shiftCovector, spacetimeMetricTensor,
covariantStressEnergyTensor, normalVector},
newMatrixRepresentation = matrixRepresentation /. (#1 -> ToExpression[#1] & ) /@
Select[Join[coordinates, {timeCoordinate}], StringQ]; newMetricMatrixRepresentation =
metricMatrixRepresentation /. (#1 -> ToExpression[#1] & ) /@ Select[Join[coordinates, {timeCoordinate}], StringQ];
newCoordinates = coordinates /. (#1 -> ToExpression[#1] & ) /@ Select[Join[coordinates, {timeCoordinate}], StringQ];
newTimeCoordinate = timeCoordinate /. (#1 -> ToExpression[#1] & ) /@ Select[Join[coordinates, {timeCoordinate}],
StringQ]; newLapseFunction = lapseFunction /. (#1 -> ToExpression[#1] & ) /@
Select[Join[coordinates, {timeCoordinate}], StringQ]; newShiftVector =
shiftVector /. (#1 -> ToExpression[#1] & ) /@ Select[Join[coordinates, {timeCoordinate}], StringQ];
shiftCovector = Normal[SparseArray[(Module[{index = #1}, index -> Total[(newMetricMatrixRepresentation[[index,#1]]*
newShiftVector[[#1]] & ) /@ Range[Length[newMetricMatrixRepresentation]]]] & ) /@
Range[Length[newMetricMatrixRepresentation]]]]; spacetimeMetricTensor =
Normal[SparseArray[Join[{{1, 1} -> Total[(newShiftVector[[#1]]*shiftCovector[[#1]] & ) /@
Range[Length[newMetricMatrixRepresentation]]] - newLapseFunction^2},
(Module[{index = #1}, {1, index + 1} -> Total[(newMetricMatrixRepresentation[[index,#1]]*newShiftVector[[
#1]] & ) /@ Range[Length[newMetricMatrixRepresentation]]]] & ) /@
Range[Length[newMetricMatrixRepresentation]],
(Module[{index = #1}, {index + 1, 1} -> Total[(newMetricMatrixRepresentation[[#1,index]]*newShiftVector[[
#1]] & ) /@ Range[Length[newMetricMatrixRepresentation]]]] & ) /@
Range[Length[newMetricMatrixRepresentation]], ({First[#1] + 1, Last[#1] + 1} -> newMetricMatrixRepresentation[[
First[#1],Last[#1]]] & ) /@ Tuples[Range[Length[newMetricMatrixRepresentation]], 2]]]];
covariantStressEnergyTensor = Normal[SparseArray[
(Module[{index = #1}, index -> Total[(spacetimeMetricTensor[[First[index],First[#1]]]*spacetimeMetricTensor[[
Last[#1],Last[index]]]*newMatrixRepresentation[[First[#1],Last[#1]]] & ) /@
Tuples[Range[Length[spacetimeMetricTensor]], 2]]] & ) /@ Tuples[Range[Length[spacetimeMetricTensor]], 2]]];
normalVector = Normal[SparseArray[
(Module[{index = #1}, index -> -Total[(newLapseFunction*Inverse[spacetimeMetricTensor][[index,#1]]*
Inactive[D][newTimeCoordinate, Join[{newTimeCoordinate}, newCoordinates][[#1]]] & ) /@ Range[
Length[spacetimeMetricTensor]]]] & ) /@ Range[Length[spacetimeMetricTensor]]]];
Total[(covariantStressEnergyTensor[[First[#1],Last[#1]]]*normalVector[[First[#1]]]*normalVector[[Last[#1]]] & ) /@
Tuples[Range[Length[spacetimeMetricTensor]], 2]] /. (ToExpression[#1] -> #1 & ) /@
Select[Join[coordinates, {timeCoordinate}], StringQ]] /; SymbolName[admDecomposition] === "ADMDecomposition" &&
SymbolName[metricTensor] === "MetricTensor" && Length[Dimensions[metricMatrixRepresentation]] == 2 &&
Length[coordinates] == Length[metricMatrixRepresentation] && BooleanQ[index1] && BooleanQ[index2] &&
Length[shiftVector] == Length[metricMatrixRepresentation] && Length[Dimensions[matrixRepresentation]] == 2 &&
Length[coordinates] + 1 == Length[matrixRepresentation]
ADMStressEnergyDecomposition[(admDecomposition_)[(metricTensor_)[metricMatrixRepresentation_List, coordinates_List,
index1_, index2_], timeCoordinate_, lapseFunction_, shiftVector_List], matrixRepresentation_List][
"MomentumDensity"] := Module[{newMatrixRepresentation, newMetricMatrixRepresentation, newCoordinates,
newTimeCoordinate, newLapseFunction, newShiftVector, shiftCovector, spacetimeMetricTensor,
covariantStressEnergyTensor, normalVector, projectionOperator, momentumCovector},
newMatrixRepresentation = matrixRepresentation /. (#1 -> ToExpression[#1] & ) /@
Select[Join[coordinates, {timeCoordinate}], StringQ]; newMetricMatrixRepresentation =
metricMatrixRepresentation /. (#1 -> ToExpression[#1] & ) /@ Select[Join[coordinates, {timeCoordinate}], StringQ];
newCoordinates = coordinates /. (#1 -> ToExpression[#1] & ) /@ Select[Join[coordinates, {timeCoordinate}], StringQ];
newTimeCoordinate = timeCoordinate /. (#1 -> ToExpression[#1] & ) /@ Select[Join[coordinates, {timeCoordinate}],
StringQ]; newLapseFunction = lapseFunction /. (#1 -> ToExpression[#1] & ) /@
Select[Join[coordinates, {timeCoordinate}], StringQ]; newShiftVector =
shiftVector /. (#1 -> ToExpression[#1] & ) /@ Select[Join[coordinates, {timeCoordinate}], StringQ];
shiftCovector = Normal[SparseArray[(Module[{index = #1}, index -> Total[(newMetricMatrixRepresentation[[index,#1]]*
newShiftVector[[#1]] & ) /@ Range[Length[newMetricMatrixRepresentation]]]] & ) /@
Range[Length[newMetricMatrixRepresentation]]]]; spacetimeMetricTensor =
Normal[SparseArray[Join[{{1, 1} -> Total[(newShiftVector[[#1]]*shiftCovector[[#1]] & ) /@
Range[Length[newMetricMatrixRepresentation]]] - newLapseFunction^2},
(Module[{index = #1}, {1, index + 1} -> Total[(newMetricMatrixRepresentation[[index,#1]]*newShiftVector[[
#1]] & ) /@ Range[Length[newMetricMatrixRepresentation]]]] & ) /@
Range[Length[newMetricMatrixRepresentation]],
(Module[{index = #1}, {index + 1, 1} -> Total[(newMetricMatrixRepresentation[[#1,index]]*newShiftVector[[
#1]] & ) /@ Range[Length[newMetricMatrixRepresentation]]]] & ) /@
Range[Length[newMetricMatrixRepresentation]], ({First[#1] + 1, Last[#1] + 1} -> newMetricMatrixRepresentation[[
First[#1],Last[#1]]] & ) /@ Tuples[Range[Length[newMetricMatrixRepresentation]], 2]]]];
covariantStressEnergyTensor = Normal[SparseArray[
(Module[{index = #1}, index -> Total[(spacetimeMetricTensor[[First[index],First[#1]]]*spacetimeMetricTensor[[
Last[#1],Last[index]]]*newMatrixRepresentation[[First[#1],Last[#1]]] & ) /@
Tuples[Range[Length[spacetimeMetricTensor]], 2]]] & ) /@ Tuples[Range[Length[spacetimeMetricTensor]], 2]]];
normalVector = Normal[SparseArray[