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SolveADMEquations.wl
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(* ::Package:: *)
SolveADMEquations[(admStressEnergyDecomposition_)[(admDecomposition_)[(metricTensor_)[metricMatrixRepresentation_List,
coordinates_List, index1_, index2_], timeCoordinate_, lapseFunction_, shiftVector_List],
matrixRepresentation_List]] :=
ADMSolution[ADMStressEnergyDecomposition[ResourceFunction["ADMDecomposition"][ResourceFunction["MetricTensor"][
metricMatrixRepresentation, coordinates, index1, index2], timeCoordinate, lapseFunction, shiftVector],
matrixRepresentation], "\[FormalCapitalLambda]"] /; SymbolName[admStressEnergyDecomposition] === "ADMStressEnergyDecomposition" &&
SymbolName[admDecomposition] === "ADMDecomposition" && SymbolName[metricTensor] === "MetricTensor" &&
Length[Dimensions[metricMatrixRepresentation]] == 2 && Length[coordinates] == Length[metricMatrixRepresentation] &&
BooleanQ[index1] && BooleanQ[index2] && Length[shiftVector] == Length[metricMatrixRepresentation] &&
Length[coordinates] + 1 == Length[matrixRepresentation]
SolveADMEquations[(admStressEnergyDecomposition_)[(admDecomposition_)[(metricTensor_)[metricMatrixRepresentation_List,
coordinates_List, index1_, index2_], timeCoordinate_, lapseFunction_, shiftVector_List],
matrixRepresentation_List], cosmologicalConstant_] :=
ADMSolution[ADMStressEnergyDecomposition[ResourceFunction["ADMDecomposition"][ResourceFunction["MetricTensor"][
metricMatrixRepresentation, coordinates, index1, index2], timeCoordinate, lapseFunction, shiftVector],
matrixRepresentation], cosmologicalConstant] /; SymbolName[admStressEnergyDecomposition] ===
"ADMStressEnergyDecomposition" && SymbolName[admDecomposition] === "ADMDecomposition" &&
SymbolName[metricTensor] === "MetricTensor" && Length[Dimensions[metricMatrixRepresentation]] == 2 &&
Length[coordinates] == Length[metricMatrixRepresentation] && BooleanQ[index1] && BooleanQ[index2] &&
Length[shiftVector] == Length[metricMatrixRepresentation] && Length[coordinates] + 1 == Length[matrixRepresentation]
ADMSolution[(admStressEnergyDecomposition_)[(admDecomposition_)[(metricTensor_)[metricMatrixRepresentation_List,
coordinates_List, index1_, index2_], timeCoordinate_, lapseFunction_, shiftVector_List],
matrixRepresentation_List], cosmologicalConstant_]["SolutionQ"] :=
Module[{newMetricMatrixRepresentation, newMatrixRepresentation, newCoordinates, newTimeCoordinate, newLapseFunction,
newShiftVector, shiftCovector, spatialChristoffelSymbols, extrinsicCurvatureTensor, mixedExtrinsicCurvatureTensor,
extrinsicCurvatureTrace, spatialRiemannTensor, spatialRicciTensor, mixedSpatialRicciTensor, spacetimeMetricTensor,
Null, covariantStressEnergyTensor, stressEnergyTrace, leftHandSide, rightHandSide, evolutionEquations},
newMetricMatrixRepresentation = metricMatrixRepresentation /. (#1 -> ToExpression[#1] & ) /@
Select[Join[coordinates, {timeCoordinate}], StringQ]; newMatrixRepresentation =
matrixRepresentation /. (#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]]]]; spatialChristoffelSymbols =
Normal[SparseArray[(Module[{index = #1}, index -> Total[((1/2)*Inverse[newMetricMatrixRepresentation][[index[[1]],
#1]]*(D[newMetricMatrixRepresentation[[#1,index[[3]]]], newCoordinates[[index[[2]]]]] +
D[newMetricMatrixRepresentation[[index[[2]],#1]], newCoordinates[[index[[3]]]]] -
D[newMetricMatrixRepresentation[[index[[2]],index[[3]]]], newCoordinates[[#1]]]) & ) /@
Range[Length[newMetricMatrixRepresentation]]]] & ) /@ Tuples[Range[Length[newMetricMatrixRepresentation]],
3]]]; extrinsicCurvatureTensor =
Normal[SparseArray[(Module[{index = #1}, index -> (1/(2*newLapseFunction))*(D[shiftCovector[[First[index]]],
newCoordinates[[Last[index]]]] - Total[(spatialChristoffelSymbols[[#1,Last[index],First[index]]]*
shiftCovector[[#1]] & ) /@ Range[Length[newMetricMatrixRepresentation]]] + D[shiftCovector[[
Last[index]]], newCoordinates[[First[index]]]] - Total[(spatialChristoffelSymbols[[#1,First[index],
Last[index]]]*shiftCovector[[#1]] & ) /@ Range[Length[newMetricMatrixRepresentation]]] -
D[newMetricMatrixRepresentation[[First[index],Last[index]]], newTimeCoordinate])] & ) /@
Tuples[Range[Length[newMetricMatrixRepresentation]], 2]]]; mixedExtrinsicCurvatureTensor =
Normal[SparseArray[(Module[{index = #1}, index -> Total[(Inverse[newMetricMatrixRepresentation][[First[index],#1]]*
extrinsicCurvatureTensor[[#1,Last[index]]] & ) /@ Range[Length[newMetricMatrixRepresentation]]]] & ) /@
Tuples[Range[Length[newMetricMatrixRepresentation]], 2]]]; extrinsicCurvatureTrace =
Total[(Inverse[newMetricMatrixRepresentation][[First[#1],Last[#1]]]*extrinsicCurvatureTensor[[First[#1],
Last[#1]]] & ) /@ Tuples[Range[Length[newMetricMatrixRepresentation]], 2]];
spatialRiemannTensor = Normal[SparseArray[
(Module[{index = #1}, index -> D[spatialChristoffelSymbols[[index[[1]],index[[2]],index[[4]]]],
newCoordinates[[index[[3]]]]] - D[spatialChristoffelSymbols[[index[[1]],index[[2]],index[[3]]]],
newCoordinates[[index[[4]]]]] + Total[(spatialChristoffelSymbols[[index[[1]],#1,index[[3]]]]*
spatialChristoffelSymbols[[#1,index[[2]],index[[4]]]] & ) /@ Range[Length[
newMetricMatrixRepresentation]]] - Total[(spatialChristoffelSymbols[[index[[1]],#1,index[[4]]]]*
spatialChristoffelSymbols[[#1,index[[2]],index[[3]]]] & ) /@ Range[Length[
newMetricMatrixRepresentation]]]] & ) /@ Tuples[Range[Length[newMetricMatrixRepresentation]], 4]]];
spatialRicciTensor = Normal[SparseArray[
(Module[{index = #1}, index -> Total[(spatialRiemannTensor[[#1,First[index],#1,Last[index]]] & ) /@
Range[Length[newMetricMatrixRepresentation]]]] & ) /@ Tuples[Range[Length[newMetricMatrixRepresentation]],
2]]]; mixedSpatialRicciTensor =
Normal[SparseArray[(Module[{index = #1}, index -> Total[(Inverse[newMetricMatrixRepresentation][[First[index],#1]]*
spatialRicciTensor[[#1,Last[index]]] & ) /@ Range[Length[newMetricMatrixRepresentation]]]] & ) /@
Tuples[Range[Length[newMetricMatrixRepresentation]], 2]]]; 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]]];
stressEnergyTrace = Total[(spacetimeMetricTensor[[First[#1],Last[#1]]]*newMatrixRepresentation[[First[#1],
Last[#1]]] & ) /@ Tuples[Range[Length[spacetimeMetricTensor]], 2]];
leftHandSide = Normal[SparseArray[(Module[{index = #1}, index -> D[mixedExtrinsicCurvatureTensor[[First[index],
Last[index]]], newTimeCoordinate]] & ) /@ Tuples[Range[Length[newMetricMatrixRepresentation]], 2]]];
rightHandSide = Normal[SparseArray[(Module[{index = #1}, index -> newLapseFunction*mixedSpatialRicciTensor[[First[
index],Last[index]]] - Total[(Module[{nestedIndex = #1}, (D[D[newLapseFunction, newCoordinates[[
Last[index]]]], newCoordinates[[nestedIndex]]] - Total[(spatialChristoffelSymbols[[#1,nestedIndex,
Last[index]]]*D[newLapseFunction, newCoordinates[[#1]]] & ) /@ Range[Length[
newMetricMatrixRepresentation]]])*Inverse[newMetricMatrixRepresentation][[nestedIndex,
First[index]]]] & ) /@ Range[Length[newMetricMatrixRepresentation]]] + newLapseFunction*
extrinsicCurvatureTrace*mixedExtrinsicCurvatureTensor[[First[index],Last[index]]] +
Total[(Module[{nestedIndex = #1}, newShiftVector[[nestedIndex]]*(D[mixedExtrinsicCurvatureTensor[[
First[index],Last[index]]], newCoordinates[[nestedIndex]]] + Total[(spatialChristoffelSymbols[[
First[index],nestedIndex,#1]]*mixedExtrinsicCurvatureTensor[[#1,Last[index]]] & ) /@
Range[Length[newMetricMatrixRepresentation]]] - Total[(spatialChristoffelSymbols[[#1,nestedIndex,
Last[index]]]*mixedExtrinsicCurvatureTensor[[First[index],#1]] & ) /@ Range[Length[
newMetricMatrixRepresentation]]])] & ) /@ Range[Length[newMetricMatrixRepresentation]]] +
Total[(Module[{nestedIndex = #1}, mixedExtrinsicCurvatureTensor[[First[index],nestedIndex]]*
(D[newShiftVector[[nestedIndex]], newCoordinates[[Last[index]]]] + Total[
(spatialChristoffelSymbols[[nestedIndex,Last[index],#1]]*newShiftVector[[#1]] & ) /@
Range[Length[newMetricMatrixRepresentation]]])] & ) /@ Range[Length[
newMetricMatrixRepresentation]]] - Total[(Module[{nestedIndex = #1}, mixedExtrinsicCurvatureTensor[[
nestedIndex,Last[index]]]*(D[newShiftVector[[First[index]]], newCoordinates[[nestedIndex]]] +
Total[(spatialChristoffelSymbols[[First[index],nestedIndex,#1]]*newShiftVector[[#1]] & ) /@
Range[Length[newMetricMatrixRepresentation]]])] & ) /@ Range[Length[
newMetricMatrixRepresentation]]] - newLapseFunction*(Total[(Module[{nestedIndex = #1},
8*Pi*covariantStressEnergyTensor[[nestedIndex + 1,Last[index] + 1]]*
Inverse[newMetricMatrixRepresentation][[nestedIndex,First[index]]]] & ) /@
Range[Length[newMetricMatrixRepresentation]]] + (4*Pi*stressEnergyTrace*KroneckerDelta[First[index],
Last[index]])/(1 - Length[spacetimeMetricTensor]/2)) - newLapseFunction*
Total[(((2*cosmologicalConstant)/(Length[newMetricMatrixRepresentation] - 1))*spacetimeMetricTensor[[#1 + 1,
Last[index] + 1]]*Inverse[newMetricMatrixRepresentation][[#1,First[index]]] & ) /@
Range[Length[newMetricMatrixRepresentation]]]] & ) /@
Tuples[Range[Length[newMetricMatrixRepresentation]], 2]]]; evolutionEquations =
FullSimplify[Thread[Catenate[leftHandSide] == Catenate[rightHandSide]] /. (ToExpression[#1] -> #1 & ) /@
Select[Join[coordinates, {timeCoordinate}], StringQ]]; If[evolutionEquations === True, True,
If[evolutionEquations === False, False, If[Length[Select[evolutionEquations, #1 === False & ]] > 0, False,
True]]]] /; SymbolName[admStressEnergyDecomposition] === "ADMStressEnergyDecomposition" &&
SymbolName[admDecomposition] === "ADMDecomposition" && SymbolName[metricTensor] === "MetricTensor" &&
Length[Dimensions[metricMatrixRepresentation]] == 2 && Length[coordinates] == Length[metricMatrixRepresentation] &&
BooleanQ[index1] && BooleanQ[index2] && Length[shiftVector] == Length[metricMatrixRepresentation] &&
Length[coordinates] + 1 == Length[matrixRepresentation]
ADMSolution[(admStressEnergyDecomposition_)[(admDecomposition_)[(metricTensor_)[metricMatrixRepresentation_List,
coordinates_List, index1_, index2_], timeCoordinate_, lapseFunction_, shiftVector_List],
matrixRepresentation_List], cosmologicalConstant_]["ExactSolutionQ"] :=
Module[{newMetricMatrixRepresentation, newMatrixRepresentation, newCoordinates, newTimeCoordinate, newLapseFunction,
newShiftVector, shiftCovector, spatialChristoffelSymbols, extrinsicCurvatureTensor, mixedExtrinsicCurvatureTensor,
extrinsicCurvatureTrace, spatialRiemannTensor, spatialRicciTensor, mixedSpatialRicciTensor, spacetimeMetricTensor,
covariantStressEnergyTensor, stressEnergyTrace, leftHandSide, rightHandSide, evolutionEquations},
newMetricMatrixRepresentation = metricMatrixRepresentation /. (#1 -> ToExpression[#1] & ) /@
Select[Join[coordinates, {timeCoordinate}], StringQ]; newMatrixRepresentation =
matrixRepresentation /. (#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]]]]; spatialChristoffelSymbols =
Normal[SparseArray[(Module[{index = #1}, index -> Total[((1/2)*Inverse[newMetricMatrixRepresentation][[index[[1]],
#1]]*(D[newMetricMatrixRepresentation[[#1,index[[3]]]], newCoordinates[[index[[2]]]]] +
D[newMetricMatrixRepresentation[[index[[2]],#1]], newCoordinates[[index[[3]]]]] -
D[newMetricMatrixRepresentation[[index[[2]],index[[3]]]], newCoordinates[[#1]]]) & ) /@
Range[Length[newMetricMatrixRepresentation]]]] & ) /@ Tuples[Range[Length[newMetricMatrixRepresentation]],
3]]]; extrinsicCurvatureTensor =
Normal[SparseArray[(Module[{index = #1}, index -> (1/(2*newLapseFunction))*(D[shiftCovector[[First[index]]],
newCoordinates[[Last[index]]]] - Total[(spatialChristoffelSymbols[[#1,Last[index],First[index]]]*
shiftCovector[[#1]] & ) /@ Range[Length[newMetricMatrixRepresentation]]] + D[shiftCovector[[
Last[index]]], newCoordinates[[First[index]]]] - Total[(spatialChristoffelSymbols[[#1,First[index],
Last[index]]]*shiftCovector[[#1]] & ) /@ Range[Length[newMetricMatrixRepresentation]]] -
D[newMetricMatrixRepresentation[[First[index],Last[index]]], newTimeCoordinate])] & ) /@
Tuples[Range[Length[newMetricMatrixRepresentation]], 2]]]; mixedExtrinsicCurvatureTensor =
Normal[SparseArray[(Module[{index = #1}, index -> Total[(Inverse[newMetricMatrixRepresentation][[First[index],#1]]*
extrinsicCurvatureTensor[[#1,Last[index]]] & ) /@ Range[Length[newMetricMatrixRepresentation]]]] & ) /@
Tuples[Range[Length[newMetricMatrixRepresentation]], 2]]]; extrinsicCurvatureTrace =
Total[(Inverse[newMetricMatrixRepresentation][[First[#1],Last[#1]]]*extrinsicCurvatureTensor[[First[#1],
Last[#1]]] & ) /@ Tuples[Range[Length[newMetricMatrixRepresentation]], 2]];
spatialRiemannTensor = Normal[SparseArray[
(Module[{index = #1}, index -> D[spatialChristoffelSymbols[[index[[1]],index[[2]],index[[4]]]],
newCoordinates[[index[[3]]]]] - D[spatialChristoffelSymbols[[index[[1]],index[[2]],index[[3]]]],
newCoordinates[[index[[4]]]]] + Total[(spatialChristoffelSymbols[[index[[1]],#1,index[[3]]]]*
spatialChristoffelSymbols[[#1,index[[2]],index[[4]]]] & ) /@ Range[Length[
newMetricMatrixRepresentation]]] - Total[(spatialChristoffelSymbols[[index[[1]],#1,index[[4]]]]*
spatialChristoffelSymbols[[#1,index[[2]],index[[3]]]] & ) /@ Range[Length[
newMetricMatrixRepresentation]]]] & ) /@ Tuples[Range[Length[newMetricMatrixRepresentation]], 4]]];
spatialRicciTensor = Normal[SparseArray[
(Module[{index = #1}, index -> Total[(spatialRiemannTensor[[#1,First[index],#1,Last[index]]] & ) /@
Range[Length[newMetricMatrixRepresentation]]]] & ) /@ Tuples[Range[Length[newMetricMatrixRepresentation]],
2]]]; mixedSpatialRicciTensor =
Normal[SparseArray[(Module[{index = #1}, index -> Total[(Inverse[newMetricMatrixRepresentation][[First[index],#1]]*
spatialRicciTensor[[#1,Last[index]]] & ) /@ Range[Length[newMetricMatrixRepresentation]]]] & ) /@
Tuples[Range[Length[newMetricMatrixRepresentation]], 2]]]; 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]]];
stressEnergyTrace = Total[(spacetimeMetricTensor[[First[#1],Last[#1]]]*newMatrixRepresentation[[First[#1],
Last[#1]]] & ) /@ Tuples[Range[Length[spacetimeMetricTensor]], 2]];
leftHandSide = Normal[SparseArray[(Module[{index = #1}, index -> D[mixedExtrinsicCurvatureTensor[[First[index],
Last[index]]], newTimeCoordinate]] & ) /@ Tuples[Range[Length[newMetricMatrixRepresentation]], 2]]];
rightHandSide = Normal[SparseArray[(Module[{index = #1}, index -> newLapseFunction*mixedSpatialRicciTensor[[First[
index],Last[index]]] - Total[(Module[{nestedIndex = #1}, (D[D[newLapseFunction, newCoordinates[[
Last[index]]]], newCoordinates[[nestedIndex]]] - Total[(spatialChristoffelSymbols[[#1,nestedIndex,
Last[index]]]*D[newLapseFunction, newCoordinates[[#1]]] & ) /@ Range[Length[
newMetricMatrixRepresentation]]])*Inverse[newMetricMatrixRepresentation][[nestedIndex,
First[index]]]] & ) /@ Range[Length[newMetricMatrixRepresentation]]] + newLapseFunction*
extrinsicCurvatureTrace*mixedExtrinsicCurvatureTensor[[First[index],Last[index]]] +
Total[(Module[{nestedIndex = #1}, newShiftVector[[nestedIndex]]*(D[mixedExtrinsicCurvatureTensor[[
First[index],Last[index]]], newCoordinates[[nestedIndex]]] + Total[(spatialChristoffelSymbols[[
First[index],nestedIndex,#1]]*mixedExtrinsicCurvatureTensor[[#1,Last[index]]] & ) /@
Range[Length[newMetricMatrixRepresentation]]] - Total[(spatialChristoffelSymbols[[#1,nestedIndex,
Last[index]]]*mixedExtrinsicCurvatureTensor[[First[index],#1]] & ) /@ Range[Length[
newMetricMatrixRepresentation]]])] & ) /@ Range[Length[newMetricMatrixRepresentation]]] +
Total[(Module[{nestedIndex = #1}, mixedExtrinsicCurvatureTensor[[First[index],nestedIndex]]*
(D[newShiftVector[[nestedIndex]], newCoordinates[[Last[index]]]] + Total[
(spatialChristoffelSymbols[[nestedIndex,Last[index],#1]]*newShiftVector[[#1]] & ) /@
Range[Length[newMetricMatrixRepresentation]]])] & ) /@ Range[Length[
newMetricMatrixRepresentation]]] - Total[(Module[{nestedIndex = #1}, mixedExtrinsicCurvatureTensor[[
nestedIndex,Last[index]]]*(D[newShiftVector[[First[index]]], newCoordinates[[nestedIndex]]] +
Total[(spatialChristoffelSymbols[[First[index],nestedIndex,#1]]*newShiftVector[[#1]] & ) /@
Range[Length[newMetricMatrixRepresentation]]])] & ) /@ Range[Length[
newMetricMatrixRepresentation]]] - newLapseFunction*(Total[(Module[{nestedIndex = #1},
8*Pi*covariantStressEnergyTensor[[nestedIndex + 1,Last[index] + 1]]*
Inverse[newMetricMatrixRepresentation][[nestedIndex,First[index]]]] & ) /@
Range[Length[newMetricMatrixRepresentation]]] + (4*Pi*stressEnergyTrace*KroneckerDelta[First[index],
Last[index]])/(1 - Length[spacetimeMetricTensor]/2)) - newLapseFunction*
Total[(((2*cosmologicalConstant)/(Length[newMetricMatrixRepresentation] - 1))*spacetimeMetricTensor[[#1 + 1,
Last[index] + 1]]*Inverse[newMetricMatrixRepresentation][[#1,First[index]]] & ) /@
Range[Length[newMetricMatrixRepresentation]]]] & ) /@
Tuples[Range[Length[newMetricMatrixRepresentation]], 2]]]; evolutionEquations =
FullSimplify[Thread[Catenate[leftHandSide] == Catenate[rightHandSide]] /. (ToExpression[#1] -> #1 & ) /@
Select[Join[coordinates, {timeCoordinate}], StringQ]]; If[evolutionEquations === True, True,
If[evolutionEquations === False, False, If[Length[Select[evolutionEquations, #1 === False & ]] > 0, False,
If[Length[DeleteDuplicates[Reverse /@ Sort /@ Select[evolutionEquations, #1 =!= True & ]]] == 0, True,
False]]]]] /; SymbolName[admStressEnergyDecomposition] === "ADMStressEnergyDecomposition" &&
SymbolName[admDecomposition] === "ADMDecomposition" && SymbolName[metricTensor] === "MetricTensor" &&
Length[Dimensions[metricMatrixRepresentation]] == 2 && Length[coordinates] == Length[metricMatrixRepresentation] &&
BooleanQ[index1] && BooleanQ[index2] && Length[shiftVector] == Length[metricMatrixRepresentation] &&
Length[coordinates] + 1 == Length[matrixRepresentation]
ADMSolution[(admStressEnergyDecomposition_)[(admDecomposition_)[(metricTensor_)[metricMatrixRepresentation_List,
coordinates_List, index1_, index2_], timeCoordinate_, lapseFunction_, shiftVector_List],
matrixRepresentation_List], cosmologicalConstant_]["FieldEquations"] :=
Module[{newMetricMatrixRepresentation, newMatrixRepresentation, newCoordinates, newTimeCoordinate, newLapseFunction,
newShiftVector, shiftCovector, spatialChristoffelSymbols, extrinsicCurvatureTensor, mixedExtrinsicCurvatureTensor,
extrinsicCurvatureTrace, spatialRiemannTensor, spatialRicciTensor, mixedSpatialRicciTensor, spacetimeMetricTensor,
covariantStressEnergyTensor, stressEnergyTrace, leftHandSide, rightHandSide, evolutionEquations},
newMetricMatrixRepresentation = metricMatrixRepresentation /. (#1 -> ToExpression[#1] & ) /@
Select[Join[coordinates, {timeCoordinate}], StringQ]; newMatrixRepresentation =
matrixRepresentation /. (#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]]]]; spatialChristoffelSymbols =
Normal[SparseArray[(Module[{index = #1}, index -> Total[((1/2)*Inverse[newMetricMatrixRepresentation][[index[[1]],
#1]]*(D[newMetricMatrixRepresentation[[#1,index[[3]]]], newCoordinates[[index[[2]]]]] +
D[newMetricMatrixRepresentation[[index[[2]],#1]], newCoordinates[[index[[3]]]]] -
D[newMetricMatrixRepresentation[[index[[2]],index[[3]]]], newCoordinates[[#1]]]) & ) /@
Range[Length[newMetricMatrixRepresentation]]]] & ) /@ Tuples[Range[Length[newMetricMatrixRepresentation]],
3]]]; extrinsicCurvatureTensor =
Normal[SparseArray[(Module[{index = #1}, index -> (1/(2*newLapseFunction))*(D[shiftCovector[[First[index]]],
newCoordinates[[Last[index]]]] - Total[(spatialChristoffelSymbols[[#1,Last[index],First[index]]]*
shiftCovector[[#1]] & ) /@ Range[Length[newMetricMatrixRepresentation]]] + D[shiftCovector[[
Last[index]]], newCoordinates[[First[index]]]] - Total[(spatialChristoffelSymbols[[#1,First[index],
Last[index]]]*shiftCovector[[#1]] & ) /@ Range[Length[newMetricMatrixRepresentation]]] -
D[newMetricMatrixRepresentation[[First[index],Last[index]]], newTimeCoordinate])] & ) /@
Tuples[Range[Length[newMetricMatrixRepresentation]], 2]]]; mixedExtrinsicCurvatureTensor =
Normal[SparseArray[(Module[{index = #1}, index -> Total[(Inverse[newMetricMatrixRepresentation][[First[index],#1]]*
extrinsicCurvatureTensor[[#1,Last[index]]] & ) /@ Range[Length[newMetricMatrixRepresentation]]]] & ) /@
Tuples[Range[Length[newMetricMatrixRepresentation]], 2]]]; extrinsicCurvatureTrace =
Total[(Inverse[newMetricMatrixRepresentation][[First[#1],Last[#1]]]*extrinsicCurvatureTensor[[First[#1],
Last[#1]]] & ) /@ Tuples[Range[Length[newMetricMatrixRepresentation]], 2]];
spatialRiemannTensor = Normal[SparseArray[
(Module[{index = #1}, index -> D[spatialChristoffelSymbols[[index[[1]],index[[2]],index[[4]]]],
newCoordinates[[index[[3]]]]] - D[spatialChristoffelSymbols[[index[[1]],index[[2]],index[[3]]]],
newCoordinates[[index[[4]]]]] + Total[(spatialChristoffelSymbols[[index[[1]],#1,index[[3]]]]*
spatialChristoffelSymbols[[#1,index[[2]],index[[4]]]] & ) /@ Range[Length[
newMetricMatrixRepresentation]]] - Total[(spatialChristoffelSymbols[[index[[1]],#1,index[[4]]]]*
spatialChristoffelSymbols[[#1,index[[2]],index[[3]]]] & ) /@ Range[Length[
newMetricMatrixRepresentation]]]] & ) /@ Tuples[Range[Length[newMetricMatrixRepresentation]], 4]]];
spatialRicciTensor = Normal[SparseArray[
(Module[{index = #1}, index -> Total[(spatialRiemannTensor[[#1,First[index],#1,Last[index]]] & ) /@
Range[Length[newMetricMatrixRepresentation]]]] & ) /@ Tuples[Range[Length[newMetricMatrixRepresentation]],
2]]]; mixedSpatialRicciTensor =
Normal[SparseArray[(Module[{index = #1}, index -> Total[(Inverse[newMetricMatrixRepresentation][[First[index],#1]]*
spatialRicciTensor[[#1,Last[index]]] & ) /@ Range[Length[newMetricMatrixRepresentation]]]] & ) /@
Tuples[Range[Length[newMetricMatrixRepresentation]], 2]]]; 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]]];
stressEnergyTrace = Total[(spacetimeMetricTensor[[First[#1],Last[#1]]]*newMatrixRepresentation[[First[#1],
Last[#1]]] & ) /@ Tuples[Range[Length[spacetimeMetricTensor]], 2]];
leftHandSide = Normal[SparseArray[(Module[{index = #1}, index -> D[mixedExtrinsicCurvatureTensor[[First[#1],
Last[#1]]], newTimeCoordinate]] & ) /@ Tuples[Range[Length[newMetricMatrixRepresentation]], 2]]];
rightHandSide = Normal[SparseArray[(Module[{index = #1}, index -> newLapseFunction*mixedSpatialRicciTensor[[First[
index],Last[index]]] - Total[(Module[{nestedIndex = #1}, (D[D[newLapseFunction, newCoordinates[[
Last[index]]]], newCoordinates[[nestedIndex]]] - Total[(spatialChristoffelSymbols[[#1,nestedIndex,
Last[index]]]*D[newLapseFunction, newCoordinates[[#1]]] & ) /@ Range[Length[
newMetricMatrixRepresentation]]])*Inverse[newMetricMatrixRepresentation][[nestedIndex,
First[index]]]] & ) /@ Range[Length[newMetricMatrixRepresentation]]] + newLapseFunction*
extrinsicCurvatureTrace*mixedExtrinsicCurvatureTensor[[First[index],Last[index]]] +
Total[(Module[{nestedIndex = #1}, newShiftVector[[nestedIndex]]*(D[mixedExtrinsicCurvatureTensor[[
First[index],Last[index]]], newCoordinates[[nestedIndex]]] + Total[(spatialChristoffelSymbols[[
First[index],nestedIndex,#1]]*mixedExtrinsicCurvatureTensor[[#1,Last[index]]] & ) /@
Range[Length[newMetricMatrixRepresentation]]] - Total[(spatialChristoffelSymbols[[#1,nestedIndex,
Last[index]]]*mixedExtrinsicCurvatureTensor[[First[index],#1]] & ) /@ Range[Length[
newMetricMatrixRepresentation]]])] & ) /@ Range[Length[newMetricMatrixRepresentation]]] +
Total[(Module[{nestedIndex = #1}, mixedExtrinsicCurvatureTensor[[First[index],nestedIndex]]*
(D[newShiftVector[[nestedIndex]], newCoordinates[[Last[index]]]] + Total[
(spatialChristoffelSymbols[[nestedIndex,Last[index],#1]]*newShiftVector[[#1]] & ) /@
Range[Length[newMetricMatrixRepresentation]]])] & ) /@ Range[Length[
newMetricMatrixRepresentation]]] - Total[(Module[{nestedIndex = #1}, mixedExtrinsicCurvatureTensor[[
nestedIndex,Last[index]]]*(D[newShiftVector[[First[index]]], newCoordinates[[nestedIndex]]] +
Total[(spatialChristoffelSymbols[[First[index],nestedIndex,#1]]*newShiftVector[[#1]] & ) /@
Range[Length[newMetricMatrixRepresentation]]])] & ) /@ Range[Length[
newMetricMatrixRepresentation]]] - newLapseFunction*(Total[(Module[{nestedIndex = #1},
8*Pi*covariantStressEnergyTensor[[nestedIndex + 1,Last[index] + 1]]*
Inverse[newMetricMatrixRepresentation][[nestedIndex,First[index]]]] & ) /@
Range[Length[newMetricMatrixRepresentation]]] + (4*Pi*stressEnergyTrace*KroneckerDelta[First[index],
Last[index]])/(1 - Length[spacetimeMetricTensor]/2)) - newLapseFunction*
Total[(((2*cosmologicalConstant)/(Length[newMetricMatrixRepresentation] - 1))*spacetimeMetricTensor[[#1 + 1,
Last[index] + 1]]*Inverse[newMetricMatrixRepresentation][[#1,First[index]]] & ) /@
Range[Length[newMetricMatrixRepresentation]]]] & ) /@
Tuples[Range[Length[newMetricMatrixRepresentation]], 2]]]; evolutionEquations =
FullSimplify[Thread[Catenate[leftHandSide] == Catenate[rightHandSide]] /. (ToExpression[#1] -> #1 & ) /@
Select[Join[coordinates, {timeCoordinate}], StringQ]]; If[evolutionEquations === True, {},
If[evolutionEquations === False, Indeterminate, If[Length[Select[evolutionEquations, #1 === False & ]] > 0,
Indeterminate, DeleteDuplicates[Reverse /@ Sort /@ Select[evolutionEquations, #1 =!= True & ]]]]]] /;
SymbolName[admStressEnergyDecomposition] === "ADMStressEnergyDecomposition" &&
SymbolName[admDecomposition] === "ADMDecomposition" && SymbolName[metricTensor] === "MetricTensor" &&
Length[Dimensions[metricMatrixRepresentation]] == 2 && Length[coordinates] == Length[metricMatrixRepresentation] &&
BooleanQ[index1] && BooleanQ[index2] && Length[shiftVector] == Length[metricMatrixRepresentation] &&
Length[coordinates] + 1 == Length[matrixRepresentation]
ADMSolution[(admStressEnergyDecomposition_)[(admDecomposition_)[(metricTensor_)[metricMatrixRepresentation_List,
coordinates_List, index1_, index2_], timeCoordinate_, lapseFunction_, shiftVector_List],
matrixRepresentation_List], cosmologicalConstant_]["EvolutionEquations"] :=
Module[{newMetricMatrixRepresentation, newMatrixRepresentation, newCoordinates, newTimeCoordinate, newLapseFunction,
newShiftVector, shiftCovector, spatialChristoffelSymbols, extrinsicCurvatureTensor, mixedExtrinsicCurvatureTensor,
extrinsicCurvatureTrace, spatialRiemannTensor, spatialRicciTensor, mixedSpatialRicciTensor, spacetimeMetricTensor,
covariantStressEnergyTensor, stressEnergyTrace, leftHandSide, rightHandSide},
newMetricMatrixRepresentation = metricMatrixRepresentation /. (#1 -> ToExpression[#1] & ) /@
Select[Join[coordinates, {timeCoordinate}], StringQ]; newMatrixRepresentation =
matrixRepresentation /. (#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]]]]; spatialChristoffelSymbols =
Normal[SparseArray[(Module[{index = #1}, index -> Total[((1/2)*Inverse[newMetricMatrixRepresentation][[index[[1]],
#1]]*(D[newMetricMatrixRepresentation[[#1,index[[3]]]], newCoordinates[[index[[2]]]]] +
D[newMetricMatrixRepresentation[[index[[2]],#1]], newCoordinates[[index[[3]]]]] -
D[newMetricMatrixRepresentation[[index[[2]],index[[3]]]], newCoordinates[[#1]]]) & ) /@
Range[Length[newMetricMatrixRepresentation]]]] & ) /@ Tuples[Range[Length[newMetricMatrixRepresentation]],
3]]]; extrinsicCurvatureTensor =
Normal[SparseArray[(Module[{index = #1}, index -> (1/(2*newLapseFunction))*(D[shiftCovector[[First[index]]],
newCoordinates[[Last[index]]]] - Total[(spatialChristoffelSymbols[[#1,Last[index],First[index]]]*
shiftCovector[[#1]] & ) /@ Range[Length[newMetricMatrixRepresentation]]] + D[shiftCovector[[
Last[index]]], newCoordinates[[First[index]]]] - Total[(spatialChristoffelSymbols[[#1,First[index],
Last[index]]]*shiftCovector[[#1]] & ) /@ Range[Length[newMetricMatrixRepresentation]]] -
D[newMetricMatrixRepresentation[[First[index],Last[index]]], newTimeCoordinate])] & ) /@
Tuples[Range[Length[newMetricMatrixRepresentation]], 2]]]; mixedExtrinsicCurvatureTensor =
Normal[SparseArray[(Module[{index = #1}, index -> Total[(Inverse[newMetricMatrixRepresentation][[First[index],#1]]*
extrinsicCurvatureTensor[[#1,Last[index]]] & ) /@ Range[Length[newMetricMatrixRepresentation]]]] & ) /@
Tuples[Range[Length[newMetricMatrixRepresentation]], 2]]]; extrinsicCurvatureTrace =
Total[(Inverse[newMetricMatrixRepresentation][[First[#1],Last[#1]]]*extrinsicCurvatureTensor[[First[#1],
Last[#1]]] & ) /@ Tuples[Range[Length[newMetricMatrixRepresentation]], 2]];
spatialRiemannTensor = Normal[SparseArray[
(Module[{index = #1}, index -> D[spatialChristoffelSymbols[[index[[1]],index[[2]],index[[4]]]],
newCoordinates[[index[[3]]]]] - D[spatialChristoffelSymbols[[index[[1]],index[[2]],index[[3]]]],
newCoordinates[[index[[4]]]]] + Total[(spatialChristoffelSymbols[[index[[1]],#1,index[[3]]]]*
spatialChristoffelSymbols[[#1,index[[2]],index[[4]]]] & ) /@ Range[Length[
newMetricMatrixRepresentation]]] - Total[(spatialChristoffelSymbols[[index[[1]],#1,index[[4]]]]*
spatialChristoffelSymbols[[#1,index[[2]],index[[3]]]] & ) /@ Range[Length[
newMetricMatrixRepresentation]]]] & ) /@ Tuples[Range[Length[newMetricMatrixRepresentation]], 4]]];
spatialRicciTensor = Normal[SparseArray[
(Module[{index = #1}, index -> Total[(spatialRiemannTensor[[#1,First[index],#1,Last[index]]] & ) /@
Range[Length[newMetricMatrixRepresentation]]]] & ) /@ Tuples[Range[Length[newMetricMatrixRepresentation]],
2]]]; mixedSpatialRicciTensor =
Normal[SparseArray[(Module[{index = #1}, index -> Total[(Inverse[newMetricMatrixRepresentation][[First[index],#1]]*
spatialRicciTensor[[#1,Last[index]]] & ) /@ Range[Length[newMetricMatrixRepresentation]]]] & ) /@
Tuples[Range[Length[newMetricMatrixRepresentation]], 2]]]; 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]]];
stressEnergyTrace = Total[(spacetimeMetricTensor[[First[#1],Last[#1]]]*newMatrixRepresentation[[First[#1],
Last[#1]]] & ) /@ Tuples[Range[Length[spacetimeMetricTensor]], 2]];
leftHandSide = Normal[SparseArray[(Module[{index = #1}, index -> D[mixedExtrinsicCurvatureTensor[[First[index],
Last[index]]], newTimeCoordinate]] & ) /@ Tuples[Range[Length[newMetricMatrixRepresentation]], 2]]];
rightHandSide = Normal[SparseArray[(Module[{index = #1}, index -> newLapseFunction*mixedSpatialRicciTensor[[First[
index],Last[index]]] - Total[(Module[{nestedIndex = #1}, (D[D[newLapseFunction, newCoordinates[[
Last[index]]]], newCoordinates[[nestedIndex]]] - Total[(spatialChristoffelSymbols[[#1,nestedIndex,
Last[index]]]*D[newLapseFunction, newCoordinates[[#1]]] & ) /@ Range[Length[
newMetricMatrixRepresentation]]])*Inverse[newMetricMatrixRepresentation][[nestedIndex,
First[index]]]] & ) /@ Range[Length[newMetricMatrixRepresentation]]] + newLapseFunction*
extrinsicCurvatureTrace*mixedExtrinsicCurvatureTensor[[First[index],Last[index]]] +
Total[(Module[{nestedIndex = #1}, newShiftVector[[nestedIndex]]*(D[mixedExtrinsicCurvatureTensor[[
First[index],Last[index]]], newCoordinates[[nestedIndex]]] + Total[(spatialChristoffelSymbols[[
First[index],nestedIndex,#1]]*mixedExtrinsicCurvatureTensor[[#1,Last[index]]] & ) /@
Range[Length[newMetricMatrixRepresentation]]] - Total[(spatialChristoffelSymbols[[#1,nestedIndex,
Last[index]]]*mixedExtrinsicCurvatureTensor[[First[index],#1]] & ) /@ Range[Length[
newMetricMatrixRepresentation]]])] & ) /@ Range[Length[newMetricMatrixRepresentation]]] +
Total[(Module[{nestedIndex = #1}, mixedExtrinsicCurvatureTensor[[First[index],nestedIndex]]*
(D[newShiftVector[[nestedIndex]], newCoordinates[[Last[index]]]] + Total[
(spatialChristoffelSymbols[[nestedIndex,Last[index],#1]]*newShiftVector[[#1]] & ) /@
Range[Length[newMetricMatrixRepresentation]]])] & ) /@ Range[Length[
newMetricMatrixRepresentation]]] - Total[(Module[{nestedIndex = #1}, mixedExtrinsicCurvatureTensor[[
nestedIndex,Last[index]]]*(D[newShiftVector[[First[index]]], newCoordinates[[nestedIndex]]] +
Total[(spatialChristoffelSymbols[[First[index],nestedIndex,#1]]*newShiftVector[[#1]] & ) /@
Range[Length[newMetricMatrixRepresentation]]])] & ) /@ Range[Length[
newMetricMatrixRepresentation]]] - newLapseFunction*(Total[(Module[{nestedIndex = #1},
8*Pi*covariantStressEnergyTensor[[nestedIndex + 1,Last[index] + 1]]*
Inverse[newMetricMatrixRepresentation][[nestedIndex,First[index]]]] & ) /@
Range[Length[newMetricMatrixRepresentation]]] + (4*Pi*stressEnergyTrace*KroneckerDelta[First[index],
Last[index]])/(1 - Length[spacetimeMetricTensor]/2)) - newLapseFunction*
Total[(((2*cosmologicalConstant)/(Length[newMetricMatrixRepresentation] - 1))*spacetimeMetricTensor[[#1 + 1,
Last[index] + 1]]*Inverse[newMetricMatrixRepresentation][[#1,First[index]]] & ) /@
Range[Length[newMetricMatrixRepresentation]]]] & ) /@
Tuples[Range[Length[newMetricMatrixRepresentation]], 2]]];
Thread[Catenate[leftHandSide] == Catenate[rightHandSide]] /. (ToExpression[#1] -> #1 & ) /@
Select[Join[coordinates, {timeCoordinate}], StringQ]] /;
SymbolName[admStressEnergyDecomposition] === "ADMStressEnergyDecomposition" &&
SymbolName[admDecomposition] === "ADMDecomposition" && SymbolName[metricTensor] === "MetricTensor" &&
Length[Dimensions[metricMatrixRepresentation]] == 2 && Length[coordinates] == Length[metricMatrixRepresentation] &&
BooleanQ[index1] && BooleanQ[index2] && Length[shiftVector] == Length[metricMatrixRepresentation] &&
Length[coordinates] + 1 == Length[matrixRepresentation]
ADMSolution[(admStressEnergyDecomposition_)[(admDecomposition_)[(metricTensor_)[metricMatrixRepresentation_List,
coordinates_List, index1_, index2_], timeCoordinate_, lapseFunction_, shiftVector_List],
matrixRepresentation_List], cosmologicalConstant_]["ReducedEvolutionEquations"] :=
Module[{newMetricMatrixRepresentation, newMatrixRepresentation, newCoordinates, newTimeCoordinate, newLapseFunction,
newShiftVector, shiftCovector, spatialChristoffelSymbols, extrinsicCurvatureTensor, mixedExtrinsicCurvatureTensor,
extrinsicCurvatureTrace, spatialRiemannTensor, spatialRicciTensor, mixedSpatialRicciTensor, spacetimeMetricTensor,
covariantStressEnergyTensor, stressEnergyTrace, leftHandSide, rightHandSide},
newMetricMatrixRepresentation = metricMatrixRepresentation /. (#1 -> ToExpression[#1] & ) /@
Select[Join[coordinates, {timeCoordinate}], StringQ]; newMatrixRepresentation =
matrixRepresentation /. (#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]]]]; spatialChristoffelSymbols =
Normal[SparseArray[(Module[{index = #1}, index -> Total[((1/2)*Inverse[newMetricMatrixRepresentation][[index[[1]],
#1]]*(D[newMetricMatrixRepresentation[[#1,index[[3]]]], newCoordinates[[index[[2]]]]] +
D[newMetricMatrixRepresentation[[index[[2]],#1]], newCoordinates[[index[[3]]]]] -
D[newMetricMatrixRepresentation[[index[[2]],index[[3]]]], newCoordinates[[#1]]]) & ) /@
Range[Length[newMetricMatrixRepresentation]]]] & ) /@ Tuples[Range[Length[newMetricMatrixRepresentation]],
3]]]; extrinsicCurvatureTensor =
Normal[SparseArray[(Module[{index = #1}, index -> (1/(2*newLapseFunction))*(D[shiftCovector[[First[index]]],
newCoordinates[[Last[index]]]] - Total[(spatialChristoffelSymbols[[#1,Last[index],First[index]]]*
shiftCovector[[#1]] & ) /@ Range[Length[newMetricMatrixRepresentation]]] + D[shiftCovector[[
Last[index]]], newCoordinates[[First[index]]]] - Total[(spatialChristoffelSymbols[[#1,First[index],
Last[index]]]*shiftCovector[[#1]] & ) /@ Range[Length[newMetricMatrixRepresentation]]] -
D[newMetricMatrixRepresentation[[First[index],Last[index]]], newTimeCoordinate])] & ) /@
Tuples[Range[Length[newMetricMatrixRepresentation]], 2]]]; mixedExtrinsicCurvatureTensor =
Normal[SparseArray[(Module[{index = #1}, index -> Total[(Inverse[newMetricMatrixRepresentation][[First[index],#1]]*
extrinsicCurvatureTensor[[#1,Last[index]]] & ) /@ Range[Length[newMetricMatrixRepresentation]]]] & ) /@
Tuples[Range[Length[newMetricMatrixRepresentation]], 2]]]; extrinsicCurvatureTrace =
Total[(Inverse[newMetricMatrixRepresentation][[First[#1],Last[#1]]]*extrinsicCurvatureTensor[[First[#1],
Last[#1]]] & ) /@ Tuples[Range[Length[newMetricMatrixRepresentation]], 2]];
spatialRiemannTensor = Normal[SparseArray[
(Module[{index = #1}, index -> D[spatialChristoffelSymbols[[index[[1]],index[[2]],index[[4]]]],
newCoordinates[[index[[3]]]]] - D[spatialChristoffelSymbols[[index[[1]],index[[2]],index[[3]]]],
newCoordinates[[index[[4]]]]] + Total[(spatialChristoffelSymbols[[index[[1]],#1,index[[3]]]]*
spatialChristoffelSymbols[[#1,index[[2]],index[[4]]]] & ) /@ Range[Length[
newMetricMatrixRepresentation]]] - Total[(spatialChristoffelSymbols[[index[[1]],#1,index[[4]]]]*
spatialChristoffelSymbols[[#1,index[[2]],index[[3]]]] & ) /@ Range[Length[
newMetricMatrixRepresentation]]]] & ) /@ Tuples[Range[Length[newMetricMatrixRepresentation]], 4]]];
spatialRicciTensor = Normal[SparseArray[
(Module[{index = #1}, index -> Total[(spatialRiemannTensor[[#1,First[index],#1,Last[index]]] & ) /@
Range[Length[newMetricMatrixRepresentation]]]] & ) /@ Tuples[Range[Length[newMetricMatrixRepresentation]],
2]]]; mixedSpatialRicciTensor =
Normal[SparseArray[(Module[{index = #1}, index -> Total[(Inverse[newMetricMatrixRepresentation][[First[index],#1]]*
spatialRicciTensor[[#1,Last[index]]] & ) /@ Range[Length[newMetricMatrixRepresentation]]]] & ) /@
Tuples[Range[Length[newMetricMatrixRepresentation]], 2]]]; 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]]];
stressEnergyTrace = Total[(spacetimeMetricTensor[[First[#1],Last[#1]]]*newMatrixRepresentation[[First[#1],
Last[#1]]] & ) /@ Tuples[Range[Length[spacetimeMetricTensor]], 2]];
leftHandSide = Normal[SparseArray[(Module[{index = #1}, index -> D[mixedExtrinsicCurvatureTensor[[First[index],
Last[index]]], newTimeCoordinate]] & ) /@ Tuples[Range[Length[newMetricMatrixRepresentation]], 2]]];
rightHandSide = Normal[SparseArray[(Module[{index = #1}, index -> newLapseFunction*mixedSpatialRicciTensor[[First[
index],Last[index]]] - Total[(Module[{nestedIndex = #1}, (D[D[newLapseFunction, newCoordinates[[
Last[index]]]], newCoordinates[[nestedIndex]]] - Total[(spatialChristoffelSymbols[[#1,nestedIndex,
Last[index]]]*D[newLapseFunction, newCoordinates[[#1]]] & ) /@ Range[Length[
newMetricMatrixRepresentation]]])*Inverse[newMetricMatrixRepresentation][[nestedIndex,
First[index]]]] & ) /@ Range[Length[newMetricMatrixRepresentation]]] + newLapseFunction*
extrinsicCurvatureTrace*mixedExtrinsicCurvatureTensor[[First[index],Last[index]]] +
Total[(Module[{nestedIndex = #1}, newShiftVector[[nestedIndex]]*(D[mixedExtrinsicCurvatureTensor[[
First[index],Last[index]]], newCoordinates[[nestedIndex]]] + Total[(spatialChristoffelSymbols[[
First[index],nestedIndex,#1]]*mixedExtrinsicCurvatureTensor[[#1,Last[index]]] & ) /@
Range[Length[newMetricMatrixRepresentation]]] - Total[(spatialChristoffelSymbols[[#1,nestedIndex,
Last[index]]]*mixedExtrinsicCurvatureTensor[[First[index],#1]] & ) /@ Range[Length[
newMetricMatrixRepresentation]]])] & ) /@ Range[Length[newMetricMatrixRepresentation]]] +
Total[(Module[{nestedIndex = #1}, mixedExtrinsicCurvatureTensor[[First[index],nestedIndex]]*
(D[newShiftVector[[nestedIndex]], newCoordinates[[Last[index]]]] + Total[
(spatialChristoffelSymbols[[nestedIndex,Last[index],#1]]*newShiftVector[[#1]] & ) /@
Range[Length[newMetricMatrixRepresentation]]])] & ) /@ Range[Length[
newMetricMatrixRepresentation]]] - Total[(Module[{nestedIndex = #1}, mixedExtrinsicCurvatureTensor[[
nestedIndex,Last[index]]]*(D[newShiftVector[[First[index]]], newCoordinates[[nestedIndex]]] +
Total[(spatialChristoffelSymbols[[First[index],nestedIndex,#1]]*newShiftVector[[#1]] & ) /@
Range[Length[newMetricMatrixRepresentation]]])] & ) /@ Range[Length[
newMetricMatrixRepresentation]]] - newLapseFunction*(Total[(Module[{nestedIndex = #1},
8*Pi*covariantStressEnergyTensor[[nestedIndex + 1,Last[index] + 1]]*
Inverse[newMetricMatrixRepresentation][[nestedIndex,First[index]]]] & ) /@
Range[Length[newMetricMatrixRepresentation]]] + (4*Pi*stressEnergyTrace*KroneckerDelta[First[index],
Last[index]])/(1 - Length[spacetimeMetricTensor]/2)) - newLapseFunction*
Total[(((2*cosmologicalConstant)/(Length[newMetricMatrixRepresentation] - 1))*spacetimeMetricTensor[[#1 + 1,
Last[index] + 1]]*Inverse[newMetricMatrixRepresentation][[#1,First[index]]] & ) /@
Range[Length[newMetricMatrixRepresentation]]]] & ) /@
Tuples[Range[Length[newMetricMatrixRepresentation]], 2]]];
FullSimplify[Thread[Catenate[leftHandSide] == Catenate[rightHandSide]] /. (ToExpression[#1] -> #1 & ) /@
Select[Join[coordinates, {timeCoordinate}], StringQ]]] /;
SymbolName[admStressEnergyDecomposition] === "ADMStressEnergyDecomposition" &&
SymbolName[admDecomposition] === "ADMDecomposition" && SymbolName[metricTensor] === "MetricTensor" &&
Length[Dimensions[metricMatrixRepresentation]] == 2 && Length[coordinates] == Length[metricMatrixRepresentation] &&
BooleanQ[index1] && BooleanQ[index2] && Length[shiftVector] == Length[metricMatrixRepresentation] &&
Length[coordinates] + 1 == Length[matrixRepresentation]
ADMSolution[(admStressEnergyDecomposition_)[(admDecomposition_)[(metricTensor_)[metricMatrixRepresentation_List,
coordinates_List, index1_, index2_], timeCoordinate_, lapseFunction_, shiftVector_List],
matrixRepresentation_List], cosmologicalConstant_]["SymbolicEvolutionEquations"] :=
Module[{newMetricMatrixRepresentation, newMatrixRepresentation, newCoordinates, newTimeCoordinate, newLapseFunction,
newShiftVector, shiftCovector, spatialChristoffelSymbols, extrinsicCurvatureTensor, mixedExtrinsicCurvatureTensor,
extrinsicCurvatureTrace, spatialRiemannTensor, spatialRicciTensor, mixedSpatialRicciTensor, spacetimeMetricTensor,
covariantStressEnergyTensor, stressEnergyTrace, leftHandSide, rightHandSide},
newMetricMatrixRepresentation = metricMatrixRepresentation /. (#1 -> ToExpression[#1] & ) /@
Select[Join[coordinates, {timeCoordinate}], StringQ]; newMatrixRepresentation =
matrixRepresentation /. (#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]]]]; spatialChristoffelSymbols =
Normal[SparseArray[(Module[{index = #1}, index -> Total[((1/2)*Inverse[newMetricMatrixRepresentation][[index[[1]],
#1]]*(Inactive[D][newMetricMatrixRepresentation[[#1,index[[3]]]], newCoordinates[[index[[2]]]]] +
Inactive[D][newMetricMatrixRepresentation[[index[[2]],#1]], newCoordinates[[index[[3]]]]] -
Inactive[D][newMetricMatrixRepresentation[[index[[2]],index[[3]]]], newCoordinates[[#1]]]) & ) /@
Range[Length[newMetricMatrixRepresentation]]]] & ) /@ Tuples[Range[Length[newMetricMatrixRepresentation]],
3]]]; extrinsicCurvatureTensor =
Normal[SparseArray[(Module[{index = #1}, index -> (1/(2*newLapseFunction))*(Inactive[D][shiftCovector[[
First[index]]], newCoordinates[[Last[index]]]] - Total[(spatialChristoffelSymbols[[#1,Last[index],
First[index]]]*shiftCovector[[#1]] & ) /@ Range[Length[newMetricMatrixRepresentation]]] +
Inactive[D][shiftCovector[[Last[index]]], newCoordinates[[First[index]]]] -
Total[(spatialChristoffelSymbols[[#1,First[index],Last[index]]]*shiftCovector[[#1]] & ) /@
Range[Length[newMetricMatrixRepresentation]]] - Inactive[D][newMetricMatrixRepresentation[[First[index],
Last[index]]], newTimeCoordinate])] & ) /@ Tuples[Range[Length[newMetricMatrixRepresentation]], 2]]];
mixedExtrinsicCurvatureTensor =
Normal[SparseArray[(Module[{index = #1}, index -> Total[(Inverse[newMetricMatrixRepresentation][[First[index],#1]]*
extrinsicCurvatureTensor[[#1,Last[index]]] & ) /@ Range[Length[newMetricMatrixRepresentation]]]] & ) /@
Tuples[Range[Length[newMetricMatrixRepresentation]], 2]]]; extrinsicCurvatureTrace =
Total[(Inverse[newMetricMatrixRepresentation][[First[#1],Last[#1]]]*extrinsicCurvatureTensor[[First[#1],
Last[#1]]] & ) /@ Tuples[Range[Length[newMetricMatrixRepresentation]], 2]];
spatialRiemannTensor = Normal[SparseArray[
(Module[{index = #1}, index -> Inactive[D][spatialChristoffelSymbols[[index[[1]],index[[2]],index[[4]]]],
newCoordinates[[index[[3]]]]] - Inactive[D][spatialChristoffelSymbols[[index[[1]],index[[2]],index[[3]]]],
newCoordinates[[index[[4]]]]] + Total[(spatialChristoffelSymbols[[index[[1]],#1,index[[3]]]]*
spatialChristoffelSymbols[[#1,index[[2]],index[[4]]]] & ) /@ Range[Length[
newMetricMatrixRepresentation]]] - Total[(spatialChristoffelSymbols[[index[[1]],#1,index[[4]]]]*
spatialChristoffelSymbols[[#1,index[[2]],index[[3]]]] & ) /@ Range[Length[
newMetricMatrixRepresentation]]]] & ) /@ Tuples[Range[Length[newMetricMatrixRepresentation]], 4]]];
spatialRicciTensor = Normal[SparseArray[
(Module[{index = #1}, index -> Total[(spatialRiemannTensor[[#1,First[index],#1,Last[index]]] & ) /@
Range[Length[newMetricMatrixRepresentation]]]] & ) /@ Tuples[Range[Length[newMetricMatrixRepresentation]],
2]]]; mixedSpatialRicciTensor =
Normal[SparseArray[(Module[{index = #1}, index -> Total[(Inverse[newMetricMatrixRepresentation][[First[index],#1]]*
spatialRicciTensor[[#1,Last[index]]] & ) /@ Range[Length[newMetricMatrixRepresentation]]]] & ) /@
Tuples[Range[Length[newMetricMatrixRepresentation]], 2]]]; 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]]];
stressEnergyTrace = Total[(spacetimeMetricTensor[[First[#1],Last[#1]]]*newMatrixRepresentation[[First[#1],
Last[#1]]] & ) /@ Tuples[Range[Length[spacetimeMetricTensor]], 2]];
leftHandSide = Normal[SparseArray[(Module[{index = #1}, index -> Inactive[D][mixedExtrinsicCurvatureTensor[[
First[index],Last[index]]], newTimeCoordinate]] & ) /@ Tuples[Range[Length[newMetricMatrixRepresentation]],
2]]]; rightHandSide = Normal[SparseArray[
(Module[{index = #1}, index -> newLapseFunction*mixedSpatialRicciTensor[[First[index],Last[index]]] -
Total[(Module[{nestedIndex = #1}, (Inactive[D][Inactive[D][newLapseFunction, newCoordinates[[Last[index]]]],
newCoordinates[[nestedIndex]]] - Total[(spatialChristoffelSymbols[[#1,nestedIndex,Last[index]]]*
Inactive[D][newLapseFunction, newCoordinates[[#1]]] & ) /@ Range[Length[
newMetricMatrixRepresentation]]])*Inverse[newMetricMatrixRepresentation][[nestedIndex,
First[index]]]] & ) /@ Range[Length[newMetricMatrixRepresentation]]] + newLapseFunction*
extrinsicCurvatureTrace*mixedExtrinsicCurvatureTensor[[First[index],Last[index]]] +
Total[(Module[{nestedIndex = #1}, newShiftVector[[nestedIndex]]*(Inactive[D][mixedExtrinsicCurvatureTensor[[
First[index],Last[index]]], newCoordinates[[nestedIndex]]] + Total[(spatialChristoffelSymbols[[
First[index],nestedIndex,#1]]*mixedExtrinsicCurvatureTensor[[#1,Last[index]]] & ) /@
Range[Length[newMetricMatrixRepresentation]]] - Total[(spatialChristoffelSymbols[[#1,nestedIndex,
Last[index]]]*mixedExtrinsicCurvatureTensor[[First[index],#1]] & ) /@ Range[Length[
newMetricMatrixRepresentation]]])] & ) /@ Range[Length[newMetricMatrixRepresentation]]] +
Total[(Module[{nestedIndex = #1}, mixedExtrinsicCurvatureTensor[[First[index],nestedIndex]]*
(Inactive[D][newShiftVector[[nestedIndex]], newCoordinates[[Last[index]]]] +
Total[(spatialChristoffelSymbols[[nestedIndex,Last[index],#1]]*newShiftVector[[#1]] & ) /@
Range[Length[newMetricMatrixRepresentation]]])] & ) /@ Range[Length[
newMetricMatrixRepresentation]]] - Total[(Module[{nestedIndex = #1}, mixedExtrinsicCurvatureTensor[[
nestedIndex,Last[index]]]*(Inactive[D][newShiftVector[[First[index]]], newCoordinates[[
nestedIndex]]] + Total[(spatialChristoffelSymbols[[First[index],nestedIndex,#1]]*newShiftVector[[
#1]] & ) /@ Range[Length[newMetricMatrixRepresentation]]])] & ) /@ Range[
Length[newMetricMatrixRepresentation]]] - newLapseFunction*(Total[(Module[{nestedIndex = #1},
8*Pi*covariantStressEnergyTensor[[nestedIndex + 1,Last[index] + 1]]*
Inverse[newMetricMatrixRepresentation][[nestedIndex,First[index]]]] & ) /@
Range[Length[newMetricMatrixRepresentation]]] + (4*Pi*stressEnergyTrace*KroneckerDelta[First[index],
Last[index]])/(1 - Length[spacetimeMetricTensor]/2)) - newLapseFunction*
Total[(((2*cosmologicalConstant)/(Length[newMetricMatrixRepresentation] - 1))*spacetimeMetricTensor[[#1 + 1,
Last[index] + 1]]*Inverse[newMetricMatrixRepresentation][[#1,First[index]]] & ) /@
Range[Length[newMetricMatrixRepresentation]]]] & ) /@
Tuples[Range[Length[newMetricMatrixRepresentation]], 2]]];
Thread[Catenate[leftHandSide] == Catenate[rightHandSide]] /. (ToExpression[#1] -> #1 & ) /@
Select[Join[coordinates, {timeCoordinate}], StringQ]] /;
SymbolName[admStressEnergyDecomposition] === "ADMStressEnergyDecomposition" &&
SymbolName[admDecomposition] === "ADMDecomposition" && SymbolName[metricTensor] === "MetricTensor" &&
Length[Dimensions[metricMatrixRepresentation]] == 2 && Length[coordinates] == Length[metricMatrixRepresentation] &&
BooleanQ[index1] && BooleanQ[index2] && Length[shiftVector] == Length[metricMatrixRepresentation] &&
Length[coordinates] + 1 == Length[matrixRepresentation]
ADMSolution[(admStressEnergyDecomposition_)[(admDecomposition_)[(metricTensor_)[metricMatrixRepresentation_List,
coordinates_List, index1_, index2_], timeCoordinate_, lapseFunction_, shiftVector_List],
matrixRepresentation_List], cosmologicalConstant_]["EnergyConservationEquation"] :=
Module[{newMatrixRepresentation, newMetricMatrixRepresentation, newCoordinates, newTimeCoordinate, newLapseFunction,
newShiftVector, shiftCovector, spatialChristoffelSymbols, extrinsicCurvatureTensor, extrinsicCurvatureTrace,
spacetimeMetricTensor, covariantStressEnergyTensor, normalVector, energyDensity, projectionOperator,
momentumCovector, momentumVector, stressTensor, contravariantStressTensor},
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]]]]; spatialChristoffelSymbols =
Normal[SparseArray[(Module[{index = #1}, index -> Total[((1/2)*Inverse[newMetricMatrixRepresentation][[index[[1]],
#1]]*(D[newMetricMatrixRepresentation[[#1,index[[3]]]], newCoordinates[[index[[2]]]]] +
D[newMetricMatrixRepresentation[[index[[2]],#1]], newCoordinates[[index[[3]]]]] -
D[newMetricMatrixRepresentation[[index[[2]],index[[3]]]], newCoordinates[[#1]]]) & ) /@
Range[Length[newMetricMatrixRepresentation]]]] & ) /@ Tuples[Range[Length[newMetricMatrixRepresentation]],
3]]]; extrinsicCurvatureTensor =
Normal[SparseArray[(Module[{index = #1}, index -> (1/(2*newLapseFunction))*(D[shiftCovector[[First[index]]],
newCoordinates[[Last[index]]]] - Total[(spatialChristoffelSymbols[[#1,Last[index],First[index]]]*
shiftCovector[[#1]] & ) /@ Range[Length[newMetricMatrixRepresentation]]] + D[shiftCovector[[
Last[index]]], newCoordinates[[First[index]]]] - Total[(spatialChristoffelSymbols[[#1,First[index],
Last[index]]]*shiftCovector[[#1]] & ) /@ Range[Length[newMetricMatrixRepresentation]]] -
D[newMetricMatrixRepresentation[[First[index],Last[index]]], newTimeCoordinate])] & ) /@
Tuples[Range[Length[newMetricMatrixRepresentation]], 2]]]; extrinsicCurvatureTrace =
Total[(Inverse[newMetricMatrixRepresentation][[First[#1],Last[#1]]]*extrinsicCurvatureTensor[[First[#1],
Last[#1]]] & ) /@ Tuples[Range[Length[newMetricMatrixRepresentation]], 2]];
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]]]];
energyDensity = Total[(covariantStressEnergyTensor[[First[#1],Last[#1]]]*normalVector[[First[#1]]]*
normalVector[[Last[#1]]] & ) /@ Tuples[Range[Length[spacetimeMetricTensor]], 2]];
projectionOperator = Normal[SparseArray[(Module[{index = #1}, index -> KroneckerDelta[First[index], Last[index]] +
Total[(normalVector[[Last[index]]]*spacetimeMetricTensor[[First[index],#1]]*normalVector[[#1]] & ) /@ Range[
Length[spacetimeMetricTensor]]]] & ) /@ Tuples[Range[Length[spacetimeMetricTensor]], 2]]];
momentumCovector = Normal[SparseArray[
(Module[{index = #1}, index -> -Total[(covariantStressEnergyTensor[[First[#1],Last[#1]]]*normalVector[[
First[#1]]]*projectionOperator[[index + 1,Last[#1]]] & ) /@ Tuples[
Range[Length[spacetimeMetricTensor]], 2]]] & ) /@ Range[Length[newMetricMatrixRepresentation]]]];
momentumVector = Normal[SparseArray[
(Module[{index = #1}, index -> Total[(Inverse[newMetricMatrixRepresentation][[index,#1]]*momentumCovector[[
#1]] & ) /@ Range[Length[newMetricMatrixRepresentation]]]] & ) /@
Range[Length[newMetricMatrixRepresentation]]]]; stressTensor =
Normal[SparseArray[(Module[{index = #1}, index -> Total[(covariantStressEnergyTensor[[First[#1],Last[#1]]]*
projectionOperator[[First[index] + 1,First[#1]]]*projectionOperator[[Last[index] + 1,Last[#1]]] & ) /@
Tuples[Range[Length[spacetimeMetricTensor]], 2]]] & ) /@
Tuples[Range[Length[newMetricMatrixRepresentation]], 2]]]; contravariantStressTensor =
Normal[SparseArray[(Module[{index = #1}, index -> Total[(Inverse[newMetricMatrixRepresentation][[First[index],
First[#1]]]*Inverse[newMetricMatrixRepresentation][[Last[#1],Last[index]]]*stressTensor[[First[#1],
Last[#1]]] & ) /@ Tuples[Range[Length[newMetricMatrixRepresentation]], 2]]] & ) /@
Tuples[Range[Length[newMetricMatrixRepresentation]], 2]]];
D[energyDensity, newTimeCoordinate] - Total[(newShiftVector[[#1]]*D[energyDensity, newCoordinates[[#1]]] & ) /@
Range[Length[newMetricMatrixRepresentation]]] + newLapseFunction*
(Total[(D[momentumVector[[#1]], newCoordinates[[#1]]] & ) /@ Range[Length[newMetricMatrixRepresentation]]] +
Total[(spatialChristoffelSymbols[[First[#1],First[#1],Last[#1]]]*momentumVector[[Last[#1]]] & ) /@
Tuples[Range[Length[newMetricMatrixRepresentation]], 2]] - extrinsicCurvatureTrace*energyDensity -
Total[(extrinsicCurvatureTensor[[First[#1],Last[#1]]]*contravariantStressTensor[[First[#1],Last[#1]]] & ) /@
Tuples[Range[Length[newMetricMatrixRepresentation]], 2]]) +
2*Total[(momentumVector[[#1]]*D[newLapseFunction, newCoordinates[[#1]]] & ) /@
Range[Length[newMetricMatrixRepresentation]]] == 0 /. (ToExpression[#1] -> #1 & ) /@
Select[Join[coordinates, {timeCoordinate}], StringQ]] /;
SymbolName[admStressEnergyDecomposition] === "ADMStressEnergyDecomposition" &&
SymbolName[admDecomposition] === "ADMDecomposition" && SymbolName[metricTensor] === "MetricTensor" &&
Length[Dimensions[metricMatrixRepresentation]] == 2 && Length[coordinates] == Length[metricMatrixRepresentation] &&
BooleanQ[index1] && BooleanQ[index2] && Length[shiftVector] == Length[metricMatrixRepresentation] &&
Length[coordinates] + 1 == Length[matrixRepresentation]
ADMSolution[(admStressEnergyDecomposition_)[(admDecomposition_)[(metricTensor_)[metricMatrixRepresentation_List,
coordinates_List, index1_, index2_], timeCoordinate_, lapseFunction_, shiftVector_List],
matrixRepresentation_List], cosmologicalConstant_]["ReducedEnergyConservationEquation"] :=
Module[{newMatrixRepresentation, newMetricMatrixRepresentation, newCoordinates, newTimeCoordinate, newLapseFunction,
newShiftVector, shiftCovector, spatialChristoffelSymbols, extrinsicCurvatureTensor, extrinsicCurvatureTrace,
spacetimeMetricTensor, covariantStressEnergyTensor, normalVector, energyDensity, projectionOperator,
momentumCovector, momentumVector, stressTensor, contravariantStressTensor},
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]]]]; spatialChristoffelSymbols =
Normal[SparseArray[(Module[{index = #1}, index -> Total[((1/2)*Inverse[newMetricMatrixRepresentation][[index[[1]],
#1]]*(D[newMetricMatrixRepresentation[[#1,index[[3]]]], newCoordinates[[index[[2]]]]] +
D[newMetricMatrixRepresentation[[index[[2]],#1]], newCoordinates[[index[[3]]]]] -
D[newMetricMatrixRepresentation[[index[[2]],index[[3]]]], newCoordinates[[#1]]]) & ) /@
Range[Length[newMetricMatrixRepresentation]]]] & ) /@ Tuples[Range[Length[newMetricMatrixRepresentation]],
3]]]; extrinsicCurvatureTensor =
Normal[SparseArray[(Module[{index = #1}, index -> (1/(2*newLapseFunction))*(D[shiftCovector[[First[index]]],
newCoordinates[[Last[index]]]] - Total[(spatialChristoffelSymbols[[#1,Last[index],First[index]]]*
shiftCovector[[#1]] & ) /@ Range[Length[newMetricMatrixRepresentation]]] + D[shiftCovector[[
Last[index]]], newCoordinates[[First[index]]]] - Total[(spatialChristoffelSymbols[[#1,First[index],
Last[index]]]*shiftCovector[[#1]] & ) /@ Range[Length[newMetricMatrixRepresentation]]] -
D[newMetricMatrixRepresentation[[First[index],Last[index]]], newTimeCoordinate])] & ) /@
Tuples[Range[Length[newMetricMatrixRepresentation]], 2]]]; extrinsicCurvatureTrace =
Total[(Inverse[newMetricMatrixRepresentation][[First[#1],Last[#1]]]*extrinsicCurvatureTensor[[First[#1],
Last[#1]]] & ) /@ Tuples[Range[Length[newMetricMatrixRepresentation]], 2]];
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]]]];
energyDensity = Total[(covariantStressEnergyTensor[[First[#1],Last[#1]]]*normalVector[[First[#1]]]*
normalVector[[Last[#1]]] & ) /@ Tuples[Range[Length[spacetimeMetricTensor]], 2]];
projectionOperator = Normal[SparseArray[(Module[{index = #1}, index -> KroneckerDelta[First[index], Last[index]] +
Total[(normalVector[[Last[index]]]*spacetimeMetricTensor[[First[index],#1]]*normalVector[[#1]] & ) /@ Range[
Length[spacetimeMetricTensor]]]] & ) /@ Tuples[Range[Length[spacetimeMetricTensor]], 2]]];
momentumCovector = Normal[SparseArray[
(Module[{index = #1}, index -> -Total[(covariantStressEnergyTensor[[First[#1],Last[#1]]]*normalVector[[
First[#1]]]*projectionOperator[[index + 1,Last[#1]]] & ) /@ Tuples[
Range[Length[spacetimeMetricTensor]], 2]]] & ) /@ Range[Length[newMetricMatrixRepresentation]]]];
momentumVector = Normal[SparseArray[
(Module[{index = #1}, index -> Total[(Inverse[newMetricMatrixRepresentation][[index,#1]]*momentumCovector[[
#1]] & ) /@ Range[Length[newMetricMatrixRepresentation]]]] & ) /@
Range[Length[newMetricMatrixRepresentation]]]]; stressTensor =
Normal[SparseArray[(Module[{index = #1}, index -> Total[(covariantStressEnergyTensor[[First[#1],Last[#1]]]*
projectionOperator[[First[index] + 1,First[#1]]]*projectionOperator[[Last[index] + 1,Last[#1]]] & ) /@
Tuples[Range[Length[spacetimeMetricTensor]], 2]]] & ) /@
Tuples[Range[Length[newMetricMatrixRepresentation]], 2]]]; contravariantStressTensor =
Normal[SparseArray[(Module[{index = #1}, index -> Total[(Inverse[newMetricMatrixRepresentation][[First[index],
First[#1]]]*Inverse[newMetricMatrixRepresentation][[Last[#1],Last[index]]]*stressTensor[[First[#1],
Last[#1]]] & ) /@ Tuples[Range[Length[newMetricMatrixRepresentation]], 2]]] & ) /@
Tuples[Range[Length[newMetricMatrixRepresentation]], 2]]];
FullSimplify[D[energyDensity, newTimeCoordinate] -
Total[(newShiftVector[[#1]]*D[energyDensity, newCoordinates[[#1]]] & ) /@
Range[Length[newMetricMatrixRepresentation]]] + newLapseFunction*
(Total[(D[momentumVector[[#1]], newCoordinates[[#1]]] & ) /@ Range[Length[newMetricMatrixRepresentation]]] +
Total[(spatialChristoffelSymbols[[First[#1],First[#1],Last[#1]]]*momentumVector[[Last[#1]]] & ) /@
Tuples[Range[Length[newMetricMatrixRepresentation]], 2]] - extrinsicCurvatureTrace*energyDensity -
Total[(extrinsicCurvatureTensor[[First[#1],Last[#1]]]*contravariantStressTensor[[First[#1],Last[#1]]] & ) /@
Tuples[Range[Length[newMetricMatrixRepresentation]], 2]]) +
2*Total[(momentumVector[[#1]]*D[newLapseFunction, newCoordinates[[#1]]] & ) /@
Range[Length[newMetricMatrixRepresentation]]] == 0 /. (ToExpression[#1] -> #1 & ) /@
Select[Join[coordinates, {timeCoordinate}], StringQ]]] /;
SymbolName[admStressEnergyDecomposition] === "ADMStressEnergyDecomposition" &&
SymbolName[admDecomposition] === "ADMDecomposition" && SymbolName[metricTensor] === "MetricTensor" &&
Length[Dimensions[metricMatrixRepresentation]] == 2 && Length[coordinates] == Length[metricMatrixRepresentation] &&
BooleanQ[index1] && BooleanQ[index2] && Length[shiftVector] == Length[metricMatrixRepresentation] &&
Length[coordinates] + 1 == Length[matrixRepresentation]
ADMSolution[(admStressEnergyDecomposition_)[(admDecomposition_)[(metricTensor_)[metricMatrixRepresentation_List,
coordinates_List, index1_, index2_], timeCoordinate_, lapseFunction_, shiftVector_List],
matrixRepresentation_List], cosmologicalConstant_]["SymbolicEnergyConservationEquation"] :=
Module[{newMatrixRepresentation, newMetricMatrixRepresentation, newCoordinates, newTimeCoordinate, newLapseFunction,
newShiftVector, shiftCovector, spatialChristoffelSymbols, extrinsicCurvatureTensor, extrinsicCurvatureTrace,
spacetimeMetricTensor, covariantStressEnergyTensor, normalVector, energyDensity, projectionOperator,
momentumCovector, momentumVector, stressTensor, contravariantStressTensor},
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]]]]; spatialChristoffelSymbols =
Normal[SparseArray[(Module[{index = #1}, index -> Total[((1/2)*Inverse[newMetricMatrixRepresentation][[index[[1]],
#1]]*(Inactive[D][newMetricMatrixRepresentation[[#1,index[[3]]]], newCoordinates[[index[[2]]]]] +
Inactive[D][newMetricMatrixRepresentation[[index[[2]],#1]], newCoordinates[[index[[3]]]]] -
Inactive[D][newMetricMatrixRepresentation[[index[[2]],index[[3]]]], newCoordinates[[#1]]]) & ) /@
Range[Length[newMetricMatrixRepresentation]]]] & ) /@ Tuples[Range[Length[newMetricMatrixRepresentation]],
3]]]; extrinsicCurvatureTensor =
Normal[SparseArray[(Module[{index = #1}, index -> (1/(2*newLapseFunction))*(Inactive[D][shiftCovector[[
First[index]]], newCoordinates[[Last[index]]]] - Total[(spatialChristoffelSymbols[[#1,Last[index],
First[index]]]*shiftCovector[[#1]] & ) /@ Range[Length[newMetricMatrixRepresentation]]] +
Inactive[D][shiftCovector[[Last[index]]], newCoordinates[[First[index]]]] -
Total[(spatialChristoffelSymbols[[#1,First[index],Last[index]]]*shiftCovector[[#1]] & ) /@
Range[Length[newMetricMatrixRepresentation]]] - Inactive[D][newMetricMatrixRepresentation[[First[index],
Last[index]]], newTimeCoordinate])] & ) /@ Tuples[Range[Length[newMetricMatrixRepresentation]], 2]]];
extrinsicCurvatureTrace = Total[(Inverse[newMetricMatrixRepresentation][[First[#1],Last[#1]]]*
extrinsicCurvatureTensor[[First[#1],Last[#1]]] & ) /@ Tuples[Range[Length[newMetricMatrixRepresentation]], 2]];
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]]]];
energyDensity = Total[(covariantStressEnergyTensor[[First[#1],Last[#1]]]*normalVector[[First[#1]]]*
normalVector[[Last[#1]]] & ) /@ Tuples[Range[Length[spacetimeMetricTensor]], 2]];
projectionOperator = Normal[SparseArray[(Module[{index = #1}, index -> KroneckerDelta[First[index], Last[index]] +
Total[(normalVector[[Last[index]]]*spacetimeMetricTensor[[First[index],#1]]*normalVector[[#1]] & ) /@ Range[
Length[spacetimeMetricTensor]]]] & ) /@ Tuples[Range[Length[spacetimeMetricTensor]], 2]]];
momentumCovector = Normal[SparseArray[
(Module[{index = #1}, index -> -Total[(covariantStressEnergyTensor[[First[#1],Last[#1]]]*normalVector[[
First[#1]]]*projectionOperator[[index + 1,Last[#1]]] & ) /@ Tuples[
Range[Length[spacetimeMetricTensor]], 2]]] & ) /@ Range[Length[newMetricMatrixRepresentation]]]];
momentumVector = Normal[SparseArray[
(Module[{index = #1}, index -> Total[(Inverse[newMetricMatrixRepresentation][[index,#1]]*momentumCovector[[
#1]] & ) /@ Range[Length[newMetricMatrixRepresentation]]]] & ) /@
Range[Length[newMetricMatrixRepresentation]]]]; stressTensor =
Normal[SparseArray[(Module[{index = #1}, index -> Total[(covariantStressEnergyTensor[[First[#1],Last[#1]]]*
projectionOperator[[First[index] + 1,First[#1]]]*projectionOperator[[Last[index] + 1,Last[#1]]] & ) /@
Tuples[Range[Length[spacetimeMetricTensor]], 2]]] & ) /@
Tuples[Range[Length[newMetricMatrixRepresentation]], 2]]]; contravariantStressTensor =
Normal[SparseArray[(Module[{index = #1}, index -> Total[(Inverse[newMetricMatrixRepresentation][[First[index],
First[#1]]]*Inverse[newMetricMatrixRepresentation][[Last[#1],Last[index]]]*stressTensor[[First[#1],
Last[#1]]] & ) /@ Tuples[Range[Length[newMetricMatrixRepresentation]], 2]]] & ) /@
Tuples[Range[Length[newMetricMatrixRepresentation]], 2]]];
Inactive[D][energyDensity, newTimeCoordinate] -
Total[(newShiftVector[[#1]]*Inactive[D][energyDensity, newCoordinates[[#1]]] & ) /@
Range[Length[newMetricMatrixRepresentation]]] + newLapseFunction*
(Total[(Inactive[D][momentumVector[[#1]], newCoordinates[[#1]]] & ) /@
Range[Length[newMetricMatrixRepresentation]]] +
Total[(spatialChristoffelSymbols[[First[#1],First[#1],Last[#1]]]*momentumVector[[Last[#1]]] & ) /@
Tuples[Range[Length[newMetricMatrixRepresentation]], 2]] - extrinsicCurvatureTrace*energyDensity -
Total[(extrinsicCurvatureTensor[[First[#1],Last[#1]]]*contravariantStressTensor[[First[#1],Last[#1]]] & ) /@
Tuples[Range[Length[newMetricMatrixRepresentation]], 2]]) +
2*Total[(momentumVector[[#1]]*Inactive[D][newLapseFunction, newCoordinates[[#1]]] & ) /@
Range[Length[newMetricMatrixRepresentation]]] == 0 /. (ToExpression[#1] -> #1 & ) /@
Select[Join[coordinates, {timeCoordinate}], StringQ]] /;
SymbolName[admStressEnergyDecomposition] === "ADMStressEnergyDecomposition" &&
SymbolName[admDecomposition] === "ADMDecomposition" && SymbolName[metricTensor] === "MetricTensor" &&
Length[Dimensions[metricMatrixRepresentation]] == 2 && Length[coordinates] == Length[metricMatrixRepresentation] &&
BooleanQ[index1] && BooleanQ[index2] && Length[shiftVector] == Length[metricMatrixRepresentation] &&
Length[coordinates] + 1 == Length[matrixRepresentation]
ADMSolution[(admStressEnergyDecomposition_)[(admDecomposition_)[(metricTensor_)[metricMatrixRepresentation_List,
coordinates_List, index1_, index2_], timeCoordinate_, lapseFunction_, shiftVector_List],
matrixRepresentation_List], cosmologicalConstant_]["MomentumConservationEquations"] :=
Module[{newMatrixRepresentation, newMetricMatrixRepresentation, newCoordinates, newTimeCoordinate, newLapseFunction,
newShiftVector, shiftCovector, spatialChristoffelSymbols, extrinsicCurvatureTensor, extrinsicCurvatureTrace,
spacetimeMetricTensor, covariantStressEnergyTensor, normalVector, energyDensity, projectionOperator,
momentumCovector, stressTensor, mixedStressTensor},
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]]]]; spatialChristoffelSymbols =
Normal[SparseArray[(Module[{index = #1}, index -> Total[((1/2)*Inverse[newMetricMatrixRepresentation][[index[[1]],
#1]]*(D[newMetricMatrixRepresentation[[#1,index[[3]]]], newCoordinates[[index[[2]]]]] +
D[newMetricMatrixRepresentation[[index[[2]],#1]], newCoordinates[[index[[3]]]]] -
D[newMetricMatrixRepresentation[[index[[2]],index[[3]]]], newCoordinates[[#1]]]) & ) /@
Range[Length[newMetricMatrixRepresentation]]]] & ) /@ Tuples[Range[Length[newMetricMatrixRepresentation]],
3]]]; extrinsicCurvatureTensor =
Normal[SparseArray[(Module[{index = #1}, index -> (1/(2*newLapseFunction))*(D[shiftCovector[[First[index]]],
newCoordinates[[Last[index]]]] - Total[(spatialChristoffelSymbols[[#1,Last[index],First[index]]]*
shiftCovector[[#1]] & ) /@ Range[Length[newMetricMatrixRepresentation]]] + D[shiftCovector[[
Last[index]]], newCoordinates[[First[index]]]] - Total[(spatialChristoffelSymbols[[#1,First[index],
Last[index]]]*shiftCovector[[#1]] & ) /@ Range[Length[newMetricMatrixRepresentation]]] -
D[newMetricMatrixRepresentation[[First[index],Last[index]]], newTimeCoordinate])] & ) /@
Tuples[Range[Length[newMetricMatrixRepresentation]], 2]]]; extrinsicCurvatureTrace =
Total[(Inverse[newMetricMatrixRepresentation][[First[#1],Last[#1]]]*extrinsicCurvatureTensor[[First[#1],
Last[#1]]] & ) /@ Tuples[Range[Length[newMetricMatrixRepresentation]], 2]];
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]]]];
energyDensity = Total[(covariantStressEnergyTensor[[First[#1],Last[#1]]]*normalVector[[First[#1]]]*
normalVector[[Last[#1]]] & ) /@ Tuples[Range[Length[spacetimeMetricTensor]], 2]];
projectionOperator = Normal[SparseArray[(Module[{index = #1}, index -> KroneckerDelta[First[index], Last[index]] +
Total[(normalVector[[Last[index]]]*spacetimeMetricTensor[[First[index],#1]]*normalVector[[#1]] & ) /@ Range[
Length[spacetimeMetricTensor]]]] & ) /@ Tuples[Range[Length[spacetimeMetricTensor]], 2]]];
momentumCovector = Normal[SparseArray[
(Module[{index = #1}, index -> -Total[(covariantStressEnergyTensor[[First[#1],Last[#1]]]*normalVector[[
First[#1]]]*projectionOperator[[index + 1,Last[#1]]] & ) /@ Tuples[
Range[Length[spacetimeMetricTensor]], 2]]] & ) /@ Range[Length[newMetricMatrixRepresentation]]]];
stressTensor = Normal[SparseArray[
(Module[{index = #1}, index -> Total[(covariantStressEnergyTensor[[First[#1],Last[#1]]]*projectionOperator[[
First[index] + 1,First[#1]]]*projectionOperator[[Last[index] + 1,Last[#1]]] & ) /@
Tuples[Range[Length[spacetimeMetricTensor]], 2]]] & ) /@
Tuples[Range[Length[newMetricMatrixRepresentation]], 2]]]; mixedStressTensor =
Normal[SparseArray[(Module[{index = #1}, index -> Total[(Inverse[newMetricMatrixRepresentation][[First[index],#1]]*
stressTensor[[#1,Last[index]]] & ) /@ Range[Length[newMetricMatrixRepresentation]]]] & ) /@
Tuples[Range[Length[newMetricMatrixRepresentation]], 2]]];
Normal[SparseArray[(Module[{index = #1}, index -> D[momentumCovector[[index]], newTimeCoordinate] -
Total[(newShiftVector[[#1]]*D[momentumCovector[[index]], newCoordinates[[#1]]] & ) /@
Range[Length[newMetricMatrixRepresentation]]] - Total[(momentumCovector[[#1]]*D[newShiftVector[[#1]],
newCoordinates[[index]]] & ) /@ Range[Length[newMetricMatrixRepresentation]]] +
newLapseFunction*(Total[(D[mixedStressTensor[[#1,index]], newCoordinates[[#1]]] & ) /@
Range[Length[newMetricMatrixRepresentation]]] + Total[(spatialChristoffelSymbols[[First[#1],First[#1],
Last[#1]]]*mixedStressTensor[[Last[#1],index]] & ) /@ Tuples[Range[Length[
newMetricMatrixRepresentation]], 2]] - Total[(spatialChristoffelSymbols[[Last[#1],First[#1],index]]*
mixedStressTensor[[First[#1],Last[#1]]] & ) /@ Tuples[Range[Length[newMetricMatrixRepresentation]],