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SolveVacuumADMEquations.wl
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(* ::Package:: *)
SolveVacuumADMEquations[(admDecomposition_)[(metricTensor_)[matrixRepresentation_List, coordinates_List, index1_,
index2_], timeCoordinate_, lapseFunction_, shiftVector_List]] :=
VacuumADMSolution[ResourceFunction["ADMDecomposition"][ResourceFunction["MetricTensor"][matrixRepresentation,
coordinates, index1, index2], timeCoordinate, lapseFunction, shiftVector], "\[FormalCapitalLambda]"] /;
SymbolName[admDecomposition] === "ADMDecomposition" && SymbolName[metricTensor] === "MetricTensor" &&
Length[Dimensions[matrixRepresentation]] == 2 && Length[coordinates] == Length[matrixRepresentation] &&
BooleanQ[index1] && BooleanQ[index2] && Length[shiftVector] == Length[matrixRepresentation]
SolveVacuumADMEquations[(admDecomposition_)[(metricTensor_)[matrixRepresentation_List, coordinates_List, index1_,
index2_], timeCoordinate_, lapseFunction_, shiftVector_List], cosmologicalConstant_] :=
VacuumADMSolution[ResourceFunction["ADMDecomposition"][ResourceFunction["MetricTensor"][matrixRepresentation,
coordinates, index1, index2], timeCoordinate, lapseFunction, shiftVector], cosmologicalConstant] /;
SymbolName[admDecomposition] === "ADMDecomposition" && SymbolName[metricTensor] === "MetricTensor" &&
Length[Dimensions[matrixRepresentation]] == 2 && Length[coordinates] == Length[matrixRepresentation] &&
BooleanQ[index1] && BooleanQ[index2] && Length[shiftVector] == Length[matrixRepresentation]
VacuumADMSolution[(admDecomposition_)[(metricTensor_)[matrixRepresentation_List, coordinates_List, index1_, index2_],
timeCoordinate_, lapseFunction_, shiftVector_List], cosmologicalConstant_]["SolutionQ"] :=
Module[{newMatrixRepresentation, newCoordinates, newTimeCoordinate, newLapseFunction, newShiftVector, shiftCovector,
spatialChristoffelSymbols, extrinsicCurvatureTensor, mixedExtrinsicCurvatureTensor, extrinsicCurvatureTrace,
spatialRiemannTensor, spatialRicciTensor, mixedSpatialRicciTensor, spacetimeMetricTensor, leftHandSide,
rightHandSide, evolutionEquations}, 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[(newMatrixRepresentation[[index,#1]]*newShiftVector[[#1]] & ) /@
Range[Length[newMatrixRepresentation]]]] & ) /@ Range[Length[newMatrixRepresentation]]]];
spatialChristoffelSymbols = Normal[SparseArray[
(Module[{index = #1}, index -> Total[((1/2)*Inverse[newMatrixRepresentation][[index[[1]],#1]]*
(D[newMatrixRepresentation[[#1,index[[3]]]], newCoordinates[[index[[2]]]]] + D[newMatrixRepresentation[[
index[[2]],#1]], newCoordinates[[index[[3]]]]] - D[newMatrixRepresentation[[index[[2]],index[[3]]]],
newCoordinates[[#1]]]) & ) /@ Range[Length[newMatrixRepresentation]]]] & ) /@
Tuples[Range[Length[newMatrixRepresentation]], 3]]]; 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[newMatrixRepresentation]]] + D[shiftCovector[[Last[index]]],
newCoordinates[[First[index]]]] - Total[(spatialChristoffelSymbols[[#1,First[index],Last[index]]]*
shiftCovector[[#1]] & ) /@ Range[Length[newMatrixRepresentation]]] - D[newMatrixRepresentation[[
First[index],Last[index]]], newTimeCoordinate])] & ) /@ Tuples[Range[Length[newMatrixRepresentation]],
2]]]; mixedExtrinsicCurvatureTensor =
Normal[SparseArray[(Module[{index = #1}, index -> Total[(Inverse[newMatrixRepresentation][[First[index],#1]]*
extrinsicCurvatureTensor[[#1,Last[index]]] & ) /@ Range[Length[newMatrixRepresentation]]]] & ) /@
Tuples[Range[Length[newMatrixRepresentation]], 2]]]; extrinsicCurvatureTrace =
Total[(Inverse[newMatrixRepresentation][[First[#1],Last[#1]]]*extrinsicCurvatureTensor[[First[#1],Last[#1]]] & ) /@
Tuples[Range[Length[newMatrixRepresentation]], 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[newMatrixRepresentation]]] -
Total[(spatialChristoffelSymbols[[index[[1]],#1,index[[4]]]]*spatialChristoffelSymbols[[#1,index[[2]],
index[[3]]]] & ) /@ Range[Length[newMatrixRepresentation]]]] & ) /@
Tuples[Range[Length[newMatrixRepresentation]], 4]]]; spatialRicciTensor =
Normal[SparseArray[(Module[{index = #1}, index -> Total[(spatialRiemannTensor[[#1,First[index],#1,
Last[index]]] & ) /@ Range[Length[newMatrixRepresentation]]]] & ) /@
Tuples[Range[Length[newMatrixRepresentation]], 2]]]; mixedSpatialRicciTensor =
Normal[SparseArray[(Module[{index = #1}, index -> Total[(Inverse[newMatrixRepresentation][[First[index],#1]]*
spatialRicciTensor[[#1,Last[index]]] & ) /@ Range[Length[newMatrixRepresentation]]]] & ) /@
Tuples[Range[Length[newMatrixRepresentation]], 2]]]; spacetimeMetricTensor =
Normal[SparseArray[Join[{{1, 1} -> Total[(newShiftVector[[#1]]*shiftCovector[[#1]] & ) /@
Range[Length[newMatrixRepresentation]]] - newLapseFunction^2},
(Module[{index = #1}, {1, index + 1} -> Total[(newMatrixRepresentation[[index,#1]]*newShiftVector[[#1]] & ) /@
Range[Length[newMatrixRepresentation]]]] & ) /@ Range[Length[newMatrixRepresentation]],
(Module[{index = #1}, {index + 1, 1} -> Total[(newMatrixRepresentation[[#1,index]]*newShiftVector[[#1]] & ) /@
Range[Length[newMatrixRepresentation]]]] & ) /@ Range[Length[newMatrixRepresentation]],
({First[#1] + 1, Last[#1] + 1} -> newMatrixRepresentation[[First[#1],Last[#1]]] & ) /@
Tuples[Range[Length[newMatrixRepresentation]], 2]]]];
leftHandSide = Normal[SparseArray[(Module[{index = #1}, index -> D[mixedExtrinsicCurvatureTensor[[First[index],
Last[index]]], newTimeCoordinate]] & ) /@ Tuples[Range[Length[newMatrixRepresentation]], 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[
newMatrixRepresentation]]])*Inverse[newMatrixRepresentation][[nestedIndex,First[index]]]] & ) /@
Range[Length[newMatrixRepresentation]]] + 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[newMatrixRepresentation]]] -
Total[(spatialChristoffelSymbols[[#1,nestedIndex,Last[index]]]*mixedExtrinsicCurvatureTensor[[
First[index],#1]] & ) /@ Range[Length[newMatrixRepresentation]]])] & ) /@ Range[
Length[newMatrixRepresentation]]] + Total[(Module[{nestedIndex = #1}, mixedExtrinsicCurvatureTensor[[
First[index],nestedIndex]]*(D[newShiftVector[[nestedIndex]], newCoordinates[[Last[index]]]] +
Total[(spatialChristoffelSymbols[[nestedIndex,Last[index],#1]]*newShiftVector[[#1]] & ) /@
Range[Length[newMatrixRepresentation]]])] & ) /@ Range[Length[newMatrixRepresentation]]] -
Total[(Module[{nestedIndex = #1}, mixedExtrinsicCurvatureTensor[[nestedIndex,Last[index]]]*
(D[newShiftVector[[First[index]]], newCoordinates[[nestedIndex]]] + Total[
(spatialChristoffelSymbols[[First[index],nestedIndex,#1]]*newShiftVector[[#1]] & ) /@
Range[Length[newMatrixRepresentation]]])] & ) /@ Range[Length[newMatrixRepresentation]]] -
newLapseFunction*Total[(((2*cosmologicalConstant)/(Length[newMatrixRepresentation] - 1))*
spacetimeMetricTensor[[#1 + 1,Last[index] + 1]]*Inverse[newMatrixRepresentation][[#1,
First[index]]] & ) /@ Range[Length[newMatrixRepresentation]]]] & ) /@
Tuples[Range[Length[newMatrixRepresentation]], 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[admDecomposition] === "ADMDecomposition" && SymbolName[metricTensor] === "MetricTensor" &&
Length[Dimensions[matrixRepresentation]] == 2 && Length[coordinates] == Length[matrixRepresentation] &&
BooleanQ[index1] && BooleanQ[index2] && Length[shiftVector] == Length[matrixRepresentation]
VacuumADMSolution[(admDecomposition_)[(metricTensor_)[matrixRepresentation_List, coordinates_List, index1_, index2_],
timeCoordinate_, lapseFunction_, shiftVector_List], cosmologicalConstant_]["ExactSolutionQ"] :=
Module[{newMatrixRepresentation, newCoordinates, newTimeCoordinate, newLapseFunction, newShiftVector, shiftCovector,
spatialChristoffelSymbols, extrinsicCurvatureTensor, mixedExtrinsicCurvatureTensor, extrinsicCurvatureTrace,
spatialRiemannTensor, spatialRicciTensor, mixedSpatialRicciTensor, spacetimeMetricTensor, leftHandSide,
rightHandSide, evolutionEquations}, 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[(newMatrixRepresentation[[index,#1]]*newShiftVector[[#1]] & ) /@
Range[Length[newMatrixRepresentation]]]] & ) /@ Range[Length[newMatrixRepresentation]]]];
spatialChristoffelSymbols = Normal[SparseArray[
(Module[{index = #1}, index -> Total[((1/2)*Inverse[newMatrixRepresentation][[index[[1]],#1]]*
(D[newMatrixRepresentation[[#1,index[[3]]]], newCoordinates[[index[[2]]]]] + D[newMatrixRepresentation[[
index[[2]],#1]], newCoordinates[[index[[3]]]]] - D[newMatrixRepresentation[[index[[2]],index[[3]]]],
newCoordinates[[#1]]]) & ) /@ Range[Length[newMatrixRepresentation]]]] & ) /@
Tuples[Range[Length[newMatrixRepresentation]], 3]]]; 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[newMatrixRepresentation]]] + D[shiftCovector[[Last[index]]],
newCoordinates[[First[index]]]] - Total[(spatialChristoffelSymbols[[#1,First[index],Last[index]]]*
shiftCovector[[#1]] & ) /@ Range[Length[newMatrixRepresentation]]] - D[newMatrixRepresentation[[
First[index],Last[index]]], newTimeCoordinate])] & ) /@ Tuples[Range[Length[newMatrixRepresentation]],
2]]]; mixedExtrinsicCurvatureTensor =
Normal[SparseArray[(Module[{index = #1}, index -> Total[(Inverse[newMatrixRepresentation][[First[index],#1]]*
extrinsicCurvatureTensor[[#1,Last[index]]] & ) /@ Range[Length[newMatrixRepresentation]]]] & ) /@
Tuples[Range[Length[newMatrixRepresentation]], 2]]]; extrinsicCurvatureTrace =
Total[(Inverse[newMatrixRepresentation][[First[#1],Last[#1]]]*extrinsicCurvatureTensor[[First[#1],Last[#1]]] & ) /@
Tuples[Range[Length[newMatrixRepresentation]], 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[newMatrixRepresentation]]] -
Total[(spatialChristoffelSymbols[[index[[1]],#1,index[[4]]]]*spatialChristoffelSymbols[[#1,index[[2]],
index[[3]]]] & ) /@ Range[Length[newMatrixRepresentation]]]] & ) /@
Tuples[Range[Length[newMatrixRepresentation]], 4]]]; spatialRicciTensor =
Normal[SparseArray[(Module[{index = #1}, index -> Total[(spatialRiemannTensor[[#1,First[index],#1,
Last[index]]] & ) /@ Range[Length[newMatrixRepresentation]]]] & ) /@
Tuples[Range[Length[newMatrixRepresentation]], 2]]]; mixedSpatialRicciTensor =
Normal[SparseArray[(Module[{index = #1}, index -> Total[(Inverse[newMatrixRepresentation][[First[index],#1]]*
spatialRicciTensor[[#1,Last[index]]] & ) /@ Range[Length[newMatrixRepresentation]]]] & ) /@
Tuples[Range[Length[newMatrixRepresentation]], 2]]]; spacetimeMetricTensor =
Normal[SparseArray[Join[{{1, 1} -> Total[(newShiftVector[[#1]]*shiftCovector[[#1]] & ) /@
Range[Length[newMatrixRepresentation]]] - newLapseFunction^2},
(Module[{index = #1}, {1, index + 1} -> Total[(newMatrixRepresentation[[index,#1]]*newShiftVector[[#1]] & ) /@
Range[Length[newMatrixRepresentation]]]] & ) /@ Range[Length[newMatrixRepresentation]],
(Module[{index = #1}, {index + 1, 1} -> Total[(newMatrixRepresentation[[#1,index]]*newShiftVector[[#1]] & ) /@
Range[Length[newMatrixRepresentation]]]] & ) /@ Range[Length[newMatrixRepresentation]],
({First[#1] + 1, Last[#1] + 1} -> newMatrixRepresentation[[First[#1],Last[#1]]] & ) /@
Tuples[Range[Length[newMatrixRepresentation]], 2]]]];
leftHandSide = Normal[SparseArray[(Module[{index = #1}, index -> D[mixedExtrinsicCurvatureTensor[[First[index],
Last[index]]], newTimeCoordinate]] & ) /@ Tuples[Range[Length[newMatrixRepresentation]], 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[
newMatrixRepresentation]]])*Inverse[newMatrixRepresentation][[nestedIndex,First[index]]]] & ) /@
Range[Length[newMatrixRepresentation]]] + 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[newMatrixRepresentation]]] -
Total[(spatialChristoffelSymbols[[#1,nestedIndex,Last[index]]]*mixedExtrinsicCurvatureTensor[[
First[index],#1]] & ) /@ Range[Length[newMatrixRepresentation]]])] & ) /@ Range[
Length[newMatrixRepresentation]]] + Total[(Module[{nestedIndex = #1}, mixedExtrinsicCurvatureTensor[[
First[index],nestedIndex]]*(D[newShiftVector[[nestedIndex]], newCoordinates[[Last[index]]]] +
Total[(spatialChristoffelSymbols[[nestedIndex,Last[index],#1]]*newShiftVector[[#1]] & ) /@
Range[Length[newMatrixRepresentation]]])] & ) /@ Range[Length[newMatrixRepresentation]]] -
Total[(Module[{nestedIndex = #1}, mixedExtrinsicCurvatureTensor[[nestedIndex,Last[index]]]*
(D[newShiftVector[[First[index]]], newCoordinates[[nestedIndex]]] + Total[
(spatialChristoffelSymbols[[First[index],nestedIndex,#1]]*newShiftVector[[#1]] & ) /@
Range[Length[newMatrixRepresentation]]])] & ) /@ Range[Length[newMatrixRepresentation]]] -
newLapseFunction*Total[(((2*cosmologicalConstant)/(Length[newMatrixRepresentation] - 1))*
spacetimeMetricTensor[[#1 + 1,Last[index] + 1]]*Inverse[newMatrixRepresentation][[#1,
First[index]]] & ) /@ Range[Length[newMatrixRepresentation]]]] & ) /@
Tuples[Range[Length[newMatrixRepresentation]], 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[admDecomposition] === "ADMDecomposition" &&
SymbolName[metricTensor] === "MetricTensor" && Length[Dimensions[matrixRepresentation]] == 2 &&
Length[coordinates] == Length[matrixRepresentation] && BooleanQ[index1] && BooleanQ[index2] &&
Length[shiftVector] == Length[matrixRepresentation]
VacuumADMSolution[(admDecomposition_)[(metricTensor_)[matrixRepresentation_List, coordinates_List, index1_, index2_],
timeCoordinate_, lapseFunction_, shiftVector_List], cosmologicalConstant_]["FieldEquations"] :=
Module[{newMatrixRepresentation, newCoordinates, newTimeCoordinate, newLapseFunction, newShiftVector, shiftCovector,
spatialChristoffelSymbols, extrinsicCurvatureTensor, mixedExtrinsicCurvatureTensor, extrinsicCurvatureTrace,
spatialRiemannTensor, spatialRicciTensor, mixedSpatialRicciTensor, spacetimeMetricTensor, leftHandSide,
rightHandSide, evolutionEquations}, 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[(newMatrixRepresentation[[index,#1]]*newShiftVector[[#1]] & ) /@
Range[Length[newMatrixRepresentation]]]] & ) /@ Range[Length[newMatrixRepresentation]]]];
spatialChristoffelSymbols = Normal[SparseArray[
(Module[{index = #1}, index -> Total[((1/2)*Inverse[newMatrixRepresentation][[index[[1]],#1]]*
(D[newMatrixRepresentation[[#1,index[[3]]]], newCoordinates[[index[[2]]]]] + D[newMatrixRepresentation[[
index[[2]],#1]], newCoordinates[[index[[3]]]]] - D[newMatrixRepresentation[[index[[2]],index[[3]]]],
newCoordinates[[#1]]]) & ) /@ Range[Length[newMatrixRepresentation]]]] & ) /@
Tuples[Range[Length[newMatrixRepresentation]], 3]]]; 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[newMatrixRepresentation]]] + D[shiftCovector[[Last[index]]],
newCoordinates[[First[index]]]] - Total[(spatialChristoffelSymbols[[#1,First[index],Last[index]]]*
shiftCovector[[#1]] & ) /@ Range[Length[newMatrixRepresentation]]] - D[newMatrixRepresentation[[
First[index],Last[index]]], newTimeCoordinate])] & ) /@ Tuples[Range[Length[newMatrixRepresentation]],
2]]]; mixedExtrinsicCurvatureTensor =
Normal[SparseArray[(Module[{index = #1}, index -> Total[(Inverse[newMatrixRepresentation][[First[index],#1]]*
extrinsicCurvatureTensor[[#1,Last[index]]] & ) /@ Range[Length[newMatrixRepresentation]]]] & ) /@
Tuples[Range[Length[newMatrixRepresentation]], 2]]]; extrinsicCurvatureTrace =
Total[(Inverse[newMatrixRepresentation][[First[#1],Last[#1]]]*extrinsicCurvatureTensor[[First[#1],Last[#1]]] & ) /@
Tuples[Range[Length[newMatrixRepresentation]], 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[newMatrixRepresentation]]] -
Total[(spatialChristoffelSymbols[[index[[1]],#1,index[[4]]]]*spatialChristoffelSymbols[[#1,index[[2]],
index[[3]]]] & ) /@ Range[Length[newMatrixRepresentation]]]] & ) /@
Tuples[Range[Length[newMatrixRepresentation]], 4]]]; spatialRicciTensor =
Normal[SparseArray[(Module[{index = #1}, index -> Total[(spatialRiemannTensor[[#1,First[index],#1,
Last[index]]] & ) /@ Range[Length[newMatrixRepresentation]]]] & ) /@
Tuples[Range[Length[newMatrixRepresentation]], 2]]]; mixedSpatialRicciTensor =
Normal[SparseArray[(Module[{index = #1}, index -> Total[(Inverse[newMatrixRepresentation][[First[index],#1]]*
spatialRicciTensor[[#1,Last[index]]] & ) /@ Range[Length[newMatrixRepresentation]]]] & ) /@
Tuples[Range[Length[newMatrixRepresentation]], 2]]]; spacetimeMetricTensor =
Normal[SparseArray[Join[{{1, 1} -> Total[(newShiftVector[[#1]]*shiftCovector[[#1]] & ) /@
Range[Length[newMatrixRepresentation]]] - newLapseFunction^2},
(Module[{index = #1}, {1, index + 1} -> Total[(newMatrixRepresentation[[index,#1]]*newShiftVector[[#1]] & ) /@
Range[Length[newMatrixRepresentation]]]] & ) /@ Range[Length[newMatrixRepresentation]],
(Module[{index = #1}, {index + 1, 1} -> Total[(newMatrixRepresentation[[#1,index]]*newShiftVector[[#1]] & ) /@
Range[Length[newMatrixRepresentation]]]] & ) /@ Range[Length[newMatrixRepresentation]],
({First[#1] + 1, Last[#1] + 1} -> newMatrixRepresentation[[First[#1],Last[#1]]] & ) /@
Tuples[Range[Length[newMatrixRepresentation]], 2]]]];
leftHandSide = Normal[SparseArray[(Module[{index = #1}, index -> D[mixedExtrinsicCurvatureTensor[[First[index],
Last[index]]], newTimeCoordinate]] & ) /@ Tuples[Range[Length[newMatrixRepresentation]], 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[
newMatrixRepresentation]]])*Inverse[newMatrixRepresentation][[nestedIndex,First[index]]]] & ) /@
Range[Length[newMatrixRepresentation]]] + 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[newMatrixRepresentation]]] -
Total[(spatialChristoffelSymbols[[#1,nestedIndex,Last[index]]]*mixedExtrinsicCurvatureTensor[[
First[index],#1]] & ) /@ Range[Length[newMatrixRepresentation]]])] & ) /@ Range[
Length[newMatrixRepresentation]]] + Total[(Module[{nestedIndex = #1}, mixedExtrinsicCurvatureTensor[[
First[index],nestedIndex]]*(D[newShiftVector[[nestedIndex]], newCoordinates[[Last[index]]]] +
Total[(spatialChristoffelSymbols[[nestedIndex,Last[index],#1]]*newShiftVector[[#1]] & ) /@
Range[Length[newMatrixRepresentation]]])] & ) /@ Range[Length[newMatrixRepresentation]]] -
Total[(Module[{nestedIndex = #1}, mixedExtrinsicCurvatureTensor[[nestedIndex,Last[index]]]*
(D[newShiftVector[[First[index]]], newCoordinates[[nestedIndex]]] + Total[
(spatialChristoffelSymbols[[First[index],nestedIndex,#1]]*newShiftVector[[#1]] & ) /@
Range[Length[newMatrixRepresentation]]])] & ) /@ Range[Length[newMatrixRepresentation]]] -
newLapseFunction*Total[(((2*cosmologicalConstant)/(Length[newMatrixRepresentation] - 1))*
spacetimeMetricTensor[[#1 + 1,Last[index] + 1]]*Inverse[newMatrixRepresentation][[#1,
First[index]]] & ) /@ Range[Length[newMatrixRepresentation]]]] & ) /@
Tuples[Range[Length[newMatrixRepresentation]], 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[admDecomposition] === "ADMDecomposition" && SymbolName[metricTensor] === "MetricTensor" &&
Length[Dimensions[matrixRepresentation]] == 2 && Length[coordinates] == Length[matrixRepresentation] &&
BooleanQ[index1] && BooleanQ[index2] && Length[shiftVector] == Length[matrixRepresentation]
VacuumADMSolution[(admDecomposition_)[(metricTensor_)[matrixRepresentation_List, coordinates_List, index1_, index2_],
timeCoordinate_, lapseFunction_, shiftVector_List], cosmologicalConstant_]["EvolutionEquations"] :=
Module[{newMatrixRepresentation, newCoordinates, newTimeCoordinate, newLapseFunction, newShiftVector, shiftCovector,
spatialChristoffelSymbols, extrinsicCurvatureTensor, mixedExtrinsicCurvatureTensor, extrinsicCurvatureTrace,
spatialRiemannTensor, spatialRicciTensor, mixedSpatialRicciTensor, spacetimeMetricTensor, leftHandSide,
rightHandSide}, 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[(newMatrixRepresentation[[index,#1]]*newShiftVector[[#1]] & ) /@
Range[Length[newMatrixRepresentation]]]] & ) /@ Range[Length[newMatrixRepresentation]]]];
spatialChristoffelSymbols = Normal[SparseArray[
(Module[{index = #1}, index -> Total[((1/2)*Inverse[newMatrixRepresentation][[index[[1]],#1]]*
(D[newMatrixRepresentation[[#1,index[[3]]]], newCoordinates[[index[[2]]]]] + D[newMatrixRepresentation[[
index[[2]],#1]], newCoordinates[[index[[3]]]]] - D[newMatrixRepresentation[[index[[2]],index[[3]]]],
newCoordinates[[#1]]]) & ) /@ Range[Length[newMatrixRepresentation]]]] & ) /@
Tuples[Range[Length[newMatrixRepresentation]], 3]]]; 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[newMatrixRepresentation]]] + D[shiftCovector[[Last[index]]],
newCoordinates[[First[index]]]] - Total[(spatialChristoffelSymbols[[#1,First[index],Last[index]]]*
shiftCovector[[#1]] & ) /@ Range[Length[newMatrixRepresentation]]] - D[newMatrixRepresentation[[
First[index],Last[index]]], newTimeCoordinate])] & ) /@ Tuples[Range[Length[newMatrixRepresentation]],
2]]]; mixedExtrinsicCurvatureTensor =
Normal[SparseArray[(Module[{index = #1}, index -> Total[(Inverse[newMatrixRepresentation][[First[index],#1]]*
extrinsicCurvatureTensor[[#1,Last[index]]] & ) /@ Range[Length[newMatrixRepresentation]]]] & ) /@
Tuples[Range[Length[newMatrixRepresentation]], 2]]]; extrinsicCurvatureTrace =
Total[(Inverse[newMatrixRepresentation][[First[#1],Last[#1]]]*extrinsicCurvatureTensor[[First[#1],Last[#1]]] & ) /@
Tuples[Range[Length[newMatrixRepresentation]], 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[newMatrixRepresentation]]] -
Total[(spatialChristoffelSymbols[[index[[1]],#1,index[[4]]]]*spatialChristoffelSymbols[[#1,index[[2]],
index[[3]]]] & ) /@ Range[Length[newMatrixRepresentation]]]] & ) /@
Tuples[Range[Length[newMatrixRepresentation]], 4]]]; spatialRicciTensor =
Normal[SparseArray[(Module[{index = #1}, index -> Total[(spatialRiemannTensor[[#1,First[index],#1,
Last[index]]] & ) /@ Range[Length[newMatrixRepresentation]]]] & ) /@
Tuples[Range[Length[newMatrixRepresentation]], 2]]]; mixedSpatialRicciTensor =
Normal[SparseArray[(Module[{index = #1}, index -> Total[(Inverse[newMatrixRepresentation][[First[index],#1]]*
spatialRicciTensor[[#1,Last[index]]] & ) /@ Range[Length[newMatrixRepresentation]]]] & ) /@
Tuples[Range[Length[newMatrixRepresentation]], 2]]]; spacetimeMetricTensor =
Normal[SparseArray[Join[{{1, 1} -> Total[(newShiftVector[[#1]]*shiftCovector[[#1]] & ) /@
Range[Length[newMatrixRepresentation]]] - newLapseFunction^2},
(Module[{index = #1}, {1, index + 1} -> Total[(newMatrixRepresentation[[index,#1]]*newShiftVector[[#1]] & ) /@
Range[Length[newMatrixRepresentation]]]] & ) /@ Range[Length[newMatrixRepresentation]],
(Module[{index = #1}, {index + 1, 1} -> Total[(newMatrixRepresentation[[#1,index]]*newShiftVector[[#1]] & ) /@
Range[Length[newMatrixRepresentation]]]] & ) /@ Range[Length[newMatrixRepresentation]],
({First[#1] + 1, Last[#1] + 1} -> newMatrixRepresentation[[First[#1],Last[#1]]] & ) /@
Tuples[Range[Length[newMatrixRepresentation]], 2]]]];
leftHandSide = Normal[SparseArray[(Module[{index = #1}, index -> D[mixedExtrinsicCurvatureTensor[[First[index],
Last[index]]], newTimeCoordinate]] & ) /@ Tuples[Range[Length[newMatrixRepresentation]], 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[
newMatrixRepresentation]]])*Inverse[newMatrixRepresentation][[nestedIndex,First[index]]]] & ) /@
Range[Length[newMatrixRepresentation]]] + 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[newMatrixRepresentation]]] -
Total[(spatialChristoffelSymbols[[#1,nestedIndex,Last[index]]]*mixedExtrinsicCurvatureTensor[[
First[index],#1]] & ) /@ Range[Length[newMatrixRepresentation]]])] & ) /@ Range[
Length[newMatrixRepresentation]]] + Total[(Module[{nestedIndex = #1}, mixedExtrinsicCurvatureTensor[[
First[index],nestedIndex]]*(D[newShiftVector[[nestedIndex]], newCoordinates[[Last[index]]]] +
Total[(spatialChristoffelSymbols[[nestedIndex,Last[index],#1]]*newShiftVector[[#1]] & ) /@
Range[Length[newMatrixRepresentation]]])] & ) /@ Range[Length[newMatrixRepresentation]]] -
Total[(Module[{nestedIndex = #1}, mixedExtrinsicCurvatureTensor[[nestedIndex,Last[index]]]*
(D[newShiftVector[[First[index]]], newCoordinates[[nestedIndex]]] + Total[
(spatialChristoffelSymbols[[First[index],nestedIndex,#1]]*newShiftVector[[#1]] & ) /@
Range[Length[newMatrixRepresentation]]])] & ) /@ Range[Length[newMatrixRepresentation]]] -
newLapseFunction*Total[(((2*cosmologicalConstant)/(Length[newMatrixRepresentation] - 1))*
spacetimeMetricTensor[[#1 + 1,Last[index] + 1]]*Inverse[newMatrixRepresentation][[#1,
First[index]]] & ) /@ Range[Length[newMatrixRepresentation]]]] & ) /@
Tuples[Range[Length[newMatrixRepresentation]], 2]]];
Thread[Catenate[leftHandSide] == Catenate[rightHandSide]] /. (ToExpression[#1] -> #1 & ) /@
Select[Join[coordinates, {timeCoordinate}], StringQ]] /; SymbolName[admDecomposition] === "ADMDecomposition" &&
SymbolName[metricTensor] === "MetricTensor" && Length[Dimensions[matrixRepresentation]] == 2 &&
Length[coordinates] == Length[matrixRepresentation] && BooleanQ[index1] && BooleanQ[index2] &&
Length[shiftVector] == Length[matrixRepresentation]
VacuumADMSolution[(admDecomposition_)[(metricTensor_)[matrixRepresentation_List, coordinates_List, index1_, index2_],
timeCoordinate_, lapseFunction_, shiftVector_List], cosmologicalConstant_]["ReducedEvolutionEquations"] :=
Module[{newMatrixRepresentation, newCoordinates, newTimeCoordinate, newLapseFunction, newShiftVector, shiftCovector,
spatialChristoffelSymbols, extrinsicCurvatureTensor, mixedExtrinsicCurvatureTensor, extrinsicCurvatureTrace,
spatialRiemannTensor, spatialRicciTensor, mixedSpatialRicciTensor, spacetimeMetricTensor, leftHandSide,
rightHandSide}, 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[(newMatrixRepresentation[[index,#1]]*newShiftVector[[#1]] & ) /@
Range[Length[newMatrixRepresentation]]]] & ) /@ Range[Length[newMatrixRepresentation]]]];
spatialChristoffelSymbols = Normal[SparseArray[
(Module[{index = #1}, index -> Total[((1/2)*Inverse[newMatrixRepresentation][[index[[1]],#1]]*
(D[newMatrixRepresentation[[#1,index[[3]]]], newCoordinates[[index[[2]]]]] + D[newMatrixRepresentation[[
index[[2]],#1]], newCoordinates[[index[[3]]]]] - D[newMatrixRepresentation[[index[[2]],index[[3]]]],
newCoordinates[[#1]]]) & ) /@ Range[Length[newMatrixRepresentation]]]] & ) /@
Tuples[Range[Length[newMatrixRepresentation]], 3]]]; 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[newMatrixRepresentation]]] + D[shiftCovector[[Last[index]]],
newCoordinates[[First[index]]]] - Total[(spatialChristoffelSymbols[[#1,First[index],Last[index]]]*
shiftCovector[[#1]] & ) /@ Range[Length[newMatrixRepresentation]]] - D[newMatrixRepresentation[[
First[index],Last[index]]], newTimeCoordinate])] & ) /@ Tuples[Range[Length[newMatrixRepresentation]],
2]]]; mixedExtrinsicCurvatureTensor =
Normal[SparseArray[(Module[{index = #1}, index -> Total[(Inverse[newMatrixRepresentation][[First[index],#1]]*
extrinsicCurvatureTensor[[#1,Last[index]]] & ) /@ Range[Length[newMatrixRepresentation]]]] & ) /@
Tuples[Range[Length[newMatrixRepresentation]], 2]]]; extrinsicCurvatureTrace =
Total[(Inverse[newMatrixRepresentation][[First[#1],Last[#1]]]*extrinsicCurvatureTensor[[First[#1],Last[#1]]] & ) /@
Tuples[Range[Length[newMatrixRepresentation]], 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[newMatrixRepresentation]]] -
Total[(spatialChristoffelSymbols[[index[[1]],#1,index[[4]]]]*spatialChristoffelSymbols[[#1,index[[2]],
index[[3]]]] & ) /@ Range[Length[newMatrixRepresentation]]]] & ) /@
Tuples[Range[Length[newMatrixRepresentation]], 4]]]; spatialRicciTensor =
Normal[SparseArray[(Module[{index = #1}, index -> Total[(spatialRiemannTensor[[#1,First[index],#1,
Last[index]]] & ) /@ Range[Length[newMatrixRepresentation]]]] & ) /@
Tuples[Range[Length[newMatrixRepresentation]], 2]]]; mixedSpatialRicciTensor =
Normal[SparseArray[(Module[{index = #1}, index -> Total[(Inverse[newMatrixRepresentation][[First[index],#1]]*
spatialRicciTensor[[#1,Last[index]]] & ) /@ Range[Length[newMatrixRepresentation]]]] & ) /@
Tuples[Range[Length[newMatrixRepresentation]], 2]]]; spacetimeMetricTensor =
Normal[SparseArray[Join[{{1, 1} -> Total[(newShiftVector[[#1]]*shiftCovector[[#1]] & ) /@
Range[Length[newMatrixRepresentation]]] - newLapseFunction^2},
(Module[{index = #1}, {1, index + 1} -> Total[(newMatrixRepresentation[[index,#1]]*newShiftVector[[#1]] & ) /@
Range[Length[newMatrixRepresentation]]]] & ) /@ Range[Length[newMatrixRepresentation]],
(Module[{index = #1}, {index + 1, 1} -> Total[(newMatrixRepresentation[[#1,index]]*newShiftVector[[#1]] & ) /@
Range[Length[newMatrixRepresentation]]]] & ) /@ Range[Length[newMatrixRepresentation]],
({First[#1] + 1, Last[#1] + 1} -> newMatrixRepresentation[[First[#1],Last[#1]]] & ) /@
Tuples[Range[Length[newMatrixRepresentation]], 2]]]];
leftHandSide = Normal[SparseArray[(Module[{index = #1}, index -> D[mixedExtrinsicCurvatureTensor[[First[index],
Last[index]]], newTimeCoordinate]] & ) /@ Tuples[Range[Length[newMatrixRepresentation]], 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[
newMatrixRepresentation]]])*Inverse[newMatrixRepresentation][[nestedIndex,First[index]]]] & ) /@
Range[Length[newMatrixRepresentation]]] + 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[newMatrixRepresentation]]] -
Total[(spatialChristoffelSymbols[[#1,nestedIndex,Last[index]]]*mixedExtrinsicCurvatureTensor[[
First[index],#1]] & ) /@ Range[Length[newMatrixRepresentation]]])] & ) /@ Range[
Length[newMatrixRepresentation]]] + Total[(Module[{nestedIndex = #1}, mixedExtrinsicCurvatureTensor[[
First[index],nestedIndex]]*(D[newShiftVector[[nestedIndex]], newCoordinates[[Last[index]]]] +
Total[(spatialChristoffelSymbols[[nestedIndex,Last[index],#1]]*newShiftVector[[#1]] & ) /@
Range[Length[newMatrixRepresentation]]])] & ) /@ Range[Length[newMatrixRepresentation]]] -
Total[(Module[{nestedIndex = #1}, mixedExtrinsicCurvatureTensor[[nestedIndex,Last[index]]]*
(D[newShiftVector[[First[index]]], newCoordinates[[nestedIndex]]] + Total[
(spatialChristoffelSymbols[[First[index],nestedIndex,#1]]*newShiftVector[[#1]] & ) /@
Range[Length[newMatrixRepresentation]]])] & ) /@ Range[Length[newMatrixRepresentation]]] -
newLapseFunction*Total[(((2*cosmologicalConstant)/(Length[newMatrixRepresentation] - 1))*
spacetimeMetricTensor[[#1 + 1,Last[index] + 1]]*Inverse[newMatrixRepresentation][[#1,
First[index]]] & ) /@ Range[Length[newMatrixRepresentation]]]] & ) /@
Tuples[Range[Length[newMatrixRepresentation]], 2]]];
FullSimplify[Thread[Catenate[leftHandSide] == Catenate[rightHandSide]] /. (ToExpression[#1] -> #1 & ) /@
Select[Join[coordinates, {timeCoordinate}], StringQ]]] /; SymbolName[admDecomposition] === "ADMDecomposition" &&
SymbolName[metricTensor] === "MetricTensor" && Length[Dimensions[matrixRepresentation]] == 2 &&
Length[coordinates] == Length[matrixRepresentation] && BooleanQ[index1] && BooleanQ[index2] &&
Length[shiftVector] == Length[matrixRepresentation]
VacuumADMSolution[(admDecomposition_)[(metricTensor_)[matrixRepresentation_List, coordinates_List, index1_, index2_],
timeCoordinate_, lapseFunction_, shiftVector_List], cosmologicalConstant_]["SymbolicEvolutionEquations"] :=
Module[{newMatrixRepresentation, newCoordinates, newTimeCoordinate, newLapseFunction, newShiftVector, shiftCovector,
spatialChristoffelSymbols, extrinsicCurvatureTensor, mixedExtrinsicCurvatureTensor, extrinsicCurvatureTrace,
spatialRiemannTensor, spatialRicciTensor, mixedSpatialRicciTensor, spacetimeMetricTensor, leftHandSide,
rightHandSide}, 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[(newMatrixRepresentation[[index,#1]]*newShiftVector[[#1]] & ) /@
Range[Length[newMatrixRepresentation]]]] & ) /@ Range[Length[newMatrixRepresentation]]]];
spatialChristoffelSymbols = Normal[SparseArray[
(Module[{index = #1}, index -> Total[((1/2)*Inverse[newMatrixRepresentation][[index[[1]],#1]]*
(Inactive[D][newMatrixRepresentation[[#1,index[[3]]]], newCoordinates[[index[[2]]]]] +
Inactive[D][newMatrixRepresentation[[index[[2]],#1]], newCoordinates[[index[[3]]]]] -
Inactive[D][newMatrixRepresentation[[index[[2]],index[[3]]]], newCoordinates[[#1]]]) & ) /@
Range[Length[newMatrixRepresentation]]]] & ) /@ Tuples[Range[Length[newMatrixRepresentation]], 3]]];
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[newMatrixRepresentation]]] + Inactive[D][shiftCovector[[
Last[index]]], newCoordinates[[First[index]]]] - Total[(spatialChristoffelSymbols[[#1,First[index],
Last[index]]]*shiftCovector[[#1]] & ) /@ Range[Length[newMatrixRepresentation]]] -
Inactive[D][newMatrixRepresentation[[First[index],Last[index]]], newTimeCoordinate])] & ) /@
Tuples[Range[Length[newMatrixRepresentation]], 2]]]; mixedExtrinsicCurvatureTensor =
Normal[SparseArray[(Module[{index = #1}, index -> Total[(Inverse[newMatrixRepresentation][[First[index],#1]]*
extrinsicCurvatureTensor[[#1,Last[index]]] & ) /@ Range[Length[newMatrixRepresentation]]]] & ) /@
Tuples[Range[Length[newMatrixRepresentation]], 2]]]; extrinsicCurvatureTrace =
Total[(Inverse[newMatrixRepresentation][[First[#1],Last[#1]]]*extrinsicCurvatureTensor[[First[#1],Last[#1]]] & ) /@
Tuples[Range[Length[newMatrixRepresentation]], 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[newMatrixRepresentation]]] - Total[(spatialChristoffelSymbols[[index[[1]],#1,index[[4]]]]*
spatialChristoffelSymbols[[#1,index[[2]],index[[3]]]] & ) /@ Range[Length[
newMatrixRepresentation]]]] & ) /@ Tuples[Range[Length[newMatrixRepresentation]], 4]]];
spatialRicciTensor = Normal[SparseArray[
(Module[{index = #1}, index -> Total[(spatialRiemannTensor[[#1,First[index],#1,Last[index]]] & ) /@
Range[Length[newMatrixRepresentation]]]] & ) /@ Tuples[Range[Length[newMatrixRepresentation]], 2]]];
mixedSpatialRicciTensor = Normal[SparseArray[
(Module[{index = #1}, index -> Total[(Inverse[newMatrixRepresentation][[First[index],#1]]*spatialRicciTensor[[#1,
Last[index]]] & ) /@ Range[Length[newMatrixRepresentation]]]] & ) /@
Tuples[Range[Length[newMatrixRepresentation]], 2]]]; spacetimeMetricTensor =
Normal[SparseArray[Join[{{1, 1} -> Total[(newShiftVector[[#1]]*shiftCovector[[#1]] & ) /@
Range[Length[newMatrixRepresentation]]] - newLapseFunction^2},
(Module[{index = #1}, {1, index + 1} -> Total[(newMatrixRepresentation[[index,#1]]*newShiftVector[[#1]] & ) /@
Range[Length[newMatrixRepresentation]]]] & ) /@ Range[Length[newMatrixRepresentation]],
(Module[{index = #1}, {index + 1, 1} -> Total[(newMatrixRepresentation[[#1,index]]*newShiftVector[[#1]] & ) /@
Range[Length[newMatrixRepresentation]]]] & ) /@ Range[Length[newMatrixRepresentation]],
({First[#1] + 1, Last[#1] + 1} -> newMatrixRepresentation[[First[#1],Last[#1]]] & ) /@
Tuples[Range[Length[newMatrixRepresentation]], 2]]]];
leftHandSide = Normal[SparseArray[(Module[{index = #1}, index -> Inactive[D][mixedExtrinsicCurvatureTensor[[
First[index],Last[index]]], newTimeCoordinate]] & ) /@ Tuples[Range[Length[newMatrixRepresentation]], 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[newMatrixRepresentation]]])*
Inverse[newMatrixRepresentation][[nestedIndex,First[index]]]] & ) /@ Range[
Length[newMatrixRepresentation]]] + 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[newMatrixRepresentation]]] -
Total[(spatialChristoffelSymbols[[#1,nestedIndex,Last[index]]]*mixedExtrinsicCurvatureTensor[[
First[index],#1]] & ) /@ Range[Length[newMatrixRepresentation]]])] & ) /@ Range[
Length[newMatrixRepresentation]]] + 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[newMatrixRepresentation]]])] & ) /@ Range[
Length[newMatrixRepresentation]]] - 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[newMatrixRepresentation]]])] & ) /@ Range[
Length[newMatrixRepresentation]]] - newLapseFunction*Total[(((2*cosmologicalConstant)/
(Length[newMatrixRepresentation] - 1))*spacetimeMetricTensor[[#1 + 1,Last[index] + 1]]*
Inverse[newMatrixRepresentation][[#1,First[index]]] & ) /@ Range[Length[
newMatrixRepresentation]]]] & ) /@ Tuples[Range[Length[newMatrixRepresentation]], 2]]];
Thread[Catenate[leftHandSide] == Catenate[rightHandSide]] /. (ToExpression[#1] -> #1 & ) /@
Select[Join[coordinates, {timeCoordinate}], StringQ]] /; SymbolName[admDecomposition] === "ADMDecomposition" &&
SymbolName[metricTensor] === "MetricTensor" && Length[Dimensions[matrixRepresentation]] == 2 &&
Length[coordinates] == Length[matrixRepresentation] && BooleanQ[index1] && BooleanQ[index2] &&
Length[shiftVector] == Length[matrixRepresentation]
VacuumADMSolution[(admDecomposition_)[(metricTensor_)[matrixRepresentation_List, coordinates_List, index1_, index2_],
timeCoordinate_, lapseFunction_, shiftVector_List], cosmologicalConstant_]["ADMDecomposition"] :=
ResourceFunction["ADMDecomposition"][ResourceFunction["MetricTensor"][matrixRepresentation, coordinates, index1,
index2], timeCoordinate, lapseFunction, shiftVector] /; SymbolName[admDecomposition] === "ADMDecomposition" &&
SymbolName[metricTensor] === "MetricTensor" && Length[Dimensions[matrixRepresentation]] == 2 &&
Length[coordinates] == Length[matrixRepresentation] && BooleanQ[index1] && BooleanQ[index2] &&
Length[shiftVector] == Length[matrixRepresentation]
VacuumADMSolution[(admDecomposition_)[(metricTensor_)[matrixRepresentation_List, coordinates_List, index1_, index2_],
timeCoordinate_, lapseFunction_, shiftVector_List], cosmologicalConstant_]["SpatialMetricTensor"] :=
ResourceFunction["MetricTensor"][matrixRepresentation, coordinates, index1, index2] /;
SymbolName[admDecomposition] === "ADMDecomposition" && SymbolName[metricTensor] === "MetricTensor" &&
Length[Dimensions[matrixRepresentation]] == 2 && Length[coordinates] == Length[matrixRepresentation] &&
BooleanQ[index1] && BooleanQ[index2] && Length[shiftVector] == Length[matrixRepresentation]
VacuumADMSolution[(admDecomposition_)[(metricTensor_)[matrixRepresentation_List, coordinates_List, index1_, index2_],
timeCoordinate_, lapseFunction_, shiftVector_List], cosmologicalConstant_]["SpacetimeMetricTensor"] :=
Module[{shiftCovector},
shiftCovector = Normal[SparseArray[(Module[{index = #1}, index -> Total[(matrixRepresentation[[index,#1]]*
shiftVector[[#1]] & ) /@ Range[Length[matrixRepresentation]]]] & ) /@
Range[Length[matrixRepresentation]]]]; ResourceFunction["MetricTensor"][
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]]]], Join[{timeCoordinate}, coordinates], True, True]] /;
SymbolName[admDecomposition] === "ADMDecomposition" && SymbolName[metricTensor] === "MetricTensor" &&
Length[Dimensions[matrixRepresentation]] == 2 && Length[coordinates] == Length[matrixRepresentation] &&
BooleanQ[index1] && BooleanQ[index2] && Length[shiftVector] == Length[matrixRepresentation]
VacuumADMSolution[(admDecomposition_)[(metricTensor_)[matrixRepresentation_List, coordinates_List, index1_, index2_],
timeCoordinate_, lapseFunction_, shiftVector_List], cosmologicalConstant_]["NormalVector"] :=
Module[{newMatrixRepresentation, newCoordinates, newTimeCoordinate, newLapseFunction, newShiftVector, shiftCovector,
spacetimeMetricTensor}, 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[(newMatrixRepresentation[[index,#1]]*newShiftVector[[#1]] & ) /@
Range[Length[newMatrixRepresentation]]]] & ) /@ Range[Length[newMatrixRepresentation]]]];
spacetimeMetricTensor = Normal[SparseArray[
Join[{{1, 1} -> Total[(newShiftVector[[#1]]*shiftCovector[[#1]] & ) /@ Range[Length[newMatrixRepresentation]]] -
newLapseFunction^2}, (Module[{index = #1}, {1, index + 1} -> Total[(newMatrixRepresentation[[index,#1]]*
newShiftVector[[#1]] & ) /@ Range[Length[newMatrixRepresentation]]]] & ) /@
Range[Length[newMatrixRepresentation]], (Module[{index = #1}, {index + 1, 1} ->
Total[(newMatrixRepresentation[[#1,index]]*newShiftVector[[#1]] & ) /@ Range[
Length[newMatrixRepresentation]]]] & ) /@ Range[Length[newMatrixRepresentation]],
({First[#1] + 1, Last[#1] + 1} -> newMatrixRepresentation[[First[#1],Last[#1]]] & ) /@
Tuples[Range[Length[newMatrixRepresentation]], 2]]]];
Normal[SparseArray[(Module[{index = #1}, index -> -Total[(newLapseFunction*Inverse[spacetimeMetricTensor][[index,
#1]]*D[newTimeCoordinate, Join[{newTimeCoordinate}, newCoordinates][[#1]]] & ) /@ Range[
Length[spacetimeMetricTensor]]]] & ) /@ Range[Length[spacetimeMetricTensor]]]] /.
(ToExpression[#1] -> #1 & ) /@ Select[Join[coordinates, {timeCoordinate}], StringQ]] /;
SymbolName[admDecomposition] === "ADMDecomposition" && SymbolName[metricTensor] === "MetricTensor" &&
Length[Dimensions[matrixRepresentation]] == 2 && Length[coordinates] == Length[matrixRepresentation] &&
BooleanQ[index1] && BooleanQ[index2] && Length[shiftVector] == Length[matrixRepresentation]
VacuumADMSolution[(admDecomposition_)[(metricTensor_)[matrixRepresentation_List, coordinates_List, index1_, index2_],
timeCoordinate_, lapseFunction_, shiftVector_List], cosmologicalConstant_]["ReducedNormalVector"] :=
Module[{newMatrixRepresentation, newCoordinates, newTimeCoordinate, newLapseFunction, newShiftVector, shiftCovector,
spacetimeMetricTensor}, 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[(newMatrixRepresentation[[index,#1]]*newShiftVector[[#1]] & ) /@
Range[Length[newMatrixRepresentation]]]] & ) /@ Range[Length[newMatrixRepresentation]]]];
spacetimeMetricTensor = Normal[SparseArray[
Join[{{1, 1} -> Total[(newShiftVector[[#1]]*shiftCovector[[#1]] & ) /@ Range[Length[newMatrixRepresentation]]] -
newLapseFunction^2}, (Module[{index = #1}, {1, index + 1} -> Total[(newMatrixRepresentation[[index,#1]]*
newShiftVector[[#1]] & ) /@ Range[Length[newMatrixRepresentation]]]] & ) /@
Range[Length[newMatrixRepresentation]], (Module[{index = #1}, {index + 1, 1} ->
Total[(newMatrixRepresentation[[#1,index]]*newShiftVector[[#1]] & ) /@ Range[
Length[newMatrixRepresentation]]]] & ) /@ Range[Length[newMatrixRepresentation]],
({First[#1] + 1, Last[#1] + 1} -> newMatrixRepresentation[[First[#1],Last[#1]]] & ) /@
Tuples[Range[Length[newMatrixRepresentation]], 2]]]];
FullSimplify[
Normal[SparseArray[(Module[{index = #1}, index -> -Total[(newLapseFunction*Inverse[spacetimeMetricTensor][[index,
#1]]*D[newTimeCoordinate, Join[{newTimeCoordinate}, newCoordinates][[#1]]] & ) /@
Range[Length[spacetimeMetricTensor]]]] & ) /@ Range[Length[spacetimeMetricTensor]]]] /.
(ToExpression[#1] -> #1 & ) /@ Select[Join[coordinates, {timeCoordinate}], StringQ]]] /;
SymbolName[admDecomposition] === "ADMDecomposition" && SymbolName[metricTensor] === "MetricTensor" &&
Length[Dimensions[matrixRepresentation]] == 2 && Length[coordinates] == Length[matrixRepresentation] &&
BooleanQ[index1] && BooleanQ[index2] && Length[shiftVector] == Length[matrixRepresentation]
VacuumADMSolution[(admDecomposition_)[(metricTensor_)[matrixRepresentation_List, coordinates_List, index1_, index2_],
timeCoordinate_, lapseFunction_, shiftVector_List], cosmologicalConstant_]["SymbolicNormalVector"] :=
Module[{newMatrixRepresentation, newCoordinates, newTimeCoordinate, newLapseFunction, newShiftVector, shiftCovector,
spacetimeMetricTensor}, 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[(newMatrixRepresentation[[index,#1]]*newShiftVector[[#1]] & ) /@
Range[Length[newMatrixRepresentation]]]] & ) /@ Range[Length[newMatrixRepresentation]]]];
spacetimeMetricTensor = Normal[SparseArray[
Join[{{1, 1} -> Total[(newShiftVector[[#1]]*shiftCovector[[#1]] & ) /@ Range[Length[newMatrixRepresentation]]] -
newLapseFunction^2}, (Module[{index = #1}, {1, index + 1} -> Total[(newMatrixRepresentation[[index,#1]]*
newShiftVector[[#1]] & ) /@ Range[Length[newMatrixRepresentation]]]] & ) /@
Range[Length[newMatrixRepresentation]], (Module[{index = #1}, {index + 1, 1} ->
Total[(newMatrixRepresentation[[#1,index]]*newShiftVector[[#1]] & ) /@ Range[
Length[newMatrixRepresentation]]]] & ) /@ Range[Length[newMatrixRepresentation]],
({First[#1] + 1, Last[#1] + 1} -> newMatrixRepresentation[[First[#1],Last[#1]]] & ) /@
Tuples[Range[Length[newMatrixRepresentation]], 2]]]];
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]]]] /.
(ToExpression[#1] -> #1 & ) /@ Select[Join[coordinates, {timeCoordinate}], StringQ]] /;
SymbolName[admDecomposition] === "ADMDecomposition" && SymbolName[metricTensor] === "MetricTensor" &&
Length[Dimensions[matrixRepresentation]] == 2 && Length[coordinates] == Length[matrixRepresentation] &&
BooleanQ[index1] && BooleanQ[index2] && Length[shiftVector] == Length[matrixRepresentation]
VacuumADMSolution[(admDecomposition_)[(metricTensor_)[matrixRepresentation_List, coordinates_List, index1_, index2_],
timeCoordinate_, lapseFunction_, shiftVector_List], cosmologicalConstant_]["TimeVector"] :=
Module[{newMatrixRepresentation, newCoordinates, newTimeCoordinate, newLapseFunction, newShiftVector, shiftCovector,
spacetimeMetricTensor, normalVector}, 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[(newMatrixRepresentation[[index,#1]]*newShiftVector[[#1]] & ) /@
Range[Length[newMatrixRepresentation]]]] & ) /@ Range[Length[newMatrixRepresentation]]]];
spacetimeMetricTensor = Normal[SparseArray[
Join[{{1, 1} -> Total[(newShiftVector[[#1]]*shiftCovector[[#1]] & ) /@ Range[Length[newMatrixRepresentation]]] -
newLapseFunction^2}, (Module[{index = #1}, {1, index + 1} -> Total[(newMatrixRepresentation[[index,#1]]*
newShiftVector[[#1]] & ) /@ Range[Length[newMatrixRepresentation]]]] & ) /@
Range[Length[newMatrixRepresentation]], (Module[{index = #1}, {index + 1, 1} ->
Total[(newMatrixRepresentation[[#1,index]]*newShiftVector[[#1]] & ) /@ Range[
Length[newMatrixRepresentation]]]] & ) /@ Range[Length[newMatrixRepresentation]],
({First[#1] + 1, Last[#1] + 1} -> newMatrixRepresentation[[First[#1],Last[#1]]] & ) /@
Tuples[Range[Length[newMatrixRepresentation]], 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]]]];
Normal[SparseArray[(Module[{index = #1}, index -> newLapseFunction*normalVector[[index]] +
Join[{0}, newShiftVector][[index]]] & ) /@ Range[Length[spacetimeMetricTensor]]]] /.
(ToExpression[#1] -> #1 & ) /@ Select[Join[coordinates, {timeCoordinate}], StringQ]] /;
SymbolName[admDecomposition] === "ADMDecomposition" && SymbolName[metricTensor] === "MetricTensor" &&
Length[Dimensions[matrixRepresentation]] == 2 && Length[coordinates] == Length[matrixRepresentation] &&
BooleanQ[index1] && BooleanQ[index2] && Length[shiftVector] == Length[matrixRepresentation]
VacuumADMSolution[(admDecomposition_)[(metricTensor_)[matrixRepresentation_List, coordinates_List, index1_, index2_],
timeCoordinate_, lapseFunction_, shiftVector_List], cosmologicalConstant_]["SymbolicTimeVector"] :=
Module[{newMatrixRepresentation, newCoordinates, newTimeCoordinate, newLapseFunction, newShiftVector, shiftCovector,
spacetimeMetricTensor, normalVector}, 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[(newMatrixRepresentation[[index,#1]]*newShiftVector[[#1]] & ) /@
Range[Length[newMatrixRepresentation]]]] & ) /@ Range[Length[newMatrixRepresentation]]]];
spacetimeMetricTensor = Normal[SparseArray[
Join[{{1, 1} -> Total[(newShiftVector[[#1]]*shiftCovector[[#1]] & ) /@ Range[Length[newMatrixRepresentation]]] -
newLapseFunction^2}, (Module[{index = #1}, {1, index + 1} -> Total[(newMatrixRepresentation[[index,#1]]*
newShiftVector[[#1]] & ) /@ Range[Length[newMatrixRepresentation]]]] & ) /@
Range[Length[newMatrixRepresentation]], (Module[{index = #1}, {index + 1, 1} ->
Total[(newMatrixRepresentation[[#1,index]]*newShiftVector[[#1]] & ) /@ Range[
Length[newMatrixRepresentation]]]] & ) /@ Range[Length[newMatrixRepresentation]],
({First[#1] + 1, Last[#1] + 1} -> newMatrixRepresentation[[First[#1],Last[#1]]] & ) /@
Tuples[Range[Length[newMatrixRepresentation]], 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]]]];
Normal[SparseArray[(Module[{index = #1}, index -> newLapseFunction*normalVector[[index]] +
Join[{0}, newShiftVector][[index]]] & ) /@ Range[Length[spacetimeMetricTensor]]]] /.
(ToExpression[#1] -> #1 & ) /@ Select[Join[coordinates, {timeCoordinate}], StringQ]] /;
SymbolName[admDecomposition] === "ADMDecomposition" && SymbolName[metricTensor] === "MetricTensor" &&
Length[Dimensions[matrixRepresentation]] == 2 && Length[coordinates] == Length[matrixRepresentation] &&
BooleanQ[index1] && BooleanQ[index2] && Length[shiftVector] == Length[matrixRepresentation]
VacuumADMSolution[(admDecomposition_)[(metricTensor_)[matrixRepresentation_List, coordinates_List, index1_, index2_],
timeCoordinate_, lapseFunction_, shiftVector_List], cosmologicalConstant_]["TimeCoordinate"] :=
timeCoordinate /; SymbolName[admDecomposition] === "ADMDecomposition" && SymbolName[metricTensor] === "MetricTensor" &&
Length[Dimensions[matrixRepresentation]] == 2 && Length[coordinates] == Length[matrixRepresentation] &&
BooleanQ[index1] && BooleanQ[index2] && Length[shiftVector] == Length[matrixRepresentation]
VacuumADMSolution[(admDecomposition_)[(metricTensor_)[matrixRepresentation_List, coordinates_List, index1_, index2_],
timeCoordinate_, lapseFunction_, shiftVector_List], cosmologicalConstant_]["SpatialCoordinates"] :=
coordinates /; SymbolName[admDecomposition] === "ADMDecomposition" && SymbolName[metricTensor] === "MetricTensor" &&
Length[Dimensions[matrixRepresentation]] == 2 && Length[coordinates] == Length[matrixRepresentation] &&
BooleanQ[index1] && BooleanQ[index2] && Length[shiftVector] == Length[matrixRepresentation]
VacuumADMSolution[(admDecomposition_)[(metricTensor_)[matrixRepresentation_List, coordinates_List, index1_, index2_],
timeCoordinate_, lapseFunction_, shiftVector_List], cosmologicalConstant_]["CoordinateOneForms"] :=
(If[Head[#1] === Subscript, Subscript[StringJoin["\[FormalD]", ToString[First[#1]]], ToString[Last[#1]]],
If[Head[#1] === Superscript, Superscript[StringJoin["\[FormalD]", ToString[First[#1]]], ToString[Last[#1]]],
StringJoin["\[FormalD]", ToString[#1]]]] & ) /@ Join[{timeCoordinate}, coordinates] /;
SymbolName[admDecomposition] === "ADMDecomposition" && SymbolName[metricTensor] === "MetricTensor" &&
Length[Dimensions[matrixRepresentation]] == 2 && Length[coordinates] == Length[matrixRepresentation] &&
BooleanQ[index1] && BooleanQ[index2] && Length[shiftVector] == Length[matrixRepresentation]
VacuumADMSolution[(admDecomposition_)[(metricTensor_)[matrixRepresentation_List, coordinates_List, index1_, index2_],
timeCoordinate_, lapseFunction_, shiftVector_List], cosmologicalConstant_]["LapseFunction"] :=
lapseFunction /; SymbolName[admDecomposition] === "ADMDecomposition" && SymbolName[metricTensor] === "MetricTensor" &&
Length[Dimensions[matrixRepresentation]] == 2 && Length[coordinates] == Length[matrixRepresentation] &&
BooleanQ[index1] && BooleanQ[index2] && Length[shiftVector] == Length[matrixRepresentation]
VacuumADMSolution[(admDecomposition_)[(metricTensor_)[matrixRepresentation_List, coordinates_List, index1_, index2_],
timeCoordinate_, lapseFunction_, shiftVector_List], cosmologicalConstant_]["ShiftVector"] :=
shiftVector /; SymbolName[admDecomposition] === "ADMDecomposition" && SymbolName[metricTensor] === "MetricTensor" &&
Length[Dimensions[matrixRepresentation]] == 2 && Length[coordinates] == Length[matrixRepresentation] &&
BooleanQ[index1] && BooleanQ[index2] && Length[shiftVector] == Length[matrixRepresentation]
VacuumADMSolution[(admDecomposition_)[(metricTensor_)[matrixRepresentation_List, coordinates_List, index1_, index2_],
timeCoordinate_, lapseFunction_, shiftVector_List], cosmologicalConstant_]["HamiltonianConstraintSatisfiedQ"] :=
Module[{newMatrixRepresentation, newCoordinates, newTimeCoordinate, newLapseFunction, newShiftVector, shiftCovector,
spatialChristoffelSymbols, extrinsicCurvatureTensor, mixedExtrinsicCurvatureTensor, extrinsicCurvatureTrace,
spatialRiemannTensor, spatialRicciTensor, spatialRicciScalar, spacetimeMetricTensor, spacetimeEinsteinTensor,
contravariantSpacetimeEinsteinTensor}, 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[(newMatrixRepresentation[[index,#1]]*newShiftVector[[#1]] & ) /@
Range[Length[newMatrixRepresentation]]]] & ) /@ Range[Length[newMatrixRepresentation]]]];
spatialChristoffelSymbols = Normal[SparseArray[
(Module[{index = #1}, index -> Total[((1/2)*Inverse[newMatrixRepresentation][[index[[1]],#1]]*
(D[newMatrixRepresentation[[#1,index[[3]]]], newCoordinates[[index[[2]]]]] + D[newMatrixRepresentation[[
index[[2]],#1]], newCoordinates[[index[[3]]]]] - D[newMatrixRepresentation[[index[[2]],index[[3]]]],
newCoordinates[[#1]]]) & ) /@ Range[Length[newMatrixRepresentation]]]] & ) /@
Tuples[Range[Length[newMatrixRepresentation]], 3]]]; 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[newMatrixRepresentation]]] + D[shiftCovector[[Last[index]]],
newCoordinates[[First[index]]]] - Total[(spatialChristoffelSymbols[[#1,First[index],Last[index]]]*
shiftCovector[[#1]] & ) /@ Range[Length[newMatrixRepresentation]]] - D[newMatrixRepresentation[[
First[index],Last[index]]], newTimeCoordinate])] & ) /@ Tuples[Range[Length[newMatrixRepresentation]],
2]]]; mixedExtrinsicCurvatureTensor =
Normal[SparseArray[(Module[{index = #1}, index -> Total[(Inverse[newMatrixRepresentation][[First[index],#1]]*
extrinsicCurvatureTensor[[#1,Last[index]]] & ) /@ Range[Length[newMatrixRepresentation]]]] & ) /@
Tuples[Range[Length[newMatrixRepresentation]], 2]]]; extrinsicCurvatureTrace =
Total[(Inverse[newMatrixRepresentation][[First[#1],Last[#1]]]*extrinsicCurvatureTensor[[First[#1],Last[#1]]] & ) /@
Tuples[Range[Length[newMatrixRepresentation]], 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[newMatrixRepresentation]]] -
Total[(spatialChristoffelSymbols[[index[[1]],#1,index[[4]]]]*spatialChristoffelSymbols[[#1,index[[2]],
index[[3]]]] & ) /@ Range[Length[newMatrixRepresentation]]]] & ) /@
Tuples[Range[Length[newMatrixRepresentation]], 4]]]; spatialRicciTensor =
Normal[SparseArray[(Module[{index = #1}, index -> Total[(spatialRiemannTensor[[#1,First[index],#1,
Last[index]]] & ) /@ Range[Length[newMatrixRepresentation]]]] & ) /@
Tuples[Range[Length[newMatrixRepresentation]], 2]]]; spatialRicciScalar =
Total[(Inverse[newMatrixRepresentation][[First[#1],Last[#1]]]*spatialRicciTensor[[First[#1],Last[#1]]] & ) /@
Tuples[Range[Length[newMatrixRepresentation]], 2]]; spacetimeMetricTensor =
Normal[SparseArray[Join[{{1, 1} -> Total[(newShiftVector[[#1]]*shiftCovector[[#1]] & ) /@
Range[Length[newMatrixRepresentation]]] - newLapseFunction^2},
(Module[{index = #1}, {1, index + 1} -> Total[(newMatrixRepresentation[[index,#1]]*newShiftVector[[#1]] & ) /@
Range[Length[newMatrixRepresentation]]]] & ) /@ Range[Length[newMatrixRepresentation]],
(Module[{index = #1}, {index + 1, 1} -> Total[(newMatrixRepresentation[[#1,index]]*newShiftVector[[#1]] & ) /@
Range[Length[newMatrixRepresentation]]]] & ) /@ Range[Length[newMatrixRepresentation]],
({First[#1] + 1, Last[#1] + 1} -> newMatrixRepresentation[[First[#1],Last[#1]]] & ) /@
Tuples[Range[Length[newMatrixRepresentation]], 2]]]]; spacetimeEinsteinTensor =
-(cosmologicalConstant*spacetimeMetricTensor); contravariantSpacetimeEinsteinTensor =
Normal[SparseArray[(Module[{index = #1}, index -> Total[(Inverse[spacetimeMetricTensor][[First[index],First[#1]]]*
Inverse[spacetimeMetricTensor][[Last[#1],Last[index]]]*spacetimeEinsteinTensor[[First[#1],
Last[#1]]] & ) /@ Tuples[Range[Length[spacetimeMetricTensor]], 2]]] & ) /@
Tuples[Range[Length[spacetimeMetricTensor]], 2]]];
FullSimplify[((spatialRicciScalar + extrinsicCurvatureTrace^2 -
Total[(mixedExtrinsicCurvatureTensor[[First[#1],Last[#1]]]*mixedExtrinsicCurvatureTensor[[Last[#1],
First[#1]]] & ) /@ Tuples[Range[Length[newMatrixRepresentation]], 2]]) -
2*newLapseFunction^2*contravariantSpacetimeEinsteinTensor[[1,1]] /. (ToExpression[#1] -> #1 & ) /@
Select[Join[coordinates, {timeCoordinate}], StringQ]) == 0] === True] /;
SymbolName[admDecomposition] === "ADMDecomposition" && SymbolName[metricTensor] === "MetricTensor" &&
Length[Dimensions[matrixRepresentation]] == 2 && Length[coordinates] == Length[matrixRepresentation] &&
BooleanQ[index1] && BooleanQ[index2] && Length[shiftVector] == Length[matrixRepresentation]
VacuumADMSolution[(admDecomposition_)[(metricTensor_)[matrixRepresentation_List, coordinates_List, index1_, index2_],
timeCoordinate_, lapseFunction_, shiftVector_List], cosmologicalConstant_]["HamiltonianConstraint"] :=
Module[{newMatrixRepresentation, newCoordinates, newTimeCoordinate, newLapseFunction, newShiftVector, shiftCovector,
spatialChristoffelSymbols, extrinsicCurvatureTensor, mixedExtrinsicCurvatureTensor, extrinsicCurvatureTrace,
spatialRiemannTensor, spatialRicciTensor, spatialRicciScalar, spacetimeMetricTensor, spacetimeEinsteinTensor,
contravariantSpacetimeEinsteinTensor}, 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[(newMatrixRepresentation[[index,#1]]*newShiftVector[[#1]] & ) /@
Range[Length[newMatrixRepresentation]]]] & ) /@ Range[Length[newMatrixRepresentation]]]];
spatialChristoffelSymbols = Normal[SparseArray[
(Module[{index = #1}, index -> Total[((1/2)*Inverse[newMatrixRepresentation][[index[[1]],#1]]*
(D[newMatrixRepresentation[[#1,index[[3]]]], newCoordinates[[index[[2]]]]] + D[newMatrixRepresentation[[
index[[2]],#1]], newCoordinates[[index[[3]]]]] - D[newMatrixRepresentation[[index[[2]],index[[3]]]],
newCoordinates[[#1]]]) & ) /@ Range[Length[newMatrixRepresentation]]]] & ) /@
Tuples[Range[Length[newMatrixRepresentation]], 3]]]; 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[newMatrixRepresentation]]] + D[shiftCovector[[Last[index]]],
newCoordinates[[First[index]]]] - Total[(spatialChristoffelSymbols[[#1,First[index],Last[index]]]*
shiftCovector[[#1]] & ) /@ Range[Length[newMatrixRepresentation]]] - D[newMatrixRepresentation[[
First[index],Last[index]]], newTimeCoordinate])] & ) /@ Tuples[Range[Length[newMatrixRepresentation]],
2]]]; mixedExtrinsicCurvatureTensor =
Normal[SparseArray[(Module[{index = #1}, index -> Total[(Inverse[newMatrixRepresentation][[First[index],#1]]*
extrinsicCurvatureTensor[[#1,Last[index]]] & ) /@ Range[Length[newMatrixRepresentation]]]] & ) /@
Tuples[Range[Length[newMatrixRepresentation]], 2]]]; extrinsicCurvatureTrace =
Total[(Inverse[newMatrixRepresentation][[First[#1],Last[#1]]]*extrinsicCurvatureTensor[[First[#1],Last[#1]]] & ) /@
Tuples[Range[Length[newMatrixRepresentation]], 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[newMatrixRepresentation]]] -
Total[(spatialChristoffelSymbols[[index[[1]],#1,index[[4]]]]*spatialChristoffelSymbols[[#1,index[[2]],
index[[3]]]] & ) /@ Range[Length[newMatrixRepresentation]]]] & ) /@
Tuples[Range[Length[newMatrixRepresentation]], 4]]]; spatialRicciTensor =
Normal[SparseArray[(Module[{index = #1}, index -> Total[(spatialRiemannTensor[[#1,First[index],#1,
Last[index]]] & ) /@ Range[Length[newMatrixRepresentation]]]] & ) /@
Tuples[Range[Length[newMatrixRepresentation]], 2]]]; spatialRicciScalar =
Total[(Inverse[newMatrixRepresentation][[First[#1],Last[#1]]]*spatialRicciTensor[[First[#1],Last[#1]]] & ) /@
Tuples[Range[Length[newMatrixRepresentation]], 2]]; spacetimeMetricTensor =
Normal[SparseArray[Join[{{1, 1} -> Total[(newShiftVector[[#1]]*shiftCovector[[#1]] & ) /@
Range[Length[newMatrixRepresentation]]] - newLapseFunction^2},
(Module[{index = #1}, {1, index + 1} -> Total[(newMatrixRepresentation[[index,#1]]*newShiftVector[[#1]] & ) /@
Range[Length[newMatrixRepresentation]]]] & ) /@ Range[Length[newMatrixRepresentation]],
(Module[{index = #1}, {index + 1, 1} -> Total[(newMatrixRepresentation[[#1,index]]*newShiftVector[[#1]] & ) /@
Range[Length[newMatrixRepresentation]]]] & ) /@ Range[Length[newMatrixRepresentation]],
({First[#1] + 1, Last[#1] + 1} -> newMatrixRepresentation[[First[#1],Last[#1]]] & ) /@
Tuples[Range[Length[newMatrixRepresentation]], 2]]]]; spacetimeEinsteinTensor =
-(cosmologicalConstant*spacetimeMetricTensor); contravariantSpacetimeEinsteinTensor =
Normal[SparseArray[(Module[{index = #1}, index -> Total[(Inverse[spacetimeMetricTensor][[First[index],First[#1]]]*
Inverse[spacetimeMetricTensor][[Last[#1],Last[index]]]*spacetimeEinsteinTensor[[First[#1],
Last[#1]]] & ) /@ Tuples[Range[Length[spacetimeMetricTensor]], 2]]] & ) /@
Tuples[Range[Length[spacetimeMetricTensor]], 2]]];
(spatialRicciScalar + extrinsicCurvatureTrace^2 - Total[(mixedExtrinsicCurvatureTensor[[First[#1],Last[#1]]]*
mixedExtrinsicCurvatureTensor[[Last[#1],First[#1]]] & ) /@ Tuples[Range[Length[newMatrixRepresentation]],
2]]) - 2*newLapseFunction^2*contravariantSpacetimeEinsteinTensor[[1,1]] /.
(ToExpression[#1] -> #1 & ) /@ Select[Join[coordinates, {timeCoordinate}], StringQ]] /;
SymbolName[admDecomposition] === "ADMDecomposition" && SymbolName[metricTensor] === "MetricTensor" &&
Length[Dimensions[matrixRepresentation]] == 2 && Length[coordinates] == Length[matrixRepresentation] &&
BooleanQ[index1] && BooleanQ[index2] && Length[shiftVector] == Length[matrixRepresentation]
VacuumADMSolution[(admDecomposition_)[(metricTensor_)[matrixRepresentation_List, coordinates_List, index1_, index2_],
timeCoordinate_, lapseFunction_, shiftVector_List], cosmologicalConstant_]["ReducedHamiltonianConstraint"] :=
Module[{newMatrixRepresentation, newCoordinates, newTimeCoordinate, newLapseFunction, newShiftVector, shiftCovector,
spatialChristoffelSymbols, extrinsicCurvatureTensor, mixedExtrinsicCurvatureTensor, extrinsicCurvatureTrace,
spatialRiemannTensor, spatialRicciTensor, spatialRicciScalar, spacetimeMetricTensor, spacetimeEinsteinTensor,
contravariantSpacetimeEinsteinTensor}, 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[(newMatrixRepresentation[[index,#1]]*newShiftVector[[#1]] & ) /@
Range[Length[newMatrixRepresentation]]]] & ) /@ Range[Length[newMatrixRepresentation]]]];
spatialChristoffelSymbols = Normal[SparseArray[
(Module[{index = #1}, index -> Total[((1/2)*Inverse[newMatrixRepresentation][[index[[1]],#1]]*
(D[newMatrixRepresentation[[#1,index[[3]]]], newCoordinates[[index[[2]]]]] + D[newMatrixRepresentation[[
index[[2]],#1]], newCoordinates[[index[[3]]]]] - D[newMatrixRepresentation[[index[[2]],index[[3]]]],
newCoordinates[[#1]]]) & ) /@ Range[Length[newMatrixRepresentation]]]] & ) /@
Tuples[Range[Length[newMatrixRepresentation]], 3]]]; 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[newMatrixRepresentation]]] + D[shiftCovector[[Last[index]]],
newCoordinates[[First[index]]]] - Total[(spatialChristoffelSymbols[[#1,First[index],Last[index]]]*
shiftCovector[[#1]] & ) /@ Range[Length[newMatrixRepresentation]]] - D[newMatrixRepresentation[[
First[index],Last[index]]], newTimeCoordinate])] & ) /@ Tuples[Range[Length[newMatrixRepresentation]],
2]]]; mixedExtrinsicCurvatureTensor =
Normal[SparseArray[(Module[{index = #1}, index -> Total[(Inverse[newMatrixRepresentation][[First[index],#1]]*
extrinsicCurvatureTensor[[#1,Last[index]]] & ) /@ Range[Length[newMatrixRepresentation]]]] & ) /@
Tuples[Range[Length[newMatrixRepresentation]], 2]]]; extrinsicCurvatureTrace =
Total[(Inverse[newMatrixRepresentation][[First[#1],Last[#1]]]*extrinsicCurvatureTensor[[First[#1],Last[#1]]] & ) /@
Tuples[Range[Length[newMatrixRepresentation]], 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[newMatrixRepresentation]]] -
Total[(spatialChristoffelSymbols[[index[[1]],#1,index[[4]]]]*spatialChristoffelSymbols[[#1,index[[2]],
index[[3]]]] & ) /@ Range[Length[newMatrixRepresentation]]]] & ) /@
Tuples[Range[Length[newMatrixRepresentation]], 4]]]; spatialRicciTensor =
Normal[SparseArray[(Module[{index = #1}, index -> Total[(spatialRiemannTensor[[#1,First[index],#1,
Last[index]]] & ) /@ Range[Length[newMatrixRepresentation]]]] & ) /@
Tuples[Range[Length[newMatrixRepresentation]], 2]]]; spatialRicciScalar =
Total[(Inverse[newMatrixRepresentation][[First[#1],Last[#1]]]*spatialRicciTensor[[First[#1],Last[#1]]] & ) /@
Tuples[Range[Length[newMatrixRepresentation]], 2]]; spacetimeMetricTensor =
Normal[SparseArray[Join[{{1, 1} -> Total[(newShiftVector[[#1]]*shiftCovector[[#1]] & ) /@
Range[Length[newMatrixRepresentation]]] - newLapseFunction^2},
(Module[{index = #1}, {1, index + 1} -> Total[(newMatrixRepresentation[[index,#1]]*newShiftVector[[#1]] & ) /@
Range[Length[newMatrixRepresentation]]]] & ) /@ Range[Length[newMatrixRepresentation]],
(Module[{index = #1}, {index + 1, 1} -> Total[(newMatrixRepresentation[[#1,index]]*newShiftVector[[#1]] & ) /@
Range[Length[newMatrixRepresentation]]]] & ) /@ Range[Length[newMatrixRepresentation]],
({First[#1] + 1, Last[#1] + 1} -> newMatrixRepresentation[[First[#1],Last[#1]]] & ) /@
Tuples[Range[Length[newMatrixRepresentation]], 2]]]]; spacetimeEinsteinTensor =
-(cosmologicalConstant*spacetimeMetricTensor); contravariantSpacetimeEinsteinTensor =
Normal[SparseArray[(Module[{index = #1}, index -> Total[(Inverse[spacetimeMetricTensor][[First[index],First[#1]]]*
Inverse[spacetimeMetricTensor][[Last[#1],Last[index]]]*spacetimeEinsteinTensor[[First[#1],
Last[#1]]] & ) /@ Tuples[Range[Length[spacetimeMetricTensor]], 2]]] & ) /@
Tuples[Range[Length[spacetimeMetricTensor]], 2]]];
FullSimplify[(spatialRicciScalar + extrinsicCurvatureTrace^2 -
Total[(mixedExtrinsicCurvatureTensor[[First[#1],Last[#1]]]*mixedExtrinsicCurvatureTensor[[Last[#1],
First[#1]]] & ) /@ Tuples[Range[Length[newMatrixRepresentation]], 2]]) -
2*newLapseFunction^2*contravariantSpacetimeEinsteinTensor[[1,1]] /. (ToExpression[#1] -> #1 & ) /@
Select[Join[coordinates, {timeCoordinate}], StringQ]]] /; SymbolName[admDecomposition] === "ADMDecomposition" &&
SymbolName[metricTensor] === "MetricTensor" && Length[Dimensions[matrixRepresentation]] == 2 &&
Length[coordinates] == Length[matrixRepresentation] && BooleanQ[index1] && BooleanQ[index2] &&
Length[shiftVector] == Length[matrixRepresentation]
VacuumADMSolution[(admDecomposition_)[(metricTensor_)[matrixRepresentation_List, coordinates_List, index1_, index2_],
timeCoordinate_, lapseFunction_, shiftVector_List], cosmologicalConstant_]["SymbolicHamiltonianConstraint"] :=
Module[{newMatrixRepresentation, newCoordinates, newTimeCoordinate, newLapseFunction, newShiftVector, shiftCovector,
spatialChristoffelSymbols, extrinsicCurvatureTensor, mixedExtrinsicCurvatureTensor, extrinsicCurvatureTrace,
spatialRiemannTensor, spatialRicciTensor, spatialRicciScalar, spacetimeMetricTensor, spacetimeEinsteinTensor,
contravariantSpacetimeEinsteinTensor}, 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[(newMatrixRepresentation[[index,#1]]*newShiftVector[[#1]] & ) /@
Range[Length[newMatrixRepresentation]]]] & ) /@ Range[Length[newMatrixRepresentation]]]];
spatialChristoffelSymbols = Normal[SparseArray[
(Module[{index = #1}, index -> Total[((1/2)*Inverse[newMatrixRepresentation][[index[[1]],#1]]*
(Inactive[D][newMatrixRepresentation[[#1,index[[3]]]], newCoordinates[[index[[2]]]]] +
Inactive[D][newMatrixRepresentation[[index[[2]],#1]], newCoordinates[[index[[3]]]]] -
Inactive[D][newMatrixRepresentation[[index[[2]],index[[3]]]], newCoordinates[[#1]]]) & ) /@
Range[Length[newMatrixRepresentation]]]] & ) /@ Tuples[Range[Length[newMatrixRepresentation]], 3]]];
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[newMatrixRepresentation]]] + Inactive[D][shiftCovector[[
Last[index]]], newCoordinates[[First[index]]]] - Total[(spatialChristoffelSymbols[[#1,First[index],
Last[index]]]*shiftCovector[[#1]] & ) /@ Range[Length[newMatrixRepresentation]]] -
Inactive[D][newMatrixRepresentation[[First[index],Last[index]]], newTimeCoordinate])] & ) /@
Tuples[Range[Length[newMatrixRepresentation]], 2]]]; mixedExtrinsicCurvatureTensor =
Normal[SparseArray[(Module[{index = #1}, index -> Total[(Inverse[newMatrixRepresentation][[First[index],#1]]*
extrinsicCurvatureTensor[[#1,Last[index]]] & ) /@ Range[Length[newMatrixRepresentation]]]] & ) /@
Tuples[Range[Length[newMatrixRepresentation]], 2]]]; extrinsicCurvatureTrace =
Total[(Inverse[newMatrixRepresentation][[First[#1],Last[#1]]]*extrinsicCurvatureTensor[[First[#1],Last[#1]]] & ) /@
Tuples[Range[Length[newMatrixRepresentation]], 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[newMatrixRepresentation]]] - Total[(spatialChristoffelSymbols[[index[[1]],#1,index[[4]]]]*
spatialChristoffelSymbols[[#1,index[[2]],index[[3]]]] & ) /@ Range[Length[
newMatrixRepresentation]]]] & ) /@ Tuples[Range[Length[newMatrixRepresentation]], 4]]];
spatialRicciTensor = Normal[SparseArray[
(Module[{index = #1}, index -> Total[(spatialRiemannTensor[[#1,First[index],#1,Last[index]]] & ) /@
Range[Length[newMatrixRepresentation]]]] & ) /@ Tuples[Range[Length[newMatrixRepresentation]], 2]]];
spatialRicciScalar = Total[(Inverse[newMatrixRepresentation][[First[#1],Last[#1]]]*spatialRicciTensor[[First[#1],
Last[#1]]] & ) /@ Tuples[Range[Length[newMatrixRepresentation]], 2]];
spacetimeMetricTensor = Normal[SparseArray[
Join[{{1, 1} -> Total[(newShiftVector[[#1]]*shiftCovector[[#1]] & ) /@ Range[Length[newMatrixRepresentation]]] -
newLapseFunction^2}, (Module[{index = #1}, {1, index + 1} -> Total[(newMatrixRepresentation[[index,#1]]*
newShiftVector[[#1]] & ) /@ Range[Length[newMatrixRepresentation]]]] & ) /@
Range[Length[newMatrixRepresentation]], (Module[{index = #1}, {index + 1, 1} ->
Total[(newMatrixRepresentation[[#1,index]]*newShiftVector[[#1]] & ) /@ Range[
Length[newMatrixRepresentation]]]] & ) /@ Range[Length[newMatrixRepresentation]],
({First[#1] + 1, Last[#1] + 1} -> newMatrixRepresentation[[First[#1],Last[#1]]] & ) /@
Tuples[Range[Length[newMatrixRepresentation]], 2]]]]; spacetimeEinsteinTensor =
-(cosmologicalConstant*spacetimeMetricTensor); contravariantSpacetimeEinsteinTensor =
Normal[SparseArray[(Module[{index = #1}, index -> Total[(Inverse[spacetimeMetricTensor][[First[index],First[#1]]]*
Inverse[spacetimeMetricTensor][[Last[#1],Last[index]]]*spacetimeEinsteinTensor[[First[#1],