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SolveVacuumEinsteinEquations.wl
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(* ::Package:: *)
SolveVacuumEinsteinEquations[(metricTensor_)[matrixRepresentation_List, coordinates_List, index1_, index2_]] :=
VacuumSolution[ResourceFunction["MetricTensor"][matrixRepresentation, coordinates, index1, index2], "\[FormalCapitalLambda]"] /;
SymbolName[metricTensor] === "MetricTensor" && Length[Dimensions[matrixRepresentation]] == 2 &&
Length[coordinates] == Length[matrixRepresentation] && BooleanQ[index1] && BooleanQ[index2]
SolveVacuumEinsteinEquations[(metricTensor_)[matrixRepresentation_List, coordinates_List, index1_, index2_],
cosmologicalConstant_] := VacuumSolution[ResourceFunction["MetricTensor"][matrixRepresentation, coordinates, index1,
index2], cosmologicalConstant] /; SymbolName[metricTensor] === "MetricTensor" &&
Length[Dimensions[matrixRepresentation]] == 2 && Length[coordinates] == Length[matrixRepresentation] &&
BooleanQ[index1] && BooleanQ[index2]
VacuumSolution[(metricTensor_)[matrixRepresentation_List, coordinates_List, index1_, index2_], cosmologicalConstant_][
"FieldEquations"] := Module[{newMatrixRepresentation, newCoordinates, christoffelSymbols, riemannTensor, ricciTensor,
ricciScalar, einsteinEquations}, newMatrixRepresentation = matrixRepresentation /.
(#1 -> ToExpression[#1] & ) /@ Select[coordinates, StringQ];
newCoordinates = coordinates /. (#1 -> ToExpression[#1] & ) /@ Select[coordinates, StringQ];
christoffelSymbols = Normal[SparseArray[
(Module[{index = #1}, index -> Total[((1/2)*Inverse[newMatrixRepresentation][[index[[1]],#1]]*
(D[newMatrixRepresentation[[#1,index[[3]]]], newCoordinates[[index[[2]]]]] + D[newMatrixRepresentation[[
index[[2]],#1]], newCoordinates[[index[[3]]]]] - D[newMatrixRepresentation[[index[[2]],index[[3]]]],
newCoordinates[[#1]]]) & ) /@ Range[Length[newMatrixRepresentation]]]] & ) /@
Tuples[Range[Length[newMatrixRepresentation]], 3]]];
riemannTensor = Normal[SparseArray[(Module[{index = #1}, index -> D[christoffelSymbols[[index[[1]],index[[2]],
index[[4]]]], newCoordinates[[index[[3]]]]] - D[christoffelSymbols[[index[[1]],index[[2]],index[[3]]]],
newCoordinates[[index[[4]]]]] + Total[(christoffelSymbols[[index[[1]],#1,index[[3]]]]*christoffelSymbols[[
#1,index[[2]],index[[4]]]] & ) /@ Range[Length[newMatrixRepresentation]]] -
Total[(christoffelSymbols[[index[[1]],#1,index[[4]]]]*christoffelSymbols[[#1,index[[2]],index[[3]]]] & ) /@
Range[Length[newMatrixRepresentation]]]] & ) /@ Tuples[Range[Length[newMatrixRepresentation]], 4]]] /.
(ToExpression[#1] -> #1 & ) /@ Select[coordinates, StringQ];
ricciTensor = Normal[SparseArray[(Module[{index = #1}, index -> Total[(riemannTensor[[#1,First[index],#1,
Last[index]]] & ) /@ Range[Length[matrixRepresentation]]]] & ) /@
Tuples[Range[Length[matrixRepresentation]], 2]]];
ricciScalar = Total[(Inverse[matrixRepresentation][[First[#1],Last[#1]]]*ricciTensor[[First[#1],Last[#1]]] & ) /@
Tuples[Range[Length[matrixRepresentation]], 2]]; einsteinEquations =
FullSimplify[Thread[Catenate[ricciTensor - (1/2)*ricciScalar*matrixRepresentation +
cosmologicalConstant*matrixRepresentation] == Catenate[ConstantArray[0, {Length[matrixRepresentation],
Length[matrixRepresentation]}]]]]; If[einsteinEquations === True, {}, If[einsteinEquations === False,
Indeterminate, If[Length[Select[einsteinEquations, #1 === False & ]] > 0, Indeterminate,
DeleteDuplicates[Reverse /@ Sort /@ Select[einsteinEquations, #1 =!= True & ]]]]]] /;
SymbolName[metricTensor] === "MetricTensor" && Length[Dimensions[matrixRepresentation]] == 2 &&
Length[coordinates] == Length[matrixRepresentation] && BooleanQ[index1] && BooleanQ[index2]
VacuumSolution[(metricTensor_)[matrixRepresentation_List, coordinates_List, index1_, index2_], cosmologicalConstant_][
"EinsteinEquations"] := Module[{newMatrixRepresentation, newCoordinates, christoffelSymbols, riemannTensor,
ricciTensor, ricciScalar}, newMatrixRepresentation = matrixRepresentation /. (#1 -> ToExpression[#1] & ) /@
Select[coordinates, StringQ]; newCoordinates = coordinates /. (#1 -> ToExpression[#1] & ) /@
Select[coordinates, StringQ]; christoffelSymbols =
Normal[SparseArray[(Module[{index = #1}, index -> Total[((1/2)*Inverse[newMatrixRepresentation][[index[[1]],#1]]*
(D[newMatrixRepresentation[[#1,index[[3]]]], newCoordinates[[index[[2]]]]] + D[newMatrixRepresentation[[
index[[2]],#1]], newCoordinates[[index[[3]]]]] - D[newMatrixRepresentation[[index[[2]],index[[3]]]],
newCoordinates[[#1]]]) & ) /@ Range[Length[newMatrixRepresentation]]]] & ) /@
Tuples[Range[Length[newMatrixRepresentation]], 3]]];
riemannTensor = Normal[SparseArray[(Module[{index = #1}, index -> D[christoffelSymbols[[index[[1]],index[[2]],
index[[4]]]], newCoordinates[[index[[3]]]]] - D[christoffelSymbols[[index[[1]],index[[2]],index[[3]]]],
newCoordinates[[index[[4]]]]] + Total[(christoffelSymbols[[index[[1]],#1,index[[3]]]]*christoffelSymbols[[
#1,index[[2]],index[[4]]]] & ) /@ Range[Length[newMatrixRepresentation]]] -
Total[(christoffelSymbols[[index[[1]],#1,index[[4]]]]*christoffelSymbols[[#1,index[[2]],index[[3]]]] & ) /@
Range[Length[newMatrixRepresentation]]]] & ) /@ Tuples[Range[Length[newMatrixRepresentation]], 4]]] /.
(ToExpression[#1] -> #1 & ) /@ Select[coordinates, StringQ];
ricciTensor = Normal[SparseArray[(Module[{index = #1}, index -> Total[(riemannTensor[[#1,First[index],#1,
Last[index]]] & ) /@ Range[Length[matrixRepresentation]]]] & ) /@
Tuples[Range[Length[matrixRepresentation]], 2]]];
ricciScalar = Total[(Inverse[matrixRepresentation][[First[#1],Last[#1]]]*ricciTensor[[First[#1],Last[#1]]] & ) /@
Tuples[Range[Length[matrixRepresentation]], 2]];
Thread[Catenate[ricciTensor - (1/2)*ricciScalar*matrixRepresentation + cosmologicalConstant*matrixRepresentation] ==
Catenate[ConstantArray[0, {Length[matrixRepresentation], Length[matrixRepresentation]}]]]] /;
SymbolName[metricTensor] === "MetricTensor" && Length[Dimensions[matrixRepresentation]] == 2 &&
Length[coordinates] == Length[matrixRepresentation] && BooleanQ[index1] && BooleanQ[index2]
VacuumSolution[(metricTensor_)[matrixRepresentation_List, coordinates_List, index1_, index2_], cosmologicalConstant_][
"ReducedEinsteinEquations"] := Module[{newMatrixRepresentation, newCoordinates, christoffelSymbols, riemannTensor,
ricciTensor, ricciScalar}, newMatrixRepresentation = matrixRepresentation /. (#1 -> ToExpression[#1] & ) /@
Select[coordinates, StringQ]; newCoordinates = coordinates /. (#1 -> ToExpression[#1] & ) /@
Select[coordinates, StringQ]; christoffelSymbols =
Normal[SparseArray[(Module[{index = #1}, index -> Total[((1/2)*Inverse[newMatrixRepresentation][[index[[1]],#1]]*
(D[newMatrixRepresentation[[#1,index[[3]]]], newCoordinates[[index[[2]]]]] + D[newMatrixRepresentation[[
index[[2]],#1]], newCoordinates[[index[[3]]]]] - D[newMatrixRepresentation[[index[[2]],index[[3]]]],
newCoordinates[[#1]]]) & ) /@ Range[Length[newMatrixRepresentation]]]] & ) /@
Tuples[Range[Length[newMatrixRepresentation]], 3]]];
riemannTensor = Normal[SparseArray[(Module[{index = #1}, index -> D[christoffelSymbols[[index[[1]],index[[2]],
index[[4]]]], newCoordinates[[index[[3]]]]] - D[christoffelSymbols[[index[[1]],index[[2]],index[[3]]]],
newCoordinates[[index[[4]]]]] + Total[(christoffelSymbols[[index[[1]],#1,index[[3]]]]*christoffelSymbols[[
#1,index[[2]],index[[4]]]] & ) /@ Range[Length[newMatrixRepresentation]]] -
Total[(christoffelSymbols[[index[[1]],#1,index[[4]]]]*christoffelSymbols[[#1,index[[2]],index[[3]]]] & ) /@
Range[Length[newMatrixRepresentation]]]] & ) /@ Tuples[Range[Length[newMatrixRepresentation]], 4]]] /.
(ToExpression[#1] -> #1 & ) /@ Select[coordinates, StringQ];
ricciTensor = Normal[SparseArray[(Module[{index = #1}, index -> Total[(riemannTensor[[#1,First[index],#1,
Last[index]]] & ) /@ Range[Length[matrixRepresentation]]]] & ) /@
Tuples[Range[Length[matrixRepresentation]], 2]]];
ricciScalar = Total[(Inverse[matrixRepresentation][[First[#1],Last[#1]]]*ricciTensor[[First[#1],Last[#1]]] & ) /@
Tuples[Range[Length[matrixRepresentation]], 2]];
FullSimplify[Thread[Catenate[ricciTensor - (1/2)*ricciScalar*matrixRepresentation +
cosmologicalConstant*matrixRepresentation] == Catenate[ConstantArray[0, {Length[matrixRepresentation],
Length[matrixRepresentation]}]]]]] /; SymbolName[metricTensor] === "MetricTensor" &&
Length[Dimensions[matrixRepresentation]] == 2 && Length[coordinates] == Length[matrixRepresentation] &&
BooleanQ[index1] && BooleanQ[index2]
VacuumSolution[(metricTensor_)[matrixRepresentation_List, coordinates_List, index1_, index2_], cosmologicalConstant_][
"SymbolicEinsteinEquations"] := Module[{newMatrixRepresentation, newCoordinates, christoffelSymbols, riemannTensor,
ricciTensor, ricciScalar}, newMatrixRepresentation = matrixRepresentation /. (#1 -> ToExpression[#1] & ) /@
Select[coordinates, StringQ]; newCoordinates = coordinates /. (#1 -> ToExpression[#1] & ) /@
Select[coordinates, StringQ]; christoffelSymbols =
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]]];
riemannTensor = Normal[SparseArray[(Module[{index = #1}, index -> Inactive[D][christoffelSymbols[[index[[1]],
index[[2]],index[[4]]]], newCoordinates[[index[[3]]]]] - Inactive[D][christoffelSymbols[[index[[1]],
index[[2]],index[[3]]]], newCoordinates[[index[[4]]]]] + Total[(christoffelSymbols[[index[[1]],#1,
index[[3]]]]*christoffelSymbols[[#1,index[[2]],index[[4]]]] & ) /@
Range[Length[newMatrixRepresentation]]] - Total[(christoffelSymbols[[index[[1]],#1,index[[4]]]]*
christoffelSymbols[[#1,index[[2]],index[[3]]]] & ) /@ Range[Length[newMatrixRepresentation]]]] & ) /@
Tuples[Range[Length[newMatrixRepresentation]], 4]]] /. (ToExpression[#1] -> #1 & ) /@
Select[coordinates, StringQ]; ricciTensor =
Normal[SparseArray[(Module[{index = #1}, index -> Total[(riemannTensor[[#1,First[index],#1,Last[index]]] & ) /@
Range[Length[matrixRepresentation]]]] & ) /@ Tuples[Range[Length[matrixRepresentation]], 2]]];
ricciScalar = Total[(Inverse[matrixRepresentation][[First[#1],Last[#1]]]*ricciTensor[[First[#1],Last[#1]]] & ) /@
Tuples[Range[Length[matrixRepresentation]], 2]];
Thread[Catenate[ricciTensor - (1/2)*ricciScalar*matrixRepresentation + cosmologicalConstant*matrixRepresentation] ==
Catenate[ConstantArray[0, {Length[matrixRepresentation], Length[matrixRepresentation]}]]]] /;
SymbolName[metricTensor] === "MetricTensor" && Length[Dimensions[matrixRepresentation]] == 2 &&
Length[coordinates] == Length[matrixRepresentation] && BooleanQ[index1] && BooleanQ[index2]
VacuumSolution[(metricTensor_)[matrixRepresentation_List, coordinates_List, index1_, index2_], cosmologicalConstant_][
"MetricTensor"] := ResourceFunction["MetricTensor"][matrixRepresentation, coordinates, index1, index2] /;
SymbolName[metricTensor] === "MetricTensor" && Length[Dimensions[matrixRepresentation]] == 2 &&
Length[coordinates] == Length[matrixRepresentation] && BooleanQ[index1] && BooleanQ[index2]
VacuumSolution[(metricTensor_)[matrixRepresentation_List, coordinates_List, index1_, index2_], cosmologicalConstant_][
"Coordinates"] := coordinates /; SymbolName[metricTensor] === "MetricTensor" &&
Length[Dimensions[matrixRepresentation]] == 2 && Length[coordinates] == Length[matrixRepresentation] &&
BooleanQ[index1] && BooleanQ[index2]
VacuumSolution[(metricTensor_)[matrixRepresentation_List, coordinates_List, index1_, index2_], 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]]]] & ) /@ coordinates /; SymbolName[metricTensor] === "MetricTensor" &&
Length[Dimensions[matrixRepresentation]] == 2 && Length[coordinates] == Length[matrixRepresentation] &&
BooleanQ[index1] && BooleanQ[index2]
VacuumSolution[(metricTensor_)[matrixRepresentation_List, coordinates_List, index1_, index2_], cosmologicalConstant_][
"SolutionQ"] := Module[{newMatrixRepresentation, newCoordinates, christoffelSymbols, riemannTensor, ricciTensor,
ricciScalar, einsteinEquations}, newMatrixRepresentation = matrixRepresentation /.
(#1 -> ToExpression[#1] & ) /@ Select[coordinates, StringQ];
newCoordinates = coordinates /. (#1 -> ToExpression[#1] & ) /@ Select[coordinates, StringQ];
christoffelSymbols = Normal[SparseArray[
(Module[{index = #1}, index -> Total[((1/2)*Inverse[newMatrixRepresentation][[index[[1]],#1]]*
(D[newMatrixRepresentation[[#1,index[[3]]]], newCoordinates[[index[[2]]]]] + D[newMatrixRepresentation[[
index[[2]],#1]], newCoordinates[[index[[3]]]]] - D[newMatrixRepresentation[[index[[2]],index[[3]]]],
newCoordinates[[#1]]]) & ) /@ Range[Length[newMatrixRepresentation]]]] & ) /@
Tuples[Range[Length[newMatrixRepresentation]], 3]]];
riemannTensor = Normal[SparseArray[(Module[{index = #1}, index -> D[christoffelSymbols[[index[[1]],index[[2]],
index[[4]]]], newCoordinates[[index[[3]]]]] - D[christoffelSymbols[[index[[1]],index[[2]],index[[3]]]],
newCoordinates[[index[[4]]]]] + Total[(christoffelSymbols[[index[[1]],#1,index[[3]]]]*christoffelSymbols[[
#1,index[[2]],index[[4]]]] & ) /@ Range[Length[newMatrixRepresentation]]] -
Total[(christoffelSymbols[[index[[1]],#1,index[[4]]]]*christoffelSymbols[[#1,index[[2]],index[[3]]]] & ) /@
Range[Length[newMatrixRepresentation]]]] & ) /@ Tuples[Range[Length[newMatrixRepresentation]], 4]]] /.
(ToExpression[#1] -> #1 & ) /@ Select[coordinates, StringQ];
ricciTensor = Normal[SparseArray[(Module[{index = #1}, index -> Total[(riemannTensor[[#1,First[index],#1,
Last[index]]] & ) /@ Range[Length[matrixRepresentation]]]] & ) /@
Tuples[Range[Length[matrixRepresentation]], 2]]];
ricciScalar = Total[(Inverse[matrixRepresentation][[First[#1],Last[#1]]]*ricciTensor[[First[#1],Last[#1]]] & ) /@
Tuples[Range[Length[matrixRepresentation]], 2]]; einsteinEquations =
FullSimplify[Thread[Catenate[ricciTensor - (1/2)*ricciScalar*matrixRepresentation +
cosmologicalConstant*matrixRepresentation] == Catenate[ConstantArray[0, {Length[matrixRepresentation],
Length[matrixRepresentation]}]]]]; If[einsteinEquations === True, True,
If[einsteinEquations === False, False, If[Length[Select[einsteinEquations, #1 === False & ]] > 0, False,
True]]]] /; SymbolName[metricTensor] === "MetricTensor" && Length[Dimensions[matrixRepresentation]] == 2 &&
Length[coordinates] == Length[matrixRepresentation] && BooleanQ[index1] && BooleanQ[index2]
VacuumSolution[(metricTensor_)[matrixRepresentation_List, coordinates_List, index1_, index2_], cosmologicalConstant_][
"ExactSolutionQ"] := Module[{newMatrixRepresentation, newCoordinates, christoffelSymbols, riemannTensor, ricciTensor,
ricciScalar, einsteinEquations}, newMatrixRepresentation = matrixRepresentation /.
(#1 -> ToExpression[#1] & ) /@ Select[coordinates, StringQ];
newCoordinates = coordinates /. (#1 -> ToExpression[#1] & ) /@ Select[coordinates, StringQ];
christoffelSymbols = Normal[SparseArray[
(Module[{index = #1}, index -> Total[((1/2)*Inverse[newMatrixRepresentation][[index[[1]],#1]]*
(D[newMatrixRepresentation[[#1,index[[3]]]], newCoordinates[[index[[2]]]]] + D[newMatrixRepresentation[[
index[[2]],#1]], newCoordinates[[index[[3]]]]] - D[newMatrixRepresentation[[index[[2]],index[[3]]]],
newCoordinates[[#1]]]) & ) /@ Range[Length[newMatrixRepresentation]]]] & ) /@
Tuples[Range[Length[newMatrixRepresentation]], 3]]];
riemannTensor = Normal[SparseArray[(Module[{index = #1}, index -> D[christoffelSymbols[[index[[1]],index[[2]],
index[[4]]]], newCoordinates[[index[[3]]]]] - D[christoffelSymbols[[index[[1]],index[[2]],index[[3]]]],
newCoordinates[[index[[4]]]]] + Total[(christoffelSymbols[[index[[1]],#1,index[[3]]]]*christoffelSymbols[[
#1,index[[2]],index[[4]]]] & ) /@ Range[Length[newMatrixRepresentation]]] -
Total[(christoffelSymbols[[index[[1]],#1,index[[4]]]]*christoffelSymbols[[#1,index[[2]],index[[3]]]] & ) /@
Range[Length[newMatrixRepresentation]]]] & ) /@ Tuples[Range[Length[newMatrixRepresentation]], 4]]] /.
(ToExpression[#1] -> #1 & ) /@ Select[coordinates, StringQ];
ricciTensor = Normal[SparseArray[(Module[{index = #1}, index -> Total[(riemannTensor[[#1,First[index],#1,
Last[index]]] & ) /@ Range[Length[matrixRepresentation]]]] & ) /@
Tuples[Range[Length[matrixRepresentation]], 2]]];
ricciScalar = Total[(Inverse[matrixRepresentation][[First[#1],Last[#1]]]*ricciTensor[[First[#1],Last[#1]]] & ) /@
Tuples[Range[Length[matrixRepresentation]], 2]]; einsteinEquations =
FullSimplify[Thread[Catenate[ricciTensor - (1/2)*ricciScalar*matrixRepresentation +
cosmologicalConstant*matrixRepresentation] == Catenate[ConstantArray[0, {Length[matrixRepresentation],
Length[matrixRepresentation]}]]]]; If[einsteinEquations === True, True,
If[einsteinEquations === False, False, If[Length[Select[einsteinEquations, #1 === False & ]] > 0, False,
If[Length[DeleteDuplicates[Reverse /@ Sort /@ Select[einsteinEquations, #1 =!= True & ]]] == 0, True,
False]]]]] /; SymbolName[metricTensor] === "MetricTensor" && Length[Dimensions[matrixRepresentation]] == 2 &&
Length[coordinates] == Length[matrixRepresentation] && BooleanQ[index1] && BooleanQ[index2]
VacuumSolution[(metricTensor_)[matrixRepresentation_List, coordinates_List, index1_, index2_], cosmologicalConstant_][
"CosmologicalConstant"] := cosmologicalConstant /; SymbolName[metricTensor] === "MetricTensor" &&
Length[Dimensions[matrixRepresentation]] == 2 && Length[coordinates] == Length[matrixRepresentation] &&
BooleanQ[index1] && BooleanQ[index2]
VacuumSolution[(metricTensor_)[matrixRepresentation_List, coordinates_List, index1_, index2_], cosmologicalConstant_][
"Dimensions"] := Length[matrixRepresentation] /; SymbolName[metricTensor] === "MetricTensor" &&
Length[Dimensions[matrixRepresentation]] == 2 && Length[coordinates] == Length[matrixRepresentation] &&
BooleanQ[index1] && BooleanQ[index2]
VacuumSolution[(metricTensor_)[matrixRepresentation_List, coordinates_List, index1_, index2_], cosmologicalConstant_][
"Signature"] := Module[{eigenvalues, positiveEigenvalues, negativeEigenvalues},
eigenvalues = Eigenvalues[matrixRepresentation]; positiveEigenvalues = Select[eigenvalues, #1 > 0 & ];
negativeEigenvalues = Select[eigenvalues, #1 < 0 & ];
If[Length[positiveEigenvalues] + Length[negativeEigenvalues] == Length[matrixRepresentation],
Join[ConstantArray[-1, Length[negativeEigenvalues]], ConstantArray[1, Length[positiveEigenvalues]]],
Indeterminate]] /; SymbolName[metricTensor] === "MetricTensor" && Length[Dimensions[matrixRepresentation]] == 2 &&
Length[coordinates] == Length[matrixRepresentation] && BooleanQ[index1] && BooleanQ[index2]
VacuumSolution[(metricTensor_)[matrixRepresentation_List, coordinates_List, index1_, index2_], cosmologicalConstant_][
"RiemannianQ"] := Module[{eigenvalues, positiveEigenvalues, negativeEigenvalues},
eigenvalues = Eigenvalues[matrixRepresentation]; positiveEigenvalues = Select[eigenvalues, #1 > 0 & ];
negativeEigenvalues = Select[eigenvalues, #1 < 0 & ];
If[Length[positiveEigenvalues] + Length[negativeEigenvalues] == Length[matrixRepresentation],
If[Length[positiveEigenvalues] == Length[matrixRepresentation] || Length[negativeEigenvalues] ==
Length[matrixRepresentation], True, False], Indeterminate]] /; SymbolName[metricTensor] === "MetricTensor" &&
Length[Dimensions[matrixRepresentation]] == 2 && Length[coordinates] == Length[matrixRepresentation] &&
BooleanQ[index1] && BooleanQ[index2]
VacuumSolution[(metricTensor_)[matrixRepresentation_List, coordinates_List, index1_, index2_], cosmologicalConstant_][
"PseudoRiemannianQ"] := Module[{eigenvalues, positiveEigenvalues, negativeEigenvalues},
eigenvalues = Eigenvalues[matrixRepresentation]; positiveEigenvalues = Select[eigenvalues, #1 > 0 & ];
negativeEigenvalues = Select[eigenvalues, #1 < 0 & ];
If[Length[positiveEigenvalues] + Length[negativeEigenvalues] == Length[matrixRepresentation],
If[Length[positiveEigenvalues] == Length[matrixRepresentation] || Length[negativeEigenvalues] ==
Length[matrixRepresentation], False, True], Indeterminate]] /; SymbolName[metricTensor] === "MetricTensor" &&
Length[Dimensions[matrixRepresentation]] == 2 && Length[coordinates] == Length[matrixRepresentation] &&
BooleanQ[index1] && BooleanQ[index2]
VacuumSolution[(metricTensor_)[matrixRepresentation_List, coordinates_List, index1_, index2_], cosmologicalConstant_][
"LorentzianQ"] := Module[{eigenvalues, positiveEigenvalues, negativeEigenvalues},
eigenvalues = Eigenvalues[matrixRepresentation]; positiveEigenvalues = Select[eigenvalues, #1 > 0 & ];
negativeEigenvalues = Select[eigenvalues, #1 < 0 & ];
If[Length[positiveEigenvalues] + Length[negativeEigenvalues] == Length[matrixRepresentation],
If[Length[positiveEigenvalues] == 1 || Length[negativeEigenvalues] == 1, True, False], Indeterminate]] /;
SymbolName[metricTensor] === "MetricTensor" && Length[Dimensions[matrixRepresentation]] == 2 &&
Length[coordinates] == Length[matrixRepresentation] && BooleanQ[index1] && BooleanQ[index2]
VacuumSolution[(metricTensor_)[matrixRepresentation_List, coordinates_List, index1_, index2_], cosmologicalConstant_][
"RiemannianConditions"] := Module[{eigenvalues, riemannianConditions},
eigenvalues = Eigenvalues[matrixRepresentation]; riemannianConditions = FullSimplify[(#1 > 0 & ) /@ eigenvalues];
If[riemannianConditions === True, {}, If[riemannianConditions === False, Indeterminate,
If[Length[Select[riemannianConditions, #1 === False & ]] > 0, Indeterminate,
DeleteDuplicates[Select[riemannianConditions, #1 =!= True & ]]]]]] /;
SymbolName[metricTensor] === "MetricTensor" && Length[Dimensions[matrixRepresentation]] == 2 &&
Length[coordinates] == Length[matrixRepresentation] && BooleanQ[index1] && BooleanQ[index2]
VacuumSolution[(metricTensor_)[matrixRepresentation_, coordinates_List, index1_, index2_], cosmologicalConstant_][
"PseudoRiemannianConditions"] := Module[{eigenvalues, pseudoRiemannianConditions},
eigenvalues = Eigenvalues[matrixRepresentation]; pseudoRiemannianConditions =
FullSimplify[(#1 != 0 & ) /@ eigenvalues]; If[pseudoRiemannianConditions === True, {},
If[pseudoRiemannianConditions === False, Indeterminate,
If[Length[Select[pseudoRiemannianConditions, #1 === False & ]] > 0, Indeterminate,
DeleteDuplicates[Reverse /@ Sort /@ Select[pseudoRiemannianConditions, #1 =!= True & ]]]]]] /;
SymbolName[metricTensor] === "MetricTensor" && Length[Dimensions[matrixRepresentation]] == 2 &&
Length[coordinates] == Length[matrixRepresentation] && BooleanQ[index1] && BooleanQ[index2]
VacuumSolution[(metricTensor_)[matrixRepresentation_List, coordinates_List, index1_, index2_], cosmologicalConstant_][
"LorentzianConditions"] := Module[{eigensystem, eigenvalues, eigenvectors, timeCoordinate, lorentzianConditions},
eigensystem = Eigensystem[matrixRepresentation]; eigenvalues = First[eigensystem]; eigenvectors = Last[eigensystem];
If[Length[Position[eigenvectors, Join[{1}, ConstantArray[0, Length[coordinates] - 1]]]] > 0,
timeCoordinate = First[First[Position[eigenvectors, Join[{1}, ConstantArray[0, Length[coordinates] - 1]]]]];
lorentzianConditions = FullSimplify[(If[#1 == timeCoordinate, eigenvalues[[#1]] < 0, eigenvalues[[#1]] > 0] & ) /@
Range[Length[eigenvalues]]]; If[lorentzianConditions === True, {}, If[lorentzianConditions === False,
Indeterminate, If[Length[Select[lorentzianConditions, #1 === False & ]] > 0, Indeterminate,
DeleteDuplicates[Select[lorentzianConditions, #1 =!= True & ]]]]], Indeterminate]] /;
SymbolName[metricTensor] === "MetricTensor" && Length[Dimensions[matrixRepresentation]] == 2 &&
Length[coordinates] == Length[matrixRepresentation] && BooleanQ[index1] && BooleanQ[index2]
VacuumSolution[(metricTensor_)[matrixRepresentation_List, coordinates_List, index1_, index2_], cosmologicalConstant_][
"Properties"] := {"FieldEquations", "EinsteinEquations", "ReducedEinsteinEquations", "SymbolicEinsteinEquations",
"MetricTensor", "Coordinates", "CoordinateOneForms", "SolutionQ", "ExactSolutionQ", "CosmologicalConstant",
"Dimensions", "Signature", "RiemannianQ", "PseudoRiemannianQ", "LorentzianQ", "RiemannianConditions",
"PseudoRiemannianConditions", "LorentzianConditions", "Properties"} /; SymbolName[metricTensor] === "MetricTensor" &&
Length[Dimensions[matrixRepresentation]] == 2 && Length[coordinates] == Length[matrixRepresentation] &&
BooleanQ[index1] && BooleanQ[index2]
VacuumSolution /: MakeBoxes[vacuumSolution:VacuumSolution[(metricTensor_)[matrixRepresentation_List, coordinates_List,
index1_, index2_], cosmologicalConstant_], format_] :=
Module[{newMatrixRepresentation, newCoordinates, christoffelSymbols, riemannTensor, ricciTensor, ricciScalar,
matrixForm, einsteinEquations, solution, exactSolution, fieldEquations, icon},
newMatrixRepresentation = matrixRepresentation /. (#1 -> ToExpression[#1] & ) /@ Select[coordinates, StringQ];
newCoordinates = coordinates /. (#1 -> ToExpression[#1] & ) /@ Select[coordinates, StringQ];
christoffelSymbols = Normal[SparseArray[
(Module[{index = #1}, index -> Total[((1/2)*Inverse[newMatrixRepresentation][[index[[1]],#1]]*
(D[newMatrixRepresentation[[#1,index[[3]]]], newCoordinates[[index[[2]]]]] + D[newMatrixRepresentation[[
index[[2]],#1]], newCoordinates[[index[[3]]]]] - D[newMatrixRepresentation[[index[[2]],index[[3]]]],
newCoordinates[[#1]]]) & ) /@ Range[Length[newMatrixRepresentation]]]] & ) /@
Tuples[Range[Length[newMatrixRepresentation]], 3]]]; riemannTensor =
Normal[SparseArray[(Module[{index = #1}, index -> D[christoffelSymbols[[index[[1]],index[[2]],index[[4]]]],
newCoordinates[[index[[3]]]]] - D[christoffelSymbols[[index[[1]],index[[2]],index[[3]]]],
newCoordinates[[index[[4]]]]] + Total[(christoffelSymbols[[index[[1]],#1,index[[3]]]]*christoffelSymbols[[
#1,index[[2]],index[[4]]]] & ) /@ Range[Length[newMatrixRepresentation]]] - Total[
(christoffelSymbols[[index[[1]],#1,index[[4]]]]*christoffelSymbols[[#1,index[[2]],index[[3]]]] & ) /@
Range[Length[newMatrixRepresentation]]]] & ) /@ Tuples[Range[Length[newMatrixRepresentation]], 4]]] /.
(ToExpression[#1] -> #1 & ) /@ Select[coordinates, StringQ];
ricciTensor = Normal[SparseArray[(Module[{index = #1}, index -> Total[(riemannTensor[[#1,First[index],#1,
Last[index]]] & ) /@ Range[Length[matrixRepresentation]]]] & ) /@
Tuples[Range[Length[matrixRepresentation]], 2]]]; ricciScalar =
Total[(Inverse[matrixRepresentation][[First[#1],Last[#1]]]*ricciTensor[[First[#1],Last[#1]]] & ) /@
Tuples[Range[Length[matrixRepresentation]], 2]];
matrixForm = ricciTensor - (1/2)*ricciScalar*matrixRepresentation + cosmologicalConstant*matrixRepresentation;
einsteinEquations = FullSimplify[Thread[Catenate[ricciTensor - (1/2)*ricciScalar*matrixRepresentation +
cosmologicalConstant*matrixRepresentation] == Catenate[ConstantArray[0, {Length[matrixRepresentation],
Length[matrixRepresentation]}]]]]; If[einsteinEquations === True, solution = True; exactSolution = True;
fieldEquations = 0, If[einsteinEquations === False, solution = False; exactSolution = False;
fieldEquations = Indeterminate, If[Length[Select[einsteinEquations, #1 === False & ]] > 0,
solution = False; exactSolution = False; fieldEquations = Indeterminate,
If[Length[DeleteDuplicates[Reverse /@ Sort /@ Select[einsteinEquations, #1 =!= True & ]]] == 0,
solution = True; exactSolution = True; fieldEquations = 0, solution = True; exactSolution = False;
fieldEquations = Length[DeleteDuplicates[Reverse /@ Sort /@ Select[einsteinEquations, #1 =!= True & ]]]]]]];
icon = MatrixPlot[matrixForm, ImageSize -> Dynamic[{Automatic, 3.5*(CurrentValue["FontCapHeight"]/
AbsoluteCurrentValue[Magnification])}], Frame -> False, FrameTicks -> None];
BoxForm`ArrangeSummaryBox["VacuumSolution", vacuumSolution, icon,
{{BoxForm`SummaryItem[{"Solution: ", solution}], BoxForm`SummaryItem[{"Exact Solution: ", exactSolution}]},
{BoxForm`SummaryItem[{"Field Equations: ", fieldEquations}], BoxForm`SummaryItem[{"Cosmological Constant: ",
cosmologicalConstant}]}}, {{}}, format, "Interpretable" -> Automatic]] /;
SymbolName[metricTensor] === "MetricTensor" && Length[Dimensions[matrixRepresentation]] == 2 &&
Length[coordinates] == Length[matrixRepresentation] && BooleanQ[index1] && BooleanQ[index2]