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SolveEinsteinEquations.wl
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(* ::Package:: *)
SolveEinsteinEquations[(stressEnergyTensor_)[(metricTensor_)[metricMatrixRepresentation_List, coordinates_List,
metricIndex1_, metricIndex2_], matrixRepresentation_List, index1_, index2_]] :=
EinsteinSolution[ResourceFunction["StressEnergyTensor"][ResourceFunction["MetricTensor"][metricMatrixRepresentation,
coordinates, metricIndex1, metricIndex2], matrixRepresentation, index1, index2], "\[FormalCapitalLambda]"] /;
SymbolName[stressEnergyTensor] === "StressEnergyTensor" && SymbolName[metricTensor] === "MetricTensor" &&
Length[Dimensions[metricMatrixRepresentation]] == 2 && Length[coordinates] == Length[metricMatrixRepresentation] &&
BooleanQ[metricIndex1] && BooleanQ[metricIndex2] && Length[Dimensions[matrixRepresentation]] == 2 &&
Length[coordinates] == Length[matrixRepresentation] && BooleanQ[index1] && BooleanQ[index2]
SolveEinsteinEquations[(stressEnergyTensor_)[(metricTensor_)[metricMatrixRepresentation_List, coordinates_List,
metricIndex1_, metricIndex2_], matrixRepresentation_List, index1_, index2_],
(metricTensor_)[newMetricMatrixRepresentation_List, newCoordinates_List, newMetricIndex1_, newMetricIndex2_]] :=
EinsteinSolution[ResourceFunction["StressEnergyTensor"][ResourceFunction["MetricTensor"][newMetricMatrixRepresentation,
newCoordinates, newMetricIndex1, newMetricIndex2], matrixRepresentation /. Thread[coordinates -> newCoordinates],
index1, index2], "\[FormalCapitalLambda]"] /; SymbolName[stressEnergyTensor] === "StressEnergyTensor" &&
SymbolName[metricTensor] === "MetricTensor" && Length[Dimensions[metricMatrixRepresentation]] == 2 &&
Length[coordinates] == Length[metricMatrixRepresentation] && BooleanQ[metricIndex1] && BooleanQ[metricIndex2] &&
Length[Dimensions[matrixRepresentation]] == 2 && Length[coordinates] == Length[matrixRepresentation] &&
BooleanQ[index1] && BooleanQ[index2] && Length[Dimensions[newMetricMatrixRepresentation]] == 2 &&
Length[newCoordinates] == Length[newMetricMatrixRepresentation] && BooleanQ[newMetricIndex1] &&
BooleanQ[newMetricIndex2] && Length[newCoordinates] == Length[matrixRepresentation]
SolveEinsteinEquations[(stressEnergyTensor_)[(metricTensor_)[metricMatrixRepresentation_List, coordinates_List,
metricIndex1_, metricIndex2_], matrixRepresentation_List, index1_, index2_], cosmologicalConstant_] :=
EinsteinSolution[ResourceFunction["StressEnergyTensor"][ResourceFunction["MetricTensor"][metricMatrixRepresentation,
coordinates, metricIndex1, metricIndex2], matrixRepresentation, index1, index2], cosmologicalConstant] /;
SymbolName[stressEnergyTensor] === "StressEnergyTensor" && SymbolName[metricTensor] === "MetricTensor" &&
Length[Dimensions[metricMatrixRepresentation]] == 2 && Length[coordinates] == Length[metricMatrixRepresentation] &&
BooleanQ[metricIndex1] && BooleanQ[metricIndex2] && Length[Dimensions[matrixRepresentation]] == 2 &&
Length[coordinates] == Length[matrixRepresentation] && BooleanQ[index1] && BooleanQ[index2]
SolveEinsteinEquations[(stressEnergyTensor_)[(metricTensor_)[metricMatrixRepresentation_List, coordinates_List,
metricIndex1_, metricIndex2_], matrixRepresentation_List, index1_, index2_],
(metricTensor_)[newMetricMatrixRepresentation_List, newCoordinates_List, newMetricIndex1_, newMetricIndex2_],
cosmologicalConstant_] := EinsteinSolution[ResourceFunction["StressEnergyTensor"][
ResourceFunction["MetricTensor"][newMetricMatrixRepresentation, newCoordinates, newMetricIndex1, newMetricIndex2],
matrixRepresentation /. Thread[coordinates -> newCoordinates], index1, index2], cosmologicalConstant] /;
SymbolName[stressEnergyTensor] === "StressEnergyTensor" && SymbolName[metricTensor] === "MetricTensor" &&
Length[Dimensions[metricMatrixRepresentation]] == 2 && Length[coordinates] == Length[metricMatrixRepresentation] &&
BooleanQ[metricIndex1] && BooleanQ[metricIndex2] && Length[Dimensions[matrixRepresentation]] == 2 &&
Length[coordinates] == Length[matrixRepresentation] && BooleanQ[index1] && BooleanQ[index2] &&
Length[Dimensions[newMetricMatrixRepresentation]] == 2 && Length[newCoordinates] ==
Length[newMetricMatrixRepresentation] && BooleanQ[newMetricIndex1] && BooleanQ[newMetricIndex2] &&
Length[newCoordinates] == Length[matrixRepresentation]
EinsteinSolution[(stressEnergyTensor_)[(metricTensor_)[metricMatrixRepresentation_List, coordinates_List, metricIndex1_,
metricIndex2_], matrixRepresentation_List, index1_, index2_], cosmologicalConstant_]["FieldEquations"] :=
Module[{newMetricMatrixRepresentation, newCoordinates, christoffelSymbols, riemannTensor, ricciTensor, ricciScalar,
tensorRepresentation, einsteinEquations}, newMetricMatrixRepresentation = metricMatrixRepresentation /.
(#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[newMetricMatrixRepresentation][[index[[1]],#1]]*
(D[newMetricMatrixRepresentation[[#1,index[[3]]]], newCoordinates[[index[[2]]]]] +
D[newMetricMatrixRepresentation[[index[[2]],#1]], newCoordinates[[index[[3]]]]] -
D[newMetricMatrixRepresentation[[index[[2]],index[[3]]]], newCoordinates[[#1]]]) & ) /@
Range[Length[newMetricMatrixRepresentation]]]] & ) /@ Tuples[Range[Length[newMetricMatrixRepresentation]],
3]]]; 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[newMetricMatrixRepresentation]]] -
Total[(christoffelSymbols[[index[[1]],#1,index[[4]]]]*christoffelSymbols[[#1,index[[2]],index[[3]]]] & ) /@
Range[Length[newMetricMatrixRepresentation]]]] & ) /@
Tuples[Range[Length[newMetricMatrixRepresentation]], 4]]] /. (ToExpression[#1] -> #1 & ) /@
Select[coordinates, StringQ]; ricciTensor =
Normal[SparseArray[(Module[{index = #1}, index -> Total[(riemannTensor[[#1,First[index],#1,Last[index]]] & ) /@
Range[Length[metricMatrixRepresentation]]]] & ) /@ Tuples[Range[Length[metricMatrixRepresentation]], 2]]];
ricciScalar = Total[(Inverse[metricMatrixRepresentation][[First[#1],Last[#1]]]*ricciTensor[[First[#1],
Last[#1]]] & ) /@ Tuples[Range[Length[metricMatrixRepresentation]], 2]];
tensorRepresentation = Normal[SparseArray[
(Module[{index = #1}, index -> Total[(metricMatrixRepresentation[[First[index],First[#1]]]*
metricMatrixRepresentation[[Last[#1],Last[index]]]*matrixRepresentation[[First[#1],Last[#1]]] & ) /@
Tuples[Range[Length[metricMatrixRepresentation]], 2]]] & ) /@
Tuples[Range[Length[metricMatrixRepresentation]], 2]]]; einsteinEquations =
FullSimplify[Thread[Catenate[ricciTensor - (1/2)*ricciScalar*metricMatrixRepresentation +
cosmologicalConstant*metricMatrixRepresentation] == Catenate[(8*Pi)*tensorRepresentation]]];
If[einsteinEquations === True, {}, If[einsteinEquations === False, Indeterminate,
If[Length[Select[einsteinEquations, #1 === False & ]] > 0, Indeterminate,
DeleteDuplicates[Reverse /@ Sort /@ Select[einsteinEquations, #1 =!= True & ]]]]]] /;
SymbolName[stressEnergyTensor] === "StressEnergyTensor" && SymbolName[metricTensor] === "MetricTensor" &&
Length[Dimensions[metricMatrixRepresentation]] == 2 && Length[coordinates] == Length[metricMatrixRepresentation] &&
BooleanQ[metricIndex1] && BooleanQ[metricIndex2] && Length[Dimensions[matrixRepresentation]] == 2 &&
Length[coordinates] == Length[matrixRepresentation] && BooleanQ[index1] && BooleanQ[index2]
EinsteinSolution[(stressEnergyTensor_)[(metricTensor_)[metricMatrixRepresentation_List, coordinates_List, metricIndex1_,
metricIndex2_], matrixRepresentation_List, index1_, index2_], cosmologicalConstant_]["EinsteinEquations"] :=
Module[{newMetricMatrixRepresentation, newCoordinates, christoffelSymbols, riemannTensor, ricciTensor, ricciScalar,
tensorRepresentation}, newMetricMatrixRepresentation = metricMatrixRepresentation /.
(#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[newMetricMatrixRepresentation][[index[[1]],#1]]*
(D[newMetricMatrixRepresentation[[#1,index[[3]]]], newCoordinates[[index[[2]]]]] +
D[newMetricMatrixRepresentation[[index[[2]],#1]], newCoordinates[[index[[3]]]]] -
D[newMetricMatrixRepresentation[[index[[2]],index[[3]]]], newCoordinates[[#1]]]) & ) /@
Range[Length[newMetricMatrixRepresentation]]]] & ) /@ Tuples[Range[Length[newMetricMatrixRepresentation]],
3]]]; 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[newMetricMatrixRepresentation]]] -
Total[(christoffelSymbols[[index[[1]],#1,index[[4]]]]*christoffelSymbols[[#1,index[[2]],index[[3]]]] & ) /@
Range[Length[newMetricMatrixRepresentation]]]] & ) /@
Tuples[Range[Length[newMetricMatrixRepresentation]], 4]]] /. (ToExpression[#1] -> #1 & ) /@
Select[coordinates, StringQ]; ricciTensor =
Normal[SparseArray[(Module[{index = #1}, index -> Total[(riemannTensor[[#1,First[index],#1,Last[index]]] & ) /@
Range[Length[metricMatrixRepresentation]]]] & ) /@ Tuples[Range[Length[metricMatrixRepresentation]], 2]]];
ricciScalar = Total[(Inverse[metricMatrixRepresentation][[First[#1],Last[#1]]]*ricciTensor[[First[#1],
Last[#1]]] & ) /@ Tuples[Range[Length[metricMatrixRepresentation]], 2]];
tensorRepresentation = Normal[SparseArray[
(Module[{index = #1}, index -> Total[(metricMatrixRepresentation[[First[index],First[#1]]]*
metricMatrixRepresentation[[Last[#1],Last[index]]]*matrixRepresentation[[First[#1],Last[#1]]] & ) /@
Tuples[Range[Length[metricMatrixRepresentation]], 2]]] & ) /@
Tuples[Range[Length[metricMatrixRepresentation]], 2]]];
Thread[Catenate[ricciTensor - (1/2)*ricciScalar*metricMatrixRepresentation + cosmologicalConstant*
metricMatrixRepresentation] == Catenate[(8*Pi)*tensorRepresentation]]] /;
SymbolName[stressEnergyTensor] === "StressEnergyTensor" && SymbolName[metricTensor] === "MetricTensor" &&
Length[Dimensions[metricMatrixRepresentation]] == 2 && Length[coordinates] == Length[metricMatrixRepresentation] &&
BooleanQ[metricIndex1] && BooleanQ[metricIndex2] && Length[Dimensions[matrixRepresentation]] == 2 &&
Length[coordinates] == Length[matrixRepresentation] && BooleanQ[index1] && BooleanQ[index2]
EinsteinSolution[(stressEnergyTensor_)[(metricTensor_)[metricMatrixRepresentation_List, coordinates_List, metricIndex1_,
metricIndex2_], matrixRepresentation_List, index1_, index2_], cosmologicalConstant_]["ReducedEinsteinEquations"] :=
Module[{newMetricMatrixRepresentation, newCoordinates, christoffelSymbols, riemannTensor, ricciTensor, ricciScalar,
tensorRepresentation}, newMetricMatrixRepresentation = metricMatrixRepresentation /.
(#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[newMetricMatrixRepresentation][[index[[1]],#1]]*
(D[newMetricMatrixRepresentation[[#1,index[[3]]]], newCoordinates[[index[[2]]]]] +
D[newMetricMatrixRepresentation[[index[[2]],#1]], newCoordinates[[index[[3]]]]] -
D[newMetricMatrixRepresentation[[index[[2]],index[[3]]]], newCoordinates[[#1]]]) & ) /@
Range[Length[newMetricMatrixRepresentation]]]] & ) /@ Tuples[Range[Length[newMetricMatrixRepresentation]],
3]]]; 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[newMetricMatrixRepresentation]]] -
Total[(christoffelSymbols[[index[[1]],#1,index[[4]]]]*christoffelSymbols[[#1,index[[2]],index[[3]]]] & ) /@
Range[Length[newMetricMatrixRepresentation]]]] & ) /@
Tuples[Range[Length[newMetricMatrixRepresentation]], 4]]] /. (ToExpression[#1] -> #1 & ) /@
Select[coordinates, StringQ]; ricciTensor =
Normal[SparseArray[(Module[{index = #1}, index -> Total[(riemannTensor[[#1,First[index],#1,Last[index]]] & ) /@
Range[Length[metricMatrixRepresentation]]]] & ) /@ Tuples[Range[Length[metricMatrixRepresentation]], 2]]];
ricciScalar = Total[(Inverse[metricMatrixRepresentation][[First[#1],Last[#1]]]*ricciTensor[[First[#1],
Last[#1]]] & ) /@ Tuples[Range[Length[metricMatrixRepresentation]], 2]];
tensorRepresentation = Normal[SparseArray[
(Module[{index = #1}, index -> Total[(metricMatrixRepresentation[[First[index],First[#1]]]*
metricMatrixRepresentation[[Last[#1],Last[index]]]*matrixRepresentation[[First[#1],Last[#1]]] & ) /@
Tuples[Range[Length[metricMatrixRepresentation]], 2]]] & ) /@
Tuples[Range[Length[metricMatrixRepresentation]], 2]]];
FullSimplify[Thread[Catenate[ricciTensor - (1/2)*ricciScalar*metricMatrixRepresentation +
cosmologicalConstant*metricMatrixRepresentation] == Catenate[(8*Pi)*tensorRepresentation]]]] /;
SymbolName[stressEnergyTensor] === "StressEnergyTensor" && SymbolName[metricTensor] === "MetricTensor" &&
Length[Dimensions[metricMatrixRepresentation]] == 2 && Length[coordinates] == Length[metricMatrixRepresentation] &&
BooleanQ[metricIndex1] && BooleanQ[metricIndex2] && Length[Dimensions[matrixRepresentation]] == 2 &&
Length[coordinates] == Length[matrixRepresentation] && BooleanQ[index1] && BooleanQ[index2]
EinsteinSolution[(stressEnergyTensor_)[(metricTensor_)[metricMatrixRepresentation_List, coordinates_List, metricIndex1_,
metricIndex2_], matrixRepresentation_List, index1_, index2_], cosmologicalConstant_][
"SymbolicEinsteinEquations"] := Module[{newMetricMatrixRepresentation, newCoordinates, christoffelSymbols,
riemannTensor, ricciTensor, ricciScalar, tensorRepresentation},
newMetricMatrixRepresentation = metricMatrixRepresentation /. (#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[newMetricMatrixRepresentation][[index[[1]],
#1]]*(Inactive[D][newMetricMatrixRepresentation[[#1,index[[3]]]], newCoordinates[[index[[2]]]]] +
Inactive[D][newMetricMatrixRepresentation[[index[[2]],#1]], newCoordinates[[index[[3]]]]] -
Inactive[D][newMetricMatrixRepresentation[[index[[2]],index[[3]]]], newCoordinates[[#1]]]) & ) /@
Range[Length[newMetricMatrixRepresentation]]]] & ) /@ Tuples[Range[Length[newMetricMatrixRepresentation]],
3]]]; 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[newMetricMatrixRepresentation]]] -
Total[(christoffelSymbols[[index[[1]],#1,index[[4]]]]*christoffelSymbols[[#1,index[[2]],index[[3]]]] & ) /@
Range[Length[newMetricMatrixRepresentation]]]] & ) /@
Tuples[Range[Length[newMetricMatrixRepresentation]], 4]]] /. (ToExpression[#1] -> #1 & ) /@
Select[coordinates, StringQ]; ricciTensor =
Normal[SparseArray[(Module[{index = #1}, index -> Total[(riemannTensor[[#1,First[index],#1,Last[index]]] & ) /@
Range[Length[metricMatrixRepresentation]]]] & ) /@ Tuples[Range[Length[metricMatrixRepresentation]], 2]]];
ricciScalar = Total[(Inverse[metricMatrixRepresentation][[First[#1],Last[#1]]]*ricciTensor[[First[#1],
Last[#1]]] & ) /@ Tuples[Range[Length[metricMatrixRepresentation]], 2]];
tensorRepresentation = Normal[SparseArray[
(Module[{index = #1}, index -> Total[(metricMatrixRepresentation[[First[index],First[#1]]]*
metricMatrixRepresentation[[Last[#1],Last[index]]]*matrixRepresentation[[First[#1],Last[#1]]] & ) /@
Tuples[Range[Length[metricMatrixRepresentation]], 2]]] & ) /@
Tuples[Range[Length[metricMatrixRepresentation]], 2]]];
Thread[Catenate[ricciTensor - (1/2)*ricciScalar*metricMatrixRepresentation + cosmologicalConstant*
metricMatrixRepresentation] == Catenate[(8*Pi)*tensorRepresentation]]] /;
SymbolName[stressEnergyTensor] === "StressEnergyTensor" && SymbolName[metricTensor] === "MetricTensor" &&
Length[Dimensions[metricMatrixRepresentation]] == 2 && Length[coordinates] == Length[metricMatrixRepresentation] &&
BooleanQ[metricIndex1] && BooleanQ[metricIndex2] && Length[Dimensions[matrixRepresentation]] == 2 &&
Length[coordinates] == Length[matrixRepresentation] && BooleanQ[index1] && BooleanQ[index2]
EinsteinSolution[(stressEnergyTensor_)[(metricTensor_)[metricMatrixRepresentation_List, coordinates_List, metricIndex1_,
metricIndex2_], matrixRepresentation_List, index1_, index2_], cosmologicalConstant_]["ContinuityEquations"] :=
Module[{newMetricMatrixRepresentation, newMatrixRepresentation, newCoordinates, christoffelSymbols},
newMetricMatrixRepresentation = metricMatrixRepresentation /. (#1 -> ToExpression[#1] & ) /@
Select[coordinates, StringQ]; 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[newMetricMatrixRepresentation][[index[[1]],#1]]*
(D[newMetricMatrixRepresentation[[#1,index[[3]]]], newCoordinates[[index[[2]]]]] +
D[newMetricMatrixRepresentation[[index[[2]],#1]], newCoordinates[[index[[3]]]]] -
D[newMetricMatrixRepresentation[[index[[2]],index[[3]]]], newCoordinates[[#1]]]) & ) /@
Range[Length[newMetricMatrixRepresentation]]]] & ) /@ Tuples[Range[Length[newMetricMatrixRepresentation]],
3]]]; (Module[{index = #1}, D[newMatrixRepresentation[[First[index],Last[index]]],
newCoordinates[[Last[index]]]] + Total[(christoffelSymbols[[First[index],Last[index],#1]]*
newMatrixRepresentation[[#1,Last[index]]] & ) /@ Range[Length[newMatrixRepresentation]]] +
Total[(christoffelSymbols[[Last[index],Last[index],#1]]*newMatrixRepresentation[[First[index],#1]] & ) /@
Range[Length[newMatrixRepresentation]]] == 0] & ) /@ Tuples[Range[Length[newMatrixRepresentation]], 2] /.
(ToExpression[#1] -> #1 & ) /@ Select[coordinates, StringQ]] /;
SymbolName[stressEnergyTensor] === "StressEnergyTensor" && SymbolName[metricTensor] === "MetricTensor" &&
Length[Dimensions[metricMatrixRepresentation]] == 2 && Length[coordinates] == Length[metricMatrixRepresentation] &&
BooleanQ[metricIndex1] && BooleanQ[metricIndex2] && Length[Dimensions[matrixRepresentation]] == 2 &&
Length[coordinates] == Length[matrixRepresentation] && BooleanQ[index1] && BooleanQ[index2]
EinsteinSolution[(stressEnergyTensor_)[(metricTensor_)[metricMatrixRepresentation_List, coordinates_List, metricIndex1_,
metricIndex2_], matrixRepresentation_List, index1_, index2_], cosmologicalConstant_][
"ReducedContinuityEquations"] := Module[{newMetricMatrixRepresentation, newMatrixRepresentation, newCoordinates,
christoffelSymbols}, newMetricMatrixRepresentation = metricMatrixRepresentation /.
(#1 -> ToExpression[#1] & ) /@ Select[coordinates, StringQ]; 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[newMetricMatrixRepresentation][[index[[1]],#1]]*
(D[newMetricMatrixRepresentation[[#1,index[[3]]]], newCoordinates[[index[[2]]]]] +
D[newMetricMatrixRepresentation[[index[[2]],#1]], newCoordinates[[index[[3]]]]] -
D[newMetricMatrixRepresentation[[index[[2]],index[[3]]]], newCoordinates[[#1]]]) & ) /@
Range[Length[newMetricMatrixRepresentation]]]] & ) /@ Tuples[Range[Length[newMetricMatrixRepresentation]],
3]]]; FullSimplify[(Module[{index = #1}, D[newMatrixRepresentation[[First[index],Last[index]]],
newCoordinates[[Last[index]]]] + Total[(christoffelSymbols[[First[index],Last[index],#1]]*
newMatrixRepresentation[[#1,Last[index]]] & ) /@ Range[Length[newMatrixRepresentation]]] +
Total[(christoffelSymbols[[Last[index],Last[index],#1]]*newMatrixRepresentation[[First[index],#1]] & ) /@
Range[Length[newMatrixRepresentation]]] == 0] & ) /@ Tuples[Range[Length[newMatrixRepresentation]], 2] /.
(ToExpression[#1] -> #1 & ) /@ Select[coordinates, StringQ]]] /;
SymbolName[stressEnergyTensor] === "StressEnergyTensor" && SymbolName[metricTensor] === "MetricTensor" &&
Length[Dimensions[metricMatrixRepresentation]] == 2 && Length[coordinates] == Length[metricMatrixRepresentation] &&
BooleanQ[metricIndex1] && BooleanQ[metricIndex2] && Length[Dimensions[matrixRepresentation]] == 2 &&
Length[coordinates] == Length[matrixRepresentation] && BooleanQ[index1] && BooleanQ[index2]
EinsteinSolution[(stressEnergyTensor_)[(metricTensor_)[metricMatrixRepresentation_List, coordinates_List, metricIndex1_,
metricIndex2_], matrixRepresentation_List, index1_, index2_], cosmologicalConstant_][
"SymbolicContinuityEquations"] := Module[{newMetricMatrixRepresentation, newMatrixRepresentation, newCoordinates,
christoffelSymbols}, newMetricMatrixRepresentation = metricMatrixRepresentation /.
(#1 -> ToExpression[#1] & ) /@ Select[coordinates, StringQ]; 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[newMetricMatrixRepresentation][[index[[1]],#1]]*
(Inactive[D][newMetricMatrixRepresentation[[#1,index[[3]]]], newCoordinates[[index[[2]]]]] +
Inactive[D][newMetricMatrixRepresentation[[index[[2]],#1]], newCoordinates[[index[[3]]]]] -
Inactive[D][newMetricMatrixRepresentation[[index[[2]],index[[3]]]], newCoordinates[[#1]]]) & ) /@
Range[Length[newMetricMatrixRepresentation]]]] & ) /@ Tuples[Range[Length[newMetricMatrixRepresentation]],
3]]]; (Module[{index = #1}, Inactive[D][newMatrixRepresentation[[First[index],Last[index]]],
newCoordinates[[Last[index]]]] + Total[(christoffelSymbols[[First[index],Last[index],#1]]*
newMatrixRepresentation[[#1,Last[index]]] & ) /@ Range[Length[newMatrixRepresentation]]] +
Total[(christoffelSymbols[[Last[index],Last[index],#1]]*newMatrixRepresentation[[First[index],#1]] & ) /@
Range[Length[newMatrixRepresentation]]] == 0] & ) /@ Tuples[Range[Length[newMatrixRepresentation]], 2] /.
(ToExpression[#1] -> #1 & ) /@ Select[coordinates, StringQ]] /;
SymbolName[stressEnergyTensor] === "StressEnergyTensor" && SymbolName[metricTensor] === "MetricTensor" &&
Length[Dimensions[metricMatrixRepresentation]] == 2 && Length[coordinates] == Length[metricMatrixRepresentation] &&
BooleanQ[metricIndex1] && BooleanQ[metricIndex2] && Length[Dimensions[matrixRepresentation]] == 2 &&
Length[coordinates] == Length[matrixRepresentation] && BooleanQ[index1] && BooleanQ[index2]
EinsteinSolution[(stressEnergyTensor_)[(metricTensor_)[metricMatrixRepresentation_List, coordinates_List, metricIndex1_,
metricIndex2_], matrixRepresentation_List, index1_, index2_], cosmologicalConstant_]["MetricTensor"] :=
ResourceFunction["MetricTensor"][metricMatrixRepresentation, coordinates, metricIndex1, metricIndex2] /;
SymbolName[stressEnergyTensor] === "StressEnergyTensor" && SymbolName[metricTensor] === "MetricTensor" &&
Length[Dimensions[metricMatrixRepresentation]] == 2 && Length[coordinates] == Length[metricMatrixRepresentation] &&
BooleanQ[metricIndex1] && BooleanQ[metricIndex2] && Length[Dimensions[matrixRepresentation]] == 2 &&
Length[coordinates] == Length[matrixRepresentation] && BooleanQ[index1] && BooleanQ[index2]
EinsteinSolution[(stressEnergyTensor_)[(metricTensor_)[metricMatrixRepresentation_List, coordinates_List, metricIndex1_,
metricIndex2_], matrixRepresentation_List, index1_, index2_], cosmologicalConstant_]["StressEnergyTensor"] :=
ResourceFunction["StressEnergyTensor"][ResourceFunction["MetricTensor"][metricMatrixRepresentation, coordinates,
metricIndex1, metricIndex2], matrixRepresentation, index1, index2] /;
SymbolName[stressEnergyTensor] === "StressEnergyTensor" && SymbolName[metricTensor] === "MetricTensor" &&
Length[Dimensions[metricMatrixRepresentation]] == 2 && Length[coordinates] == Length[metricMatrixRepresentation] &&
BooleanQ[metricIndex1] && BooleanQ[metricIndex2] && Length[Dimensions[matrixRepresentation]] == 2 &&
Length[coordinates] == Length[matrixRepresentation] && BooleanQ[index1] && BooleanQ[index2]
EinsteinSolution[(stressEnergyTensor_)[(metricTensor_)[metricMatrixRepresentation_List, coordinates_List, metricIndex1_,
metricIndex2_], matrixRepresentation_List, index1_, index2_], cosmologicalConstant_]["Coordinates"] :=
coordinates /; SymbolName[stressEnergyTensor] === "StressEnergyTensor" &&
SymbolName[metricTensor] === "MetricTensor" && Length[Dimensions[metricMatrixRepresentation]] == 2 &&
Length[coordinates] == Length[metricMatrixRepresentation] && BooleanQ[metricIndex1] && BooleanQ[metricIndex2] &&
Length[Dimensions[matrixRepresentation]] == 2 && Length[coordinates] == Length[matrixRepresentation] &&
BooleanQ[index1] && BooleanQ[index2]
EinsteinSolution[(stressEnergyTensor_)[(metricTensor_)[metricMatrixRepresentation_List, coordinates_List, metricIndex1_,
metricIndex2_], matrixRepresentation_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[stressEnergyTensor] === "StressEnergyTensor" &&
SymbolName[metricTensor] === "MetricTensor" && Length[Dimensions[metricMatrixRepresentation]] == 2 &&
Length[coordinates] == Length[metricMatrixRepresentation] && BooleanQ[metricIndex1] && BooleanQ[metricIndex2] &&
Length[Dimensions[matrixRepresentation]] == 2 && Length[coordinates] == Length[matrixRepresentation] &&
BooleanQ[index1] && BooleanQ[index2]
EinsteinSolution[(stressEnergyTensor_)[(metricTensor_)[metricMatrixRepresentation_List, coordinates_List, metricIndex1_,
metricIndex2_], matrixRepresentation_List, index1_, index2_], cosmologicalConstant_]["SolutionQ"] :=
Module[{newMetricMatrixRepresentation, newCoordinates, christoffelSymbols, riemannTensor, ricciTensor, ricciScalar,
tensorRepresentation, einsteinEquations}, newMetricMatrixRepresentation = metricMatrixRepresentation /.
(#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[newMetricMatrixRepresentation][[index[[1]],#1]]*
(D[newMetricMatrixRepresentation[[#1,index[[3]]]], newCoordinates[[index[[2]]]]] +
D[newMetricMatrixRepresentation[[index[[2]],#1]], newCoordinates[[index[[3]]]]] -
D[newMetricMatrixRepresentation[[index[[2]],index[[3]]]], newCoordinates[[#1]]]) & ) /@
Range[Length[newMetricMatrixRepresentation]]]] & ) /@ Tuples[Range[Length[newMetricMatrixRepresentation]],
3]]]; 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[newMetricMatrixRepresentation]]] -
Total[(christoffelSymbols[[index[[1]],#1,index[[4]]]]*christoffelSymbols[[#1,index[[2]],index[[3]]]] & ) /@
Range[Length[newMetricMatrixRepresentation]]]] & ) /@
Tuples[Range[Length[newMetricMatrixRepresentation]], 4]]] /. (ToExpression[#1] -> #1 & ) /@
Select[coordinates, StringQ]; ricciTensor =
Normal[SparseArray[(Module[{index = #1}, index -> Total[(riemannTensor[[#1,First[index],#1,Last[index]]] & ) /@
Range[Length[metricMatrixRepresentation]]]] & ) /@ Tuples[Range[Length[metricMatrixRepresentation]], 2]]];
ricciScalar = Total[(Inverse[metricMatrixRepresentation][[First[#1],Last[#1]]]*ricciTensor[[First[#1],
Last[#1]]] & ) /@ Tuples[Range[Length[metricMatrixRepresentation]], 2]];
tensorRepresentation = Normal[SparseArray[
(Module[{index = #1}, index -> Total[(metricMatrixRepresentation[[First[index],First[#1]]]*
metricMatrixRepresentation[[Last[#1],Last[index]]]*matrixRepresentation[[First[#1],Last[#1]]] & ) /@
Tuples[Range[Length[metricMatrixRepresentation]], 2]]] & ) /@
Tuples[Range[Length[metricMatrixRepresentation]], 2]]]; einsteinEquations =
FullSimplify[Thread[Catenate[ricciTensor - (1/2)*ricciScalar*metricMatrixRepresentation +
cosmologicalConstant*metricMatrixRepresentation] == Catenate[(8*Pi)*tensorRepresentation]]];
If[einsteinEquations === True, True, If[einsteinEquations === False, False,
If[Length[Select[einsteinEquations, #1 === False & ]] > 0, False, True]]]] /;
SymbolName[stressEnergyTensor] === "StressEnergyTensor" && SymbolName[metricTensor] === "MetricTensor" &&
Length[Dimensions[metricMatrixRepresentation]] == 2 && Length[coordinates] == Length[metricMatrixRepresentation] &&
BooleanQ[metricIndex1] && BooleanQ[metricIndex2] && Length[Dimensions[matrixRepresentation]] == 2 &&
Length[coordinates] == Length[matrixRepresentation] && BooleanQ[index1] && BooleanQ[index2]
EinsteinSolution[(stressEnergyTensor_)[(metricTensor_)[metricMatrixRepresentation_List, coordinates_List, metricIndex1_,
metricIndex2_], matrixRepresentation_List, index1_, index2_], cosmologicalConstant_]["ExactSolutionQ"] :=
Module[{newMetricMatrixRepresentation, newCoordinates, christoffelSymbols, riemannTensor, ricciTensor, ricciScalar,
tensorRepresentation, einsteinEquations}, newMetricMatrixRepresentation = metricMatrixRepresentation /.
(#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[newMetricMatrixRepresentation][[index[[1]],#1]]*
(D[newMetricMatrixRepresentation[[#1,index[[3]]]], newCoordinates[[index[[2]]]]] +
D[newMetricMatrixRepresentation[[index[[2]],#1]], newCoordinates[[index[[3]]]]] -
D[newMetricMatrixRepresentation[[index[[2]],index[[3]]]], newCoordinates[[#1]]]) & ) /@
Range[Length[newMetricMatrixRepresentation]]]] & ) /@ Tuples[Range[Length[newMetricMatrixRepresentation]],
3]]]; 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[newMetricMatrixRepresentation]]] -
Total[(christoffelSymbols[[index[[1]],#1,index[[4]]]]*christoffelSymbols[[#1,index[[2]],index[[3]]]] & ) /@
Range[Length[newMetricMatrixRepresentation]]]] & ) /@
Tuples[Range[Length[newMetricMatrixRepresentation]], 4]]] /. (ToExpression[#1] -> #1 & ) /@
Select[coordinates, StringQ]; ricciTensor =
Normal[SparseArray[(Module[{index = #1}, index -> Total[(riemannTensor[[#1,First[index],#1,Last[index]]] & ) /@
Range[Length[metricMatrixRepresentation]]]] & ) /@ Tuples[Range[Length[metricMatrixRepresentation]], 2]]];
ricciScalar = Total[(Inverse[metricMatrixRepresentation][[First[#1],Last[#1]]]*ricciTensor[[First[#1],
Last[#1]]] & ) /@ Tuples[Range[Length[metricMatrixRepresentation]], 2]];
tensorRepresentation = Normal[SparseArray[
(Module[{index = #1}, index -> Total[(metricMatrixRepresentation[[First[index],First[#1]]]*
metricMatrixRepresentation[[Last[#1],Last[index]]]*matrixRepresentation[[First[#1],Last[#1]]] & ) /@
Tuples[Range[Length[metricMatrixRepresentation]], 2]]] & ) /@
Tuples[Range[Length[metricMatrixRepresentation]], 2]]]; einsteinEquations =
FullSimplify[Thread[Catenate[ricciTensor - (1/2)*ricciScalar*metricMatrixRepresentation +
cosmologicalConstant*metricMatrixRepresentation] == Catenate[(8*Pi)*tensorRepresentation]]];
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[stressEnergyTensor] === "StressEnergyTensor" &&
SymbolName[metricTensor] === "MetricTensor" && Length[Dimensions[metricMatrixRepresentation]] == 2 &&
Length[coordinates] == Length[metricMatrixRepresentation] && BooleanQ[metricIndex1] && BooleanQ[metricIndex2] &&
Length[Dimensions[matrixRepresentation]] == 2 && Length[coordinates] == Length[matrixRepresentation] &&
BooleanQ[index1] && BooleanQ[index2]
EinsteinSolution[(stressEnergyTensor_)[(metricTensor_)[metricMatrixRepresentation_List, coordinates_List, metricIndex1_,
metricIndex2_], matrixRepresentation_List, index1_, index2_], cosmologicalConstant_]["CosmologicalConstant"] :=
cosmologicalConstant /; SymbolName[stressEnergyTensor] === "StressEnergyTensor" &&
SymbolName[metricTensor] === "MetricTensor" && Length[Dimensions[metricMatrixRepresentation]] == 2 &&
Length[coordinates] == Length[metricMatrixRepresentation] && BooleanQ[metricIndex1] && BooleanQ[metricIndex2] &&
Length[Dimensions[matrixRepresentation]] == 2 && Length[coordinates] == Length[matrixRepresentation] &&
BooleanQ[index1] && BooleanQ[index2]
EinsteinSolution[(stressEnergyTensor_)[(metricTensor_)[metricMatrixRepresentation_List, coordinates_List, metricIndex1_,
metricIndex2_], matrixRepresentation_List, index1_, index2_], cosmologicalConstant_]["Dimensions"] :=
Length[metricMatrixRepresentation] /; SymbolName[stressEnergyTensor] === "StressEnergyTensor" &&
SymbolName[metricTensor] === "MetricTensor" && Length[Dimensions[metricMatrixRepresentation]] == 2 &&
Length[coordinates] == Length[metricMatrixRepresentation] && BooleanQ[metricIndex1] && BooleanQ[metricIndex2] &&
Length[Dimensions[matrixRepresentation]] == 2 && Length[coordinates] == Length[matrixRepresentation] &&
BooleanQ[index1] && BooleanQ[index2]
EinsteinSolution[(stressEnergyTensor_)[(metricTensor_)[metricMatrixRepresentation_List, coordinates_List, metricIndex1_,
metricIndex2_], matrixRepresentation_List, index1_, index2_], cosmologicalConstant_]["Signature"] :=
Module[{eigenvalues, positiveEigenvalues, negativeEigenvalues}, eigenvalues = Eigenvalues[metricMatrixRepresentation];
positiveEigenvalues = Select[eigenvalues, #1 > 0 & ]; negativeEigenvalues = Select[eigenvalues, #1 < 0 & ];
If[Length[positiveEigenvalues] + Length[negativeEigenvalues] == Length[metricMatrixRepresentation],
Join[ConstantArray[-1, Length[negativeEigenvalues]], ConstantArray[1, Length[positiveEigenvalues]]],
Indeterminate]] /; SymbolName[stressEnergyTensor] === "StressEnergyTensor" &&
SymbolName[metricTensor] === "MetricTensor" && Length[Dimensions[metricMatrixRepresentation]] == 2 &&
Length[coordinates] == Length[metricMatrixRepresentation] && BooleanQ[metricIndex1] && BooleanQ[metricIndex2] &&
Length[Dimensions[matrixRepresentation]] == 2 && Length[coordinates] == Length[matrixRepresentation] &&
BooleanQ[index1] && BooleanQ[index2]
EinsteinSolution[(stressEnergyTensor_)[(metricTensor_)[metricMatrixRepresentation_List, coordinates_List, metricIndex1_,
metricIndex2_], matrixRepresentation_List, index1_, index2_], cosmologicalConstant_]["RiemannianQ"] :=
Module[{eigenvalues, positiveEigenvalues, negativeEigenvalues}, eigenvalues = Eigenvalues[metricMatrixRepresentation];
positiveEigenvalues = Select[eigenvalues, #1 > 0 & ]; negativeEigenvalues = Select[eigenvalues, #1 < 0 & ];
If[Length[positiveEigenvalues] + Length[negativeEigenvalues] == Length[matrixRepresentation],
If[Length[positiveEigenvalues] == Length[metricMatrixRepresentation] || Length[negativeEigenvalues] ==
Length[metricMatrixRepresentation], True, False], Indeterminate]] /;
SymbolName[stressEnergyTensor] === "StressEnergyTensor" && SymbolName[metricTensor] === "MetricTensor" &&
Length[Dimensions[metricMatrixRepresentation]] == 2 && Length[coordinates] == Length[metricMatrixRepresentation] &&
BooleanQ[metricIndex1] && BooleanQ[metricIndex2] && Length[Dimensions[matrixRepresentation]] == 2 &&
Length[coordinates] == Length[matrixRepresentation] && BooleanQ[index1] && BooleanQ[index2]
EinsteinSolution[(stressEnergyTensor_)[(metricTensor_)[metricMatrixRepresentation_List, coordinates_List, metricIndex1_,
metricIndex2_], matrixRepresentation_List, index1_, index2_], cosmologicalConstant_]["PseudoRiemannianQ"] :=
Module[{eigenvalues, positiveEigenvalues, negativeEigenvalues}, eigenvalues = Eigenvalues[metricMatrixRepresentation];
positiveEigenvalues = Select[eigenvalues, #1 > 0 & ]; negativeEigenvalues = Select[eigenvalues, #1 < 0 & ];
If[Length[positiveEigenvalues] + Length[negativeEigenvalues] == Length[metricMatrixRepresentation],
If[Length[positiveEigenvalues] == Length[metricMatrixRepresentation] || Length[negativeEigenvalues] ==
Length[metricMatrixRepresentation], False, True], Indeterminate]] /;
SymbolName[stressEnergyTensor] === "StressEnergyTensor" && SymbolName[metricTensor] === "MetricTensor" &&
Length[Dimensions[metricMatrixRepresentation]] == 2 && Length[coordinates] == Length[metricMatrixRepresentation] &&
BooleanQ[metricIndex1] && BooleanQ[metricIndex2] && Length[Dimensions[matrixRepresentation]] == 2 &&
Length[coordinates] == Length[matrixRepresentation] && BooleanQ[index1] && BooleanQ[index2]
EinsteinSolution[(stressEnergyTensor_)[(metricTensor_)[metricMatrixRepresentation_List, coordinates_List, metricIndex1_,
metricIndex2_], matrixRepresentation_List, index1_, index2_], cosmologicalConstant_]["LorentzianQ"] :=
Module[{eigenvalues, positiveEigenvalues, negativeEigenvalues}, eigenvalues = Eigenvalues[metricMatrixRepresentation];
positiveEigenvalues = Select[eigenvalues, #1 > 0 & ]; negativeEigenvalues = Select[eigenvalues, #1 < 0 & ];
If[Length[positiveEigenvalues] + Length[negativeEigenvalues] == Length[metricMatrixRepresentation],
If[Length[positiveEigenvalues] == 1 || Length[negativeEigenvalues] == 1, True, False], Indeterminate]] /;
SymbolName[stressEnergyTensor] === "StressEnergyTensor" && SymbolName[metricTensor] === "MetricTensor" &&
Length[Dimensions[metricMatrixRepresentation]] == 2 && Length[coordinates] == Length[metricMatrixRepresentation] &&
BooleanQ[metricIndex1] && BooleanQ[metricIndex2] && Length[Dimensions[matrixRepresentation]] == 2 &&
Length[coordinates] == Length[matrixRepresentation] && BooleanQ[index1] && BooleanQ[index2]
EinsteinSolution[(stressEnergyTensor_)[(metricTensor_)[metricMatrixRepresentation_List, coordinates_List, metricIndex1_,
metricIndex2_], matrixRepresentation_List, index1_, index2_], cosmologicalConstant_]["RiemannianConditions"] :=
Module[{eigenvalues, riemannianConditions}, eigenvalues = Eigenvalues[metricMatrixRepresentation];
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[stressEnergyTensor] === "StressEnergyTensor" && SymbolName[metricTensor] === "MetricTensor" &&
Length[Dimensions[metricMatrixRepresentation]] == 2 && Length[coordinates] == Length[metricMatrixRepresentation] &&
BooleanQ[metricIndex1] && BooleanQ[metricIndex2] && Length[Dimensions[matrixRepresentation]] == 2 &&
Length[coordinates] == Length[matrixRepresentation] && BooleanQ[index1] && BooleanQ[index2]
EinsteinSolution[(stressEnergyTensor_)[(metricTensor_)[metricMatrixRepresentation_List, coordinates_List, metricIndex1_,
metricIndex2_], matrixRepresentation_List, index1_, index2_], cosmologicalConstant_][
"PseudoRiemannianConditions"] := Module[{eigenvalues, pseudoRiemannianConditions},
eigenvalues = Eigenvalues[metricMatrixRepresentation]; 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[stressEnergyTensor] === "StressEnergyTensor" && SymbolName[metricTensor] === "MetricTensor" &&
Length[Dimensions[metricMatrixRepresentation]] == 2 && Length[coordinates] == Length[metricMatrixRepresentation] &&
BooleanQ[metricIndex1] && BooleanQ[metricIndex2] && Length[Dimensions[matrixRepresentation]] == 2 &&
Length[coordinates] == Length[matrixRepresentation] && BooleanQ[index1] && BooleanQ[index2]
EinsteinSolution[(stressEnergyTensor_)[(metricTensor_)[metricMatrixRepresentation_List, coordinates_List, metricIndex1_,
metricIndex2_], matrixRepresentation_List, index1_, index2_], cosmologicalConstant_]["LorentzianConditions"] :=
Module[{eigensystem, eigenvalues, eigenvectors, timeCoordinate, lorentzianConditions},
eigensystem = Eigensystem[metricMatrixRepresentation]; 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[stressEnergyTensor] === "StressEnergyTensor" && SymbolName[metricTensor] === "MetricTensor" &&
Length[Dimensions[metricMatrixRepresentation]] == 2 && Length[coordinates] == Length[metricMatrixRepresentation] &&
BooleanQ[metricIndex1] && BooleanQ[metricIndex2] && Length[Dimensions[matrixRepresentation]] == 2 &&
Length[coordinates] == Length[matrixRepresentation] && BooleanQ[index1] && BooleanQ[index2]
EinsteinSolution[(stressEnergyTensor_)[(metricTensor_)[metricMatrixRepresentation_List, coordinates_List, metricIndex1_,
metricIndex2_], matrixRepresentation_List, index1_, index2_], cosmologicalConstant_]["Properties"] :=
{"FieldEquations", "EinsteinEquations", "ReducedEinsteinEquations", "SymbolicEinsteinEquations", "ContinuityEquations",
"ReducedContinuityEquations", "SymbolicContinuityEquations", "MetricTensor", "StressEnergyTensor", "Coordinates",
"CoordinateOneForms", "SolutionQ", "ExactSolutionQ", "CosmologicalConstant", "Dimensions", "Signature",
"RiemannianQ", "PseudoRiemannianQ", "LorentzianQ", "RiemannianConditions", "PseudoRiemannianConditions",
"LorentzianConditions", "Properties"} /; SymbolName[stressEnergyTensor] === "StressEnergyTensor" &&
SymbolName[metricTensor] === "MetricTensor" && Length[Dimensions[metricMatrixRepresentation]] == 2 &&
Length[coordinates] == Length[metricMatrixRepresentation] && BooleanQ[metricIndex1] && BooleanQ[metricIndex2] &&
Length[Dimensions[matrixRepresentation]] == 2 && Length[coordinates] == Length[matrixRepresentation] &&
BooleanQ[index1] && BooleanQ[index2]
EinsteinSolution /:
MakeBoxes[einsteinSolution:EinsteinSolution[(stressEnergyTensor_)[(metricTensor_)[metricMatrixRepresentation_List,
coordinates_List, metricIndex1_, metricIndex2_], matrixRepresentation_List, index1_, index2_],
cosmologicalConstant_], format_] := Module[{newMetricMatrixRepresentation, newCoordinates, christoffelSymbols,
riemannTensor, ricciTensor, ricciScalar, matrixForm, tensorRepresentation, einsteinEquations, solution,
exactSolution, fieldEquations, icon}, newMetricMatrixRepresentation = metricMatrixRepresentation /.
(#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[newMetricMatrixRepresentation][[index[[1]],#1]]*
(D[newMetricMatrixRepresentation[[#1,index[[3]]]], newCoordinates[[index[[2]]]]] +
D[newMetricMatrixRepresentation[[index[[2]],#1]], newCoordinates[[index[[3]]]]] -
D[newMetricMatrixRepresentation[[index[[2]],index[[3]]]], newCoordinates[[#1]]]) & ) /@ Range[
Length[newMetricMatrixRepresentation]]]] & ) /@ Tuples[Range[Length[newMetricMatrixRepresentation]],
3]]]; 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[newMetricMatrixRepresentation]]] - Total[
(christoffelSymbols[[index[[1]],#1,index[[4]]]]*christoffelSymbols[[#1,index[[2]],index[[3]]]] & ) /@
Range[Length[newMetricMatrixRepresentation]]]] & ) /@
Tuples[Range[Length[newMetricMatrixRepresentation]], 4]]] /. (ToExpression[#1] -> #1 & ) /@
Select[coordinates, StringQ]; ricciTensor =
Normal[SparseArray[(Module[{index = #1}, index -> Total[(riemannTensor[[#1,First[index],#1,Last[index]]] & ) /@
Range[Length[metricMatrixRepresentation]]]] & ) /@ Tuples[Range[Length[metricMatrixRepresentation]], 2]]];
ricciScalar = Total[(Inverse[metricMatrixRepresentation][[First[#1],Last[#1]]]*ricciTensor[[First[#1],
Last[#1]]] & ) /@ Tuples[Range[Length[metricMatrixRepresentation]], 2]];
matrixForm = ricciTensor - (1/2)*ricciScalar*metricMatrixRepresentation + cosmologicalConstant*
metricMatrixRepresentation; tensorRepresentation =
Normal[SparseArray[(Module[{index = #1}, index -> Total[(metricMatrixRepresentation[[First[index],First[#1]]]*
metricMatrixRepresentation[[Last[#1],Last[index]]]*matrixRepresentation[[First[#1],Last[#1]]] & ) /@
Tuples[Range[Length[metricMatrixRepresentation]], 2]]] & ) /@
Tuples[Range[Length[metricMatrixRepresentation]], 2]]]; einsteinEquations =
FullSimplify[Thread[Catenate[ricciTensor - (1/2)*ricciScalar*metricMatrixRepresentation +
cosmologicalConstant*metricMatrixRepresentation] == Catenate[(8*Pi)*tensorRepresentation]]];
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["EinsteinSolution", einsteinSolution, icon,
{{BoxForm`SummaryItem[{"Solution: ", solution}], BoxForm`SummaryItem[{"Exact Solution: ", exactSolution}]},
{BoxForm`SummaryItem[{"Field Equations: ", fieldEquations}], BoxForm`SummaryItem[{"Cosmological Constant: ",
cosmologicalConstant}]}}, {{}}, format, "Interpretable" -> Automatic]] /;
SymbolName[stressEnergyTensor] === "StressEnergyTensor" && SymbolName[metricTensor] === "MetricTensor" &&
Length[Dimensions[metricMatrixRepresentation]] == 2 && Length[coordinates] == Length[metricMatrixRepresentation] &&
BooleanQ[metricIndex1] && BooleanQ[metricIndex2] && Length[Dimensions[matrixRepresentation]] == 2 &&
Length[coordinates] == Length[matrixRepresentation] && BooleanQ[index1] && BooleanQ[index2]