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SolveElectrovacuumEinsteinEquations.wl
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(* ::Package:: *)
SolveElectrovacuumEinsteinEquations[(electromagneticTensor_)[(metricTensor_)[matrixRepresentation_List, coordinates_List,
metricIndex1_, metricIndex2_], electromagneticPotential_List, vacuumPermeability_, index1_, index2_]] :=
ElectrovacuumSolution[ElectromagneticTensor[ResourceFunction["MetricTensor"][matrixRepresentation, coordinates,
metricIndex1, metricIndex2], electromagneticPotential, vacuumPermeability, index1, index2], "\[FormalCapitalLambda]"] /;
SymbolName[electromagneticTensor] === "ElectromagneticTensor" && SymbolName[metricTensor] === "MetricTensor" &&
Length[Dimensions[matrixRepresentation]] == 2 && Length[coordinates] == Length[matrixRepresentation] &&
BooleanQ[metricIndex1] && BooleanQ[metricIndex2] && Length[electromagneticPotential] ==
Length[matrixRepresentation] && BooleanQ[index1] && BooleanQ[index2]
SolveElectrovacuumEinsteinEquations[(electromagneticTensor_)[(metricTensor_)[matrixRepresentation_List, coordinates_List,
metricIndex1_, metricIndex2_], electromagneticPotential_List, vacuumPermeability_, index1_, index2_],
(metricTensor_)[newMatrixRepresentation_List, newCoordinates_List, newMetricIndex1_, newMetricIndex2_]] :=
ElectrovacuumSolution[ElectromagneticTensor[ResourceFunction["MetricTensor"][newMatrixRepresentation, newCoordinates,
newMetricIndex1, newMetricIndex2], electromagneticPotential /. Thread[coordinates -> newCoordinates],
vacuumPermeability /. Thread[coordinates -> newCoordinates], index1, index2], "\[FormalCapitalLambda]"] /;
SymbolName[electromagneticTensor] === "ElectromagneticTensor" && SymbolName[metricTensor] === "MetricTensor" &&
Length[Dimensions[matrixRepresentation]] == 2 && Length[coordinates] == Length[matrixRepresentation] &&
BooleanQ[metricIndex1] && BooleanQ[metricIndex2] && Length[electromagneticPotential] ==
Length[matrixRepresentation] && BooleanQ[index1] && BooleanQ[index2] &&
Length[Dimensions[newMatrixRepresentation]] == 2 && Length[newCoordinates] == Length[newMatrixRepresentation] &&
BooleanQ[newMetricIndex1] && BooleanQ[newMetricIndex2] && Length[electromagneticPotential] ==
Length[newMatrixRepresentation]
SolveElectrovacuumEinsteinEquations[(electromagneticTensor_)[(metricTensor_)[matrixRepresentation_List, coordinates_List,
metricIndex1_, metricIndex2_], electromagneticPotential_List, vacuumPermeability_, index1_, index2_],
cosmologicalConstant_] := ElectrovacuumSolution[ElectromagneticTensor[ResourceFunction["MetricTensor"][
matrixRepresentation, coordinates, metricIndex1, metricIndex2], electromagneticPotential, vacuumPermeability,
index1, index2], cosmologicalConstant] /; SymbolName[electromagneticTensor] === "ElectromagneticTensor" &&
SymbolName[metricTensor] === "MetricTensor" && Length[Dimensions[matrixRepresentation]] == 2 &&
Length[coordinates] == Length[matrixRepresentation] && BooleanQ[metricIndex1] && BooleanQ[metricIndex2] &&
Length[electromagneticPotential] == Length[matrixRepresentation] && BooleanQ[index1] && BooleanQ[index2]
SolveElectrovacuumEinsteinEquations[(electromagneticTensor_)[(metricTensor_)[matrixRepresentation_List, coordinates_List,
metricIndex1_, metricIndex2_], electromagneticPotential_List, vacuumPermeability_, index1_, index2_],
(metricTensor_)[newMatrixRepresentation_List, newCoordinates_List, newMetricIndex1_, newMetricIndex2_],
cosmologicalConstant_] := ElectrovacuumSolution[ElectromagneticTensor[ResourceFunction["MetricTensor"][
newMatrixRepresentation, newCoordinates, newMetricIndex1, newMetricIndex2],
electromagneticPotential /. Thread[coordinates -> newCoordinates], vacuumPermeability /.
Thread[coordinates -> newCoordinates], index1, index2], cosmologicalConstant] /;
SymbolName[electromagneticTensor] === "ElectromagneticTensor" && SymbolName[metricTensor] === "MetricTensor" &&
Length[Dimensions[matrixRepresentation]] == 2 && Length[coordinates] == Length[matrixRepresentation] &&
BooleanQ[metricIndex1] && BooleanQ[metricIndex2] && Length[electromagneticPotential] ==
Length[matrixRepresentation] && BooleanQ[index1] && BooleanQ[index2] &&
Length[Dimensions[newMatrixRepresentation]] == 2 && Length[newCoordinates] == Length[newMatrixRepresentation] &&
BooleanQ[newMetricIndex1] && BooleanQ[newMetricIndex2] && Length[electromagneticPotential] ==
Length[newMatrixRepresentation]
ElectrovacuumSolution[(electromagneticTensor_)[(metricTensor_)[matrixRepresentation_List, coordinates_List,
metricIndex1_, metricIndex2_], electromagneticPotential_List, vacuumPermeability_, index1_, index2_],
cosmologicalConstant_]["FieldEquations"] :=
Module[{newMatrixRepresentation, newElectromagneticPotential, newCoordinates, christoffelSymbols, riemannTensor,
ricciTensor, ricciScalar, covariantElectromagneticPotential, tensorRepresentation,
contravariantElectromagneticTensor, mixedElectromagneticTensor, stressEnergyTensor, covariantStressEnergyTensor,
einsteinEquations}, newMatrixRepresentation = matrixRepresentation /. (#1 -> ToExpression[#1] & ) /@
Select[coordinates, StringQ]; newElectromagneticPotential = electromagneticPotential /.
(#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]]; covariantElectromagneticPotential =
Normal[SparseArray[(Module[{index = #1}, index -> Total[(newMatrixRepresentation[[index,#1]]*
newElectromagneticPotential[[#1]] & ) /@ Range[Length[newElectromagneticPotential]]]] & ) /@
Range[Length[newElectromagneticPotential]]]]; tensorRepresentation =
Normal[SparseArray[(#1 -> D[covariantElectromagneticPotential[[Last[#1]]], newCoordinates[[First[#1]]]] -
D[covariantElectromagneticPotential[[First[#1]]], newCoordinates[[Last[#1]]]] & ) /@
Tuples[Range[Length[newElectromagneticPotential]], 2]]] /. (ToExpression[#1] -> #1 & ) /@
Select[coordinates, StringQ]; contravariantElectromagneticTensor =
Normal[SparseArray[(Module[{index = #1}, index -> Total[(Inverse[matrixRepresentation][[First[index],First[#1]]]*
Inverse[matrixRepresentation][[Last[#1],Last[index]]]*tensorRepresentation[[First[#1],Last[#1]]] & ) /@
Tuples[Range[Length[electromagneticPotential]], 2]]] & ) /@ Tuples[Range[Length[electromagneticPotential]],
2]]]; mixedElectromagneticTensor =
Normal[SparseArray[(Module[{index = #1}, index -> Total[(Inverse[matrixRepresentation][[First[index],#1]]*
tensorRepresentation[[#1,Last[index]]] & ) /@ Range[Length[electromagneticPotential]]]] & ) /@
Tuples[Range[Length[electromagneticPotential]], 2]]]; stressEnergyTensor =
(1/vacuumPermeability)*Normal[SparseArray[
(Module[{index = #1}, index -> Total[(contravariantElectromagneticTensor[[First[index],#1]]*
mixedElectromagneticTensor[[Last[index],#1]] & ) /@ Range[Length[electromagneticPotential]]] -
(1/4)*Total[(Inverse[matrixRepresentation][[First[index],Last[index]]]*tensorRepresentation[[First[#1],
Last[#1]]]*contravariantElectromagneticTensor[[First[#1],Last[#1]]] & ) /@
Tuples[Range[Length[electromagneticPotential]], 2]]] & ) /@
Tuples[Range[Length[electromagneticPotential]], 2]]]; covariantStressEnergyTensor =
Normal[SparseArray[(Module[{index = #1}, index -> Total[(matrixRepresentation[[First[index],First[#1]]]*
matrixRepresentation[[Last[#1],Last[index]]]*stressEnergyTensor[[First[#1],Last[#1]]] & ) /@
Tuples[Range[Length[matrixRepresentation]], 2]]] & ) /@ Tuples[Range[Length[matrixRepresentation]], 2]]];
einsteinEquations = FullSimplify[Thread[Catenate[ricciTensor - (1/2)*ricciScalar*matrixRepresentation +
cosmologicalConstant*matrixRepresentation] == Catenate[(8*Pi)*covariantStressEnergyTensor]]];
If[einsteinEquations === True, {}, If[einsteinEquations === False, Indeterminate,
If[Length[Select[einsteinEquations, #1 === False & ]] > 0, Indeterminate,
DeleteDuplicates[Reverse /@ Sort /@ Select[einsteinEquations, #1 =!= True & ]]]]]] /;
SymbolName[electromagneticTensor] === "ElectromagneticTensor" && SymbolName[metricTensor] === "MetricTensor" &&
Length[Dimensions[matrixRepresentation]] == 2 && Length[coordinates] == Length[matrixRepresentation] &&
BooleanQ[metricIndex1] && BooleanQ[metricIndex2] && Length[electromagneticPotential] ==
Length[matrixRepresentation] && BooleanQ[index1] && BooleanQ[index2]
ElectrovacuumSolution[(electromagneticTensor_)[(metricTensor_)[matrixRepresentation_List, coordinates_List,
metricIndex1_, metricIndex2_], electromagneticPotential_List, vacuumPermeability_, index1_, index2_],
cosmologicalConstant_]["EinsteinEquations"] :=
Module[{newMatrixRepresentation, newElectromagneticPotential, newCoordinates, christoffelSymbols, riemannTensor,
ricciTensor, ricciScalar, covariantElectromagneticPotential, tensorRepresentation,
contravariantElectromagneticTensor, mixedElectromagneticTensor, stressEnergyTensor, covariantStressEnergyTensor},
newMatrixRepresentation = matrixRepresentation /. (#1 -> ToExpression[#1] & ) /@ Select[coordinates, StringQ];
newElectromagneticPotential = electromagneticPotential /. (#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]]; covariantElectromagneticPotential =
Normal[SparseArray[(Module[{index = #1}, index -> Total[(newMatrixRepresentation[[index,#1]]*
newElectromagneticPotential[[#1]] & ) /@ Range[Length[newElectromagneticPotential]]]] & ) /@
Range[Length[newElectromagneticPotential]]]]; tensorRepresentation =
Normal[SparseArray[(#1 -> D[covariantElectromagneticPotential[[Last[#1]]], newCoordinates[[First[#1]]]] -
D[covariantElectromagneticPotential[[First[#1]]], newCoordinates[[Last[#1]]]] & ) /@
Tuples[Range[Length[newElectromagneticPotential]], 2]]] /. (ToExpression[#1] -> #1 & ) /@
Select[coordinates, StringQ]; contravariantElectromagneticTensor =
Normal[SparseArray[(Module[{index = #1}, index -> Total[(Inverse[matrixRepresentation][[First[index],First[#1]]]*
Inverse[matrixRepresentation][[Last[#1],Last[index]]]*tensorRepresentation[[First[#1],Last[#1]]] & ) /@
Tuples[Range[Length[electromagneticPotential]], 2]]] & ) /@ Tuples[Range[Length[electromagneticPotential]],
2]]]; mixedElectromagneticTensor =
Normal[SparseArray[(Module[{index = #1}, index -> Total[(Inverse[matrixRepresentation][[First[index],#1]]*
tensorRepresentation[[#1,Last[index]]] & ) /@ Range[Length[electromagneticPotential]]]] & ) /@
Tuples[Range[Length[electromagneticPotential]], 2]]]; stressEnergyTensor =
(1/vacuumPermeability)*Normal[SparseArray[
(Module[{index = #1}, index -> Total[(contravariantElectromagneticTensor[[First[index],#1]]*
mixedElectromagneticTensor[[Last[index],#1]] & ) /@ Range[Length[electromagneticPotential]]] -
(1/4)*Total[(Inverse[matrixRepresentation][[First[index],Last[index]]]*tensorRepresentation[[First[#1],
Last[#1]]]*contravariantElectromagneticTensor[[First[#1],Last[#1]]] & ) /@
Tuples[Range[Length[electromagneticPotential]], 2]]] & ) /@
Tuples[Range[Length[electromagneticPotential]], 2]]]; covariantStressEnergyTensor =
Normal[SparseArray[(Module[{index = #1}, index -> Total[(matrixRepresentation[[First[index],First[#1]]]*
matrixRepresentation[[Last[#1],Last[index]]]*stressEnergyTensor[[First[#1],Last[#1]]] & ) /@
Tuples[Range[Length[matrixRepresentation]], 2]]] & ) /@ Tuples[Range[Length[matrixRepresentation]], 2]]];
Thread[Catenate[ricciTensor - (1/2)*ricciScalar*matrixRepresentation + cosmologicalConstant*matrixRepresentation] ==
Catenate[(8*Pi)*covariantStressEnergyTensor]]] /; SymbolName[electromagneticTensor] === "ElectromagneticTensor" &&
SymbolName[metricTensor] === "MetricTensor" && Length[Dimensions[matrixRepresentation]] == 2 &&
Length[coordinates] == Length[matrixRepresentation] && BooleanQ[metricIndex1] && BooleanQ[metricIndex2] &&
Length[electromagneticPotential] == Length[matrixRepresentation] && BooleanQ[index1] && BooleanQ[index2]
ElectrovacuumSolution[(electromagneticTensor_)[(metricTensor_)[matrixRepresentation_List, coordinates_List,
metricIndex1_, metricIndex2_], electromagneticPotential_List, vacuumPermeability_, index1_, index2_],
cosmologicalConstant_]["ReducedEinsteinEquations"] :=
Module[{newMatrixRepresentation, newElectromagneticPotential, newCoordinates, christoffelSymbols, riemannTensor,
ricciTensor, ricciScalar, covariantElectromagneticPotential, tensorRepresentation,
contravariantElectromagneticTensor, mixedElectromagneticTensor, stressEnergyTensor, covariantStressEnergyTensor},
newMatrixRepresentation = matrixRepresentation /. (#1 -> ToExpression[#1] & ) /@ Select[coordinates, StringQ];
newElectromagneticPotential = electromagneticPotential /. (#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]]; covariantElectromagneticPotential =
Normal[SparseArray[(Module[{index = #1}, index -> Total[(newMatrixRepresentation[[index,#1]]*
newElectromagneticPotential[[#1]] & ) /@ Range[Length[newElectromagneticPotential]]]] & ) /@
Range[Length[newElectromagneticPotential]]]]; tensorRepresentation =
Normal[SparseArray[(#1 -> D[covariantElectromagneticPotential[[Last[#1]]], newCoordinates[[First[#1]]]] -
D[covariantElectromagneticPotential[[First[#1]]], newCoordinates[[Last[#1]]]] & ) /@
Tuples[Range[Length[newElectromagneticPotential]], 2]]] /. (ToExpression[#1] -> #1 & ) /@
Select[coordinates, StringQ]; contravariantElectromagneticTensor =
Normal[SparseArray[(Module[{index = #1}, index -> Total[(Inverse[matrixRepresentation][[First[index],First[#1]]]*
Inverse[matrixRepresentation][[Last[#1],Last[index]]]*tensorRepresentation[[First[#1],Last[#1]]] & ) /@
Tuples[Range[Length[electromagneticPotential]], 2]]] & ) /@ Tuples[Range[Length[electromagneticPotential]],
2]]]; mixedElectromagneticTensor =
Normal[SparseArray[(Module[{index = #1}, index -> Total[(Inverse[matrixRepresentation][[First[index],#1]]*
tensorRepresentation[[#1,Last[index]]] & ) /@ Range[Length[electromagneticPotential]]]] & ) /@
Tuples[Range[Length[electromagneticPotential]], 2]]]; stressEnergyTensor =
(1/vacuumPermeability)*Normal[SparseArray[
(Module[{index = #1}, index -> Total[(contravariantElectromagneticTensor[[First[index],#1]]*
mixedElectromagneticTensor[[Last[index],#1]] & ) /@ Range[Length[electromagneticPotential]]] -
(1/4)*Total[(Inverse[matrixRepresentation][[First[index],Last[index]]]*tensorRepresentation[[First[#1],
Last[#1]]]*contravariantElectromagneticTensor[[First[#1],Last[#1]]] & ) /@
Tuples[Range[Length[electromagneticPotential]], 2]]] & ) /@
Tuples[Range[Length[electromagneticPotential]], 2]]]; covariantStressEnergyTensor =
Normal[SparseArray[(Module[{index = #1}, index -> Total[(matrixRepresentation[[First[index],First[#1]]]*
matrixRepresentation[[Last[#1],Last[index]]]*stressEnergyTensor[[First[#1],Last[#1]]] & ) /@
Tuples[Range[Length[matrixRepresentation]], 2]]] & ) /@ Tuples[Range[Length[matrixRepresentation]], 2]]];
FullSimplify[Thread[Catenate[ricciTensor - (1/2)*ricciScalar*matrixRepresentation +
cosmologicalConstant*matrixRepresentation] == Catenate[(8*Pi)*covariantStressEnergyTensor]]]] /;
SymbolName[electromagneticTensor] === "ElectromagneticTensor" && SymbolName[metricTensor] === "MetricTensor" &&
Length[Dimensions[matrixRepresentation]] == 2 && Length[coordinates] == Length[matrixRepresentation] &&
BooleanQ[metricIndex1] && BooleanQ[metricIndex2] && Length[electromagneticPotential] ==
Length[matrixRepresentation] && BooleanQ[index1] && BooleanQ[index2]
ElectrovacuumSolution[(electromagneticTensor_)[(metricTensor_)[matrixRepresentation_List, coordinates_List,
metricIndex1_, metricIndex2_], electromagneticPotential_List, vacuumPermeability_, index1_, index2_],
cosmologicalConstant_]["SymbolicEinsteinEquations"] :=
Module[{newMatrixRepresentation, newElectromagneticPotential, newCoordinates, christoffelSymbols, riemannTensor,
ricciTensor, ricciScalar, covariantElectromagneticPotential, tensorRepresentation,
contravariantElectromagneticTensor, mixedElectromagneticTensor, stressEnergyTensor, covariantStressEnergyTensor},
newMatrixRepresentation = matrixRepresentation /. (#1 -> ToExpression[#1] & ) /@ Select[coordinates, StringQ];
newElectromagneticPotential = electromagneticPotential /. (#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]]; covariantElectromagneticPotential =
Normal[SparseArray[(Module[{index = #1}, index -> Total[(newMatrixRepresentation[[index,#1]]*
newElectromagneticPotential[[#1]] & ) /@ Range[Length[newElectromagneticPotential]]]] & ) /@
Range[Length[newElectromagneticPotential]]]]; tensorRepresentation =
Normal[SparseArray[(#1 -> Inactive[D][covariantElectromagneticPotential[[Last[#1]]], newCoordinates[[First[#1]]]] -
Inactive[D][covariantElectromagneticPotential[[First[#1]]], newCoordinates[[Last[#1]]]] & ) /@
Tuples[Range[Length[newElectromagneticPotential]], 2]]] /. (ToExpression[#1] -> #1 & ) /@
Select[coordinates, StringQ]; contravariantElectromagneticTensor =
Normal[SparseArray[(Module[{index = #1}, index -> Total[(Inverse[matrixRepresentation][[First[index],First[#1]]]*
Inverse[matrixRepresentation][[Last[#1],Last[index]]]*tensorRepresentation[[First[#1],Last[#1]]] & ) /@
Tuples[Range[Length[electromagneticPotential]], 2]]] & ) /@ Tuples[Range[Length[electromagneticPotential]],
2]]]; mixedElectromagneticTensor =
Normal[SparseArray[(Module[{index = #1}, index -> Total[(Inverse[matrixRepresentation][[First[index],#1]]*
tensorRepresentation[[#1,Last[index]]] & ) /@ Range[Length[electromagneticPotential]]]] & ) /@
Tuples[Range[Length[electromagneticPotential]], 2]]]; stressEnergyTensor =
(1/vacuumPermeability)*Normal[SparseArray[
(Module[{index = #1}, index -> Total[(contravariantElectromagneticTensor[[First[index],#1]]*
mixedElectromagneticTensor[[Last[index],#1]] & ) /@ Range[Length[electromagneticPotential]]] -
(1/4)*Total[(Inverse[matrixRepresentation][[First[index],Last[index]]]*tensorRepresentation[[First[#1],
Last[#1]]]*contravariantElectromagneticTensor[[First[#1],Last[#1]]] & ) /@
Tuples[Range[Length[electromagneticPotential]], 2]]] & ) /@
Tuples[Range[Length[electromagneticPotential]], 2]]]; covariantStressEnergyTensor =
Normal[SparseArray[(Module[{index = #1}, index -> Total[(matrixRepresentation[[First[index],First[#1]]]*
matrixRepresentation[[Last[#1],Last[index]]]*stressEnergyTensor[[First[#1],Last[#1]]] & ) /@
Tuples[Range[Length[matrixRepresentation]], 2]]] & ) /@ Tuples[Range[Length[matrixRepresentation]], 2]]];
Thread[Catenate[ricciTensor - (1/2)*ricciScalar*matrixRepresentation + cosmologicalConstant*matrixRepresentation] ==
Catenate[(8*Pi)*covariantStressEnergyTensor]]] /; SymbolName[electromagneticTensor] === "ElectromagneticTensor" &&
SymbolName[metricTensor] === "MetricTensor" && Length[Dimensions[matrixRepresentation]] == 2 &&
Length[coordinates] == Length[matrixRepresentation] && BooleanQ[metricIndex1] && BooleanQ[metricIndex2] &&
Length[electromagneticPotential] == Length[matrixRepresentation] && BooleanQ[index1] && BooleanQ[index2]
ElectrovacuumSolution[(electromagneticTensor_)[(metricTensor_)[matrixRepresentation_List, coordinates_List,
metricIndex1_, metricIndex2_], electromagneticPotential_List, vacuumPermeability_, index1_, index2_],
cosmologicalConstant_]["ContinuityEquations"] :=
Module[{newMatrixRepresentation, newElectromagneticPotential, newCoordinates, christoffelSymbols,
covariantElectromagneticPotential, tensorRepresentation, contravariantElectromagneticTensor,
mixedElectromagneticTensor, stressEnergyTensor},
newMatrixRepresentation = matrixRepresentation /. (#1 -> ToExpression[#1] & ) /@ Select[coordinates, StringQ];
newElectromagneticPotential = electromagneticPotential /. (#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]]]; covariantElectromagneticPotential =
Normal[SparseArray[(Module[{index = #1}, index -> Total[(newMatrixRepresentation[[index,#1]]*
newElectromagneticPotential[[#1]] & ) /@ Range[Length[newElectromagneticPotential]]]] & ) /@
Range[Length[newElectromagneticPotential]]]]; tensorRepresentation =
Normal[SparseArray[(#1 -> D[covariantElectromagneticPotential[[Last[#1]]], newCoordinates[[First[#1]]]] -
D[covariantElectromagneticPotential[[First[#1]]], newCoordinates[[Last[#1]]]] & ) /@
Tuples[Range[Length[newElectromagneticPotential]], 2]]]; contravariantElectromagneticTensor =
Normal[SparseArray[(Module[{index = #1}, index -> Total[(Inverse[newMatrixRepresentation][[First[index],First[#1]]]*
Inverse[newMatrixRepresentation][[Last[#1],Last[index]]]*tensorRepresentation[[First[#1],
Last[#1]]] & ) /@ Tuples[Range[Length[newElectromagneticPotential]], 2]]] & ) /@
Tuples[Range[Length[newElectromagneticPotential]], 2]]]; mixedElectromagneticTensor =
Normal[SparseArray[(Module[{index = #1}, index -> Total[(Inverse[newMatrixRepresentation][[First[index],#1]]*
tensorRepresentation[[#1,Last[index]]] & ) /@ Range[Length[newElectromagneticPotential]]]] & ) /@
Tuples[Range[Length[newElectromagneticPotential]], 2]]]; stressEnergyTensor =
(1/vacuumPermeability)*Normal[SparseArray[
(Module[{index = #1}, index -> Total[(contravariantElectromagneticTensor[[First[index],#1]]*
mixedElectromagneticTensor[[Last[index],#1]] & ) /@ Range[Length[newElectromagneticPotential]]] -
(1/4)*Total[(Inverse[newMatrixRepresentation][[First[index],Last[index]]]*tensorRepresentation[[First[#1],
Last[#1]]]*contravariantElectromagneticTensor[[First[#1],Last[#1]]] & ) /@
Tuples[Range[Length[newElectromagneticPotential]], 2]]] & ) /@
Tuples[Range[Length[newElectromagneticPotential]], 2]]];
(Module[{index = #1}, Total[(Module[{nestedIndex = #1}, D[stressEnergyTensor[[nestedIndex,index]],
newCoordinates[[nestedIndex]]] + Total[(christoffelSymbols[[nestedIndex,nestedIndex,#1]]*
stressEnergyTensor[[#1,index]] & ) /@ Range[Length[newElectromagneticPotential]]] + Total[
(christoffelSymbols[[index,nestedIndex,#1]]*stressEnergyTensor[[nestedIndex,#1]] & ) /@
Range[Length[newElectromagneticPotential]]]] & ) /@ Range[Length[newElectromagneticPotential]]] ==
0] & ) /@ Range[Length[newElectromagneticPotential]] /. (ToExpression[#1] -> #1 & ) /@
Select[coordinates, StringQ]] /; SymbolName[electromagneticTensor] === "ElectromagneticTensor" &&
SymbolName[metricTensor] === "MetricTensor" && Length[Dimensions[matrixRepresentation]] == 2 &&
Length[coordinates] == Length[matrixRepresentation] && BooleanQ[metricIndex1] && BooleanQ[metricIndex2] &&
Length[electromagneticPotential] == Length[matrixRepresentation] && BooleanQ[index1] && BooleanQ[index2]
ElectrovacuumSolution[(electromagneticTensor_)[(metricTensor_)[matrixRepresentation_List, coordinates_List,
metricIndex1_, metricIndex2_], electromagneticPotential_List, vacuumPermeability_, index1_, index2_],
cosmologicalConstant_]["ReducedContinuityEquations"] :=
Module[{newMatrixRepresentation, newElectromagneticPotential, newCoordinates, christoffelSymbols,
covariantElectromagneticPotential, tensorRepresentation, contravariantElectromagneticTensor,
mixedElectromagneticTensor, stressEnergyTensor},
newMatrixRepresentation = matrixRepresentation /. (#1 -> ToExpression[#1] & ) /@ Select[coordinates, StringQ];
newElectromagneticPotential = electromagneticPotential /. (#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]]]; covariantElectromagneticPotential =
Normal[SparseArray[(Module[{index = #1}, index -> Total[(newMatrixRepresentation[[index,#1]]*
newElectromagneticPotential[[#1]] & ) /@ Range[Length[newElectromagneticPotential]]]] & ) /@
Range[Length[newElectromagneticPotential]]]]; tensorRepresentation =
Normal[SparseArray[(#1 -> D[covariantElectromagneticPotential[[Last[#1]]], newCoordinates[[First[#1]]]] -
D[covariantElectromagneticPotential[[First[#1]]], newCoordinates[[Last[#1]]]] & ) /@
Tuples[Range[Length[newElectromagneticPotential]], 2]]]; contravariantElectromagneticTensor =
Normal[SparseArray[(Module[{index = #1}, index -> Total[(Inverse[newMatrixRepresentation][[First[index],First[#1]]]*
Inverse[newMatrixRepresentation][[Last[#1],Last[index]]]*tensorRepresentation[[First[#1],
Last[#1]]] & ) /@ Tuples[Range[Length[newElectromagneticPotential]], 2]]] & ) /@
Tuples[Range[Length[newElectromagneticPotential]], 2]]]; mixedElectromagneticTensor =
Normal[SparseArray[(Module[{index = #1}, index -> Total[(Inverse[newMatrixRepresentation][[First[index],#1]]*
tensorRepresentation[[#1,Last[index]]] & ) /@ Range[Length[newElectromagneticPotential]]]] & ) /@
Tuples[Range[Length[newElectromagneticPotential]], 2]]]; stressEnergyTensor =
(1/vacuumPermeability)*Normal[SparseArray[
(Module[{index = #1}, index -> Total[(contravariantElectromagneticTensor[[First[index],#1]]*
mixedElectromagneticTensor[[Last[index],#1]] & ) /@ Range[Length[newElectromagneticPotential]]] -
(1/4)*Total[(Inverse[newMatrixRepresentation][[First[index],Last[index]]]*tensorRepresentation[[First[#1],
Last[#1]]]*contravariantElectromagneticTensor[[First[#1],Last[#1]]] & ) /@
Tuples[Range[Length[newElectromagneticPotential]], 2]]] & ) /@
Tuples[Range[Length[newElectromagneticPotential]], 2]]];
FullSimplify[(Module[{index = #1}, Total[(Module[{nestedIndex = #1}, D[stressEnergyTensor[[nestedIndex,index]],
newCoordinates[[nestedIndex]]] + Total[(christoffelSymbols[[nestedIndex,nestedIndex,#1]]*
stressEnergyTensor[[#1,index]] & ) /@ Range[Length[newElectromagneticPotential]]] +
Total[(christoffelSymbols[[index,nestedIndex,#1]]*stressEnergyTensor[[nestedIndex,#1]] & ) /@
Range[Length[newElectromagneticPotential]]]] & ) /@ Range[Length[newElectromagneticPotential]]] ==
0] & ) /@ Range[Length[newElectromagneticPotential]] /. (ToExpression[#1] -> #1 & ) /@
Select[coordinates, StringQ]]] /; SymbolName[electromagneticTensor] === "ElectromagneticTensor" &&
SymbolName[metricTensor] === "MetricTensor" && Length[Dimensions[matrixRepresentation]] == 2 &&
Length[coordinates] == Length[matrixRepresentation] && BooleanQ[metricIndex1] && BooleanQ[metricIndex2] &&
Length[electromagneticPotential] == Length[matrixRepresentation] && BooleanQ[index1] && BooleanQ[index2]
ElectrovacuumSolution[(electromagneticTensor_)[(metricTensor_)[matrixRepresentation_List, coordinates_List,
metricIndex1_, metricIndex2_], electromagneticPotential_List, vacuumPermeability_, index1_, index2_],
cosmologicalConstant_]["SymbolicContinuityEquations"] :=
Module[{newMatrixRepresentation, newElectromagneticPotential, newCoordinates, christoffelSymbols,
covariantElectromagneticPotential, tensorRepresentation, contravariantElectromagneticTensor,
mixedElectromagneticTensor, stressEnergyTensor},
newMatrixRepresentation = matrixRepresentation /. (#1 -> ToExpression[#1] & ) /@ Select[coordinates, StringQ];
newElectromagneticPotential = electromagneticPotential /. (#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]]];
covariantElectromagneticPotential =
Normal[SparseArray[(Module[{index = #1}, index -> Total[(newMatrixRepresentation[[index,#1]]*
newElectromagneticPotential[[#1]] & ) /@ Range[Length[newElectromagneticPotential]]]] & ) /@
Range[Length[newElectromagneticPotential]]]]; tensorRepresentation =
Normal[SparseArray[(#1 -> Inactive[D][covariantElectromagneticPotential[[Last[#1]]], newCoordinates[[First[#1]]]] -
Inactive[D][covariantElectromagneticPotential[[First[#1]]], newCoordinates[[Last[#1]]]] & ) /@
Tuples[Range[Length[newElectromagneticPotential]], 2]]]; contravariantElectromagneticTensor =
Normal[SparseArray[(Module[{index = #1}, index -> Total[(Inverse[newMatrixRepresentation][[First[index],First[#1]]]*
Inverse[newMatrixRepresentation][[Last[#1],Last[index]]]*tensorRepresentation[[First[#1],
Last[#1]]] & ) /@ Tuples[Range[Length[newElectromagneticPotential]], 2]]] & ) /@
Tuples[Range[Length[newElectromagneticPotential]], 2]]]; mixedElectromagneticTensor =
Normal[SparseArray[(Module[{index = #1}, index -> Total[(Inverse[newMatrixRepresentation][[First[index],#1]]*
tensorRepresentation[[#1,Last[index]]] & ) /@ Range[Length[newElectromagneticPotential]]]] & ) /@
Tuples[Range[Length[newElectromagneticPotential]], 2]]]; stressEnergyTensor =
(1/vacuumPermeability)*Normal[SparseArray[
(Module[{index = #1}, index -> Total[(contravariantElectromagneticTensor[[First[index],#1]]*
mixedElectromagneticTensor[[Last[index],#1]] & ) /@ Range[Length[newElectromagneticPotential]]] -
(1/4)*Total[(Inverse[newMatrixRepresentation][[First[index],Last[index]]]*tensorRepresentation[[First[#1],
Last[#1]]]*contravariantElectromagneticTensor[[First[#1],Last[#1]]] & ) /@
Tuples[Range[Length[newElectromagneticPotential]], 2]]] & ) /@
Tuples[Range[Length[newElectromagneticPotential]], 2]]];
(Module[{index = #1}, Total[(Module[{nestedIndex = #1}, Inactive[D][stressEnergyTensor[[nestedIndex,index]],
newCoordinates[[nestedIndex]]] + Total[(christoffelSymbols[[nestedIndex,nestedIndex,#1]]*
stressEnergyTensor[[#1,index]] & ) /@ Range[Length[newElectromagneticPotential]]] + Total[
(christoffelSymbols[[index,nestedIndex,#1]]*stressEnergyTensor[[nestedIndex,#1]] & ) /@
Range[Length[newElectromagneticPotential]]]] & ) /@ Range[Length[newElectromagneticPotential]]] ==
0] & ) /@ Range[Length[newElectromagneticPotential]] /. (ToExpression[#1] -> #1 & ) /@
Select[coordinates, StringQ]] /; SymbolName[electromagneticTensor] === "ElectromagneticTensor" &&
SymbolName[metricTensor] === "MetricTensor" && Length[Dimensions[matrixRepresentation]] == 2 &&
Length[coordinates] == Length[matrixRepresentation] && BooleanQ[metricIndex1] && BooleanQ[metricIndex2] &&
Length[electromagneticPotential] == Length[matrixRepresentation] && BooleanQ[index1] && BooleanQ[index2]
ElectrovacuumSolution[(electromagneticTensor_)[(metricTensor_)[matrixRepresentation_List, coordinates_List,
metricIndex1_, metricIndex2_], electromagneticPotential_List, vacuumPermeability_, index1_, index2_],
cosmologicalConstant_]["InhomogeneousMaxwellEquations"] :=
Module[{newMatrixRepresentation, newElectromagneticPotential, newCoordinates, christoffelSymbols,
covariantElectromagneticPotential, tensorRepresentation, contravariantElectromagneticTensor,
electromagneticDisplacementTensor, currentDensity},
newMatrixRepresentation = matrixRepresentation /. (#1 -> ToExpression[#1] & ) /@ Select[coordinates, StringQ];
newElectromagneticPotential = electromagneticPotential /. (#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[newElectromagneticPotential]]]] & ) /@
Tuples[Range[Length[newMatrixRepresentation]], 3]]]; covariantElectromagneticPotential =
Normal[SparseArray[(Module[{index = #1}, index -> Total[(newMatrixRepresentation[[index,#1]]*
newElectromagneticPotential[[#1]] & ) /@ Range[Length[newElectromagneticPotential]]]] & ) /@
Range[Length[newElectromagneticPotential]]]]; tensorRepresentation =
Normal[SparseArray[(#1 -> D[covariantElectromagneticPotential[[Last[#1]]], newCoordinates[[First[#1]]]] -
D[covariantElectromagneticPotential[[First[#1]]], newCoordinates[[Last[#1]]]] & ) /@
Tuples[Range[Length[newElectromagneticPotential]], 2]]]; contravariantElectromagneticTensor =
Normal[SparseArray[(Module[{index = #1}, index -> Total[(Inverse[newMatrixRepresentation][[First[index],First[#1]]]*
Inverse[newMatrixRepresentation][[Last[#1],Last[index]]]*tensorRepresentation[[First[#1],
Last[#1]]] & ) /@ Tuples[Range[Length[newElectromagneticPotential]], 2]]] & ) /@
Tuples[Range[Length[newElectromagneticPotential]], 2]]]; electromagneticDisplacementTensor =
(Sqrt[-Det[newMatrixRepresentation]]/vacuumPermeability)*contravariantElectromagneticTensor;
currentDensity = Normal[SparseArray[
(Module[{index = #1}, index -> Total[(D[electromagneticDisplacementTensor[[index,#1]], newCoordinates[[
#1]]] & ) /@ Range[Length[newElectromagneticPotential]]]] & ) /@
Range[Length[newElectromagneticPotential]]]];
(Module[{index = #1}, Total[(Module[{nestedIndex = #1}, D[contravariantElectromagneticTensor[[index,nestedIndex]],
newCoordinates[[nestedIndex]]] + Total[(christoffelSymbols[[index,nestedIndex,#1]]*
contravariantElectromagneticTensor[[#1,nestedIndex]] & ) /@ Range[Length[
newElectromagneticPotential]]] + Total[(christoffelSymbols[[nestedIndex,nestedIndex,#1]]*
contravariantElectromagneticTensor[[index,#1]] & ) /@ Range[Length[
newElectromagneticPotential]]]] & ) /@ Range[Length[newElectromagneticPotential]]] ==
vacuumPermeability*currentDensity[[index]]] & ) /@ Range[Length[newElectromagneticPotential]] /.
(ToExpression[#1] -> #1 & ) /@ Select[coordinates, StringQ]] /;
SymbolName[electromagneticTensor] === "ElectromagneticTensor" && SymbolName[metricTensor] === "MetricTensor" &&
Length[Dimensions[matrixRepresentation]] == 2 && Length[coordinates] == Length[matrixRepresentation] &&
BooleanQ[metricIndex1] && BooleanQ[metricIndex2] && Length[electromagneticPotential] ==
Length[matrixRepresentation] && BooleanQ[index1] && BooleanQ[index2]
ElectrovacuumSolution[(electromagneticTensor_)[(metricTensor_)[matrixRepresentation_List, coordinates_List,
metricIndex1_, metricIndex2_], electromagneticPotential_List, vacuumPermeability_, index1_, index2_],
cosmologicalConstant_]["ReducedInhomogeneousMaxwellEquations"] :=
Module[{newMatrixRepresentation, newElectromagneticPotential, newCoordinates, christoffelSymbols,
covariantElectromagneticPotential, tensorRepresentation, contravariantElectromagneticTensor,
electromagneticDisplacementTensor, currentDensity},
newMatrixRepresentation = matrixRepresentation /. (#1 -> ToExpression[#1] & ) /@ Select[coordinates, StringQ];
newElectromagneticPotential = electromagneticPotential /. (#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]]]; covariantElectromagneticPotential =
Normal[SparseArray[(Module[{index = #1}, index -> Total[(newMatrixRepresentation[[index,#1]]*
newElectromagneticPotential[[#1]] & ) /@ Range[Length[newElectromagneticPotential]]]] & ) /@
Range[Length[newElectromagneticPotential]]]]; tensorRepresentation =
Normal[SparseArray[(#1 -> D[covariantElectromagneticPotential[[Last[#1]]], newCoordinates[[First[#1]]]] -
D[covariantElectromagneticPotential[[First[#1]]], newCoordinates[[Last[#1]]]] & ) /@
Tuples[Range[Length[newElectromagneticPotential]], 2]]]; contravariantElectromagneticTensor =
Normal[SparseArray[(Module[{index = #1}, index -> Total[(Inverse[newMatrixRepresentation][[First[index],First[#1]]]*
Inverse[newMatrixRepresentation][[Last[#1],Last[index]]]*tensorRepresentation[[First[#1],
Last[#1]]] & ) /@ Tuples[Range[Length[newElectromagneticPotential]], 2]]] & ) /@
Tuples[Range[Length[newElectromagneticPotential]], 2]]]; electromagneticDisplacementTensor =
(Sqrt[-Det[newMatrixRepresentation]]/vacuumPermeability)*contravariantElectromagneticTensor;
currentDensity = Normal[SparseArray[
(Module[{index = #1}, index -> Total[(D[electromagneticDisplacementTensor[[index,#1]], newCoordinates[[
#1]]] & ) /@ Range[Length[newElectromagneticPotential]]]] & ) /@
Range[Length[newElectromagneticPotential]]]];
FullSimplify[(Module[{index = #1}, Total[(Module[{nestedIndex = #1}, D[contravariantElectromagneticTensor[[index,
nestedIndex]], newCoordinates[[nestedIndex]]] + Total[(christoffelSymbols[[index,nestedIndex,#1]]*
contravariantElectromagneticTensor[[#1,nestedIndex]] & ) /@ Range[Length[
newElectromagneticPotential]]] + Total[(christoffelSymbols[[nestedIndex,nestedIndex,#1]]*
contravariantElectromagneticTensor[[index,#1]] & ) /@ Range[Length[
newElectromagneticPotential]]]] & ) /@ Range[Length[newElectromagneticPotential]]] ==
vacuumPermeability*currentDensity[[index]]] & ) /@ Range[Length[newElectromagneticPotential]] /.
(ToExpression[#1] -> #1 & ) /@ Select[coordinates, StringQ]]] /;
SymbolName[electromagneticTensor] === "ElectromagneticTensor" && SymbolName[metricTensor] === "MetricTensor" &&
Length[Dimensions[matrixRepresentation]] == 2 && Length[coordinates] == Length[matrixRepresentation] &&
BooleanQ[metricIndex1] && BooleanQ[metricIndex2] && Length[electromagneticPotential] ==
Length[matrixRepresentation] && BooleanQ[index1] && BooleanQ[index2]
ElectrovacuumSolution[(electromagneticTensor_)[(metricTensor_)[matrixRepresentation_List, coordinates_List,
metricIndex1_, metricIndex2_], electromagneticPotential_List, vacuumPermeability_, index1_, index2_],
cosmologicalConstant_]["ChargeConservationEquation"] :=
Module[{newMatrixRepresentation, newElectromagneticPotential, newCoordinates, christoffelSymbols,
covariantElectromagneticPotential, tensorRepresentation, electromagneticDisplacementTensor, currentDensity},
newMatrixRepresentation = matrixRepresentation /. (#1 -> ToExpression[#1] & ) /@ Select[coordinates, StringQ];
newElectromagneticPotential = electromagneticPotential /. (#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]]]; covariantElectromagneticPotential =
Normal[SparseArray[(Module[{index = #1}, index -> Total[(newMatrixRepresentation[[index,#1]]*
newElectromagneticPotential[[#1]] & ) /@ Range[Length[newElectromagneticPotential]]]] & ) /@
Range[Length[newElectromagneticPotential]]]]; tensorRepresentation =
Normal[SparseArray[(#1 -> D[covariantElectromagneticPotential[[Last[#1]]], newCoordinates[[First[#1]]]] -
D[covariantElectromagneticPotential[[First[#1]]], newCoordinates[[Last[#1]]]] & ) /@
Tuples[Range[Length[newElectromagneticPotential]], 2]]]; electromagneticDisplacementTensor =
(Sqrt[-Det[newMatrixRepresentation]]/vacuumPermeability)*
Normal[SparseArray[(Module[{index = #1}, index -> Total[(Inverse[newMatrixRepresentation][[First[index],
First[#1]]]*Inverse[newMatrixRepresentation][[Last[#1],Last[index]]]*tensorRepresentation[[First[#1],
Last[#1]]] & ) /@ Tuples[Range[Length[newElectromagneticPotential]], 2]]] & ) /@
Tuples[Range[Length[newElectromagneticPotential]], 2]]];
currentDensity = Normal[SparseArray[
(Module[{index = #1}, index -> Total[(D[electromagneticDisplacementTensor[[index,#1]], newCoordinates[[
#1]]] & ) /@ Range[Length[newElectromagneticPotential]]]] & ) /@
Range[Length[newElectromagneticPotential]]]];
Total[(Module[{index = #1}, D[currentDensity[[index]], newCoordinates[[index]]] +
Total[(christoffelSymbols[[index,index,#1]]*currentDensity[[#1]] & ) /@ Range[Length[
newElectromagneticPotential]]]] & ) /@ Range[Length[newElectromagneticPotential]]] == 0 /.
(ToExpression[#1] & ) /@ Select[coordinates, StringQ]] /;
SymbolName[electromagneticTensor] === "ElectromagneticTensor" && SymbolName[metricTensor] === "MetricTensor" &&
Length[Dimensions[matrixRepresentation]] == 2 && Length[coordinates] == Length[matrixRepresentation] &&
BooleanQ[metricIndex1] && BooleanQ[metricIndex2] && Length[electromagneticPotential] ==
Length[matrixRepresentation] && BooleanQ[index1] && BooleanQ[index2]
ElectrovacuumSolution[(electromagneticTensor_)[(metricTensor_)[matrixRepresentation_List, coordinates_List,
metricIndex1_, metricIndex2_], electromagneticPotential_List, vacuumPermeability_, index1_, index2_],
cosmologicalConstant_]["ReducedChargeConservationEquation"] :=
Module[{newMatrixRepresentation, newElectromagneticPotential, newCoordinates, christoffelSymbols,
covariantElectromagneticPotential, tensorRepresentation, electromagneticDisplacementTensor, currentDensity},
newMatrixRepresentation = matrixRepresentation /. (#1 -> ToExpression[#1] & ) /@ Select[coordinates, StringQ];
newElectromagneticPotential = electromagneticPotential /. (#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]]]; covariantElectromagneticPotential =
Normal[SparseArray[(Module[{index = #1}, index -> Total[(newMatrixRepresentation[[index,#1]]*
newElectromagneticPotential[[#1]] & ) /@ Range[Length[newElectromagneticPotential]]]] & ) /@
Range[Length[newElectromagneticPotential]]]]; tensorRepresentation =
Normal[SparseArray[(#1 -> D[covariantElectromagneticPotential[[Last[#1]]], newCoordinates[[First[#1]]]] -
D[covariantElectromagneticPotential[[First[#1]]], newCoordinates[[Last[#1]]]] & ) /@
Tuples[Range[Length[newElectromagneticPotential]], 2]]]; electromagneticDisplacementTensor =
(Sqrt[-Det[newMatrixRepresentation]]/vacuumPermeability)*
Normal[SparseArray[(Module[{index = #1}, index -> Total[(Inverse[newMatrixRepresentation][[First[index],
First[#1]]]*Inverse[newMatrixRepresentation][[Last[#1],Last[index]]]*tensorRepresentation[[First[#1],
Last[#1]]] & ) /@ Tuples[Range[Length[newElectromagneticPotential]], 2]]] & ) /@
Tuples[Range[Length[newElectromagneticPotential]], 2]]];
currentDensity = Normal[SparseArray[
(Module[{index = #1}, index -> Total[(D[electromagneticDisplacementTensor[[index,#1]], newCoordinates[[
#1]]] & ) /@ Range[Length[newElectromagneticPotential]]]] & ) /@
Range[Length[newElectromagneticPotential]]]];
FullSimplify[Total[(Module[{index = #1}, D[currentDensity[[index]], newCoordinates[[index]]] +
Total[(christoffelSymbols[[index,index,#1]]*currentDensity[[#1]] & ) /@ Range[
Length[newElectromagneticPotential]]]] & ) /@ Range[Length[newElectromagneticPotential]]] == 0 /.
(ToExpression[#1] -> #1 & ) /@ Select[coordinates, StringQ]]] /;
SymbolName[electromagneticTensor] === "ElectromagneticTensor" && SymbolName[metricTensor] === "MetricTensor" &&
Length[Dimensions[matrixRepresentation]] == 2 && Length[coordinates] == Length[matrixRepresentation] &&
BooleanQ[metricIndex1] && BooleanQ[metricIndex2] && Length[electromagneticPotential] ==
Length[matrixRepresentation] && BooleanQ[index1] && BooleanQ[index2]
ElectrovacuumSolution[(electromagneticTensor_)[(metricTensor_)[matrixRepresentation_List, coordinates_List,
metricIndex1_, metricIndex2_], electromagneticPotential_List, vacuumPermeability_, index1_, index2_],
cosmologicalConstant_]["MetricTensor"] := ResourceFunction["MetricTensor"][matrixRepresentation, coordinates,
metricIndex1, metricIndex2] /; SymbolName[electromagneticTensor] === "ElectromagneticTensor" &&
SymbolName[metricTensor] === "MetricTensor" && Length[Dimensions[matrixRepresentation]] == 2 &&
Length[coordinates] == Length[matrixRepresentation] && BooleanQ[metricIndex1] && BooleanQ[metricIndex2] &&
Length[electromagneticPotential] == Length[matrixRepresentation] && BooleanQ[index1] && BooleanQ[index2]
ElectrovacuumSolution[(electromagneticTensor_)[(metricTensor_)[matrixRepresentation_List, coordinates_List,
metricIndex1_, metricIndex2_], electromagneticPotential_List, vacuumPermeability_, index1_, index2_],
cosmologicalConstant_]["ElectromagneticTensor"] :=
ElectromagneticTensor[ResourceFunction["MetricTensor"][matrixRepresentation, coordinates, metricIndex1, metricIndex2],
electromagneticPotential, vacuumPermeability, index1, index2] /;
SymbolName[electromagneticTensor] === "ElectromagneticTensor" && SymbolName[metricTensor] === "MetricTensor" &&
Length[Dimensions[matrixRepresentation]] == 2 && Length[coordinates] == Length[matrixRepresentation] &&
BooleanQ[metricIndex1] && BooleanQ[metricIndex2] && Length[electromagneticPotential] ==
Length[matrixRepresentation] && BooleanQ[index1] && BooleanQ[index2]
ElectrovacuumSolution[(electromagneticTensor_)[(metricTensor_)[matrixRepresentation_List, coordinates_List,
metricIndex1_, metricIndex2_], electromagneticPotential_List, vacuumPermeability_, index1_, index2_],
cosmologicalConstant_]["ElectromagneticStressEnergyTensor"] :=
Module[{newMatrixRepresentation, newElectromagneticPotential, newCoordinates, covariantElectromagneticPotential,
tensorRepresentation, contravariantElectromagneticTensor, mixedElectromagneticTensor, stressEnergyTensor},
newMatrixRepresentation = matrixRepresentation /. (#1 -> ToExpression[#1] & ) /@ Select[coordinates, StringQ];
newElectromagneticPotential = electromagneticPotential /. (#1 -> ToExpression[#1] & ) /@
Select[coordinates, StringQ]; newCoordinates = coordinates /. (#1 -> ToExpression[#1] & ) /@
Select[coordinates, StringQ]; covariantElectromagneticPotential =
Normal[SparseArray[(Module[{index = #1}, index -> Total[(newMatrixRepresentation[[index,#1]]*
newElectromagneticPotential[[#1]] & ) /@ Range[Length[newElectromagneticPotential]]]] & ) /@
Range[Length[newElectromagneticPotential]]]]; tensorRepresentation =
Normal[SparseArray[(#1 -> D[covariantElectromagneticPotential[[Last[#1]]], newCoordinates[[First[#1]]]] -
D[covariantElectromagneticPotential[[First[#1]]], newCoordinates[[Last[#1]]]] & ) /@
Tuples[Range[Length[newElectromagneticPotential]], 2]]] /. (ToExpression[#1] -> #1 & ) /@
Select[coordinates, StringQ]; contravariantElectromagneticTensor =
Normal[SparseArray[(Module[{index = #1}, index -> Total[(Inverse[matrixRepresentation][[First[index],First[#1]]]*
Inverse[matrixRepresentation][[Last[#1],Last[index]]]*tensorRepresentation[[First[#1],Last[#1]]] & ) /@
Tuples[Range[Length[electromagneticPotential]], 2]]] & ) /@ Tuples[Range[Length[electromagneticPotential]],
2]]]; mixedElectromagneticTensor =
Normal[SparseArray[(Module[{index = #1}, index -> Total[(Inverse[matrixRepresentation][[First[index],#1]]*
tensorRepresentation[[#1,Last[index]]] & ) /@ Range[Length[electromagneticPotential]]]] & ) /@
Tuples[Range[Length[electromagneticPotential]], 2]]]; stressEnergyTensor =
(1/vacuumPermeability)*Normal[SparseArray[
(Module[{index = #1}, index -> Total[(contravariantElectromagneticTensor[[First[index],#1]]*
mixedElectromagneticTensor[[Last[index],#1]] & ) /@ Range[Length[electromagneticPotential]]] -
(1/4)*Total[(Inverse[matrixRepresentation][[First[index],Last[index]]]*tensorRepresentation[[First[#1],
Last[#1]]]*contravariantElectromagneticTensor[[First[#1],Last[#1]]] & ) /@
Tuples[Range[Length[electromagneticPotential]], 2]]] & ) /@
Tuples[Range[Length[electromagneticPotential]], 2]]]; ResourceFunction["StressEnergyTensor"][
ResourceFunction["MetricTensor"][matrixRepresentation, coordinates, metricIndex1, metricIndex2],
stressEnergyTensor, False, False]] /; SymbolName[electromagneticTensor] === "ElectromagneticTensor" &&
SymbolName[metricTensor] === "MetricTensor" && Length[Dimensions[matrixRepresentation]] == 2 &&
Length[coordinates] == Length[matrixRepresentation] && BooleanQ[metricIndex1] && BooleanQ[metricIndex2] &&
Length[electromagneticPotential] == Length[matrixRepresentation] && BooleanQ[index1] && BooleanQ[index2]
ElectrovacuumSolution[(electromagneticTensor_)[(metricTensor_)[matrixRepresentation_List, coordinates_List,
metricIndex1_, metricIndex2_], electromagneticPotential_List, vacuumPermeability_, index1_, index2_],
cosmologicalConstant_]["Coordinates"] :=
coordinates /; SymbolName[electromagneticTensor] === "ElectromagneticTensor" &&
SymbolName[metricTensor] === "MetricTensor" && Length[Dimensions[matrixRepresentation]] == 2 &&
Length[coordinates] == Length[matrixRepresentation] && BooleanQ[metricIndex1] && BooleanQ[metricIndex2] &&
Length[electromagneticPotential] == Length[matrixRepresentation] && BooleanQ[index1] && BooleanQ[index2]
ElectrovacuumSolution[(electromagneticTensor_)[(metricTensor_)[matrixRepresentation_List, coordinates_List,
metricIndex1_, metricIndex2_], electromagneticPotential_List, vacuumPermeablity_, 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[electromagneticTensor] === "ElectromagneticTensor" && SymbolName[metricTensor] === "MetricTensor" &&
Length[Dimensions[matrixRepresentation]] == 2 && Length[coordinates] == Length[matrixRepresentation] &&
BooleanQ[metricIndex1] && BooleanQ[metricIndex2] && Length[electromagneticPotential] ==
Length[matrixRepresentation] && BooleanQ[index1] && BooleanQ[index2]
ElectrovacuumSolution[(electromagneticTensor_)[(metricTensor_)[matrixRepresentation_List, coordinates_List,
metricIndex1_, metricIndex2_], electromagneticPotential_List, vacuumPermeability_, index1_, index2_],
cosmologicalConstant_]["SolutionQ"] := Module[{newMatrixRepresentation, newElectromagneticPotential, newCoordinates,
christoffelSymbols, riemannTensor, ricciTensor, ricciScalar, covariantElectromagneticPotential,
tensorRepresentation, contravariantElectromagneticTensor, mixedElectromagneticTensor, stressEnergyTensor,
covariantStressEnergyTensor, einsteinEquations},
newMatrixRepresentation = matrixRepresentation /. (#1 -> ToExpression[#1] & ) /@ Select[coordinates, StringQ];
newElectromagneticPotential = electromagneticPotential /. (#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]]; covariantElectromagneticPotential =
Normal[SparseArray[(Module[{index = #1}, index -> Total[(newMatrixRepresentation[[index,#1]]*
newElectromagneticPotential[[#1]] & ) /@ Range[Length[newElectromagneticPotential]]]] & ) /@
Range[Length[newElectromagneticPotential]]]]; tensorRepresentation =
Normal[SparseArray[(#1 -> D[covariantElectromagneticPotential[[Last[#1]]], newCoordinates[[First[#1]]]] -
D[covariantElectromagneticPotential[[First[#1]]], newCoordinates[[Last[#1]]]] & ) /@
Tuples[Range[Length[newElectromagneticPotential]], 2]]] /. (ToExpression[#1] -> #1 & ) /@
Select[coordinates, StringQ]; contravariantElectromagneticTensor =
Normal[SparseArray[(Module[{index = #1}, index -> Total[(Inverse[matrixRepresentation][[First[index],First[#1]]]*
Inverse[matrixRepresentation][[Last[#1],Last[index]]]*tensorRepresentation[[First[#1],Last[#1]]] & ) /@
Tuples[Range[Length[electromagneticPotential]], 2]]] & ) /@ Tuples[Range[Length[electromagneticPotential]],
2]]]; mixedElectromagneticTensor =
Normal[SparseArray[(Module[{index = #1}, index -> Total[(Inverse[matrixRepresentation][[First[index],#1]]*
tensorRepresentation[[#1,Last[index]]] & ) /@ Range[Length[electromagneticPotential]]]] & ) /@
Tuples[Range[Length[electromagneticPotential]], 2]]]; stressEnergyTensor =
(1/vacuumPermeability)*Normal[SparseArray[
(Module[{index = #1}, index -> Total[(contravariantElectromagneticTensor[[First[index],#1]]*
mixedElectromagneticTensor[[Last[index],#1]] & ) /@ Range[Length[electromagneticPotential]]] -
(1/4)*Total[(Inverse[matrixRepresentation][[First[index],Last[index]]]*tensorRepresentation[[First[#1],
Last[#1]]]*contravariantElectromagneticTensor[[First[#1],Last[#1]]] & ) /@
Tuples[Range[Length[electromagneticPotential]], 2]]] & ) /@
Tuples[Range[Length[electromagneticPotential]], 2]]]; covariantStressEnergyTensor =
Normal[SparseArray[(Module[{index = #1}, index -> Total[(matrixRepresentation[[First[index],First[#1]]]*
matrixRepresentation[[Last[#1],Last[index]]]*stressEnergyTensor[[First[#1],Last[#1]]] & ) /@
Tuples[Range[Length[matrixRepresentation]], 2]]] & ) /@ Tuples[Range[Length[matrixRepresentation]], 2]]];
einsteinEquations = FullSimplify[Thread[Catenate[ricciTensor - (1/2)*ricciScalar*matrixRepresentation +
cosmologicalConstant*matrixRepresentation] == Catenate[(8*Pi)*covariantStressEnergyTensor]]];
If[einsteinEquations === True, True, If[einsteinEquations === False, False,
If[Length[Select[einsteinEquations, #1 === False & ]] > 0, False, True]]]] /;
SymbolName[electromagneticTensor] === "ElectromagneticTensor" && SymbolName[metricTensor] === "MetricTensor" &&
Length[Dimensions[matrixRepresentation]] == 2 && Length[coordinates] == Length[matrixRepresentation] &&
BooleanQ[metricIndex1] && BooleanQ[metricIndex2] && Length[electromagneticPotential] ==
Length[matrixRepresentation] && BooleanQ[index1] && BooleanQ[index2]
ElectrovacuumSolution[(electromagneticTensor_)[(metricTensor_)[matrixRepresentation_List, coordinates_List,
metricIndex1_, metricIndex2_], electromagneticPotential_List, vacuumPermeability_, index1_, index2_],
cosmologicalConstant_]["ExactSolutionQ"] :=
Module[{newMatrixRepresentation, newElectromagneticPotential, newCoordinates, christoffelSymbols, riemannTensor,
ricciTensor, ricciScalar, covariantElectromagneticPotential, tensorRepresentation,
contravariantElectromagneticTensor, mixedElectromagneticTensor, stressEnergyTensor, covariantStressEnergyTensor,
einsteinEquations}, newMatrixRepresentation = matrixRepresentation /. (#1 -> ToExpression[#1] & ) /@
Select[coordinates, StringQ]; newElectromagneticPotential = electromagneticPotential /.
(#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]]; covariantElectromagneticPotential =
Normal[SparseArray[(Module[{index = #1}, index -> Total[(newMatrixRepresentation[[index,#1]]*
newElectromagneticPotential[[#1]] & ) /@ Range[Length[newElectromagneticPotential]]]] & ) /@
Range[Length[newElectromagneticPotential]]]]; tensorRepresentation =
Normal[SparseArray[(#1 -> D[covariantElectromagneticPotential[[Last[#1]]], newCoordinates[[First[#1]]]] -
D[covariantElectromagneticPotential[[First[#1]]], newCoordinates[[Last[#1]]]] & ) /@
Tuples[Range[Length[newElectromagneticPotential]], 2]]] /. (ToExpression[#1] -> #1 & ) /@
Select[coordinates, StringQ]; contravariantElectromagneticTensor =
Normal[SparseArray[(Module[{index = #1}, index -> Total[(Inverse[matrixRepresentation][[First[index],First[#1]]]*
Inverse[matrixRepresentation][[Last[#1],Last[index]]]*tensorRepresentation[[First[#1],Last[#1]]] & ) /@
Tuples[Range[Length[electromagneticPotential]], 2]]] & ) /@ Tuples[Range[Length[electromagneticPotential]],
2]]]; mixedElectromagneticTensor =
Normal[SparseArray[(Module[{index = #1}, index -> Total[(Inverse[matrixRepresentation][[First[index],#1]]*
tensorRepresentation[[#1,Last[index]]] & ) /@ Range[Length[electromagneticPotential]]]] & ) /@
Tuples[Range[Length[electromagneticPotential]], 2]]]; stressEnergyTensor =
(1/vacuumPermeability)*Normal[SparseArray[
(Module[{index = #1}, index -> Total[(contravariantElectromagneticTensor[[First[index],#1]]*
mixedElectromagneticTensor[[Last[index],#1]] & ) /@ Range[Length[electromagneticPotential]]] -
(1/4)*Total[(Inverse[matrixRepresentation][[First[index],Last[index]]]*tensorRepresentation[[First[#1],
Last[#1]]]*contravariantElectromagneticTensor[[First[#1],Last[#1]]] & ) /@
Tuples[Range[Length[electromagneticPotential]], 2]]] & ) /@
Tuples[Range[Length[electromagneticPotential]], 2]]]; covariantStressEnergyTensor =
Normal[SparseArray[(Module[{index = #1}, index -> Total[(matrixRepresentation[[First[index],First[#1]]]*
matrixRepresentation[[Last[#1],Last[index]]]*stressEnergyTensor[[First[#1],Last[#1]]] & ) /@
Tuples[Range[Length[matrixRepresentation]], 2]]] & ) /@ Tuples[Range[Length[matrixRepresentation]], 2]]];
einsteinEquations = FullSimplify[Thread[Catenate[ricciTensor - (1/2)*ricciScalar*matrixRepresentation +
cosmologicalConstant*matrixRepresentation] == Catenate[(8*Pi)*covariantStressEnergyTensor]]];
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[electromagneticTensor] === "ElectromagneticTensor" &&
SymbolName[metricTensor] === "MetricTensor" && Length[Dimensions[matrixRepresentation]] == 2 &&
Length[coordinates] == Length[matrixRepresentation] && BooleanQ[metricIndex1] && BooleanQ[metricIndex2] &&
Length[electromagneticPotential] == Length[matrixRepresentation] && BooleanQ[index1] && BooleanQ[index2]
ElectrovacuumSolution[(electromagneticTensor_)[(metricTensor_)[matrixRepresentation_List, coordinates_List,
metricIndex1_, metricIndex2_], electromagneticPotential_List, vacuumPermeability_, index1_, index2_],
cosmologicalConstant_]["CosmologicalConstant"] := cosmologicalConstant /;
SymbolName[electromagneticTensor] === "ElectromagneticTensor" && SymbolName[metricTensor] === "MetricTensor" &&
Length[Dimensions[matrixRepresentation]] == 2 && Length[coordinates] == Length[matrixRepresentation] &&
BooleanQ[metricIndex1] && BooleanQ[metricIndex2] && Length[electromagneticPotential] ==
Length[matrixRepresentation] && BooleanQ[index1] && BooleanQ[index2]
ElectrovacuumSolution[(electromagneticTensor_)[(metricTensor_)[matrixRepresentation_List, coordinates_List,
metricIndex1_, metricIndex2_], electromagneticPotential_List, vacuumPermeablity_, index1_, index2_],
cosmologicalConstant_]["Dimensions"] := Length[matrixRepresentation] /;
SymbolName[electromagneticTensor] === "ElectromagneticTensor" && SymbolName[metricTensor] === "MetricTensor" &&
Length[Dimensions[matrixRepresentation]] == 2 && Length[coordinates] == Length[matrixRepresentation] &&
BooleanQ[metricIndex1] && BooleanQ[metricIndex2] && Length[electromagneticPotential] ==
Length[matrixRepresentation] && BooleanQ[index1] && BooleanQ[index2]
ElectrovacuumSolution[(electromagneticTensor_)[(metricTensor_)[matrixRepresentation_List, coordinates_List,
metricIndex1_, metricIndex2_], electromagneticPotential_List, vacuumPermeability_, 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[electromagneticTensor] === "ElectromagneticTensor" &&
SymbolName[metricTensor] === "MetricTensor" && Length[Dimensions[matrixRepresentation]] == 2 &&
Length[coordinates] == Length[matrixRepresentation] && BooleanQ[metricIndex1] && BooleanQ[metricIndex2] &&
Length[electromagneticPotential] == Length[matrixRepresentation] && BooleanQ[index1] && BooleanQ[index2]
ElectrovacuumSolution[(electromagneticTensor_)[(metricTensor_)[matrixRepresentation_List, coordinates_List,
metricIndex1_, metricIndex2_], electromagneticPotential_List, vacuumPermeability_, 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[electromagneticTensor] === "ElectromagneticTensor" && SymbolName[metricTensor] === "MetricTensor" &&
Length[Dimensions[matrixRepresentation]] == 2 && Length[coordinates] == Length[matrixRepresentation] &&
BooleanQ[metricIndex1] && BooleanQ[metricIndex2] && Length[electromagneticPotential] ==
Length[matrixRepresentation] && BooleanQ[index1] && BooleanQ[index2]
ElectrovacuumSolution /:
MakeBoxes[electrovacuumSolution:ElectrovacuumSolution[(electromagneticTensor_)[
(metricTensor_)[matrixRepresentation_List, coordinates_List, metricIndex1_, metricIndex2_],
electromagneticPotential_List, vacuumPermeability_, index1_, index2_], cosmologicalConstant_], format_] :=
Module[{newMatrixRepresentation, newElectromagneticPotential, newCoordinates, christoffelSymbols, riemannTensor,
ricciTensor, ricciScalar, matrixForm, covariantElectromagneticPotential, tensorRepresentation,
contravariantElectromagneticTensor, mixedElectromagneticTensor, stressEnergyTensor, covariantStressEnergyTensor,
einsteinEquations, solution, exactSolution, fieldEquations, icon},
newMatrixRepresentation = matrixRepresentation /. (#1 -> ToExpression[#1] & ) /@ Select[coordinates, StringQ];
newElectromagneticPotential = electromagneticPotential /. (#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;
covariantElectromagneticPotential = Normal[SparseArray[
(Module[{index = #1}, index -> Total[(newMatrixRepresentation[[index,#1]]*newElectromagneticPotential[[
#1]] & ) /@ Range[Length[newElectromagneticPotential]]]] & ) /@
Range[Length[newElectromagneticPotential]]]]; tensorRepresentation =
Normal[SparseArray[(#1 -> D[covariantElectromagneticPotential[[Last[#1]]], newCoordinates[[First[#1]]]] -
D[covariantElectromagneticPotential[[First[#1]]], newCoordinates[[Last[#1]]]] & ) /@
Tuples[Range[Length[newElectromagneticPotential]], 2]]] /. (ToExpression[#1] -> #1 & ) /@
Select[coordinates, StringQ]; contravariantElectromagneticTensor =
Normal[SparseArray[(Module[{index = #1}, index -> Total[(Inverse[matrixRepresentation][[First[index],First[#1]]]*
Inverse[matrixRepresentation][[Last[#1],Last[index]]]*tensorRepresentation[[First[#1],Last[#1]]] & ) /@
Tuples[Range[Length[electromagneticPotential]], 2]]] & ) /@
Tuples[Range[Length[electromagneticPotential]], 2]]]; mixedElectromagneticTensor =
Normal[SparseArray[(Module[{index = #1}, index -> Total[(Inverse[matrixRepresentation][[First[index],#1]]*
tensorRepresentation[[#1,Last[index]]] & ) /@ Range[Length[electromagneticPotential]]]] & ) /@
Tuples[Range[Length[electromagneticPotential]], 2]]]; stressEnergyTensor =
(1/vacuumPermeability)*Normal[SparseArray[
(Module[{index = #1}, index -> Total[(contravariantElectromagneticTensor[[First[index],#1]]*
mixedElectromagneticTensor[[Last[index],#1]] & ) /@ Range[Length[electromagneticPotential]]] - (1/4)*
Total[(Inverse[matrixRepresentation][[First[index],Last[index]]]*tensorRepresentation[[First[#1],
Last[#1]]]*contravariantElectromagneticTensor[[First[#1],Last[#1]]] & ) /@
Tuples[Range[Length[electromagneticPotential]], 2]]] & ) /@
Tuples[Range[Length[electromagneticPotential]], 2]]]; covariantStressEnergyTensor =
Normal[SparseArray[(Module[{index = #1}, index -> Total[(matrixRepresentation[[First[index],First[#1]]]*
matrixRepresentation[[Last[#1],Last[index]]]*stressEnergyTensor[[First[#1],Last[#1]]] & ) /@ Tuples[
Range[Length[matrixRepresentation]], 2]]] & ) /@ Tuples[Range[Length[matrixRepresentation]], 2]]];
einsteinEquations = FullSimplify[Thread[Catenate[ricciTensor - (1/2)*ricciScalar*matrixRepresentation +
cosmologicalConstant*matrixRepresentation] == Catenate[(8*Pi)*covariantStressEnergyTensor]]];
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["ElectrovacuumSolution", electrovacuumSolution, icon,
{{BoxForm`SummaryItem[{"Solution: ", solution}], BoxForm`SummaryItem[{"Exact Solution: ", exactSolution}]},
{BoxForm`SummaryItem[{"Field Equations: ", fieldEquations}], BoxForm`SummaryItem[{"Cosmological Constant: ",
cosmologicalConstant}]}}, {{}}, format, "Interpretable" -> Automatic]] /;
SymbolName[electromagneticTensor] === "ElectromagneticTensor" && SymbolName[metricTensor] === "MetricTensor" &&
Length[Dimensions[matrixRepresentation]] == 2 && Length[coordinates] == Length[matrixRepresentation] &&
BooleanQ[metricIndex1] && BooleanQ[metricIndex2] && Length[electromagneticPotential] ==
Length[matrixRepresentation] && BooleanQ[index1] && BooleanQ[index2]