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ExtrinsicCurvatureTensor.wl
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(* ::Package:: *)
ExtrinsicCurvatureTensor[(admDecomposition_)[(metricTensor_)[matrixRepresentation_List, coordinates_List, index1_,
index2_], timeCoordinate_, lapseFunction_, shiftVector_List]] :=
ExtrinsicCurvatureTensor[ResourceFunction["ADMDecomposition"][ResourceFunction["MetricTensor"][matrixRepresentation,
coordinates, index1, index2], timeCoordinate, lapseFunction, shiftVector], True, True] /;
SymbolName[admDecomposition] === "ADMDecomposition" && SymbolName[metricTensor] === "MetricTensor" &&
Length[Dimensions[matrixRepresentation]] == 2 && Length[coordinates] == Length[matrixRepresentation] &&
BooleanQ[index1] && BooleanQ[index2] && Length[shiftVector] == Length[matrixRepresentation]
ExtrinsicCurvatureTensor[(admDecomposition_)[(metricTensor_)[matrixRepresentation_List, coordinates_List, index1_,
index2_], timeCoordinate_, lapseFunction_, shiftVector_List], newTimeCoordinate_] :=
ExtrinsicCurvatureTensor[ResourceFunction["ADMDecomposition"][ResourceFunction["MetricTensor"][
matrixRepresentation /. timeCoordinate -> newTimeCoordinate, coordinates, index1, index2], newTimeCoordinate,
lapseFunction /. timeCoordinate -> newTimeCoordinate, shiftVector /. timeCoordinate -> newTimeCoordinate], True,
True] /; SymbolName[admDecomposition] === "ADMDecomposition" && SymbolName[metricTensor] === "MetricTensor" &&
Length[Dimensions[matrixRepresentation]] == 2 && Length[coordinates] == Length[matrixRepresentation] &&
BooleanQ[index1] && BooleanQ[index2] && Length[shiftVector] == Length[matrixRepresentation]
ExtrinsicCurvatureTensor[(admDecomposition_)[(metricTensor_)[matrixRepresentation_List, coordinates_List, metricIndex1_,
metricIndex2_], timeCoordinate_, lapseFunction_, shiftVector_List], newTimeCoordinate_, index1_, index2_] :=
ExtrinsicCurvatureTensor[ResourceFunction["ADMDecomposition"][ResourceFunction["MetricTensor"][
matrixRepresentation /. timeCoordinate -> newTimeCoordinate, coordinates, metricIndex1, metricIndex2],
newTimeCoordinate, lapseFunction /. timeCoordinate -> newTimeCoordinate,
shiftVector /. timeCoordinate -> newTimeCoordinate], index1, index2] /;
SymbolName[admDecomposition] === "ADMDecomposition" && SymbolName[metricTensor] === "MetricTensor" &&
Length[Dimensions[matrixRepresentation]] == 2 && Length[coordinates] == Length[matrixRepresentation] &&
BooleanQ[metricIndex1] && BooleanQ[metricIndex2] && Length[shiftVector] == Length[matrixRepresentation] &&
BooleanQ[index1] && BooleanQ[index2]
ExtrinsicCurvatureTensor[(admDecomposition_)[(metricTensor_)[matrixRepresentation_List, coordinates_List, index1_,
index2_], timeCoordinate_, lapseFunction_, shiftVector_List], newLapseFunction_, newShiftVector_List] :=
ExtrinsicCurvatureTensor[ResourceFunction["ADMDecomposition"][ResourceFunction["MetricTensor"][matrixRepresentation,
coordinates, index1, index2], timeCoordinate, newLapseFunction, newShiftVector], True, True] /;
SymbolName[admDecomposition] === "ADMDecomposition" && SymbolName[metricTensor] === "MetricTensor" &&
Length[Dimensions[matrixRepresentation]] == 2 && Length[coordinates] == Length[matrixRepresentation] &&
BooleanQ[index1] && BooleanQ[index2] && Length[shiftVector] == Length[matrixRepresentation] &&
Length[newShiftVector] == Length[matrixRepresentation]
ExtrinsicCurvatureTensor[(admDecomposition_)[(metricTensor_)[matrixRepresentation_List, coordinates_List, metricIndex1_,
metricIndex2_], timeCoordinate_, lapseFunction_, shiftVector_List], newLapseFunction_, newShiftVector_, index1_,
index2_] := ExtrinsicCurvatureTensor[ResourceFunction["ADMDecomposition"][ResourceFunction["MetricTensor"][
matrixRepresentation, coordinates, metricIndex1, metricIndex2], timeCoordinate, newLapseFunction, newShiftVector],
index1, index2] /; SymbolName[admDecomposition] === "ADMDecomposition" &&
SymbolName[metricTensor] === "MetricTensor" && Length[Dimensions[matrixRepresentation]] == 2 &&
Length[coordinates] == Length[matrixRepresentation] && BooleanQ[metricIndex1] && BooleanQ[metricIndex2] &&
Length[shiftVector] == Length[matrixRepresentation] && Length[newShiftVector] == Length[matrixRepresentation] &&
BooleanQ[index1] && BooleanQ[index2]
ExtrinsicCurvatureTensor[(admDecomposition_)[(metricTensor_)[matrixRepresentation_List, coordinates_List, index1_,
index2_], timeCoordinate_, lapseFunction_, shiftVector_List], newTimeCoordinate_, newLapseFunction_,
newShiftVector_List] := ExtrinsicCurvatureTensor[ResourceFunction["ADMDecomposition"][
ResourceFunction["MetricTensor"][matrixRepresentation /. timeCoordinate -> newTimeCoordinate, coordinates, index1,
index2], newTimeCoordinate, newLapseFunction, newShiftVector], True, True] /;
SymbolName[admDecomposition] === "ADMDecomposition" && SymbolName[metricTensor] === "MetricTensor" &&
Length[Dimensions[matrixRepresentation]] == 2 && Length[coordinates] == Length[matrixRepresentation] &&
BooleanQ[index1] && BooleanQ[index2] && Length[shiftVector] == Length[matrixRepresentation] &&
Length[newShiftVector] == Length[matrixRepresentation]
ExtrinsicCurvatureTensor[(admDecomposition_)[(metricTensor_)[matrixRepresentation_List, coordinates_List, metricIndex1_,
metricIndex2_], timeCoordinate_, lapseFunction_, shiftVector_List], newTimeCoordinate_, newLapseFunction_,
newShiftVector_List, index1_, index2_] :=
ExtrinsicCurvatureTensor[ResourceFunction["ADMDecomposition"][ResourceFunction["MetricTensor"][
matrixRepresentation /. timeCoordinate -> newTimeCoordinate, coordinates, metricIndex1, metricIndex2],
newTimeCoordinate, newLapseFunction, newShiftVector], index1, index2] /;
SymbolName[admDecomposition] === "ADMDecomposition" && SymbolName[metricTensor] === "MetricTensor" &&
Length[Dimensions[matrixRepresentation]] == 2 && Length[coordinates] == Length[matrixRepresentation] &&
BooleanQ[metricIndex1] && BooleanQ[metricIndex2] && Length[shiftVector] == Length[matrixRepresentation] &&
Length[newShiftVector] == Length[matrixRepresentation] && BooleanQ[index1] && BooleanQ[index2]
ExtrinsicCurvatureTensor[(admDecomposition_)[(metricTensor_)[matrixRepresentation_List, coordinates_List, metricIndex1_,
metricIndex2_], timeCoordinate_, lapseFunction_, shiftVector_List], index1_, index2_]["MatrixRepresentation"] :=
Module[{newMatrixRepresentation, newCoordinates, newTimeCoordinate, newLapseFunction, newShiftVector, shiftCovector,
spatialChristoffelSymbols, extrinsicCurvatureTensor},
newMatrixRepresentation = matrixRepresentation /. (#1 -> ToExpression[#1] & ) /@
Select[Join[coordinates, {timeCoordinate}], StringQ]; newCoordinates =
coordinates /. (#1 -> ToExpression[#1] & ) /@ Select[Join[coordinates, {timeCoordinate}], StringQ];
newTimeCoordinate = timeCoordinate /. (#1 -> ToExpression[#1] & ) /@ Select[Join[coordinates, {timeCoordinate}],
StringQ]; newLapseFunction = lapseFunction /. (#1 -> ToExpression[#1] & ) /@
Select[Join[coordinates, {timeCoordinate}], StringQ]; newShiftVector =
shiftVector /. (#1 -> ToExpression[#1] & ) /@ Select[Join[coordinates, {timeCoordinate}], StringQ];
shiftCovector = Normal[SparseArray[
(Module[{index = #1}, index -> Total[(newMatrixRepresentation[[index,#1]]*newShiftVector[[#1]] & ) /@
Range[Length[newMatrixRepresentation]]]] & ) /@ Range[Length[newMatrixRepresentation]]]];
spatialChristoffelSymbols = Normal[SparseArray[
(Module[{index = #1}, index -> Total[((1/2)*Inverse[newMatrixRepresentation][[index[[1]],#1]]*
(D[newMatrixRepresentation[[#1,index[[3]]]], newCoordinates[[index[[2]]]]] + D[newMatrixRepresentation[[
index[[2]],#1]], newCoordinates[[index[[3]]]]] - D[newMatrixRepresentation[[index[[2]],index[[3]]]],
newCoordinates[[#1]]]) & ) /@ Range[Length[newMatrixRepresentation]]]] & ) /@
Tuples[Range[Length[newMatrixRepresentation]], 3]]]; extrinsicCurvatureTensor =
Normal[SparseArray[(Module[{index = #1}, index -> (1/(2*newLapseFunction))*(D[shiftCovector[[First[index]]],
newCoordinates[[Last[index]]]] - Total[(spatialChristoffelSymbols[[#1,Last[index],First[index]]]*
shiftCovector[[#1]] & ) /@ Range[Length[newMatrixRepresentation]]] + D[shiftCovector[[Last[index]]],
newCoordinates[[First[index]]]] - Total[(spatialChristoffelSymbols[[#1,First[index],Last[index]]]*
shiftCovector[[#1]] & ) /@ Range[Length[newMatrixRepresentation]]] - D[newMatrixRepresentation[[
First[index],Last[index]]], newTimeCoordinate])] & ) /@ Tuples[Range[Length[newMatrixRepresentation]],
2]]] /. (ToExpression[#1] -> #1 & ) /@ Select[Join[coordinates, {timeCoordinate}], StringQ];
If[index1 === True && index2 === True, extrinsicCurvatureTensor, If[index1 === False && index2 === False,
Normal[SparseArray[(Module[{index = #1}, index -> Total[(Inverse[matrixRepresentation][[First[index],First[#1]]]*
Inverse[matrixRepresentation][[Last[#1],Last[index]]]*extrinsicCurvatureTensor[[First[#1],
Last[#1]]] & ) /@ Tuples[Range[Length[matrixRepresentation]], 2]]] & ) /@
Tuples[Range[Length[matrixRepresentation]], 2]]], If[index1 === True && index2 === False,
Normal[SparseArray[(Module[{index = #1}, index -> Total[(Inverse[matrixRepresentation][[#1,Last[index]]]*
extrinsicCurvatureTensor[[First[index],#1]] & ) /@ Range[Length[matrixRepresentation]]]] & ) /@
Tuples[Range[Length[matrixRepresentation]], 2]]], If[index1 === False && index2 === True,
Normal[SparseArray[(Module[{index = #1}, index -> Total[(Inverse[matrixRepresentation][[First[index],#1]]*
extrinsicCurvatureTensor[[#1,Last[index]]] & ) /@ Range[Length[matrixRepresentation]]]] & ) /@
Tuples[Range[Length[matrixRepresentation]], 2]]], Indeterminate]]]]] /;
SymbolName[admDecomposition] == "ADMDecomposition" && SymbolName[metricTensor] === "MetricTensor" &&
Length[Dimensions[matrixRepresentation]] == 2 && Length[coordinates] == Length[matrixRepresentation] &&
BooleanQ[metricIndex1] && BooleanQ[metricIndex2] && Length[shiftVector] == Length[matrixRepresentation] &&
BooleanQ[index1] && BooleanQ[index2]
ExtrinsicCurvatureTensor[(admDecomposition_)[(metricTensor_)[matrixRepresentation_List, coordinates_List, metricIndex1_,
metricIndex2_], timeCoordinate_, lapseFunction_, shiftVector_List], index1_, index2_][
"ReducedMatrixRepresentation"] := Module[{newMatrixRepresentation, newCoordinates, newTimeCoordinate,
newLapseFunction, newShiftVector, shiftCovector, spatialChristoffelSymbols, extrinsicCurvatureTensor},
newMatrixRepresentation = matrixRepresentation /. (#1 -> ToExpression[#1] & ) /@
Select[Join[coordinates, {timeCoordinate}], StringQ]; newCoordinates =
coordinates /. (#1 -> ToExpression[#1] & ) /@ Select[Join[coordinates, {timeCoordinate}], StringQ];
newTimeCoordinate = timeCoordinate /. (#1 -> ToExpression[#1] & ) /@ Select[Join[coordinates, {timeCoordinate}],
StringQ]; newLapseFunction = lapseFunction /. (#1 -> ToExpression[#1] & ) /@
Select[Join[coordinates, {timeCoordinate}], StringQ]; newShiftVector =
shiftVector /. (#1 -> ToExpression[#1] & ) /@ Select[Join[coordinates, {timeCoordinate}], StringQ];
shiftCovector = Normal[SparseArray[
(Module[{index = #1}, index -> Total[(newMatrixRepresentation[[index,#1]]*newShiftVector[[#1]] & ) /@
Range[Length[newMatrixRepresentation]]]] & ) /@ Range[Length[newMatrixRepresentation]]]];
spatialChristoffelSymbols = Normal[SparseArray[
(Module[{index = #1}, index -> Total[((1/2)*Inverse[newMatrixRepresentation][[index[[1]],#1]]*
(D[newMatrixRepresentation[[#1,index[[3]]]], newCoordinates[[index[[2]]]]] + D[newMatrixRepresentation[[
index[[2]],#1]], newCoordinates[[index[[3]]]]] - D[newMatrixRepresentation[[index[[2]],index[[3]]]],
newCoordinates[[#1]]]) & ) /@ Range[Length[newMatrixRepresentation]]]] & ) /@
Tuples[Range[Length[newMatrixRepresentation]], 3]]]; extrinsicCurvatureTensor =
Normal[SparseArray[(Module[{index = #1}, index -> (1/(2*newLapseFunction))*(D[shiftCovector[[First[index]]],
newCoordinates[[Last[index]]]] - Total[(spatialChristoffelSymbols[[#1,Last[index],First[index]]]*
shiftCovector[[#1]] & ) /@ Range[Length[newMatrixRepresentation]]] + D[shiftCovector[[Last[index]]],
newCoordinates[[First[index]]]] - Total[(spatialChristoffelSymbols[[#1,First[index],Last[index]]]*
shiftCovector[[#1]] & ) /@ Range[Length[newMatrixRepresentation]]] - D[newMatrixRepresentation[[
First[index],Last[index]]], newTimeCoordinate])] & ) /@ Tuples[Range[Length[newMatrixRepresentation]],
2]]] /. (ToExpression[#1] -> #1 & ) /@ Select[Join[coordinates, {timeCoordinate}], StringQ];
If[index1 === True && index2 === True, FullSimplify[extrinsicCurvatureTensor],
If[index1 === False && index2 === False, FullSimplify[
Normal[SparseArray[(Module[{index = #1}, index -> Total[(Inverse[matrixRepresentation][[First[index],First[#1]]]*
Inverse[matrixRepresentation][[Last[#1],Last[index]]]*extrinsicCurvatureTensor[[First[#1],
Last[#1]]] & ) /@ Tuples[Range[Length[matrixRepresentation]], 2]]] & ) /@
Tuples[Range[Length[matrixRepresentation]], 2]]]], If[index1 === True && index2 === False,
FullSimplify[Normal[SparseArray[(Module[{index = #1}, index -> Total[(Inverse[matrixRepresentation][[#1,
Last[index]]]*extrinsicCurvatureTensor[[First[index],#1]] & ) /@ Range[Length[
matrixRepresentation]]]] & ) /@ Tuples[Range[Length[matrixRepresentation]], 2]]]],
If[index1 === False && index2 === True, FullSimplify[
Normal[SparseArray[(Module[{index = #1}, index -> Total[(Inverse[matrixRepresentation][[First[index],#1]]*
extrinsicCurvatureTensor[[#1,Last[index]]] & ) /@ Range[Length[matrixRepresentation]]]] & ) /@
Tuples[Range[Length[matrixRepresentation]], 2]]]], Indeterminate]]]]] /;
SymbolName[admDecomposition] === "ADMDecomposition" && SymbolName[metricTensor] === "MetricTensor" &&
Length[Dimensions[matrixRepresentation]] == 2 && Length[coordinates] == Length[matrixRepresentation] &&
BooleanQ[metricIndex1] && BooleanQ[metricIndex2] && Length[shiftVector] == Length[matrixRepresentation] &&
BooleanQ[index1] && BooleanQ[index2]
ExtrinsicCurvatureTensor[(admDecomposition_)[(metricTensor_)[matrixRepresentation_List, coordinates_List, metricIndex1_,
metricIndex2_], timeCoordinate_, lapseFunction_, shiftVector_List], index1_, index2_][
"SymbolicMatrixRepresentation"] := Module[{newMatrixRepresentation, newCoordinates, newTimeCoordinate,
newLapseFunction, newShiftVector, shiftCovector, spatialChristoffelSymbols, extrinsicCurvatureTensor},
newMatrixRepresentation = matrixRepresentation /. (#1 -> ToExpression[#1] & ) /@
Select[Join[coordinates, {timeCoordinate}], StringQ]; newCoordinates =
coordinates /. (#1 -> ToExpression[#1] & ) /@ Select[Join[coordinates, {timeCoordinate}], StringQ];
newTimeCoordinate = timeCoordinate /. (#1 -> ToExpression[#1] & ) /@ Select[Join[coordinates, {timeCoordinate}],
StringQ]; newLapseFunction = lapseFunction /. (#1 -> ToExpression[#1] & ) /@
Select[Join[coordinates, {timeCoordinate}], StringQ]; newShiftVector =
shiftVector /. (#1 -> ToExpression[#1] & ) /@ Select[Join[coordinates, {timeCoordinate}], StringQ];
shiftCovector = Normal[SparseArray[
(Module[{index = #1}, index -> Total[(newMatrixRepresentation[[index,#1]]*newShiftVector[[#1]] & ) /@
Range[Length[newMatrixRepresentation]]]] & ) /@ Range[Length[newMatrixRepresentation]]]];
spatialChristoffelSymbols = Normal[SparseArray[
(Module[{index = #1}, index -> Total[((1/2)*Inverse[newMatrixRepresentation][[index[[1]],#1]]*
(Inactive[D][newMatrixRepresentation[[#1,index[[3]]]], newCoordinates[[index[[2]]]]] +
Inactive[D][newMatrixRepresentation[[index[[2]],#1]], newCoordinates[[index[[3]]]]] -
Inactive[D][newMatrixRepresentation[[index[[2]],index[[3]]]], newCoordinates[[#1]]]) & ) /@
Range[Length[newMatrixRepresentation]]]] & ) /@ Tuples[Range[Length[newMatrixRepresentation]], 3]]];
extrinsicCurvatureTensor =
Normal[SparseArray[(Module[{index = #1}, index -> (1/(2*newLapseFunction))*(Inactive[D][shiftCovector[[
First[index]]], newCoordinates[[Last[index]]]] - Total[(spatialChristoffelSymbols[[#1,Last[index],
First[index]]]*shiftCovector[[#1]] & ) /@ Range[Length[newMatrixRepresentation]]] + Inactive[D][
shiftCovector[[Last[index]]], newCoordinates[[First[index]]]] - Total[
(spatialChristoffelSymbols[[#1,First[index],Last[index]]]*shiftCovector[[#1]] & ) /@
Range[Length[newMatrixRepresentation]]] - Inactive[D][newMatrixRepresentation[[First[index],
Last[index]]], newTimeCoordinate])] & ) /@ Tuples[Range[Length[newMatrixRepresentation]], 2]]] /.
(ToExpression[#1] -> #1 & ) /@ Select[Join[coordinates, {timeCoordinate}], StringQ];
If[index1 === True && index2 === True, extrinsicCurvatureTensor, If[index1 === False && index2 === False,
Normal[SparseArray[(Module[{index = #1}, index -> Total[(Inverse[matrixRepresentation][[First[index],First[#1]]]*
Inverse[matrixRepresentation][[Last[#1],Last[index]]]*extrinsicCurvatureTensor[[First[#1],
Last[#1]]] & ) /@ Tuples[Range[Length[matrixRepresentation]], 2]]] & ) /@
Tuples[Range[Length[matrixRepresentation]], 2]]], If[index1 === True && index2 === False,
Normal[SparseArray[(Module[{index = #1}, index -> Total[(Inverse[matrixRepresentation][[#1,Last[index]]]*
extrinsicCurvatureTensor[[First[index],#1]] & ) /@ Range[Length[matrixRepresentation]]]] & ) /@
Tuples[Range[Length[matrixRepresentation]], 2]]], If[index1 === False && index2 === True,
Normal[SparseArray[(Module[{index = #1}, index -> Total[(Inverse[matrixRepresentation][[First[index],#1]]*
extrinsicCurvatureTensor[[#1,Last[index]]] & ) /@ Range[Length[matrixRepresentation]]]] & ) /@
Tuples[Range[Length[matrixRepresentation]], 2]]], Indeterminate]]]]] /;
SymbolName[admDecomposition] === "ADMDecomposition" && SymbolName[metricTensor] === "MetricTensor" &&
Length[Dimensions[matrixRepresentation]] == 2 && Length[coordinates] == Length[matrixRepresentation] &&
BooleanQ[metricIndex1] && BooleanQ[metricIndex2] && Length[shiftVector] == Length[matrixRepresentation] &&
BooleanQ[index1] && BooleanQ[index2]
ExtrinsicCurvatureTensor[(admDecomposition_)[(metricTensor_)[matrixRepresentation_List, coordinates_List, metricIndex1_,
metricIndex2_], timeCoordinate_, lapseFunction_, shiftVector_List], index1_, index2_]["Trace"] :=
Module[{newMatrixRepresentation, newCoordinates, newTimeCoordinate, newLapseFunction, newShiftVector, shiftCovector,
spatialChristoffelSymbols, extrinsicCurvatureTensor},
newMatrixRepresentation = matrixRepresentation /. (#1 -> ToExpression[#1] & ) /@
Select[Join[coordinates, {timeCoordinate}], StringQ]; newCoordinates =
coordinates /. (#1 -> ToExpression[#1] & ) /@ Select[Join[coordinates, {timeCoordinate}], StringQ];
newTimeCoordinate = timeCoordinate /. (#1 -> ToExpression[#1] & ) /@ Select[Join[coordinates, {timeCoordinate}],
StringQ]; newLapseFunction = lapseFunction /. (#1 -> ToExpression[#1] & ) /@
Select[Join[coordinates, {timeCoordinate}], StringQ]; newShiftVector =
shiftVector /. (#1 -> ToExpression[#1] & ) /@ Select[Join[coordinates, {timeCoordinate}], StringQ];
shiftCovector = Normal[SparseArray[
(Module[{index = #1}, index -> Total[(newMatrixRepresentation[[index,#1]]*newShiftVector[[#1]] & ) /@
Range[Length[newMatrixRepresentation]]]] & ) /@ Range[Length[newMatrixRepresentation]]]];
spatialChristoffelSymbols = Normal[SparseArray[
(Module[{index = #1}, index -> Total[((1/2)*Inverse[newMatrixRepresentation][[index[[1]],#1]]*
(D[newMatrixRepresentation[[#1,index[[3]]]], newCoordinates[[index[[2]]]]] + D[newMatrixRepresentation[[
index[[2]],#1]], newCoordinates[[index[[3]]]]] - D[newMatrixRepresentation[[index[[2]],index[[3]]]],
newCoordinates[[#1]]]) & ) /@ Range[Length[newMatrixRepresentation]]]] & ) /@
Tuples[Range[Length[newMatrixRepresentation]], 3]]]; extrinsicCurvatureTensor =
Normal[SparseArray[(Module[{index = #1}, index -> (1/(2*newLapseFunction))*(D[shiftCovector[[First[index]]],
newCoordinates[[Last[index]]]] - Total[(spatialChristoffelSymbols[[#1,Last[index],First[index]]]*
shiftCovector[[#1]] & ) /@ Range[Length[newMatrixRepresentation]]] + D[shiftCovector[[Last[index]]],
newCoordinates[[First[index]]]] - Total[(spatialChristoffelSymbols[[#1,First[index],Last[index]]]*
shiftCovector[[#1]] & ) /@ Range[Length[newMatrixRepresentation]]] - D[newMatrixRepresentation[[
First[index],Last[index]]], newTimeCoordinate])] & ) /@ Tuples[Range[Length[newMatrixRepresentation]],
2]]] /. (ToExpression[#1] -> #1 & ) /@ Select[Join[coordinates, {timeCoordinate}], StringQ];
Total[(Inverse[matrixRepresentation][[First[#1],Last[#1]]]*extrinsicCurvatureTensor[[First[#1],Last[#1]]] & ) /@
Tuples[Range[Length[matrixRepresentation]], 2]]] /; SymbolName[admDecomposition] === "ADMDecomposition" &&
SymbolName[metricTensor] === "MetricTensor" && Length[Dimensions[matrixRepresentation]] == 2 &&
Length[coordinates] == Length[matrixRepresentation] && BooleanQ[metricIndex1] && BooleanQ[metricIndex2] &&
Length[shiftVector] == Length[matrixRepresentation] && BooleanQ[index1] && BooleanQ[index2]
ExtrinsicCurvatureTensor[(admDecomposition_)[(metricTensor_)[matrixRepresentation_List, coordinates_List, metricIndex1_,
metricIndex2_], timeCoordinate_, lapseFunction_, shiftVector_List], index1_, index2_]["ReducedTrace"] :=
Module[{newMatrixRepresentation, newCoordinates, newTimeCoordinate, newLapseFunction, newShiftVector, shiftCovector,
spatialChristoffelSymbols, extrinsicCurvatureTensor},
newMatrixRepresentation = matrixRepresentation /. (#1 -> ToExpression[#1] & ) /@
Select[Join[coordinates, {timeCoordinate}], StringQ]; newCoordinates =
coordinates /. (#1 -> ToExpression[#1] & ) /@ Select[Join[coordinates, {timeCoordinate}], StringQ];
newTimeCoordinate = timeCoordinate /. (#1 -> ToExpression[#1] & ) /@ Select[Join[coordinates, {timeCoordinate}],
StringQ]; newLapseFunction = lapseFunction /. (#1 -> ToExpression[#1] & ) /@
Select[Join[coordinates, {timeCoordinate}], StringQ]; newShiftVector =
shiftVector /. (#1 -> ToExpression[#1] & ) /@ Select[Join[coordinates, {timeCoordinate}], StringQ];
shiftCovector = Normal[SparseArray[
(Module[{index = #1}, index -> Total[(newMatrixRepresentation[[index,#1]]*newShiftVector[[#1]] & ) /@
Range[Length[newMatrixRepresentation]]]] & ) /@ Range[Length[newMatrixRepresentation]]]];
spatialChristoffelSymbols = Normal[SparseArray[
(Module[{index = #1}, index -> Total[((1/2)*Inverse[newMatrixRepresentation][[index[[1]],#1]]*
(D[newMatrixRepresentation[[#1,index[[3]]]], newCoordinates[[index[[2]]]]] + D[newMatrixRepresentation[[
index[[2]],#1]], newCoordinates[[index[[3]]]]] - D[newMatrixRepresentation[[index[[2]],index[[3]]]],
newCoordinates[[#1]]]) & ) /@ Range[Length[newMatrixRepresentation]]]] & ) /@
Tuples[Range[Length[newMatrixRepresentation]], 3]]]; extrinsicCurvatureTensor =
Normal[SparseArray[(Module[{index = #1}, index -> (1/(2*newLapseFunction))*(D[shiftCovector[[First[index]]],
newCoordinates[[Last[index]]]] - Total[(spatialChristoffelSymbols[[#1,Last[index],First[index]]]*
shiftCovector[[#1]] & ) /@ Range[Length[newMatrixRepresentation]]] + D[shiftCovector[[Last[index]]],
newCoordinates[[First[index]]]] - Total[(spatialChristoffelSymbols[[#1,First[index],Last[index]]]*
shiftCovector[[#1]] & ) /@ Range[Length[newMatrixRepresentation]]] - D[newMatrixRepresentation[[
First[index],Last[index]]], newTimeCoordinate])] & ) /@ Tuples[Range[Length[newMatrixRepresentation]],
2]]] /. (ToExpression[#1] -> #1 & ) /@ Select[Join[coordinates, {timeCoordinate}], StringQ];
FullSimplify[Total[(Inverse[matrixRepresentation][[First[#1],Last[#1]]]*extrinsicCurvatureTensor[[First[#1],
Last[#1]]] & ) /@ Tuples[Range[Length[matrixRepresentation]], 2]]]] /;
SymbolName[admDecomposition] === "ADMDecomposition" && SymbolName[metricTensor] === "MetricTensor" &&
Length[Dimensions[matrixRepresentation]] == 2 && Length[coordinates] == Length[matrixRepresentation] &&
BooleanQ[metricIndex1] && BooleanQ[metricIndex2] && Length[shiftVector] == Length[matrixRepresentation] &&
BooleanQ[index1] && BooleanQ[index2]
ExtrinsicCurvatureTensor[(admDecomposition_)[(metricTensor_)[matrixRepresentation_List, coordinates_List, metricIndex1_,
metricIndex2_], timeCoordinate_, lapseFunction_, shiftVector_List], index1_, index2_]["SymbolicTrace"] :=
Module[{newMatrixRepresentation, newCoordinates, newTimeCoordinate, newLapseFunction, newShiftVector, shiftCovector,
spatialChristoffelSymbols, extrinsicCurvatureTensor},
newMatrixRepresentation = matrixRepresentation /. (#1 -> ToExpression[#1] & ) /@
Select[Join[coordinates, {timeCoordinate}], StringQ]; newCoordinates =
coordinates /. (#1 -> ToExpression[#1] & ) /@ Select[Join[coordinates, {timeCoordinate}], StringQ];
newTimeCoordinate = timeCoordinate /. (#1 -> ToExpression[#1] & ) /@ Select[Join[coordinates, {timeCoordinate}],
StringQ]; newLapseFunction = lapseFunction /. (#1 -> ToExpression[#1] & ) /@
Select[Join[coordinates, {timeCoordinate}], StringQ]; newShiftVector =
shiftVector /. (#1 -> ToExpression[#1] & ) /@ Select[Join[coordinates, {timeCoordinate}], StringQ];
shiftCovector = Normal[SparseArray[
(Module[{index = #1}, index -> Total[(newMatrixRepresentation[[index,#1]]*newShiftVector[[#1]] & ) /@
Range[Length[newMatrixRepresentation]]]] & ) /@ Range[Length[newMatrixRepresentation]]]];
spatialChristoffelSymbols = Normal[SparseArray[
(Module[{index = #1}, index -> Total[((1/2)*Inverse[newMatrixRepresentation][[index[[1]],#1]]*
(Inactive[D][newMatrixRepresentation[[#1,index[[3]]]], newCoordinates[[index[[2]]]]] +
Inactive[D][newMatrixRepresentation[[index[[2]],#1]], newCoordinates[[index[[3]]]]] -
Inactive[D][newMatrixRepresentation[[index[[2]],index[[3]]]], newCoordinates[[#1]]]) & ) /@
Range[Length[newMatrixRepresentation]]]] & ) /@ Tuples[Range[Length[newMatrixRepresentation]], 3]]];
extrinsicCurvatureTensor =
Normal[SparseArray[(Module[{index = #1}, index -> (1/(2*newLapseFunction))*(Inactive[D][shiftCovector[[
First[index]]], newCoordinates[[Last[index]]]] - Total[(spatialChristoffelSymbols[[#1,Last[index],
First[index]]]*shiftCovector[[#1]] & ) /@ Range[Length[newMatrixRepresentation]]] + Inactive[D][
shiftCovector[[Last[index]]], newCoordinates[[First[index]]]] - Total[
(spatialChristoffelSymbols[[#1,First[index],Last[index]]]*shiftCovector[[#1]] & ) /@
Range[Length[newMatrixRepresentation]]] - Inactive[D][newMatrixRepresentation[[First[index],
Last[index]]], newTimeCoordinate])] & ) /@ Tuples[Range[Length[newMatrixRepresentation]], 2]]] /.
(ToExpression[#1] -> #1 & ) /@ Select[Join[coordinates, {timeCoordinate}], StringQ];
Total[(Inverse[matrixRepresentation][[First[#1],Last[#1]]]*extrinsicCurvatureTensor[[First[#1],Last[#1]]] & ) /@
Tuples[Range[Length[matrixRepresentation]], 2]]] /; SymbolName[admDecomposition] === "ADMDecomposition" &&
SymbolName[metricTensor] === "MetricTensor" && Length[Dimensions[matrixRepresentation]] == 2 &&
Length[coordinates] == Length[matrixRepresentation] && BooleanQ[metricIndex1] && BooleanQ[metricIndex2] &&
Length[shiftVector] == Length[matrixRepresentation] && BooleanQ[index1] && BooleanQ[index2]
ExtrinsicCurvatureTensor[(admDecomposition_)[(metricTensor_)[matrixRepresentation_List, coordinates_List, metricIndex1_,
metricIndex2_], timeCoordinate_, lapseFunction_, shiftVector_List], index1_, index2_]["ADMDecomposition"] :=
ResourceFunction["ADMDecomposition"][ResourceFunction["MetricTensor"][matrixRepresentation, coordinates, metricIndex1,
metricIndex2], timeCoordinate, lapseFunction, shiftVector] /; SymbolName[admDecomposition] === "ADMDecomposition" &&
SymbolName[metricTensor] === "MetricTensor" && Length[Dimensions[matrixRepresentation]] == 2 &&
Length[coordinates] == Length[matrixRepresentation] && BooleanQ[metricIndex1] && BooleanQ[metricIndex2] &&
Length[shiftVector] == Length[matrixRepresentation] && BooleanQ[index1] && BooleanQ[index2]
ExtrinsicCurvatureTensor[(admDecomposition_)[(metricTensor_)[matrixRepresentation_List, coordinates_List, metricIndex1_,
metricIndex2_], timeCoordinate_, lapseFunction_, shiftVector_List], index1_, index2_]["SpatialMetricTensor"] :=
ResourceFunction["MetricTensor"][matrixRepresentation, coordinates, metricIndex1, metricIndex2] /;
SymbolName[admDecomposition] === "ADMDecomposition" && SymbolName[metricTensor] === "MetricTensor" &&
Length[Dimensions[matrixRepresentation]] == 2 && Length[coordinates] == Length[matrixRepresentation] &&
BooleanQ[metricIndex1] && BooleanQ[metricIndex2] && Length[shiftVector] == Length[matrixRepresentation] &&
BooleanQ[index1] && BooleanQ[index2]
ExtrinsicCurvatureTensor[(admDecomposition_)[(metricTensor_)[matrixRepresentation_List, coordinates_List, metricIndex1_,
metricIndex2_], timeCoordinate_, lapseFunction_, shiftVector_List], index1_, index2_]["SpacetimeMetricTensor"] :=
Module[{shiftCovector},
shiftCovector = Normal[SparseArray[(Module[{index = #1}, index -> Total[(matrixRepresentation[[index,#1]]*
shiftVector[[#1]] & ) /@ Range[Length[matrixRepresentation]]]] & ) /@
Range[Length[matrixRepresentation]]]]; ResourceFunction["MetricTensor"][
Normal[SparseArray[Join[{{1, 1} -> Total[(shiftVector[[#1]]*shiftCovector[[#1]] & ) /@
Range[Length[matrixRepresentation]]] - lapseFunction^2},
(Module[{index = #1}, {1, index + 1} -> Total[(matrixRepresentation[[index,#1]]*shiftVector[[#1]] & ) /@ Range[
Length[matrixRepresentation]]]] & ) /@ Range[Length[matrixRepresentation]],
(Module[{index = #1}, {index + 1, 1} -> Total[(matrixRepresentation[[#1,index]]*shiftVector[[#1]] & ) /@ Range[
Length[matrixRepresentation]]]] & ) /@ Range[Length[matrixRepresentation]],
({First[#1] + 1, Last[#1] + 1} -> matrixRepresentation[[First[#1],Last[#1]]] & ) /@
Tuples[Range[Length[matrixRepresentation]], 2]]]], Join[{timeCoordinate}, coordinates], True, True]] /;
SymbolName[admDecomposition] === "ADMDecomposition" && SymbolName[metricTensor] === "MetricTensor" &&
Length[Dimensions[matrixRepresentation]] == 2 && Length[coordinates] == Length[matrixRepresentation] &&
BooleanQ[metricIndex1] && BooleanQ[metricIndex2] && Length[shiftVector] == Length[matrixRepresentation] &&
BooleanQ[index1] && BooleanQ[index2]
ExtrinsicCurvatureTensor[(admDecomposition_)[(metricTensor_)[matrixRepresentation_List, coordinates_List, metricIndex1_,
metricIndex2_], timeCoordinate_, lapseFunction_, shiftVector_List], index1_, index2_]["NormalVector"] :=
Module[{newMatrixRepresentation, newCoordinates, newTimeCoordinate, newLapseFunction, newShiftVector, shiftCovector,
spacetimeMetricTensor}, newMatrixRepresentation = matrixRepresentation /. (#1 -> ToExpression[#1] & ) /@
Select[Join[coordinates, {timeCoordinate}], StringQ]; newCoordinates =
coordinates /. (#1 -> ToExpression[#1] & ) /@ Select[Join[coordinates, {timeCoordinate}], StringQ];
newTimeCoordinate = timeCoordinate /. (#1 -> ToExpression[#1] & ) /@ Select[Join[coordinates, {timeCoordinate}],
StringQ]; newLapseFunction = lapseFunction /. (#1 -> ToExpression[#1] & ) /@
Select[Join[coordinates, {timeCoordinate}], StringQ]; newShiftVector =
shiftVector /. (#1 -> ToExpression[#1] & ) /@ Select[Join[coordinates, {timeCoordinate}], StringQ];
shiftCovector = Normal[SparseArray[
(Module[{index = #1}, index -> Total[(newMatrixRepresentation[[index,#1]]*newShiftVector[[#1]] & ) /@
Range[Length[newMatrixRepresentation]]]] & ) /@ Range[Length[newMatrixRepresentation]]]];
spacetimeMetricTensor = Normal[SparseArray[
Join[{{1, 1} -> Total[(newShiftVector[[#1]]*shiftCovector[[#1]] & ) /@ Range[Length[newMatrixRepresentation]]] -
newLapseFunction^2}, (Module[{index = #1}, {1, index + 1} -> Total[(newMatrixRepresentation[[index,#1]]*
newShiftVector[[#1]] & ) /@ Range[Length[newMatrixRepresentation]]]] & ) /@
Range[Length[newMatrixRepresentation]], (Module[{index = #1}, {index + 1, 1} ->
Total[(newMatrixRepresentation[[#1,index]]*newShiftVector[[#1]] & ) /@ Range[
Length[newMatrixRepresentation]]]] & ) /@ Range[Length[newMatrixRepresentation]],
({First[#1] + 1, Last[#1] + 1} -> newMatrixRepresentation[[First[#1],Last[#1]]] & ) /@
Tuples[Range[Length[newMatrixRepresentation]], 2]]]];
Normal[SparseArray[(Module[{index = #1}, index -> -Total[(newLapseFunction*Inverse[spacetimeMetricTensor][[index,
#1]]*D[newTimeCoordinate, Join[{newTimeCoordinate}, newCoordinates][[#1]]] & ) /@ Range[
Length[spacetimeMetricTensor]]]] & ) /@ Range[Length[spacetimeMetricTensor]]]] /.
(ToExpression[#1] -> #1 & ) /@ Select[Join[coordinates, {timeCoordinate}], StringQ]] /;
SymbolName[admDecomposition] === "ADMDecomposition" && SymbolName[metricTensor] === "MetricTensor" &&
Length[Dimensions[matrixRepresentation]] == 2 && Length[coordinates] == Length[matrixRepresentation] &&
BooleanQ[metricIndex1] && BooleanQ[metricIndex2] && Length[shiftVector] == Length[matrixRepresentation] &&
BooleanQ[index1] && BooleanQ[index2]
ExtrinsicCurvatureTensor[(admDecomposition_)[(metricTensor_)[matrixRepresentation_List, coordinates_List, metricIndex1_,
metricIndex2_], timeCoordinate_, lapseFunction_, shiftVector_List], index1_, index2_]["ReducedNormalVector"] :=
Module[{newMatrixRepresentation, newCoordinates, newTimeCoordinate, newLapseFunction, newShiftVector, shiftCovector,
spacetimeMetricTensor}, newMatrixRepresentation = matrixRepresentation /. (#1 -> ToExpression[#1] & ) /@
Select[Join[coordinates, {timeCoordinate}], StringQ]; newCoordinates =
coordinates /. (#1 -> ToExpression[#1] & ) /@ Select[Join[coordinates, {timeCoordinate}], StringQ];
newTimeCoordinate = timeCoordinate /. (#1 -> ToExpression[#1] & ) /@ Select[Join[coordinates, {timeCoordinate}],
StringQ]; newLapseFunction = lapseFunction /. (#1 -> ToExpression[#1] & ) /@
Select[Join[coordinates, {timeCoordinate}], StringQ]; newShiftVector =
shiftVector /. (#1 -> ToExpression[#1] & ) /@ Select[Join[coordinates, {timeCoordinate}], StringQ];
shiftCovector = Normal[SparseArray[
(Module[{index = #1}, index -> Total[(newMatrixRepresentation[[index,#1]]*newShiftVector[[#1]] & ) /@
Range[Length[newMatrixRepresentation]]]] & ) /@ Range[Length[newMatrixRepresentation]]]];
spacetimeMetricTensor = Normal[SparseArray[
Join[{{1, 1} -> Total[(newShiftVector[[#1]]*shiftCovector[[#1]] & ) /@ Range[Length[newMatrixRepresentation]]] -
newLapseFunction^2}, (Module[{index = #1}, {1, index + 1} -> Total[(newMatrixRepresentation[[index,#1]]*
newShiftVector[[#1]] & ) /@ Range[Length[newMatrixRepresentation]]]] & ) /@
Range[Length[newMatrixRepresentation]], (Module[{index = #1}, {index + 1, 1} ->
Total[(newMatrixRepresentation[[#1,index]]*newShiftVector[[#1]] & ) /@ Range[
Length[newMatrixRepresentation]]]] & ) /@ Range[Length[newMatrixRepresentation]],
({First[#1] + 1, Last[#1] + 1} -> newMatrixRepresentation[[First[#1],Last[#1]]] & ) /@
Tuples[Range[Length[newMatrixRepresentation]], 2]]]];
FullSimplify[
Normal[SparseArray[(Module[{index = #1}, index -> -Total[(newLapseFunction*Inverse[spacetimeMetricTensor][[index,
#1]]*D[newTimeCoordinate, Join[{newTimeCoordinate}, newCoordinates][[#1]]] & ) /@
Range[Length[spacetimeMetricTensor]]]] & ) /@ Range[Length[spacetimeMetricTensor]]]] /.
(ToExpression[#1] -> #1 & ) /@ Select[Join[coordinates, {timeCoordinate}], StringQ]]] /;
SymbolName[admDecomposition] === "ADMDecomposition" && SymbolName[metricTensor] === "MetricTensor" &&
Length[Dimensions[matrixRepresentation]] == 2 && Length[coordinates] == Length[matrixRepresentation] &&
BooleanQ[metricIndex1] && BooleanQ[metricIndex2] && Length[shiftVector] == Length[matrixRepresentation] &&
BooleanQ[index1] && BooleanQ[index2]
ExtrinsicCurvatureTensor[(admDecomposition_)[(metricTensor_)[matrixRepresentation_List, coordinates_List, metricIndex1_,
metricIndex2_], timeCoordinate_, lapseFunction_, shiftVector_List], index1_, index2_]["SymbolicNormalVector"] :=
Module[{newMatrixRepresentation, newCoordinates, newTimeCoordinate, newLapseFunction, newShiftVector, shiftCovector,
spacetimeMetricTensor}, newMatrixRepresentation = matrixRepresentation /. (#1 -> ToExpression[#1] & ) /@
Select[Join[coordinates, {timeCoordinate}], StringQ]; newCoordinates =
coordinates /. (#1 -> ToExpression[#1] & ) /@ Select[Join[coordinates, {timeCoordinate}], StringQ];
newTimeCoordinate = timeCoordinate /. (#1 -> ToExpression[#1] & ) /@ Select[Join[coordinates, {timeCoordinate}],
StringQ]; newLapseFunction = lapseFunction /. (#1 -> ToExpression[#1] & ) /@
Select[Join[coordinates, {timeCoordinate}], StringQ]; newShiftVector =
shiftVector /. (#1 -> ToExpression[#1] & ) /@ Select[Join[coordinates, {timeCoordinate}], StringQ];
shiftCovector = Normal[SparseArray[
(Module[{index = #1}, index -> Total[(newMatrixRepresentation[[index,#1]]*newShiftVector[[#1]] & ) /@
Range[Length[newMatrixRepresentation]]]] & ) /@ Range[Length[newMatrixRepresentation]]]];
spacetimeMetricTensor = Normal[SparseArray[
Join[{{1, 1} -> Total[(newShiftVector[[#1]]*shiftCovector[[#1]] & ) /@ Range[Length[newMatrixRepresentation]]] -
newLapseFunction^2}, (Module[{index = #1}, {1, index + 1} -> Total[(newMatrixRepresentation[[index,#1]]*
newShiftVector[[#1]] & ) /@ Range[Length[newMatrixRepresentation]]]] & ) /@
Range[Length[newMatrixRepresentation]], (Module[{index = #1}, {index + 1, 1} ->
Total[(newMatrixRepresentation[[#1,index]]*newShiftVector[[#1]] & ) /@ Range[
Length[newMatrixRepresentation]]]] & ) /@ Range[Length[newMatrixRepresentation]],
({First[#1] + 1, Last[#1] + 1} -> newMatrixRepresentation[[First[#1],Last[#1]]] & ) /@
Tuples[Range[Length[newMatrixRepresentation]], 2]]]];
Normal[SparseArray[(Module[{index = #1}, index -> -Total[(newLapseFunction*Inverse[spacetimeMetricTensor][[index,
#1]]*Inactive[D][newTimeCoordinate, Join[{newTimeCoordinate}, newCoordinates][[#1]]] & ) /@ Range[
Length[spacetimeMetricTensor]]]] & ) /@ Range[Length[spacetimeMetricTensor]]]] /.
(ToExpression[#1] -> #1 & ) /@ Select[Join[coordinates, {timeCoordinate}], StringQ]] /;
SymbolName[admDecomposition] === "ADMDecomposition" && SymbolName[metricTensor] === "MetricTensor" &&
Length[Dimensions[matrixRepresentation]] == 2 && Length[coordinates] == Length[matrixRepresentation] &&
BooleanQ[metricIndex1] && BooleanQ[metricIndex2] && Length[shiftVector] == Length[matrixRepresentation] &&
BooleanQ[index1] && BooleanQ[index2]
ExtrinsicCurvatureTensor[(admDecomposition_)[(metricTensor_)[matrixRepresentation_List, coordinates_List, metricIndex1_,
metricIndex2_], timeCoordinate_, lapseFunction_, shiftVector_List], index1_, index2_]["TimeVector"] :=
Module[{newMatrixRepresentation, newCoordinates, newTimeCoordinate, newLapseFunction, newShiftVector, shiftCovector,
spacetimeMetricTensor, normalVector}, newMatrixRepresentation = matrixRepresentation /.
(#1 -> ToExpression[#1] & ) /@ Select[Join[coordinates, {timeCoordinate}], StringQ];
newCoordinates = coordinates /. (#1 -> ToExpression[#1] & ) /@ Select[Join[coordinates, {timeCoordinate}], StringQ];
newTimeCoordinate = timeCoordinate /. (#1 -> ToExpression[#1] & ) /@ Select[Join[coordinates, {timeCoordinate}],
StringQ]; newLapseFunction = lapseFunction /. (#1 -> ToExpression[#1] & ) /@
Select[Join[coordinates, {timeCoordinate}], StringQ]; newShiftVector =
shiftVector /. (#1 -> ToExpression[#1] & ) /@ Select[Join[coordinates, {timeCoordinate}], StringQ];
shiftCovector = Normal[SparseArray[
(Module[{index = #1}, index -> Total[(newMatrixRepresentation[[index,#1]]*newShiftVector[[#1]] & ) /@
Range[Length[newMatrixRepresentation]]]] & ) /@ Range[Length[newMatrixRepresentation]]]];
spacetimeMetricTensor = Normal[SparseArray[
Join[{{1, 1} -> Total[(newShiftVector[[#1]]*shiftCovector[[#1]] & ) /@ Range[Length[newMatrixRepresentation]]] -
newLapseFunction^2}, (Module[{index = #1}, {1, index + 1} -> Total[(newMatrixRepresentation[[index,#1]]*
newShiftVector[[#1]] & ) /@ Range[Length[newMatrixRepresentation]]]] & ) /@
Range[Length[newMatrixRepresentation]], (Module[{index = #1}, {index + 1, 1} ->
Total[(newMatrixRepresentation[[#1,index]]*newShiftVector[[#1]] & ) /@ Range[
Length[newMatrixRepresentation]]]] & ) /@ Range[Length[newMatrixRepresentation]],
({First[#1] + 1, Last[#1] + 1} -> newMatrixRepresentation[[First[#1],Last[#1]]] & ) /@
Tuples[Range[Length[newMatrixRepresentation]], 2]]]];
normalVector = Normal[SparseArray[
(Module[{index = #1}, index -> -Total[(newLapseFunction*Inverse[spacetimeMetricTensor][[index,#1]]*
D[newTimeCoordinate, Join[{newTimeCoordinate}, newCoordinates][[#1]]] & ) /@ Range[
Length[spacetimeMetricTensor]]]] & ) /@ Range[Length[spacetimeMetricTensor]]]];
Normal[SparseArray[(Module[{index = #1}, index -> newLapseFunction*normalVector[[index]] +
Join[{0}, newShiftVector][[index]]] & ) /@ Range[Length[spacetimeMetricTensor]]]] /.
(ToExpression[#1] -> #1 & ) /@ Select[Join[coordinates, {timeCoordinate}], StringQ]] /;
SymbolName[admDecomposition] === "ADMDecomposition" && SymbolName[metricTensor] === "MetricTensor" &&
Length[Dimensions[matrixRepresentation]] == 2 && Length[coordinates] == Length[matrixRepresentation] &&
BooleanQ[metricIndex1] && BooleanQ[metricIndex2] && Length[shiftVector] == Length[matrixRepresentation] &&
BooleanQ[index1] && BooleanQ[index2]
ExtrinsicCurvatureTensor[(admDecomposition_)[(metricTensor_)[matrixRepresentation_List, coordinates_List, metricIndex1_,
metricIndex2_], timeCoordinate_, lapseFunction_, shiftVector_List], index1_, index2_]["SymbolicTimeVector"] :=
Module[{newMatrixRepresentation, newCoordinates, newTimeCoordinate, newLapseFunction, newShiftVector, shiftCovector,
spacetimeMetricTensor, normalVector}, newMatrixRepresentation = matrixRepresentation /.
(#1 -> ToExpression[#1] & ) /@ Select[Join[coordinates, {timeCoordinate}], StringQ];
newCoordinates = coordinates /. (#1 -> ToExpression[#1] & ) /@ Select[Join[coordinates, {timeCoordinate}], StringQ];
newTimeCoordinate = timeCoordinate /. (#1 -> ToExpression[#1] & ) /@ Select[Join[coordinates, {timeCoordinate}],
StringQ]; newLapseFunction = lapseFunction /. (#1 -> ToExpression[#1] & ) /@
Select[Join[coordinates, {timeCoordinate}], StringQ]; newShiftVector =
shiftVector /. (#1 -> ToExpression[#1] & ) /@ Select[Join[coordinates, {timeCoordinate}], StringQ];
shiftCovector = Normal[SparseArray[
(Module[{index = #1}, index -> Total[(newMatrixRepresentation[[index,#1]]*newShiftVector[[#1]] & ) /@
Range[Length[newMatrixRepresentation]]]] & ) /@ Range[Length[newMatrixRepresentation]]]];
spacetimeMetricTensor = Normal[SparseArray[
Join[{{1, 1} -> Total[(newShiftVector[[#1]]*shiftCovector[[#1]] & ) /@ Range[Length[newMatrixRepresentation]]] -
newLapseFunction^2}, (Module[{index = #1}, {1, index + 1} -> Total[(newMatrixRepresentation[[index,#1]]*
newShiftVector[[#1]] & ) /@ Range[Length[newMatrixRepresentation]]]] & ) /@
Range[Length[newMatrixRepresentation]], (Module[{index = #1}, {index + 1, 1} ->
Total[(newMatrixRepresentation[[#1,index]]*newShiftVector[[#1]] & ) /@ Range[
Length[newMatrixRepresentation]]]] & ) /@ Range[Length[newMatrixRepresentation]],
({First[#1] + 1, Last[#1] + 1} -> newMatrixRepresentation[[First[#1],Last[#1]]] & ) /@
Tuples[Range[Length[newMatrixRepresentation]], 2]]]];
normalVector = Normal[SparseArray[
(Module[{index = #1}, index -> -Total[(newLapseFunction*Inverse[spacetimeMetricTensor][[index,#1]]*
Inactive[D][newTimeCoordinate, Join[{newTimeCoordinate}, newCoordinates][[#1]]] & ) /@ Range[
Length[spacetimeMetricTensor]]]] & ) /@ Range[Length[spacetimeMetricTensor]]]];
Normal[SparseArray[(Module[{index = #1}, index -> newLapseFunction*normalVector[[index]] +
Join[{0}, newShiftVector][[index]]] & ) /@ Range[Length[spacetimeMetricTensor]]]] /.
(ToExpression[#1] -> #1 & ) /@ Select[Join[coordinates, {timeCoordinate}], StringQ]] /;
SymbolName[admDecomposition] === "ADMDecomposition" && SymbolName[metricTensor] === "MetricTensor" &&
Length[Dimensions[matrixRepresentation]] == 2 && Length[coordinates] == Length[matrixRepresentation] &&
BooleanQ[metricIndex1] && BooleanQ[metricIndex2] && Length[shiftVector] == Length[matrixRepresentation] &&
BooleanQ[index1] && BooleanQ[index2]
ExtrinsicCurvatureTensor[(admDecomposition_)[(metricTensor_)[matrixRepresentation_List, coordinates_List, metricIndex1_,
metricIndex2_], timeCoordinate_, lapseFunction_, shiftVector_List], index1_, index2_]["TimeCoordinate"] :=
timeCoordinate /; SymbolName[admDecomposition] === "ADMDecomposition" && SymbolName[metricTensor] === "MetricTensor" &&
Length[Dimensions[matrixRepresentation]] == 2 && Length[coordinates] == Length[matrixRepresentation] &&
BooleanQ[metricIndex1] && BooleanQ[metricIndex2] && Length[shiftVector] == Length[matrixRepresentation] &&
BooleanQ[index1] && BooleanQ[index2]
ExtrinsicCurvatureTensor[(admDecomposition_)[(metricTensor_)[matrixRepresentation_List, coordinates_List, metricIndex1_,
metricIndex2_], timeCoordinate_, lapseFunction_, shiftVector_List], index1_, index2_]["SpatialCoordinates"] :=
coordinates /; SymbolName[admDecomposition] === "ADMDecomposition" && SymbolName[metricTensor] === "MetricTensor" &&
Length[Dimensions[matrixRepresentation]] == 2 && Length[coordinates] == Length[matrixRepresentation] &&
BooleanQ[metricIndex1] && BooleanQ[metricIndex2] && Length[shiftVector] == Length[matrixRepresentation] &&
BooleanQ[index1] && BooleanQ[index2]
ExtrinsicCurvatureTensor[(admDecomposition_)[(metricTensor_)[matrixRepresentation_List, coordinates_List, metricIndex1_,
metricIndex2_], timeCoordinate_, lapseFunction_, shiftVector_List], index1_, index2_]["CoordinateOneForms"] :=
(If[Head[#1] === Subscript, Subscript[StringJoin["\[FormalD]", ToString[First[#1]]], ToString[Last[#1]]],
If[Head[#1] === Superscript, Superscript[StringJoin["\[FormalD]", ToString[First[#1]]], ToString[Last[#1]]],
StringJoin["\[FormalD]", ToString[#1]]]] & ) /@ Join[{timeCoordinate}, coordinates] /;
SymbolName[admDecomposition] === "ADMDecomposition" && SymbolName[metricTensor] === "MetricTensor" &&
Length[Dimensions[matrixRepresentation]] == 2 && Length[coordinates] == Length[matrixRepresentation] &&
BooleanQ[metricIndex1] && BooleanQ[metricIndex2] && Length[shiftVector] == Length[matrixRepresentation] &&
BooleanQ[index1] && BooleanQ[index2]
ExtrinsicCurvatureTensor[(admDecomposition_)[(metricTensor_)[matrixRepresentation_List, coordinates_List, metricIndex1_,
metricIndex2_], timeCoordinate_, lapseFunction_, shiftVector_List], index1_, index2_]["LapseFunction"] :=
lapseFunction /; SymbolName[admDecomposition] === "ADMDecomposition" && SymbolName[metricTensor] === "MetricTensor" &&
Length[Dimensions[matrixRepresentation]] == 2 && Length[coordinates] == Length[matrixRepresentation] &&
BooleanQ[metricIndex1] && BooleanQ[metricIndex2] && Length[shiftVector] == Length[matrixRepresentation] &&
BooleanQ[index1] && BooleanQ[index2]
ExtrinsicCurvatureTensor[(admDecomposition_)[(metricTensor_)[matrixRepresentation_List, coordinates_List, metricIndex1_,
metricIndex2_], timeCoordinate_, lapseFunction_, shiftVector_List], index1_, index2_]["ShiftVector"] :=
shiftVector /; SymbolName[admDecomposition] === "ADMDecomposition" && SymbolName[metricTensor] === "MetricTensor" &&
Length[Dimensions[matrixRepresentation]] == 2 && Length[coordinates] == Length[matrixRepresentation] &&
BooleanQ[metricIndex1] && BooleanQ[metricIndex2] && Length[shiftVector] == Length[matrixRepresentation] &&
BooleanQ[index1] && BooleanQ[index2]
ExtrinsicCurvatureTensor[(admDecomposition_)[(metricTensor_)[matrixRepresentation_List, coordinates_List, metricIndex1_,
metricIndex2_], timeCoordinate_, lapseFunction_, shiftVector_List], index1_, index2_]["Indices"] :=
{index1, index2} /; SymbolName[admDecomposition] === "ADMDecomposition" &&
SymbolName[metricTensor] === "MetricTensor" && Length[Dimensions[matrixRepresentation]] == 2 &&
Length[coordinates] == Length[matrixRepresentation] && BooleanQ[metricIndex1] && BooleanQ[metricIndex2] &&
Length[shiftVector] == Length[matrixRepresentation] && BooleanQ[index1] && BooleanQ[index2]
ExtrinsicCurvatureTensor[(admDecomposition_)[(metricTensor_)[matrixRepresentation_List, coordinates_List, metricIndex1_,
metricIndex2_], timeCoordinate_, lapseFunction_, shiftVector_List], index1_, index2_]["CovariantQ"] :=
If[index1 === True && index2 === True, True, False] /; SymbolName[admDecomposition] === "ADMDecomposition" &&
SymbolName[metricTensor] === "MetricTensor" && Length[Dimensions[matrixRepresentation]] == 2 &&
Length[coordinates] == Length[matrixRepresentation] && BooleanQ[metricIndex1] && BooleanQ[metricIndex2] &&
Length[shiftVector] == Length[matrixRepresentation] && BooleanQ[index1] && BooleanQ[index2]
ExtrinsicCurvatureTensor[(admDecomposition_)[(metricTensor_)[matrixRepresentation_List, coordinates_List, metricIndex1_,
metricIndex2_], timeCoordinate_, lapseFunction_, shiftVector_List], index1_, index2_]["ContravariantQ"] :=
If[index1 === False && index2 === False, True, False] /; SymbolName[admDecomposition] === "ADMDecomposition" &&
SymbolName[metricTensor] === "MetricTensor" && Length[Dimensions[matrixRepresentation]] == 2 &&
Length[coordinates] == Length[matrixRepresentation] && BooleanQ[metricIndex1] && BooleanQ[metricIndex2] &&
Length[shiftVector] == Length[matrixRepresentation] && BooleanQ[index1] && BooleanQ[index2]
ExtrinsicCurvatureTensor[(admDecomposition_)[(metricTensor_)[matrixRepresentation_List, coordinates_List, metricIndex1_,
metricIndex2_], timeCoordinate_, lapseFunction_, shiftVector_List], index1_, index2_]["MixedQ"] :=
If[(index1 === True && index2 === False) || (index1 === False && index2 === True), True, False] /;
SymbolName[admDecomposition] === "ADMDecomposition" && SymbolName[metricTensor] === "MetricTensor" &&
Length[Dimensions[matrixRepresentation]] == 2 && Length[coordinates] == Length[matrixRepresentation] &&
BooleanQ[metricIndex1] && BooleanQ[metricIndex2] && Length[shiftVector] == Length[matrixRepresentation] &&
BooleanQ[index1] && BooleanQ[index2]
ExtrinsicCurvatureTensor[(admDecomposition_)[(metricTensor_)[matrixRepresentation_List, coordinates_List, metricIndex1_,
metricIndex2_], timeCoordinate_, lapseFunction_, shiftVector_List], index1_, index2_]["Symbol"] :=
If[index1 === True && index2 === True, Subscript["\[FormalCapitalK]", "\[FormalMu]\[FormalNu]"], If[index1 === False && index2 === False,
Superscript["\[FormalCapitalK]", "\[FormalMu]\[FormalNu]"], If[index1 === True && index2 === False, Subsuperscript["\[FormalCapitalK]", "\[FormalMu]", "\[FormalNu]"],
If[index1 === False && index2 === True, Subsuperscript["\[FormalCapitalK]", "\[FormalNu]", "\[FormalMu]"], Indeterminate]]]] /;
SymbolName[admDecomposition] === "ADMDecomposition" && SymbolName[metricTensor] === "MetricTensor" &&
Length[Dimensions[matrixRepresentation]] == 2 && Length[coordinates] == Length[matrixRepresentation] &&
BooleanQ[metricIndex1] && BooleanQ[metricIndex2] && Length[shiftVector] == Length[matrixRepresentation] &&
BooleanQ[index1] && BooleanQ[index2]
ExtrinsicCurvatureTensor[(admDecomposition_)[(metricTensor_)[matrixRepresentation_List, coordinates_List, metricIndex1_,
metricIndex2_], timeCoordinate_, lapseFunction_, shiftVector_List], index1_, index2_]["ExtrinsicallyFlatQ"] :=
Module[{newMatrixRepresentation, newCoordinates, newTimeCoordinate, newLapseFunction, newShiftVector, shiftCovector,
spatialChristoffelSymbols, extrinsicCurvatureTensor, fieldEquations},
newMatrixRepresentation = matrixRepresentation /. (#1 -> ToExpression[#1] & ) /@
Select[Join[coordinates, {timeCoordinate}], StringQ]; newCoordinates =
coordinates /. (#1 -> ToExpression[#1] & ) /@ Select[Join[coordinates, {timeCoordinate}], StringQ];
newTimeCoordinate = timeCoordinate /. (#1 -> ToExpression[#1] & ) /@ Select[Join[coordinates, {timeCoordinate}],
StringQ]; newLapseFunction = lapseFunction /. (#1 -> ToExpression[#1] & ) /@
Select[Join[coordinates, {timeCoordinate}], StringQ]; newShiftVector =
shiftVector /. (#1 -> ToExpression[#1] & ) /@ Select[Join[coordinates, {timeCoordinate}], StringQ];
shiftCovector = Normal[SparseArray[
(Module[{index = #1}, index -> Total[(newMatrixRepresentation[[index,#1]]*newShiftVector[[#1]] & ) /@
Range[Length[newMatrixRepresentation]]]] & ) /@ Range[Length[newMatrixRepresentation]]]];
spatialChristoffelSymbols = Normal[SparseArray[
(Module[{index = #1}, index -> Total[((1/2)*Inverse[newMatrixRepresentation][[index[[1]],#1]]*
(D[newMatrixRepresentation[[#1,index[[3]]]], newCoordinates[[index[[2]]]]] + D[newMatrixRepresentation[[
index[[2]],#1]], newCoordinates[[index[[3]]]]] - D[newMatrixRepresentation[[index[[2]],index[[3]]]],
newCoordinates[[#1]]]) & ) /@ Range[Length[newMatrixRepresentation]]]] & ) /@
Tuples[Range[Length[newMatrixRepresentation]], 3]]]; extrinsicCurvatureTensor =
Normal[SparseArray[(Module[{index = #1}, index -> (1/(2*newLapseFunction))*(D[shiftCovector[[First[index]]],
newCoordinates[[Last[index]]]] - Total[(spatialChristoffelSymbols[[#1,Last[index],First[index]]]*
shiftCovector[[#1]] & ) /@ Range[Length[newMatrixRepresentation]]] + D[shiftCovector[[Last[index]]],
newCoordinates[[First[index]]]] - Total[(spatialChristoffelSymbols[[#1,First[index],Last[index]]]*
shiftCovector[[#1]] & ) /@ Range[Length[newMatrixRepresentation]]] - D[newMatrixRepresentation[[
First[index],Last[index]]], newTimeCoordinate])] & ) /@ Tuples[Range[Length[newMatrixRepresentation]],
2]]] /. (ToExpression[#1] -> #1 & ) /@ Select[Join[coordinates, {timeCoordinate}], StringQ];
fieldEquations = FullSimplify[Thread[Catenate[extrinsicCurvatureTensor] ==
Catenate[ConstantArray[0, {Length[matrixRepresentation], Length[matrixRepresentation]}]]]];
If[fieldEquations === True, True, If[fieldEquations === False, False,
If[Length[Select[fieldEquations, #1 === True & ]] == Length[matrixRepresentation]*Length[matrixRepresentation],
True, False]]]] /; SymbolName[admDecomposition] === "ADMDecomposition" &&
SymbolName[metricTensor] === "MetricTensor" && Length[Dimensions[matrixRepresentation]] == 2 &&
Length[coordinates] == Length[matrixRepresentation] && BooleanQ[metricIndex1] && BooleanQ[metricIndex2] &&
Length[shiftVector] == Length[matrixRepresentation] && BooleanQ[index1] && BooleanQ[index2]
ExtrinsicCurvatureTensor[(admDecomposition_)[(metricTensor_)[matrixRepresentation_List, coordinates_List, metricIndex1_,
metricIndex2_], timeCoordinate_, lapseFunction_, shiftVector_List], index1_, index2_][
"VanishingExtrinsicTraceQ"] := Module[{newMatrixRepresentation, newCoordinates, newTimeCoordinate, newLapseFunction,
newShiftVector, shiftCovector, spatialChristoffelSymbols, extrinsicCurvatureTensor},
newMatrixRepresentation = matrixRepresentation /. (#1 -> ToExpression[#1] & ) /@
Select[Join[coordinates, {timeCoordinate}], StringQ]; newCoordinates =
coordinates /. (#1 -> ToExpression[#1] & ) /@ Select[Join[coordinates, {timeCoordinate}], StringQ];
newTimeCoordinate = timeCoordinate /. (#1 -> ToExpression[#1] & ) /@ Select[Join[coordinates, {timeCoordinate}],
StringQ]; newLapseFunction = lapseFunction /. (#1 -> ToExpression[#1] & ) /@
Select[Join[coordinates, {timeCoordinate}], StringQ]; newShiftVector =
shiftVector /. (#1 -> ToExpression[#1] & ) /@ Select[Join[coordinates, {timeCoordinate}], StringQ];
shiftCovector = Normal[SparseArray[
(Module[{index = #1}, index -> Total[(newMatrixRepresentation[[index,#1]]*newShiftVector[[#1]] & ) /@
Range[Length[newMatrixRepresentation]]]] & ) /@ Range[Length[newMatrixRepresentation]]]];
spatialChristoffelSymbols = Normal[SparseArray[
(Module[{index = #1}, index -> Total[((1/2)*Inverse[newMatrixRepresentation][[index[[1]],#1]]*
(D[newMatrixRepresentation[[#1,index[[3]]]], newCoordinates[[index[[2]]]]] + D[newMatrixRepresentation[[
index[[2]],#1]], newCoordinates[[index[[3]]]]] - D[newMatrixRepresentation[[index[[2]],index[[3]]]],
newCoordinates[[#1]]]) & ) /@ Range[Length[newMatrixRepresentation]]]] & ) /@
Tuples[Range[Length[newMatrixRepresentation]], 3]]]; extrinsicCurvatureTensor =
Normal[SparseArray[(Module[{index = #1}, index -> (1/(2*newLapseFunction))*(D[shiftCovector[[First[index]]],
newCoordinates[[Last[index]]]] - Total[(spatialChristoffelSymbols[[#1,Last[index],First[index]]]*
shiftCovector[[#1]] & ) /@ Range[Length[newMatrixRepresentation]]] + D[shiftCovector[[Last[index]]],
newCoordinates[[First[index]]]] - Total[(spatialChristoffelSymbols[[#1,First[index],Last[index]]]*
shiftCovector[[#1]] & ) /@ Range[Length[newMatrixRepresentation]]] - D[newMatrixRepresentation[[
First[index],Last[index]]], newTimeCoordinate])] & ) /@ Tuples[Range[Length[newMatrixRepresentation]],
2]]] /. (ToExpression[#1] -> #1 & ) /@ Select[Join[coordinates, {timeCoordinate}], StringQ];
FullSimplify[Total[(Inverse[matrixRepresentation][[First[#1],Last[#1]]]*extrinsicCurvatureTensor[[First[#1],
Last[#1]]] & ) /@ Tuples[Range[Length[matrixRepresentation]], 2]] == 0] === True] /;
SymbolName[admDecomposition] === "ADMDecomposition" && SymbolName[metricTensor] === "MetricTensor" &&
Length[Dimensions[matrixRepresentation]] == 2 && Length[coordinates] == Length[matrixRepresentation] &&
BooleanQ[metricIndex1] && BooleanQ[metricIndex2] && Length[shiftVector] == Length[matrixRepresentation] &&
BooleanQ[index1] && BooleanQ[index2]
ExtrinsicCurvatureTensor[(admDecomposition_)[(metricTensor_)[matrixRepresentation_List, coordinates_List, metricIndex1_,
metricIndex2_], timeCoordinate_, lapseFunction_, shiftVector_List], index1_, index2_][
"ExtrinsicallyFlatConditions"] := Module[{newMatrixRepresentation, newCoordinates, newTimeCoordinate,
newLapseFunction, newShiftVector, shiftCovector, spatialChristoffelSymbols, extrinsicCurvatureTensor,
fieldEquations}, newMatrixRepresentation = matrixRepresentation /. (#1 -> ToExpression[#1] & ) /@
Select[Join[coordinates, {timeCoordinate}], StringQ]; newCoordinates =
coordinates /. (#1 -> ToExpression[#1] & ) /@ Select[Join[coordinates, {timeCoordinate}], StringQ];
newTimeCoordinate = timeCoordinate /. (#1 -> ToExpression[#1] & ) /@ Select[Join[coordinates, {timeCoordinate}],
StringQ]; newLapseFunction = lapseFunction /. (#1 -> ToExpression[#1] & ) /@
Select[Join[coordinates, {timeCoordinate}], StringQ]; newShiftVector =
shiftVector /. (#1 -> ToExpression[#1] & ) /@ Select[Join[coordinates, {timeCoordinate}], StringQ];
shiftCovector = Normal[SparseArray[
(Module[{index = #1}, index -> Total[(newMatrixRepresentation[[index,#1]]*newShiftVector[[#1]] & ) /@
Range[Length[newMatrixRepresentation]]]] & ) /@ Range[Length[newMatrixRepresentation]]]];
spatialChristoffelSymbols = Normal[SparseArray[
(Module[{index = #1}, index -> Total[((1/2)*Inverse[newMatrixRepresentation][[index[[1]],#1]]*
(D[newMatrixRepresentation[[#1,index[[3]]]], newCoordinates[[index[[2]]]]] + D[newMatrixRepresentation[[
index[[2]],#1]], newCoordinates[[index[[3]]]]] - D[newMatrixRepresentation[[index[[2]],index[[3]]]],
newCoordinates[[#1]]]) & ) /@ Range[Length[newMatrixRepresentation]]]] & ) /@
Tuples[Range[Length[newMatrixRepresentation]], 3]]]; extrinsicCurvatureTensor =
Normal[SparseArray[(Module[{index = #1}, index -> (1/(2*newLapseFunction))*(D[shiftCovector[[First[index]]],
newCoordinates[[Last[index]]]] - Total[(spatialChristoffelSymbols[[#1,Last[index],First[index]]]*
shiftCovector[[#1]] & ) /@ Range[Length[newMatrixRepresentation]]] + D[shiftCovector[[Last[index]]],
newCoordinates[[First[index]]]] - Total[(spatialChristoffelSymbols[[#1,First[index],Last[index]]]*
shiftCovector[[#1]] & ) /@ Range[Length[newMatrixRepresentation]]] - D[newMatrixRepresentation[[
First[index],Last[index]]], newTimeCoordinate])] & ) /@ Tuples[Range[Length[newMatrixRepresentation]],
2]]] /. (ToExpression[#1] -> #1 & ) /@ Select[Join[coordinates, {timeCoordinate}], StringQ];
fieldEquations = FullSimplify[Thread[Catenate[extrinsicCurvatureTensor] ==
Catenate[ConstantArray[0, {Length[matrixRepresentation], Length[matrixRepresentation]}]]]];
If[fieldEquations === True, {}, If[fieldEquations === False, Indeterminate,
If[Length[Select[fieldEquations, #1 === False & ]] > 0, Indeterminate,
DeleteDuplicates[Reverse /@ Sort /@ Select[fieldEquations, #1 =!= True & ]]]]]] /;
SymbolName[admDecomposition] === "ADMDecomposition" && SymbolName[metricTensor] === "MetricTensor" &&
Length[Dimensions[matrixRepresentation]] == 2 && Length[coordinates] == Length[matrixRepresentation] &&
BooleanQ[metricIndex1] && BooleanQ[metricIndex2] && Length[shiftVector] == Length[matrixRepresentation] &&
BooleanQ[index1] && BooleanQ[index2]
ExtrinsicCurvatureTensor[(admDecomposition_)[(metricTensor_)[matrixRepresentation_List, coordinates_List, metricIndex1_,
metricIndex2_], timeCoordinate_, lapseFunction_, shiftVector_List], index1_, index2_][
"VanishingExtrinsicTraceCondition"] := Module[{newMatrixRepresentation, newCoordinates, newTimeCoordinate,
newLapseFunction, newShiftVector, shiftCovector, spatialChristoffelSymbols, extrinsicCurvatureTensor,
fieldEquation}, newMatrixRepresentation = matrixRepresentation /. (#1 -> ToExpression[#1] & ) /@
Select[Join[coordinates, {timeCoordinate}], StringQ]; newCoordinates =
coordinates /. (#1 -> ToExpression[#1] & ) /@ Select[Join[coordinates, {timeCoordinate}], StringQ];
newTimeCoordinate = timeCoordinate /. (#1 -> ToExpression[#1] & ) /@ Select[Join[coordinates, {timeCoordinate}],
StringQ]; newLapseFunction = lapseFunction /. (#1 -> ToExpression[#1] & ) /@
Select[Join[coordinates, {timeCoordinate}], StringQ]; newShiftVector =
shiftVector /. (#1 -> ToExpression[#1] & ) /@ Select[Join[coordinates, {timeCoordinate}], StringQ];
shiftCovector = Normal[SparseArray[
(Module[{index = #1}, index -> Total[(newMatrixRepresentation[[index,#1]]*newShiftVector[[#1]] & ) /@
Range[Length[newMatrixRepresentation]]]] & ) /@ Range[Length[newMatrixRepresentation]]]];
spatialChristoffelSymbols = Normal[SparseArray[
(Module[{index = #1}, index -> Total[((1/2)*Inverse[newMatrixRepresentation][[index[[1]],#1]]*
(D[newMatrixRepresentation[[#1,index[[3]]]], newCoordinates[[index[[2]]]]] + D[newMatrixRepresentation[[
index[[2]],#1]], newCoordinates[[index[[3]]]]] - D[newMatrixRepresentation[[index[[2]],index[[3]]]],
newCoordinates[[#1]]]) & ) /@ Range[Length[newMatrixRepresentation]]]] & ) /@
Tuples[Range[Length[newMatrixRepresentation]], 3]]]; extrinsicCurvatureTensor =
Normal[SparseArray[(Module[{index = #1}, index -> (1/(2*newLapseFunction))*(D[shiftCovector[[First[index]]],
newCoordinates[[Last[index]]]] - Total[(spatialChristoffelSymbols[[#1,Last[index],First[index]]]*
shiftCovector[[#1]] & ) /@ Range[Length[newMatrixRepresentation]]] + D[shiftCovector[[Last[index]]],
newCoordinates[[First[index]]]] - Total[(spatialChristoffelSymbols[[#1,First[index],Last[index]]]*
shiftCovector[[#1]] & ) /@ Range[Length[newMatrixRepresentation]]] - D[newMatrixRepresentation[[
First[index],Last[index]]], newTimeCoordinate])] & ) /@ Tuples[Range[Length[newMatrixRepresentation]],
2]]] /. (ToExpression[#1] -> #1 & ) /@ Select[Join[coordinates, {timeCoordinate}], StringQ];
fieldEquation = FullSimplify[Total[(Inverse[matrixRepresentation][[First[#1],Last[#1]]]*
extrinsicCurvatureTensor[[First[#1],Last[#1]]] & ) /@ Tuples[Range[Length[matrixRepresentation]], 2]] == 0];
If[fieldEquation === False, Indeterminate, fieldEquation]] /; SymbolName[admDecomposition] === "ADMDecomposition" &&
SymbolName[metricTensor] === "MetricTensor" && Length[Dimensions[matrixRepresentation]] == 2 &&
Length[coordinates] == Length[matrixRepresentation] && BooleanQ[metricIndex1] && BooleanQ[metricIndex2] &&
Length[shiftVector] == Length[matrixRepresentation] && BooleanQ[index1] && BooleanQ[index2]
ExtrinsicCurvatureTensor[(admDecomposition_)[(metricTensor_)[matrixRepresentation_List, coordinates_List, metricIndex1_,
metricIndex2_], timeCoordinate_, lapseFunction_, shiftVector_List], index1_, index2_]["CovariantDerivatives"] :=
Module[{newMatrixRepresentation, newCoordinates, newTimeCoordinate, newLapseFunction, newShiftVector, shiftCovector,
spatialChristoffelSymbols, extrinsicCurvatureTensor, newExtrinsicCurvatureTensor},
newMatrixRepresentation = matrixRepresentation /. (#1 -> ToExpression[#1] & ) /@
Select[Join[coordinates, {timeCoordinate}], StringQ]; newCoordinates =
coordinates /. (#1 -> ToExpression[#1] & ) /@ Select[Join[coordinates, {timeCoordinate}], StringQ];
newTimeCoordinate = timeCoordinate /. (#1 -> ToExpression[#1] & ) /@ Select[Join[coordinates, {timeCoordinate}],
StringQ]; newLapseFunction = lapseFunction /. (#1 -> ToExpression[#1] & ) /@
Select[Join[coordinates, {timeCoordinate}], StringQ]; newShiftVector =
shiftVector /. (#1 -> ToExpression[#1] & ) /@ Select[Join[coordinates, {timeCoordinate}], StringQ];
shiftCovector = Normal[SparseArray[
(Module[{index = #1}, index -> Total[(newMatrixRepresentation[[index,#1]]*newShiftVector[[#1]] & ) /@
Range[Length[newMatrixRepresentation]]]] & ) /@ Range[Length[newMatrixRepresentation]]]];
spatialChristoffelSymbols = Normal[SparseArray[
(Module[{index = #1}, index -> Total[((1/2)*Inverse[newMatrixRepresentation][[index[[1]],#1]]*
(D[newMatrixRepresentation[[#1,index[[3]]]], newCoordinates[[index[[2]]]]] + D[newMatrixRepresentation[[
index[[2]],#1]], newCoordinates[[index[[3]]]]] - D[newMatrixRepresentation[[index[[2]],index[[3]]]],
newCoordinates[[#1]]]) & ) /@ Range[Length[newMatrixRepresentation]]]] & ) /@
Tuples[Range[Length[newMatrixRepresentation]], 3]]]; extrinsicCurvatureTensor =
Normal[SparseArray[(Module[{index = #1}, index -> (1/(2*newLapseFunction))*(D[shiftCovector[[First[index]]],
newCoordinates[[Last[index]]]] - Total[(spatialChristoffelSymbols[[#1,Last[index],First[index]]]*
shiftCovector[[#1]] & ) /@ Range[Length[newMatrixRepresentation]]] + D[shiftCovector[[Last[index]]],
newCoordinates[[First[index]]]] - Total[(spatialChristoffelSymbols[[#1,First[index],Last[index]]]*
shiftCovector[[#1]] & ) /@ Range[Length[newMatrixRepresentation]]] - D[newMatrixRepresentation[[
First[index],Last[index]]], newTimeCoordinate])] & ) /@ Tuples[Range[Length[newMatrixRepresentation]],
2]]]; If[index1 === True && index2 === True, newExtrinsicCurvatureTensor = extrinsicCurvatureTensor;
Association[(Module[{index = #1}, StringJoin[ToString[Subscript["\[Del]", ToString[newCoordinates[[index[[1]]]],
StandardForm]], StandardForm], ToString[Subscript["\[FormalCapitalK]", StringJoin[ToString[newCoordinates[[
index[[2]]]], StandardForm], ToString[newCoordinates[[index[[3]]]], StandardForm]]], StandardForm]] ->
D[newExtrinsicCurvatureTensor[[index[[2]],index[[3]]]], newCoordinates[[index[[1]]]]] -
Total[(spatialChristoffelSymbols[[#1,index[[1]],index[[2]]]]*newExtrinsicCurvatureTensor[[#1,
index[[3]]]] & ) /@ Range[Length[newMatrixRepresentation]]] -
Total[(spatialChristoffelSymbols[[#1,index[[1]],index[[3]]]]*newExtrinsicCurvatureTensor[[index[[2]],
#1]] & ) /@ Range[Length[newMatrixRepresentation]]]] & ) /@
Tuples[Range[Length[newMatrixRepresentation]], 3] /. (ToExpression[#1] -> #1 & ) /@
Select[Join[coordinates, {timeCoordinate}], StringQ]], If[index1 === False && index2 === False,
newExtrinsicCurvatureTensor = Normal[SparseArray[(Module[{index = #1}, index -> Total[
(Inverse[newMatrixRepresentation][[First[index],First[#1]]]*Inverse[newMatrixRepresentation][[Last[#1],
Last[index]]]*extrinsicCurvatureTensor[[First[#1],Last[#1]]] & ) /@ Tuples[
Range[Length[newMatrixRepresentation]], 2]]] & ) /@ Tuples[Range[Length[newMatrixRepresentation]],
2]]]; Association[(Module[{index = #1}, StringJoin[ToString[Subscript["\[Del]", ToString[newCoordinates[[
index[[1]]]], StandardForm]], StandardForm], ToString[Superscript["\[FormalCapitalK]", StringJoin[
ToString[newCoordinates[[index[[2]]]], StandardForm], ToString[newCoordinates[[index[[3]]]],
StandardForm]]], StandardForm]] -> D[newExtrinsicCurvatureTensor[[index[[2]],index[[3]]]],
newCoordinates[[index[[1]]]]] + Total[(spatialChristoffelSymbols[[index[[2]],index[[1]],#1]]*
newExtrinsicCurvatureTensor[[#1,index[[3]]]] & ) /@ Range[Length[newMatrixRepresentation]]] + Total[
(spatialChristoffelSymbols[[index[[3]],index[[1]],#1]]*newExtrinsicCurvatureTensor[[index[[2]],
#1]] & ) /@ Range[Length[newMatrixRepresentation]]]] & ) /@
Tuples[Range[Length[newMatrixRepresentation]], 3] /. (ToExpression[#1] -> #1 & ) /@
Select[Join[coordinates, {timeCoordinate}], StringQ]], If[index1 === True && index2 === False,
newExtrinsicCurvatureTensor = Normal[SparseArray[(Module[{index = #1}, index ->
Total[(Inverse[newMatrixRepresentation][[#1,Last[index]]]*extrinsicCurvatureTensor[[First[index],
#1]] & ) /@ Range[Length[newMatrixRepresentation]]]] & ) /@
Tuples[Range[Length[newMatrixRepresentation]], 2]]]; Association[
(Module[{index = #1}, StringJoin[ToString[Subscript["\[Del]", ToString[newCoordinates[[index[[1]]]],
StandardForm]], StandardForm], ToString[Subsuperscript["\[FormalCapitalK]", ToString[newCoordinates[[index[[2]]]],
StandardForm], ToString[newCoordinates[[index[[3]]]], StandardForm]], StandardForm]] ->
D[newExtrinsicCurvatureTensor[[index[[2]],index[[3]]]], newCoordinates[[index[[1]]]]] +
Total[(spatialChristoffelSymbols[[index[[3]],index[[1]],#1]]*newExtrinsicCurvatureTensor[[index[[2]],
#1]] & ) /@ Range[Length[newMatrixRepresentation]]] - Total[(spatialChristoffelSymbols[[#1,
index[[1]],index[[2]]]]*newExtrinsicCurvatureTensor[[#1,index[[3]]]] & ) /@
Range[Length[newMatrixRepresentation]]]] & ) /@ Tuples[Range[Length[newMatrixRepresentation]], 3] /.
(ToExpression[#1] -> #1 & ) /@ Select[Join[coordinates, {timeCoordinate}], StringQ]],
If[index1 === False && index2 === True, newExtrinsicCurvatureTensor =
Normal[SparseArray[(Module[{index = #1}, index -> Total[(Inverse[newMatrixRepresentation][[First[index],#1]]*
extrinsicCurvatureTensor[[#1,Last[index]]] & ) /@ Range[Length[newMatrixRepresentation]]]] & ) /@
Tuples[Range[Length[newMatrixRepresentation]], 2]]]; Association[
(Module[{index = #1}, StringJoin[ToString[Subscript["\[Del]", ToString[newCoordinates[[index[[1]]]],
StandardForm]], StandardForm], ToString[Subsuperscript["\[FormalCapitalK]", ToString[newCoordinates[[index[[3]]]],
StandardForm], ToString[newCoordinates[[index[[2]]]], StandardForm]], StandardForm]] ->
D[newExtrinsicCurvatureTensor[[index[[2]],index[[3]]]], newCoordinates[[index[[1]]]]] +
Total[(spatialChristoffelSymbols[[index[[2]],index[[1]],#1]]*newExtrinsicCurvatureTensor[[#1,
index[[3]]]] & ) /@ Range[Length[newMatrixRepresentation]]] - Total[
(spatialChristoffelSymbols[[#1,index[[1]],index[[3]]]]*newExtrinsicCurvatureTensor[[index[[2]],
#1]] & ) /@ Range[Length[newMatrixRepresentation]]]] & ) /@
Tuples[Range[Length[newMatrixRepresentation]], 3] /. (ToExpression[#1] -> #1 & ) /@
Select[Join[coordinates, {timeCoordinate}], StringQ]]]]]]] /;
SymbolName[admDecomposition] === "ADMDecomposition" && SymbolName[metricTensor] === "MetricTensor" &&
Length[Dimensions[matrixRepresentation]] == 2 && Length[coordinates] == Length[matrixRepresentation] &&
BooleanQ[metricIndex1] && BooleanQ[metricIndex2] && Length[shiftVector] == Length[matrixRepresentation] &&
BooleanQ[index1] && BooleanQ[index2]
ExtrinsicCurvatureTensor[(admDecomposition_)[(metricTensor_)[matrixRepresentation_List, coordinates_List, metricIndex1_,
metricIndex2_], timeCoordinate_, lapseFunction_, shiftVector_List], index1_, index2_][
"ReducedCovariantDerivatives"] := Module[{newMatrixRepresentation, newCoordinates, newTimeCoordinate,
newLapseFunction, newShiftVector, shiftCovector, spatialChristoffelSymbols, extrinsicCurvatureTensor,
newExtrinsicCurvatureTensor}, newMatrixRepresentation = matrixRepresentation /.
(#1 -> ToExpression[#1] & ) /@ Select[Join[coordinates, {timeCoordinate}], StringQ];
newCoordinates = coordinates /. (#1 -> ToExpression[#1] & ) /@ Select[Join[coordinates, {timeCoordinate}], StringQ];
newTimeCoordinate = timeCoordinate /. (#1 -> ToExpression[#1] & ) /@ Select[Join[coordinates, {timeCoordinate}],
StringQ]; newLapseFunction = lapseFunction /. (#1 -> ToExpression[#1] & ) /@
Select[Join[coordinates, {timeCoordinate}], StringQ]; newShiftVector =
shiftVector /. (#1 -> ToExpression[#1] & ) /@ Select[Join[coordinates, {timeCoordinate}], StringQ];
shiftCovector = Normal[SparseArray[
(Module[{index = #1}, index -> Total[(newMatrixRepresentation[[index,#1]]*newShiftVector[[#1]] & ) /@
Range[Length[newMatrixRepresentation]]]] & ) /@ Range[Length[newMatrixRepresentation]]]];
spatialChristoffelSymbols = Normal[SparseArray[
(Module[{index = #1}, index -> Total[((1/2)*Inverse[newMatrixRepresentation][[index[[1]],#1]]*
(D[newMatrixRepresentation[[#1,index[[3]]]], newCoordinates[[index[[2]]]]] + D[newMatrixRepresentation[[
index[[2]],#1]], newCoordinates[[index[[3]]]]] - D[newMatrixRepresentation[[index[[2]],index[[3]]]],
newCoordinates[[#1]]]) & ) /@ Range[Length[newMatrixRepresentation]]]] & ) /@
Tuples[Range[Length[newMatrixRepresentation]], 3]]]; extrinsicCurvatureTensor =
Normal[SparseArray[(Module[{index = #1}, index -> (1/(2*newLapseFunction))*(D[shiftCovector[[First[index]]],
newCoordinates[[Last[index]]]] - Total[(spatialChristoffelSymbols[[#1,Last[index],First[index]]]*
shiftCovector[[#1]] & ) /@ Range[Length[newMatrixRepresentation]]] + D[shiftCovector[[Last[index]]],
newCoordinates[[First[index]]]] - Total[(spatialChristoffelSymbols[[#1,First[index],Last[index]]]*
shiftCovector[[#1]] & ) /@ Range[Length[newMatrixRepresentation]]] - D[newMatrixRepresentation[[
First[index],Last[index]]], newTimeCoordinate])] & ) /@ Tuples[Range[Length[newMatrixRepresentation]],
2]]]; If[index1 === True && index2 === True, newExtrinsicCurvatureTensor = extrinsicCurvatureTensor;
Association[(Module[{index = #1}, StringJoin[ToString[Subscript["\[Del]", ToString[newCoordinates[[index[[1]]]],
StandardForm]], StandardForm], ToString[Subscript["\[FormalCapitalK]", StringJoin[ToString[newCoordinates[[
index[[2]]]], StandardForm], ToString[newCoordinates[[index[[3]]]], StandardForm]]], StandardForm]] ->
FullSimplify[D[newExtrinsicCurvatureTensor[[index[[2]],index[[3]]]], newCoordinates[[index[[1]]]]] - Total[
(spatialChristoffelSymbols[[#1,index[[1]],index[[2]]]]*newExtrinsicCurvatureTensor[[#1,
index[[3]]]] & ) /@ Range[Length[newMatrixRepresentation]]] - Total[
(spatialChristoffelSymbols[[#1,index[[1]],index[[3]]]]*newExtrinsicCurvatureTensor[[index[[2]],
#1]] & ) /@ Range[Length[newMatrixRepresentation]]]]] & ) /@
Tuples[Range[Length[newMatrixRepresentation]], 3] /. (ToExpression[#1] -> #1 & ) /@
Select[Join[coordinates, {timeCoordinate}], StringQ]], If[index1 === False && index2 === False,
newExtrinsicCurvatureTensor = Normal[SparseArray[(Module[{index = #1}, index -> Total[
(Inverse[newMatrixRepresentation][[First[index],First[#1]]]*Inverse[newMatrixRepresentation][[Last[#1],
Last[index]]]*extrinsicCurvatureTensor[[First[#1],Last[#1]]] & ) /@ Tuples[
Range[Length[newMatrixRepresentation]], 2]]] & ) /@ Tuples[Range[Length[newMatrixRepresentation]],
2]]]; Association[(Module[{index = #1}, StringJoin[ToString[Subscript["\[Del]", ToString[newCoordinates[[
index[[1]]]], StandardForm]], StandardForm], ToString[Superscript["\[FormalCapitalK]", StringJoin[
ToString[newCoordinates[[index[[2]]]], StandardForm], ToString[newCoordinates[[index[[3]]]],
StandardForm]]], StandardForm]] -> FullSimplify[D[newExtrinsicCurvatureTensor[[index[[2]],
index[[3]]]], newCoordinates[[index[[1]]]]] + Total[(spatialChristoffelSymbols[[index[[2]],index[[1]],
#1]]*newExtrinsicCurvatureTensor[[#1,index[[3]]]] & ) /@ Range[Length[newMatrixRepresentation]]] +
Total[(spatialChristoffelSymbols[[index[[3]],index[[1]],#1]]*newExtrinsicCurvatureTensor[[index[[2]],
#1]] & ) /@ Range[Length[newMatrixRepresentation]]]]] & ) /@
Tuples[Range[Length[newMatrixRepresentation]], 3] /. (ToExpression[#1] -> #1 & ) /@
Select[Join[coordinates, {timeCoordinate}], StringQ]], If[index1 === True && index2 === False,
newExtrinsicCurvatureTensor = Normal[SparseArray[(Module[{index = #1}, index ->
Total[(Inverse[newMatrixRepresentation][[#1,Last[index]]]*extrinsicCurvatureTensor[[First[index],
#1]] & ) /@ Range[Length[newMatrixRepresentation]]]] & ) /@
Tuples[Range[Length[newMatrixRepresentation]], 2]]]; Association[
(Module[{index = #1}, StringJoin[ToString[Subscript["\[Del]", ToString[newCoordinates[[index[[1]]]],
StandardForm]], StandardForm], ToString[Subsuperscript["\[FormalCapitalK]", ToString[newCoordinates[[index[[2]]]],
StandardForm], ToString[newCoordinates[[index[[3]]]], StandardForm]], StandardForm]] -> FullSimplify[
D[newExtrinsicCurvatureTensor[[index[[2]],index[[3]]]], newCoordinates[[index[[1]]]]] +
Total[(spatialChristoffelSymbols[[index[[3]],index[[1]],#1]]*newExtrinsicCurvatureTensor[[index[[2]],
#1]] & ) /@ Range[Length[newMatrixRepresentation]]] - Total[(spatialChristoffelSymbols[[#1,
index[[1]],index[[2]]]]*newExtrinsicCurvatureTensor[[#1,index[[3]]]] & ) /@
Range[Length[newMatrixRepresentation]]]]] & ) /@ Tuples[Range[Length[newMatrixRepresentation]], 3] /.
(ToExpression[#1] -> #1 & ) /@ Select[Join[coordinates, {timeCoordinate}], StringQ]],
If[index1 === False && index2 === True, newExtrinsicCurvatureTensor =
Normal[SparseArray[(Module[{index = #1}, index -> Total[(Inverse[newMatrixRepresentation][[First[index],#1]]*
extrinsicCurvatureTensor[[#1,Last[index]]] & ) /@ Range[Length[newMatrixRepresentation]]]] & ) /@
Tuples[Range[Length[newMatrixRepresentation]], 2]]]; Association[
(Module[{index = #1}, StringJoin[ToString[Subscript["\[Del]", ToString[newCoordinates[[index[[1]]]],
StandardForm]], StandardForm], ToString[Subsuperscript["\[FormalCapitalK]", ToString[newCoordinates[[index[[3]]]],
StandardForm], ToString[newCoordinates[[index[[2]]]], StandardForm]], StandardForm]] ->
FullSimplify[D[newExtrinsicCurvatureTensor[[index[[2]],index[[3]]]], newCoordinates[[index[[1]]]]] +
Total[(spatialChristoffelSymbols[[index[[2]],index[[1]],#1]]*newExtrinsicCurvatureTensor[[#1,
index[[3]]]] & ) /@ Range[Length[newMatrixRepresentation]]] - Total[
(spatialChristoffelSymbols[[#1,index[[1]],index[[3]]]]*newExtrinsicCurvatureTensor[[index[[2]],
#1]] & ) /@ Range[Length[newMatrixRepresentation]]]]] & ) /@
Tuples[Range[Length[newMatrixRepresentation]], 3] /. (ToExpression[#1] -> #1 & ) /@
Select[Join[coordinates, {timeCoordinate}], StringQ]]]]]]] /;
SymbolName[admDecomposition] === "ADMDecomposition" && SymbolName[metricTensor] === "MetricTensor" &&
Length[Dimensions[matrixRepresentation]] == 2 && Length[coordinates] == Length[matrixRepresentation] &&
BooleanQ[metricIndex1] && BooleanQ[metricIndex2] && Length[shiftVector] == Length[matrixRepresentation] &&
BooleanQ[index1] && BooleanQ[index2]
ExtrinsicCurvatureTensor[(admDecomposition_)[(metricTensor_)[matrixRepresentation_List, coordinates_List, metricIndex1_,
metricIndex2_], timeCoordinate_, lapseFunction_, shiftVector_List], index1_, index2_][
"SymbolicCovariantDerivatives"] := Module[{newMatrixRepresentation, newCoordinates, newTimeCoordinate,
newLapseFunction, newShiftVector, shiftCovector, spatialChristoffelSymbols, extrinsicCurvatureTensor,
newExtrinsicCurvatureTensor}, newMatrixRepresentation = matrixRepresentation /.
(#1 -> ToExpression[#1] & ) /@ Select[Join[coordinates, {timeCoordinate}], StringQ];
newCoordinates = coordinates /. (#1 -> ToExpression[#1] & ) /@ Select[Join[coordinates, {timeCoordinate}], StringQ];
newTimeCoordinate = timeCoordinate /. (#1 -> ToExpression[#1] & ) /@ Select[Join[coordinates, {timeCoordinate}],
StringQ]; newLapseFunction = lapseFunction /. (#1 -> ToExpression[#1] & ) /@
Select[Join[coordinates, {timeCoordinate}], StringQ]; newShiftVector =
shiftVector /. (#1 -> ToExpression[#1] & ) /@ Select[Join[coordinates, {timeCoordinate}], StringQ];
shiftCovector = Normal[SparseArray[
(Module[{index = #1}, index -> Total[(newMatrixRepresentation[[index,#1]]*newShiftVector[[#1]] & ) /@
Range[Length[newMatrixRepresentation]]]] & ) /@ Range[Length[newMatrixRepresentation]]]];
spatialChristoffelSymbols = Normal[SparseArray[
(Module[{index = #1}, index -> Total[((1/2)*Inverse[newMatrixRepresentation][[index[[1]],#1]]*
(Inactive[D][newMatrixRepresentation[[#1,index[[3]]]], newCoordinates[[index[[2]]]]] +
Inactive[D][newMatrixRepresentation[[index[[2]],#1]], newCoordinates[[index[[3]]]]] -
Inactive[D][newMatrixRepresentation[[index[[2]],index[[3]]]], newCoordinates[[#1]]]) & ) /@
Range[Length[newMatrixRepresentation]]]] & ) /@ Tuples[Range[Length[newMatrixRepresentation]], 3]]];
extrinsicCurvatureTensor = Normal[SparseArray[
(Module[{index = #1}, index -> (1/(2*newLapseFunction))*(Inactive[D][shiftCovector[[First[index]]],
newCoordinates[[Last[index]]]] - Total[(spatialChristoffelSymbols[[#1,Last[index],First[index]]]*
shiftCovector[[#1]] & ) /@ Range[Length[newMatrixRepresentation]]] + Inactive[D][shiftCovector[[
Last[index]]], newCoordinates[[First[index]]]] - Total[(spatialChristoffelSymbols[[#1,First[index],
Last[index]]]*shiftCovector[[#1]] & ) /@ Range[Length[newMatrixRepresentation]]] -
Inactive[D][newMatrixRepresentation[[First[index],Last[index]]], newTimeCoordinate])] & ) /@
Tuples[Range[Length[newMatrixRepresentation]], 2]]]; If[index1 === True && index2 === True,
newExtrinsicCurvatureTensor = extrinsicCurvatureTensor;
Association[(Module[{index = #1}, StringJoin[ToString[Subscript["\[Del]", ToString[newCoordinates[[index[[1]]]],
StandardForm]], StandardForm], ToString[Subscript["\[FormalCapitalK]", StringJoin[ToString[newCoordinates[[
index[[2]]]], StandardForm], ToString[newCoordinates[[index[[3]]]], StandardForm]]], StandardForm]] ->
Inactive[D][newExtrinsicCurvatureTensor[[index[[2]],index[[3]]]], newCoordinates[[index[[1]]]]] -
Total[(spatialChristoffelSymbols[[#1,index[[1]],index[[2]]]]*newExtrinsicCurvatureTensor[[#1,
index[[3]]]] & ) /@ Range[Length[newMatrixRepresentation]]] -
Total[(spatialChristoffelSymbols[[#1,index[[1]],index[[3]]]]*newExtrinsicCurvatureTensor[[index[[2]],
#1]] & ) /@ Range[Length[newMatrixRepresentation]]]] & ) /@
Tuples[Range[Length[newMatrixRepresentation]], 3] /. (ToExpression[#1] -> #1 & ) /@
Select[Join[coordinates, {timeCoordinate}], StringQ]], If[index1 === False && index2 === False,
newExtrinsicCurvatureTensor = Normal[SparseArray[(Module[{index = #1}, index -> Total[
(Inverse[newMatrixRepresentation][[First[index],First[#1]]]*Inverse[newMatrixRepresentation][[Last[#1],
Last[index]]]*extrinsicCurvatureTensor[[First[#1],Last[#1]]] & ) /@ Tuples[
Range[Length[newMatrixRepresentation]], 2]]] & ) /@ Tuples[Range[Length[newMatrixRepresentation]],
2]]]; Association[(Module[{index = #1}, StringJoin[ToString[Subscript["\[Del]", ToString[newCoordinates[[
index[[1]]]], StandardForm]], StandardForm], ToString[Superscript["\[FormalCapitalK]", StringJoin[
ToString[newCoordinates[[index[[2]]]], StandardForm], ToString[newCoordinates[[index[[3]]]],
StandardForm]]], StandardForm]] -> Inactive[D][newExtrinsicCurvatureTensor[[index[[2]],index[[3]]]],
newCoordinates[[index[[1]]]]] + Total[(spatialChristoffelSymbols[[index[[2]],index[[1]],#1]]*
newExtrinsicCurvatureTensor[[#1,index[[3]]]] & ) /@ Range[Length[newMatrixRepresentation]]] + Total[
(spatialChristoffelSymbols[[index[[3]],index[[1]],#1]]*newExtrinsicCurvatureTensor[[index[[2]],
#1]] & ) /@ Range[Length[newMatrixRepresentation]]]] & ) /@
Tuples[Range[Length[newMatrixRepresentation]], 3] /. (ToExpression[#1] -> #1 & ) /@
Select[Join[coordinates, {timeCoordinate}], StringQ]], If[index1 === True && index2 === False,
newExtrinsicCurvatureTensor = Normal[SparseArray[(Module[{index = #1}, index ->
Total[(Inverse[newMatrixRepresentation][[#1,Last[index]]]*extrinsicCurvatureTensor[[First[index],
#1]] & ) /@ Range[Length[newMatrixRepresentation]]]] & ) /@
Tuples[Range[Length[newMatrixRepresentation]], 2]]]; Association[
(Module[{index = #1}, StringJoin[ToString[Subscript["\[Del]", ToString[newCoordinates[[index[[1]]]],
StandardForm]], StandardForm], ToString[Subsuperscript["\[FormalCapitalK]", ToString[newCoordinates[[index[[2]]]],
StandardForm], ToString[newCoordinates[[index[[3]]]], StandardForm]], StandardForm]] ->
Inactive[D][newExtrinsicCurvatureTensor[[index[[2]],index[[3]]]], newCoordinates[[index[[1]]]]] +
Total[(spatialChristoffelSymbols[[index[[3]],index[[1]],#1]]*newExtrinsicCurvatureTensor[[index[[2]],
#1]] & ) /@ Range[Length[newMatrixRepresentation]]] - Total[(spatialChristoffelSymbols[[#1,
index[[1]],index[[2]]]]*newExtrinsicCurvatureTensor[[#1,index[[3]]]] & ) /@
Range[Length[newMatrixRepresentation]]]] & ) /@ Tuples[Range[Length[newMatrixRepresentation]], 3] /.
(ToExpression[#1] -> #1 & ) /@ Select[Join[coordinates, {timeCoordinate}], StringQ]],
If[index1 === False && index2 === True, newExtrinsicCurvatureTensor =
Normal[SparseArray[(Module[{index = #1}, index -> Total[(Inverse[newMatrixRepresentation][[First[index],#1]]*
extrinsicCurvatureTensor[[#1,Last[index]]] & ) /@ Range[Length[newMatrixRepresentation]]]] & ) /@
Tuples[Range[Length[newMatrixRepresentation]], 2]]]; Association[
(Module[{index = #1}, StringJoin[ToString[Subscript["\[Del]", ToString[newCoordinates[[index[[1]]]],
StandardForm]], StandardForm], ToString[Subsuperscript["\[FormalCapitalK]", ToString[newCoordinates[[index[[3]]]],
StandardForm], ToString[newCoordinates[[index[[2]]]], StandardForm]], StandardForm]] ->
Inactive[D][newExtrinsicCurvatureTensor[[index[[2]],index[[3]]]], newCoordinates[[index[[1]]]]] +
Total[(spatialChristoffelSymbols[[index[[2]],index[[1]],#1]]*newExtrinsicCurvatureTensor[[#1,
index[[3]]]] & ) /@ Range[Length[newMatrixRepresentation]]] - Total[
(spatialChristoffelSymbols[[#1,index[[1]],index[[3]]]]*newExtrinsicCurvatureTensor[[index[[2]],
#1]] & ) /@ Range[Length[newMatrixRepresentation]]]] & ) /@
Tuples[Range[Length[newMatrixRepresentation]], 3] /. (ToExpression[#1] -> #1 & ) /@
Select[Join[coordinates, {timeCoordinate}], StringQ]]]]]]] /;
SymbolName[admDecomposition] === "ADMDecomposition" && SymbolName[metricTensor] === "MetricTensor" &&
Length[Dimensions[matrixRepresentation]] == 2 && Length[coordinates] == Length[matrixRepresentation] &&
BooleanQ[metricIndex1] && BooleanQ[metricIndex2] && Length[shiftVector] == Length[matrixRepresentation] &&
BooleanQ[index1] && BooleanQ[index2]
ExtrinsicCurvatureTensor[(admDecomposition_)[(metricTensor_)[matrixRepresentation_List, coordinates_List, metricIndex1_,
metricIndex2_], timeCoordinate_, lapseFunction_, shiftVector_List], index1_, index2_]["Dimensions"] :=
Length[matrixRepresentation] + 1 /; SymbolName[admDecomposition] === "ADMDecomposition" &&
SymbolName[metricTensor] === "MetricTensor" && Length[Dimensions[matrixRepresentation]] == 2 &&
Length[coordinates] == Length[matrixRepresentation] && BooleanQ[metricIndex1] && BooleanQ[metricIndex2] &&
Length[shiftVector] == Length[matrixRepresentation] && BooleanQ[index1] && BooleanQ[index2]
ExtrinsicCurvatureTensor[(admDecomposition_)[(metricTensor_)[matrixRepresentation_List, coordinates_List, metricIndex1_,
metricIndex2_], timeCoordinate_, lapseFunction_, shiftVector_List], index1_, index2_]["SymmetricQ"] :=
Module[{newMatrixRepresentation, newCoordinates, newTimeCoordinate, newLapseFunction, newShiftVector, shiftCovector,
spatialChristoffelSymbols}, newMatrixRepresentation = matrixRepresentation /. (#1 -> ToExpression[#1] & ) /@
Select[Join[coordinates, {timeCoordinate}], StringQ]; newCoordinates =
coordinates /. (#1 -> ToExpression[#1] & ) /@ Select[Join[coordinates, {timeCoordinate}], StringQ];
newTimeCoordinate = timeCoordinate /. (#1 -> ToExpression[#1] & ) /@ Select[Join[coordinates, {timeCoordinate}],
StringQ]; newLapseFunction = lapseFunction /. (#1 -> ToExpression[#1] & ) /@
Select[Join[coordinates, {timeCoordinate}], StringQ]; newShiftVector =
shiftVector /. (#1 -> ToExpression[#1] & ) /@ Select[Join[coordinates, {timeCoordinate}], StringQ];
shiftCovector = Normal[SparseArray[
(Module[{index = #1}, index -> Total[(newMatrixRepresentation[[index,#1]]*newShiftVector[[#1]] & ) /@
Range[Length[newMatrixRepresentation]]]] & ) /@ Range[Length[newMatrixRepresentation]]]];
spatialChristoffelSymbols = Normal[SparseArray[
(Module[{index = #1}, index -> Total[((1/2)*Inverse[newMatrixRepresentation][[index[[1]],#1]]*
(D[newMatrixRepresentation[[#1,index[[3]]]], newCoordinates[[index[[2]]]]] + D[newMatrixRepresentation[[
index[[2]],#1]], newCoordinates[[index[[3]]]]] - D[newMatrixRepresentation[[index[[2]],index[[3]]]],
newCoordinates[[#1]]]) & ) /@ Range[Length[newMatrixRepresentation]]]] & ) /@
Tuples[Range[Length[newMatrixRepresentation]], 3]]]; SymmetricMatrixQ[
Normal[SparseArray[(Module[{index = #1}, index -> (1/(2*newLapseFunction))*(D[shiftCovector[[First[index]]],
newCoordinates[[Last[index]]]] - Total[(spatialChristoffelSymbols[[#1,Last[index],First[index]]]*
shiftCovector[[#1]] & ) /@ Range[Length[newMatrixRepresentation]]] + D[shiftCovector[[Last[index]]],
newCoordinates[[First[index]]]] - Total[(spatialChristoffelSymbols[[#1,First[index],Last[index]]]*
shiftCovector[[#1]] & ) /@ Range[Length[newMatrixRepresentation]]] - D[newMatrixRepresentation[[
First[index],Last[index]]], newTimeCoordinate])] & ) /@ Tuples[Range[Length[newMatrixRepresentation]],
2]]] /. (ToExpression[#1] -> #1 & ) /@ Select[Join[coordinates, {timeCoordinate}], StringQ]]] /;
SymbolName[admDecomposition] === "ADMDecomposition" && SymbolName[metricTensor] === "MetricTensor" &&
Length[Dimensions[matrixRepresentation]] == 2 && Length[coordinates] == Length[matrixRepresentation] &&
BooleanQ[metricIndex1] && BooleanQ[metricIndex2] && Length[shiftVector] == Length[matrixRepresentation] &&
BooleanQ[index1] && BooleanQ[index2]
ExtrinsicCurvatureTensor[(admDecomposition_)[(metricTensor_)[matrixRepresentation_List, coordinates_List, metricIndex1_,
metricIndex2_], timeCoordinate_, lapseFunction_, shiftVector_List], index1_, index2_]["DiagonalQ"] :=
Module[{newMatrixRepresentation, newCoordinates, newTimeCoordinate, newLapseFunction, newShiftVector, shiftCovector,
spatialChristoffelSymbols}, newMatrixRepresentation = matrixRepresentation /. (#1 -> ToExpression[#1] & ) /@
Select[Join[coordinates, {timeCoordinate}], StringQ]; newCoordinates =
coordinates /. (#1 -> ToExpression[#1] & ) /@ Select[Join[coordinates, {timeCoordinate}], StringQ];
newTimeCoordinate = timeCoordinate /. (#1 -> ToExpression[#1] & ) /@ Select[Join[coordinates, {timeCoordinate}],
StringQ]; newLapseFunction = lapseFunction /. (#1 -> ToExpression[#1] & ) /@
Select[Join[coordinates, {timeCoordinate}], StringQ]; newShiftVector =
shiftVector /. (#1 -> ToExpression[#1] & ) /@ Select[Join[coordinates, {timeCoordinate}], StringQ];
shiftCovector = Normal[SparseArray[
(Module[{index = #1}, index -> Total[(newMatrixRepresentation[[index,#1]]*newShiftVector[[#1]] & ) /@
Range[Length[newMatrixRepresentation]]]] & ) /@ Range[Length[newMatrixRepresentation]]]];
spatialChristoffelSymbols = Normal[SparseArray[
(Module[{index = #1}, index -> Total[((1/2)*Inverse[newMatrixRepresentation][[index[[1]],#1]]*
(D[newMatrixRepresentation[[#1,index[[3]]]], newCoordinates[[index[[2]]]]] + D[newMatrixRepresentation[[
index[[2]],#1]], newCoordinates[[index[[3]]]]] - D[newMatrixRepresentation[[index[[2]],index[[3]]]],
newCoordinates[[#1]]]) & ) /@ Range[Length[newMatrixRepresentation]]]] & ) /@
Tuples[Range[Length[newMatrixRepresentation]], 3]]];
DiagonalMatrixQ[Normal[SparseArray[(Module[{index = #1}, index -> (1/(2*newLapseFunction))*
(D[shiftCovector[[First[index]]], newCoordinates[[Last[index]]]] - Total[
(spatialChristoffelSymbols[[#1,Last[index],First[index]]]*shiftCovector[[#1]] & ) /@
Range[Length[newMatrixRepresentation]]] + D[shiftCovector[[Last[index]]], newCoordinates[[
First[index]]]] - Total[(spatialChristoffelSymbols[[#1,First[index],Last[index]]]*shiftCovector[[
#1]] & ) /@ Range[Length[newMatrixRepresentation]]] - D[newMatrixRepresentation[[First[index],
Last[index]]], newTimeCoordinate])] & ) /@ Tuples[Range[Length[newMatrixRepresentation]], 2]]] /.
(ToExpression[#1] -> #1 & ) /@ Select[Join[coordinates, {timeCoordinate}], StringQ]]] /;
SymbolName[admDecomposition] === "ADMDecomposition" && SymbolName[metricTensor] === "MetricTensor" &&
Length[Dimensions[matrixRepresentation]] == 2 && Length[coordinates] == Length[matrixRepresentation] &&
BooleanQ[metricIndex1] && BooleanQ[metricIndex2] && Length[shiftVector] == Length[matrixRepresentation] &&
BooleanQ[index1] && BooleanQ[index2]