-
Notifications
You must be signed in to change notification settings - Fork 13
/
Copy pathBachTensor.wl
286 lines (285 loc) · 28.8 KB
/
BachTensor.wl
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
(* ::Package:: *)
BachTensor[(metricTensor_)[matrixRepresentation_List, coordinates_List, index1_, index2_]] :=
BachTensor[ResourceFunction["MetricTensor"][matrixRepresentation, coordinates, index1, index2], True, True] /;
SymbolName[metricTensor] === "MetricTensor" && Length[Dimensions[matrixRepresentation]] == 2 &&
Length[coordinates] == Length[matrixRepresentation] && BooleanQ[index1] && BooleanQ[index2]
BachTensor[(metricTensor_)[matrixRepresentation_List, coordinates_List, index1_, index2_], newCoordinates_List] :=
BachTensor[ResourceFunction["MetricTensor"][matrixRepresentation /. Thread[coordinates -> newCoordinates],
newCoordinates, index1, index2], True, True] /; SymbolName[metricTensor] === "MetricTensor" &&
Length[Dimensions[matrixRepresentation]] == 2 && Length[coordinates] == Length[matrixRepresentation] &&
Length[newCoordinates] == Length[matrixRepresentation] && BooleanQ[index1] && BooleanQ[index2]
BachTensor[(metricTensor_)[matrixRepresentation_List, coordinates_List, metricIndex1_, metricIndex2_],
newCoordinates_List, index1_, index2_] :=
BachTensor[ResourceFunction["MetricTensor"][matrixRepresentation /. Thread[coordinates -> newCoordinates],
newCoordinates, metricIndex1, metricIndex2], index1, index2] /; SymbolName[metricTensor] === "MetricTensor" &&
Length[Dimensions[matrixRepresentation]] == 2 && Length[coordinates] == Length[matrixRepresentation] &&
Length[newCoordinates] == Length[matrixRepresentation] && BooleanQ[metricIndex1] && BooleanQ[metricIndex2] &&
BooleanQ[index1] && BooleanQ[index2]
BachTensor[(metricTensor_)[matrixRepresentation_List, coordinates_List, metricIndex1_, metricIndex2_], index1_, index2_][
"MatrixRepresentation"] := Module[{newMatrixRepresentation, newCoordinates, christoffelSymbols, riemannTensor,
covariantRiemannTensor, ricciTensor, ricciScalar, weylTensor, mixedWeylTensor, schoutenTensor, covariantDerivatives,
bachTensor}, newMatrixRepresentation = matrixRepresentation /. (#1 -> ToExpression[#1] & ) /@
Select[coordinates, StringQ]; newCoordinates = coordinates /. (#1 -> ToExpression[#1] & ) /@
Select[coordinates, StringQ]; christoffelSymbols =
Normal[SparseArray[(Module[{index = #1}, index -> Total[((1/2)*Inverse[newMatrixRepresentation][[index[[1]],#1]]*
(D[newMatrixRepresentation[[#1,index[[3]]]], newCoordinates[[index[[2]]]]] + D[newMatrixRepresentation[[
index[[2]],#1]], newCoordinates[[index[[3]]]]] - D[newMatrixRepresentation[[index[[2]],index[[3]]]],
newCoordinates[[#1]]]) & ) /@ Range[Length[newMatrixRepresentation]]]] & ) /@
Tuples[Range[Length[newMatrixRepresentation]], 3]]];
riemannTensor = Normal[SparseArray[(Module[{index = #1}, index -> D[christoffelSymbols[[index[[1]],index[[2]],index[[
4]]]], newCoordinates[[index[[3]]]]] - D[christoffelSymbols[[index[[1]],index[[2]],index[[3]]]],
newCoordinates[[index[[4]]]]] + Total[(christoffelSymbols[[index[[1]],#1,index[[3]]]]*christoffelSymbols[[
#1,index[[2]],index[[4]]]] & ) /@ Range[Length[newMatrixRepresentation]]] -
Total[(christoffelSymbols[[index[[1]],#1,index[[4]]]]*christoffelSymbols[[#1,index[[2]],index[[3]]]] & ) /@
Range[Length[newMatrixRepresentation]]]] & ) /@ Tuples[Range[Length[newMatrixRepresentation]], 4]]];
covariantRiemannTensor = Normal[SparseArray[
(Module[{index = #1}, index -> Total[(newMatrixRepresentation[[index[[1]],#1]]*riemannTensor[[#1,index[[2]],
index[[3]],index[[4]]]] & ) /@ Range[Length[newMatrixRepresentation]]]] & ) /@
Tuples[Range[Length[newMatrixRepresentation]], 4]]];
ricciTensor = Normal[SparseArray[(Module[{index = #1}, index -> Total[(riemannTensor[[#1,First[index],#1,
Last[index]]] & ) /@ Range[Length[newMatrixRepresentation]]]] & ) /@
Tuples[Range[Length[newMatrixRepresentation]], 2]]];
ricciScalar = Total[(Inverse[newMatrixRepresentation][[First[#1],Last[#1]]]*ricciTensor[[First[#1],Last[#1]]] & ) /@
Tuples[Range[Length[newMatrixRepresentation]], 2]];
weylTensor = Normal[SparseArray[(Module[{index = #1}, index -> covariantRiemannTensor[[index[[1]],index[[2]],
index[[3]],index[[4]]]] + (1/(Length[newMatrixRepresentation] - 2))*(ricciTensor[[index[[1]],index[[4]]]]*
newMatrixRepresentation[[index[[2]],index[[3]]]] - ricciTensor[[index[[1]],index[[3]]]]*
newMatrixRepresentation[[index[[2]],index[[4]]]] + ricciTensor[[index[[2]],index[[3]]]]*
newMatrixRepresentation[[index[[1]],index[[4]]]] - ricciTensor[[index[[2]],index[[4]]]]*
newMatrixRepresentation[[index[[1]],index[[3]]]]) + (1/((Length[newMatrixRepresentation] - 1)*
(Length[newMatrixRepresentation] - 2)))*(ricciScalar*(newMatrixRepresentation[[index[[1]],index[[3]]]]*
newMatrixRepresentation[[index[[2]],index[[4]]]] - newMatrixRepresentation[[index[[1]],index[[4]]]]*
newMatrixRepresentation[[index[[2]],index[[3]]]]))] & ) /@
Tuples[Range[Length[newMatrixRepresentation]], 4]]]; mixedWeylTensor =
Normal[SparseArray[(Module[{index = #1}, index -> Total[(Inverse[newMatrixRepresentation][[index[[2]],#1[[1]]]]*
Inverse[newMatrixRepresentation][[#1[[2]],index[[4]]]]*weylTensor[[index[[1]],#1[[1]],index[[3]],
#1[[2]]]] & ) /@ Tuples[Range[Length[newMatrixRepresentation]], 2]]] & ) /@
Tuples[Range[Length[newMatrixRepresentation]], 4]]]; schoutenTensor = (1/(Length[newMatrixRepresentation] - 2))*
(ricciTensor - (ricciScalar/(2*(Length[newMatrixRepresentation] - 1)))*newMatrixRepresentation);
covariantDerivatives = Normal[SparseArray[
(Module[{index = #1}, index -> D[schoutenTensor[[index[[2]],index[[3]]]], newCoordinates[[index[[1]]]]] -
Total[(christoffelSymbols[[#1,index[[1]],index[[2]]]]*schoutenTensor[[#1,index[[3]]]] & ) /@ Range[
Length[newMatrixRepresentation]]] - Total[(christoffelSymbols[[#1,index[[1]],index[[3]]]]*
schoutenTensor[[index[[2]],#1]] & ) /@ Range[Length[newMatrixRepresentation]]]] & ) /@
Tuples[Range[Length[newMatrixRepresentation]], 3]]];
bachTensor = Normal[SparseArray[(Module[{index = #1}, index -> Total[(schoutenTensor[[First[#1],Last[#1]]]*
mixedWeylTensor[[First[index],First[#1],Last[index],Last[#1]]] & ) /@ Tuples[
Range[Length[newMatrixRepresentation]], 2]] + Total[(Module[{nestedIndex = #1},
Inverse[newMatrixRepresentation][[First[nestedIndex],Last[nestedIndex]]]*(D[covariantDerivatives[[
First[nestedIndex],First[index],Last[index]]], newCoordinates[[Last[nestedIndex]]]] -
Total[(christoffelSymbols[[#1,Last[nestedIndex],First[nestedIndex]]]*covariantDerivatives[[#1,
First[index],Last[index]]] & ) /@ Range[Length[newMatrixRepresentation]]] -
Total[(christoffelSymbols[[#1,Last[nestedIndex],First[index]]]*covariantDerivatives[[
First[nestedIndex],#1,Last[index]]] & ) /@ Range[Length[newMatrixRepresentation]]] -
Total[(christoffelSymbols[[#1,Last[nestedIndex],Last[index]]]*covariantDerivatives[[First[
nestedIndex],First[index],#1]] & ) /@ Range[Length[newMatrixRepresentation]]])] & ) /@
Tuples[Range[Length[newMatrixRepresentation]], 2]] - Total[(Module[{nestedIndex = #1},
Inverse[newMatrixRepresentation][[First[nestedIndex],Last[nestedIndex]]]*(D[covariantDerivatives[[
First[index],Last[index],First[nestedIndex]]], newCoordinates[[Last[nestedIndex]]]] -
Total[(christoffelSymbols[[#1,Last[nestedIndex],First[index]]]*covariantDerivatives[[#1,Last[index],
First[nestedIndex]]] & ) /@ Range[Length[newMatrixRepresentation]]] - Total[
(christoffelSymbols[[#1,Last[nestedIndex],Last[index]]]*covariantDerivatives[[First[index],#1,
First[nestedIndex]]] & ) /@ Range[Length[newMatrixRepresentation]]] - Total[
(christoffelSymbols[[#1,Last[nestedIndex],First[nestedIndex]]]*covariantDerivatives[[First[index],
Last[index],#1]] & ) /@ Range[Length[newMatrixRepresentation]]])] & ) /@
Tuples[Range[Length[newMatrixRepresentation]], 2]]] & ) /@ Tuples[Range[Length[newMatrixRepresentation]],
2]]] /. (ToExpression[#1] -> #1 & ) /@ Select[coordinates, StringQ]; If[index1 === True && index2 === True,
bachTensor, If[index1 === False && index2 === False,
Normal[SparseArray[(Module[{index = #1}, index -> Total[(Inverse[matrixRepresentation][[First[index],First[#1]]]*
Inverse[matrixRepresentation][[Last[#1],Last[index]]]*bachTensor[[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]]]*bachTensor[[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]]*
bachTensor[[#1,Last[index]]] & ) /@ Range[Length[matrixRepresentation]]]] & ) /@
Tuples[Range[Length[matrixRepresentation]], 2]]], Indeterminate]]]]] /;
SymbolName[metricTensor] === "MetricTensor" && Length[Dimensions[matrixRepresentation]] == 2 &&
Length[coordinates] == Length[matrixRepresentation] && BooleanQ[metricIndex1] && BooleanQ[metricIndex2] &&
BooleanQ[index1] && BooleanQ[index2]
BachTensor[(metricTensor_)[matrixRepresentation_List, coordinates_List, metricIndex1_, metricIndex2_], index1_, index2_][
"ReducedMatrixRepresentation"] := Module[{newMatrixRepresentation, newCoordinates, christoffelSymbols, riemannTensor,
covariantRiemannTensor, ricciTensor, ricciScalar, weylTensor, mixedWeylTensor, schoutenTensor, covariantDerivatives,
bachTensor}, newMatrixRepresentation = matrixRepresentation /. (#1 -> ToExpression[#1] & ) /@
Select[coordinates, StringQ]; newCoordinates = coordinates /. (#1 -> ToExpression[#1] & ) /@
Select[coordinates, StringQ]; christoffelSymbols =
Normal[SparseArray[(Module[{index = #1}, index -> Total[((1/2)*Inverse[newMatrixRepresentation][[index[[1]],#1]]*
(D[newMatrixRepresentation[[#1,index[[3]]]], newCoordinates[[index[[2]]]]] + D[newMatrixRepresentation[[
index[[2]],#1]], newCoordinates[[index[[3]]]]] - D[newMatrixRepresentation[[index[[2]],index[[3]]]],
newCoordinates[[#1]]]) & ) /@ Range[Length[newMatrixRepresentation]]]] & ) /@
Tuples[Range[Length[newMatrixRepresentation]], 3]]];
riemannTensor = Normal[SparseArray[(Module[{index = #1}, index -> D[christoffelSymbols[[index[[1]],index[[2]],index[[
4]]]], newCoordinates[[index[[3]]]]] - D[christoffelSymbols[[index[[1]],index[[2]],index[[3]]]],
newCoordinates[[index[[4]]]]] + Total[(christoffelSymbols[[index[[1]],#1,index[[3]]]]*christoffelSymbols[[
#1,index[[2]],index[[4]]]] & ) /@ Range[Length[newMatrixRepresentation]]] -
Total[(christoffelSymbols[[index[[1]],#1,index[[4]]]]*christoffelSymbols[[#1,index[[2]],index[[3]]]] & ) /@
Range[Length[newMatrixRepresentation]]]] & ) /@ Tuples[Range[Length[newMatrixRepresentation]], 4]]];
covariantRiemannTensor = Normal[SparseArray[
(Module[{index = #1}, index -> Total[(newMatrixRepresentation[[index[[1]],#1]]*riemannTensor[[#1,index[[2]],
index[[3]],index[[4]]]] & ) /@ Range[Length[newMatrixRepresentation]]]] & ) /@
Tuples[Range[Length[newMatrixRepresentation]], 4]]];
ricciTensor = Normal[SparseArray[(Module[{index = #1}, index -> Total[(riemannTensor[[#1,First[index],#1,
Last[index]]] & ) /@ Range[Length[newMatrixRepresentation]]]] & ) /@
Tuples[Range[Length[newMatrixRepresentation]], 2]]];
ricciScalar = Total[(Inverse[newMatrixRepresentation][[First[#1],Last[#1]]]*ricciTensor[[First[#1],Last[#1]]] & ) /@
Tuples[Range[Length[newMatrixRepresentation]], 2]];
weylTensor = Normal[SparseArray[(Module[{index = #1}, index -> covariantRiemannTensor[[index[[1]],index[[2]],
index[[3]],index[[4]]]] + (1/(Length[newMatrixRepresentation] - 2))*(ricciTensor[[index[[1]],index[[4]]]]*
newMatrixRepresentation[[index[[2]],index[[3]]]] - ricciTensor[[index[[1]],index[[3]]]]*
newMatrixRepresentation[[index[[2]],index[[4]]]] + ricciTensor[[index[[2]],index[[3]]]]*
newMatrixRepresentation[[index[[1]],index[[4]]]] - ricciTensor[[index[[2]],index[[4]]]]*
newMatrixRepresentation[[index[[1]],index[[3]]]]) + (1/((Length[newMatrixRepresentation] - 1)*
(Length[newMatrixRepresentation] - 2)))*(ricciScalar*(newMatrixRepresentation[[index[[1]],index[[3]]]]*
newMatrixRepresentation[[index[[2]],index[[4]]]] - newMatrixRepresentation[[index[[1]],index[[4]]]]*
newMatrixRepresentation[[index[[2]],index[[3]]]]))] & ) /@
Tuples[Range[Length[newMatrixRepresentation]], 4]]]; mixedWeylTensor =
Normal[SparseArray[(Module[{index = #1}, index -> Total[(Inverse[newMatrixRepresentation][[index[[2]],#1[[1]]]]*
Inverse[newMatrixRepresentation][[#1[[2]],index[[4]]]]*weylTensor[[index[[1]],#1[[1]],index[[3]],
#1[[2]]]] & ) /@ Tuples[Range[Length[newMatrixRepresentation]], 2]]] & ) /@
Tuples[Range[Length[newMatrixRepresentation]], 4]]]; schoutenTensor = (1/(Length[newMatrixRepresentation] - 2))*
(ricciTensor - (ricciScalar/(2*(Length[newMatrixRepresentation] - 1)))*newMatrixRepresentation);
covariantDerivatives = Normal[SparseArray[
(Module[{index = #1}, index -> D[schoutenTensor[[index[[2]],index[[3]]]], newCoordinates[[index[[1]]]]] -
Total[(christoffelSymbols[[#1,index[[1]],index[[2]]]]*schoutenTensor[[#1,index[[3]]]] & ) /@ Range[
Length[newMatrixRepresentation]]] - Total[(christoffelSymbols[[#1,index[[1]],index[[3]]]]*
schoutenTensor[[index[[2]],#1]] & ) /@ Range[Length[newMatrixRepresentation]]]] & ) /@
Tuples[Range[Length[newMatrixRepresentation]], 3]]];
bachTensor = Normal[SparseArray[(Module[{index = #1}, index -> Total[(schoutenTensor[[First[#1],Last[#1]]]*
mixedWeylTensor[[First[index],First[#1],Last[index],Last[#1]]] & ) /@ Tuples[
Range[Length[newMatrixRepresentation]], 2]] + Total[(Module[{nestedIndex = #1},
Inverse[newMatrixRepresentation][[First[nestedIndex],Last[nestedIndex]]]*(D[covariantDerivatives[[
First[nestedIndex],First[index],Last[index]]], newCoordinates[[Last[nestedIndex]]]] -
Total[(christoffelSymbols[[#1,Last[nestedIndex],First[nestedIndex]]]*covariantDerivatives[[#1,
First[index],Last[index]]] & ) /@ Range[Length[newMatrixRepresentation]]] -
Total[(christoffelSymbols[[#1,Last[nestedIndex],First[index]]]*covariantDerivatives[[
First[nestedIndex],#1,Last[index]]] & ) /@ Range[Length[newMatrixRepresentation]]] -
Total[(christoffelSymbols[[#1,Last[nestedIndex],Last[index]]]*covariantDerivatives[[First[
nestedIndex],First[index],#1]] & ) /@ Range[Length[newMatrixRepresentation]]])] & ) /@
Tuples[Range[Length[newMatrixRepresentation]], 2]] - Total[(Module[{nestedIndex = #1},
Inverse[newMatrixRepresentation][[First[nestedIndex],Last[nestedIndex]]]*(D[covariantDerivatives[[
First[index],Last[index],First[nestedIndex]]], newCoordinates[[Last[nestedIndex]]]] -
Total[(christoffelSymbols[[#1,Last[nestedIndex],First[index]]]*covariantDerivatives[[#1,Last[index],
First[nestedIndex]]] & ) /@ Range[Length[newMatrixRepresentation]]] - Total[
(christoffelSymbols[[#1,Last[nestedIndex],Last[index]]]*covariantDerivatives[[First[index],#1,
First[nestedIndex]]] & ) /@ Range[Length[newMatrixRepresentation]]] - Total[
(christoffelSymbols[[#1,Last[nestedIndex],First[nestedIndex]]]*covariantDerivatives[[First[index],
Last[index],#1]] & ) /@ Range[Length[newMatrixRepresentation]]])] & ) /@
Tuples[Range[Length[newMatrixRepresentation]], 2]]] & ) /@ Tuples[Range[Length[newMatrixRepresentation]],
2]]] /. (ToExpression[#1] -> #1 & ) /@ Select[coordinates, StringQ]; If[index1 === True && index2 === True,
FullSimplify[bachTensor], 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]]]*bachTensor[[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]]]*bachTensor[[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]]*bachTensor[[#1,Last[index]]] & ) /@ Range[Length[matrixRepresentation]]]] & ) /@
Tuples[Range[Length[matrixRepresentation]], 2]]]], Indeterminate]]]]] /;
SymbolName[metricTensor] === "MetricTensor" && Length[Dimensions[matrixRepresentation]] == 2 &&
Length[coordinates] == Length[matrixRepresentation] && BooleanQ[metricIndex1] && BooleanQ[metricIndex2] &&
BooleanQ[index1] && BooleanQ[index2]
BachTensor /: MakeBoxes[bachTensor:BachTensor[(metricTensor_)[matrixRepresentation_List, coordinates_List, metricIndex1_,
metricIndex2_], index1_, index2_], format_] :=
Module[{newMatrixRepresentation, newCoordinates, christoffelSymbols, riemannTensor, covariantRiemannTensor,
ricciTensor, ricciScalar, weylTensor, mixedWeylTensor, schoutenTensor, covariantDerivatives, tensorRepresentation,
matrixForm, type, symbol, dimensions, eigenvalues, positiveEigenvalues, negativeEigenvalues, signature, icon},
newMatrixRepresentation = matrixRepresentation /. (#1 -> ToExpression[#1] & ) /@ Select[coordinates, StringQ];
newCoordinates = coordinates /. (#1 -> ToExpression[#1] & ) /@ Select[coordinates, StringQ];
christoffelSymbols = Normal[SparseArray[
(Module[{index = #1}, index -> Total[((1/2)*Inverse[newMatrixRepresentation][[index[[1]],#1]]*
(D[newMatrixRepresentation[[#1,index[[3]]]], newCoordinates[[index[[2]]]]] + D[newMatrixRepresentation[[
index[[2]],#1]], newCoordinates[[index[[3]]]]] - D[newMatrixRepresentation[[index[[2]],index[[3]]]],
newCoordinates[[#1]]]) & ) /@ Range[Length[newMatrixRepresentation]]]] & ) /@
Tuples[Range[Length[newMatrixRepresentation]], 3]]]; riemannTensor =
Normal[SparseArray[(Module[{index = #1}, index -> D[christoffelSymbols[[index[[1]],index[[2]],index[[4]]]],
newCoordinates[[index[[3]]]]] - D[christoffelSymbols[[index[[1]],index[[2]],index[[3]]]], newCoordinates[[
index[[4]]]]] + Total[(christoffelSymbols[[index[[1]],#1,index[[3]]]]*christoffelSymbols[[#1,index[[2]],
index[[4]]]] & ) /@ Range[Length[newMatrixRepresentation]]] -
Total[(christoffelSymbols[[index[[1]],#1,index[[4]]]]*christoffelSymbols[[#1,index[[2]],index[[3]]]] & ) /@
Range[Length[newMatrixRepresentation]]]] & ) /@ Tuples[Range[Length[newMatrixRepresentation]], 4]]];
covariantRiemannTensor = Normal[SparseArray[
(Module[{index = #1}, index -> Total[(newMatrixRepresentation[[index[[1]],#1]]*riemannTensor[[#1,index[[2]],
index[[3]],index[[4]]]] & ) /@ Range[Length[newMatrixRepresentation]]]] & ) /@
Tuples[Range[Length[newMatrixRepresentation]], 4]]];
ricciTensor = Normal[SparseArray[(Module[{index = #1}, index -> Total[(riemannTensor[[#1,First[index],#1,
Last[index]]] & ) /@ Range[Length[newMatrixRepresentation]]]] & ) /@
Tuples[Range[Length[newMatrixRepresentation]], 2]]];
ricciScalar = Total[(Inverse[newMatrixRepresentation][[First[#1],Last[#1]]]*ricciTensor[[First[#1],
Last[#1]]] & ) /@ Tuples[Range[Length[newMatrixRepresentation]], 2]];
weylTensor = Normal[SparseArray[(Module[{index = #1}, index -> covariantRiemannTensor[[index[[1]],index[[2]],index[[
3]],index[[4]]]] + (1/(Length[newMatrixRepresentation] - 2))*(ricciTensor[[index[[1]],index[[4]]]]*
newMatrixRepresentation[[index[[2]],index[[3]]]] - ricciTensor[[index[[1]],index[[3]]]]*
newMatrixRepresentation[[index[[2]],index[[4]]]] + ricciTensor[[index[[2]],index[[3]]]]*
newMatrixRepresentation[[index[[1]],index[[4]]]] - ricciTensor[[index[[2]],index[[4]]]]*
newMatrixRepresentation[[index[[1]],index[[3]]]]) + (1/((Length[newMatrixRepresentation] - 1)*
(Length[newMatrixRepresentation] - 2)))*(ricciScalar*(newMatrixRepresentation[[index[[1]],index[[3]]]]*
newMatrixRepresentation[[index[[2]],index[[4]]]] - newMatrixRepresentation[[index[[1]],index[[4]]]]*
newMatrixRepresentation[[index[[2]],index[[3]]]]))] & ) /@
Tuples[Range[Length[newMatrixRepresentation]], 4]]]; mixedWeylTensor =
Normal[SparseArray[(Module[{index = #1}, index -> Total[(Inverse[newMatrixRepresentation][[index[[2]],#1[[1]]]]*
Inverse[newMatrixRepresentation][[#1[[2]],index[[4]]]]*weylTensor[[index[[1]],#1[[1]],index[[3]],
#1[[2]]]] & ) /@ Tuples[Range[Length[newMatrixRepresentation]], 2]]] & ) /@
Tuples[Range[Length[newMatrixRepresentation]], 4]]]; schoutenTensor = (1/(Length[newMatrixRepresentation] - 2))*
(ricciTensor - (ricciScalar/(2*(Length[newMatrixRepresentation] - 1)))*newMatrixRepresentation);
covariantDerivatives = Normal[SparseArray[
(Module[{index = #1}, index -> D[schoutenTensor[[index[[2]],index[[3]]]], newCoordinates[[index[[1]]]]] -
Total[(christoffelSymbols[[#1,index[[1]],index[[2]]]]*schoutenTensor[[#1,index[[3]]]] & ) /@
Range[Length[newMatrixRepresentation]]] - Total[(christoffelSymbols[[#1,index[[1]],index[[3]]]]*
schoutenTensor[[index[[2]],#1]] & ) /@ Range[Length[newMatrixRepresentation]]]] & ) /@
Tuples[Range[Length[newMatrixRepresentation]], 3]]]; tensorRepresentation =
Normal[SparseArray[(Module[{index = #1}, index -> Total[(schoutenTensor[[First[#1],Last[#1]]]*mixedWeylTensor[[
First[index],First[#1],Last[index],Last[#1]]] & ) /@ Tuples[Range[Length[newMatrixRepresentation]],
2]] + Total[(Module[{nestedIndex = #1}, Inverse[newMatrixRepresentation][[First[nestedIndex],
Last[nestedIndex]]]*(D[covariantDerivatives[[First[nestedIndex],First[index],Last[index]]],
newCoordinates[[Last[nestedIndex]]]] - Total[(christoffelSymbols[[#1,Last[nestedIndex],
First[nestedIndex]]]*covariantDerivatives[[#1,First[index],Last[index]]] & ) /@
Range[Length[newMatrixRepresentation]]] - Total[(christoffelSymbols[[#1,Last[nestedIndex],
First[index]]]*covariantDerivatives[[First[nestedIndex],#1,Last[index]]] & ) /@
Range[Length[newMatrixRepresentation]]] - Total[(christoffelSymbols[[#1,Last[nestedIndex],
Last[index]]]*covariantDerivatives[[First[nestedIndex],First[index],#1]] & ) /@
Range[Length[newMatrixRepresentation]]])] & ) /@ Tuples[Range[Length[newMatrixRepresentation]],
2]] - Total[(Module[{nestedIndex = #1}, Inverse[newMatrixRepresentation][[First[nestedIndex],
Last[nestedIndex]]]*(D[covariantDerivatives[[First[index],Last[index],First[nestedIndex]]],
newCoordinates[[Last[nestedIndex]]]] - Total[(christoffelSymbols[[#1,Last[nestedIndex],
First[index]]]*covariantDerivatives[[#1,Last[index],First[nestedIndex]]] & ) /@
Range[Length[newMatrixRepresentation]]] - Total[(christoffelSymbols[[#1,Last[nestedIndex],
Last[index]]]*covariantDerivatives[[First[index],#1,First[nestedIndex]]] & ) /@
Range[Length[newMatrixRepresentation]]] - Total[(christoffelSymbols[[#1,Last[nestedIndex],
First[nestedIndex]]]*covariantDerivatives[[First[index],Last[index],#1]] & ) /@
Range[Length[newMatrixRepresentation]]])] & ) /@ Tuples[Range[Length[newMatrixRepresentation]],
2]]] & ) /@ Tuples[Range[Length[newMatrixRepresentation]], 2]]] /. (ToExpression[#1] -> #1 & ) /@
Select[coordinates, StringQ]; If[index1 === True && index2 === True, matrixForm = tensorRepresentation;
type = "Covariant"; symbol = Subscript["\[FormalCapitalB]", "\[FormalMu]\[FormalNu]"], If[index1 === False && index2 === False,
matrixForm = Normal[SparseArray[(Module[{index = #1}, index -> Total[(Inverse[matrixRepresentation][[First[index],
Last[#1]]]*Inverse[matrixRepresentation][[Last[#1],Last[index]]]*tensorRepresentation[[First[#1],
Last[#1]]] & ) /@ Tuples[Range[Length[matrixRepresentation]], 2]]] & ) /@
Tuples[Range[Length[matrixRepresentation]], 2]]]; type = "Contravariant";
symbol = Superscript["\[FormalCapitalB]", "\[FormalMu]\[FormalNu]"], If[index1 === True && index2 === False,
matrixForm = Normal[SparseArray[(Module[{index = #1}, index -> Total[(Inverse[matrixRepresentation][[#1,
Last[index]]]*tensorRepresentation[[First[index],#1]] & ) /@ Range[Length[
matrixRepresentation]]]] & ) /@ Tuples[Range[Length[matrixRepresentation]], 2]]]; type = "Mixed";
symbol = Subsuperscript["\[FormalCapitalB]", "\[FormalMu]", "\[FormalNu]"], If[index1 === False && index2 === True,
matrixForm = Normal[SparseArray[(Module[{index = #1}, index -> Total[(Inverse[matrixRepresentation][[
First[index],#1]]*tensorRepresentation[[#1,Last[index]]] & ) /@ Range[Length[
matrixRepresentation]]]] & ) /@ Tuples[Range[Length[matrixRepresentation]], 2]]]; type = "Mixed";
symbol = Subsuperscript["\[FormalCapitalB]", "\[FormalNu]", "\[FormalMu]"], matrixForm = ConstantArray[Indeterminate,
{Length[matrixRepresentation], Length[matrixRepresentation]}]; type = Indeterminate;
symbol = Indeterminate]]]]; dimensions = Length[matrixRepresentation];
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], signature = "Riemannian", If[Length[positiveEigenvalues] == 1 ||
Length[negativeEigenvalues] == 1, signature = "Lorentzian", signature = "Pseudo-Riemannian"]],
signature = Indeterminate]; icon = MatrixPlot[matrixForm, ImageSize ->
Dynamic[{Automatic, 3.5*(CurrentValue["FontCapHeight"]/AbsoluteCurrentValue[Magnification])}], Frame -> False,
FrameTicks -> None]; BoxForm`ArrangeSummaryBox["BachTensor", bachTensor, icon,
{{BoxForm`SummaryItem[{"Type: ", type}], BoxForm`SummaryItem[{"Symbol: ", symbol}]},
{BoxForm`SummaryItem[{"Dimensions: ", dimensions}], BoxForm`SummaryItem[{"Signature: ", signature}]}},
{{BoxForm`SummaryItem[{"Coordinates: ", coordinates}]}}, format, "Interpretable" -> Automatic]] /;
SymbolName[metricTensor] === "MetricTensor" && Length[Dimensions[matrixRepresentation]] == 2 &&
Length[coordinates] == Length[matrixRepresentation] && BooleanQ[metricIndex1] && BooleanQ[metricIndex2] &&
BooleanQ[index1] && BooleanQ[index2]