-
Notifications
You must be signed in to change notification settings - Fork 13
/
Copy pathStressEnergyTensor.wl
2508 lines (2507 loc) · 246 KB
/
StressEnergyTensor.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
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
497
498
499
500
501
502
503
504
505
506
507
508
509
510
511
512
513
514
515
516
517
518
519
520
521
522
523
524
525
526
527
528
529
530
531
532
533
534
535
536
537
538
539
540
541
542
543
544
545
546
547
548
549
550
551
552
553
554
555
556
557
558
559
560
561
562
563
564
565
566
567
568
569
570
571
572
573
574
575
576
577
578
579
580
581
582
583
584
585
586
587
588
589
590
591
592
593
594
595
596
597
598
599
600
601
602
603
604
605
606
607
608
609
610
611
612
613
614
615
616
617
618
619
620
621
622
623
624
625
626
627
628
629
630
631
632
633
634
635
636
637
638
639
640
641
642
643
644
645
646
647
648
649
650
651
652
653
654
655
656
657
658
659
660
661
662
663
664
665
666
667
668
669
670
671
672
673
674
675
676
677
678
679
680
681
682
683
684
685
686
687
688
689
690
691
692
693
694
695
696
697
698
699
700
701
702
703
704
705
706
707
708
709
710
711
712
713
714
715
716
717
718
719
720
721
722
723
724
725
726
727
728
729
730
731
732
733
734
735
736
737
738
739
740
741
742
743
744
745
746
747
748
749
750
751
752
753
754
755
756
757
758
759
760
761
762
763
764
765
766
767
768
769
770
771
772
773
774
775
776
777
778
779
780
781
782
783
784
785
786
787
788
789
790
791
792
793
794
795
796
797
798
799
800
801
802
803
804
805
806
807
808
809
810
811
812
813
814
815
816
817
818
819
820
821
822
823
824
825
826
827
828
829
830
831
832
833
834
835
836
837
838
839
840
841
842
843
844
845
846
847
848
849
850
851
852
853
854
855
856
857
858
859
860
861
862
863
864
865
866
867
868
869
870
871
872
873
874
875
876
877
878
879
880
881
882
883
884
885
886
887
888
889
890
891
892
893
894
895
896
897
898
899
900
901
902
903
904
905
906
907
908
909
910
911
912
913
914
915
916
917
918
919
920
921
922
923
924
925
926
927
928
929
930
931
932
933
934
935
936
937
938
939
940
941
942
943
944
945
946
947
948
949
950
951
952
953
954
955
956
957
958
959
960
961
962
963
964
965
966
967
968
969
970
971
972
973
974
975
976
977
978
979
980
981
982
983
984
985
986
987
988
989
990
991
992
993
994
995
996
997
998
999
1000
(* ::Package:: *)
StressEnergyTensor[] := {"Symmetric", "SymmetricField", "Asymmetric", "AsymmetricField", "PerfectFluid",
"PerfectFluidField", "Dust", "DustField", "Radiation", "RadiationField", "ElectromagneticField", "MassiveScalarField"}
StressEnergyTensor[dimensionCount_Integer] := StressEnergyTensor[ResourceFunction["MetricTensor"][
DiagonalMatrix[Join[{-1}, ConstantArray[1, dimensionCount - 1]]],
Join[{"\[FormalT]"}, (Superscript["\[FormalX]", ToString[#1]] & ) /@ Range[dimensionCount - 1]], True, True],
Normal[SparseArray[(#1 -> Superscript["\[FormalCapitalT]", Sort[#1]] & ) /@ Tuples[Range[dimensionCount], 2]]], False, False]
StressEnergyTensor[dimensionCount_Integer, (metricTensor_)[matrixRepresentation_List, coordinates_List, metricIndex1_,
metricIndex2_]] := StressEnergyTensor[ResourceFunction["MetricTensor"][matrixRepresentation, coordinates,
metricIndex1, metricIndex2], Normal[SparseArray[(#1 -> Superscript["\[FormalCapitalT]", Sort[#1]] & ) /@
Tuples[Range[dimensionCount], 2]]], False, False] /; SymbolName[metricTensor] === "MetricTensor" &&
Length[Dimensions[matrixRepresentation]] == 2 && Length[coordinates] == Length[matrixRepresentation] &&
BooleanQ[metricIndex1] && BooleanQ[metricIndex2] && Length[matrixRepresentation] == dimensionCount
StressEnergyTensor[dimensionCount_Integer, index1_, index2_] :=
StressEnergyTensor[ResourceFunction["MetricTensor"][DiagonalMatrix[Join[{-1}, ConstantArray[1, dimensionCount - 1]]],
Join[{"\[FormalT]"}, (Superscript["\[FormalX]", ToString[#1]] & ) /@ Range[dimensionCount - 1]], True, True],
Normal[SparseArray[(#1 -> Superscript["\[FormalCapitalT]", Sort[#1]] & ) /@ Tuples[Range[dimensionCount], 2]]], index1, index2] /;
BooleanQ[index1] && BooleanQ[index2]
StressEnergyTensor[dimensionCount_Integer, (metricTensor_)[matrixRepresentation_List, coordinates_List, metricIndex1_,
metricIndex2_], index1_, index2_] := StressEnergyTensor[ResourceFunction["MetricTensor"][matrixRepresentation,
coordinates, metricIndex1, metricIndex2], Normal[SparseArray[(#1 -> Superscript["\[FormalCapitalT]", Sort[#1]] & ) /@
Tuples[Range[dimensionCount], 2]]], index1, index2] /; SymbolName[metricTensor] === "MetricTensor" &&
Length[Dimensions[matrixRepresentation]] == 2 && Length[coordinates] == Length[matrixRepresentation] &&
BooleanQ[metricIndex1] && BooleanQ[metricIndex2] && BooleanQ[index1] && BooleanQ[index2] &&
Length[matrixRepresentation] == dimensionCount
StressEnergyTensor["Symmetric"] := StressEnergyTensor[ResourceFunction["MetricTensor"][
DiagonalMatrix[Join[{-1}, ConstantArray[1, 3]]], Join[{"\[FormalT]"}, (Superscript["\[FormalX]", ToString[#1]] & ) /@ Range[3]],
True, True], Normal[SparseArray[(#1 -> Superscript["\[FormalCapitalT]", Sort[#1]] & ) /@ Tuples[Range[4], 2]]], False, False]
StressEnergyTensor["Symmetric", (metricTensor_)[matrixRepresentation_List, coordinates_List, metricIndex1_,
metricIndex2_]] := StressEnergyTensor[ResourceFunction["MetricTensor"][matrixRepresentation, coordinates,
metricIndex1, metricIndex2], Normal[SparseArray[(#1 -> Superscript["\[FormalCapitalT]", Sort[#1]] & ) /@ Tuples[Range[4], 2]]],
False, False] /; SymbolName[metricTensor] === "MetricTensor" && Length[Dimensions[matrixRepresentation]] == 2 &&
Length[coordinates] == Length[matrixRepresentation] && BooleanQ[metricIndex1] && BooleanQ[metricIndex2] &&
Length[matrixRepresentation] == 4
StressEnergyTensor["Symmetric", index1_, index2_] :=
StressEnergyTensor[ResourceFunction["MetricTensor"][DiagonalMatrix[Join[{-1}, ConstantArray[1, 3]]],
Join[{"\[FormalT]"}, (Superscript["\[FormalX]", ToString[#1]] & ) /@ Range[3]], True, True],
Normal[SparseArray[(#1 -> Superscript["\[FormalCapitalT]", Sort[#1]] & ) /@ Tuples[Range[4], 2]]], index1, index2] /;
BooleanQ[index1] && BooleanQ[index2]
StressEnergyTensor["Symmetric", (metricTensor_)[matrixRepresentation_List, coordinates_List, metricIndex1_,
metricIndex2_], index1_, index2_] := StressEnergyTensor[ResourceFunction["MetricTensor"][matrixRepresentation,
coordinates, metricIndex1, metricIndex2], Normal[SparseArray[(#1 -> Superscript["\[FormalCapitalT]", Sort[#1]] & ) /@
Tuples[Range[4], 2]]], index1, index2] /; SymbolName[metricTensor] === "MetricTensor" &&
Length[Dimensions[matrixRepresentation]] == 2 && Length[coordinates] == Length[matrixRepresentation] &&
BooleanQ[metricIndex1] && BooleanQ[metricIndex2] && BooleanQ[index1] && BooleanQ[index2] &&
Length[matrixRepresentation] == 4
StressEnergyTensor[{"Symmetric", dimensionCount_Integer}] :=
StressEnergyTensor[ResourceFunction["MetricTensor"][DiagonalMatrix[Join[{-1}, ConstantArray[1, dimensionCount - 1]]],
Join[{"\[FormalT]"}, (Superscript["\[FormalX]", ToString[#1]] & ) /@ Range[dimensionCount - 1]], True, True],
Normal[SparseArray[(#1 -> Superscript["\[FormalCapitalT]", Sort[#1]] & ) /@ Tuples[Range[dimensionCount], 2]]], False, False]
StressEnergyTensor[{"Symmetric", dimensionCount_Integer}, (metricTensor_)[matrixRepresentation_List, coordinates_List,
metricIndex1_, metricIndex2_]] := StressEnergyTensor[ResourceFunction["MetricTensor"][matrixRepresentation,
coordinates, metricIndex1, metricIndex2], Normal[SparseArray[(#1 -> Superscript["\[FormalCapitalT]", Sort[#1]] & ) /@
Tuples[Range[dimensionCount], 2]]], False, False] /; SymbolName[metricTensor] === "MetricTensor" &&
Length[Dimensions[matrixRepresentation]] == 2 && Length[coordinates] == Length[matrixRepresentation] &&
BooleanQ[metricIndex1] && BooleanQ[metricIndex2] && Length[matrixRepresentation] == dimensionCount
StressEnergyTensor[{"Symmetric", dimensionCount_Integer}, index1_, index2_] :=
StressEnergyTensor[ResourceFunction["MetricTensor"][DiagonalMatrix[Join[{-1}, ConstantArray[1, dimensionCount - 1]]],
Join[{"\[FormalT]"}, (Superscript["\[FormalX]", ToString[#1]] & ) /@ Range[dimensionCount - 1]], True, True],
Normal[SparseArray[(#1 -> Superscript["\[FormalCapitalT]", Sort[#1]] & ) /@ Tuples[Range[dimensionCount], 2]]], index1, index2] /;
BooleanQ[index1] && BooleanQ[index2]
StressEnergyTensor[{"Symmetric", dimensionCount_Integer}, (metricTensor_)[matrixRepresentation_List, coordinates_List,
metricIndex1_, metricIndex2_], index1_, index2_] :=
StressEnergyTensor[ResourceFunction["MetricTensor"][matrixRepresentation, coordinates, metricIndex1, metricIndex2],
Normal[SparseArray[(#1 -> Superscript["\[FormalCapitalT]", Sort[#1]] & ) /@ Tuples[Range[dimensionCount], 2]]], index1, index2] /;
SymbolName[metricTensor] === "MetricTensor" && Length[Dimensions[matrixRepresentation]] == 2 &&
Length[coordinates] == Length[matrixRepresentation] && BooleanQ[metricIndex1] && BooleanQ[metricIndex2] &&
BooleanQ[index1] && BooleanQ[index2] && Length[matrixRepresentation] == dimensionCount
StressEnergyTensor["SymmetricField"] := StressEnergyTensor[ResourceFunction["MetricTensor"][
DiagonalMatrix[Join[{-1}, ConstantArray[1, 3]]], Join[{"\[FormalT]"}, (Superscript["\[FormalX]", ToString[#1]] & ) /@ Range[3]],
True, True],
Normal[SparseArray[(Module[{index = #1}, index -> Superscript["\[FormalCapitalT]", Sort[index]] @@
Join[{"\[FormalT]"}, (Superscript["\[FormalX]", ToString[#1]] & ) /@ Range[3]]] & ) /@ Tuples[Range[4], 2]]], False, False]
StressEnergyTensor["SymmetricField", (metricTensor_)[matrixRepresentation_List, coordinates_List, metricIndex1_,
metricIndex2_]] := StressEnergyTensor[ResourceFunction["MetricTensor"][matrixRepresentation, coordinates,
metricIndex1, metricIndex2],
Normal[SparseArray[(Module[{index = #1}, index -> Superscript["\[FormalCapitalT]", Sort[index]] @@ coordinates] & ) /@
Tuples[Range[4], 2]]], False, False] /; SymbolName[metricTensor] === "MetricTensor" &&
Length[Dimensions[matrixRepresentation]] == 2 && Length[coordinates] == Length[matrixRepresentation] &&
BooleanQ[metricIndex1] && BooleanQ[metricIndex2] && Length[matrixRepresentation] == 4
StressEnergyTensor["SymmetricField", index1_, index2_] :=
StressEnergyTensor[ResourceFunction["MetricTensor"][DiagonalMatrix[Join[{-1}, ConstantArray[1, 3]]],
Join[{"\[FormalT]"}, (Superscript["\[FormalX]", ToString[#1]] & ) /@ Range[3]], True, True],
Normal[SparseArray[(Module[{index = #1}, index -> Superscript["\[FormalCapitalT]", Sort[index]] @@
Join[{"\[FormalT]"}, (Superscript["\[FormalX]", ToString[#1]] & ) /@ Range[3]]] & ) /@ Tuples[Range[4], 2]]], index1,
index2] /; BooleanQ[index1] && BooleanQ[index2]
StressEnergyTensor["SymmetricField", (metricTensor_)[matrixRepresentation_List, coordinates_List, metricIndex1_,
metricIndex2_], index1_, index2_] := StressEnergyTensor[ResourceFunction["MetricTensor"][matrixRepresentation,
coordinates, metricIndex1, metricIndex2],
Normal[SparseArray[(Module[{index = #1}, index -> Superscript["\[FormalCapitalT]", Sort[index]] @@ coordinates] & ) /@
Tuples[Range[4], 2]]], index1, index2] /; SymbolName[metricTensor] === "MetricTensor" &&
Length[Dimensions[matrixRepresentation]] == 2 && Length[coordinates] == Length[matrixRepresentation] &&
BooleanQ[metricIndex1] && BooleanQ[metricIndex2] && BooleanQ[index1] && BooleanQ[index2] &&
Length[matrixRepresentation] == 4
StressEnergyTensor[{"SymmetricField", dimensionCount_Integer}] :=
StressEnergyTensor[ResourceFunction["MetricTensor"][DiagonalMatrix[Join[{-1}, ConstantArray[1, dimensionCount - 1]]],
Join[{"\[FormalT]"}, (Superscript["\[FormalX]", ToString[#1]] & ) /@ Range[dimensionCount - 1]], True, True],
Normal[SparseArray[(Module[{index = #1}, index -> Superscript["\[FormalCapitalT]", Sort[index]] @@
Join[{"\[FormalT]"}, (Superscript["\[FormalX]", ToString[#1]] & ) /@ Range[dimensionCount - 1]]] & ) /@
Tuples[Range[dimensionCount], 2]]], False, False]
StressEnergyTensor[{"SymmetricField", dimensionCount_Integer}, (metricTensor_)[matrixRepresentation_List,
coordinates_List, metricIndex1_, metricIndex2_]] :=
StressEnergyTensor[ResourceFunction["MetricTensor"][matrixRepresentation, coordinates, metricIndex1, metricIndex2],
Normal[SparseArray[(Module[{index = #1}, index -> Superscript["\[FormalCapitalT]", Sort[index]] @@ coordinates] & ) /@
Tuples[Range[dimensionCount], 2]]], False, False] /; SymbolName[metricTensor] === "MetricTensor" &&
Length[Dimensions[matrixRepresentation]] == 2 && Length[coordinates] == Length[matrixRepresentation] &&
BooleanQ[metricIndex1] && BooleanQ[metricIndex2] && Length[matrixRepresentation] == dimensionCount
StressEnergyTensor[{"SymmetricField", dimensionCount_Integer}, index1_, index2_] :=
StressEnergyTensor[ResourceFunction["MetricTensor"][DiagonalMatrix[Join[{-1}, ConstantArray[1, dimensionCount - 1]]],
Join[{"\[FormalT]"}, (Superscript["\[FormalX]", ToString[#1]] & ) /@ Range[dimensionCount - 1]], True, True],
Normal[SparseArray[(Module[{index = #1}, index -> Superscript["\[FormalCapitalT]", Sort[index]] @@
Join[{"\[FormalT]"}, (Superscript["\[FormalX]", ToString[#1]] & ) /@ Range[dimensionCount - 1]]] & ) /@
Tuples[Range[dimensionCount], 2]]], index1, index2] /; BooleanQ[index1] && BooleanQ[index2]
StressEnergyTensor[{"SymmetricField", dimensionCount_Integer}, (metricTensor_)[matrixRepresentation_List,
coordinates_List, metricIndex1_, metricIndex2_], index1_, index2_] :=
StressEnergyTensor[ResourceFunction["MetricTensor"][matrixRepresentation, coordinates, metricIndex1, metricIndex2],
Normal[SparseArray[(Module[{index = #1}, index -> Superscript["\[FormalCapitalT]", Sort[index]] @@ coordinates] & ) /@
Tuples[Range[dimensionCount], 2]]], index1, index2] /; SymbolName[metricTensor] === "MetricTensor" &&
Length[Dimensions[matrixRepresentation]] == 2 && Length[coordinates] == Length[matrixRepresentation] &&
BooleanQ[metricIndex1] && BooleanQ[metricIndex2] && BooleanQ[index1] && BooleanQ[index2] &&
Length[matrixRepresentation] == dimensionCount
StressEnergyTensor["Asymmetric"] := StressEnergyTensor[ResourceFunction["MetricTensor"][
DiagonalMatrix[Join[{-1}, ConstantArray[1, 3]]], Join[{"\[FormalT]"}, (Superscript["\[FormalX]", ToString[#1]] & ) /@ Range[3]],
True, True], Normal[SparseArray[(#1 -> Superscript["\[FormalCapitalT]", #1] & ) /@ Tuples[Range[4], 2]]], False, False]
StressEnergyTensor["Asymmetric", (metricTensor_)[matrixRepresentation_List, coordinates_List, metricIndex1_,
metricIndex2_]] := StressEnergyTensor[ResourceFunction["MetricTensor"][matrixRepresentation, coordinates,
metricIndex1, metricIndex2], Normal[SparseArray[(#1 -> Superscript["\[FormalCapitalT]", #1] & ) /@ Tuples[Range[4], 2]]], False,
False] /; SymbolName[metricTensor] === "MetricTensor" && Length[Dimensions[matrixRepresentation]] == 2 &&
Length[coordinates] == Length[matrixRepresentation] && BooleanQ[metricIndex1] && BooleanQ[metricIndex2] &&
Length[matrixRepresentation] == 4
StressEnergyTensor["Asymmetric", index1_, index2_] :=
StressEnergyTensor[ResourceFunction["MetricTensor"][DiagonalMatrix[Join[{-1}, ConstantArray[1, 3]]],
Join[{"\[FormalT]"}, (Superscript["\[FormalX]", ToString[#1]] & ) /@ Range[3]], True, True],
Normal[SparseArray[(#1 -> Superscript["\[FormalCapitalT]", #1] & ) /@ Tuples[Range[4], 2]]], index1, index2] /;
BooleanQ[index1] && BooleanQ[index2]
StressEnergyTensor["Asymmetric", (metricTensor_)[matrixRepresentation_List, coordinates_List, metricIndex1_,
metricIndex2_], index1_, index2_] := StressEnergyTensor[ResourceFunction["MetricTensor"][matrixRepresentation,
coordinates, metricIndex1, metricIndex2], Normal[SparseArray[(#1 -> Superscript["\[FormalCapitalT]", #1] & ) /@
Tuples[Range[4], 2]]], index1, index2] /; SymbolName[metricTensor] === "MetricTensor" &&
Length[Dimensions[matrixRepresentation]] == 2 && Length[coordinates] == Length[matrixRepresentation] &&
BooleanQ[metricIndex1] && BooleanQ[metricIndex2] && BooleanQ[index1] && BooleanQ[index2] &&
Length[matrixRepresentation] == 4
StressEnergyTensor[{"Asymmetric", dimensionCount_Integer}] :=
StressEnergyTensor[ResourceFunction["MetricTensor"][DiagonalMatrix[Join[{-1}, ConstantArray[1, dimensionCount - 1]]],
Join[{"\[FormalT]"}, (Superscript["\[FormalX]", ToString[#1]] & ) /@ Range[dimensionCount - 1]], True, True],
Normal[SparseArray[(#1 -> Superscript["\[FormalCapitalT]", #1] & ) /@ Tuples[Range[dimensionCount], 2]]], False, False]
StressEnergyTensor[{"Asymmetric", dimensionCount_Integer}, (metricTensor_)[matrixRepresentation_List, coordinates_List,
metricIndex1_, metricIndex2_]] := StressEnergyTensor[ResourceFunction["MetricTensor"][matrixRepresentation,
coordinates, metricIndex1, metricIndex2], Normal[SparseArray[(#1 -> Superscript["\[FormalCapitalT]", #1] & ) /@
Tuples[Range[dimensionCount], 2]]], False, False] /; SymbolName[metricTensor] === "MetricTensor" &&
Length[Dimensions[matrixRepresentation]] == 2 && Length[coordinates] == Length[matrixRepresentation] &&
BooleanQ[metricIndex1] && BooleanQ[metricIndex2] && Length[matrixRepresentation] == dimensionCount
StressEnergyTensor[{"Asymmetric", dimensionCount_Integer}, index1_, index2_] :=
StressEnergyTensor[ResourceFunction["MetricTensor"][DiagonalMatrix[Join[{-1}, ConstantArray[1, dimensionCount - 1]]],
Join[{"\[FormalT]"}, (Superscript["\[FormalX]", ToString[#1]] & ) /@ Range[dimensionCount - 1]], True, True],
Normal[SparseArray[(#1 -> Superscript["\[FormalCapitalT]", #1] & ) /@ Tuples[Range[dimensionCount], 2]]], index1, index2] /;
BooleanQ[index1] && BooleanQ[index2]
StressEnergyTensor[{"Asymmetric", dimensionCount_Integer}, (metricTensor_)[matrixRepresentation_List, coordinates_List,
metricIndex1_, metricIndex2_], index1_, index2_] :=
StressEnergyTensor[ResourceFunction["MetricTensor"][matrixRepresentation, coordinates, metricIndex1, metricIndex2],
Normal[SparseArray[(#1 -> Superscript["\[FormalCapitalT]", #1] & ) /@ Tuples[Range[dimensionCount], 2]]], index1, index2] /;
SymbolName[metricTensor] === "MetricTensor" && Length[Dimensions[matrixRepresentation]] == 2 &&
Length[coordinates] == Length[matrixRepresentation] && BooleanQ[metricIndex1] && BooleanQ[metricIndex2] &&
BooleanQ[index1] && BooleanQ[index2] && Length[matrixRepresentation] == dimensionCount
StressEnergyTensor["AsymmetricField"] := StressEnergyTensor[ResourceFunction["MetricTensor"][
DiagonalMatrix[Join[{-1}, ConstantArray[1, 3]]], Join[{"\[FormalT]"}, (Superscript["\[FormalX]", ToString[#1]] & ) /@ Range[3]],
True, True],
Normal[SparseArray[(Module[{index = #1}, index -> Superscript["\[FormalCapitalT]", index] @@
Join[{"\[FormalT]"}, (Superscript["\[FormalX]", ToString[#1]] & ) /@ Range[3]]] & ) /@ Tuples[Range[4], 2]]], False, False]
StressEnergyTensor["AsymmetricField", (metricTensor_)[matrixRepresentation_List, coordinates_List, metricIndex1_,
metricIndex2_]] := StressEnergyTensor[ResourceFunction["MetricTensor"][matrixRepresentation, coordinates,
metricIndex1, metricIndex2],
Normal[SparseArray[(Module[{index = #1}, index -> Superscript["\[FormalCapitalT]", index] @@ coordinates] & ) /@
Tuples[Range[4], 2]]], False, False] /; SymbolName[metricTensor] === "MetricTensor" &&
Length[Dimensions[matrixRepresentation]] == 2 && Length[coordinates] == Length[matrixRepresentation] &&
BooleanQ[metricIndex1] && BooleanQ[metricIndex2] && Length[matrixRepresentation] == 4
StressEnergyTensor["AsymmetricField", index1_, index2_] :=
StressEnergyTensor[ResourceFunction["MetricTensor"][DiagonalMatrix[Join[{-1}, ConstantArray[1, 3]]],
Join[{"\[FormalT]"}, (Superscript["\[FormalX]", ToString[#1]] & ) /@ Range[3]], True, True],
Normal[SparseArray[(Module[{index = #1}, index -> Superscript["\[FormalCapitalT]", index] @@
Join[{"\[FormalT]"}, (Superscript["\[FormalX]", ToString[#1]] & ) /@ Range[3]]] & ) /@ Tuples[Range[4], 2]]], index1,
index2] /; BooleanQ[index1] && BooleanQ[index2]
StressEnergyTensor["AsymmetricField", (metricTensor_)[matrixRepresentation_List, coordinates_List, metricIndex1_,
metricIndex2_], index1_, index2_] := StressEnergyTensor[ResourceFunction["MetricTensor"][matrixRepresentation,
coordinates, metricIndex1, metricIndex2],
Normal[SparseArray[(Module[{index = #1}, index -> Superscript["\[FormalCapitalT]", index] @@ coordinates] & ) /@
Tuples[Range[4], 2]]], index1, index2] /; SymbolName[metricTensor] === "MetricTensor" &&
Length[Dimensions[matrixRepresentation]] == 2 && Length[coordinates] == Length[matrixRepresentation] &&
BooleanQ[metricIndex1] && BooleanQ[metricIndex2] && BooleanQ[index1] && BooleanQ[index2] &&
Length[matrixRepresentation] == 4
StressEnergyTensor[{"AsymmetricField", dimensionCount_Integer}] :=
StressEnergyTensor[ResourceFunction["MetricTensor"][DiagonalMatrix[Join[{-1}, ConstantArray[1, dimensionCount - 1]]],
Join[{"\[FormalT]"}, (Superscript["\[FormalX]", ToString[#1]] & ) /@ Range[dimensionCount - 1]], True, True],
Normal[SparseArray[(Module[{index = #1}, index -> Superscript["\[FormalCapitalT]", index] @@
Join[{"\[FormalT]"}, (Superscript["\[FormalX]", ToString[#1]] & ) /@ Range[dimensionCount - 1]]] & ) /@
Tuples[Range[dimensionCount], 2]]], False, False]
StressEnergyTensor[{"AsymmetricField", dimensionCount_Integer}, (metricTensor_)[matrixRepresentation_List,
coordinates_List, metricIndex1_, metricIndex2_]] :=
StressEnergyTensor[ResourceFunction["MetricTensor"][matrixRepresentation, coordinates, metricIndex1, metricIndex2],
Normal[SparseArray[(Module[{index = #1}, index -> Superscript["\[FormalCapitalT]", index] @@ coordinates] & ) /@
Tuples[Range[dimensionCount], 2]]], False, False] /; SymbolName[metricTensor] === "MetricTensor" &&
Length[Dimensions[matrixRepresentation]] == 2 && Length[coordinates] == Length[matrixRepresentation] &&
BooleanQ[metricIndex1] && BooleanQ[metricIndex2] && Length[matrixRepresentation] == dimensionCount
StressEnergyTensor[{"AsymmetricField", dimensionCount_Integer}, index1_, index2_] :=
StressEnergyTensor[ResourceFunction["MetricTensor"][DiagonalMatrix[Join[{-1}, ConstantArray[1, dimensionCount - 1]]],
Join[{"\[FormalT]"}, (Superscript["\[FormalX]", ToString[#1]] & ) /@ Range[dimensionCount - 1]], True, True],
Normal[SparseArray[(Module[{index = #1}, index -> Superscript["\[FormalCapitalT]", index] @@
Join[{"\[FormalT]"}, (Superscript["\[FormalX]", ToString[#1]] & ) /@ Range[dimensionCount - 1]]] & ) /@
Tuples[Range[dimensionCount], 2]]], index1, index2] /; BooleanQ[index1] && BooleanQ[index2]
StressEnergyTensor[{"AsymmetricField", dimensionCount_Integer}, (metricTensor_)[matrixRepresentation_List,
coordinates_List, metricIndex1_, metricIndex2_], index1_, index2_] :=
StressEnergyTensor[ResourceFunction["MetricTensor"][matrixRepresentation, coordinates, metricIndex1, metricIndex2],
Normal[SparseArray[(Module[{index = #1}, index -> Superscript["\[FormalCapitalT]", index] @@ coordinates] & ) /@
Tuples[Range[dimensionCount], 2]]], index1, index2] /; SymbolName[metricTensor] === "MetricTensor" &&
Length[Dimensions[matrixRepresentation]] == 2 && Length[coordinates] == Length[matrixRepresentation] &&
BooleanQ[metricIndex1] && BooleanQ[metricIndex2] && BooleanQ[index1] && BooleanQ[index2] &&
Length[matrixRepresentation] == dimensionCount
StressEnergyTensor["PerfectFluid"] := StressEnergyTensor[ResourceFunction["MetricTensor"][
DiagonalMatrix[Join[{-1}, ConstantArray[1, 3]]], Join[{"\[FormalT]"}, (Superscript["\[FormalX]", ToString[#1]] & ) /@ Range[3]],
True, True],
Normal[SparseArray[(#1 -> ("\[FormalRho]" + "\[FormalCapitalP]")*Superscript["\[FormalU]", ToString[First[#1]]]*Superscript["\[FormalU]",
ToString[Last[#1]]] + "\[FormalCapitalP]"*Inverse[DiagonalMatrix[Join[{-1}, ConstantArray[1, 3]]]][[First[#1],
Last[#1]]] & ) /@ Tuples[Range[4], 2]]], False, False]
StressEnergyTensor["PerfectFluid", (metricTensor_)[matrixRepresentation_List, coordinates_List, metricIndex1_,
metricIndex2_]] := StressEnergyTensor[ResourceFunction["MetricTensor"][matrixRepresentation, coordinates,
metricIndex1, metricIndex2],
Normal[SparseArray[(#1 -> ("\[FormalRho]" + "\[FormalCapitalP]")*Superscript["\[FormalU]", ToString[First[#1]]]*Superscript["\[FormalU]",
ToString[Last[#1]]] + "\[FormalCapitalP]"*Inverse[matrixRepresentation][[First[#1],Last[#1]]] & ) /@
Tuples[Range[Length[matrixRepresentation]], 2]]], False, False] /; SymbolName[metricTensor] === "MetricTensor" &&
Length[Dimensions[matrixRepresentation]] == 2 && Length[coordinates] == Length[matrixRepresentation] &&
BooleanQ[metricIndex1] && BooleanQ[metricIndex2]
StressEnergyTensor["PerfectFluid", index1_, index2_] :=
StressEnergyTensor[ResourceFunction["MetricTensor"][DiagonalMatrix[Join[{-1}, ConstantArray[1, 3]]],
Join[{"\[FormalT]"}, (Superscript["\[FormalX]", ToString[#1]] & ) /@ Range[3]], True, True],
Normal[SparseArray[(#1 -> ("\[FormalRho]" + "\[FormalCapitalP]")*Superscript["\[FormalU]", ToString[First[#1]]]*Superscript["\[FormalU]",
ToString[Last[#1]]] + "\[FormalCapitalP]"*Inverse[DiagonalMatrix[Join[{-1}, ConstantArray[1, 3]]]][[First[#1],
Last[#1]]] & ) /@ Tuples[Range[4], 2]]], index1, index2] /; BooleanQ[index1] && BooleanQ[index2]
StressEnergyTensor["PerfectFluid", (metricTensor_)[matrixRepresentation_List, coordinates_List, metricIndex1_,
metricIndex2_], index1_, index2_] := StressEnergyTensor[ResourceFunction["MetricTensor"][matrixRepresentation,
coordinates, metricIndex1, metricIndex1],
Normal[SparseArray[(#1 -> ("\[FormalRho]" + "\[FormalCapitalP]")*Superscript["\[FormalU]", ToString[First[#1]]]*Superscript["\[FormalU]",
ToString[Last[#1]]] + "\[FormalCapitalP]"*Inverse[matrixRepresentation][[First[#1],Last[#1]]] & ) /@
Tuples[Range[Length[matrixRepresentation]], 2]]], index1, index2] /;
SymbolName[metricTensor] === "MetricTensor" && Length[Dimensions[matrixRepresentation]] == 2 &&
Length[coordinates] == Length[matrixRepresentation] && BooleanQ[metricIndex1] && BooleanQ[metricIndex2] &&
BooleanQ[index1] && BooleanQ[index2]
StressEnergyTensor[{"PerfectFluid", fourVelocity_List}] :=
StressEnergyTensor[ResourceFunction["MetricTensor"][
DiagonalMatrix[Join[{-1}, ConstantArray[1, Length[fourVelocity] - 1]]],
Join[{"\[FormalT]"}, (Superscript["\[FormalX]", ToString[#1]] & ) /@ Range[Length[fourVelocity] - 1]], True, True],
Normal[SparseArray[(#1 -> ("\[FormalRho]" + "\[FormalCapitalP]")*fourVelocity[[First[#1]]]*fourVelocity[[Last[#1]]] +
"\[FormalCapitalP]"*Inverse[DiagonalMatrix[Join[{-1}, ConstantArray[1, Length[fourVelocity] - 1]]]][[First[#1],
Last[#1]]] & ) /@ Tuples[Range[Length[fourVelocity]], 2]]], False, False]
StressEnergyTensor[{"PerfectFluid", fourVelocity_List}, (metricTensor_)[matrixRepresentation_List, coordinates_List,
metricIndex1_, metricIndex2_]] := StressEnergyTensor[ResourceFunction["MetricTensor"][matrixRepresentation,
coordinates, metricIndex1, metricIndex2],
Normal[SparseArray[(#1 -> ("\[FormalRho]" + "\[FormalCapitalP]")*fourVelocity[[First[#1]]]*fourVelocity[[Last[#1]]] +
"\[FormalCapitalP]"*Inverse[matrixRepresentation][[First[#1],Last[#1]]] & ) /@ Tuples[Range[Length[matrixRepresentation]],
2]]], False, False] /; SymbolName[metricTensor] === "MetricTensor" && Length[Dimensions[matrixRepresentation]] ==
2 && Length[coordinates] == Length[matrixRepresentation] && BooleanQ[metricIndex1] && BooleanQ[metricIndex2] &&
Length[fourVelocity] == Length[matrixRepresentation]
StressEnergyTensor[{"PerfectFluid", fourVelocity_List}, index1_, index2_] :=
StressEnergyTensor[ResourceFunction["MetricTensor"][DiagonalMatrix[
Join[{-1}, ConstantArray[1, Length[fourVelocity] - 1]]], Join[{"\[FormalT]"}, (Superscript["\[FormalX]", ToString[#1]] & ) /@
Range[Length[fourVelocity] - 1]], True, True],
Normal[SparseArray[(#1 -> ("\[FormalRho]" + "\[FormalCapitalP]")*fourVelocity[[First[#1]]]*fourVelocity[[Last[#1]]] +
"\[FormalCapitalP]"*Inverse[DiagonalMatrix[Join[{-1}, ConstantArray[1, Length[fourVelocity] - 1]]]][[First[#1],
Last[#1]]] & ) /@ Tuples[Range[Length[fourVelocity]], 2]]], index1, index2] /;
BooleanQ[index1] && BooleanQ[index2]
StressEnergyTensor[{"PerfectFluid", fourVelocity_List}, (metricTensor_)[matrixRepresentation_List, coordinates_List,
metricIndex1_, metricIndex2_], index1_, index2_] :=
StressEnergyTensor[ResourceFunction["MetricTensor"][matrixRepresentation, coordinates, metricIndex1, metricIndex2],
Normal[SparseArray[(#1 -> ("\[FormalRho]" + "\[FormalCapitalP]")*fourVelocity[[First[#1]]]*fourVelocity[[Last[#1]]] +
"\[FormalCapitalP]"*Inverse[matrixRepresentation][[First[#1],Last[#1]]] & ) /@ Tuples[Range[Length[matrixRepresentation]],
2]]], index1, index2] /; SymbolName[metricTensor] === "MetricTensor" &&
Length[Dimensions[matrixRepresentation]] == 2 && Length[coordinates] == Length[matrixRepresentation] &&
BooleanQ[metricIndex1] && BooleanQ[metricIndex2] && BooleanQ[index1] && BooleanQ[index2] &&
Length[fourVelocity] == Length[matrixRepresentation]
StressEnergyTensor[{"PerfectFluid", fourVelocity_List, density_}] :=
StressEnergyTensor[ResourceFunction["MetricTensor"][
DiagonalMatrix[Join[{-1}, ConstantArray[1, Length[fourVelocity] - 1]]],
Join[{"\[FormalT]"}, (Superscript["\[FormalX]", ToString[#1]] & ) /@ Range[Length[fourVelocity] - 1]], True, True],
Normal[SparseArray[(#1 -> (density + "\[FormalCapitalP]")*fourVelocity[[First[#1]]]*fourVelocity[[Last[#1]]] +
"\[FormalCapitalP]"*Inverse[DiagonalMatrix[Join[{-1}, ConstantArray[1, Length[fourVelocity] - 1]]]][[First[#1],
Last[#1]]] & ) /@ Tuples[Range[Length[fourVelocity]], 2]]], False, False]
StressEnergyTensor[{"PerfectFluid", fourVelocity_List, density_}, (metricTensor_)[matrixRepresentation_List,
coordinates_List, metricIndex1_, metricIndex2_]] :=
StressEnergyTensor[ResourceFunction["MetricTensor"][matrixRepresentation, coordinates, metricIndex1, metricIndex2],
Normal[SparseArray[(#1 -> (density + "\[FormalCapitalP]")*fourVelocity[[First[#1]]]*fourVelocity[[Last[#1]]] +
"\[FormalCapitalP]"*Inverse[matrixRepresentation][[First[#1],Last[#1]]] & ) /@ Tuples[Range[Length[matrixRepresentation]],
2]]], False, False] /; SymbolName[metricTensor] === "MetricTensor" && Length[Dimensions[matrixRepresentation]] ==
2 && Length[coordinates] == Length[matrixRepresentation] && BooleanQ[metricIndex1] && BooleanQ[metricIndex2] &&
Length[fourVelocity] == Length[matrixRepresentation]
StressEnergyTensor[{"PerfectFluid", fourVelocity_List, density_}, index1_, index2_] :=
StressEnergyTensor[ResourceFunction["MetricTensor"][DiagonalMatrix[
Join[{-1}, ConstantArray[1, Length[fourVelocity] - 1]]], Join[{"\[FormalT]"}, (Superscript["\[FormalX]", ToString[#1]] & ) /@
Range[Length[fourVelocity] - 1]], True, True],
Normal[SparseArray[(#1 -> (density + "\[FormalCapitalP]")*fourVelocity[[First[#1]]]*fourVelocity[[Last[#1]]] +
"\[FormalCapitalP]"*Inverse[DiagonalMatrix[Join[{-1}, ConstantArray[1, Length[fourVelocity] - 1]]]][[First[#1],
Last[#1]]] & ) /@ Tuples[Range[Length[fourVelocity]], 2]]], index1, index2] /;
BooleanQ[index1] && BooleanQ[index2]
StressEnergyTensor[{"PerfectFluid", fourVelocity_List, density_}, (metricTensor_)[matrixRepresentation_List,
coordinates_List, metricIndex1_, metricIndex2_], index1_, index2_] :=
StressEnergyTensor[ResourceFunction["MetricTensor"][matrixRepresentation, coordinates, metricIndex1, metricIndex2],
Normal[SparseArray[(#1 -> (density + "\[FormalCapitalP]")*fourVelocity[[First[#1]]]*fourVelocity[[Last[#1]]] +
"\[FormalCapitalP]"*Inverse[matrixRepresentation][[First[#1],Last[#1]]] & ) /@ Tuples[Range[Length[matrixRepresentation]],
2]]], index1, index2] /; SymbolName[metricTensor] === "MetricTensor" &&
Length[Dimensions[matrixRepresentation]] == 2 && Length[coordinates] == Length[matrixRepresentation] &&
BooleanQ[metricIndex1] && BooleanQ[metricIndex2] && BooleanQ[index1] && BooleanQ[index2] &&
Length[fourVelocity] == Length[matrixRepresentation]
StressEnergyTensor[{"PerfectFluid", fourVelocity_List, density_, pressure_}] :=
StressEnergyTensor[ResourceFunction["MetricTensor"][
DiagonalMatrix[Join[{-1}, ConstantArray[1, Length[fourVelocity] - 1]]],
Join[{"\[FormalT]"}, (Superscript["\[FormalX]", ToString[#1]] & ) /@ Range[Length[fourVelocity] - 1]], True, True],
Normal[SparseArray[(#1 -> (density + pressure)*fourVelocity[[First[#1]]]*fourVelocity[[Last[#1]]] +
pressure*Inverse[DiagonalMatrix[Join[{-1}, ConstantArray[1, Length[fourVelocity] - 1]]]][[First[#1],
Last[#1]]] & ) /@ Tuples[Range[Length[fourVelocity]], 2]]], False, False]
StressEnergyTensor[{"PerfectFluid", fourVelocity_List, density_, pressure_},
(metricTensor_)[matrixRepresentation_List, coordinates_List, metricIndex1_, metricIndex2_]] :=
StressEnergyTensor[ResourceFunction["MetricTensor"][matrixRepresentation, coordinates, metricIndex1, metricIndex2],
Normal[SparseArray[(#1 -> (density + pressure)*fourVelocity[[First[#1]]]*fourVelocity[[Last[#1]]] +
pressure*Inverse[matrixRepresentation][[First[#1],Last[#1]]] & ) /@ Tuples[Range[Length[matrixRepresentation]],
2]]], False, False] /; SymbolName[metricTensor] === "MetricTensor" && Length[Dimensions[matrixRepresentation]] ==
2 && Length[coordinates] == Length[matrixRepresentation] && BooleanQ[metricIndex1] && BooleanQ[metricIndex2] &&
Length[fourVelocity] == Length[matrixRepresentation]
StressEnergyTensor[{"PerfectFluid", fourVelocity_List, density_, pressure_}, index1_, index2_] :=
StressEnergyTensor[ResourceFunction["MetricTensor"][DiagonalMatrix[
Join[{-1}, ConstantArray[1, Length[fourVelocity] - 1]]], Join[{"\[FormalT]"}, (Superscript["\[FormalX]", ToString[#1]] & ) /@
Range[Length[fourVelocity] - 1]], True, True],
Normal[SparseArray[(#1 -> (density + pressure)*fourVelocity[[First[#1]]]*fourVelocity[[Last[#1]]] +
pressure*Inverse[DiagonalMatrix[Join[{-1}, ConstantArray[1, Length[fourVelocity] - 1]]]][[First[#1],
Last[#1]]] & ) /@ Tuples[Range[Length[fourVelocity]], 2]]], index1, index2] /;
BooleanQ[index1] && BooleanQ[index2]
StressEnergyTensor[{"PerfectFluid", fourVelocity_List, density_, pressure_},
(metricTensor_)[matrixRepresentation_List, coordinates_List, metricIndex1_, metricIndex2_], index1_, index2_] :=
StressEnergyTensor[ResourceFunction["MetricTensor"][matrixRepresentation, coordinates, metricIndex1, metricIndex2],
Normal[SparseArray[(#1 -> (density + pressure)*fourVelocity[[First[#1]]]*fourVelocity[[Last[#1]]] +
pressure*Inverse[matrixRepresentation][[First[#1],Last[#1]]] & ) /@ Tuples[Range[Length[matrixRepresentation]],
2]]], index1, index2] /; SymbolName[metricTensor] === "MetricTensor" &&
Length[Dimensions[matrixRepresentation]] == 2 && Length[coordinates] == Length[matrixRepresentation] &&
BooleanQ[metricIndex1] && BooleanQ[metricIndex2] && BooleanQ[index1] && BooleanQ[index2] &&
Length[fourVelocity] == Length[matrixRepresentation]
StressEnergyTensor["PerfectFluidField"] := StressEnergyTensor[ResourceFunction["MetricTensor"][
DiagonalMatrix[Join[{-1}, ConstantArray[1, 3]]], Join[{"\[FormalT]"}, (Superscript["\[FormalX]", ToString[#1]] & ) /@ Range[3]],
True, True],
Normal[SparseArray[
(Module[{index = #1}, index -> ("\[FormalRho]" @@ Join[{"\[FormalT]"}, (Superscript["\[FormalX]", ToString[#1]] & ) /@ Range[3]] +
"\[FormalCapitalP]" @@ Join[{"\[FormalT]"}, (Superscript["\[FormalX]", ToString[#1]] & ) /@ Range[3]])*
Superscript["\[FormalU]", ToString[First[index]]] @@ Join[{"\[FormalT]"}, (Superscript["\[FormalX]", ToString[#1]] & ) /@ Range[3]]*
Superscript["\[FormalU]", ToString[Last[index]]] @@ Join[{"\[FormalT]"}, (Superscript["\[FormalX]", ToString[#1]] & ) /@
Range[3]] + "\[FormalCapitalP]" @@ Join[{"\[FormalT]"}, (Superscript["\[FormalX]", ToString[#1]] & ) /@ Range[3]]*
Inverse[DiagonalMatrix[Join[{-1}, ConstantArray[1, 3]]]][[First[index],Last[index]]]] & ) /@
Tuples[Range[4], 2]]], False, False]
StressEnergyTensor["PerfectFluidField", (metricTensor_)[matrixRepresentation_List, coordinates_List, metricIndex1_,
metricIndex2_]] := StressEnergyTensor[ResourceFunction["MetricTensor"][matrixRepresentation, coordinates,
metricIndex1, metricIndex2],
Normal[SparseArray[(Module[{index = #1}, index -> ("\[FormalRho]" @@ coordinates + "\[FormalCapitalP]" @@ coordinates)*
Superscript["\[FormalU]", ToString[First[index]]] @@ coordinates*Superscript["\[FormalU]", ToString[Last[index]]] @@
coordinates + "\[FormalCapitalP]" @@ coordinates*Inverse[matrixRepresentation][[First[index],Last[index]]]] & ) /@
Tuples[Range[Length[matrixRepresentation]], 2]]], False, False] /; SymbolName[metricTensor] === "MetricTensor" &&
Length[Dimensions[matrixRepresentation]] == 2 && Length[coordinates] == Length[matrixRepresentation] &&
BooleanQ[metricIndex1] && BooleanQ[metricIndex2]
StressEnergyTensor["PerfectFluidField", index1_, index2_] :=
StressEnergyTensor[ResourceFunction["MetricTensor"][DiagonalMatrix[Join[{-1}, ConstantArray[1, 3]]],
Join[{"\[FormalT]"}, (Superscript["\[FormalX]", ToString[#1]] & ) /@ Range[3]], True, True],
Normal[SparseArray[(Module[{index = #1}, index -> ("\[FormalRho]" @@ Join[{"\[FormalT]"}, (Superscript["\[FormalX]", ToString[#1]] & ) /@
Range[3]] + "\[FormalCapitalP]" @@ Join[{"\[FormalT]"}, (Superscript["\[FormalX]", ToString[#1]] & ) /@ Range[3]])*
Superscript["\[FormalU]", ToString[First[index]]] @@ Join[{"\[FormalT]"}, (Superscript["\[FormalX]", ToString[#1]] & ) /@ Range[
3]]*Superscript["\[FormalU]", ToString[Last[index]]] @@ Join[{"\[FormalT]"}, (Superscript["\[FormalX]", ToString[#1]] & ) /@
Range[3]] + "\[FormalCapitalP]" @@ Join[{"\[FormalT]"}, (Superscript["\[FormalX]", ToString[#1]] & ) /@ Range[3]]*
Inverse[DiagonalMatrix[Join[{-1}, ConstantArray[1, 3]]]][[First[index],Last[index]]]] & ) /@
Tuples[Range[4], 2]]], index1, index2] /; BooleanQ[index1] && BooleanQ[index2]
StressEnergyTensor["PerfectFluidField", (metricTensor_)[matrixRepresentation_List, coordinates_List, metricIndex1_,
metricIndex2_], index1_, index2_] := StressEnergyTensor[ResourceFunction["MetricTensor"][matrixRepresentation,
coordinates, metricIndex1, metricIndex2],
Normal[SparseArray[(Module[{index = #1}, index -> ("\[FormalRho]" @@ coordinates + "\[FormalCapitalP]" @@ coordinates)*
Superscript["\[FormalU]", ToString[First[index]]] @@ coordinates*Superscript["\[FormalU]", ToString[Last[index]]] @@
coordinates + "\[FormalCapitalP]" @@ coordinates*Inverse[matrixRepresentation][[First[index],Last[index]]]] & ) /@
Tuples[Range[Length[matrixRepresentation]], 2]]], index1, index2] /;
SymbolName[metricTensor] === "MetricTensor" && Length[Dimensions[matrixRepresentation]] == 2 &&
Length[coordinates] == Length[matrixRepresentation] && BooleanQ[metricIndex1] && BooleanQ[metricIndex2] &&
BooleanQ[index1] && BooleanQ[index2]
StressEnergyTensor[{"PerfectFluidField", fourVelocity_List}] :=
StressEnergyTensor[ResourceFunction["MetricTensor"][
DiagonalMatrix[Join[{-1}, ConstantArray[1, Length[fourVelocity] - 1]]],
Join[{"\[FormalT]"}, (Superscript["\[FormalX]", ToString[#1]] & ) /@ Range[Length[fourVelocity] - 1]], True, True],
Normal[SparseArray[
(Module[{index = #1}, index -> ("\[FormalRho]" @@ Join[{"\[FormalT]"}, (Superscript["\[FormalX]", ToString[#1]] & ) /@ Range[
Length[fourVelocity] - 1]] + "\[FormalCapitalP]" @@ Join[{"\[FormalT]"}, (Superscript["\[FormalX]", ToString[#1]] & ) /@ Range[
Length[fourVelocity] - 1]])*fourVelocity[[First[index]]] @@ Join[{"\[FormalT]"},
(Superscript["\[FormalX]", ToString[#1]] & ) /@ Range[Length[fourVelocity] - 1]]*fourVelocity[[Last[index]]] @@
Join[{"\[FormalT]"}, (Superscript["\[FormalX]", ToString[#1]] & ) /@ Range[Length[fourVelocity] - 1]] +
"\[FormalCapitalP]" @@ Join[{"\[FormalT]"}, (Superscript["\[FormalX]", ToString[#1]] & ) /@ Range[Length[fourVelocity] - 1]]*
Inverse[DiagonalMatrix[Join[{-1}, ConstantArray[1, Length[fourVelocity] - 1]]]][[First[index],
Last[index]]]] & ) /@ Tuples[Range[Length[fourVelocity]], 2]]], False, False]
StressEnergyTensor[{"PerfectFluidField", fourVelocity_List}, (metricTensor_)[matrixRepresentation_List, coordinates_List,
metricIndex1_, metricIndex2_]] := StressEnergyTensor[ResourceFunction["MetricTensor"][matrixRepresentation,
coordinates, metricIndex1, metricIndex2],
Normal[SparseArray[(Module[{index = #1}, index -> ("\[FormalRho]" @@ coordinates + "\[FormalCapitalP]" @@ coordinates)*
fourVelocity[[First[index]]] @@ coordinates*fourVelocity[[Last[index]]] @@ coordinates +
"\[FormalCapitalP]" @@ coordinates*Inverse[matrixRepresentation][[First[index],Last[index]]]] & ) /@
Tuples[Range[Length[matrixRepresentation]], 2]]], False, False] /; SymbolName[metricTensor] === "MetricTensor" &&
Length[Dimensions[matrixRepresentation]] == 2 && Length[coordinates] == Length[matrixRepresentation] &&
BooleanQ[metricIndex1] && BooleanQ[metricIndex2] && Length[fourVelocity] == Length[matrixRepresentation]
StressEnergyTensor[{"PerfectFluidField", fourVelocity_List}, index1_, index2_] :=
StressEnergyTensor[ResourceFunction["MetricTensor"][DiagonalMatrix[
Join[{-1}, ConstantArray[1, Length[fourVelocity] - 1]]], Join[{"\[FormalT]"}, (Superscript["\[FormalX]", ToString[#1]] & ) /@
Range[Length[fourVelocity] - 1]], True, True],
Normal[SparseArray[(Module[{index = #1}, index -> ("\[FormalRho]" @@ Join[{"\[FormalT]"}, (Superscript["\[FormalX]", ToString[#1]] & ) /@
Range[Length[fourVelocity] - 1]] + "\[FormalCapitalP]" @@ Join[{"\[FormalT]"}, (Superscript["\[FormalX]", ToString[#1]] & ) /@
Range[Length[fourVelocity] - 1]])*fourVelocity[[First[index]]] @@ Join[{"\[FormalT]"},
(Superscript["\[FormalX]", ToString[#1]] & ) /@ Range[Length[fourVelocity] - 1]]*fourVelocity[[Last[index]]] @@
Join[{"\[FormalT]"}, (Superscript["\[FormalX]", ToString[#1]] & ) /@ Range[Length[fourVelocity] - 1]] +
"\[FormalCapitalP]" @@ Join[{"\[FormalT]"}, (Superscript["\[FormalX]", ToString[#1]] & ) /@ Range[Length[fourVelocity] - 1]]*
Inverse[DiagonalMatrix[Join[{-1}, ConstantArray[1, Length[fourVelocity] - 1]]]][[First[index],
Last[index]]]] & ) /@ Tuples[Range[Length[fourVelocity]], 2]]], index1, index2] /;
BooleanQ[index1] && BooleanQ[index2]
StressEnergyTensor[{"PerfectFluidField", fourVelocity_List}, (metricTensor_)[matrixRepresentation_List, coordinates_List,
metricIndex1_, metricIndex2_], index1_, index2_] :=
StressEnergyTensor[ResourceFunction["MetricTensor"][matrixRepresentation, coordinates, metricIndex1, metricIndex2],
Normal[SparseArray[(Module[{index = #1}, index -> ("\[FormalRho]" @@ coordinates + "\[FormalCapitalP]" @@ coordinates)*
fourVelocity[[First[index]]] @@ coordinates*fourVelocity[[Last[index]]] @@ coordinates +
"\[FormalCapitalP]" @@ coordinates*Inverse[matrixRepresentation][[First[index],Last[index]]]] & ) /@
Tuples[Range[Length[matrixRepresentation]], 2]]], index1, index2] /;
SymbolName[metricTensor] === "MetricTensor" && Length[Dimensions[matrixRepresentation]] == 2 &&
Length[coordinates] == Length[matrixRepresentation] && BooleanQ[metricIndex1] && BooleanQ[metricIndex2] &&
BooleanQ[index1] && BooleanQ[index2] && Length[fourVelocity] == Length[matrixRepresentation]
StressEnergyTensor[{"PerfectFluidField", fourVelocity_List, density_}] :=
StressEnergyTensor[ResourceFunction["MetricTensor"][
DiagonalMatrix[Join[{-1}, ConstantArray[1, Length[fourVelocity] - 1]]],
Join[{"\[FormalT]"}, (Superscript["\[FormalX]", ToString[#1]] & ) /@ Range[Length[fourVelocity] - 1]], True, True],
Normal[SparseArray[
(Module[{index = #1}, index -> (density @@ Join[{"\[FormalT]"}, (Superscript["\[FormalX]", ToString[#1]] & ) /@ Range[
Length[fourVelocity] - 1]] + "\[FormalCapitalP]" @@ Join[{"\[FormalT]"}, (Superscript["\[FormalX]", ToString[#1]] & ) /@ Range[
Length[fourVelocity] - 1]])*fourVelocity[[First[index]]] @@ Join[{"\[FormalT]"},
(Superscript["\[FormalX]", ToString[#1]] & ) /@ Range[Length[fourVelocity] - 1]]*fourVelocity[[Last[index]]] @@
Join[{"\[FormalT]"}, (Superscript["\[FormalX]", ToString[#1]] & ) /@ Range[Length[fourVelocity] - 1]] +
"\[FormalCapitalP]" @@ Join[{"\[FormalT]"}, (Superscript["\[FormalX]", ToString[#1]] & ) /@ Range[Length[fourVelocity] - 1]]*
Inverse[DiagonalMatrix[Join[{-1}, ConstantArray[1, Length[fourVelocity] - 1]]]][[First[index],
Last[index]]]] & ) /@ Tuples[Range[Length[fourVelocity]], 2]]], False, False]
StressEnergyTensor[{"PerfectFluidField", fourVelocity_List, density_}, (metricTensor_)[matrixRepresentation_List,
coordinates_List, metricIndex1_, metricIndex2_]] :=
StressEnergyTensor[ResourceFunction["MetricTensor"][matrixRepresentation, coordinates, metricIndex1, metricIndex2],
Normal[SparseArray[(Module[{index = #1}, index -> (density @@ coordinates + "\[FormalCapitalP]" @@ coordinates)*
fourVelocity[[First[index]]] @@ coordinates*fourVelocity[[Last[index]]] @@ coordinates +
"\[FormalCapitalP]" @@ coordinates*Inverse[matrixRepresentation][[First[index],Last[index]]]] & ) /@
Tuples[Range[Length[matrixRepresentation]], 2]]], False, False] /; SymbolName[metricTensor] === "MetricTensor" &&
Length[Dimensions[matrixRepresentation]] == 2 && Length[coordinates] == Length[matrixRepresentation] &&
BooleanQ[metricIndex1] && BooleanQ[metricIndex2] && Length[fourVelocity] == Length[matrixRepresentation]
StressEnergyTensor[{"PerfectFluidField", fourVelocity_List, density_}, index1_, index2_] :=
StressEnergyTensor[ResourceFunction["MetricTensor"][DiagonalMatrix[
Join[{-1}, ConstantArray[1, Length[fourVelocity] - 1]]], Join[{"\[FormalT]"}, (Superscript["\[FormalX]", ToString[#1]] & ) /@
Range[Length[fourVelocity] - 1]], True, True],
Normal[SparseArray[(Module[{index = #1}, index -> (density @@ Join[{"\[FormalT]"}, (Superscript["\[FormalX]", ToString[#1]] & ) /@
Range[Length[fourVelocity] - 1]] + "\[FormalCapitalP]" @@ Join[{"\[FormalT]"}, (Superscript["\[FormalX]", ToString[#1]] & ) /@
Range[Length[fourVelocity] - 1]])*fourVelocity[[First[index]]] @@ Join[{"\[FormalT]"},
(Superscript["\[FormalX]", ToString[#1]] & ) /@ Range[Length[fourVelocity] - 1]]*fourVelocity[[Last[index]]] @@
Join[{"\[FormalT]"}, (Superscript["\[FormalX]", ToString[#1]] & ) /@ Range[Length[fourVelocity] - 1]] +
"\[FormalCapitalP]" @@ Join[{"\[FormalT]"}, (Superscript["\[FormalX]", ToString[#1]] & ) /@ Range[Length[fourVelocity] - 1]]*
Inverse[DiagonalMatrix[Join[{-1}, ConstantArray[1, Length[fourVelocity] - 1]]]][[First[index],
Last[index]]]] & ) /@ Tuples[Range[Length[fourVelocity]], 2]]], index1, index2] /;
BooleanQ[index1] && BooleanQ[index2]
StressEnergyTensor[{"PerfectFluidField", fourVelocity_List, density_}, (metricTensor_)[matrixRepresentation_List,
coordinates_List, metricIndex1_, metricIndex2_], index1_, index2_] :=
StressEnergyTensor[ResourceFunction["MetricTensor"][matrixRepresentation, coordinates, metricIndex1, metricIndex2],
Normal[SparseArray[(Module[{index = #1}, index -> (density @@ coordinates + "\[FormalCapitalP]" @@ coordinates)*
fourVelocity[[First[index]]] @@ coordinates*fourVelocity[[Last[index]]] @@ coordinates +
"\[FormalCapitalP]" @@ coordinates*Inverse[matrixRepresentation][[First[index],Last[index]]]] & ) /@
Tuples[Range[Length[matrixRepresentation]], 2]]], index1, index2] /;
SymbolName[metricTensor] === "MetricTensor" && Length[Dimensions[matrixRepresentation]] == 2 &&
Length[coordinates] == Length[matrixRepresentation] && BooleanQ[metricIndex1] && BooleanQ[metricIndex2] &&
BooleanQ[index1] && BooleanQ[index2] && Length[fourVelocity] == Length[matrixRepresentation]
StressEnergyTensor[{"PerfectFluidField", fourVelocity_List, density_, pressure_}] :=
StressEnergyTensor[ResourceFunction["MetricTensor"][
DiagonalMatrix[Join[{-1}, ConstantArray[1, Length[fourVelocity] - 1]]],
Join[{"\[FormalT]"}, (Superscript["\[FormalX]", ToString[#1]] & ) /@ Range[Length[fourVelocity] - 1]], True, True],
Normal[SparseArray[
(Module[{index = #1}, index -> (density @@ Join[{"\[FormalT]"}, (Superscript["\[FormalX]", ToString[#1]] & ) /@ Range[
Length[fourVelocity] - 1]] + pressure @@ Join[{"\[FormalT]"}, (Superscript["\[FormalX]", ToString[#1]] & ) /@ Range[
Length[fourVelocity] - 1]])*fourVelocity[[First[index]]] @@ Join[{"\[FormalT]"},
(Superscript["\[FormalX]", ToString[#1]] & ) /@ Range[Length[fourVelocity] - 1]]*fourVelocity[[Last[index]]] @@
Join[{"\[FormalT]"}, (Superscript["\[FormalX]", ToString[#1]] & ) /@ Range[Length[fourVelocity] - 1]] +
pressure @@ Join[{"\[FormalT]"}, (Superscript["\[FormalX]", ToString[#1]] & ) /@ Range[Length[fourVelocity] - 1]]*
Inverse[DiagonalMatrix[Join[{-1}, ConstantArray[1, Length[fourVelocity] - 1]]]][[First[index],
Last[index]]]] & ) /@ Tuples[Range[Length[fourVelocity]], 2]]], False, False]
StressEnergyTensor[{"PerfectFluidField", fourVelocity_List, density_, pressure_},
(metricTensor_)[matrixRepresentation_List, coordinates_List, metricIndex1_, metricIndex2_]] :=
StressEnergyTensor[ResourceFunction["MetricTensor"][matrixRepresentation, coordinates, metricIndex1, metricIndex2],
Normal[SparseArray[(Module[{index = #1}, index -> (density @@ coordinates + pressure @@ coordinates)*
fourVelocity[[First[index]]] @@ coordinates*fourVelocity[[Last[index]]] @@ coordinates +
pressure @@ coordinates*Inverse[matrixRepresentation][[First[index],Last[index]]]] & ) /@
Tuples[Range[Length[matrixRepresentation]], 2]]], False, False] /; SymbolName[metricTensor] === "MetricTensor" &&
Length[Dimensions[matrixRepresentation]] == 2 && Length[coordinates] == Length[matrixRepresentation] &&
BooleanQ[metricIndex1] && BooleanQ[metricIndex2] && Length[fourVelocity] == Length[matrixRepresentation]
StressEnergyTensor[{"PerfectFluidField", fourVelocity_List, density_, pressure_}, index1_, index2_] :=
StressEnergyTensor[ResourceFunction["MetricTensor"][DiagonalMatrix[
Join[{-1}, ConstantArray[1, Length[fourVelocity] - 1]]], Join[{"\[FormalT]"}, (Superscript["\[FormalX]", ToString[#1]] & ) /@
Range[Length[fourVelocity] - 1]], True, True],
Normal[SparseArray[(Module[{index = #1}, index -> (density @@ Join[{"\[FormalT]"}, (Superscript["\[FormalX]", ToString[#1]] & ) /@
Range[Length[fourVelocity] - 1]] + pressure @@ Join[{"\[FormalT]"}, (Superscript["\[FormalX]", ToString[#1]] & ) /@
Range[Length[fourVelocity] - 1]])*fourVelocity[[First[index]]] @@ Join[{"\[FormalT]"},
(Superscript["\[FormalX]", ToString[#1]] & ) /@ Range[Length[fourVelocity] - 1]]*fourVelocity[[Last[index]]] @@
Join[{"\[FormalT]"}, (Superscript["\[FormalX]", ToString[#1]] & ) /@ Range[Length[fourVelocity] - 1]] +
pressure @@ Join[{"\[FormalT]"}, (Superscript["\[FormalX]", ToString[#1]] & ) /@ Range[Length[fourVelocity] - 1]]*
Inverse[DiagonalMatrix[Join[{-1}, ConstantArray[1, Length[fourVelocity] - 1]]]][[First[index],
Last[index]]]] & ) /@ Tuples[Range[Length[fourVelocity]], 2]]], index1, index2] /;
BooleanQ[index1] && BooleanQ[index2]
StressEnergyTensor[{"PerfectFluidField", fourVelocity_List, density_, pressure_},
(metricTensor_)[matrixRepresentation_List, coordinates_List, metricIndex1_, metricIndex2_], index1_, index2_] :=
StressEnergyTensor[ResourceFunction["MetricTensor"][matrixRepresentation, coordinates, metricIndex1, metricIndex2],
Normal[SparseArray[(Module[{index = #1}, index -> (density @@ coordinates + pressure @@ coordinates)*
fourVelocity[[First[index]]] @@ coordinates*fourVelocity[[Last[index]]] @@ coordinates +
pressure @@ coordinates*Inverse[matrixRepresentation][[First[index],Last[index]]]] & ) /@
Tuples[Range[Length[matrixRepresentation]], 2]]], index1, index2] /;
SymbolName[metricTensor] === "MetricTensor" && Length[Dimensions[matrixRepresentation]] == 2 &&
Length[coordinates] == Length[matrixRepresentation] && BooleanQ[metricIndex1] && BooleanQ[metricIndex2] &&
BooleanQ[index1] && BooleanQ[index2] && Length[fourVelocity] == Length[matrixRepresentation]
StressEnergyTensor["Dust"] := StressEnergyTensor[ResourceFunction["MetricTensor"][
DiagonalMatrix[Join[{-1}, ConstantArray[1, 3]]], Join[{"\[FormalT]"}, (Superscript["\[FormalX]", ToString[#1]] & ) /@ Range[3]],
True, True], Normal[SparseArray[(#1 -> "\[FormalRho]"*Superscript["\[FormalU]", ToString[First[#1]]]*
Superscript["\[FormalU]", ToString[Last[#1]]] & ) /@ Tuples[Range[4], 2]]], False, False]
StressEnergyTensor["Dust", (metricTensor_)[matrixRepresentation_List, coordinates_List, metricIndex1_, metricIndex2_]] :=
StressEnergyTensor[ResourceFunction["MetricTensor"][matrixRepresentation, coordinates, metricIndex1, metricIndex2],
Normal[SparseArray[(#1 -> "\[FormalRho]"*Superscript["\[FormalU]", ToString[First[#1]]]*Superscript["\[FormalU]", ToString[Last[#1]]] & ) /@
Tuples[Range[Length[matrixRepresentation]], 2]]], False, False] /; SymbolName[metricTensor] === "MetricTensor" &&
Length[Dimensions[matrixRepresentation]] == 2 && Length[coordinates] == Length[matrixRepresentation] &&
BooleanQ[metricIndex1] && BooleanQ[metricIndex2]
StressEnergyTensor["Dust", index1_, index2_] :=
StressEnergyTensor[ResourceFunction["MetricTensor"][DiagonalMatrix[Join[{-1}, ConstantArray[1, 3]]],
Join[{"\[FormalT]"}, (Superscript["\[FormalX]", ToString[#1]] & ) /@ Range[3]], True, True],
Normal[SparseArray[(#1 -> "\[FormalRho]"*Superscript["\[FormalU]", ToString[First[#1]]]*Superscript["\[FormalU]", ToString[Last[#1]]] & ) /@
Tuples[Range[4], 2]]], index1, index2] /; BooleanQ[index1] && BooleanQ[index2]
StressEnergyTensor["Dust", (metricTensor_)[matrixRepresentation_List, coordinates_List, metricIndex1_, metricIndex2_],
index1_, index2_] := StressEnergyTensor[ResourceFunction["MetricTensor"][matrixRepresentation, coordinates,
metricIndex1, metricIndex2],
Normal[SparseArray[(#1 -> "\[FormalRho]"*Superscript["\[FormalU]", ToString[First[#1]]]*Superscript["\[FormalU]", ToString[Last[#1]]] & ) /@
Tuples[Range[Length[matrixRepresentation]], 2]]], index1, index2] /;
SymbolName[metricTensor] === "MetricTensor" && Length[Dimensions[matrixRepresentation]] == 2 &&
Length[coordinates] == Length[matrixRepresentation] && BooleanQ[metricIndex1] && BooleanQ[metricIndex2] &&
BooleanQ[index1] && BooleanQ[index2]
StressEnergyTensor[{"Dust", fourVelocity_List}] := StressEnergyTensor[ResourceFunction["MetricTensor"][
DiagonalMatrix[Join[{-1}, ConstantArray[1, Length[fourVelocity] - 1]]],
Join[{"\[FormalT]"}, (Superscript["\[FormalX]", ToString[#1]] & ) /@ Range[Length[fourVelocity] - 1]], True, True],
Normal[SparseArray[(#1 -> "\[FormalRho]"*fourVelocity[[First[#1]]]*fourVelocity[[Last[#1]]] & ) /@
Tuples[Range[Length[fourVelocity]], 2]]], False, False]
StressEnergyTensor[{"Dust", fourVelocity_List}, (metricTensor_)[matrixRepresentation_List, coordinates_List,
metricIndex1_, metricIndex2_]] := StressEnergyTensor[ResourceFunction["MetricTensor"][matrixRepresentation,
coordinates, metricIndex1, metricIndex2],
Normal[SparseArray[(#1 -> "\[FormalRho]"*fourVelocity[[First[#1]]]*fourVelocity[[Last[#1]]] & ) /@
Tuples[Range[Length[matrixRepresentation]], 2]]], False, False] /; SymbolName[metricTensor] === "MetricTensor" &&
Length[Dimensions[matrixRepresentation]] == 2 && Length[coordinates] == Length[matrixRepresentation] &&
BooleanQ[metricIndex1] && BooleanQ[metricIndex2] && Length[fourVelocity] == Length[matrixRepresentation]
StressEnergyTensor[{"Dust", fourVelocity_List}, index1_, index2_] :=
StressEnergyTensor[ResourceFunction["MetricTensor"][DiagonalMatrix[
Join[{-1}, ConstantArray[1, Length[fourVelocity] - 1]]], Join[{"\[FormalT]"}, (Superscript["\[FormalX]", ToString[#1]] & ) /@
Range[Length[fourVelocity] - 1]], True, True],
Normal[SparseArray[(#1 -> "\[FormalRho]"*fourVelocity[[First[#1]]]*fourVelocity[[Last[#1]]] & ) /@
Tuples[Range[Length[fourVelocity]], 2]]], index1, index2] /; BooleanQ[index1] && BooleanQ[index2]
StressEnergyTensor[{"Dust", fourVelocity_List}, (metricTensor_)[matrixRepresentation_List, coordinates_List,
metricIndex1_, metricIndex2_], index1_, index2_] :=
StressEnergyTensor[ResourceFunction["MetricTensor"][matrixRepresentation, coordinates, metricIndex1, metricIndex2],
Normal[SparseArray[(#1 -> "\[FormalRho]"*fourVelocity[[First[#1]]]*fourVelocity[[Last[#1]]] & ) /@
Tuples[Range[Length[matrixRepresentation]], 2]]], index1, index2] /;
SymbolName[metricTensor] === "MetricTensor" && Length[Dimensions[matrixRepresentation]] == 2 &&
Length[coordinates] == Length[matrixRepresentation] && BooleanQ[metricIndex1] && BooleanQ[metricIndex2] &&
BooleanQ[index1] && BooleanQ[index2] && Length[fourVelocity] == Length[matrixRepresentation]
StressEnergyTensor[{"Dust", fourVelocity_List, density_}] :=
StressEnergyTensor[ResourceFunction["MetricTensor"][
DiagonalMatrix[Join[{-1}, ConstantArray[1, Length[fourVelocity] - 1]]],
Join[{"\[FormalT]"}, (Superscript["\[FormalX]", ToString[#1]] & ) /@ Range[Length[fourVelocity] - 1]], True, True],
Normal[SparseArray[(#1 -> density*fourVelocity[[First[#1]]]*fourVelocity[[Last[#1]]] & ) /@
Tuples[Range[Length[fourVelocity]], 2]]], False, False]
StressEnergyTensor[{"Dust", fourVelocity_List, density_}, (metricTensor_)[matrixRepresentation_List, coordinates_List,
metricIndex1_, metricIndex2_]] := StressEnergyTensor[ResourceFunction["MetricTensor"][matrixRepresentation,
coordinates, metricIndex1, metricIndex2],
Normal[SparseArray[(#1 -> density*fourVelocity[[First[#1]]]*fourVelocity[[Last[#1]]] & ) /@
Tuples[Range[Length[matrixRepresentation]], 2]]], False, False] /; SymbolName[metricTensor] === "MetricTensor" &&
Length[Dimensions[matrixRepresentation]] == 2 && Length[coordinates] == Length[matrixRepresentation] &&
BooleanQ[metricIndex1] && BooleanQ[metricIndex2] && Length[fourVelocity] == Length[matrixRepresentation]
StressEnergyTensor[{"Dust", fourVelocity_List, density_}, index1_, index2_] :=
StressEnergyTensor[ResourceFunction["MetricTensor"][DiagonalMatrix[
Join[{-1}, ConstantArray[1, Length[fourVelocity] - 1]]], Join[{"\[FormalT]"}, (Superscript["\[FormalX]", ToString[#1]] & ) /@
Range[Length[fourVelocity] - 1]], True, True],
Normal[SparseArray[(#1 -> density*fourVelocity[[First[#1]]]*fourVelocity[[Last[#1]]] & ) /@
Tuples[Range[Length[fourVelocity]], 2]]], index1, index2] /; BooleanQ[index1] && BooleanQ[index2]
StressEnergyTensor[{"Dust", fourVelocity_List, density_}, (metricTensor_)[matrixRepresentation_List, coordinates_List,
metricIndex1_, metricIndex2_], index1_, index2_] :=
StressEnergyTensor[ResourceFunction["MetricTensor"][matrixRepresentation, coordinates, metricIndex1, metricIndex2],
Normal[SparseArray[(#1 -> density*fourVelocity[[First[#1]]]*fourVelocity[[Last[#1]]] & ) /@
Tuples[Range[Length[matrixRepresentation]], 2]]], index1, index2] /;
SymbolName[metricTensor] === "MetricTensor" && Length[Dimensions[matrixRepresentation]] == 2 &&
Length[coordinates] == Length[matrixRepresentation] && BooleanQ[metricIndex1] && BooleanQ[metricIndex2] &&
BooleanQ[index1] && BooleanQ[index2] && Length[fourVelocity] == Length[matrixRepresentation]
StressEnergyTensor["DustField"] := StressEnergyTensor[ResourceFunction["MetricTensor"][
DiagonalMatrix[Join[{-1}, ConstantArray[1, 3]]], Join[{"\[FormalT]"}, (Superscript["\[FormalX]", ToString[#1]] & ) /@ Range[3]],
True, True],
Normal[SparseArray[(Module[{index = #1}, index -> "\[FormalRho]" @@ Join[{"\[FormalT]"}, (Superscript["\[FormalX]", ToString[#1]] & ) /@
Range[3]]*Superscript["\[FormalU]", ToString[First[index]]] @@ Join[{"\[FormalT]"},
(Superscript["\[FormalX]", ToString[#1]] & ) /@ Range[3]]*Superscript["\[FormalU]", ToString[Last[index]]] @@
Join[{"\[FormalT]"}, (Superscript["\[FormalX]", ToString[#1]] & ) /@ Range[3]]] & ) /@ Tuples[Range[4], 2]]], False, False]
StressEnergyTensor["DustField", (metricTensor_)[matrixRepresentation_List, coordinates_List, metricIndex1_,
metricIndex2_]] := StressEnergyTensor[ResourceFunction["MetricTensor"][matrixRepresentation, coordinates,
metricIndex1, metricIndex2],
Normal[SparseArray[(Module[{index = #1}, index -> "\[FormalRho]" @@ coordinates*Superscript["\[FormalU]", ToString[First[index]]] @@
coordinates*Superscript["\[FormalU]", ToString[Last[index]]] @@ coordinates] & ) /@
Tuples[Range[Length[matrixRepresentation]], 2]]], False, False] /; SymbolName[metricTensor] === "MetricTensor" &&
Length[Dimensions[matrixRepresentation]] == 2 && Length[coordinates] == Length[matrixRepresentation] &&
BooleanQ[metricIndex1] && BooleanQ[metricIndex2]
StressEnergyTensor["DustField", index1_, index2_] :=
StressEnergyTensor[ResourceFunction["MetricTensor"][DiagonalMatrix[Join[{-1}, ConstantArray[1, 3]]],
Join[{"\[FormalT]"}, (Superscript["\[FormalX]", ToString[#1]] & ) /@ Range[3]], True, True],
Normal[SparseArray[(Module[{index = #1}, index -> "\[FormalRho]" @@ Join[{"\[FormalT]"}, (Superscript["\[FormalX]", ToString[#1]] & ) /@
Range[3]]*Superscript["\[FormalU]", ToString[First[index]]] @@ Join[{"\[FormalT]"},
(Superscript["\[FormalX]", ToString[#1]] & ) /@ Range[3]]*Superscript["\[FormalU]", ToString[Last[index]]] @@
Join[{"\[FormalT]"}, (Superscript["\[FormalX]", ToString[#1]] & ) /@ Range[3]]] & ) /@ Tuples[Range[4], 2]]], index1,
index2] /; BooleanQ[index1] && BooleanQ[index2]
StressEnergyTensor["DustField", (metricTensor_)[matrixRepresentation_List, coordinates_List, metricIndex1_,
metricIndex2_], index1_, index2_] := StressEnergyTensor[ResourceFunction["MetricTensor"][matrixRepresentation,
coordinates, metricIndex1, metricIndex2],
Normal[SparseArray[(Module[{index = #1}, index -> "\[FormalRho]" @@ coordinates*Superscript["\[FormalU]", ToString[First[index]]] @@
coordinates*Superscript["\[FormalU]", ToString[Last[index]]] @@ coordinates] & ) /@
Tuples[Range[Length[matrixRepresentation]], 2]]], index1, index2] /;
SymbolName[metricTensor] === "MetricTensor" && Length[Dimensions[matrixRepresentation]] == 2 &&
Length[coordinates] == Length[matrixRepresentation] && BooleanQ[metricIndex1] && BooleanQ[metricIndex2] &&
BooleanQ[index1] && BooleanQ[index2]
StressEnergyTensor[{"DustField", fourVelocity_List}] :=
StressEnergyTensor[ResourceFunction["MetricTensor"][
DiagonalMatrix[Join[{-1}, ConstantArray[1, Length[fourVelocity] - 1]]],
Join[{"\[FormalT]"}, (Superscript["\[FormalX]", ToString[#1]] & ) /@ Range[Length[fourVelocity] - 1]], True, True],
Normal[SparseArray[(Module[{index = #1}, index -> "\[FormalRho]" @@ Join[{"\[FormalT]"}, (Superscript["\[FormalX]", ToString[#1]] & ) /@
Range[Length[fourVelocity] - 1]]*fourVelocity[[First[index]]] @@ Join[{"\[FormalT]"},
(Superscript["\[FormalX]", ToString[#1]] & ) /@ Range[Length[fourVelocity] - 1]]*fourVelocity[[Last[index]]] @@
Join[{"\[FormalT]"}, (Superscript["\[FormalX]", ToString[#1]] & ) /@ Range[Length[fourVelocity] - 1]]] & ) /@
Tuples[Range[Length[fourVelocity]], 2]]], False, False]
StressEnergyTensor[{"DustField", fourVelocity_List}, (metricTensor_)[matrixRepresentation_List, coordinates_List,
metricIndex1_, metricIndex2_]] := StressEnergyTensor[ResourceFunction["MetricTensor"][matrixRepresentation,
coordinates, metricIndex1, metricIndex2],
Normal[SparseArray[(Module[{index = #1}, index -> "\[FormalRho]" @@ coordinates*fourVelocity[[First[index]]] @@ coordinates*
fourVelocity[[Last[index]]] @@ coordinates] & ) /@ Tuples[Range[Length[matrixRepresentation]], 2]]], False,
False] /; SymbolName[metricTensor] === "MetricTensor" && Length[Dimensions[matrixRepresentation]] == 2 &&
Length[coordinates] == Length[matrixRepresentation] && BooleanQ[metricIndex1] && BooleanQ[metricIndex2] &&
Length[fourVelocity] == Length[matrixRepresentation]
StressEnergyTensor[{"DustField", fourVelocity_List}, index1_, index2_] :=
StressEnergyTensor[ResourceFunction["MetricTensor"][DiagonalMatrix[
Join[{-1}, ConstantArray[1, Length[fourVelocity] - 1]]], Join[{"\[FormalT]"}, (Superscript["\[FormalX]", ToString[#1]] & ) /@
Range[Length[fourVelocity] - 1]], True, True],
Normal[SparseArray[(Module[{index = #1}, index -> "\[FormalRho]" @@ Join[{"\[FormalT]"}, (Superscript["\[FormalX]", ToString[#1]] & ) /@
Range[Length[fourVelocity] - 1]]*fourVelocity[[First[index]]] @@ Join[{"\[FormalT]"},
(Superscript["\[FormalX]", ToString[#1]] & ) /@ Range[Length[fourVelocity] - 1]]*fourVelocity[[Last[index]]] @@
Join[{"\[FormalT]"}, (Superscript["\[FormalX]", ToString[#1]] & ) /@ Range[Length[fourVelocity] - 1]]] & ) /@
Tuples[Range[Length[fourVelocity]], 2]]], index1, index2] /; BooleanQ[index1] && BooleanQ[index2]
StressEnergyTensor[{"DustField", fourVelocity_List}, (metricTensor_)[matrixRepresentation_List, coordinates_List,
metricIndex1_, metricIndex2_], index1_, index2_] :=
StressEnergyTensor[ResourceFunction["MetricTensor"][matrixRepresentation, coordinates, metricIndex1, metricIndex2],
Normal[SparseArray[(Module[{index = #1}, index -> "\[FormalRho]" @@ coordinates*fourVelocity[[First[index]]] @@ coordinates*
fourVelocity[[Last[index]]] @@ coordinates] & ) /@ Tuples[Range[Length[matrixRepresentation]], 2]]], index1,
index2] /; SymbolName[metricTensor] === "MetricTensor" && Length[Dimensions[matrixRepresentation]] == 2 &&
Length[coordinates] == Length[matrixRepresentation] && BooleanQ[metricIndex1] && BooleanQ[metricIndex2] &&
BooleanQ[index1] && BooleanQ[index2] && Length[fourVelocity] == Length[matrixRepresentation]
StressEnergyTensor[{"DustField", fourVelocity_List, density_}] :=
StressEnergyTensor[ResourceFunction["MetricTensor"][
DiagonalMatrix[Join[{-1}, ConstantArray[1, Length[fourVelocity] - 1]]],
Join[{"\[FormalT]"}, (Superscript["\[FormalX]", ToString[#1]] & ) /@ Range[Length[fourVelocity] - 1]], True, True],
Normal[SparseArray[(Module[{index = #1}, index -> density @@ Join[{"\[FormalT]"}, (Superscript["\[FormalX]", ToString[#1]] & ) /@
Range[Length[fourVelocity] - 1]]*fourVelocity[[First[index]]] @@ Join[{"\[FormalT]"},
(Superscript["\[FormalX]", ToString[#1]] & ) /@ Range[Length[fourVelocity] - 1]]*fourVelocity[[Last[index]]] @@
Join[{"\[FormalT]"}, (Superscript["\[FormalX]", ToString[#1]] & ) /@ Range[Length[fourVelocity] - 1]]] & ) /@
Tuples[Range[Length[fourVelocity]], 2]]], False, False]
StressEnergyTensor[{"DustField", fourVelocity_List, density_}, (metricTensor_)[matrixRepresentation_List,
coordinates_List, metricIndex1_, metricIndex2_]] :=
StressEnergyTensor[ResourceFunction["MetricTensor"][matrixRepresentation, coordinates, metricIndex1, metricIndex2],
Normal[SparseArray[(Module[{index = #1}, index -> density @@ coordinates*fourVelocity[[First[index]]] @@ coordinates*
fourVelocity[[Last[index]]] @@ coordinates] & ) /@ Tuples[Range[Length[matrixRepresentation]], 2]]], False,
False] /; SymbolName[metricTensor] === "MetricTensor" && Length[Dimensions[matrixRepresentation]] == 2 &&
Length[coordinates] == Length[matrixRepresentation] && BooleanQ[metricIndex1] && BooleanQ[metricIndex2] &&
Length[fourVelocity] == Length[matrixRepresentation]
StressEnergyTensor[{"DustField", fourVelocity_List, density_}, index1_, index2_] :=
StressEnergyTensor[ResourceFunction["MetricTensor"][DiagonalMatrix[
Join[{-1}, ConstantArray[1, Length[fourVelocity] - 1]]], Join[{"\[FormalT]"}, (Superscript["\[FormalX]", ToString[#1]] & ) /@
Range[Length[fourVelocity] - 1]], True, True],
Normal[SparseArray[(Module[{index = #1}, index -> density @@ Join[{"\[FormalT]"}, (Superscript["\[FormalX]", ToString[#1]] & ) /@
Range[Length[fourVelocity] - 1]]*fourVelocity[[First[index]]] @@ Join[{"\[FormalT]"},
(Superscript["\[FormalX]", ToString[#1]] & ) /@ Range[Length[fourVelocity] - 1]]*fourVelocity[[Last[index]]] @@
Join[{"\[FormalT]"}, (Superscript["\[FormalX]", ToString[#1]] & ) /@ Range[Length[fourVelocity] - 1]]] & ) /@
Tuples[Range[Length[fourVelocity]], 2]]], index1, index2] /; BooleanQ[index1] && BooleanQ[index2]
StressEnergyTensor[{"DustField", fourVelocity_List, density_}, (metricTensor_)[matrixRepresentation_List,
coordinates_List, metricIndex1_, metricIndex2_], index1_, index2_] :=
StressEnergyTensor[ResourceFunction["MetricTensor"][matrixRepresentation, coordinates, metricIndex1, metricIndex2],
Normal[SparseArray[(Module[{index = #1}, index -> density @@ coordinates*fourVelocity[[First[index]]] @@ coordinates*
fourVelocity[[Last[index]]] @@ coordinates] & ) /@ Tuples[Range[Length[matrixRepresentation]], 2]]], index1,
index2] /; SymbolName[metricTensor] === "MetricTensor" && Length[Dimensions[matrixRepresentation]] == 2 &&
Length[coordinates] == Length[matrixRepresentation] && BooleanQ[metricIndex1] && BooleanQ[metricIndex2] &&
BooleanQ[index1] && BooleanQ[index2] && Length[fourVelocity] == Length[matrixRepresentation]
StressEnergyTensor["Radiation"] := StressEnergyTensor[ResourceFunction["MetricTensor"][
DiagonalMatrix[Join[{-1}, ConstantArray[1, 3]]], Join[{"\[FormalT]"}, (Superscript["\[FormalX]", ToString[#1]] & ) /@ Range[3]],
True, True],
Normal[SparseArray[(#1 -> (4*"\[FormalCapitalP]")*Superscript["\[FormalU]", ToString[First[#1]]]*Superscript["\[FormalU]", ToString[Last[#1]]] +
"\[FormalCapitalP]"*Inverse[DiagonalMatrix[Join[{-1}, ConstantArray[1, 3]]]][[First[#1],Last[#1]]] & ) /@
Tuples[Range[4], 2]]], False, False]
StressEnergyTensor["Radiation", (metricTensor_)[matrixRepresentation_List, coordinates_List, metricIndex1_,
metricIndex2_]] := StressEnergyTensor[ResourceFunction["MetricTensor"][matrixRepresentation, coordinates,
metricIndex1, metricIndex2],
Normal[SparseArray[(#1 -> (Length[matrixRepresentation]*"\[FormalCapitalP]")*Superscript["\[FormalU]", ToString[First[#1]]]*
Superscript["\[FormalU]", ToString[Last[#1]]] + "\[FormalCapitalP]"*Inverse[matrixRepresentation][[First[#1],Last[#1]]] & ) /@
Tuples[Range[Length[matrixRepresentation]], 2]]], False, False] /; SymbolName[metricTensor] === "MetricTensor" &&
Length[Dimensions[matrixRepresentation]] == 2 && Length[coordinates] == Length[matrixRepresentation] &&
BooleanQ[metricIndex1] && BooleanQ[metricIndex2]
StressEnergyTensor["Radiation", index1_, index2_] :=
StressEnergyTensor[ResourceFunction["MetricTensor"][DiagonalMatrix[Join[{-1}, ConstantArray[1, 3]]],
Join[{"\[FormalT]"}, (Superscript["\[FormalX]", ToString[#1]] & ) /@ Range[3]], True, True],
Normal[SparseArray[(#1 -> (4*"\[FormalCapitalP]")*Superscript["\[FormalU]", ToString[First[#1]]]*Superscript["\[FormalU]", ToString[Last[#1]]] +
"\[FormalCapitalP]"*Inverse[DiagonalMatrix[Join[{-1}, ConstantArray[1, 3]]]][[First[#1],Last[#1]]] & ) /@
Tuples[Range[4], 2]]], index1, index2] /; BooleanQ[index1] && BooleanQ[index2]
StressEnergyTensor["Radiation", (metricTensor_)[matrixRepresentation_List, coordinates_List, metricIndex1_,
metricIndex2_], index1_, index2_] := StressEnergyTensor[ResourceFunction["MetricTensor"][matrixRepresentation,
coordinates, metricIndex1, metricIndex2],
Normal[SparseArray[(#1 -> (Length[matrixRepresentation]*"\[FormalCapitalP]")*Superscript["\[FormalU]", ToString[First[#1]]]*
Superscript["\[FormalU]", ToString[Last[#1]]] + "\[FormalCapitalP]"*Inverse[matrixRepresentation][[First[#1],Last[#1]]] & ) /@
Tuples[Range[Length[matrixRepresentation]], 2]]], index1, index2] /;
SymbolName[metricTensor] === "MetricTensor" && Length[Dimensions[matrixRepresentation]] == 2 &&
Length[coordinates] == Length[matrixRepresentation] && BooleanQ[metricIndex1] && BooleanQ[metricIndex2] &&
BooleanQ[index1] && BooleanQ[index2]
StressEnergyTensor[{"Radiation", fourVelocity_List}] :=
StressEnergyTensor[ResourceFunction["MetricTensor"][
DiagonalMatrix[Join[{-1}, ConstantArray[1, Length[fourVelocity] - 1]]],
Join[{"\[FormalT]"}, (Superscript["\[FormalX]", ToString[#1]] & ) /@ Range[Length[fourVelocity] - 1]], True, True],
Normal[SparseArray[(#1 -> (Length[fourVelocity]*"\[FormalCapitalP]")*fourVelocity[[First[#1]]]*fourVelocity[[Last[#1]]] +
"\[FormalCapitalP]"*Inverse[DiagonalMatrix[Join[{-1}, ConstantArray[1, Length[fourVelocity] - 1]]]][[First[#1],
Last[#1]]] & ) /@ Tuples[Range[Length[fourVelocity]], 2]]], False, False]
StressEnergyTensor[{"Radiation", fourVelocity_List}, (metricTensor_)[matrixRepresentation_List, coordinates_List,
metricIndex1_, metricIndex2_]] := StressEnergyTensor[ResourceFunction["MetricTensor"][matrixRepresentation,
coordinates, metricIndex1, metricIndex2],
Normal[SparseArray[(#1 -> (Length[matrixRepresentation]*"\[FormalCapitalP]")*fourVelocity[[First[#1]]]*fourVelocity[[Last[#1]]] +
"\[FormalCapitalP]"*Inverse[matrixRepresentation][[First[#1],Last[#1]]] & ) /@ Tuples[Range[Length[matrixRepresentation]],
2]]], False, False] /; SymbolName[metricTensor] === "MetricTensor" && Length[Dimensions[matrixRepresentation]] ==
2 && Length[coordinates] == Length[matrixRepresentation] && BooleanQ[metricIndex1] && BooleanQ[metricIndex2] &&
Length[fourVelocity] == Length[matrixRepresentation]
StressEnergyTensor[{"Radiation", fourVelocity_List}, index1_, index2_] :=
StressEnergyTensor[ResourceFunction["MetricTensor"][DiagonalMatrix[
Join[{-1}, ConstantArray[1, Length[fourVelocity] - 1]]], Join[{"\[FormalT]"}, (Superscript["\[FormalX]", ToString[#1]] & ) /@
Range[Length[fourVelocity] - 1]], True, True],
Normal[SparseArray[(#1 -> (Length[fourVelocity]*"\[FormalCapitalP]")*fourVelocity[[First[#1]]]*fourVelocity[[Last[#1]]] +
"\[FormalCapitalP]"*Inverse[DiagonalMatrix[Join[{-1}, ConstantArray[1, Length[fourVelocity] - 1]]]][[First[#1],
Last[#1]]] & ) /@ Tuples[Range[Length[fourVelocity]], 2]]], index1, index2] /;
BooleanQ[index1] && BooleanQ[index2]
StressEnergyTensor[{"Radiation", fourVelocity_List}, (metricTensor_)[matrixRepresentation_List, coordinates_List,
metricIndex1_, metricIndex2_], index1_, index2_] :=
StressEnergyTensor[ResourceFunction["MetricTensor"][matrixRepresentation, coordinates, metricIndex1, metricIndex2],
Normal[SparseArray[(#1 -> (Length[matrixRepresentation]*"\[FormalCapitalP]")*fourVelocity[[First[#1]]]*fourVelocity[[Last[#1]]] +
"\[FormalCapitalP]"*Inverse[matrixRepresentation][[First[#1],Last[#1]]] & ) /@ Tuples[Range[Length[matrixRepresentation]],
2]]], index1, index2] /; SymbolName[metricTensor] === "MetricTensor" &&
Length[Dimensions[matrixRepresentation]] == 2 && Length[coordinates] == Length[matrixRepresentation] &&
BooleanQ[metricIndex1] && BooleanQ[metricIndex2] && BooleanQ[index1] && BooleanQ[index2] &&
Length[fourVelocity] == Length[matrixRepresentation]
StressEnergyTensor[{"Radiation", fourVelocity_List, pressure_}] :=
StressEnergyTensor[ResourceFunction["MetricTensor"][
DiagonalMatrix[Join[{-1}, ConstantArray[1, Length[fourVelocity] - 1]]],
Join[{"\[FormalT]"}, (Superscript["\[FormalX]", ToString[#1]] & ) /@ Range[Length[fourVelocity] - 1]], True, True],
Normal[SparseArray[(#1 -> (Length[fourVelocity]*pressure)*fourVelocity[[First[#1]]]*fourVelocity[[Last[#1]]] +
pressure*Inverse[DiagonalMatrix[Join[{-1}, ConstantArray[1, Length[fourVelocity] - 1]]]][[First[#1],
Last[#1]]] & ) /@ Tuples[Range[Length[fourVelocity]], 2]]], False, False]
StressEnergyTensor[{"Radiation", fourVelocity_List, pressure_}, (metricTensor_)[matrixRepresentation_List,
coordinates_List, metricIndex1_, metricIndex2_]] :=
StressEnergyTensor[ResourceFunction["MetricTensor"][matrixRepresentation, coordinates, metricIndex1, metricIndex2],
Normal[SparseArray[(#1 -> (Length[matrixRepresentation]*pressure)*fourVelocity[[First[#1]]]*
fourVelocity[[Last[#1]]] + pressure*Inverse[matrixRepresentation][[First[#1],Last[#1]]] & ) /@
Tuples[Range[Length[matrixRepresentation]], 2]]], False, False] /; SymbolName[metricTensor] === "MetricTensor" &&
Length[Dimensions[matrixRepresentation]] == 2 && Length[coordinates] == Length[matrixRepresentation] &&
BooleanQ[metricIndex1] && BooleanQ[metricIndex2] && Length[fourVelocity] == Length[matrixRepresentation]
StressEnergyTensor[{"Radiation", fourVelocity_List, pressure_}, index1_, index2_] :=
StressEnergyTensor[ResourceFunction["MetricTensor"][DiagonalMatrix[
Join[{-1}, ConstantArray[1, Length[fourVelocity] - 1]]], Join[{"\[FormalT]"}, (Superscript["\[FormalX]", ToString[#1]] & ) /@
Range[Length[fourVelocity] - 1]], True, True],
Normal[SparseArray[(#1 -> (Length[fourVelocity]*pressure)*fourVelocity[[First[#1]]]*fourVelocity[[Last[#1]]] +
pressure*Inverse[DiagonalMatrix[Join[{-1}, ConstantArray[1, Length[fourVelocity] - 1]]]][[First[#1],
Last[#1]]] & ) /@ Tuples[Range[Length[fourVelocity]], 2]]], index1, index2] /;
BooleanQ[index1] && BooleanQ[index2]
StressEnergyTensor[{"Radiation", fourVelocity_List, pressure_}, (metricTensor_)[matrixRepresentation_List,
coordinates_List, metricIndex1_, metricIndex2_], index1_, index2_] :=
StressEnergyTensor[ResourceFunction["MetricTensor"][matrixRepresentation, coordinates, metricIndex1, metricIndex2],
Normal[SparseArray[(#1 -> (Length[matrixRepresentation]*pressure)*fourVelocity[[First[#1]]]*
fourVelocity[[Last[#1]]] + pressure*Inverse[matrixRepresentation][[First[#1],Last[#1]]] & ) /@
Tuples[Range[Length[matrixRepresentation]], 2]]], index1, index2] /;
SymbolName[metricTensor] === "MetricTensor" && Length[Dimensions[matrixRepresentation]] == 2 &&
Length[coordinates] == Length[matrixRepresentation] && BooleanQ[metricIndex1] && BooleanQ[metricIndex2] &&
BooleanQ[index1] && BooleanQ[index2] && Length[fourVelocity] == Length[matrixRepresentation]
StressEnergyTensor["RadiationField"] := StressEnergyTensor[ResourceFunction["MetricTensor"][
DiagonalMatrix[Join[{-1}, ConstantArray[1, 3]]], Join[{"\[FormalT]"}, (Superscript["\[FormalX]", ToString[#1]] & ) /@ Range[3]],
True, True],
Normal[SparseArray[
(Module[{index = #1}, index -> (4*"\[FormalCapitalP]" @@ Join[{"\[FormalT]"}, (Superscript["\[FormalX]", ToString[#1]] & ) /@ Range[3]])*
Superscript["\[FormalU]", ToString[First[index]]] @@ Join[{"\[FormalT]"}, (Superscript["\[FormalX]", ToString[#1]] & ) /@ Range[3]]*
Superscript["\[FormalU]", ToString[Last[index]]] @@ Join[{"\[FormalT]"}, (Superscript["\[FormalX]", ToString[#1]] & ) /@
Range[3]] + "\[FormalCapitalP]" @@ Join[{"\[FormalT]"}, (Superscript["\[FormalX]", ToString[#1]] & ) /@ Range[3]]*
Inverse[DiagonalMatrix[Join[{-1}, ConstantArray[1, 3]]]][[First[index],Last[index]]]] & ) /@
Tuples[Range[4], 2]]], False, False]
StressEnergyTensor["RadiationField", (metricTensor_)[matrixRepresentation_List, coordinates_List, metricIndex1_,
metricIndex2_]] := StressEnergyTensor[ResourceFunction["MetricTensor"][matrixRepresentation, coordinates,
metricIndex1, metricIndex2],
Normal[SparseArray[(Module[{index = #1}, index -> (Length[matrixRepresentation]*"\[FormalCapitalP]" @@ coordinates)*
Superscript["\[FormalU]", ToString[First[index]]] @@ coordinates*Superscript["\[FormalU]", ToString[Last[index]]] @@
coordinates + "\[FormalCapitalP]" @@ coordinates*Inverse[matrixRepresentation][[First[index],Last[index]]]] & ) /@
Tuples[Range[Length[matrixRepresentation]], 2]]], False, False] /; SymbolName[metricTensor] === "MetricTensor" &&
Length[Dimensions[matrixRepresentation]] == 2 && Length[coordinates] == Length[matrixRepresentation] &&
BooleanQ[metricIndex1] && BooleanQ[metricIndex2]
StressEnergyTensor["RadiationField", index1_, index2_] :=
StressEnergyTensor[ResourceFunction["MetricTensor"][DiagonalMatrix[Join[{-1}, ConstantArray[1, 3]]],
Join[{"\[FormalT]"}, (Superscript["\[FormalX]", ToString[#1]] & ) /@ Range[3]], True, True],
Normal[SparseArray[(Module[{index = #1}, index -> (4*"\[FormalCapitalP]" @@ Join[{"\[FormalT]"}, (Superscript["\[FormalX]", ToString[#1]] & ) /@
Range[3]])*Superscript["\[FormalU]", ToString[First[index]]] @@ Join[{"\[FormalT]"},
(Superscript["\[FormalX]", ToString[#1]] & ) /@ Range[3]]*Superscript["\[FormalU]", ToString[Last[index]]] @@
Join[{"\[FormalT]"}, (Superscript["\[FormalX]", ToString[#1]] & ) /@ Range[3]] +
"\[FormalCapitalP]" @@ Join[{"\[FormalT]"}, (Superscript["\[FormalX]", ToString[#1]] & ) /@ Range[3]]*
Inverse[DiagonalMatrix[Join[{-1}, ConstantArray[1, 3]]]][[First[index],Last[index]]]] & ) /@
Tuples[Range[4], 2]]], index1, index2] /; BooleanQ[index1] && BooleanQ[index2]
StressEnergyTensor["RadiationField", (metricTensor_)[matrixRepresentation_List, coordinates_List, metricIndex1_,
metricIndex2_], index1_, index2_] := StressEnergyTensor[ResourceFunction["MetricTensor"][matrixRepresentation,
coordinates, metricIndex1, metricIndex2],
Normal[SparseArray[(Module[{index = #1}, index -> (Length[matrixRepresentation]*"\[FormalCapitalP]" @@ coordinates)*
Superscript["\[FormalU]", ToString[First[index]]] @@ coordinates*Superscript["\[FormalU]", ToString[Last[index]]] @@
coordinates + "\[FormalCapitalP]" @@ coordinates*Inverse[matrixRepresentation][[First[index],Last[index]]]] & ) /@
Tuples[Range[Length[matrixRepresentation]], 2]]], index1, index2] /;
SymbolName[metricTensor] === "MetricTensor" && Length[Dimensions[matrixRepresentation]] == 2 &&
Length[coordinates] == Length[matrixRepresentation] && BooleanQ[metricIndex1] && BooleanQ[metricIndex2] &&
BooleanQ[index1] && BooleanQ[index2]
StressEnergyTensor[{"RadiationField", fourVelocity_List}] :=
StressEnergyTensor[ResourceFunction["MetricTensor"][
DiagonalMatrix[Join[{-1}, ConstantArray[1, Length[fourVelocity] - 1]]],
Join[{"\[FormalT]"}, (Superscript["\[FormalX]", ToString[#1]] & ) /@ Range[Length[fourVelocity] - 1]], True, True],
Normal[SparseArray[
(Module[{index = #1}, index -> (Length[fourVelocity]*"\[FormalCapitalP]" @@ Join[{"\[FormalT]"}, (Superscript["\[FormalX]", ToString[#1]] & ) /@
Range[Length[fourVelocity] - 1]])*fourVelocity[[First[index]]] @@ Join[{"\[FormalT]"},
(Superscript["\[FormalX]", ToString[#1]] & ) /@ Range[Length[fourVelocity] - 1]]*fourVelocity[[Last[index]]] @@
Join[{"\[FormalT]"}, (Superscript["\[FormalX]", ToString[#1]] & ) /@ Range[Length[fourVelocity] - 1]] +
"\[FormalCapitalP]" @@ Join[{"\[FormalT]"}, (Superscript["\[FormalX]", ToString[#1]] & ) /@ Range[Length[fourVelocity] - 1]]*
Inverse[DiagonalMatrix[Join[{-1}, ConstantArray[1, Length[fourVelocity] - 1]]]][[First[index],
Last[index]]]] & ) /@ Tuples[Range[Length[fourVelocity]], 2]]], False, False]
StressEnergyTensor[{"RadiationField", fourVelocity_List}, (metricTensor_)[matrixRepresentation_List, coordinates_List,
metricIndex1_, metricIndex2_]] := StressEnergyTensor[ResourceFunction["MetricTensor"][matrixRepresentation,
coordinates, metricIndex1, metricIndex2],
Normal[SparseArray[(Module[{index = #1}, index -> (Length[matrixRepresentation]*"\[FormalCapitalP]" @@ coordinates)*
fourVelocity[[First[index]]] @@ coordinates*fourVelocity[[Last[index]]] @@ coordinates +
"\[FormalCapitalP]" @@ coordinates*Inverse[matrixRepresentation][[First[index],Last[index]]]] & ) /@
Tuples[Range[Length[matrixRepresentation]], 2]]], False, False] /; SymbolName[metricTensor] === "MetricTensor" &&
Length[Dimensions[matrixRepresentation]] == 2 && Length[coordinates] == Length[matrixRepresentation] &&
BooleanQ[metricIndex1] && BooleanQ[metricIndex2] && Length[fourVelocity] == Length[matrixRepresentation]
StressEnergyTensor[{"RadiationField", fourVelocity_List}, index1_, index2_] :=
StressEnergyTensor[ResourceFunction["MetricTensor"][DiagonalMatrix[
Join[{-1}, ConstantArray[1, Length[fourVelocity] - 1]]], Join[{"\[FormalT]"}, (Superscript["\[FormalX]", ToString[#1]] & ) /@
Range[Length[fourVelocity] - 1]], True, True],
Normal[SparseArray[
(Module[{index = #1}, index -> (Length[fourVelocity]*"\[FormalCapitalP]" @@ Join[{"\[FormalT]"}, (Superscript["\[FormalX]",
ToString[#1]] & ) /@ Range[Length[fourVelocity] - 1]])*fourVelocity[[First[index]]] @@
Join[{"\[FormalT]"}, (Superscript["\[FormalX]", ToString[#1]] & ) /@ Range[Length[fourVelocity] - 1]]*
fourVelocity[[Last[index]]] @@ Join[{"\[FormalT]"}, (Superscript["\[FormalX]", ToString[#1]] & ) /@ Range[
Length[fourVelocity] - 1]] + "\[FormalCapitalP]" @@ Join[{"\[FormalT]"}, (Superscript["\[FormalX]", ToString[#1]] & ) /@ Range[
Length[fourVelocity] - 1]]*Inverse[DiagonalMatrix[Join[{-1}, ConstantArray[1, Length[fourVelocity] -
1]]]][[First[index],Last[index]]]] & ) /@ Tuples[Range[Length[fourVelocity]], 2]]], index1, index2] /;
BooleanQ[index1] && BooleanQ[index2]
StressEnergyTensor[{"RadiationField", fourVelocity_List}, (metricTensor_)[matrixRepresentation_List, coordinates_List,
metricIndex1_, metricIndex2_], index1_, index2_] :=
StressEnergyTensor[ResourceFunction["MetricTensor"][matrixRepresentation, coordinates, metricIndex1, metricIndex2],
Normal[SparseArray[(Module[{index = #1}, index -> (Length[matrixRepresentation]*"\[FormalCapitalP]" @@ coordinates)*
fourVelocity[[First[index]]] @@ coordinates*fourVelocity[[Last[index]]] @@ coordinates +
"\[FormalCapitalP]" @@ coordinates*Inverse[matrixRepresentation][[First[index],Last[index]]]] & ) /@
Tuples[Range[Length[matrixRepresentation]], 2]]], index1, index2] /;
SymbolName[metricTensor] === "MetricTensor" && Length[Dimensions[matrixRepresentation]] == 2 &&
Length[coordinates] == Length[matrixRepresentation] && BooleanQ[metricIndex1] && BooleanQ[metricIndex2] &&
BooleanQ[index1] && BooleanQ[index2] && Length[fourVelocity] == Length[matrixRepresentation]
StressEnergyTensor[{"RadiationField", fourVelocity_List, pressure_}] :=
StressEnergyTensor[ResourceFunction["MetricTensor"][
DiagonalMatrix[Join[{-1}, ConstantArray[1, Length[fourVelocity] - 1]]],
Join[{"\[FormalT]"}, (Superscript["\[FormalX]", ToString[#1]] & ) /@ Range[Length[fourVelocity] - 1]], True, True],
Normal[SparseArray[
(Module[{index = #1}, index -> (Length[fourVelocity]*pressure @@ Join[{"\[FormalT]"}, (Superscript["\[FormalX]",
ToString[#1]] & ) /@ Range[Length[fourVelocity] - 1]])*fourVelocity[[First[index]]] @@
Join[{"\[FormalT]"}, (Superscript["\[FormalX]", ToString[#1]] & ) /@ Range[Length[fourVelocity] - 1]]*
fourVelocity[[Last[index]]] @@ Join[{"\[FormalT]"}, (Superscript["\[FormalX]", ToString[#1]] & ) /@
Range[Length[fourVelocity] - 1]] + pressure @@ Join[{"\[FormalT]"}, (Superscript["\[FormalX]", ToString[#1]] & ) /@
Range[Length[fourVelocity] - 1]]*Inverse[DiagonalMatrix[Join[{-1}, ConstantArray[1, Length[fourVelocity] -
1]]]][[First[index],Last[index]]]] & ) /@ Tuples[Range[Length[fourVelocity]], 2]]], False, False]
StressEnergyTensor[{"RadiationField", fourVelocity_List, pressure_}, (metricTensor_)[matrixRepresentation_List,
coordinates_List, metricIndex1_, metricIndex2_]] :=
StressEnergyTensor[ResourceFunction["MetricTensor"][matrixRepresentation, coordinates, metricIndex1, metricIndex2],
Normal[SparseArray[(Module[{index = #1}, index -> (Length[matrixRepresentation]*pressure @@ coordinates)*
fourVelocity[[First[index]]] @@ coordinates*fourVelocity[[Last[index]]] @@ coordinates +
pressure @@ coordinates*Inverse[matrixRepresentation][[First[index],Last[index]]]] & ) /@
Tuples[Range[Length[matrixRepresentation]], 2]]], False, False] /; SymbolName[metricTensor] === "MetricTensor" &&
Length[Dimensions[matrixRepresentation]] == 2 && Length[coordinates] == Length[matrixRepresentation] &&
BooleanQ[metricIndex1] && BooleanQ[metricIndex2] && Length[fourVelocity] == Length[matrixRepresentation]
StressEnergyTensor[{"RadiationField", fourVelocity_List, pressure_}, index1_, index2_] :=
StressEnergyTensor[ResourceFunction["MetricTensor"][DiagonalMatrix[
Join[{-1}, ConstantArray[1, Length[fourVelocity] - 1]]], Join[{"\[FormalT]"}, (Superscript["\[FormalX]", ToString[#1]] & ) /@
Range[Length[fourVelocity] - 1]], True, True],
Normal[SparseArray[
(Module[{index = #1}, index -> (Length[fourVelocity]*pressure @@ Join[{"\[FormalT]"}, (Superscript["\[FormalX]",
ToString[#1]] & ) /@ Range[Length[fourVelocity] - 1]])*fourVelocity[[First[index]]] @@
Join[{"\[FormalT]"}, (Superscript["\[FormalX]", ToString[#1]] & ) /@ Range[Length[fourVelocity] - 1]]*
fourVelocity[[Last[index]]] @@ Join[{"\[FormalT]"}, (Superscript["\[FormalX]", ToString[#1]] & ) /@ Range[
Length[fourVelocity] - 1]] + pressure @@ Join[{"\[FormalT]"}, (Superscript["\[FormalX]", ToString[#1]] & ) /@ Range[
Length[fourVelocity] - 1]]*Inverse[DiagonalMatrix[Join[{-1}, ConstantArray[1, Length[fourVelocity] -
1]]]][[First[index],Last[index]]]] & ) /@ Tuples[Range[Length[fourVelocity]], 2]]], index1, index2] /;
BooleanQ[index1] && BooleanQ[index2]
StressEnergyTensor[{"RadiationField", fourVelocity_List, pressure_}, (metricTensor_)[matrixRepresentation_List,
coordinates_List, metricIndex1_, metricIndex2_], index1_, index2_] :=
StressEnergyTensor[ResourceFunction["MetricTensor"][matrixRepresentation, coordinates, metricIndex1, metricIndex2],
Normal[SparseArray[(Module[{index = #1}, index -> (Length[matrixRepresentation]*pressure @@ coordinates)*
fourVelocity[[First[index]]] @@ coordinates*fourVelocity[[Last[index]]] @@ coordinates +
pressure @@ coordinates*Inverse[matrixRepresentation][[First[index],Last[index]]]] & ) /@
Tuples[Range[Length[matrixRepresentation]], 2]]], index1, index2] /;
SymbolName[metricTensor] === "MetricTensor" && Length[Dimensions[matrixRepresentation]] == 2 &&
Length[coordinates] == Length[matrixRepresentation] && BooleanQ[metricIndex1] && BooleanQ[metricIndex2] &&
BooleanQ[index1] && BooleanQ[index2] && Length[fourVelocity] == Length[matrixRepresentation]
StressEnergyTensor["ElectromagneticField"] := Module[{matrixRepresentation, electromagneticPotential, coordinates,
newMatrixRepresentation, newElectromagneticPotential, newCoordinates, covariantElectromagneticPotential,
electromagneticTensor, contravariantElectromagneticTensor, mixedElectromagneticTensor},
matrixRepresentation = DiagonalMatrix[Join[{-1}, ConstantArray[1, 3]]];
electromagneticPotential = Join[{"\[FormalCapitalPhi]" @@ Join[{"\[FormalT]"}, (Superscript["\[FormalX]", ToString[#1]] & ) /@ Range[3]]},
(Module[{index = #1}, Superscript["\[FormalCapitalA]", ToString[index]] @@ Join[{"\[FormalT]"}, (Superscript["\[FormalX]", ToString[#1]] & ) /@
Range[3]]] & ) /@ Range[Length[matrixRepresentation] - 1]];
coordinates = Join[{"\[FormalT]"}, (Superscript["\[FormalX]", ToString[#1]] & ) /@ Range[3]];
newMatrixRepresentation = matrixRepresentation /. (#1 -> ToExpression[#1] & ) /@ Select[coordinates, StringQ];
newElectromagneticPotential = electromagneticPotential /. (#1 -> ToExpression[#1] & ) /@
Select[coordinates, StringQ]; newCoordinates = coordinates /. (#1 -> ToExpression[#1] & ) /@
Select[coordinates, StringQ]; covariantElectromagneticPotential =
Normal[SparseArray[(Module[{index = #1}, index -> Total[(newMatrixRepresentation[[index,#1]]*
newElectromagneticPotential[[#1]] & ) /@ Range[Length[newElectromagneticPotential]]]] & ) /@
Range[Length[newElectromagneticPotential]]]]; electromagneticTensor =
Normal[SparseArray[(#1 -> D[covariantElectromagneticPotential[[Last[#1]]], newCoordinates[[First[#1]]]] -
D[covariantElectromagneticPotential[[First[#1]]], newCoordinates[[Last[#1]]]] & ) /@
Tuples[Range[Length[newElectromagneticPotential]], 2]]] /. (ToExpression[#1] -> #1 & ) /@
Select[coordinates, StringQ]; contravariantElectromagneticTensor =
Normal[SparseArray[(Module[{index = #1}, index -> Total[(Inverse[matrixRepresentation][[First[index],First[#1]]]*
Inverse[matrixRepresentation][[Last[#1],Last[index]]]*electromagneticTensor[[First[#1],Last[#1]]] & ) /@
Tuples[Range[Length[electromagneticPotential]], 2]]] & ) /@ Tuples[Range[Length[electromagneticPotential]],
2]]]; mixedElectromagneticTensor =
Normal[SparseArray[(Module[{index = #1}, index -> Total[(Inverse[matrixRepresentation][[First[index],#1]]*
electromagneticTensor[[#1,Last[index]]] & ) /@ Range[Length[electromagneticPotential]]]] & ) /@
Tuples[Range[Length[electromagneticPotential]], 2]]]; StressEnergyTensor[
ResourceFunction["MetricTensor"][matrixRepresentation, coordinates, True, True],
(1/Subscript["\[FormalMu]", "0"])*Normal[SparseArray[
(Module[{index = #1}, index -> Total[(contravariantElectromagneticTensor[[First[index],#1]]*
mixedElectromagneticTensor[[Last[index],#1]] & ) /@ Range[Length[electromagneticPotential]]] -
(1/4)*Total[(Inverse[matrixRepresentation][[First[index],Last[index]]]*electromagneticTensor[[First[#1],
Last[#1]]]*contravariantElectromagneticTensor[[First[#1],Last[#1]]] & ) /@
Tuples[Range[Length[electromagneticPotential]], 2]]] & ) /@
Tuples[Range[Length[electromagneticPotential]], 2]]], False, False]]
StressEnergyTensor["ElectromagneticField", (metricTensor_)[matrixRepresentation_List, coordinates_List, metricIndex1_,
metricIndex2_]] := Module[{electromagneticPotential, newMatrixRepresentation, newElectromagneticPotential,
newCoordinates, covariantElectromagneticPotential, electromagneticTensor, contravariantElectromagneticTensor,
mixedElectromagneticTensor}, electromagneticPotential = Join[{"\[FormalCapitalPhi]" @@ coordinates},
(Superscript["\[FormalCapitalA]", ToString[#1]] @@ coordinates & ) /@ Range[Length[matrixRepresentation] - 1]];
newMatrixRepresentation = matrixRepresentation /. (#1 -> ToExpression[#1] & ) /@ Select[coordinates, StringQ];
newElectromagneticPotential = electromagneticPotential /. (#1 -> ToExpression[#1] & ) /@
Select[coordinates, StringQ]; newCoordinates = coordinates /. (#1 -> ToExpression[#1] & ) /@
Select[coordinates, StringQ]; covariantElectromagneticPotential =
Normal[SparseArray[(Module[{index = #1}, index -> Total[(newMatrixRepresentation[[index,#1]]*
newElectromagneticPotential[[#1]] & ) /@ Range[Length[newElectromagneticPotential]]]] & ) /@
Range[Length[newElectromagneticPotential]]]]; electromagneticTensor =
Normal[SparseArray[(#1 -> D[covariantElectromagneticPotential[[Last[#1]]], newCoordinates[[First[#1]]]] -
D[covariantElectromagneticPotential[[First[#1]]], newCoordinates[[Last[#1]]]] & ) /@
Tuples[Range[Length[newElectromagneticPotential]], 2]]] /. (ToExpression[#1] -> #1 & ) /@
Select[coordinates, StringQ]; contravariantElectromagneticTensor =
Normal[SparseArray[(Module[{index = #1}, index -> Total[(Inverse[matrixRepresentation][[First[index],First[#1]]]*
Inverse[matrixRepresentation][[Last[#1],Last[index]]]*electromagneticTensor[[First[#1],Last[#1]]] & ) /@
Tuples[Range[Length[electromagneticPotential]], 2]]] & ) /@ Tuples[Range[Length[electromagneticPotential]],
2]]]; mixedElectromagneticTensor =
Normal[SparseArray[(Module[{index = #1}, index -> Total[(Inverse[matrixRepresentation][[First[index],#1]]*
electromagneticTensor[[#1,Last[index]]] & ) /@ Range[Length[electromagneticPotential]]]] & ) /@
Tuples[Range[Length[electromagneticPotential]], 2]]]; StressEnergyTensor[ResourceFunction["MetricTensor"][
matrixRepresentation, coordinates, metricIndex1, metricIndex2], (1/Subscript["\[FormalMu]", "0"])*
Normal[SparseArray[(Module[{index = #1}, index -> Total[(contravariantElectromagneticTensor[[First[index],#1]]*
mixedElectromagneticTensor[[Last[index],#1]] & ) /@ Range[Length[electromagneticPotential]]] -
(1/4)*Total[(Inverse[matrixRepresentation][[First[index],Last[index]]]*electromagneticTensor[[First[#1],
Last[#1]]]*contravariantElectromagneticTensor[[First[#1],Last[#1]]] & ) /@
Tuples[Range[Length[electromagneticPotential]], 2]]] & ) /@
Tuples[Range[Length[electromagneticPotential]], 2]]], False, False]] /;
SymbolName[metricTensor] === "MetricTensor" && Length[Dimensions[matrixRepresentation]] == 2 &&
Length[coordinates] == Length[matrixRepresentation] && BooleanQ[metricIndex1] && BooleanQ[metricIndex2]
StressEnergyTensor["ElectromagneticField", index1_, index2_] :=
Module[{matrixRepresentation, electromagneticPotential, coordinates, newMatrixRepresentation,
newElectromagneticPotential, newCoordinates, covariantElectromagneticPotential, electromagneticTensor,
contravariantElectromagneticTensor, mixedElectromagneticTensor},
matrixRepresentation = DiagonalMatrix[Join[{-1}, ConstantArray[1, 3]]]; electromagneticPotential =
Join[{"\[FormalCapitalPhi]" @@ Join[{"\[FormalT]"}, (Superscript["\[FormalX]", ToString[#1]] & ) /@ Range[3]]},
(Module[{index = #1}, Superscript["\[FormalCapitalA]", ToString[index]] @@ Join[{"\[FormalT]"},
(Superscript["\[FormalX]", ToString[#1]] & ) /@ Range[3]]] & ) /@ Range[Length[matrixRepresentation] - 1]];
coordinates = Join[{"\[FormalT]"}, (Superscript["\[FormalX]", ToString[#1]] & ) /@ Range[3]];
newMatrixRepresentation = matrixRepresentation /. (#1 -> ToExpression[#1] & ) /@ Select[coordinates, StringQ];
newElectromagneticPotential = electromagneticPotential /. (#1 -> ToExpression[#1] & ) /@
Select[coordinates, StringQ]; newCoordinates = coordinates /. (#1 -> ToExpression[#1] & ) /@
Select[coordinates, StringQ]; covariantElectromagneticPotential =
Normal[SparseArray[(Module[{index = #1}, index -> Total[(newMatrixRepresentation[[index,#1]]*
newElectromagneticPotential[[#1]] & ) /@ Range[Length[newElectromagneticPotential]]]] & ) /@
Range[Length[newElectromagneticPotential]]]]; electromagneticTensor =
Normal[SparseArray[(#1 -> D[covariantElectromagneticPotential[[Last[#1]]], newCoordinates[[First[#1]]]] -
D[covariantElectromagneticPotential[[First[#1]]], newCoordinates[[Last[#1]]]] & ) /@
Tuples[Range[Length[newElectromagneticPotential]], 2]]] /. (ToExpression[#1] -> #1 & ) /@
Select[coordinates, StringQ]; contravariantElectromagneticTensor =