-
Notifications
You must be signed in to change notification settings - Fork 0
/
Copy pathcyl-generic-static.m
585 lines (276 loc) · 23.7 KB
/
cyl-generic-static.m
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
CompSimp[a_] := Together[Expand[a /. CompSimpRules]]
CompSimpOptions := {SetOptions[Together, Trig -> False], SetOptions[Expand, Trig -> False]}
CompSimpRules = {Cos[theta]^2 -> 1 - Sin[theta]^2, Cos[theta]^4 -> (1 - Sin[theta]^2)^2}
Evaluate[CompSimpOptions]
Off[MetricgFlag]
Print["MetricgFlag has been turned off."]
Rmsign = Rmsign
Rcsign = Rcsign
Dimension = Dimension
MatrixMetricgLower =
Table[Metricg[-i999, -j999], {i999, Dimension}, {j999, Dimension}]
MatrixMetricgUpper =
Inverse[Table[Metricg[-i999, -j999], {i999, Dimension}, {j999, Dimension}]]
Detg = Det[Table[Metricg[-i999, -j999], {i999, Dimension}, {j999, Dimension}]]
Null
Metricg[a_, b_] := Metricg[b, a] /; IndicesAndNotOrderedQ[{a, b}]
AffineG[a_, b_, c_] := AffineG[a, c, b] /; IndicesAndNotOrderedQ[{b, c}]
RicciR[a_, b_] := RicciR[b, a] /; IndicesAndNotOrderedQ[{a, b}]
RicciR[a_, b_] := CompSimp[Sum[Metricg[b, s199]*RicciR[a, -s199], {s199, 1, Dimension}]] /; NegIntegerQ[a] && PosIntegerQ[b]
RicciR[a_, b_] := CompSimp[Sum[Metricg[a, s299]*Metricg[b, s199]*RicciR[-s299, -s199], {s199, 1, Dimension}, {s299, 1, Dimension}]] /; PosIntegerQ[a] && PosIntegerQ[b]
ScalarR = Sum[Metricg[i999, i999]*RicciR[-i999, -i999], {i999, 1, Dimension}] + 2*Sum[Metricg[j999, k999]*RicciR[-j999, -k999], {j999, 1, Dimension}, {k999, 1 + j999, Dimension}]
(* Done *)
CompSimp[a_] := Together[Expand[a /. CompSimpRules]]
CompSimpOptions := {SetOptions[Together, Trig -> False], SetOptions[Expand, Trig -> False]}
CompSimpRules = {Cos[theta]^2 -> 1 - Sin[theta]^2, Cos[theta]^4 -> (1 - Sin[theta]^2)^2}
Evaluate[CompSimpOptions]
Off[MetricgFlag]
Print["MetricgFlag has been turned off."]
Rmsign = Rmsign
Rcsign = Rcsign
Dimension = Dimension
MatrixMetricgLower =
Table[Metricg[-i999, -j999], {i999, Dimension}, {j999, Dimension}]
MatrixMetricgUpper =
Inverse[Table[Metricg[-i999, -j999], {i999, Dimension}, {j999, Dimension}]]
Detg = Det[Table[Metricg[-i999, -j999], {i999, Dimension}, {j999, Dimension}]]
Null
Metricg[a_, b_] := Metricg[b, a] /; IndicesAndNotOrderedQ[{a, b}]
AffineG[a_, b_, c_] := AffineG[a, c, b] /; IndicesAndNotOrderedQ[{b, c}]
RicciR[a_, b_] := RicciR[b, a] /; IndicesAndNotOrderedQ[{a, b}]
RicciR[a_, b_] := CompSimp[Sum[Metricg[b, s199]*RicciR[a, -s199], {s199, 1, Dimension}]] /; NegIntegerQ[a] && PosIntegerQ[b]
RicciR[a_, b_] := CompSimp[Sum[Metricg[a, s299]*Metricg[b, s199]*RicciR[-s299, -s199], {s199, 1, Dimension}, {s299, 1, Dimension}]] /; PosIntegerQ[a] && PosIntegerQ[b]
ScalarR = Sum[Metricg[i999, i999]*RicciR[-i999, -i999], {i999, 1, Dimension}] + 2*Sum[Metricg[j999, k999]*RicciR[-j999, -k999], {j999, 1, Dimension}, {k999, 1 + j999, Dimension}]
(* Done *)
CompSimp[a_] := Together[Expand[a /. CompSimpRules]]
CompSimpOptions := {SetOptions[Together, Trig -> False], SetOptions[Expand, Trig -> False]}
CompSimpRules = {Cos[theta]^2 -> 1 - Sin[theta]^2, Cos[theta]^4 -> (1 - Sin[theta]^2)^2}
Evaluate[CompSimpOptions]
Off[MetricgFlag]
Print["MetricgFlag has been turned off."]
Rmsign = Rmsign
Rcsign = Rcsign
Dimension = Dimension
MatrixMetricgLower =
Table[Metricg[-i999, -j999], {i999, Dimension}, {j999, Dimension}]
MatrixMetricgUpper =
Inverse[Table[Metricg[-i999, -j999], {i999, Dimension}, {j999, Dimension}]]
Detg = Det[Table[Metricg[-i999, -j999], {i999, Dimension}, {j999, Dimension}]]
Null
Metricg[a_, b_] := Metricg[b, a] /; IndicesAndNotOrderedQ[{a, b}]
AffineG[a_, b_, c_] := AffineG[a, c, b] /; IndicesAndNotOrderedQ[{b, c}]
RicciR[a_, b_] := RicciR[b, a] /; IndicesAndNotOrderedQ[{a, b}]
RicciR[a_, b_] := CompSimp[Sum[Metricg[b, s199]*RicciR[a, -s199], {s199, 1, Dimension}]] /; NegIntegerQ[a] && PosIntegerQ[b]
RicciR[a_, b_] := CompSimp[Sum[Metricg[a, s299]*Metricg[b, s199]*RicciR[-s299, -s199], {s199, 1, Dimension}, {s299, 1, Dimension}]] /; PosIntegerQ[a] && PosIntegerQ[b]
ScalarR = Sum[Metricg[i999, i999]*RicciR[-i999, -i999], {i999, 1, Dimension}] + 2*Sum[Metricg[j999, k999]*RicciR[-j999, -k999], {j999, 1, Dimension}, {k999, 1 + j999, Dimension}]
(* Done *)
CompSimp[a_] := Together[Expand[a /. CompSimpRules]]
CompSimpOptions := {SetOptions[Together, Trig -> False], SetOptions[Expand, Trig -> False]}
CompSimpRules = {Cos[theta]^2 -> 1 - Sin[theta]^2, Cos[theta]^4 -> (1 - Sin[theta]^2)^2}
Evaluate[CompSimpOptions]
Off[MetricgFlag]
Print["MetricgFlag has been turned off."]
Rmsign = Rmsign
Rcsign = Rcsign
Dimension = Dimension
MatrixMetricgLower =
Table[Metricg[-i999, -j999], {i999, Dimension}, {j999, Dimension}]
MatrixMetricgUpper =
Inverse[Table[Metricg[-i999, -j999], {i999, Dimension}, {j999, Dimension}]]
Detg = Det[Table[Metricg[-i999, -j999], {i999, Dimension}, {j999, Dimension}]]
Null
Metricg[a_, b_] := Metricg[b, a] /; IndicesAndNotOrderedQ[{a, b}]
AffineG[a_, b_, c_] := AffineG[a, c, b] /; IndicesAndNotOrderedQ[{b, c}]
RicciR[a_, b_] := RicciR[b, a] /; IndicesAndNotOrderedQ[{a, b}]
RicciR[a_, b_] := CompSimp[Sum[Metricg[b, s199]*RicciR[a, -s199], {s199, 1, Dimension}]] /; NegIntegerQ[a] && PosIntegerQ[b]
RicciR[a_, b_] := CompSimp[Sum[Metricg[a, s299]*Metricg[b, s199]*RicciR[-s299, -s199], {s199, 1, Dimension}, {s299, 1, Dimension}]] /; PosIntegerQ[a] && PosIntegerQ[b]
ScalarR = Sum[Metricg[i999, i999]*RicciR[-i999, -i999], {i999, 1, Dimension}] + 2*Sum[Metricg[j999, k999]*RicciR[-j999, -k999], {j999, 1, Dimension}, {k999, 1 + j999, Dimension}]
(* Done *)
CompSimp[a_] := Together[Expand[a //. CompSimpRules]]
CompSimpOptions := {SetOptions[Together, Trig -> False], SetOptions[Expand, Trig -> False]}
CompSimpRules = {}
Evaluate[CompSimpOptions]
Off[MetricgFlag]
Print["MetricgFlag has been turned off."]
Rmsign = 1
Rcsign = 1
Dimension = 4
MatrixMetricgLower =
{{-E^(2*(-\[Lambda][\[Rho], z] + \[Nu][\[Rho], z])), 0, 0, 0}, {0, -E^(2*(-\[Lambda][\[Rho], z] + \[Nu][\[Rho], z])), 0, 0}, {0, 0, -(\[Rho]^2/E^(2*\[Lambda][\[Rho], z])), 0}, {0, 0, 0, E^(2*\[Lambda][\[Rho], z])}}
MatrixMetricgUpper =
{{-E^(2*\[Lambda][\[Rho], z] - 2*\[Nu][\[Rho], z]), 0, 0, 0}, {0, -E^(2*\[Lambda][\[Rho], z] - 2*\[Nu][\[Rho], z]), 0, 0}, {0, 0, -(E^(2*\[Lambda][\[Rho], z])/\[Rho]^2), 0}, {0, 0, 0, E^(-2*\[Lambda][\[Rho], z])}}
Detg = -(E^(-4*\[Lambda][\[Rho], z] + 4*\[Nu][\[Rho], z])*\[Rho]^2)
x[1] = \[Rho]
x[2] = z
x[3] = \[Phi]
x[4] = t
Metricg[-4, -4] = E^(2*\[Lambda][\[Rho], z])
Metricg[-4, -3] = 0
Metricg[-4, -2] = 0
Metricg[-4, -1] = 0
Metricg[-4, 1] = 0
Metricg[-4, 2] = 0
Metricg[-4, 3] = 0
Metricg[-4, 4] = 1
Metricg[-3, -3] = -(\[Rho]^2/E^(2*\[Lambda][\[Rho], z]))
Metricg[-3, -2] = 0
Metricg[-3, -1] = 0
Metricg[-3, 1] = 0
Metricg[-3, 2] = 0
Metricg[-3, 3] = 1
Metricg[-3, 4] = 0
Metricg[-2, -2] = -E^(2*(-\[Lambda][\[Rho], z] + \[Nu][\[Rho], z]))
Metricg[-2, -1] = 0
Metricg[-2, 1] = 0
Metricg[-2, 2] = 1
Metricg[-2, 3] = 0
Metricg[-2, 4] = 0
Metricg[-1, -1] = -E^(2*(-\[Lambda][\[Rho], z] + \[Nu][\[Rho], z]))
Metricg[-1, 1] = 1
Metricg[-1, 2] = 0
Metricg[-1, 3] = 0
Metricg[-1, 4] = 0
Metricg[1, 1] = -E^(2*\[Lambda][\[Rho], z] - 2*\[Nu][\[Rho], z])
Metricg[1, 2] = 0
Metricg[1, 3] = 0
Metricg[1, 4] = 0
Metricg[2, 2] = -E^(2*\[Lambda][\[Rho], z] - 2*\[Nu][\[Rho], z])
Metricg[2, 3] = 0
Metricg[2, 4] = 0
Metricg[3, 3] = -(E^(2*\[Lambda][\[Rho], z])/\[Rho]^2)
Metricg[3, 4] = 0
Metricg[4, 4] = E^(-2*\[Lambda][\[Rho], z])
Metricg[a_, b_] := Metricg[b, a] /; IndicesAndNotOrderedQ[{a, b}]
AffineG[1, -4, -4] = E^(4*\[Lambda][\[Rho], z] - 2*\[Nu][\[Rho], z])*Derivative[1, 0][\[Lambda]][\[Rho], z]
AffineG[1, -4, -3] = 0
AffineG[1, -4, -2] = 0
AffineG[1, -4, -1] = 0
AffineG[1, -3, -3] = (\[Rho]*(-1 + \[Rho]*Derivative[1, 0][\[Lambda]][\[Rho], z]))/E^(2*\[Nu][\[Rho], z])
AffineG[1, -3, -2] = 0
AffineG[1, -3, -1] = 0
AffineG[1, -2, -2] = Derivative[1, 0][\[Lambda]][\[Rho], z] - Derivative[1, 0][\[Nu]][\[Rho], z]
AffineG[1, -2, -1] = -Derivative[0, 1][\[Lambda]][\[Rho], z] + Derivative[0, 1][\[Nu]][\[Rho], z]
AffineG[1, -1, -1] = -Derivative[1, 0][\[Lambda]][\[Rho], z] + Derivative[1, 0][\[Nu]][\[Rho], z]
AffineG[2, -4, -4] = E^(4*\[Lambda][\[Rho], z] - 2*\[Nu][\[Rho], z])*Derivative[0, 1][\[Lambda]][\[Rho], z]
AffineG[2, -4, -3] = 0
AffineG[2, -4, -2] = 0
AffineG[2, -4, -1] = 0
AffineG[2, -3, -3] = (\[Rho]^2*Derivative[0, 1][\[Lambda]][\[Rho], z])/E^(2*\[Nu][\[Rho], z])
AffineG[2, -3, -2] = 0
AffineG[2, -3, -1] = 0
AffineG[2, -2, -2] = -Derivative[0, 1][\[Lambda]][\[Rho], z] + Derivative[0, 1][\[Nu]][\[Rho], z]
AffineG[2, -2, -1] = -Derivative[1, 0][\[Lambda]][\[Rho], z] + Derivative[1, 0][\[Nu]][\[Rho], z]
AffineG[2, -1, -1] = Derivative[0, 1][\[Lambda]][\[Rho], z] - Derivative[0, 1][\[Nu]][\[Rho], z]
AffineG[3, -4, -4] = 0
AffineG[3, -4, -3] = 0
AffineG[3, -4, -2] = 0
AffineG[3, -4, -1] = 0
AffineG[3, -3, -3] = 0
AffineG[3, -3, -2] = -Derivative[0, 1][\[Lambda]][\[Rho], z]
AffineG[3, -3, -1] = (1 - \[Rho]*Derivative[1, 0][\[Lambda]][\[Rho], z])/\[Rho]
AffineG[3, -2, -2] = 0
AffineG[3, -2, -1] = 0
AffineG[3, -1, -1] = 0
AffineG[4, -4, -4] = 0
AffineG[4, -4, -3] = 0
AffineG[4, -4, -2] = Derivative[0, 1][\[Lambda]][\[Rho], z]
AffineG[4, -4, -1] = Derivative[1, 0][\[Lambda]][\[Rho], z]
AffineG[4, -3, -3] = 0
AffineG[4, -3, -2] = 0
AffineG[4, -3, -1] = 0
AffineG[4, -2, -2] = 0
AffineG[4, -2, -1] = 0
AffineG[4, -1, -1] = 0
AffineG[a_, b_, c_] := AffineG[a, c, b] /; IndicesAndNotOrderedQ[{b, c}]
RiemannR[-4, -3, -4, -3] = E^(2*\[Lambda][\[Rho], z] - 2*\[Nu][\[Rho], z])*\[Rho]*(\[Rho]*Derivative[0, 1][\[Lambda]][\[Rho], z]^2 - Derivative[1, 0][\[Lambda]][\[Rho], z] + \[Rho]*Derivative[1, 0][\[Lambda]][\[Rho], z]^2)
RiemannR[-4, -3, -4, -2] = 0
RiemannR[-4, -3, -4, -1] = 0
RiemannR[-4, -3, -3, -2] = 0
RiemannR[-4, -3, -3, -1] = 0
RiemannR[-4, -3, -2, -1] = 0
RiemannR[-4, -2, -4, -2] = -(E^(2*\[Lambda][\[Rho], z])*(2*Derivative[0, 1][\[Lambda]][\[Rho], z]^2 - Derivative[0, 1][\[Lambda]][\[Rho], z]*Derivative[0, 1][\[Nu]][\[Rho], z] + Derivative[0, 2][\[Lambda]][\[Rho], z] - Derivative[1, 0][\[Lambda]][\[Rho], z]^2 + Derivative[1, 0][\[Lambda]][\[Rho], z]*Derivative[1, 0][\[Nu]][\[Rho], z]))
RiemannR[-4, -2, -4, -1] = -(E^(2*\[Lambda][\[Rho], z])*(3*Derivative[0, 1][\[Lambda]][\[Rho], z]*Derivative[1, 0][\[Lambda]][\[Rho], z] - Derivative[0, 1][\[Nu]][\[Rho], z]*Derivative[1, 0][\[Lambda]][\[Rho], z] - Derivative[0, 1][\[Lambda]][\[Rho], z]*Derivative[1, 0][\[Nu]][\[Rho], z] + Derivative[1, 1][\[Lambda]][\[Rho], z]))
RiemannR[-4, -2, -3, -2] = 0
RiemannR[-4, -2, -3, -1] = 0
RiemannR[-4, -2, -2, -1] = 0
RiemannR[-4, -1, -4, -1] = E^(2*\[Lambda][\[Rho], z])*(Derivative[0, 1][\[Lambda]][\[Rho], z]^2 - Derivative[0, 1][\[Lambda]][\[Rho], z]*Derivative[0, 1][\[Nu]][\[Rho], z] - 2*Derivative[1, 0][\[Lambda]][\[Rho], z]^2 + Derivative[1, 0][\[Lambda]][\[Rho], z]*Derivative[1, 0][\[Nu]][\[Rho], z] - Derivative[2, 0][\[Lambda]][\[Rho], z])
RiemannR[-4, -1, -3, -2] = 0
RiemannR[-4, -1, -3, -1] = 0
RiemannR[-4, -1, -2, -1] = 0
RiemannR[-3, -2, -3, -2] = (\[Rho]*(\[Rho]*Derivative[0, 1][\[Lambda]][\[Rho], z]*Derivative[0, 1][\[Nu]][\[Rho], z] - \[Rho]*Derivative[0, 2][\[Lambda]][\[Rho], z] - Derivative[1, 0][\[Lambda]][\[Rho], z] + \[Rho]*Derivative[1, 0][\[Lambda]][\[Rho], z]^2 + Derivative[1, 0][\[Nu]][\[Rho], z] - \[Rho]*Derivative[1, 0][\[Lambda]][\[Rho], z]*Derivative[1, 0][\[Nu]][\[Rho], z]))/E^(2*\[Lambda][\[Rho], z])
RiemannR[-3, -2, -3, -1] = (\[Rho]*(-Derivative[0, 1][\[Nu]][\[Rho], z] - \[Rho]*Derivative[0, 1][\[Lambda]][\[Rho], z]*Derivative[1, 0][\[Lambda]][\[Rho], z] + \[Rho]*Derivative[0, 1][\[Nu]][\[Rho], z]*Derivative[1, 0][\[Lambda]][\[Rho], z] + \[Rho]*Derivative[0, 1][\[Lambda]][\[Rho], z]*Derivative[1, 0][\[Nu]][\[Rho], z] - \[Rho]*Derivative[1, 1][\[Lambda]][\[Rho], z]))/E^(2*\[Lambda][\[Rho], z])
RiemannR[-3, -2, -2, -1] = 0
RiemannR[-3, -1, -3, -1] = (\[Rho]*(\[Rho]*Derivative[0, 1][\[Lambda]][\[Rho], z]^2 - \[Rho]*Derivative[0, 1][\[Lambda]][\[Rho], z]*Derivative[0, 1][\[Nu]][\[Rho], z] - Derivative[1, 0][\[Lambda]][\[Rho], z] - Derivative[1, 0][\[Nu]][\[Rho], z] + \[Rho]*Derivative[1, 0][\[Lambda]][\[Rho], z]*Derivative[1, 0][\[Nu]][\[Rho], z] - \[Rho]*Derivative[2, 0][\[Lambda]][\[Rho], z]))/E^(2*\[Lambda][\[Rho], z])
RiemannR[-3, -1, -2, -1] = 0
RiemannR[-2, -1, -2, -1] = -(E^(-2*\[Lambda][\[Rho], z] + 2*\[Nu][\[Rho], z])*(Derivative[0, 2][\[Lambda]][\[Rho], z] - Derivative[0, 2][\[Nu]][\[Rho], z] + Derivative[2, 0][\[Lambda]][\[Rho], z] - Derivative[2, 0][\[Nu]][\[Rho], z]))
RiemannR[i_, j_, k_, l_] := RiemannR[k, l, i, j] /; i == k && IndicesAndNotOrderedQ[{j, l}]
RiemannR[i_, j_, k_, l_] := 0 /; i == j || k == l
RiemannR[a_, b_, c_, d_] := -RiemannR[b, a, c, d] /; IndicesAndNotOrderedQ[{a, b}]
RiemannR[a_, b_, c_, d_] := -RiemannR[a, b, d, c] /; IndicesAndNotOrderedQ[{c, d}]
RiemannR[a_, b_, c_, d_] := RiemannR[c, d, a, b] /; IndicesAndNotOrderedQ[{a, c}]
RiemannR[a_, b_, c_, d_] := CompSimp[Sum[Metricg[d, s199]*RiemannR[a, b, c, -s199], {s199, 1, Dimension}]] /; NegIntegerQ[a] && NegIntegerQ[b] && NegIntegerQ[c] && PosIntegerQ[d]
RiemannR[a_, b_, c_, d_] := CompSimp[Sum[Metricg[c, s299]*Metricg[d, s199]*RiemannR[a, b, -s299, -s199], {s199, 1, Dimension}, {s299, 1, Dimension}]] /; NegIntegerQ[a] && NegIntegerQ[b] && PosIntegerQ[c] && PosIntegerQ[d]
RiemannR[a_, b_, c_, d_] := CompSimp[Sum[Metricg[b, s299]*Metricg[d, s199]*RiemannR[a, -s299, c, -s199], {s199, 1, Dimension}, {s299, 1, Dimension}]] /; NegIntegerQ[a] && PosIntegerQ[b] && NegIntegerQ[c] && PosIntegerQ[d]
RiemannR[a_, b_, c_, d_] := CompSimp[Sum[Metricg[b, s399]*Metricg[c, s299]*Metricg[d, s199]*RiemannR[a, -s399, -s299, -s199], {s199, 1, Dimension}, {s299, 1, Dimension}, {s399, 1, Dimension}]] /; NegIntegerQ[a] && PosIntegerQ[b] && PosIntegerQ[c] && PosIntegerQ[d]
RiemannR[a_, b_, c_, d_] := CompSimp[Sum[Metricg[a, s499]*Metricg[b, s399]*Metricg[c, s299]*Metricg[d, s199]*RiemannR[-s499, -s399, -s299, -s199], {s199, 1, Dimension}, {s299, 1, Dimension}, {s399, 1, Dimension}, {s499, 1, Dimension}]] /; PosIntegerQ[a] && PosIntegerQ[b] && PosIntegerQ[c] && PosIntegerQ[d]
RiemannR[a_, b_, c_, d_] := CompSimp[Sum[Metricg[b, s199]*RiemannR[a, -s199, c, d], {s199, 1, Dimension}]] /; NegIntegerQ[a] && PosIntegerQ[b] && NegIntegerQ[c] && NegIntegerQ[d]
RicciR[-4, -4] = (E^(4*\[Lambda][\[Rho], z] - 2*\[Nu][\[Rho], z])*(\[Rho]*Derivative[0, 2][\[Lambda]][\[Rho], z] + Derivative[1, 0][\[Lambda]][\[Rho], z] + \[Rho]*Derivative[2, 0][\[Lambda]][\[Rho], z]))/\[Rho]
RicciR[-4, -3] = 0
RicciR[-4, -2] = 0
RicciR[-4, -1] = 0
RicciR[-3, -3] = (\[Rho]*(\[Rho]*Derivative[0, 2][\[Lambda]][\[Rho], z] + Derivative[1, 0][\[Lambda]][\[Rho], z] + \[Rho]*Derivative[2, 0][\[Lambda]][\[Rho], z]))/E^(2*\[Nu][\[Rho], z])
RicciR[-3, -2] = 0
RicciR[-3, -1] = 0
RicciR[-2, -2] = (-2*\[Rho]*Derivative[0, 1][\[Lambda]][\[Rho], z]^2 + \[Rho]*Derivative[0, 2][\[Lambda]][\[Rho], z] - \[Rho]*Derivative[0, 2][\[Nu]][\[Rho], z] + Derivative[1, 0][\[Lambda]][\[Rho], z] - Derivative[1, 0][\[Nu]][\[Rho], z] + \[Rho]*Derivative[2, 0][\[Lambda]][\[Rho], z] - \[Rho]*Derivative[2, 0][\[Nu]][\[Rho], z])/\[Rho]
RicciR[-2, -1] = (Derivative[0, 1][\[Nu]][\[Rho], z] - 2*\[Rho]*Derivative[0, 1][\[Lambda]][\[Rho], z]*Derivative[1, 0][\[Lambda]][\[Rho], z])/\[Rho]
RicciR[-1, -1] = (\[Rho]*Derivative[0, 2][\[Lambda]][\[Rho], z] - \[Rho]*Derivative[0, 2][\[Nu]][\[Rho], z] + Derivative[1, 0][\[Lambda]][\[Rho], z] - 2*\[Rho]*Derivative[1, 0][\[Lambda]][\[Rho], z]^2 + Derivative[1, 0][\[Nu]][\[Rho], z] + \[Rho]*Derivative[2, 0][\[Lambda]][\[Rho], z] - \[Rho]*Derivative[2, 0][\[Nu]][\[Rho], z])/\[Rho]
RicciR[a_, b_] := RicciR[b, a] /; IndicesAndNotOrderedQ[{a, b}]
RicciR[a_, b_] := CompSimp[Sum[Metricg[b, s199]*RicciR[a, -s199], {s199, 1, Dimension}]] /; NegIntegerQ[a] && PosIntegerQ[b]
RicciR[a_, b_] := CompSimp[Sum[Metricg[a, s299]*Metricg[b, s199]*RicciR[-s299, -s199], {s199, 1, Dimension}, {s299, 1, Dimension}]] /; PosIntegerQ[a] && PosIntegerQ[b]
ScalarR = (2*E^(2*\[Lambda][\[Rho], z] - 2*\[Nu][\[Rho], z])*(\[Rho]*Derivative[0, 1][\[Lambda]][\[Rho], z]^2 - \[Rho]*Derivative[0, 2][\[Lambda]][\[Rho], z] + \[Rho]*Derivative[0, 2][\[Nu]][\[Rho], z] - Derivative[1, 0][\[Lambda]][\[Rho], z] + \[Rho]*Derivative[1, 0][\[Lambda]][\[Rho], z]^2 - \[Rho]*Derivative[2, 0][\[Lambda]][\[Rho], z] + \[Rho]*Derivative[2, 0][\[Nu]][\[Rho], z]))/\[Rho]
EinsteinG[-4, -4] = -((E^(4*\[Lambda][\[Rho], z] - 2*\[Nu][\[Rho], z])*(\[Rho]*Derivative[0, 1][\[Lambda]][\[Rho], z]^2 - 2*\[Rho]*Derivative[0, 2][\[Lambda]][\[Rho], z] + \[Rho]*Derivative[0, 2][\[Nu]][\[Rho], z] - 2*Derivative[1, 0][\[Lambda]][\[Rho], z] + \[Rho]*Derivative[1, 0][\[Lambda]][\[Rho], z]^2 - 2*\[Rho]*Derivative[2, 0][\[Lambda]][\[Rho], z] + \[Rho]*Derivative[2, 0][\[Nu]][\[Rho], z]))/\[Rho])
EinsteinG[-4, -3] = 0
EinsteinG[-4, -2] = 0
EinsteinG[-4, -1] = 0
EinsteinG[-3, -3] = (\[Rho]^2*(Derivative[0, 1][\[Lambda]][\[Rho], z]^2 + Derivative[0, 2][\[Nu]][\[Rho], z] + Derivative[1, 0][\[Lambda]][\[Rho], z]^2 + Derivative[2, 0][\[Nu]][\[Rho], z]))/E^(2*\[Nu][\[Rho], z])
EinsteinG[-3, -2] = 0
EinsteinG[-3, -1] = 0
EinsteinG[-2, -2] = (-(\[Rho]*Derivative[0, 1][\[Lambda]][\[Rho], z]^2) + \[Rho]*Derivative[1, 0][\[Lambda]][\[Rho], z]^2 - Derivative[1, 0][\[Nu]][\[Rho], z])/\[Rho]
EinsteinG[-2, -1] = (Derivative[0, 1][\[Nu]][\[Rho], z] - 2*\[Rho]*Derivative[0, 1][\[Lambda]][\[Rho], z]*Derivative[1, 0][\[Lambda]][\[Rho], z])/\[Rho]
EinsteinG[-1, -1] = (\[Rho]*Derivative[0, 1][\[Lambda]][\[Rho], z]^2 - \[Rho]*Derivative[1, 0][\[Lambda]][\[Rho], z]^2 + Derivative[1, 0][\[Nu]][\[Rho], z])/\[Rho]
EinsteinG[i_, j_] := EinsteinG[j, i] /; IndicesAndNotOrderedQ[{i, j}]
EinsteinG[a_, b_] := CompSimp[Sum[Metricg[b, s199]*EinsteinG[a, -s199], {s199, 1, Dimension}]] /; NegIntegerQ[a] && PosIntegerQ[b]
EinsteinG[a_, b_] := CompSimp[Sum[Metricg[a, s299]*Metricg[b, s199]*EinsteinG[-s299, -s199], {s199, 1, Dimension}, {s299, 1, Dimension}]] /; PosIntegerQ[a] && PosIntegerQ[b]
WeylC[-4, -3, -4, -3] = (E^(2*\[Lambda][\[Rho], z] - 2*\[Nu][\[Rho], z])*\[Rho]*(2*\[Rho]*Derivative[0, 1][\[Lambda]][\[Rho], z]^2 + \[Rho]*Derivative[0, 2][\[Lambda]][\[Rho], z] - \[Rho]*Derivative[0, 2][\[Nu]][\[Rho], z] - 2*Derivative[1, 0][\[Lambda]][\[Rho], z] + 2*\[Rho]*Derivative[1, 0][\[Lambda]][\[Rho], z]^2 + \[Rho]*Derivative[2, 0][\[Lambda]][\[Rho], z] - \[Rho]*Derivative[2, 0][\[Nu]][\[Rho], z]))/3
WeylC[-4, -3, -4, -2] = 0
WeylC[-4, -3, -4, -1] = 0
WeylC[-4, -3, -3, -2] = 0
WeylC[-4, -3, -3, -1] = 0
WeylC[-4, -3, -2, -1] = 0
WeylC[-4, -2, -4, -2] = -(E^(2*\[Lambda][\[Rho], z])*(8*\[Rho]*Derivative[0, 1][\[Lambda]][\[Rho], z]^2 - 6*\[Rho]*Derivative[0, 1][\[Lambda]][\[Rho], z]*Derivative[0, 1][\[Nu]][\[Rho], z] + 4*\[Rho]*Derivative[0, 2][\[Lambda]][\[Rho], z] - \[Rho]*Derivative[0, 2][\[Nu]][\[Rho], z] - 2*Derivative[1, 0][\[Lambda]][\[Rho], z] - 4*\[Rho]*Derivative[1, 0][\[Lambda]][\[Rho], z]^2 - 3*Derivative[1, 0][\[Nu]][\[Rho], z] + 6*\[Rho]*Derivative[1, 0][\[Lambda]][\[Rho], z]*Derivative[1, 0][\[Nu]][\[Rho], z] - 2*\[Rho]*Derivative[2, 0][\[Lambda]][\[Rho], z] - \[Rho]*Derivative[2, 0][\[Nu]][\[Rho], z]))/(6*\[Rho])
WeylC[-4, -2, -4, -1] = (E^(2*\[Lambda][\[Rho], z])*(-Derivative[0, 1][\[Nu]][\[Rho], z] - 4*\[Rho]*Derivative[0, 1][\[Lambda]][\[Rho], z]*Derivative[1, 0][\[Lambda]][\[Rho], z] + 2*\[Rho]*Derivative[0, 1][\[Nu]][\[Rho], z]*Derivative[1, 0][\[Lambda]][\[Rho], z] + 2*\[Rho]*Derivative[0, 1][\[Lambda]][\[Rho], z]*Derivative[1, 0][\[Nu]][\[Rho], z] - 2*\[Rho]*Derivative[1, 1][\[Lambda]][\[Rho], z]))/(2*\[Rho])
WeylC[-4, -2, -3, -2] = 0
WeylC[-4, -2, -3, -1] = 0
WeylC[-4, -2, -2, -1] = 0
WeylC[-4, -1, -4, -1] = (E^(2*\[Lambda][\[Rho], z])*(4*\[Rho]*Derivative[0, 1][\[Lambda]][\[Rho], z]^2 - 6*\[Rho]*Derivative[0, 1][\[Lambda]][\[Rho], z]*Derivative[0, 1][\[Nu]][\[Rho], z] + 2*\[Rho]*Derivative[0, 2][\[Lambda]][\[Rho], z] + \[Rho]*Derivative[0, 2][\[Nu]][\[Rho], z] + 2*Derivative[1, 0][\[Lambda]][\[Rho], z] - 8*\[Rho]*Derivative[1, 0][\[Lambda]][\[Rho], z]^2 - 3*Derivative[1, 0][\[Nu]][\[Rho], z] + 6*\[Rho]*Derivative[1, 0][\[Lambda]][\[Rho], z]*Derivative[1, 0][\[Nu]][\[Rho], z] - 4*\[Rho]*Derivative[2, 0][\[Lambda]][\[Rho], z] + \[Rho]*Derivative[2, 0][\[Nu]][\[Rho], z]))/(6*\[Rho])
WeylC[-4, -1, -3, -2] = 0
WeylC[-4, -1, -3, -1] = 0
WeylC[-4, -1, -2, -1] = 0
WeylC[-3, -2, -3, -2] = -(\[Rho]*(4*\[Rho]*Derivative[0, 1][\[Lambda]][\[Rho], z]^2 - 6*\[Rho]*Derivative[0, 1][\[Lambda]][\[Rho], z]*Derivative[0, 1][\[Nu]][\[Rho], z] + 2*\[Rho]*Derivative[0, 2][\[Lambda]][\[Rho], z] + \[Rho]*Derivative[0, 2][\[Nu]][\[Rho], z] + 2*Derivative[1, 0][\[Lambda]][\[Rho], z] - 8*\[Rho]*Derivative[1, 0][\[Lambda]][\[Rho], z]^2 - 3*Derivative[1, 0][\[Nu]][\[Rho], z] + 6*\[Rho]*Derivative[1, 0][\[Lambda]][\[Rho], z]*Derivative[1, 0][\[Nu]][\[Rho], z] - 4*\[Rho]*Derivative[2, 0][\[Lambda]][\[Rho], z] + \[Rho]*Derivative[2, 0][\[Nu]][\[Rho], z]))/(6*E^(2*\[Lambda][\[Rho], z]))
WeylC[-3, -2, -3, -1] = (\[Rho]*(-Derivative[0, 1][\[Nu]][\[Rho], z] - 4*\[Rho]*Derivative[0, 1][\[Lambda]][\[Rho], z]*Derivative[1, 0][\[Lambda]][\[Rho], z] + 2*\[Rho]*Derivative[0, 1][\[Nu]][\[Rho], z]*Derivative[1, 0][\[Lambda]][\[Rho], z] + 2*\[Rho]*Derivative[0, 1][\[Lambda]][\[Rho], z]*Derivative[1, 0][\[Nu]][\[Rho], z] - 2*\[Rho]*Derivative[1, 1][\[Lambda]][\[Rho], z]))/(2*E^(2*\[Lambda][\[Rho], z]))
WeylC[-3, -2, -2, -1] = 0
WeylC[-3, -1, -3, -1] = (\[Rho]*(8*\[Rho]*Derivative[0, 1][\[Lambda]][\[Rho], z]^2 - 6*\[Rho]*Derivative[0, 1][\[Lambda]][\[Rho], z]*Derivative[0, 1][\[Nu]][\[Rho], z] + 4*\[Rho]*Derivative[0, 2][\[Lambda]][\[Rho], z] - \[Rho]*Derivative[0, 2][\[Nu]][\[Rho], z] - 2*Derivative[1, 0][\[Lambda]][\[Rho], z] - 4*\[Rho]*Derivative[1, 0][\[Lambda]][\[Rho], z]^2 - 3*Derivative[1, 0][\[Nu]][\[Rho], z] + 6*\[Rho]*Derivative[1, 0][\[Lambda]][\[Rho], z]*Derivative[1, 0][\[Nu]][\[Rho], z] - 2*\[Rho]*Derivative[2, 0][\[Lambda]][\[Rho], z] - \[Rho]*Derivative[2, 0][\[Nu]][\[Rho], z]))/(6*E^(2*\[Lambda][\[Rho], z]))
WeylC[-3, -1, -2, -1] = 0
WeylC[-2, -1, -2, -1] = -(E^(-2*\[Lambda][\[Rho], z] + 2*\[Nu][\[Rho], z])*(2*\[Rho]*Derivative[0, 1][\[Lambda]][\[Rho], z]^2 + \[Rho]*Derivative[0, 2][\[Lambda]][\[Rho], z] - \[Rho]*Derivative[0, 2][\[Nu]][\[Rho], z] - 2*Derivative[1, 0][\[Lambda]][\[Rho], z] + 2*\[Rho]*Derivative[1, 0][\[Lambda]][\[Rho], z]^2 + \[Rho]*Derivative[2, 0][\[Lambda]][\[Rho], z] - \[Rho]*Derivative[2, 0][\[Nu]][\[Rho], z]))/(3*\[Rho])
WeylC[i_, j_, k_, l_] := 0 /; i == j || k == l
WeylC[i_, j_, k_, l_] := -WeylC[i, j, l, k] /; IndicesAndNotOrderedQ[{k, l}]
WeylC[i_, j_, k_, l_] := -WeylC[j, i, k, l] /; IndicesAndNotOrderedQ[{i, j}]
WeylC[i_, j_, k_, l_] := WeylC[k, l, i, j] /; IndicesAndNotOrderedQ[{i, k}]
WeylC[i_, j_, k_, l_] := WeylC[k, l, i, j] /; i == k && IndicesAndNotOrderedQ[{j, l}]
WeylC[a_, b_, c_, d_] := CompSimp[Sum[Metricg[d, s199]*WeylC[a, b, c, -s199], {s199, 1, Dimension}]] /; NegIntegerQ[a] && NegIntegerQ[b] && NegIntegerQ[c] && PosIntegerQ[d]
WeylC[a_, b_, c_, d_] := CompSimp[Sum[Metricg[c, s299]*Metricg[d, s199]*WeylC[a, b, -s299, -s199], {s199, 1, Dimension}, {s299, 1, Dimension}]] /; NegIntegerQ[a] && NegIntegerQ[b] && PosIntegerQ[c] && PosIntegerQ[d]
WeylC[a_, b_, c_, d_] := CompSimp[Sum[Metricg[b, s299]*Metricg[d, s199]*WeylC[a, -s299, c, -s199], {s199, 1, Dimension}, {s299, 1, Dimension}]] /; NegIntegerQ[a] && PosIntegerQ[b] && NegIntegerQ[c] && PosIntegerQ[d]
WeylC[a_, b_, c_, d_] := CompSimp[Sum[Metricg[b, s399]*Metricg[c, s299]*Metricg[d, s199]*WeylC[a, -s399, -s299, -s199], {s199, 1, Dimension}, {s299, 1, Dimension}, {s399, 1, Dimension}]] /; NegIntegerQ[a] && PosIntegerQ[b] && PosIntegerQ[c] && PosIntegerQ[d]
WeylC[a_, b_, c_, d_] := CompSimp[Sum[Metricg[a, s499]*Metricg[b, s399]*Metricg[c, s299]*Metricg[d, s199]*WeylC[-s499, -s399, -s299, -s199], {s199, 1, Dimension}, {s299, 1, Dimension}, {s399, 1, Dimension}, {s499, 1, Dimension}]] /; PosIntegerQ[a] && PosIntegerQ[b] && PosIntegerQ[c] && PosIntegerQ[d]
WeylC[a_, b_, c_, d_] := CompSimp[Sum[Metricg[b, s199]*RiemannR[a, -s199, c, d], {s199, 1, Dimension}]] /; NegIntegerQ[a] && PosIntegerQ[b] && NegIntegerQ[c] && NegIntegerQ[d]
(* Done *)