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
|
! $Header$
#include "cctk.h"
module adm_metric
use tensor
implicit none
private
public calc_4metric
public calc_4metricderivs
public calc_4metricderivs2
public calc_3metric
public calc_3metricderivs
public calc_3metricdot
public calc_extcurv
public calc_3metricdot_simple
public calc_3metricderivdot_simple
public calc_extcurvdot_simple
public calc_3metricdot2_simple
contains
subroutine calc_4metric (gg, alfa, beta, g4)
CCTK_REAL, intent(in) :: gg(3,3), alfa, beta(3)
CCTK_REAL, intent(out) :: g4(0:3,0:3)
CCTK_REAL :: betal(3)
! ds^2 = -alpha^2 dt^2 + g_ij (dx^i + beta^i dt) (dx^j + beta^j dt)
betal = matmul(gg, beta)
g4(0 ,0 ) = -alfa**2 + sum(betal*beta)
g4(1:3,0 ) = betal
g4(0 ,1:3) = betal
g4(1:3,1:3) = gg
end subroutine calc_4metric
subroutine calc_4metricderivs (gg,alfa,beta, dgg,dalfa,dbeta, &
gg_dot,alfa_dot,beta_dot, g4,dg4)
CCTK_REAL, intent(in) :: gg(3,3),alfa,beta(3)
CCTK_REAL, intent(in) :: dgg(3,3,3),dalfa(3),dbeta(3,3)
CCTK_REAL, intent(in) :: gg_dot(3,3),alfa_dot,beta_dot(3)
CCTK_REAL, intent(out) :: g4(0:3,0:3),dg4(0:3,0:3,0:3)
CCTK_REAL :: d4gg(3,3,0:3),d4alfa(0:3),d4beta(3,0:3)
CCTK_REAL :: betal(3),d4betal(3,0:3)
integer :: i,j
! 4-metric
forall (i=1:3)
betal(i) = sum(gg(i,:) * beta(:))
end forall
g4(0 ,0 ) = -alfa**2 + sum(betal(:) * beta(:))
g4(1:3,0 ) = betal
g4(0 ,1:3) = betal
g4(1:3,1:3) = gg
! derivatives
d4gg (:,:,0 ) = gg_dot(:,:)
d4gg (:,:,1:3) = dgg(:,:,:)
d4alfa( 0 ) = alfa_dot
d4alfa( 1:3) = dalfa(:)
d4beta(:, 0 ) = beta_dot(:)
d4beta(:, 1:3) = dbeta(:,:)
forall (i=1:3, j=0:3)
d4betal(i,j) = sum(d4gg(i,:,j) * beta(:) + gg(i,:) * d4beta(:,j))
end forall
forall (i=0:3)
dg4(0 ,0 ,i) = - 2 * alfa * d4alfa(i) &
& + sum(d4betal(:,i) * beta(:) + betal(:) * d4beta(:,i))
dg4(1:3,0 ,i) = d4betal(:,i)
dg4(0 ,1:3,i) = d4betal(:,i)
dg4(1:3,1:3,i) = d4gg(:,:,i)
end forall
end subroutine calc_4metricderivs
subroutine calc_4metricderivs2 (gg,alfa,beta, dgg,dalfa,dbeta, &
ddgg,ddalfa,ddbeta, gg_dot,alfa_dot,beta_dot, &
gg_dot2,alfa_dot2,beta_dot2, dgg_dot,dalfa_dot,dbeta_dot, g4,dg4,ddg4)
CCTK_REAL, intent(in) :: gg(3,3),alfa,beta(3)
CCTK_REAL, intent(in) :: dgg(3,3,3),dalfa(3),dbeta(3,3)
CCTK_REAL, intent(in) :: ddgg(3,3,3,3),ddalfa(3,3),ddbeta(3,3,3)
CCTK_REAL, intent(in) :: gg_dot(3,3),alfa_dot,beta_dot(3)
CCTK_REAL, intent(in) :: gg_dot2(3,3),alfa_dot2,beta_dot2(3)
CCTK_REAL, intent(in) :: dgg_dot(3,3,3),dalfa_dot(3),dbeta_dot(3,3)
CCTK_REAL, intent(out) :: g4(0:3,0:3),dg4(0:3,0:3,0:3)
CCTK_REAL, intent(out) :: ddg4(0:3,0:3,0:3,0:3)
CCTK_REAL :: d4gg(3,3,0:3),d4alfa(0:3),d4beta(3,0:3)
CCTK_REAL :: dd4gg(3,3,0:3,0:3),dd4alfa(0:3,0:3),dd4beta(3,0:3,0:3)
CCTK_REAL :: betal(3),d4betal(3,0:3),dd4betal(3,0:3,0:3)
integer :: i,j,k
! 4-metric
forall (i=1:3)
betal(i) = sum(gg(i,:) * beta(:))
end forall
g4(0 ,0 ) = -alfa**2 + sum(betal(:) * beta(:))
g4(1:3,0 ) = betal
g4(0 ,1:3) = betal
g4(1:3,1:3) = gg
! first derivatives
d4gg (:,:,0 ) = gg_dot(:,:)
d4gg (:,:,1:3) = dgg(:,:,:)
d4alfa( 0 ) = alfa_dot
d4alfa( 1:3) = dalfa(:)
d4beta(:, 0 ) = beta_dot(:)
d4beta(:, 1:3) = dbeta(:,:)
forall (i=1:3, j=0:3)
d4betal(i,j) = sum(d4gg(i,:,j) * beta(:) + gg(i,:) * d4beta(:,j))
end forall
forall (i=0:3)
dg4(0 ,0 ,i) = - 2 * alfa * d4alfa(i) &
& + sum(d4betal(:,i) * beta(:) + betal(:) * d4beta(:,i))
dg4(1:3,0 ,i) = d4betal(:,i)
dg4(0 ,1:3,i) = d4betal(:,i)
dg4(1:3,1:3,i) = d4gg(:,:,i)
end forall
! second derivatives
dd4gg (:,:,0 ,0 ) = gg_dot2(:,:)
dd4gg (:,:,1:3,0 ) = dgg_dot(:,:,:)
dd4gg (:,:,0 ,1:3) = dgg_dot(:,:,:)
dd4gg (:,:,1:3,1:3) = ddgg(:,:,:,:)
dd4alfa( 0 ,0 ) = alfa_dot2
dd4alfa( 1:3,0 ) = dalfa_dot(:)
dd4alfa( 0 ,1:3) = dalfa_dot(:)
dd4alfa( 1:3,1:3) = ddalfa(:,:)
dd4beta(:, 0 ,0 ) = beta_dot2(:)
dd4beta(:, 1:3,0 ) = dbeta_dot(:,:)
dd4beta(:, 0 ,1:3) = dbeta_dot(:,:)
dd4beta(:, 1:3,1:3) = ddbeta(:,:,:)
! betal(i) = gg(i,m) * beta(m)
! d4betal(i,j) = d4gg(i,m,j) * beta(m) + gg(i,m) * d4beta(m,j)
! dd4betal(i,j,k) = dd4gg(i,m,j,k) * beta(m) + d4gg(i,m,j) * d4beta(m,k)
! + d4gg(i,m,k) * d4beta(m,j) + gg(i,m) * dd4beta(m,j,k)
forall (i=1:3, j=0:3, k=0:3)
dd4betal(i,j,k) = sum(+ dd4gg(i,:,j,k) * beta(:) &
& + d4gg(i,:,j) * d4beta(:,k) &
& + d4gg(i,:,k) * d4beta(:,j) &
& + gg(i,:) * dd4beta(:,j,k))
end forall
! g4(0 ,0 ) = -alfa**2 + sum(betal*beta)
! g4(1:3,0 ) = betal
! g4(0 ,1:3) = betal
! g4(1:3,1:3) = gg
! dg4(0 ,0 ,i) = -2*alfa*d4alfa(i) &
! & + d4betal(m,i)*beta(m) + betal(m)*d4beta(m,i)
! dg4(1:3,0 ,i) = d4betal(:,i)
! dg4(0 ,1:3,i) = d4betal(:,i)
! dg4(1:3,1:3,i) = d4gg(:,:,i)
forall (i=0:3, j=0:3)
ddg4(0 ,0 ,i,j) = - 2 * d4alfa(j) * d4alfa(i) &
& - 2 * alfa * dd4alfa(i,j) &
& + sum(+ dd4betal(:,i,j) * beta(:) &
& + d4betal(:,i) * d4beta(:,j) &
& + d4betal(:,j) * d4beta(:,i) &
& + betal(:) * dd4beta(:,i,j))
ddg4(1:3,0 ,i,j) = dd4betal(:,i,j)
ddg4(0 ,1:3,i,j) = dd4betal(:,i,j)
ddg4(1:3,1:3,i,j) = dd4gg(:,:,i,j)
end forall
end subroutine calc_4metricderivs2
subroutine calc_3metric (g4, gg, alfa, beta)
CCTK_REAL, intent(in) :: g4(0:3,0:3)
CCTK_REAL, intent(out) :: gg(3,3), alfa, beta(3)
CCTK_REAL :: betal(3)
CCTK_REAL :: dtg, gu(3,3)
! ds^2 = -alpha^2 dt^2 + g_ij (dx^i + beta^i dt) (dx^j + beta^j dt)
betal = g4(1:3,0)
gg = g4(1:3,1:3)
call calc_det (gg, dtg)
call calc_inv (gg, dtg, gu)
beta = matmul(gu, betal)
alfa = sqrt(sum(betal*beta) - g4(0,0))
end subroutine calc_3metric
subroutine calc_3metricderivs (g4,dg4, gg,alfa,beta, dgg,dalfa,dbeta, &
gg_dot,alfa_dot,beta_dot)
CCTK_REAL, intent(in) :: g4(0:3,0:3),dg4(0:3,0:3,0:3)
CCTK_REAL, intent(out) :: gg(3,3),alfa,beta(3)
CCTK_REAL, intent(out) :: dgg(3,3,3),dalfa(3),dbeta(3,3)
CCTK_REAL, intent(out) :: gg_dot(3,3),alfa_dot,beta_dot(3)
CCTK_REAL :: betal(3),d4betal(3,0:3)
CCTK_REAL :: dtg,gu(3,3),dgu(3,3,3),gu_dot(3,3)
CCTK_REAL :: d4gg(3,3,0:3),d4gu(3,3,0:3)
CCTK_REAL :: d4alfa(0:3),d4beta(3,0:3)
integer :: i,j
! ds^2 = -alpha^2 dt^2 + g_ij (dx^i + beta^i dt) (dx^j + beta^j dt)
gg = g4(1:3,1:3)
call calc_det (gg, dtg)
call calc_inv (gg, dtg, gu)
betal = g4(1:3,0)
beta = matmul(gu, betal)
alfa = sqrt(sum(betal*beta) - g4(0,0))
forall (i=0:3)
d4gg(:,:,i) = dg4(1:3,1:3,i)
end forall
gg_dot = d4gg(:,:,0)
dgg = d4gg(:,:,1:3)
call calc_invderiv (gu, dgg, dgu)
call calc_invdot (gu, gg_dot, gu_dot)
d4gu(:,:,0) = gu_dot
d4gu(:,:,1:3) = dgu
forall (i=0:3)
d4betal(:,i) = dg4(1:3,0,i)
end forall
forall (i=0:3, j=1:3)
d4beta(j,i) = sum(d4gu(j,:,i)*betal) + sum(gu(j,:)*d4betal(:,i))
end forall
forall (i=0:3)
d4alfa(i) = 1/(2*alfa) &
* (sum(d4betal(:,i)*beta) + sum(betal*d4beta(:,i)) - dg4(0,0,i))
end forall
alfa_dot = d4alfa(0)
dalfa = d4alfa(1:3)
beta_dot = d4beta(:,0)
dbeta = d4beta(:,1:3)
end subroutine calc_3metricderivs
subroutine calc_3metricdot (gg, dgg, kk, alfa, beta, dbeta, gg_dot)
CCTK_REAL, intent(in) :: gg(3,3), dgg(3,3,3)
CCTK_REAL, intent(in) :: kk(3,3)
CCTK_REAL, intent(in) :: alfa
CCTK_REAL, intent(in) :: beta(3), dbeta(3,3)
CCTK_REAL, intent(out) :: gg_dot(3,3)
integer :: i,j,k
! d/dt g_ij = -2 alpha K_ij + g_kj beta^k,i + g_ik beta^k,j + beta^k g_ij,k
do i=1,3
do j=1,3
gg_dot(i,j) = -2*alfa * kk(i,j)
do k=1,3
gg_dot(i,j) = gg_dot(i,j) &
+ gg(k,j) * dbeta(k,i) + gg(i,k) * dbeta(k,j) &
+ beta(k) * dgg(i,j,k)
end do
end do
end do
end subroutine calc_3metricdot
subroutine calc_extcurv (gg, dgg, gg_dot, alfa, beta, dbeta, kk)
CCTK_REAL, intent(in) :: gg(3,3), dgg(3,3,3), gg_dot(3,3)
CCTK_REAL, intent(in) :: alfa
CCTK_REAL, intent(in) :: beta(3), dbeta(3,3)
CCTK_REAL, intent(out) :: kk(3,3)
integer :: i,j,k
! d/dt g_ij = -2 alpha K_ij + g_kj beta^k,i + g_ik beta^k,j + beta^k g_ij,k
do i=1,3
do j=1,3
kk(i,j) = - gg_dot(i,j)
do k=1,3
kk(i,j) = kk(i,j) &
+ gg(k,j) * dbeta(k,i) + gg(i,k) * dbeta(k,j) &
+ beta(k) * dgg(i,j,k)
end do
kk(i,j) = kk(i,j) / (2*alfa)
end do
end do
end subroutine calc_extcurv
subroutine calc_3metricdot_simple (kk, gg_dot)
CCTK_REAL, intent(in) :: kk(3,3)
CCTK_REAL, intent(out) :: gg_dot(3,3)
integer :: i,j
! d/dt g_ij = -2 K_ij
do i=1,3
do j=1,3
gg_dot(i,j) = -2 * kk(i,j)
end do
end do
end subroutine calc_3metricdot_simple
subroutine calc_3metricderivdot_simple (dkk, dgg_dot)
CCTK_REAL, intent(in) :: dkk(3,3,3)
CCTK_REAL, intent(out) :: dgg_dot(3,3,3)
integer :: i,j,k
! d/dt g_ij,k = -2 K_ij,k
do i=1,3
do j=1,3
do k=1,3
dgg_dot(i,j,k) = -2 * dkk(i,j,k)
end do
end do
end do
end subroutine calc_3metricderivdot_simple
subroutine calc_extcurvdot_simple (gu,ri, kk, kk_dot)
CCTK_REAL, intent(in) :: gu(3,3)
CCTK_REAL, intent(in) :: ri(3,3)
CCTK_REAL, intent(in) :: kk(3,3)
CCTK_REAL, intent(out) :: kk_dot(3,3)
integer :: i,j,l,m
! d/dt K_ij = R_ij - 2 K_il g^lm K_mj + g^lm K_lm K_ij
do i=1,3
do j=1,3
kk_dot(i,j) = ri(i,j)
do l=1,3
do m=1,3
kk_dot(i,j) = kk_dot(i,j) + gu(l,m) &
* (-2 * kk(i,l) * kk(m,j) + kk(l,m) * kk(i,j))
end do
end do
end do
end do
end subroutine calc_extcurvdot_simple
subroutine calc_3metricdot2_simple (kk_dot, gg_dot2)
CCTK_REAL, intent(in) :: kk_dot(3,3)
CCTK_REAL, intent(out) :: gg_dot2(3,3)
integer :: i,j
! d/dt g_ij = -2 K_ij
! d^2/dt^2 g_ij = -2 d/dt K_ij
do i=1,3
do j=1,3
gg_dot2(i,j) = -2 * kk_dot(i,j)
end do
end do
end subroutine calc_3metricdot2_simple
end module adm_metric
|
