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
|
! $Header$
#include "cctk.h"
#undef DEBUG
module ricci4
implicit none
private
public calc_4connections
public calc_4connectionderivs
public calc_4ricci
public calc_4riemann
public calc_4weyl
contains
subroutine calc_4connections (gu, dg, gamma)
CCTK_REAL, intent(in) :: gu(4,4), dg(4,4,4)
CCTK_REAL, intent(out) :: gamma(4,4,4)
integer :: i,j,k,l
! Gamma^i_jk = 1/2 g^il (g_lj,k + g_lk,j - g_jk,l)
do i=1,4
do j=1,4
do k=1,4
gamma(i,j,k) = 0
do l=1,4
gamma(i,j,k) = gamma(i,j,k) + 0.5d0 * gu(i,l) &
* (dg(l,j,k) + dg(l,k,j) - dg(j,k,l))
end do
end do
end do
end do
end subroutine calc_4connections
subroutine calc_4connectionderivs (gu, dgg, dgu, ddgg, dgamma)
CCTK_REAL, intent(in) :: gu(4,4), dgg(4,4,4), dgu(4,4,4), ddgg(4,4,4,4)
CCTK_REAL, intent(out) :: dgamma(4,4,4,4)
integer :: i,j,k,l,m
! Gamma^i_jk,l = 1/2 g^im,l (g_mj,k + g_mk,j - g_jk,m)
! + 1/2 g^im (g_mj,kl + g_mk,jl - g_jk,ml)
do i=1,4
do j=1,4
do k=1,4
do l=1,4
dgamma(i,j,k,l) = 0
do m=1,4
dgamma(i,j,k,l) = dgamma(i,j,k,l) &
+ 0.5d0 * dgu(i,m,l) * (dgg(m,j,k) + dgg(m,k,j) - dgg(j,k,m)) &
+ 0.5d0 * gu(i,m) * (ddgg(m,j,k,l) + ddgg(m,k,j,l) - ddgg(j,k,m,l))
end do
end do
end do
end do
end do
end subroutine calc_4connectionderivs
subroutine calc_4ricci (gamma, dgamma, ri)
CCTK_REAL, intent(in) :: gamma(4,4,4), dgamma(4,4,4,4)
CCTK_REAL, intent(out) :: ri(4,4)
#ifdef DEBUG
CCTK_REAL :: nrm, cnt
#endif
integer :: i,j,k,l
#if 1
! R_ij = Gamma^k_ij,k - Gamma^k_ik,j
! + Gamma^k_lk Gamma^l_ij - Gamma^k_lj Gamma^l_ki
do i=1,4
do j=1,4
ri(i,j) = 0
do k=1,4
ri(i,j) = ri(i,j) + dgamma(k,i,j,k) - dgamma(k,i,k,j)
do l=1,4
ri(i,j) = ri(i,j) + gamma(k,l,k) * gamma(l,i,j) &
& - gamma(k,l,j) * gamma(l,k,i)
end do
end do
end do
end do
#else
CCTK_REAL :: rmu(4,4,4,4)
integer :: m
do i=1,4
do j=1,4
do k=1,4
do l=1,4
rmu(i,j,k,l) = dgamma(i,j,l,k) - dgamma(i,j,k,l)
do m=1,4
rmu(i,j,k,l) = rmu(i,j,k,l) &
+ gamma(m,j,l) * gamma(i,m,k) &
- gamma(m,j,k) * gamma(i,m,l)
end do
end do
end do
end do
end do
do i=1,4
do j=1,4
ri(i,j) = 0
do k=1,4
ri(i,j) = ri(i,j) + rmu(k,i,k,j)
end do
end do
end do
#endif
#ifdef DEBUG
! check symmetries
nrm = 0
cnt = 0
do i=1,4
do j=1,4
nrm = nrm + (ri(i,j) - ri(j,i))**2
cnt = cnt + ri(i,j)**2
end do
end do
if (nrm > 1.0e-12 * cnt) call CCTK_WARN (0, "4-Ricci tensor is not symmetric")
#endif
end subroutine calc_4ricci
subroutine calc_4riemann (gg, gamma, dgamma, rm)
CCTK_REAL, intent(in) :: gg(4,4), gamma(4,4,4), dgamma(4,4,4,4)
CCTK_REAL, intent(out) :: rm(4,4,4,4)
CCTK_REAL :: rmu(4,4,4,4)
#ifdef DEBUG
CCTK_REAL :: nrm, cnt
#endif
integer :: i,j,k,l,m
#if 1
! R^i_jkl = + Gamma^i_jl,k - Gamma^i_jk,l
! + Gamma^m_jl Gamma^i_mk - Gamma^m_jk Gamma^i_ml
! (Wald, 3.4, eqn. (3.4.4), p. 48)
! Note that Wald writes R_lkj^i, i.e., we have the order of the
! indices exactly reversed. Due to the Riemann tensor's various
! symmetries, this does not change anything.
do i=1,4
do j=1,4
do k=1,4
do l=1,4
rmu(i,j,k,l) = dgamma(i,j,l,k) - dgamma(i,j,k,l)
do m=1,4
rmu(i,j,k,l) = rmu(i,j,k,l) &
+ gamma(m,j,l) * gamma(i,m,k) &
- gamma(m,j,k) * gamma(i,m,l)
end do
end do
end do
end do
end do
do i=1,4
do j=1,4
do k=1,4
do l=1,4
rm(i,j,k,l) = 0
do m=1,4
rm(i,j,k,l) = rm(i,j,k,l) + gg(i,m) * rmu(m,j,k,l)
end do
end do
end do
end do
end do
#else
! R_ijk^l = + Gamma^l_ik,j - Gamma^l_jk,i
! + Gamma^m_ik Gamma^l_mj - Gamma^m_jk Gamma^l_mi
! (Wald, 3.4, eqn. (3.4.4), p. 48)
do i=1,4
do j=1,4
do k=1,4
do l=1,4
rmu(i,j,k,l) = + dgamma(l,i,k,j) - dgamma(l,j,k,i)
do m=1,4
rmu(i,j,k,l) = rmu(i,j,k,l) &
+ gamma(m,i,k) * gamma(l,m,j) &
- gamma(m,j,k) * gamma(l,m,i)
end do
end do
end do
end do
end do
do i=1,4
do j=1,4
do k=1,4
do l=1,4
rm(i,j,k,l) = 0
do m=1,4
rm(i,j,k,l) = rm(i,j,k,l) + rmu(i,j,k,m) * gg(m,l)
end do
end do
end do
end do
end do
#endif
#ifdef DEBUG
! check symmetries
nrm = 0
cnt = 0
do i=1,4
do j=1,4
do k=1,4
do l=1,4
nrm = nrm + (rm(i,j,k,l) + rm(i,j,l,k))**2
nrm = nrm + (rm(i,j,k,l) - rm(k,l,i,j))**2
nrm = nrm + (rm(i,j,k,l) + rm(j,i,k,l))**2
nrm = nrm + (rm(i,j,k,l) + rm(j,k,i,l) + rm(k,i,j,l))**2
cnt = cnt + rm(i,j,k,l)**2
end do
end do
end do
end do
if (nrm > 1.0e-12 * cnt) call CCTK_WARN (0, "4-Riemann tensor is not symmetric")
#endif
end subroutine calc_4riemann
subroutine calc_4weyl (gg, rm, ri, rsc, we)
CCTK_REAL, parameter :: one=1, half=one/2, third=one/3
CCTK_REAL, intent(in) :: gg(4,4), rm(4,4,4,4), ri(4,4), rsc
CCTK_REAL, intent(out) :: we(4,4,4,4)
#ifdef DEBUG
CCTK_REAL :: nrm, cnt
#endif
integer :: i,j,k,l
! R_ijkl = C_ijkl + 2/(n-2) (g_i[k R_l]j - g_j[k R_l]i)
! - 2/(n-1)(n-2) R g_i[k g_l]j
! (Wald, 3.2, eqn. (3.2.28), p. 40)
do i=1,4
do j=1,4
do k=1,4
do l=1,4
we(i,j,k,l) = rm(i,j,k,l) &
- ( (gg(i,k) * ri(l,j) - gg(i,l) * ri(k,j)) &
& - (gg(j,k) * ri(l,i) - gg(j,l) * ri(k,i))) &
+ third * rsc * (gg(i,k) * gg(l,j) - gg(i,l) * gg(k,j))
end do
end do
end do
end do
#ifdef DEBUG
! check symmetries
nrm = 0
cnt = 0
do i=1,4
do j=1,4
do k=1,4
do l=1,4
nrm = nrm + (we(i,j,k,l) + we(i,j,l,k))**2
nrm = nrm + (we(i,j,k,l) - we(k,l,i,j))**2
nrm = nrm + (we(i,j,k,l) + we(j,i,k,l))**2
nrm = nrm + (we(i,j,k,l) + we(j,k,i,l) + we(k,i,j,l))**2
cnt = cnt + we(i,j,k,l)**2
end do
end do
end do
end do
if (nrm > 1.0e-12 * cnt) call CCTK_WARN (0, "4-Weyl tensor is not symmetric")
#endif
end subroutine calc_4weyl
end module ricci4
|
