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
|
SetEnhancedTimes[False];
SetSourceLanguage["C"];
(****************************************************************************
Derivatives
****************************************************************************)
derivatives =
{
PDstandard2nd[i_] -> StandardCenteredDifferenceOperator[1,1,i],
PDstandard2nd[i_, i_] -> StandardCenteredDifferenceOperator[2,1,i],
PDstandard2nd[i_, j_] -> StandardCenteredDifferenceOperator[1,1,i] *
StandardCenteredDifferenceOperator[1,1,j],
PDstandard4th[i_] -> StandardCenteredDifferenceOperator[1,2,i],
PDstandard4th[i_, i_] -> StandardCenteredDifferenceOperator[2,2,i],
PDstandard4th[i_, j_] -> StandardCenteredDifferenceOperator[1,2,i] *
StandardCenteredDifferenceOperator[1,2,j],
PDstandard[i_] -> StandardCenteredDifferenceOperator[1,fdOrder/2,i],
PDstandard[i_, i_] -> StandardCenteredDifferenceOperator[2,fdOrder/2,i],
PDstandard[i_, j_] -> StandardCenteredDifferenceOperator[1,fdOrder/2,i] *
StandardCenteredDifferenceOperator[1,fdOrder/2,j]
};
(*PD = PDstandard2nd;*)
(****************************************************************************
Tensors
****************************************************************************)
(* Register all the tensors that will be used with TensorTools *)
Map[DefineTensor,
{
R, gamma,g, gInv, k, ltet, n, rm, im, rmbar, imbar, tn, va, vb, vc,
wa, wb, wc, ea, eb, ec,
Ro, Rojo, R4p, Psi0r, Psi0i, Psi1r, Psi1i, Psi2r, Psi2i, Psi3r,
Psi3i, Psi4r,Psi4i, curvIr
}];
(* Psi0,2,4 behave as (pseudo)scalars *)
SetTensorAttribute[Psi0r, TensorParity, +1];
SetTensorAttribute[Psi2r, TensorParity, +1];
SetTensorAttribute[Psi4r, TensorParity, +1];
SetTensorAttribute[Psi0i, TensorParity, -1];
SetTensorAttribute[Psi2i, TensorParity, -1];
SetTensorAttribute[Psi4i, TensorParity, -1];
(* these are 'special' and require a patch to the symmetry thorns *)
SetTensorAttribute[Psi1r, TensorManualCartesianParities, {1,1,-1}];
SetTensorAttribute[Psi3r, TensorManualCartesianParities, {-1,-1,1}];
SetTensorAttribute[Psi1i, TensorManualCartesianParities, {-1,-1,1}];
SetTensorAttribute[Psi3i, TensorManualCartesianParities, {-1,-1,1}];
(* Register the TensorTools symmetries (this is very simplistic) *)
Map[AssertSymmetricDecreasing,
{
k[la,lb], g[la,lb]
}];
AssertSymmetricDecreasing[gamma[ua,lb,lc], lb, lc];
(* Determinants of the metrics in terms of their components
(Mathematica symbolic expressions) *)
gDet = Det[MatrixOfComponents[g[la,lb]]];
(****************************************************************************
Groups
****************************************************************************)
(* Cactus group definitions *)
scalars = {Psi0r, Psi0i, Psi1r, Psi1i, Psi2r, Psi2i, Psi3r, Psi3i, Psi4r,Psi4i, curvIr};
scalarGroups = Map[CreateGroupFromTensor, scalars];
(* We need extra timelevels so that interpolation onto the extraction sphere
works properly with mesh refinement *)
scalarGroups = Map[AddGroupExtra[#, Timelevels -> 3] &, scalarGroups];
admGroups =
{{"admbase::metric", {gxx,gxy,gxz,gyy,gyz,gzz}},
{"admbase::curv", {kxx,kxy,kxz,kyy,kyz,kzz}},
{"admbase::lapse", {alp}},
{"admbase::shift", {betax,betay,betaz}}};
declaredGroups = scalarGroups;
declaredGroupNames = Map[First, declaredGroups];
groups = Join[declaredGroups, admGroups];
(****************************************************************************
Shorthands
****************************************************************************)
shorthands =
{
gamma[ua,lb,lc], R[la,lb,lc,ld], invdetg, detg, third, detgmthirdroot,
gInv[ua,ub], Ro[la,lb,lc], Rojo[la,lb], R4p[li,lj,lk,ll],
omega11, omega22, omega33, omega12, omega13, omega23, va[ua], vb[ua], vc[ua],
wa[ua], wb[ua], wc[ua], ea[ua], eb[ua], ec[ua],
tn[ua], nn, ltet[ua],n[ua],rm[ua],im[ua],rmbar[ua],imbar[ua], isqrt2, xmoved,
ymoved, zmoved
};
k11=kxx; k21=kxy; k22=kyy; k31=kxz; k32=kyz; k33=kzz;
g11=gxx; g21=gxy; g22=gyy; g31=gxz; g32=gyz; g33=gzz;
realParameters = {{Name -> offset, Default -> 10^(-15)},xorig,yorig,zorig};
PsisCalc[fdOrder_, PD_] :=
{
Name -> "psis_calc_" <> fdOrder,
Where -> Interior,
After -> "ADMBase_SetADMVars",
ConditionalOnKeyword -> {"fd_order", fdOrder},
Shorthands -> shorthands,
Equations ->
{
detg -> gDet,
invdetg -> 1 / detg,
gInv[ua,ub] -> invdetg gDet MatrixInverse[g[ua,ub]],
gamma[ua, lb, lc] -> 1/2 gInv[ua,ud] (PD[g[lb,ld], lc] +
PD[g[lc,ld], lb] - PD[g[lb,lc],ld]),
(****************************************************************************
Offset the origin
****************************************************************************)
xmoved -> x - xorig,
ymoved -> y - yorig,
zmoved -> z - zorig,
(****************************************************************************
Compute the local tetrad
****************************************************************************)
va1 -> -ymoved, va2 -> xmoved+offset, va3 -> 0,
vb1 -> xmoved+offset, vb2 -> ymoved, vb3 -> zmoved,
vc[ua] -> Sqrt[detg] gInv[ua,ud] Eps[ld,lb,lc] va[ub] vb[uc],
(* Orthonormalize using Gram-Schmidt*)
wa[ua] -> va[ua],
omega11 -> wa[ua] wa[ub] g[la,lb],
ea[ua] -> wa[ua] / Sqrt[omega11],
omega12 -> ea[ua] vb[ub] g[la,lb],
wb[ua] -> vb[ua] - omega12 ea[ua],
omega22 -> wb[ua] wb[ub] g[la,lb],
eb[ua] -> wb[ua]/Sqrt[omega22],
omega13 -> ea[ua] vc[ub] g[la,lb],
omega23 -> eb[ua] vc[ub] g[la,lb],
wc[ua] -> vc[ua] - omega13 ea[ua] - omega23 eb[ua],
omega33 -> wc[ua] wc[ub] g[la,lb],
ec[ua] -> wc[ua]/Sqrt[omega33],
(* Create Spatial Portion of Null Tetrad *)
isqrt2 -> 0.7071067811865475244,
ltet[ua] -> isqrt2 eb[ua],
n[ua] -> - isqrt2 eb[ua],
rm[ua] -> isqrt2 ec[ua],
im[ua] -> isqrt2 ea[ua],
rmbar[ua] -> isqrt2 ec[ua],
imbar[ua] -> -isqrt2 ea[ua],
(* nn here is the projection of both l^a and n^a with u^a (the time-like unit
vector normal to the hypersurface). We do NOT save the t component of the
tetrads in this code to avoid unnecessary factors of lapse and shift. *)
nn -> isqrt2,
(****************************************************************************
Compute the NP pseudoscalars
****************************************************************************)
(* Calculate the relevant Riemann Quantities *)
(* The 3-Riemann *)
R[la,lb,lc,ld] -> 1/2 ( PD[g[la,ld],lc,lb] + PD[g[lb,lc],ld,la] )
- 1/2 ( PD[g[la,lc],lb,ld] + PD[g[lb,ld],la,lc] )
+ g[lj,le] gamma[uj,lb,lc] gamma[ue,la,ld]
- g[lj,le] gamma[uj,lb,ld] gamma[ue,la,lc],
(* The 4-Riemann projected into the slice on all its indices.
The Gauss equation. *)
R4p[li,lj,lk,ll] -> R[li,lj,lk,ll] +
2 AntiSymmetrize[k[li,lk] k[ll,lj], lk, ll],
(* The 4-Riemann projected in the unit normal direction on one
index, then into the slice on the remaining indices. The Codazzi
equation. *)
Ro[lj,lk,ll] -> - 2 AntiSymmetrize[ PD[k[lj,lk],ll], lk,ll]
- 2 AntiSymmetrize[ gamma[up,lj,lk] k[ll,lp], lk,ll],
(* The 4-Riemann projected in the unit normal direction on two
indices, and into the slice on the remaining two. *)
Rojo[lj,ll] -> gInv[uc,ud] (R[lj,lc,ll,ld] )
- k[lj,lp] gInv[up,ud] k[ld,ll]
+ k[lc,ld] gInv[uc,ud] k[lj,ll],
(* Calculate End Quantities
NOTE: In writing this, I assume m[0]=0!! to save lots of work *)
Psi4r -> R4p[li,lj,lk,ll] n[ui] n[uk]
( rmbar[uj] rmbar[ul] - imbar[uj] imbar[ul] )
+ 2 Ro[lj,lk,ll] n[uk] nn
( rmbar[uj] rmbar[ul] - imbar[uj] imbar[ul] )
+ Rojo[lj,ll] nn nn ( rmbar[uj] rmbar[ul] - imbar[uj] imbar[ul]
(* + terms in mbar^0 == 0*) ),
Psi4i -> R4p[la,lb,lc,ld] n[ua] n[uc] ( - rm[ub] im[ud] - im[ub] rm[ud] )
+ 2 Ro[la,lb,lc] n[ub] nn ( - rm[ua] im[uc] - im[ua] rm[uc] )
+ Rojo[la,lb] nn nn ( - rm[ua] im[ub] - im[ua] rm[ub] ) ,
Psi3r -> R4p[la,lb,lc,ld] ltet[ua] n[ub] rm[uc] n[ud]
+ Ro[la,lb,lc] ( nn (n[ua]-ltet[ua]) rm[ub] n[uc]
- nn rm[ua] ltet[ub] n[uc] )
- Rojo[la,lb] nn (n[ua]-ltet[ua]) nn rm[ub],
Psi3i -> - R4p[la,lb,lc,ld] ltet[ua] n[ub] im[uc] n[ud]
- Ro[la,lb,lc] ( nn (n[ua]-ltet[ua]) im[ub] n[uc]
- nn im[ua] ltet[ub] n[uc] )
+ Rojo[la,lb] nn (n[ua]-ltet[ua]) nn im[ub],
Psi2r -> R4p[la,lb,lc,ld] ltet[ua] n[ud] (rm[ub] rm[uc] + im[ub] im[uc])
+ Ro[la,lb,lc] nn ( n[uc] (rm[ua] rm[ub] + im[ua] im[ub])
- ltet[ub] (rm[ua] rm[uc] + im[ua] im[uc]) )
- Rojo[la,lb] nn nn (rm[ua] rm[ub] + im[ua] im[ub]),
Psi2i -> R4p[la,lb,lc,ld] ltet[ua] n[ud] (im[ub] rm[uc] - rm[ub] im[uc])
+ Ro[la,lb,lc] nn ( n[uc] (im[ua] rm[ub] - rm[ua] im[ub])
- ltet[ub] (rm[ua] im[uc] - im[ua] rm[uc]) )
- Rojo[la,lb] nn nn (im[ua] rm[ub] - rm[ua] im[ub]),
Psi1r -> R4p[la,lb,lc,ld] n[ua] ltet[ub] rm[uc] ltet[ud]
+ Ro[la,lb,lc] ( nn ltet[ua] rm[ub] ltet[uc]
- nn rm[ua] n[ub] ltet[uc]
- nn n[ua] rm[ub] ltet[uc] )
+ Rojo[la,lb] nn nn ( n[ua] rm[ub] - ltet[ua] rm[ub] ),
Psi1i -> R4p[la,lb,lc,ld] n[ua] ltet[ub] im[uc] ltet[ud]
+ Ro[la,lb,lc] ( nn ltet[ua] im[ub] ltet[uc]
- nn im[ua] n[ub] ltet[uc]
- nn n[ua] im[ub] ltet[uc] )
+ Rojo[la,lb] nn nn ( n[ua] im[ub] - ltet[ua] im[ub] ),
Psi0r -> R4p[la,lb,lc,ld] ltet[ua] ltet[uc] (rm[ub] rm[ud] - im[ub] im[ud])
+ 2 Ro[la,lb,lc] nn ltet[ub] (rm[ua] rm[uc] - im[ua] im[uc])
+ Rojo[la,lb] nn nn (rm[ua] rm[ub] - im[ua] im[ub]),
Psi0i -> R4p[la,lb,lc,ld] ltet[ua] ltet[uc] (rm[ub] im[ud] + im[ub] rm[ud])
+ 2 Ro[la,lb,lc] nn ltet[ub] (rm[ua] im[uc] + im[ua] rm[uc])
+ Rojo[la,lb] nn nn (rm[ua] im[ub] + im[ua] rm[ub]),
curvIr -> ComplexExpand[Re[3 (Psi2r+I Psi2i)^2 - 4 (Psi1r+I Psi1i) (Psi3r + I Psi3i) + (Psi4r + I Psi4i) (Psi0r + I Psi0i)]]
}
};
(****************************************************************************
Construct the thorn
****************************************************************************)
fdOrderParam =
{
Name -> "fd_order",
Default -> "Nth",
AllowedValues -> {"Nth", "2nd", "4th"}
};
keywordParameters =
{
fdOrderParam
};
intParameters =
{
{
Name -> fdOrder,
Default -> 2,
AllowedValues -> {2,4,6,8}
}
};
calculations =
{
PsisCalc["Nth", PDstandard],
PsisCalc["2nd", PDstandard2nd],
PsisCalc["4th", PDstandard4th]
};
CreateKrancThornTT[groups, ".", "WeylScal4",
Calculations -> calculations,
DeclaredGroups -> declaredGroupNames,
PartialDerivatives -> derivatives,
UseLoopControl -> True,
KeywordParameters -> keywordParameters,
RealParameters -> realParameters,
IntParameters -> intParameters,
InheritedImplementations -> {"admbase", "methodoflines"},
UseJacobian -> True];
|
