Line data Source code
1 0 : // Distributed under the MIT License.
2 : // See LICENSE.txt for details.
3 :
4 : #pragma once
5 :
6 : #include <complex>
7 : #include <cstddef>
8 : #include <limits>
9 : #include <memory>
10 : #include <vector>
11 :
12 : #include "DataStructures/SpinWeighted.hpp"
13 : #include "Evolution/Systems/Cce/AnalyticSolutions/SphericalMetricData.hpp"
14 : #include "Evolution/Systems/Cce/AnalyticSolutions/WorldtubeData.hpp"
15 : #include "Evolution/Systems/Cce/Initialize/InitializeJ.hpp"
16 : #include "Evolution/Systems/Cce/Tags.hpp"
17 : #include "Options/StdComplex.hpp"
18 : #include "Options/String.hpp"
19 : #include "Utilities/Gsl.hpp"
20 : #include "Utilities/Serialization/CharmPupable.hpp"
21 : #include "Utilities/TMPL.hpp"
22 :
23 : /// \cond
24 : class DataVector;
25 : class ComplexDataVector;
26 : /// \endcond
27 :
28 : namespace Cce::Solutions {
29 : namespace LinearizedBondiSachs_detail {
30 : namespace InitializeJ {
31 : // First hypersurface Initialization for the
32 : // `Cce::Solutions::LinearizedBondiSachs` analytic solution.
33 : //
34 : // This initialization procedure should not be used except when the
35 : // `Cce::Solutions::LinearizedBondiSachs` analytic solution is used,
36 : // as a consequence, this initial data generator is deliberately not
37 : // option-creatable; it should only be obtained from the `get_initialize_j`
38 : // function of `Cce::InitializeJ::LinearizedBondiSachs`.
39 : struct LinearizedBondiSachs : ::Cce::InitializeJ::InitializeJ<false> {
40 : WRAPPED_PUPable_decl_template(LinearizedBondiSachs); // NOLINT
41 : explicit LinearizedBondiSachs(CkMigrateMessage* /*unused*/) {}
42 :
43 : LinearizedBondiSachs() = default;
44 :
45 : LinearizedBondiSachs(double start_time, double frequency,
46 : std::complex<double> c_2a, std::complex<double> c_2b,
47 : std::complex<double> c_3a, std::complex<double> c_3b);
48 :
49 : std::unique_ptr<InitializeJ> get_clone() const override;
50 :
51 : void operator()(
52 : gsl::not_null<Scalar<SpinWeighted<ComplexDataVector, 2>>*> j,
53 : gsl::not_null<tnsr::i<DataVector, 3>*> cartesian_cauchy_coordinates,
54 : gsl::not_null<
55 : tnsr::i<DataVector, 2, ::Frame::Spherical<::Frame::Inertial>>*>
56 : angular_cauchy_coordinates,
57 : const Scalar<SpinWeighted<ComplexDataVector, 2>>& boundary_j,
58 : const Scalar<SpinWeighted<ComplexDataVector, 2>>& boundary_dr_j,
59 : const Scalar<SpinWeighted<ComplexDataVector, 0>>& r,
60 : const Scalar<SpinWeighted<ComplexDataVector, 0>>& beta, size_t l_max,
61 : size_t number_of_radial_points,
62 : gsl::not_null<Parallel::NodeLock*> hdf5_lock) const override;
63 :
64 : void pup(PUP::er& /*p*/) override;
65 :
66 : private:
67 : std::complex<double> c_2a_ = std::numeric_limits<double>::signaling_NaN();
68 : std::complex<double> c_2b_ = std::numeric_limits<double>::signaling_NaN();
69 : std::complex<double> c_3a_ = std::numeric_limits<double>::signaling_NaN();
70 : std::complex<double> c_3b_ = std::numeric_limits<double>::signaling_NaN();
71 : double frequency_ = std::numeric_limits<double>::signaling_NaN();
72 : double time_ = std::numeric_limits<double>::signaling_NaN();
73 : };
74 : } // namespace InitializeJ
75 :
76 : // combine the (2,2) and (3,3) modes to collocation values for
77 : // `bondi_quantity`
78 : template <int Spin, typename FactorType>
79 : void assign_components_from_l_factors(
80 : gsl::not_null<SpinWeighted<ComplexDataVector, Spin>*> bondi_quantity,
81 : const FactorType& l_2_factor, const FactorType& l_3_factor, size_t l_max,
82 : double frequency, double time);
83 :
84 : // combine the (2,2) and (3,3) modes to time derivative collocation values for
85 : // `bondi_quantity`
86 : template <int Spin>
87 : void assign_du_components_from_l_factors(
88 : gsl::not_null<SpinWeighted<ComplexDataVector, Spin>*> du_bondi_quantity,
89 : const std::complex<double>& l_2_factor,
90 : const std::complex<double>& l_3_factor, size_t l_max, double frequency,
91 : double time);
92 : } // namespace LinearizedBondiSachs_detail
93 :
94 : /*!
95 : * \brief Computes the analytic data for a Linearized solution to the
96 : * Bondi-Sachs equations described in \cite Barkett2019uae.
97 : *
98 : * \details The solution represented by this function is generated with only
99 : * \f$(2,\pm2)\f$ and \f$(3,\pm3)\f$ modes, and is constructed according to the
100 : * linearized solution documented in Section VI of \cite Barkett2019uae. For
101 : * this solution, we impose additional restrictions that the linearized solution
102 : * be asymptotically flat so that it is compatible with the gauge
103 : * transformations performed in the SpECTRE regularity-preserving CCE. Using the
104 : * notation of \cite Barkett2019uae, we set:
105 : *
106 : * \f{align*}{
107 : * B_2 &= B_3 = 0\\
108 : * C_{2b} &= 3 C_{2a} / \nu^2\\
109 : * C_{3b} &= -3 i C_{3a} / \nu^3
110 : * \f}
111 : *
112 : * where \f$C_{2a}\f$ and \f$C_{3a}\f$ may be specified freely and are taken via
113 : * input option `InitialModes`.
114 : */
115 1 : struct LinearizedBondiSachs : public SphericalMetricData {
116 0 : struct InitialModes {
117 0 : using type = std::array<std::complex<double>, 2>;
118 0 : static constexpr Options::String help{
119 : "The initial modes of the Robinson-Trautman scalar"};
120 : };
121 0 : struct ExtractionRadius {
122 0 : using type = double;
123 0 : static constexpr Options::String help{
124 : "The extraction radius of the spherical solution"};
125 0 : static type lower_bound() { return 0.0; }
126 : };
127 0 : struct Frequency {
128 0 : using type = double;
129 0 : static constexpr Options::String help{
130 : "The frequency of the linearized modes."};
131 0 : static type lower_bound() { return 0.0; }
132 : };
133 :
134 0 : static constexpr Options::String help{
135 : "A linearized Bondi-Sachs analytic solution"};
136 :
137 0 : using options = tmpl::list<InitialModes, ExtractionRadius, Frequency>;
138 :
139 0 : WRAPPED_PUPable_decl_template(LinearizedBondiSachs); // NOLINT
140 :
141 0 : explicit LinearizedBondiSachs(CkMigrateMessage* msg)
142 : : SphericalMetricData(msg) {}
143 :
144 : // clang doesn't manage to use = default correctly in this case
145 : // NOLINTNEXTLINE(hicpp-use-equals-default,modernize-use-equals-default)
146 0 : LinearizedBondiSachs() {}
147 :
148 0 : LinearizedBondiSachs(
149 : const std::array<std::complex<double>, 2>& mode_constants,
150 : double extraction_radius, double frequency);
151 :
152 0 : std::unique_ptr<WorldtubeData> get_clone() const override;
153 :
154 0 : void pup(PUP::er& p) override;
155 :
156 0 : std::unique_ptr<Cce::InitializeJ::InitializeJ<false>> get_initialize_j(
157 : double start_time) const override;
158 :
159 : protected:
160 : /// A no-op as the linearized solution does not have substantial shared
161 : /// computation to prepare before the separate component calculations.
162 1 : void prepare_solution(const size_t /*output_l_max*/,
163 : const double /*time*/) const override {}
164 :
165 : /*!
166 : * \brief Computes the linearized solution for \f$J\f$.
167 : *
168 : * \details The linearized solution for \f$J\f$ is given by
169 : * \cite Barkett2019uae,
170 : *
171 : * \f[
172 : * J = \sqrt{12} ({}_2 Y_{2\,2} + {}_2 Y_{2\, -2})
173 : * \mathrm{Re}(J_2(r) e^{i \nu u})
174 : * + \sqrt{60} ({}_2 Y_{3\,3} - {}_2 Y_{3\, -3})
175 : * \mathrm{Re}(J_3(r) e^{i \nu u}),
176 : * \f]
177 : *
178 : * where
179 : *
180 : * \f{align*}{
181 : * J_2(r) &= \frac{C_{2a}}{4 r} - \frac{C_{2b}}{12 r^3}, \\
182 : * J_3(r) &= \frac{C_{3a}}{10 r} - \frac{i \nu C_{3 b}}{6 r^3}
183 : * - \frac{C_{3 b}}{4 r^4}.
184 : * \f}
185 : */
186 1 : void linearized_bondi_j(
187 : gsl::not_null<SpinWeighted<ComplexDataVector, 2>*> bondi_j, size_t l_max,
188 : double time) const;
189 :
190 : /*!
191 : * \brief Compute the linearized solution for \f$U\f$
192 : *
193 : * \details The linearized solution for \f$U\f$ is given by
194 : * \cite Barkett2019uae,
195 : *
196 : * \f[
197 : * U = \sqrt{3} ({}_1 Y_{2\,2} + {}_1 Y_{2\, -2})
198 : * \mathrm{Re}(U_2(r) e^{i \nu u})
199 : * + \sqrt{6} ({}_1 Y_{3\,3} - {}_1 Y_{3\, -3})
200 : * \mathrm{Re}(U_3(r) e^{i \nu u}),
201 : * \f]
202 : *
203 : * where
204 : *
205 : * \f{align*}{
206 : * U_2(r) &= \frac{C_{2a}}{2 r^2} + \frac{i \nu C_{2 b}}{3 r^3}
207 : * + \frac{C_{2b}}{4 r^4} \\
208 : * U_3(r) &= \frac{C_{3a}}{2 r^2} - \frac{2 \nu^2 C_{3b}}{3 r^3}
209 : * + \frac{5 i \nu C_{3b}}{4 r^4} + \frac{C_{3 b}}{r^5}
210 : * \f}
211 : */
212 1 : void linearized_bondi_u(
213 : gsl::not_null<SpinWeighted<ComplexDataVector, 1>*> bondi_u, size_t l_max,
214 : double time) const;
215 :
216 : /*!
217 : * \brief Computes the linearized solution for \f$W\f$.
218 : *
219 : * \details The linearized solution for \f$W\f$ is given by
220 : * \cite Barkett2019uae,
221 : *
222 : * \f[
223 : * W = \frac{1}{\sqrt{2}} ({}_0 Y_{2\,2} + {}_0 Y_{2\, -2})
224 : * \mathrm{Re}(W_2(r) e^{i \nu u})
225 : * + \frac{1}{\sqrt{2}} ({}_0 Y_{3\,3} - {}_0 Y_{3\, -3})
226 : * \mathrm{Re}(W_3(r) e^{i \nu u}),
227 : * \f]
228 : *
229 : * where
230 : *
231 : * \f{align*}{
232 : * W_2(r) &= - \frac{\nu^2 C_{2b}}{r^2} + \frac{i \nu C_{2 b}}{r^3}
233 : * + \frac{C_{2b}}{2 r^4}, \\
234 : * W_3(r) &= -\frac{2 i \nu^3 C_{3b}}{r^2} - \frac{4 i \nu^2 C_{3b}}{r^3}
235 : * + \frac{5 \nu C_{3b}}{2 r^4} + \frac{3 C_{3b}}{r^5}.
236 : * \f}
237 : */
238 1 : void linearized_bondi_w(
239 : gsl::not_null<SpinWeighted<ComplexDataVector, 0>*> bondi_w, size_t l_max,
240 : double time) const;
241 :
242 : /*!
243 : * \brief Computes the linearized solution for \f$\partial_r J\f$.
244 : *
245 : * \details The linearized solution for \f$\partial_r J\f$ is given by
246 : * \cite Barkett2019uae,
247 : *
248 : * \f[
249 : * \partial_r J = \sqrt{12} ({}_2 Y_{2\,2} + {}_2 Y_{2\, -2})
250 : * \mathrm{Re}(\partial_r J_2(r) e^{i \nu u})
251 : * + \sqrt{60} ({}_2 Y_{3\,3} - {}_2 Y_{3\, -3})
252 : * \mathrm{Re}(\partial_r J_3(r) e^{i \nu u}),
253 : * \f]
254 : *
255 : * where
256 : *
257 : * \f{align*}{
258 : * \partial_r J_2(r) &= - \frac{C_{2a}}{4 r^2} + \frac{C_{2b}}{4 r^4}, \\
259 : * \partial_r J_3(r) &= -\frac{C_{3a}}{10 r^2} + \frac{i \nu C_{3 b}}{2 r^4}
260 : * + \frac{C_{3 b}}{r^5}.
261 : * \f}
262 : */
263 1 : void linearized_dr_bondi_j(
264 : gsl::not_null<SpinWeighted<ComplexDataVector, 2>*> dr_bondi_j,
265 : size_t l_max, double time) const;
266 :
267 : /*!
268 : * \brief Compute the linearized solution for \f$\partial_r U\f$
269 : *
270 : * \details The linearized solution for \f$\partial_r U\f$ is given by
271 : * \cite Barkett2019uae,
272 : *
273 : * \f[
274 : * \partial_r U = \sqrt{3} ({}_1 Y_{2\,2} + {}_1 Y_{2\, -2})
275 : * \mathrm{Re}(\partial_r U_2(r) e^{i \nu u})
276 : * + \sqrt{6} ({}_1 Y_{3\,3} - {}_1 Y_{3\, -3})
277 : * \mathrm{Re}(\partial_r U_3(r) e^{i \nu u}),
278 : * \f]
279 : *
280 : * where
281 : *
282 : * \f{align*}{
283 : * \partial_r U_2(r) &= -\frac{C_{2a}}{r^3} - \frac{i \nu C_{2 b}}{r^4}
284 : * - \frac{C_{2b}}{r^5} \\
285 : * \partial_r U_3(r) &= -\frac{C_{3a}}{r^3} + \frac{2 \nu^2 C_{3b}}{r^4}
286 : * - \frac{5 i \nu C_{3b}}{r^5} - \frac{5 C_{3 b}}{r^6}
287 : * \f}
288 : */
289 1 : void linearized_dr_bondi_u(
290 : gsl::not_null<SpinWeighted<ComplexDataVector, 1>*> dr_bondi_u,
291 : size_t l_max, double time) const;
292 :
293 : /*!
294 : * \brief Computes the linearized solution for \f$\partial_r W\f$.
295 : *
296 : * \details The linearized solution for \f$W\f$ is given by
297 : * \cite Barkett2019uae,
298 : *
299 : * \f[
300 : * \partial_r W = \frac{1}{\sqrt{2}} ({}_0 Y_{2\,2} + {}_0 Y_{2\, -2})
301 : * \mathrm{Re}(\partial_r W_2(r) e^{i \nu u})
302 : * + \frac{1}{\sqrt{2}} ({}_0 Y_{3\,3} - {}_0 Y_{3\, -3})
303 : * \mathrm{Re}(\partial_r W_3(r) e^{i \nu u}),
304 : * \f]
305 : *
306 : * where
307 : *
308 : * \f{align*}{
309 : * \partial_r W_2(r) &= \frac{2 \nu^2 C_{2b}}{r^3}
310 : * - \frac{3 i \nu C_{2 b}}{r^4} - \frac{2 C_{2b}}{r^5}, \\
311 : * \partial_r W_3(r) &= \frac{4 i \nu^3 C_{3b}}{r^3}
312 : * + \frac{12 i \nu^2 C_{3b}}{r^4} - \frac{10 \nu C_{3b}}{r^5}
313 : * - \frac{15 C_{3b}}{r^6}.
314 : * \f}
315 : */
316 1 : void linearized_dr_bondi_w(
317 : gsl::not_null<SpinWeighted<ComplexDataVector, 0>*> dr_bondi_w,
318 : size_t l_max, double time) const;
319 :
320 : /*!
321 : * \brief Computes the linearized solution for \f$\partial_u J\f$.
322 : *
323 : * \details The linearized solution for \f$\partial_u J\f$ is given by
324 : * \cite Barkett2019uae,
325 : *
326 : * \f[
327 : * \partial_u J = \sqrt{12} ({}_2 Y_{2\,2} + {}_2 Y_{2\, -2})
328 : * \mathrm{Re}(i \nu J_2(r) e^{i \nu u})
329 : * + \sqrt{60} ({}_2 Y_{3\,3} - {}_2 Y_{3\, -3})
330 : * \mathrm{Re}(i \nu J_3(r) e^{i \nu u}),
331 : * \f]
332 : *
333 : * where
334 : *
335 : * \f{align*}{
336 : * J_2(r) &= \frac{C_{2a}}{4 r} - \frac{C_{2b}}{12 r^3}, \\
337 : * J_3(r) &= \frac{C_{3a}}{10 r} - \frac{i \nu C_{3 b}}{6 r^3}
338 : * - \frac{C_{3 b}}{4 r^4}.
339 : * \f}
340 : */
341 1 : void linearized_du_bondi_j(
342 : gsl::not_null<SpinWeighted<ComplexDataVector, 2>*> du_bondi_j,
343 : size_t l_max, double time) const;
344 :
345 : /*!
346 : * \brief Compute the linearized solution for \f$\partial_u U\f$
347 : *
348 : * \details The linearized solution for \f$U\f$ is given by
349 : * \cite Barkett2019uae,
350 : *
351 : * \f[
352 : * \partial_u U = \sqrt{3} ({}_2 Y_{2\,2} + {}_2 Y_{2\, -2})
353 : * \mathrm{Re}(i \nu U_2(r) e^{i \nu u})
354 : * + \sqrt{6} ({}_2 Y_{3\,3} - {}_2 Y_{3\, -3})
355 : * \mathrm{Re}(i \nu U_3(r) e^{i \nu u}),
356 : * \f]
357 : *
358 : * where
359 : *
360 : * \f{align*}{
361 : * U_2(r) &= \frac{C_{2a}}{2 r^2} + \frac{i \nu C_{2 b}}{3 r^3}
362 : * + \frac{C_{2b}}{4 r^4} \\
363 : * U_3(r) &= \frac{C_{3a}}{2 r^2} - \frac{2 \nu^2 C_{3b}}{3 r^3}
364 : * + \frac{5 i \nu C_{3b}}{4 r^4} + \frac{C_{3 b}}{r^5}
365 : * \f}
366 : */
367 1 : void linearized_du_bondi_u(
368 : gsl::not_null<SpinWeighted<ComplexDataVector, 1>*> du_bondi_u,
369 : size_t l_max, double time) const;
370 :
371 : /*!
372 : * \brief Computes the linearized solution for \f$\partial_u W\f$.
373 : *
374 : * \details The linearized solution for \f$\partial_u W\f$ is given by
375 : * \cite Barkett2019uae,
376 : *
377 : * \f[
378 : * \partial_u W = \frac{1}{\sqrt{2}} ({}_1 Y_{2\,2} + {}_1 Y_{2\, -2})
379 : * \mathrm{Re}(i \nu W_2(r) e^{i \nu u})
380 : * + \frac{1}{\sqrt{2}} ({}_1 Y_{3\,3} - {}_1 Y_{3\, -3})
381 : * \mathrm{Re}(i \nu W_3(r) e^{i \nu u}),
382 : * \f]
383 : *
384 : * where
385 : *
386 : * \f{align*}{
387 : * W_2(r) &= \frac{\nu^2 C_{2b}}{r^2} + \frac{i \nu C_{2 b}}{r^3}
388 : * + \frac{C_{2b}}{2 r^4}, \\
389 : * W_3(r) &= \frac{2 i \nu^3 C_{3b}}{r^2} - \frac{4 i \nu^2 C_{3b}}{r^3}
390 : * + \frac{5 \nu C_{3b}}{2 r^4} + \frac{3 C_{3b}}{r^5}.
391 : * \f}
392 : */
393 1 : void linearized_du_bondi_w(
394 : gsl::not_null<SpinWeighted<ComplexDataVector, 0>*> du_bondi_w,
395 : size_t l_max, double time) const;
396 :
397 : /*!
398 : * \brief Compute the spherical coordinate metric from the linearized
399 : * Bondi-Sachs system.
400 : *
401 : * \details This function dispatches to the individual computations in this
402 : * class to determine the Bondi-Sachs scalars for the linearized solution.
403 : * Once the scalars are determined, the metric is assembled via (note
404 : * \f$\beta = 0\f$ in this solution)
405 : *
406 : * \f{align*}{
407 : * ds^2 =& - ((1 + r W) - r^2 h_{A B} U^A U^B) (dt - dr)^2
408 : * - 2 (dt - dr) dr \\
409 : * &- 2 r^2 h_{A B} U^B (dt - dr) dx^A + r^2 h_{A B} dx^A dx^B,
410 : * \f}
411 : *
412 : * where indices with capital letters refer to angular coordinates and the
413 : * angular tensors may be written in terms of spin-weighted scalars. Doing so
414 : * gives the metric components,
415 : *
416 : * \f{align*}{
417 : * g_{t t} &= -\left(1 + r W
418 : * - r^2 \Re\left(\bar J U^2 + K U \bar U\right)\right)\\
419 : * g_{t r} &= -1 - g_{t t}\\
420 : * g_{r r} &= 2 + g_{t t}\\
421 : * g_{t \theta} &= r^2 \Re\left(K U + J \bar U\right)\\
422 : * g_{t \phi} &= r^2 \Im\left(K U + J \bar U\right)\\
423 : * g_{r \theta} &= -g_{t \theta}\\
424 : * g_{r \phi} &= -g_{t \phi}\\
425 : * g_{\theta \theta} &= r^2 \Re\left(J + K\right)\\
426 : * g_{\theta \phi} &= r^2 \Im\left(J\right)\\
427 : * g_{\phi \phi} &= r^2 \Re\left(K - J\right),
428 : * \f}
429 : *
430 : * and all other components are zero.
431 : */
432 1 : void spherical_metric(
433 : gsl::not_null<
434 : tnsr::aa<DataVector, 3, ::Frame::Spherical<::Frame::Inertial>>*>
435 : spherical_metric,
436 : size_t l_max, double time) const override;
437 :
438 : /*!
439 : * \brief Compute the radial derivative of the spherical coordinate metric
440 : * from the linearized Bondi-Sachs system.
441 : *
442 : * \details This function dispatches to the individual computations in this
443 : * class to determine the Bondi-Sachs scalars for the linearized solution.
444 : * Once the scalars are determined, the radial derivative of the metric is
445 : * assembled via (note \f$\beta = 0\f$ in this solution)
446 : *
447 : * \f{align*}{
448 : * \partial_r g_{a b} dx^a dx^b =& - (W + r \partial_r W
449 : * - 2 r h_{A B} U^A U^B - r^2 (\partial_r h_{A B}) U^A U^B
450 : * - 2 r^2 h_{A B} U^A \partial_r U^B) (dt - dr)^2 \\
451 : * &- (4 r h_{A B} U^B + 2 r^2 ((\partial_r h_{A B}) U^B
452 : * + h_{AB} \partial_r U^B) ) (dt - dr) dx^A
453 : * + (2 r h_{A B} + r^2 \partial_r h_{A B}) dx^A dx^B,
454 : * \f}
455 : *
456 : * where indices with capital letters refer to angular coordinates and the
457 : * angular tensors may be written in terms of spin-weighted scalars. Doing so
458 : * gives the metric components,
459 : *
460 : * \f{align*}{
461 : * \partial_r g_{t t} &= -\left( W + r \partial_r W
462 : * - 2 r \Re\left(\bar J U^2 + K U \bar U\right)
463 : * - r^2 \partial_r \Re\left(\bar J U^2 + K U \bar U\right)\right) \\
464 : * \partial_r g_{t r} &= -\partial_r g_{t t}\\
465 : * \partial_r g_{t \theta} &= 2 r \Re\left(K U + J \bar U\right)
466 : * + r^2 \partial_r \Re\left(K U + J \bar U\right) \\
467 : * \partial_r g_{t \phi} &= 2r \Im\left(K U + J \bar U\right)
468 : * + r^2 \partial_r \Im\left(K U + J \bar U\right) \\
469 : * \partial_r g_{r r} &= \partial_r g_{t t}\\
470 : * \partial_r g_{r \theta} &= -\partial_r g_{t \theta}\\
471 : * \partial_r g_{r \phi} &= -\partial_r g_{t \phi}\\
472 : * \partial_r g_{\theta \theta} &= 2 r \Re\left(J + K\right)
473 : * + r^2 \Re\left(\partial_r J + \partial_r K\right) \\
474 : * \partial_r g_{\theta \phi} &= 2 r \Im\left(J\right)
475 : * + r^2 \Im\left(\partial_r J\right)\\
476 : * \partial_r g_{\phi \phi} &= 2 r \Re\left(K - J\right)
477 : * + r^2 \Re\left(\partial_r K - \partial_r J\right),
478 : * \f}
479 : *
480 : * and all other components are zero.
481 : */
482 1 : void dr_spherical_metric(
483 : gsl::not_null<
484 : tnsr::aa<DataVector, 3, ::Frame::Spherical<::Frame::Inertial>>*>
485 : dr_spherical_metric,
486 : size_t l_max, double time) const override;
487 :
488 : /*!
489 : * \brief Compute the time derivative of the spherical coordinate metric from
490 : * the linearized Bondi-Sachs system.
491 : *
492 : * \details This function dispatches to the individual computations in this
493 : * class to determine the Bondi-Sachs scalars for the linearized solution.
494 : * Once the scalars are determined, the metric is assembled via (note
495 : * \f$\beta = 0\f$ in this solution, and note that we take coordinate
496 : * \f$t=u\f$ in converting to the Cartesian coordinates)
497 : *
498 : * \f{align*}{
499 : * \partial_t g_{a b} dx^a dx^b =& - (r \partial_u W
500 : * - r^2 \partial_u h_{A B} U^A U^B
501 : * - 2 r^2 h_{A B} U^B \partial_u U^A) (dt - dr)^2 \\
502 : * &- 2 r^2 (\partial_u h_{A B} U^B + h_{A B} \partial_u U^B) (dt - dr) dx^A
503 : * + r^2 \partial_u h_{A B} dx^A dx^B,
504 : * \f}
505 : *
506 : * where indices with capital letters refer to angular coordinates and the
507 : * angular tensors may be written in terms of spin-weighted scalars. Doing so
508 : * gives the metric components,
509 : *
510 : * \f{align*}{
511 : * \partial_t g_{t t} &= -\left(r \partial_u W
512 : * - r^2 \partial_u \Re\left(\bar J U^2 + K U \bar U\right)\right)\\
513 : * \partial_t g_{t r} &= -\partial_t g_{t t}\\
514 : * \partial_t g_{t \theta} &= r^2 \partial_u \Re\left(K U + J \bar U\right)\\
515 : * \partial_t g_{t \phi} &= r^2 \partial_u \Im\left(K U + J \bar U\right)\\
516 : * \partial_t g_{r r} &= \partial_t g_{t t}\\
517 : * \partial_t g_{r \theta} &= -\partial_t g_{t \theta}\\
518 : * \partial_t g_{r \phi} &= -\partial_t g_{t \phi}\\
519 : * \partial_t g_{\theta \theta} &= r^2 \Re\left(\partial_u J
520 : * + \partial_u K\right)\\
521 : * \partial_t g_{\theta \phi} &= r^2 \Im\left(\partial_u J\right)\\
522 : * \partial_t g_{\phi \phi} &= r^2 \Re\left(\partial_u K
523 : * - \partial_u J\right),
524 : * \f}
525 : *
526 : * and all other components are zero.
527 : */
528 1 : void dt_spherical_metric(
529 : gsl::not_null<
530 : tnsr::aa<DataVector, 3, ::Frame::Spherical<::Frame::Inertial>>*>
531 : dt_spherical_metric,
532 : size_t l_max, double time) const override;
533 :
534 0 : using WorldtubeData::variables_impl;
535 :
536 1 : using SphericalMetricData::variables_impl;
537 :
538 : /*!
539 : * \brief Determines the News function from the linearized solution
540 : * parameters.
541 : *
542 : * \details The News is determined from the formula given in
543 : * \cite Barkett2019uae,
544 : *
545 : * \f{align*}{
546 : * N = \frac{1}{2 \sqrt{3}} ({}_{-2} Y_{2\, 2} + {}_{-2} Y_{2\,-2})
547 : * \Re\left(i \nu^3 C_{2 b} e^{i \nu u}\right)
548 : * + \frac{1}{\sqrt{15}} ({}_{-2} Y_{3\, 3} - {}_{-2} Y_{3\, -3})
549 : * \Re\left(- \nu^4 C_{3 b} e^{i \nu u} \right)
550 : * \f}
551 : */
552 1 : void variables_impl(
553 : gsl::not_null<Scalar<SpinWeighted<ComplexDataVector, -2>>*> news,
554 : size_t l_max, double time,
555 : tmpl::type_<Tags::News> /*meta*/) const override;
556 :
557 0 : std::complex<double> c_2a_ = std::numeric_limits<double>::signaling_NaN();
558 0 : std::complex<double> c_3a_ = std::numeric_limits<double>::signaling_NaN();
559 0 : std::complex<double> c_2b_ = std::numeric_limits<double>::signaling_NaN();
560 0 : std::complex<double> c_3b_ = std::numeric_limits<double>::signaling_NaN();
561 :
562 0 : double frequency_ = 0.0;
563 : };
564 : } // namespace Cce::Solutions
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