Line data Source code
1 0 : // Distributed under the MIT License.
2 : // See LICENSE.txt for details.
3 :
4 : #pragma once
5 :
6 : #include <algorithm>
7 : #include <array>
8 : #include <cstddef>
9 : #include <pup.h>
10 :
11 : #include "DataStructures/CachedTempBuffer.hpp"
12 : #include "DataStructures/Tags/TempTensor.hpp"
13 : #include "DataStructures/Tensor/TypeAliases.hpp"
14 : #include "NumericalAlgorithms/LinearOperators/PartialDerivatives.hpp"
15 : #include "Options/Context.hpp"
16 : #include "Options/String.hpp"
17 : #include "PointwiseFunctions/AnalyticSolutions/AnalyticSolution.hpp"
18 : #include "PointwiseFunctions/AnalyticSolutions/GeneralRelativity/Solutions.hpp"
19 : #include "PointwiseFunctions/GeneralRelativity/TagsDeclarations.hpp"
20 : #include "Utilities/ContainerHelpers.hpp"
21 : #include "Utilities/ForceInline.hpp"
22 : #include "Utilities/Gsl.hpp"
23 : #include "Utilities/MakeArray.hpp"
24 : #include "Utilities/TMPL.hpp"
25 : #include "Utilities/TaggedTuple.hpp"
26 :
27 : /// \cond
28 : namespace Tags {
29 : template <typename Tag>
30 : struct dt;
31 : } // namespace Tags
32 : namespace gsl {
33 : template <class T>
34 : class not_null;
35 : } // namespace gsl
36 : /// \endcond
37 :
38 : namespace gr {
39 : namespace Solutions {
40 :
41 : /*!
42 : * \brief Kerr black hole in Spherical Kerr-Schild coordinates
43 : *
44 : * ## Introduction
45 : *
46 : * Given a Kerr-Schild (KS) black hole system, we denote the coordinate system
47 : * using \f$\{t,x,y,z\}\f$. In the transformed system, Spherical Kerr-Schild
48 : * (Spherical KS), we denote the coordinate system using
49 : * \f$\{t,\bar{x},\bar{y},\bar{z}\}\f$. Further, when considering indexed
50 : * objects, we will use Greek and Latin indices with the standard convention
51 : * \f$(\mu=0,1,2,3\f$ and \f$i=1,2,3\f$ respectively), but we will use a bar to
52 : * denote that a given index is in reference to the Spherical KS coordinate
53 : * system (i.e. \f$i\f$ vs. \f$\bar{\imath}\f$ and \f$\mu\f$ vs.
54 : * \f$\bar{\mu}\f$).
55 : *
56 : * ## Spin in the z direction
57 : *
58 : * ### The Transformation
59 : *
60 : * The Boyer-Lindquist radius for KS with its spin in the \f$z\f$ direction is
61 : * defined by
62 : *
63 : * \f{align}{
64 : * \frac{x^2 + y^2}{r^2 + a^2} + \frac{z^2}{r^2} = 1,
65 : * \f}
66 : *
67 : * or equivalently,
68 : *
69 : * \f{align}{
70 : * r^2 &= \frac{1}{2}\left(x^2+y^2+z^2-a^2\right) +
71 : * \left(\frac{1}{4}\left(x^2+y^2+z^2-a^2\right)^2 + a^2z^2\right)^{1/2}.
72 : * \f}
73 : *
74 : * The Spherical KS coordinates
75 : *
76 : * \f{align}{
77 : * \vec{\bar{x}} = x^{\bar{\imath}} = x_{\bar{\imath}} =
78 : * (\bar{x},\bar{y},\bar{z}),
79 : * \f}
80 : *
81 : * and KS coordinates
82 : *
83 : * \f{align}{
84 : * \vec{x} = x^{i} = x_{i} = (x,y,z),
85 : * \f}
86 : *
87 : * are related by
88 : *
89 : * \f{align}{
90 : * \left(\frac{\bar{x}}{r},\frac{\bar{y}}{r},\frac{\bar{z}}{r}\right) \equiv
91 : * \left(\frac{x}{\rho},\frac{y}{\rho},\frac{z}{r}\right),
92 : * \f}
93 : *
94 : * where we have defined
95 : *
96 : * \f{align}{
97 : * \rho^2 \equiv r^2 + a^2. \label{eq:rho}
98 : * \f}
99 : *
100 : * Therefore, we have that \f$r\f$ satisfies the equation for a sphere in
101 : * Spherical KS
102 : *
103 : * \f{align}{
104 : * r^2 = \vec{\bar{x}}\cdot\vec{\bar{x}} = \bar{x}^2 + \bar{y}^2 +
105 : * \bar{z}^2.
106 : * \f}
107 : *
108 : * It is clear to see that the Spherical KS radius coincides with the
109 : * Boyer-Lindquist radius.
110 : *
111 : * ## Spin in an Arbitrary Direction
112 : *
113 : * Given that the remaining important quantities take forms that are easily
114 : * specialized to the \f$z\f$ spin case, we instead focus on the general case
115 : * for brevity.
116 : *
117 : * ### The Transformation
118 : *
119 : * The Boyer-Lindquist radius for KS with spin in an arbitrary direction is
120 : * defined by
121 : *
122 : * \f{align}{
123 : * r^2 = \frac{1}{2}\left(\vec{x}\cdot\vec{x}-a^2\right) +
124 : * \left(\frac{1}{4}\left(\vec{x}\cdot\vec{x}-a^2\right)^2 +
125 : * \left(\vec{a}\cdot\vec{x}\right)^2\right)^{1/2}.
126 : * \f}
127 : *
128 : * Then, defining two transformation matrices \f$Q^{\bar{\imath}}{}_{j}\f$ and
129 : * \f$P^{j}{}_{\bar{\imath}}\f$ as
130 : *
131 : * \f{align}{
132 : * Q^{\bar{\imath}}{}_{j} &= \frac{r}{\rho}\delta^{\bar{\imath}}{}_{j} +
133 : * \frac{1}{(\rho + r)\rho}a^{\bar{\imath}}a_{j}, \\
134 : * P^{j}{}_{\bar{\imath}} &= \frac{\rho}{r}\delta^{j}{}_{\bar{\imath}} -
135 : * \frac{1}{(\rho + r)r}a^{j}a_{\bar{\imath}},
136 : * \f}
137 : *
138 : * where the definition of \f$\rho\f$ is identical to Eq. \f$(\ref{eq:rho})\f$,
139 : * such that
140 : *
141 : * \f{align}{
142 : * Q^{\bar{\imath}}{}_{j}x^{j} &= x^{\bar{\imath}}, \\
143 : * P^{j}{}_{\bar{\imath}}x^{\bar{\imath}} &= x^{j}.
144 : * \f}
145 : *
146 : * We again recover that \f$r\f$ satisfies the equation for a sphere in
147 : * Spherical KS
148 : *
149 : * \f{align}{
150 : * r^2 = \vec{\bar{x}}\cdot\vec{\bar{x}} = \bar{x}^2+\bar{y}^2+\bar{z}^2,
151 : * \f}
152 : *
153 : * and we also have that
154 : *
155 : * \f{align}{
156 : * \vec{a}\cdot\vec{\bar{x}} = \vec{a}\cdot\vec{x}.
157 : * \f}
158 : *
159 : * Note that \f$Q^{\bar{\imath}}{}_{ j}\f$ and \f$P^{j}{}_{\bar{\imath}}\f$
160 : * satisfy
161 : *
162 : * \f{align}{
163 : * P^{i}{}_{\bar{\jmath}}\,Q^{\bar{\jmath}}{}_{ k} = \delta^{i}{}_{ k}.
164 : * \f}
165 : *
166 : * The Jacobian is then given by
167 : *
168 : * \f{align}{
169 : * T^{i}{}_{\bar{\jmath}} = \partial_{\bar{\jmath}}\,x^{i} =
170 : * \frac{\partial x^i}{\partial x^{\bar{\jmath}}},
171 : * \f}
172 : *
173 : * while its inverse is given by
174 : *
175 : * \f{align}{
176 : * S^{\bar{\jmath}}{}_{ i} = \partial_{i}\,x^{\bar{\jmath}} =
177 : * \frac{\partial x^{\bar{\jmath}}}{\partial x^i},
178 : * \f}
179 : *
180 : * which in turn satisfies
181 : *
182 : * \f{align}{
183 : * T^{i}{}_{\bar{\jmath}}\,S^{\bar{\jmath}}{}_{ k} &= \delta^i_{\; k}.
184 : * \f}
185 : *
186 : * ### The Metric
187 : *
188 : * A KS coordinate system is defined by
189 : *
190 : * \f{align}{
191 : * g_{\mu\nu} = \eta_{\mu\nu} + 2Hl_{\mu}l_{\nu},
192 : * \f}
193 : *
194 : * where \f$H\f$ is a scalar function of the coordinates, \f$\eta_{\mu\nu}\f$ is
195 : * the Minkowski metric, and \f$l^\mu\f$ is a null vector. Note that the inverse
196 : * of the spacetime metric is given by
197 : *
198 : * \f{align}{
199 : * g^{\mu\nu} = \eta^{\mu\nu} - 2Hl^{\mu}l^{\nu}.
200 : * \f}
201 : *
202 : * The scalar function \f$H\f$ takes the form
203 : *
204 : * \f{align}{
205 : * H = \frac{Mr^3}{r^4 + \left(\vec{a}\cdot\vec{x}\right)^{2}},
206 : * \f}
207 : *
208 : * where \f$M\f$ is the mass, while the spatial part of the null vector takes
209 : * the form
210 : *
211 : * \f{align}{
212 : * l_{i} = l^{i} = \frac{r\vec{x} - \vec{a}\times\vec{x} +
213 : * \frac{(\vec{a}\cdot\vec{x})\vec{a}}{r}}{\rho^2}.
214 : * \f}
215 : *
216 : * Note that the full spacetime form of the null vector is
217 : *
218 : * \f{align}{
219 : * l_{\mu} &= (-1,l_{i}), & l^{\mu} &= (1,l^{i}).
220 : * \f}
221 : *
222 : * Transforming the KS spatial metric then yields the following Spherical KS
223 : * spatial metric
224 : *
225 : * \f{align}{
226 : * \gamma_{\bar{\imath}\bar{\jmath}} &=
227 : * \gamma_{mn}T^{m}{}_{\bar{\imath}}\,T^{n}{}_{\bar{\jmath}}, \nonumber \\
228 : * &= \eta_{mn}T^{m}{}_{\bar{\imath}}\,T^{n}{}_{\bar{\jmath}} +
229 : * 2Hl_{m}l_{n}T^{m}{}_{\bar{\imath}}\,T^{n}{}_{\bar{\jmath}}, \nonumber \\
230 : * &= \eta_{\bar{\imath}\bar{\jmath}} + 2Hl_{\bar{\imath}}l_{\bar{\jmath}}.
231 : * \f}
232 : *
233 : * The transformed spacetime Minkowski metric is given by
234 : *
235 : * \f{align}{
236 : * \eta_{\bar{\mu}\bar{\nu}} = (-1)\otimes\eta_{\bar{\imath}\bar{\jmath}},
237 : * \f}
238 : *
239 : * and the transformed spacetime null vector is given by
240 : *
241 : * \f{align}{
242 : * l_{\bar{\mu}} &= (-1,l_{\bar{\imath}}), &
243 : * l^{\bar{\mu}} &= (1,l^{\bar{\imath}}).
244 : * \f}
245 : *
246 : * Therefore, the Spherical KS spacetime metric is
247 : *
248 : * \f{align}{
249 : * g_{\bar{\mu}\bar{\nu}} = \eta_{\bar{\mu}\bar{\nu}} +
250 : * 2Hl_{\bar{\mu}}l_{\bar{\nu}}.
251 : * \f}
252 : *
253 : * Further, we have that the lapse in Spherical KS is given by
254 : *
255 : * \f{align}{
256 : * \alpha = \left(1 + 2H\right)^{-1/2},
257 : * \f}
258 : *
259 : * and the shift in Spherical KS by
260 : *
261 : * \f{align}{
262 : * \beta^{\bar{\imath}} &= -\frac{2Hl^{t}l^{\bar{\imath}}}{1 + 2Hl^{t}l^{t}} =
263 : * -2H\alpha^{2}l^{t}l^{\bar{\imath}}, & \beta_{\bar{\imath}} &=
264 : * -2Hl_{t}l_{\bar{\imath}}.
265 : * \f}
266 : *
267 : * ### Derivatives
268 : *
269 : * The derivatives of the preceding quantities are
270 : *
271 : * \f{align}{
272 : * \frac{\partial r}{\partial x^{i}} &= \frac{r^{2}x_{i} +
273 : * \left(\vec{a}\cdot\vec{x}\right)a_{i}}{rs}, \\
274 : * \partial_{\bar{\imath}}H &= HT^{m}{}_{\bar{\imath}}
275 : * \left[\frac{3}{r}\frac{\partial r}{\partial x^{m}} -
276 : * \frac{4r^{3}\frac{\partial r}{\partial x^{m}} +
277 : * 2\left(\vec{a}\cdot\vec{x}\right)a_{m}}{r^{4} +
278 : * \left(\vec{a}\cdot\vec{x}\right)^{2}}\right], \\
279 : * \partial_{\bar{\jmath}}l^{\bar{\imath}} &=
280 : * \partial_{\bar{\jmath}}\left(l_{k}T^{k}{}_{\bar{\imath}}\right),
281 : * \nonumber \\
282 : * &= T^{k}{}_{\bar{\imath}}T^{m}{}_{\bar{\jmath}}\frac{1}{\rho^{2}}
283 : * \left[\left(x_{k} - 2rl_{k} -
284 : * \frac{\left(\vec{a}\cdot\vec{x}\right)a_{k}}{r^{2}}\right)
285 : * \frac{\partial r}{\partial x^{m}} + r\delta_{km} + \frac{a_{k}a_{m}}{r} +
286 : * \epsilon^{kmn}a_{n}\right] +
287 : * l_{k}\partial_{\bar{\jmath}}T^{k}{}_{\bar{\imath}}, \\
288 : * \partial_{\bar{k}}\gamma_{\bar{\imath}\bar{\jmath}} &=
289 : * 2l_{\bar{\imath}}l_{\bar{\jmath}}\partial_{\bar{k}}H +
290 : * 4Hl_{(\bar{\imath}}\partial_{\bar{k}}l_{\bar{\jmath})} +
291 : * T^{m}{}_{\bar{\jmath}}\partial_{\bar{k}}T^{m}{}_{\bar{\imath}} +
292 : * T^{m}{}_{\bar{\imath}}\partial_{\bar{k}}T^{m}{}_{\bar{\jmath}}, \\
293 : * \partial_{\bar{k}}\alpha &= -\left(1+2H\right)^{-3/2}\partial_{\bar{k}}H =
294 : * -\alpha^{3}\partial_{\bar{k}}H, \\
295 : * \partial_{\bar{k}}\beta^{i} &=
296 : * 2\alpha^{2}\left[l^{\bar{\imath}}\partial_{\bar{k}}H +
297 : * H\left(S^{\bar{\imath}}{}_{j}S^{\bar{m}}{}_{n}
298 : * \delta_{nj}\partial_{\bar{k}}l_{\bar{m}} +
299 : * S^{\bar{\imath}}{}_{j}l_{\bar{m}}\partial_{\bar{k}}S^{\bar{m}}{}_{n}
300 : * \delta_{nj} + S^{\bar{m}}{}_{n}\delta_{nj}l_{\bar{m}}
301 : * \partial_{\bar{k}}S^{\bar{\imath}}{}_{j}\right)\right] -
302 : * 4Hl^{\bar{\imath}}\alpha^{4}\partial_{\bar{k}}H,
303 : * \f}
304 : *
305 : * where we have defined \f$s\f$ as
306 : *
307 : * \f{align}{
308 : * s &\equiv r^{2} + \frac{\left(\vec{a}\cdot\vec{x}\right)^{2}}{r^{2}}.
309 : * \label{eq: s_number}
310 : * \f}
311 : *
312 : * ## Code
313 : *
314 : * While the previous sections described the relevant physical quantities, the
315 : * actual files make use of internally defined objects to ease the computation
316 : * of the Jacobian, the inverse Jacobian, and their corresponding derivatives.
317 : * Therefore, we now list these intermediary objects as well as how they
318 : * construct the aforementioned physical quantities with the appropriate
319 : * definition found in the code (all for arbitrary spin).
320 : *
321 : * ### Helper Matrices
322 : *
323 : * The intermediary objects used to define the various Jacobian objects, the
324 : * so-called "helper matrices", are defined below.
325 : *
326 : * \f{align}{
327 : * F^{i}{}_{\bar{k}} &\equiv
328 : * -\frac{1}{\rho r^{3}}\left(a^{2}\delta^{i}{}_{\bar{k}} -
329 : * a^{i}a_{\bar{k}}\right), \\
330 : * \left(G_1\right)^{\bar{\imath}}{}_{ \bar{m}} &\equiv
331 : * \frac{1}{\rho^{2}r}\left(a^{2}\delta^{\bar{\imath}}{}_{\bar{m}} -
332 : * a^{\bar{\imath}}a_{\bar{m}}\right), \\
333 : * \left(G_2\right)^{\bar{n}}{}_{ j} &\equiv
334 : * \frac{\rho^{2}}{sr}Q^{\bar{n}}{}_{j}, \\
335 : * D^{i}{}_{\bar{m}} &\equiv
336 : * \frac{1}{\rho^{3}r}\left(a^{2}\delta^{i}{}_{\bar{m}} -
337 : * a^{i}a_{\bar{m}}\right), \\
338 : * C^{i}{}_{\bar{m}} &\equiv D^{i}{}_{\bar{m}} - 3F^{i}{}_{\bar{m}} =
339 : * \frac{1}{\rho r}\left(\frac{1}{\rho^{2}} +
340 : * \frac{3}{r^{2}}\right)\left(a^{2}\delta^{i}{}_{\bar{m}} -
341 : * a^{i}a_{\bar{m}}\right), \\
342 : * \left(E_1\right)^{i}{}_{\bar{m}} &\equiv
343 : * -\frac{1}{\rho^{2}}\left(\frac{1}{r^{2}} +
344 : * \frac{2}{\rho^{2}}\right)\left(a^{2}\delta^{i}{}_{\bar{m}} -
345 : * a^{i}a_{\bar{m}}\right), \nonumber \\
346 : * &= -\frac{\left(\rho^{2} +
347 : * 2r^{2}\right)}{r^{2}\rho^{4}}\left(a^{2}\delta^{i}{}_{\bar{m}} -
348 : * a^{i}a_{\bar{m}}\right), \\
349 : * \left(E_2\right)^{\bar{n}}{}_{ j} &\equiv \left[-\frac{a^{2}}{\rho^{2}r} -
350 : * \frac{2}{s}\left(r - \frac{\left(\vec{a}\cdot\vec{x}\right)^{2}}{r^{3}}
351 : * \right)\right]\cdot\left(G_2\right)^{\bar{n}}{}_{ j} +
352 : * \frac{1}{s}P^{\bar{n}}{}_{ j},
353 : * \f}
354 : *
355 : * where \f$s\f$ is defined identically to Eq. \f$(\ref{eq: s_number})\f$.
356 : *
357 : * ### Physical Quantities
358 : *
359 : * Below are the definitions for how we construct the Jacobian, inverse
360 : * Jacobian, derivative of the Jacobian, and derivative of the inverse Jacobian
361 : * in the code using the helper matrices.
362 : *
363 : * \f{align}{
364 : * T^{i}{}_{\bar{\jmath}} &= P^{i}{}_{\bar{\jmath}} +
365 : * F^{i}{}_{\bar{k}}x^{\bar{k}}x_{\bar{\jmath}}, \\
366 : * S^{\bar{\imath}}{}_{ j} &= Q^{\bar{\imath}}{}_{ j} +
367 : * \left(G_1\right)^{\bar{\imath}}{}_{\bar{m}}x^{\bar{m}}x_{\bar{n}}
368 : * \left(G_2\right)^{\bar{n}}{}_{ j}, \\
369 : * \partial_{\bar{k}}T^{i}{}_{\bar{\jmath}} &=
370 : * F^{i}{}_{\bar{\jmath}}x_{\bar{k}} + F^{i}{}_{\bar{k}}x_{\bar{\jmath}} +
371 : * F^{i}{}_{\bar{m}}x^{\bar{m}}\delta_{jk} +
372 : * C^{i}{}_{\bar{m}}\frac{x_{\bar{k}}x^{\bar{m}}x_{\bar{\jmath}}}{r^{2}}, \\
373 : * \partial_{\bar{k}}S^{\;\bar{\imath}}{}_{ j} &=
374 : * D^{\bar{\imath}}{}_{j}x_{\bar{k}} +
375 : * \left(G_1\right)^{\bar{\imath}}{}_{\bar{k}}x_{\bar{n}}
376 : * \left(G_2\right)^{\bar{n}}{}_{j} +
377 : * \left(G_1\right)^{\bar{\imath}}{}_{\bar{m}}x^{\bar{m}}
378 : * \left(G_2\right)^{\bar{n}}{}_{j}\delta_{\bar{n}\bar{k}} \nonumber \\
379 : * &\quad +
380 : * \left(E_1\right)^{i}{}_{\bar{m}}\frac{x_{\bar{k}}x^{\bar{m}}x_{\bar{n}}}{r}
381 : * \left(G_2\right)^{\bar{n}}{}_{j} +
382 : * \left(G_1\right)^{\bar{i}}{}_{\bar{m}}
383 : * \frac{x_{\bar{k}}x^{\bar{m}}x_{\bar{n}}}{r}\left(E_2\right)^{\bar{n}}{}_{j}
384 : * - \left(G_1\right)^{\bar{\imath}}{}_{\bar{m}}x^{\bar{m}}x_{\bar{n}}
385 : * \left(G_2\right)^{\bar{n}}{}_{j}
386 : * \frac{2\left(\vec{a}\cdot\vec{x}\right)}{sr^{2}}a_{\bar{k}}.
387 : * \f}
388 : */
389 1 : class SphericalKerrSchild : public AnalyticSolution<3_st>,
390 : public MarkAsAnalyticSolution {
391 : public:
392 0 : struct Mass {
393 0 : using type = double;
394 0 : static constexpr Options::String help = {"Mass of the black hole"};
395 0 : static type lower_bound() { return 0.; }
396 : };
397 0 : struct Spin {
398 0 : using type = std::array<double, volume_dim>;
399 0 : static constexpr Options::String help = {
400 : "The [x,y,z] dimensionless spin of the black hole"};
401 : };
402 0 : struct Center {
403 0 : using type = std::array<double, volume_dim>;
404 0 : static constexpr Options::String help = {
405 : "The [x,y,z] center of the black hole"};
406 : };
407 0 : using options = tmpl::list<Mass, Spin, Center>;
408 0 : static constexpr Options::String help{
409 : "Black hole in Spherical Kerr-Schild coordinates"};
410 :
411 : template <typename DataType, typename Frame = Frame::Inertial>
412 0 : using tags = tmpl::flatten<tmpl::list<
413 : AnalyticSolution<3_st>::tags<DataType, Frame>,
414 : gr::Tags::DerivDetSpatialMetric<DataType, 3, Frame>,
415 : gr::Tags::TraceExtrinsicCurvature<DataType>,
416 : gr::Tags::SpatialChristoffelFirstKind<DataType, 3, Frame>,
417 : gr::Tags::SpatialChristoffelSecondKind<DataType, 3, Frame>,
418 : gr::Tags::TraceSpatialChristoffelSecondKind<DataType, 3, Frame>>>;
419 :
420 0 : SphericalKerrSchild(double mass, Spin::type dimensionless_spin,
421 : Center::type center,
422 : const Options::Context& context = {});
423 :
424 0 : explicit SphericalKerrSchild(CkMigrateMessage* /*unused*/);
425 :
426 0 : SphericalKerrSchild() = default;
427 0 : SphericalKerrSchild(const SphericalKerrSchild& /*rhs*/) = default;
428 0 : SphericalKerrSchild& operator=(const SphericalKerrSchild& /*rhs*/) = default;
429 0 : SphericalKerrSchild(SphericalKerrSchild&& /*rhs*/) = default;
430 0 : SphericalKerrSchild& operator=(SphericalKerrSchild&& /*rhs*/) = default;
431 0 : ~SphericalKerrSchild() = default;
432 :
433 : // NOLINTNEXTLINE(google-runtime-references)
434 0 : void pup(PUP::er& p);
435 :
436 0 : SPECTRE_ALWAYS_INLINE double mass() const { return mass_; }
437 0 : SPECTRE_ALWAYS_INLINE const std::array<double, volume_dim>& center() const {
438 : return center_;
439 : }
440 : SPECTRE_ALWAYS_INLINE const std::array<double, volume_dim>&
441 0 : dimensionless_spin() const {
442 : return dimensionless_spin_;
443 : }
444 :
445 0 : struct internal_tags {
446 : template <typename DataType, typename Frame = ::Frame::Inertial>
447 0 : using x_minus_center = ::Tags::TempI<0, 3, Frame, DataType>;
448 : template <typename DataType>
449 0 : using r_squared = ::Tags::TempScalar<1, DataType>;
450 : template <typename DataType>
451 0 : using r = ::Tags::TempScalar<2, DataType>;
452 : template <typename DataType>
453 0 : using rho = ::Tags::TempScalar<3, DataType>;
454 : template <typename DataType, typename Frame = ::Frame::Inertial>
455 0 : using helper_matrix_F = ::Tags::TempIj<4, 3, Frame, DataType>;
456 : template <typename DataType, typename Frame = ::Frame::Inertial>
457 0 : using transformation_matrix_P = ::Tags::TempIj<5, 3, Frame, DataType>;
458 : template <typename DataType, typename Frame = ::Frame::Inertial>
459 0 : using jacobian = ::Tags::TempIj<6, 3, Frame, DataType>;
460 : template <typename DataType, typename Frame = ::Frame::Inertial>
461 0 : using helper_matrix_D = ::Tags::TempIj<7, 3, Frame, DataType>;
462 : template <typename DataType, typename Frame = ::Frame::Inertial>
463 0 : using helper_matrix_C = ::Tags::TempIj<8, 3, Frame, DataType>;
464 : template <typename DataType, typename Frame = ::Frame::Inertial>
465 0 : using deriv_jacobian = ::Tags::TempiJk<9, 3, Frame, DataType>;
466 : template <typename DataType, typename Frame = ::Frame::Inertial>
467 0 : using transformation_matrix_Q = ::Tags::TempIj<10, 3, Frame, DataType>;
468 : template <typename DataType, typename Frame = ::Frame::Inertial>
469 0 : using helper_matrix_G1 = ::Tags::TempIj<11, 3, Frame, DataType>;
470 : template <typename DataType>
471 0 : using a_dot_x = ::Tags::TempScalar<12, DataType>;
472 : template <typename DataType>
473 0 : using s_number = ::Tags::TempScalar<13, DataType>;
474 : template <typename DataType, typename Frame = ::Frame::Inertial>
475 0 : using helper_matrix_G2 = ::Tags::TempIj<14, 3, Frame, DataType>;
476 : template <typename DataType, typename Frame = ::Frame::Inertial>
477 0 : using G1_dot_x = ::Tags::TempI<15, 3, Frame, DataType>;
478 : template <typename DataType, typename Frame = ::Frame::Inertial>
479 0 : using G2_dot_x = ::Tags::Tempi<16, 3, Frame, DataType>;
480 : template <typename DataType, typename Frame = ::Frame::Inertial>
481 0 : using inv_jacobian = ::Tags::TempIj<17, 3, Frame, DataType>;
482 : template <typename DataType, typename Frame = ::Frame::Inertial>
483 0 : using helper_matrix_E1 = ::Tags::TempIj<18, 3, Frame, DataType>;
484 : template <typename DataType, typename Frame = ::Frame::Inertial>
485 0 : using helper_matrix_E2 = ::Tags::TempIj<19, 3, Frame, DataType>;
486 : template <typename DataType, typename Frame = ::Frame::Inertial>
487 0 : using deriv_inv_jacobian = ::Tags::TempiJk<20, 3, Frame, DataType>;
488 : template <typename DataType>
489 0 : using H = ::Tags::TempScalar<21, DataType>;
490 : template <typename DataType, typename Frame = ::Frame::Inertial>
491 0 : using kerr_schild_x = ::Tags::TempI<22, 3, Frame, DataType>;
492 : template <typename DataType, typename Frame = ::Frame::Inertial>
493 0 : using a_cross_x = ::Tags::TempI<23, 3, Frame, DataType>;
494 : template <typename DataType, typename Frame = ::Frame::Inertial>
495 0 : using kerr_schild_l = ::Tags::TempI<24, 3, Frame, DataType>;
496 : template <typename DataType, typename Frame = ::Frame::Inertial>
497 0 : using l_lower = ::Tags::Tempi<25, 4, Frame, DataType>;
498 : template <typename DataType, typename Frame = ::Frame::Inertial>
499 0 : using l_upper = ::Tags::TempI<26, 4, Frame, DataType>;
500 : template <typename DataType, typename Frame = ::Frame::Inertial>
501 0 : using deriv_r = ::Tags::TempI<27, 3, Frame, DataType>;
502 : template <typename DataType, typename Frame = ::Frame::Inertial>
503 0 : using deriv_H = ::Tags::TempI<28, 4, Frame, DataType>;
504 : template <typename DataType, typename Frame = ::Frame::Inertial>
505 0 : using kerr_schild_deriv_l = ::Tags::Tempij<29, 4, Frame, DataType>;
506 : template <typename DataType, typename Frame = ::Frame::Inertial>
507 0 : using deriv_l = ::Tags::Tempij<30, 4, Frame, DataType>;
508 : template <typename DataType>
509 0 : using lapse_squared = ::Tags::TempScalar<31, DataType>;
510 : template <typename DataType>
511 0 : using deriv_lapse_multiplier = ::Tags::TempScalar<32, DataType>;
512 : template <typename DataType>
513 0 : using shift_multiplier = ::Tags::TempScalar<33, DataType>;
514 : };
515 :
516 : template <typename DataType, typename Frame = ::Frame::Inertial>
517 0 : using CachedBuffer = CachedTempBuffer<
518 : internal_tags::x_minus_center<DataType, Frame>,
519 : internal_tags::r_squared<DataType>, internal_tags::r<DataType>,
520 : internal_tags::rho<DataType>,
521 : internal_tags::helper_matrix_F<DataType, Frame>,
522 : internal_tags::transformation_matrix_P<DataType, Frame>,
523 : internal_tags::jacobian<DataType, Frame>,
524 : internal_tags::helper_matrix_D<DataType, Frame>,
525 : internal_tags::helper_matrix_C<DataType, Frame>,
526 : internal_tags::deriv_jacobian<DataType, Frame>,
527 : internal_tags::transformation_matrix_Q<DataType, Frame>,
528 : internal_tags::helper_matrix_G1<DataType, Frame>,
529 : internal_tags::a_dot_x<DataType>, internal_tags::s_number<DataType>,
530 : internal_tags::helper_matrix_G2<DataType, Frame>,
531 : internal_tags::G1_dot_x<DataType, Frame>,
532 : internal_tags::G2_dot_x<DataType, Frame>,
533 : internal_tags::inv_jacobian<DataType, Frame>,
534 : internal_tags::helper_matrix_E1<DataType, Frame>,
535 : internal_tags::helper_matrix_E2<DataType, Frame>,
536 : internal_tags::deriv_inv_jacobian<DataType, Frame>,
537 : internal_tags::H<DataType>, internal_tags::kerr_schild_x<DataType, Frame>,
538 : internal_tags::a_cross_x<DataType, Frame>,
539 : internal_tags::kerr_schild_l<DataType, Frame>,
540 : internal_tags::l_lower<DataType, Frame>,
541 : internal_tags::l_upper<DataType, Frame>,
542 : internal_tags::deriv_r<DataType, Frame>,
543 : internal_tags::deriv_H<DataType, Frame>,
544 : internal_tags::kerr_schild_deriv_l<DataType, Frame>,
545 : internal_tags::deriv_l<DataType, Frame>,
546 : internal_tags::lapse_squared<DataType>, gr::Tags::Lapse<DataType>,
547 : internal_tags::deriv_lapse_multiplier<DataType>,
548 : internal_tags::shift_multiplier<DataType>,
549 : gr::Tags::Shift<DataType, 3, Frame>, DerivShift<DataType, Frame>,
550 : gr::Tags::SpatialMetric<DataType, 3, Frame>,
551 : DerivSpatialMetric<DataType, Frame>,
552 : ::Tags::dt<gr::Tags::SpatialMetric<DataType, 3, Frame>>,
553 : gr::Tags::ExtrinsicCurvature<DataType, 3, Frame>,
554 : gr::Tags::InverseSpatialMetric<DataType, 3, Frame>,
555 : gr::Tags::SpatialChristoffelFirstKind<DataType, 3, Frame>,
556 : gr::Tags::SpatialChristoffelSecondKind<DataType, 3, Frame>>;
557 :
558 : // forward-declaration needed.
559 : template <typename DataType, typename Frame>
560 : class IntermediateVars;
561 :
562 : template <typename DataType, typename Frame = Frame::Inertial>
563 0 : using allowed_tags =
564 : tmpl::push_back<tags<DataType, Frame>,
565 : typename internal_tags::inv_jacobian<DataType, Frame>>;
566 :
567 : template <typename DataType, typename Frame, typename... Tags>
568 0 : tuples::TaggedTuple<Tags...> variables(
569 : const tnsr::I<DataType, volume_dim, Frame>& x, double /*t*/,
570 : tmpl::list<Tags...> /*meta*/) const {
571 : static_assert(
572 : tmpl2::flat_all_v<
573 : tmpl::list_contains_v<allowed_tags<DataType, Frame>, Tags>...>,
574 : "At least one of the requested tags is not supported. The requested "
575 : "tags are listed as template parameters of the `variables` function.");
576 : IntermediateVars<DataType, Frame> cache(get_size(*x.begin()));
577 : IntermediateComputer<DataType, Frame> computer(*this, x);
578 : return {cache.get_var(computer, Tags{})...};
579 : }
580 :
581 : template <typename DataType, typename Frame, typename... Tags>
582 0 : tuples::TaggedTuple<Tags...> variables(
583 : const tnsr::I<DataType, volume_dim, Frame>& x, double /*t*/,
584 : tmpl::list<Tags...> /*meta*/,
585 : gsl::not_null<IntermediateVars<DataType, Frame>*> cache) const {
586 : static_assert(
587 : tmpl2::flat_all_v<
588 : tmpl::list_contains_v<allowed_tags<DataType, Frame>, Tags>...>,
589 : "At least one of the requested tags is not supported. The requested "
590 : "tags are listed as template parameters of the `variables` function.");
591 : if (cache->number_of_grid_points() != get_size(*x.begin())) {
592 : *cache = IntermediateVars<DataType, Frame>(get_size(*x.begin()));
593 : }
594 : IntermediateComputer<DataType, Frame> computer(*this, x);
595 : return {cache->get_var(computer, Tags{})...};
596 : }
597 :
598 : template <typename DataType, typename Frame = ::Frame::Inertial>
599 0 : class IntermediateComputer {
600 : public:
601 0 : using CachedBuffer = SphericalKerrSchild::CachedBuffer<DataType, Frame>;
602 :
603 0 : IntermediateComputer(const SphericalKerrSchild& solution,
604 : const tnsr::I<DataType, 3, Frame>& x);
605 :
606 : // spin_a_and_squared(const SphericalKerrSchild& solution);
607 :
608 0 : void operator()(
609 : const gsl::not_null<tnsr::I<DataType, 3, Frame>*> x_minus_center,
610 : const gsl::not_null<CachedBuffer*> /*cache*/,
611 : internal_tags::x_minus_center<DataType, Frame> /*meta*/) const;
612 :
613 0 : void operator()(const gsl::not_null<Scalar<DataType>*> r_squared,
614 : const gsl::not_null<CachedBuffer*> cache,
615 : internal_tags::r_squared<DataType> /*meta*/) const;
616 :
617 0 : void operator()(const gsl::not_null<Scalar<DataType>*> r,
618 : const gsl::not_null<CachedBuffer*> cache,
619 : internal_tags::r<DataType> /*meta*/) const;
620 :
621 0 : void operator()(const gsl::not_null<Scalar<DataType>*> rho,
622 : const gsl::not_null<CachedBuffer*> cache,
623 : internal_tags::rho<DataType> /*meta*/) const;
624 :
625 0 : void operator()(
626 : const gsl::not_null<tnsr::Ij<DataType, 3, Frame>*> helper_matrix_F,
627 : const gsl::not_null<CachedBuffer*> cache,
628 : internal_tags::helper_matrix_F<DataType, Frame> /*meta*/) const;
629 :
630 0 : void operator()(
631 : const gsl::not_null<tnsr::Ij<DataType, 3, Frame>*>
632 : transformation_matrix_P,
633 : const gsl::not_null<CachedBuffer*> cache,
634 : internal_tags::transformation_matrix_P<DataType, Frame> /*meta*/) const;
635 :
636 0 : void operator()(const gsl::not_null<tnsr::Ij<DataType, 3, Frame>*> jacobian,
637 : const gsl::not_null<CachedBuffer*> cache,
638 : internal_tags::jacobian<DataType, Frame> /*meta*/) const;
639 :
640 0 : void operator()(
641 : const gsl::not_null<tnsr::Ij<DataType, 3, Frame>*> helper_matrix_D,
642 : const gsl::not_null<CachedBuffer*> cache,
643 : internal_tags::helper_matrix_D<DataType, Frame> /*meta*/) const;
644 :
645 0 : void operator()(
646 : const gsl::not_null<tnsr::Ij<DataType, 3, Frame>*> helper_matrix_C,
647 : const gsl::not_null<CachedBuffer*> cache,
648 : internal_tags::helper_matrix_C<DataType, Frame> /*meta*/) const;
649 :
650 0 : void operator()(
651 : const gsl::not_null<tnsr::iJk<DataType, 3, Frame>*> deriv_jacobian,
652 : const gsl::not_null<CachedBuffer*> cache,
653 : internal_tags::deriv_jacobian<DataType, Frame> /*meta*/) const;
654 :
655 0 : void operator()(
656 : const gsl::not_null<tnsr::Ij<DataType, 3, Frame>*>
657 : transformation_matrix_Q,
658 : const gsl::not_null<CachedBuffer*> cache,
659 : internal_tags::transformation_matrix_Q<DataType, Frame> /*meta*/) const;
660 :
661 0 : void operator()(
662 : const gsl::not_null<tnsr::Ij<DataType, 3, Frame>*> helper_matrix_G1,
663 : const gsl::not_null<CachedBuffer*> cache,
664 : internal_tags::helper_matrix_G1<DataType, Frame> /*meta*/) const;
665 :
666 0 : void operator()(const gsl::not_null<Scalar<DataType>*> a_dot_x,
667 : const gsl::not_null<CachedBuffer*> cache,
668 : internal_tags::a_dot_x<DataType> /*meta*/) const;
669 :
670 0 : void operator()(const gsl::not_null<Scalar<DataType>*> s_number,
671 : const gsl::not_null<CachedBuffer*> cache,
672 : internal_tags::s_number<DataType> /*meta*/) const;
673 :
674 0 : void operator()(
675 : const gsl::not_null<tnsr::Ij<DataType, 3, Frame>*> helper_matrix_G2,
676 : const gsl::not_null<CachedBuffer*> cache,
677 : internal_tags::helper_matrix_G2<DataType, Frame> /*meta*/) const;
678 :
679 0 : void operator()(const gsl::not_null<tnsr::I<DataType, 3, Frame>*> G1_dot_x,
680 : const gsl::not_null<CachedBuffer*> cache,
681 : internal_tags::G1_dot_x<DataType, Frame> /*meta*/) const;
682 :
683 0 : void operator()(const gsl::not_null<tnsr::i<DataType, 3, Frame>*> G2_dot_x,
684 : const gsl::not_null<CachedBuffer*> cache,
685 : internal_tags::G2_dot_x<DataType, Frame> /*meta*/) const;
686 :
687 0 : void operator()(
688 : const gsl::not_null<tnsr::Ij<DataType, 3, Frame>*> inv_jacobian,
689 : const gsl::not_null<CachedBuffer*> cache,
690 : internal_tags::inv_jacobian<DataType, Frame> /*meta*/) const;
691 :
692 0 : void operator()(
693 : const gsl::not_null<tnsr::Ij<DataType, 3, Frame>*> helper_matrix_E1,
694 : const gsl::not_null<CachedBuffer*> cache,
695 : internal_tags::helper_matrix_E1<DataType, Frame> /*meta*/) const;
696 :
697 0 : void operator()(
698 : const gsl::not_null<tnsr::Ij<DataType, 3, Frame>*> helper_matrix_E2,
699 : const gsl::not_null<CachedBuffer*> cache,
700 : internal_tags::helper_matrix_E2<DataType, Frame> /*meta*/) const;
701 :
702 0 : void operator()(
703 : const gsl::not_null<tnsr::iJk<DataType, 3, Frame>*> deriv_inv_jacobian,
704 : const gsl::not_null<CachedBuffer*> cache,
705 : internal_tags::deriv_inv_jacobian<DataType, Frame> /*meta*/) const;
706 :
707 0 : void operator()(const gsl::not_null<Scalar<DataType>*> H,
708 : const gsl::not_null<CachedBuffer*> cache,
709 : internal_tags::H<DataType> /*meta*/) const;
710 :
711 0 : void operator()(
712 : const gsl::not_null<tnsr::I<DataType, 3, Frame>*> kerr_schild_x,
713 : const gsl::not_null<CachedBuffer*> /*cache*/,
714 : internal_tags::kerr_schild_x<DataType, Frame> /*meta*/) const;
715 :
716 0 : void operator()(const gsl::not_null<tnsr::I<DataType, 3, Frame>*> a_cross_x,
717 : const gsl::not_null<CachedBuffer*> cache,
718 : internal_tags::a_cross_x<DataType, Frame> /*meta*/) const;
719 :
720 0 : void operator()(
721 : const gsl::not_null<tnsr::I<DataType, 3, Frame>*> kerr_schild_l,
722 : const gsl::not_null<CachedBuffer*> cache,
723 : internal_tags::kerr_schild_l<DataType, Frame> /*meta*/) const;
724 :
725 0 : void operator()(const gsl::not_null<tnsr::i<DataType, 4, Frame>*> l_lower,
726 : const gsl::not_null<CachedBuffer*> cache,
727 : internal_tags::l_lower<DataType, Frame> /*meta*/) const;
728 :
729 0 : void operator()(const gsl::not_null<tnsr::I<DataType, 4, Frame>*> l_upper,
730 : const gsl::not_null<CachedBuffer*> cache,
731 : internal_tags::l_upper<DataType, Frame> /*meta*/) const;
732 :
733 0 : void operator()(const gsl::not_null<tnsr::I<DataType, 3, Frame>*> deriv_r,
734 : const gsl::not_null<CachedBuffer*> cache,
735 : internal_tags::deriv_r<DataType, Frame> /*meta*/) const;
736 :
737 0 : void operator()(const gsl::not_null<tnsr::I<DataType, 4, Frame>*> deriv_H,
738 : const gsl::not_null<CachedBuffer*> cache,
739 : internal_tags::deriv_H<DataType, Frame> /*meta*/) const;
740 :
741 0 : void operator()(
742 : const gsl::not_null<tnsr::ij<DataType, 4, Frame>*> kerr_schild_deriv_l,
743 : const gsl::not_null<CachedBuffer*> cache,
744 : internal_tags::kerr_schild_deriv_l<DataType, Frame> /*meta*/) const;
745 :
746 0 : void operator()(const gsl::not_null<tnsr::ij<DataType, 4, Frame>*> deriv_l,
747 : const gsl::not_null<CachedBuffer*> cache,
748 : internal_tags::deriv_l<DataType, Frame> /*meta*/) const;
749 :
750 0 : void operator()(const gsl::not_null<Scalar<DataType>*> lapse_squared,
751 : const gsl::not_null<CachedBuffer*> cache,
752 : internal_tags::lapse_squared<DataType> /*meta*/) const;
753 :
754 0 : void operator()(const gsl::not_null<Scalar<DataType>*> lapse,
755 : const gsl::not_null<CachedBuffer*> cache,
756 : gr::Tags::Lapse<DataType> /*meta*/) const;
757 :
758 0 : void operator()(
759 : const gsl::not_null<Scalar<DataType>*> deriv_lapse_multiplier,
760 : const gsl::not_null<CachedBuffer*> cache,
761 : internal_tags::deriv_lapse_multiplier<DataType> /*meta*/) const;
762 :
763 0 : void operator()(const gsl::not_null<Scalar<DataType>*> shift_multiplier,
764 : const gsl::not_null<CachedBuffer*> cache,
765 : internal_tags::shift_multiplier<DataType> /*meta*/) const;
766 :
767 0 : void operator()(const gsl::not_null<tnsr::I<DataType, 3, Frame>*> shift,
768 : const gsl::not_null<CachedBuffer*> cache,
769 : gr::Tags::Shift<DataType, 3, Frame> /*meta*/) const;
770 :
771 0 : void operator()(
772 : const gsl::not_null<tnsr::iJ<DataType, 3, Frame>*> deriv_shift,
773 : const gsl::not_null<CachedBuffer*> cache,
774 : DerivShift<DataType, Frame> /*meta*/) const;
775 :
776 0 : void operator()(
777 : const gsl::not_null<tnsr::ii<DataType, 3, Frame>*> spatial_metric,
778 : const gsl::not_null<CachedBuffer*> cache,
779 : gr::Tags::SpatialMetric<DataType, 3, Frame> /*meta*/) const;
780 :
781 0 : void operator()(
782 : const gsl::not_null<tnsr::II<DataType, 3, Frame>*>
783 : inverse_spatial_metric,
784 : const gsl::not_null<CachedBuffer*> cache,
785 : gr::Tags::InverseSpatialMetric<DataType, 3, Frame> /*meta*/) const;
786 :
787 0 : void operator()(const gsl::not_null<tnsr::ijj<DataType, 3, Frame>*>
788 : deriv_spatial_metric,
789 : const gsl::not_null<CachedBuffer*> cache,
790 : DerivSpatialMetric<DataType, Frame> /*meta*/) const;
791 :
792 0 : void operator()(
793 : const gsl::not_null<tnsr::ii<DataType, 3, Frame>*> dt_spatial_metric,
794 : const gsl::not_null<CachedBuffer*> cache,
795 : ::Tags::dt<gr::Tags::SpatialMetric<DataType, 3, Frame>> /*meta*/) const;
796 :
797 0 : void operator()(
798 : const gsl::not_null<tnsr::ii<DataType, 3, Frame>*> extrinsic_curvature,
799 : const gsl::not_null<CachedBuffer*> cache,
800 : gr::Tags::ExtrinsicCurvature<DataType, 3, Frame> /*meta*/) const;
801 :
802 0 : void operator()(
803 : const gsl::not_null<tnsr::ijj<DataType, 3, Frame>*>
804 : christoffel_first_kind,
805 : const gsl::not_null<CachedBuffer*> cache,
806 : gr::Tags::SpatialChristoffelFirstKind<DataType, 3, Frame> /*meta*/)
807 : const;
808 :
809 0 : void operator()(
810 : const gsl::not_null<tnsr::Ijj<DataType, 3, Frame>*>
811 : christoffel_second_kind,
812 : const gsl::not_null<CachedBuffer*> cache,
813 : gr::Tags::SpatialChristoffelSecondKind<DataType, 3, Frame> /*meta*/)
814 : const;
815 :
816 : private:
817 0 : const SphericalKerrSchild& solution_;
818 0 : const tnsr::I<DataType, 3, Frame>& x_;
819 : // Here null_vector_0 is simply -1, but if you have a boosted solution,
820 : // then null_vector_0 can be something different, so we leave it coded
821 : // in instead of eliminating it.
822 0 : static constexpr double null_vector_0_ = -1.0;
823 : };
824 :
825 : template <typename DataType, typename Frame = ::Frame::Inertial>
826 0 : class IntermediateVars : public CachedBuffer<DataType, Frame> {
827 : public:
828 0 : using CachedBuffer = SphericalKerrSchild::CachedBuffer<DataType, Frame>;
829 : using CachedBuffer::CachedBuffer;
830 1 : using CachedBuffer::get_var;
831 :
832 0 : tnsr::i<DataType, 3, Frame> get_var(
833 : const IntermediateComputer<DataType, Frame>& computer,
834 : DerivLapse<DataType, Frame> /*meta*/);
835 :
836 0 : Scalar<DataType> get_var(
837 : const IntermediateComputer<DataType, Frame>& computer,
838 : ::Tags::dt<gr::Tags::Lapse<DataType>> /*meta*/);
839 :
840 0 : tnsr::I<DataType, 3, Frame> get_var(
841 : const IntermediateComputer<DataType, Frame>& computer,
842 : ::Tags::dt<gr::Tags::Shift<DataType, 3, Frame>> /*meta*/);
843 :
844 0 : Scalar<DataType> get_var(
845 : const IntermediateComputer<DataType, Frame>& computer,
846 : gr::Tags::SqrtDetSpatialMetric<DataType> /*meta*/);
847 :
848 0 : tnsr::i<DataType, 3, Frame> get_var(
849 : const IntermediateComputer<DataType, Frame>& computer,
850 : gr::Tags::DerivDetSpatialMetric<DataType, 3, Frame> /*meta*/);
851 :
852 0 : Scalar<DataType> get_var(
853 : const IntermediateComputer<DataType, Frame>& computer,
854 : gr::Tags::TraceExtrinsicCurvature<DataType> /*meta*/);
855 :
856 0 : tnsr::I<DataType, 3, Frame> get_var(
857 : const IntermediateComputer<DataType, Frame>& computer,
858 : gr::Tags::TraceSpatialChristoffelSecondKind<DataType, 3,
859 : Frame> /*meta*/);
860 :
861 : private:
862 : // Here null_vector_0 is simply -1, but if you have a boosted solution,
863 : // then null_vector_0 can be something different, so we leave it coded
864 : // in instead of eliminating it.
865 0 : static constexpr double null_vector_0_ = -1.0;
866 : };
867 :
868 : private:
869 0 : double mass_{std::numeric_limits<double>::signaling_NaN()};
870 0 : std::array<double, volume_dim> dimensionless_spin_ =
871 : make_array<volume_dim>(std::numeric_limits<double>::signaling_NaN());
872 0 : std::array<double, volume_dim> center_ =
873 : make_array<volume_dim>(std::numeric_limits<double>::signaling_NaN());
874 : };
875 :
876 0 : bool operator==(const SphericalKerrSchild& lhs, const SphericalKerrSchild& rhs);
877 :
878 0 : bool operator!=(const SphericalKerrSchild& lhs, const SphericalKerrSchild& rhs);
879 :
880 : } // namespace Solutions
881 : } // namespace gr
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